Pacific Journal of Mathematics Vol 207 Issue 1, Nov

Volume 207

No. 1

November 2002

Pacific Journal of Mathematics

PACIFIC JOURNAL OF MATHEMATICS

Pacific Journal of Mathematics

2002 Vol. 207, No. 1 Volume 207

No. 1

November 2002

PACIFIC JOURNAL OF MATHEMATICS http://www.pjmath.org Founded in 1951 by E. F. Beckenbach (1906–1982)

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PACIFIC JOURNAL OF MATHEMATICS Vol. 207, No. 1, 2002

FUNDAMENTAL SOLUTIONS OF INVARIANT DIFFERENTIAL OPERATORS ON A SEMISIMPLE LIE GROUP II Guillermo Ames Let G be a linear connected semisimple Lie group. We denote by U (g)K the algebra of left invariant differential operators on G that are also right invariant by K, and Z(U (g)K ) denotes center of U (g)K . In this paper we give a sufficient condition for a differential operator P ∈ Z(U (g)K ) to have a fundamental solution on G. This result extends the same one obtained previously for real rank one Lie groups and groups with only one conjugacy class of Cartan subgroups.

1. Introduction. Let G be a linear connected semisimple Lie group. The algebra of left invariant differential operators on G is canonically identified with the universal algebra U(g). The operators of the center Z(g) are the bi-invariant differential operators on G. More generally, we consider the algebra U(g)K of right K-invariant differential operators in U(g), where K is a maximal compact subgroup of G. Z(U(g)K ) will denote its center. We denote by D(G) the space of C ∞ functions with compact support. The dual D0 (G) of continuous linear functionals in D(G) is the space of distributions of G. An operator P in U(g) acts on D0 (G) in the following way: P E(f ) = E(P t f ), where P t ∈ U(g) is such that, if dx is a Haar measure on G, Z Z P f (x)g(x)dx = f (x)P t g(x)dx. G

G

Xt

In addition, if X ∈ g, = −X, so the P 7→ P t is the anti-automorphism of U(g) extending −Id of g. Also, this map preserves the subalgebras Z(g) and U(g)K . Definition 1. A distribution E ∈ D0 (G) is a fundamental solution of a differential operator P ∈ U(g) if P E = δ, where δ(f ) = f (1); and E is a parametrix of P if P E − δ ∈ C ∞ (G). 1

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GUILLERMO AMES

In this paper we extend the main result of [1] for a connected semisimple Lie group with a simply connected complexification. We start by recalling this result as it was stated in [1]. Remember that in general Z(U(g)K ) ' Z(g) ⊗ Z(k) (Knop’s Theorem [9]). Given h0 and t0 Cartan subalgebras of g0 and k0 respectively, we will denote γhG and γtK the Harish-Chandra homomorphisms of Z(g) and Z(k) with respect to the subalgebras h and t: Then we have Z(g) × Z(k) ↓⊗ '

Z(U(g)K ) −→ Z(g) ⊗ Z(k)

G ×γ K γh t

−→

G ⊗γ K γh t

−→

U(h)W × U(t)WK ↓⊗ i⊗i

U(h)W ⊗ U(t)WK −→ U(h ⊕ t).

Therefore, by the way of the homomorphisms described above, we can associate to P ∈ Z(U(g)K ) a differential operator (γhG ⊗ γtK )(P ) in the group H × T , where H and T are the respective Cartan subgroups of G and K with Lie algebras h0 and t0 . We say that a Cartan subgroup H of G is fundamental if it contains a G-conjugate of a Cartan subgroup of K. All fundamental Cartan subgroups of G are conjugate. We will denote γ G = γhG , where h0 is the Lie subalgebra of a fundamental Cartan subgroup H. Because in K all Cartan subgroups are conjugate, we put γ K = γtK . We can now state our main result: Theorem 1.1. Let G be a connected semisimple Lie group with a simply connected complexification. Let H be a fundamental Cartan subgroup of G and P ∈ Z(U(g)K ). If γ G ⊗ γ K (P ) has a fundamental solution in H × T , then P has a fundamental solution in G. When P is a bi-invariant operator, we obtain a complete proof of the Theorem announced in [2] (see [3] for another proof of this result): Corollary 1.2 (Benabdallah-Rouvière). Let P ∈ Z(g). If γ G (P ) has a fundamental solution in H, then P has a fundamental solution in G. The proof of Theorem 1.1 goes by explicit construction of the fundamental solution of P , using the Plancherel formula as the main tool. 2. Preliminaries. In this section we fix notation and summarize some known facts about representation theory that will be needed through this paper. We refer to [1] for any unexplained notation. 2.1. Notation. Let G be a linear connected semisimple Lie group, g0 its Lie algebra. The complexification of any real Lie algebra will be denoted without the subscript.

FUNDAMENTAL SOLUTIONS ...

3

We will assume that G has a simply connected complexification, that is, G is the analytic subgroup corresponding to g0 of the simply connected complex group with Lie algebra g. θ will denote a Cartan involution in either g0 , g or G. Let g0 = k0 ⊕ p0 be a Cartan decomposition of g0 with respect to θ; that is, k0 = {X ∈ g0 : θX = X} and p0 = {X ∈ g0 : θX = −X}. If K is the analytic subgroup of G with Lie algebra k0 , K is a maximally compact subgroup of G. We fix t0 a Cartan subalgebra of k0 coming from a maximal torus T of K. On g we define the inner product, (X, Y ) = −B(X, JθY ), where B is the Killing form of g and J is conjugation with respect to g0 . H Let H be any θ-stable Cartan subgroup. Then h0 = bH 0 ⊕ a0 , with H H b0 ⊆ k0 , a0 ⊆ p0 . We can associate to H a cuspidal parabolic subgroup QH with Langlands decomposition M H AH N H . The group M H is a reductive Lie group with compact Cartan subgroup H B . To avoid overloading the notation, we will not use superscripts for these subgroups when working with a fixed H or whenever is clear from the context. 2.2. Discrete series of M (cf. [8, XII.8]). Let M0 be the identity component of M , ZM the center of M , and define M # = M0 ZM . If α ∈ ∆(g0 , h0 ) is a real root, let γα = exp 2πiHα , where Hα ∈ a is the standard co-root. If F (B) is the group generated by all the γα , then F (B) is a finite abelian subgroup of ZM . It also holds M # = M0 F (B). Discrete series representation of M # are parametrized by pairs (λ, χ) such that: 1) λ is a discrete series parameter of M0 ; that is, λ ∈ ib00 is nonsingular and λ − ρM is analytically integral, or equivalently, λ is analytically integral, because ρM is. ˆ 2) χ ∈ F (B). 3) χ = eλ−ρM on B ∩ F (B). We denote π # (λ, χ) the respective discrete series representation of M # , which has the following properties: 1) π # (λ, χ) has infinitesimal character λ. 2) π # (λ, χ) ' π # (λ0 , χ0 ) if and only if χ = χ0 and λ0 = wλ for some w ∈ W (B0 , M0 ). Discrete series representations of M are exactly the representations # π(λ, χ) = IndM M # π (λ, χ).

Properties 1 and 2 above also hold for π(λ, χ). As in the connected case, λ is called the Harish-Chandra parameter of π(λ, χ) (or π # (λ, χ)). We will denote Sd (M ) the set of all pairs (λ, χ) that

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GUILLERMO AMES

defines a discrete series representation of M and Sd (M0 ) the set of HarishChandra parameters λ. 2.3. H-series of G. Given π(λ, χ) a discrete series representation of M , ν ∈ a0 a complex linear functional on a, the H-series of G consists of the representations ν π(H, λ, χ, ν) = IndG M AN (π(λ, χ) ⊗ e ⊗ 1) # ν = IndG M # AN (π (λ, χ) ⊗ e ⊗ 1).

π(H, λ, χ, ν) has infinitesimal character λ + ν relative to h0 . We will denote its global character by Θ(H, λ, χ, ν). The unitary H-series is the subset when ν takes pure imaginary values on a0 . 2.4. Plancherel formula. We can now write down the Plancherel formula for semisimple groups. Let Car(G) denote a set of representatives of θ-stables Cartan subgroups. Theorem 2.1. There is a nonnegative function m(H, λ, χ, ν) defined in Sd (M ) × ia00 such that   Z X X  (1) δ = Θ(H, λ, χ, ν)m(H, λ, χ, ν) dν  . H∈Car(G)

0 (λ,χ)∈Sd (M ) ν∈ia0

The function m(H, λ, χ, ν) has the following properties: (i) For each (λ, χ) ∈ Sd (M ), m(H, λ, χ, ν) is the restriction to ia00 of a meromorphic function on a0 without poles on ia00 . (ii) Exist a positive constant C and a positive integer l such that for all (λ, χ) ∈ Sd (M ), ν ∈ ia00 , we have |m(H, λ, χ, ν)| ≤ C(1 + |λ|2 )l (1 + |ν|2 )l . We refer to [5, Theorem 6.17] for a more general and explicit statement of the above formula, and to [6, Lemma 3.3] for the inequality (ii). 3. Action of P on characters. If π is an admissible representation with global character Θπ and infinitesimal character χπ , and P ∈ Z(g) is a bi-invariant differential operator, then P Θπ = χπ (P ) Θπ ([8, Prop. 10.24]). In [1] we prove a similar result for an operator P ∈ Z(U(g)K ). We recall ˆ is an irreducible unitary representation of it here. If E ∈ D0 (G) and τ ∈ K τ K, then E denotes its isotypic component. Proposition 3.1. If π is an admissible representation with infinitesimal character χπ and global character Θπ , and P is a differential operator in Z(U(g)K ), then P Θτπ = (χπ ⊗ χτ )(P ) Θτπ .


5

4. Inversion of infinitesimal characters. Let H ∈ Car(G), P ∈ Z(U(g)K ), λ ∈ b0 , µ ∈ t0 . We will set (2) P (H, λ, µ, ν) = (χλ+ν ⊗ χµ )(P ) = γhG ⊗ γ K (P ) (λ + ν, µ). ˆ with infinitesimal So if π(H, λ, χ, ν) is an H-series representation, τ ∈ K 0 K character µτ ∈ it0 , and P ∈ Z(U(g) ), we denote (3) P (H, λ, τ, ν) = P (H, λ, µτ , ν) = γhG ⊗ γ K (P ) (λ + ν, µτ ). Notice that this equation is independent of χ. With this notation, Proposition 3.1 can be written (4)

P Θ(H, λ, χ, ν) = P (H, λ, τ, ν)Θτ (H, λ, χ, ν).

Observe that P (H, λ, τ, ν) is a polynomial function of ν ∈ a0 , and we will denote it P (H, λ, τ ). Also, if P ∈ Z(g) or in Z(k), the expression P (H, λ, τ, ν) simplifies to P (H, λ, ν) or P (τ ), respectively. Proposition 4.1. Given P ∈ Z(U(g)K ) and H ∈ Car(G), if γhG ⊗ γ K (P ) has a fundamental solution on H0 × T = A × B0 × T , then there exist a constant C and a positive integer k such that kP (H, λ, τ )k ≥

C (1 +

|λ|2 )k

(1 + |τ |2 )k

ˆ ∀ (λ, τ ) ∈ Sd (M0 ) × K. Proof. This follows directly from (3) and Theorem 4.1 in [1].

Proposition 4.1 allows us to invert infinitesimal characters of the fundamental series, but we need to do it simultaneously for every H-series, H ∈ Car(G). Then we will need the following: Proposition 4.2. Let H, J ∈ Car(G), H fundamental, P ∈ Z(U(g)K ). If (γhG ⊗ γ K )(P ) has a fundamental solution on H × T , then (γjG ⊗ γ K )(P ) has a fundamental solution on J0 × T . Proof. According to Theorem 4.1 in [1], if H ∈ Car(G), then (γhG ⊗ γ K )(P ) has a fundamental solution on H0 × T if and only if (5)

kP (H, λ, µ)k ≥

C (1 +

|λ|2 )k

(1 + |µ|2 )k

ˆ0 × Tˆ. ∀ (λ, µ) ∈ B

So we just need to verify (5) for J0 × T provided that it is satisfied for H × T with H fundamental. Let Γ be a set of strongly orthogonal noncompact imaginary roots such G that bJ is obtained from b by Cayley transform cΓ . Then γ G = c−1 Γ ◦ γj .

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GUILLERMO AMES

ˆ Given λJ ∈ Sd (M0J ), its extension by 0 to ib00 defines a character in B J 0 We write λ for this extension, so cΓ (λ) = λ . Then, if ν ∈ a , P (H, λ, µ)(ν) = γ G ⊗ γ K (P ) (λ + ν, µ) = γjG ⊗ γ K (P ) (cΓ (λ + ν), µ) = γjG ⊗ γ K (P ) (λJ + ν, µ)

1

.

= P (J, λJ , µ)(ν). If P has order m in U(g), P (H, λ, µ) and P (J, λJ , µ) are polynomial functions of order ≤ m on a0 and (aJ )0 respectively, and their norms are the norms of the vectors in Cm+1 formed with their coefficients, then, by Schwarz inequality, kP (H, λ, µ)k ≤ kP (J, λJ , µ)k; so by hypothesis e C

kP (J, λJ , µ)k ≥ (1 +

|λJ |2 )k

(1 +

|µ|2 )k

∀ (λJ , µ) ∈ Jˆ0 × Tˆ.

5. Inversion of global characters. One important step in building the fundamental solution of P is the construction of distributions Rπ such that P Rπ = Θπ for each representation π that appears in the Plancherel formula. In this section we will define these distributions Rπ . First we state an analog of Proposition 6.3 in [1] for the H-series. We notice that the proof goes exactly the same way. Proposition 5.1. Let H ∈ Car(G). There exist Z ∈ Z(g), a positive constant C, an ε > 0 and a positive integer k such that |Z(H, λ, ν+z)| ≥ C(1+|λ|2 )k (1+|ν|2 )k ∀λ ∈ Sd (M0 ), ν ∈ ia00 , z ∈ a0 , |z| < ε. 1

Although this is a well-known result, we include a sketch of the proof here, since we don’t have a reference. Let T be a torus, and T1 ⊆ T any subtorus. Then there is a vector space V and a lattice Λ such that T = V/Λ. Also T1 = V1 /Λ1 , where V1 is a subspace of V and Λ1 = Λ ∩ V1 . It suffices to find another subtorus T2 such that T = T1 × T2 , and for that it suffices to find a sublattice Λ2 such that Λ = Λ1 ⊕ Λ2 . Let’s see first that Λ/Λ1 is torsion free: let η ∈ Λ and suppose that kη ∈ Λ1 for some positive integer k. Then η = (1/k)kη ∈ V1 so η ∈ Λ1 . Now if {ξ1 , . . . , ξn1 } and {η1 + Λ1 , . . . , ηn2 + Λ1 } are generating sets as free abelian groups of Λ1 and Λ/Λ1 respectively, then it’s easy to see that Λ = Z[ξi , ηj ], and we set Λ2 = Z[ηj ]. I wish to thank Prof. Joseph Wolf for giving me this sketch.


7

We are now in position to define the distributions R(H, λ, χ, ν) for P ∈ Z(U(g)K ) satisfying the hypothesis of Theorem 1.1. Let ε > 0 be given by Proposition 5.1, m the order of P and let Φ ∈ C ∞ (Pol0 (m) × C) be the nonnegative function given by [1, Lemma 6.6]. ˆ and a fixed ν ∈ a0 , we put Given (λ, χ) ∈ Sd (M ), τ ∈ K, (6)

P ν (H, λ, τ )(z) = P (H, λ, τ )(ν + z) = P (H, λ, τ, ν + z).

Finally, if dz is Lebesgue measure in a0 , f ∈ D(G), we define XZ Θτ (H, λ, χ, ν + z) (7) R(H, λ, χ, ν) = Φ(P ν (H, λ, τ ), z) dz. ν (H, λ, τ )(z) P |z| 0 which only depends on the support of f and a differential operator Dk ∈ Z(U(g)K ) such that ∀ (λ, χ) ∈ Sd (M ), ν ∈ ia00 |R(H, λ, χ, ν)(f )| ≤

C (1 +

|ν|2 )k (1

+ |λ|2 )k

kDk f kL2 (G) .

Proof. Although the argument is the same as [1, Proposition 6.8] for M AN minimal parabolic, we will sketch it here in order to verify that every step of the argument still applies for M AN any cuspidal parabolic. (i) is clear, combining (4) with [1, Lemma 6.6 (iii)] and the fact that P (H, λ, τ, ν + z) is holomorphic in z. Let’s see that (7) defines a distribution: According to [1, Lemma 6.6 (iv)] and [1, Lemma 6.7] together with Proposition 4.2, it holds, for all |z| < ε and some e k, Φ(P ν (H, λ, τ ), z) e ek 2 m 2 k (8) 1 + |τ |2 . P ν (H, λ, τ )(z) ≤ C1 (1 + |ν| ) 1 + |λ| On the other hand, if Z ∈ Z(g) and Ω ∈ Z(k) are given by Proposition 5.1 and [1, Lemma 6.1] respectively, and if s1 and s2 are positive integers, we have, for some positive integer k, C2 |Θ(H, λ, χ, ν + z) (Z t )s1 (Ωt )s2 f | τ (9) |Θ (H, λ, χ, ν + z)(f )| ≤ . ((1 + |λ|2 )(1 + |ν|2 ))s1 −k (1 + |τ |2 )s2 e ⊆ G be a compact subset. Note that for H-series representations Let K the multiplicity nτ of any K-type satisfies nτ ≤ dim τ ([8, p. 207]); so

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GUILLERMO AMES

e ∈ Z(k) and a constant C3 independent of λ ∈ by [1, Lemma 6.5] exist Ω 0 Sd (M0 ), ν, z ∈ a such that Z 1/2 2 2 e |Ωf (g)| kπ(H, λ, χ, ν + z)(g)k dg . |Θ(H, λ, χ, ν + z)(f )| ≤ C3 G

On the other hand, given ϕ in the space where π(H, λ, χ, ν + z) acts, if a(g) is the A-component of g in the KM AN decomposition, then (cf. [8, p. 169]), (π(H, λ, χ, ν + z)(g)ϕ) (k) = e−z log a(g and taking A =

sup

|e−z log a(g

−1 k)

−1 k)

(π(H, λ, χ, ν)(g)ϕ) (k),

| and Bλ,ν = sup kπ(H, λ, χ, ν)(g)k,

e g∈K,k∈K,|z| e k + 1/2 dim K. Hence Rσ,ν 2 )s2 −e k (1 + |τ | ˆ τ ∈K is a finite order distribution. To see (ii), just observe that Bλ,ν = 1 if ν ∈ ia00 , so given k if we take e t )s1 (Ωt )s2 with the s2 chosen above and s1 ≥ k + k + max(e Dk = Ω(Z k, m), (11) becomes |R(H, λ, χ, ν)(f )| ≤

C (1 +

|ν|2 )k (1

+ |λ|2 )k

kDk f kL2 (G)

e with C depending only K.

6. Demonstration of Theorem 1.1. Now we are ready to complete the proof of Theorem 1.1 with the explicit construction of the fundamental solution of P . Proposition 6.1. Let P ∈ Z(U(g)K ). Suppose that for each H ∈ Car(G) there exist a constant C and a positive integer k such that C ˆ (12) kP (H, λ, τ )k ≥ ∀ (λ, τ ) ∈ Sd (M0 ) × K. k (1 + |λ|2 ) (1 + |τ |2 )k


9

If R(H, λ, χ, ν) are the distributions defined by (7), then the map R defined by   Z X X  R= R(H, λ, χ, ν) m(H, λ, χ, ν) dν  (13) H∈Car(G)

0 (λ,χ)∈Sd (M ) ν∈ia0

is a finite order distribution which is a fundamental solution of P . Remark. If P ∈ Z(U(g)K ) is such that γ G ⊗ γ K (P ) has a fundamental solution in H × T , Propositions 4.1 and 4.2 imply (12) for all H ∈ Car(G), so Theorem 1.1 is a direct consequence of Proposition 6.1. Proof. Equality P R = δ is clear by Plancherel formula (Theorem 2.1) and because P R(H, λ, χ, ν) = Θ(H, λ, χ, ν); it only remains to prove that R is a finite order distribution. For that we will prove that each of the following are finite order distributions: Z X RH = R(H, λ, χ, ν) m(H, λ, χ, ν) dν. 0 (λ,χ)∈Sd (M ) ν∈ia0

e be a compact subset and f ∈ D e (G); for each positive integer k let Let K K Dk ∈ Z(U(g)K ) be given by Proposition 5.2 (ii); then, using Theorem 2.1 (ii),   ! Z X 1 1  kDk f kL2 (G) . |RH (f )| ≤ C1 C2 2 k−l1 (1 + |λ|2 )k−l2 ν∈ia00 (1 + |ν| ) λ∈Sd (M0 )

Choosing k large enough so that the sum and the integral are finite, we e obtain an operator D ∈ Z(U(g)K ) and a constant C depending only on K such that |RH (f )| ≤ CkDf kL2 (G) , and this proves that RH is a distribution of finite order less or equal that the order of D. 7. Final remarks. 7.1. P-convexity of G. Suppose that P ∈ Z(U(g)K ) satisfies the conditions of Proposition 6.1. The existence of fundamental solution of P implies that the differential equation P u = f has a solution u ∈ C ∞ (G) for all f ∈ D(G). Now, in order to guarantee the solvability of P u = f when f ∈ C ∞ (G), it is necessary to analyze the P -convexity of G. We have done this already in [1, §8]. We just have to notice that that argument applies for general linear semisimple groups, since Johnson’s injectivity criterion (Theorems 5.1 and 5.2 in [7]) holds for these groups.

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GUILLERMO AMES

7.2. Casimir operator. Let Ω ∈ Z(g) be the Casimir operator of G. As it was remarked in [2], Corollary 1.2 provides a fundamental solution of Ω only if G doesn’t have a compact Cartan subgroup. For the sake of completeness, we recall the argument. If h is a Cartan subalgebra of g, and if λ ∈ h0 , then χλ (Ω) = B(λ, λ) − ˆ0 , ν ∈ ia0 , then B(ρ, ρ); in particular, if H ∈ Car(G), h = b ⊕ a, λ ∈ B 0 (14)

Ω(H, λ, ν) = χλ+ν (Ω) = |λ|2 − |ν|2 − |ρ|2 ,

ˆ0 , and γ G (Ω) has a fundamental so if a 6= 0, kΩ(H, λ)k ≥ 1 for all λ ∈ B h solution on H0 . Now if G has a compact Cartan subgroup B (15)

Ω(B, λ) = |λ|2 − |ρ|2 .

Since GC is simply connected, λ = ρ is a character of B, and consequently we cannot apply Corollary 1.2 in this case. 7.3. Operators of Z(k). Let’s consider P ∈ Z(k) ⊂ Z(U(g)K ). Then P can be seen as a differential operator acting on both G and K. Now, according to [4, Theorem I], P has a fundamental solution on K if and only if there exists C > 0, k ∈ N such that (16)

|P (τ )| ≥

C (1 + |τ |2 )k

ˆ ∀ τ ∈ K.

We are in position to prove the following: Proposition 7.1. If P ∈ Z(k) ⊂ Z(U(g)K ) has a fundamental solution on K, then it has one in G. Proof. All we have to do at this point is notice that inequality (16) is nothing else than (12) for P ∈ Z(k), so we can apply Proposition 6.1 directly. Let’s finish analyzing the case of ΩK ∈ Z(k) the Casimir of K. In this case we have (see [1, §11]) γa ⊗ χτ (ΩK ) = χτ (ΩK ) = |λτ |2 − |ρK |2 . So if we choose τ such that λτ = ρK , by [1, Prop. 11.1], ΩK doesn’t have a parametrix. However, if we take ΩK + C such that C > |ρK |2 , then (ΩK + C)(H, λ, µ, ν) = χµ (ΩK + C) ≥ Constant and Theorem 1.1 applies, therefore ΩK +C does have a fundamental solution.


11

Acknowledgments. I wish to thank Prof. Joseph Wolf, who is my advisor as a postdoctoral fellow in Berkeley, for all his support during this period; and particularly for his help with many topics of this paper. I also want to thank my friend Dr. Leandro Cagliero for his unconditional will to spend his time to read, discuss, and help me with this work. Finally, I’d like to thank the Mathematics Department of the University of California at Berkeley. References [1] G. Ames, Fundamental solutions of invariant differential operators on a semisimple Lie group, Pacific J. Math., 196(1) (2000), 17-44, MR 2002h:22013. [2] A. Benabdallah and F. Rouvière, Solvability of bi-invariant operators on a semisimple Lie group, C.R. Acad. Sc. Paris, Serie I, 298(17) (1984), 405-408, MR 85j:22017, Zbl 0564.22013. ´ [3] A. Bouaziz and N. Kamoun, Equations différentielles sur les groupes et algèbres de Lie réductifs, Ann. Inst. Fourier, 50(6) (2000), 1799-1857, MR 2002i:22013, Zbl 0970.22009. [4] A. Cerezo and F. Rouvière, Solution élémentaire dún opérateur invariant ` a gauche ´ Norm. Sup., 2 (1969), 561-581, sur un groupe de Lie réel compact, Ann. Scient. Ec. MR 42 #6869, Zbl 0191.43801. [5] R. Herb and J.A. Wolf, The Plancherel theorem for semisimple Lie groups, Compositio Mathematica, 57 (1986), 271–355, MR 87h:22020, Zbl 0587.22005. [6]

, Rapidly decreasing functions on general semisimple groups, Compositio Mathematica, 58 (1986), 73-110, MR 87f:22014, Zbl 0587.22006.

[7] K. Johnson, Differential equations and an analog to Paley-Wiener theorem for linear semisimple Lie groups, Nagoya Math. Journal, 64 (1976), 17-29, MR 58 #1025, Zbl 0329.43008. [8] A. Knapp, Representation Theory of Semisimple Groups, Princeton University Press, 1986, MR 87j:22022, Zbl 0604.22001. [9] F. Knop, Harish-Chandra homomorphisms for reductive group actions, Ann. of Math., 140 (1994), 253-288, MR 95h:14045, Zbl 0828.22017. Received March 1, 2001 and revised May 30, 2001. This work was supported by CONICET. Department of Mathematics University of California Berkeley, CA 94720 E-mail address: [email protected]


FREE STOCHASTIC MEASURES VIA NONCROSSING PARTITIONS II Michael Anshelevich We show that for stochastic processes with freely independent increments, the partition-dependent stochastic measures can be expressed purely in terms of the higher stochastic measures and the higher diagonal measures of the original process.

1. Introduction. Starting with an operator-valued stochastic process with freely independent increments X(t), in [A] we defined two families {Prπ } and {Stπ } indexed by set partitions. These objects give a precise meaning to the following heuristic expressions. For a partition π = (B1 , B2 , . . . , Bn ) ∈ P(k), temporarily denote by c(i) the number of the class Bc(i) to which i belongs. Then, heuristically, Z Prπ (t) = dX(sc(1) )dX(sc(2) ) . . . dX(sc(k) ) [0,t)n

and Z Stπ (t) =

[0,t)n all si ’s distinct

dX(sc(1) )dX(sc(2) ) . . . dX(sc(k) ).

In particular, denote by ψk and ∆k the higher stochastic measures and the higher diagonal measures, defined, respectively, by Z ψk (t) = dX(s1 )dX(s2 ) . . . dX(sk ) k [0,t) all si ’s distinct

and Z

(dX(s))k .

∆k (t) = [0,t)

Rigorous definitions of all these objects in terms of Riemann sums are given below. These definitions were motivated by [RW], where corresponding objects were defined for the usual Lévy processes. There is a number of differences between the classical and the free case. First, the free increments property implies that Stπ = 0 unless π is a noncrossing partition. Second, the point of the analysis of [RW] was that while we are really interested in the stochastic measures Stπ , notably ψk , these are rather hard to define or 13

14

M. ANSHELEVICH

to handle. However, by the use of Möbius inversion these can be expressed through the Prπ . It is easy to see that if the increments of the process X commute, in the defining expression for Prπ all the terms corresponding to the same class can be collected together, and the result is just a product measure over the classes of the partition, Prπ = ∆B1 ∆B2 . . . ∆Bn . So in this way stochastic measures St can be defined using ordinary product measures. This fact is a consequence of the commutativity of the increments of the process; in the free probability case the operators do not commute, and unless the classes of the partition π are just intervals we cannot expect Prπ to be a product measure; indeed a counterexample was given in [A]. In this paper we show that while we cannot expect nice factorization properties in the general case of noncommuting variables, the free independence of the increments does imply a product-like property. Namely, by an argument similar to the above one, if the increments of the process commute, then Stπ = ψk (∆B1 , ∆B2 , . . . , ∆Bn ). This property certainly does not hold either if the increments do not commute, but if the increments are freely independent it can be modified as follows. In a noncrossing partition, one distinguishes classes which are inner, or covered by some other classes, and outer. For example, in the noncrossing partition ((1, 6, 7)(2, 5)(3)(4)(8)(9, 10)), the classes {(2, 5), (3), (4)} are inner while the classes {(1, 6, 7), (8), (9, 10)} are outer. For a partition with only outer classes, which therefore have to be intervals, the product decomposition of Prπ and the above decomposition of Stπ hold even in the noncommutative case. We show in the main theorem of this paper that the inner classes, while making a complicated contribution to Prπ , make only scalar contributions to Stπ , and those contributions commute with everything. We also use the opportunity to make extensions of the definitions of [A] in various directions. The objects Stπ and Prπ were defined by limits of Riemann-like sums with uniform subdivisions. In this paper we extend that definition to arbitrary subdivisions. We also make some preliminary steps towards defining multi-dimensional free stochastic measures. 2. Preliminaries. This paper is a sequel to [A]; see that paper for all the definitions that are not explicitly provided here. 2.1. Notation. Denote by [n] the set {1, 2, . . . , n}. For two vectors X = (X1 , X2 , . . . , Xk ) and Z = (Z1 , Z2 , . . . , Zk−1 ) denote X ◦ Z = X1 Z1 X2 Z2 . . . Xk−1 Zk−1 Xk . For a collection of vectors {Xi }ni=1 , denote by (X1 , X2 , . . . , Xn ) their concatenation.

FREE STOCHASTIC MEASURES II

15

n o (i) yj and a multi-index v = (v1 , v2 , . . . , vn ), Q (j) we will throughout the paper use the notation yv to denote nj=1 yvj . For a family of functions {Fj }, where Fj is a function of j arguments, v a vector with k components, and B ⊂ [k], denote F (v) = Fk (v) and For a collection of objects

F (B; v) = F|B| (vi(1) , vi(2) , . . . , vi(|B|) ), where B = (i(1), i(2), . . . , i(|B|)). In particular, using this notation y(B;v ) = Q (i) i∈B yvi . 2.2. Partitions. Denote by P(k) and NC (k) the lattice of all set partitions of the set [k] and its sub-lattice of noncrossing partitions. Let ˆ0 and ˆ1 be the smallest and the largest elements in the lattice ordering, namely ˆ0 = ((1), (2), . . . , (k)) and ˆ 1 = (1, 2, . . . , k). Denote by ∧ the join operation in the lattices. For π ∈ NC (k), denote by K(π) its Kreweras complement. For π ∈ P(n), define π op ∈ P(n) to be π taken in the opposite order, i.e., π op

π

i ∼ j ⇔ (n − i + 1) ∼ (n − j + 1). For π ∈ P(n), σ ∈ P(k), define π + σ ∈ P(n + k) by π+σ

π

σ

i ∼ j ⇔ ((i, j ≤ n, i ∼ j) or (i, j > n, (i − n) ∼ (j − n))). 2.3. Free cumulants. All the operators involved will live in an ambient noncommutative probability space (A, ϕ), where A is a finite von Neumann algebra, and ϕ is a faithful normal tracial state on it. Let A = (A1 , A2 , . . . , Ak ) be a k-tuple of self-adjoint operators in (A, ϕ). Denote their joint moments by M (A) = ϕ[A1 A2 . . . Ak ]. For a noncrossing partition π, denote using the above notation Y Mπ (A) = M (B; A). B∈π

Also define the combinatorial R-transform, or the collection of joint free cumulants R(A): Denoting Y Rπ (A) = R(B; A), B∈π

the functional R is determined inductively by X M (A) = Rσ (A), σ∈NC (k)

or more generally by Mπ (A) =

X σ∈NC (k) σ≤π

Rσ (A).

16

M. ANSHELEVICH

Any such relation can be inverted by using Möbius inversion, so we also get X Rπ (A) = Möb(σ, π)Mσ (A), σ∈NC (k) σ≤π

where Möb is the relative Möbius function on the lattice of noncrossing parQ titions. In particular, since |Mπ (A)| ≤ ki=1 kAi k and Möb(π, σ), |NC (k)| are products of Catalan numbers and so are bounded in norm by 4k , we Q conclude that |Rπ (A)| ≤ 16k ki=1 kAi k. Finally, the relation between the free cumulants and free independence is expressed in the “mixed cumulants are zero” condition: R(A) = 0 whenever some Ai , Aj are freely independent. Definition 1. Let X = (X (1) , X (2) , . . . , X (k) ) be a k-tuple of free stochastic measures with distributions µ(1) , µ(2) , . . . , µ(k) . Here, µ(i) is a freely infinitely divisible distribution with compact support, and X (i) is an operator-valued measure on R that is self-adjoint, additive, stationary, and has freely independent increments. We say that the k-tuple is consistent if the following extra conditions are satisfied. 1) Free increments: For a family of disjoint intervals {Ii }ni=1 and a multi n index u of length n, the family X (ui ) (Ii ) i=1 is a freely independent family. 2) Stationarity: For an interval I and a multi-index u, ϕ[X (u1 ) (I)X (u2 ) (I) . . . X (un ) (I)] depends only on u and |I|. 3) Continuity: For a fixed u, the function |I| 7→ ϕ[X (u1 ) (I)X (u2 ) (I) . . . X (un ) (I)] is continuous. Note that these conditions together imply that for an arbitrary collection (u2 ) (un ) of intervals {Ii }ni=1 , ϕ[X (u1 ) (I1 )X T (I2 ) . . . X (In )] depends only on u and the sizes of all elements of i∈G Ii : G ⊂ [n] . Note also that by definition and Möbius inversion, the stationarity and continuity properties apply not just to M , but to Mπ , R, Rπ as well. Remark 1. It is easy to see that a k-tuple X is consistent if X (i) = X k for all i, or if the family X (i) i=1 is a freely independent family. More examples are given in Lemma 10 and in Remark 5. Fix a consistent k-tuple of free stochastic measures X. For t > 0, denote = X (i) ([0, t)). Throughout most of the paper, we will consider t = 1, in which case we will omit t from the notation; in particular we will denote X (i) ([0, 1)) simply by X (i) . Let S = (I1 , I2 , . . . , IN ) be a subdivision of the

X (i) (t)


17

interval [0, t) into N = |S| disjoint half-open intervals, listed in increasing (i) order. Denote δ(S) = max1≤i≤N |Ii |. Let Xj (S) = X (i) (Ij ). In the future we will frequently omit the dependence on S and N in the notation. Notation 2. For any set G and a partition π ∈ P(k), denote o n π Gkπ = v ∈ Gk : i ∼ j ⇔ vi = vj and n o π Gk≥π = v ∈ Gk : i ∼ j ⇒ vi = vj . Denote X

Stπ (X, S) =

Xv (S)

v ∈[N ]kπ and X

Prπ (X, S) = v

Xv (S).

∈[N ]k≥π

Definition 3. Define the free stochastic and product measures depending on a partition to be the limits along the net of subdivisions of the interval [0, t) Stπ (X, t) = lim Stπ (X, S), δ(S)→0

Prπ (X, t) = lim Prπ (X, S). δ(S)→0

In particular, let the higher diagonal measure be ∆(X, t) = Stˆ1 (X, t) = Prˆ1 (X, t), and the k-dimensional free stochastic measure be ψ(X, t) = Stˆ0 (X, t). If X (i) = X for all i, we denote ∆(X, t) by ∆k (t) and ψ(X, t) by ψk (t). Here the limits are taken in the operator norm; the proof of their existence is part of the arguments in the next section. Lemma 1. For an arbitrary family of intervals {Ji ⊂ [0, 1)}ni=1 , a multiindex u, and π ∈ NC (k), Rπ (X (u1 ) (J1 ), X (u2 ) (J2 ), . . . , X (un ) (Jn )) ! Y \ Ji Rπ (X (u1 ) , X (u2 ) , . . . , X (un ) ). = B∈π i∈B

In particular, for a subdivision S = (I1 , I2 , . . . , IN ) of [0, 1), (u ) (u ) (u ) Rπ Xj 1 , Xj 2 , . . . , Xj n = |Ij ||π| Rπ X (u1 ) , X (u2 ) , . . . , X (un ) .

18

M. ANSHELEVICH

Proof. The second statement follows from the first one with all intervals Ji = Ij with a fixed j. For the first statement, it suffices to show that R(X (u1 ) (J1 ), X (u2 ) (J2 ), . . . , X (un ) (Jn )) n \ = Ji R(X (u1 ) , X (u2 ) , . . . , X (un ) ). i=1

Moreover, since each X (j) is an additive process with freely independent increments, Rπ (A1 , A2 , . . . , An ) is multi-linear in its arguments, and all mixed cumulants are equal to 0, it suffices to show that (1) R X (u1 ) (I), X (u2 ) (I), . . . , X (un ) (I) = |I| R X (u1 ) , X (u2 ) , . . . , X (un ) T with I = ni=1 Ji . First suppose that I = Ij , one of the intervals in a uniform i subdivision, with Ii = [ i−1 N , N ). Then using the same properties as above, X R X (u1 ) , X (u2 ) , . . . , X (un ) = R Xv(u1 1 ) , Xv(u2 2 ) , . . . , Xv(unn ) v ∈[N ]n N X (u ) (u ) (u ) = R Xi 1 , X i 2 , . . . , X i n i=1

(u ) (u ) (u ) = N R Xj 1 , X j 2 , . . . , X j n , where in the last equality we have used that fact that by stationarity of X, (u ) (u ) (u ) R(Xi 1 , Xi 2 , . . . , Xi n ) does not depend on i. Therefore (u ) (u ) (u ) R Xj 1 , Xj 2 , . . . , Xj n = N −1 R X (u1 ) , X (u2 ) , . . . , X (un ) = |Ij | R X (u1 ) , X (u2 ) , . . . , X (un ) . By stationarity, it follows that Equation (1) holds for |I| = 1/N and consequently for any rational |I|. The result follows for general I by the continuity assumption on X. This was the main fact used in the proofs of [A], and so the results from that paper carry over to the consistent k-tuples of free stochastic measures. Note that all of these results were proven for uniform subdivisions. However, their proofs carry over to the more general definitions of this paper without difficulty; see Lemma 2 for an example of a computation. We list some of those results, with the numbering from [A]. 1) Stπ (X) = 0 unless π is noncrossing (Theorem 1). 2) ϕ[Stπ (X)] = Rπ (X) (Corollary 2).


19

3) If π contains an inner singleton and X is centered, then Stπ (X) = 0 (Proposition 1). 4) If Z is freely independent from the free stochastic measure X, then N X

lim δ(S)→0

Xi ZXi = ϕ[Z]∆2 (X)

i=1

(Corollary 13). 5) The limit defining Stπ exists in the norm topology if the corresponding limit exists for the free Poisson process. In particular, for any consistent k-tuple X of free stochastic measures, ∆(X) is well-defined. In fact, the argument in [A] needs to be modified (Pr should be used in place of St); such a modification is contained in Lemma 7 of this paper. Results 2 and 3 are consequences of, and result 4 is parallel to, the following theorem. Main Theorem. Let π be a noncrossing partition of [k] with o(π) outer classes B1 , B2 , . . . , Bo(π) and i(π) inner classes C1 , C2 , . . . , Ci(π) . Then i(π)

Stπ (X) =

Y

R(Ci ; X) · ψ(∆(B1 ; X), ∆(B2 ; X), . . . , ∆(Bo(π) ; X)).

i=1

Remark 2. The distinction between the inner and the outer classes of a noncrossing partition was noted in [BLS]. It would be interesting to see what the relation is between the conditionally free cumulants of that paper (which are scalar-valued) and our Stπ ; cf. also [M]. Example 3. Let π be as in the theorem, and {X(t)} be the free Brownian motion. Then the free cumulants of µt are r1 (t) = 0, r2 (t) = t, rn (t) = 0 for n > 2, and the diagonal measures of X are ∆1 (t) = X(t), ∆2 (t) = t, ∆n (t) = 0 for n > 2. Therefore the theorem states that   0 if π contains a class of more than 2 elements, Stπ (X, t) = 0 if π contains an inner singleton,   i(π) #{Bj :|Bj |=2} t t ψ#{Bj :|Bj |=1} (t) otherwise ( t#{V ∈π:|V |=2} ψ#{V ∈π:|V |=1} (t) if ∀i, j, |Ci | = 2, |Bj | = 1, 2, = 0 otherwise. Example 4. Let π be as in the theorem, and {X(t)} be the free Poisson process. Then for n ≥ 1, rn (t) = t, ∆n (t) = X(t). Therefore the theorem states that Stπ (t) = ti(π) ψo(π) (t).

20

M. ANSHELEVICH

3. Proof of the theorem. We start the analysis with a single free Poisson stochastic measure. It has the following remarkable representation: One can take I 7→ X(I) to be sp(I)s, where s is a variable with a semicircular distribution freely independent from p(I), and I 7→ p(I) is a projection-valued measure, such that disjoint intervals correspond to orthogonal projections, and ϕ[p(I)] = |I|. Lemma 2. Given a subdivision S = (I1 , I2 , . . . , IN ) of [0, 1), let {pi }N i=1 be orthogonal projections adding up to 1 with ϕ[pi ] = |Ii |. Let {Zi,j }i∈[N ],j∈[k] be a family of operators (dependent on S) such that for each i, the family {Zi,j }j∈[k] is freely independent from pi . In addition, assume that for each i, at least one of Zi,j is centered, and that for all i, j, S, kZi,j k < c/16. Then N X

lim δ(S)→0

pi Zi,1 pi Zi,2 . . . Zi,k pi = 0,

i=1

where the limit is taken in the operator norm. Proof. Denote A(S) =

N X

pi Zi,1 pi Zi,2 . . . Zi,k pi =

i=1

N X

(pi , . . . , pi ) ◦ Zi ,

i=1

where Zi = (Zi,1 , . . . , Zi,k ). Then ∗ n

(A(S)A(S) ) = where (Zi

)∗

=

N X

(pi , . . . , pi ) ◦ (Zi , (Zi )∗ , . . . , Zi , (Zi )∗ ),

i=1 ∗ ). ∗ (Zi,k , . . . , Zi,1 ∗ n

Then as in [A, Theorem 3],

ϕ[(A(S)A(S) ) ] =

N X

X

RK(π) (Zi , (Zi )∗ , . . . , Zi , (Zi )∗ ) · Mπ (pi , pi , . . . , pi ).

i=1 π∈NC (2nk)

At least one Zi,j is centered, so for partitions that contribute to the sum, at least n elements do not belong to singleton classes of K(π). Therefore |K(π)| ≤ 2nk − n and |π| ≥ (n + 1) (since |π| + |K(π)| = 2nk + 1). Thus ϕ[(A(S)A(S)∗ )n ] ≤

N X

X

c2nk ϕ[pi ]|π| ≤ 42nk c2nk δ(S)n .

i=1 π∈NC (2nk) |π|≥n+1

Therefore kA(S)k2n < δ(S)1/2k 4k ck , and so kA(S)k < δ(S)1/2k (4c)k , which converges to 0 as δ(S) → 0.


21

Lemma 3. Let I → 7 X(I) = sp(I)s be a free Poisson stochastic measure (see [A]), so that Xi = spi s with pi as in Lemma 2. Let Zi,j be as in Lemma 2. Then N X

lim δ(S)→0

Xi Zi,1 Xi Zi,2 . . . Zi,k Xi = 0.

i=1

Proof. By the free independence assumption on {Zi,j }, the joint distribution of {Xi , Zi,1 , Zi,2 , . . . , Zi,k } is, for each i, entirely determined by the distribution of Xi and the joint distribution of {Zi,j }kj=1 . These distributions, and hence the conclusion of the lemma, are not changed if we assume in addition that the family {Zi,j }kj=1 is freely independent from s. (Xi , . . . , Xi ) ◦ Zi = (spi s, . . . , spi s) ◦ Zi = s((pi , . . . , pi ) ◦ (sZi s))s, where sZi s denotes the vector Zi with each term multiplied by s on both sides. Since s and Zi are freely independent, if Zi,j is centered then so is sZi,j s. Then Lemma 2 implies the result. Lemma 4. Let π ∈ NC (k) have only one outer class B consisting of n + 1 elements. That is, π = {(u0 = 1, u1 , u2 , . . . , un = k), π(1), π(2), . . . , π(n)} , where π(j) is supported on Cj = [uj−1 + 1, uj − 1]. Let X be a free Poisson stochastic measure. Then for X with all X (i) = X, Stπ (X) =

n Y

Rπ(j) (Cj ; X) · ∆(B; X).

j=1

Proof. Let Zi,j (N ) =

N X i=1

(u0 )

Xi

(u1 )

Zi,1 Xi

P

|C | X(Cj ;v ) . v ∈([N ]\{i})π(j)j

(un )

Zi,2 . . . Zi,n Xi

=

N X X

(u0 )

Xi

(un )

, . . . , Xi

◦ Z(G),

i=1 G⊂[n]

where Z(G) is a vector of length n such that ( Zi,j − ϕ[Zi,j ], j ∈ G, Z(G)j = ϕ[Zi,j ], j 6∈ G. For G 6= ∅, at least one of the Z(G)j is centered. So Lemma 3 applies and the limit, as δ(S) goes to 0, of the appropriate term is 0. On the other hand, it follows from [A, Corollary 2] that for any i the limit of ϕ[Zi,j ] is

22

M. ANSHELEVICH

Rπj (Cj ; X). We conclude that, denoting ı = (i, i, . . . , i), lim δ(S)→0

N X

(u ) (u ) (u ) Xi 0 Zi,1 Xi 1 Zi,2 . . . Zi,n Xi n

=

i=1

=

n Y j=1 n Y

Rπ(j) (X) · lim

δ(S)→0

N X

X(B;ı)

i=1

Rπ(j) (X) · ∆(B; X),

j=1

where ∆(B; X) is well-defined for the free Poisson stochastic measure by [A, Corollary 4]. On the other hand, N X

(u0 )

Xi

(u1 )

Zi,1 Xi

(un )

Zi,2 . . . Zi,n Xi

=

X

Stσ (X, S),

σ

i=1

where the sum is taken over all partitions σ of [k] which contain the class B and such that for all j, the restriction of σ to Cj is π(j). The only noncrossing partition satisfying these requirements is π, so X lim Stσ (X, S) = Stπ (X). δ(S)→0

σ

Notation 4. Let π ∈ NC (k). Then π can be written as π = (B1 (π), B2 (π), . . . , Bo(π) (π), I1 (π), I2 (π), . . . , Io(π) (π)). Here, {Bi (π)} are outer classes of π, listed in increasing order. Denote Bi = {j|∃a, b ∈ Bi : a ≤ j ≤ b}, the subset covered by Bi , and let Ii (π) be the restriction of π to the set Bi \Bi strictly covered by Bi . Denote by Ii0 (π) the So(π) noncrossing partition (Bi (π), Ii (π)). Finally, denote C(π) = [k]\ i=1 Bi , and let I(π) be the partition consisting of all inner classes of π, i.e., the restriction of π to C(π). Lemma 5. With the above notation, for a consistent k-tuple X of free stoQo(π) chastic measures, Prπ (X) = i=1 PrIi0 (π) (X). Proof. Such a product decomposition is valid for any subdivision S.

Lemma 6. Let X be a consistent k-tuple of free stochastic measures. Then: 1) The measures Prπ (X) and Stπ (X) are related as follows: For π ∈ NC (k), X Prπ (X) = Stσ (X), σ∈NC (k) σ≥π

Stπ (X) =

X σ∈NC (k) σ≥π

Möb(π, σ)Prσ (X).


23

2) Let π1 , π2 , . . . , πn be noncrossing partitions such that π = π1 + π2 + · · · + πn ∈ NC (k). For each i, identify πi with a sub-partition of π, and let Ci be the support of πi in [k]. Denote τ ∈ NC (k) the partition (C1 , C2 , . . . , Cn ). Then n Y i=1

X

Stπi (Ci ; X) =

Stσ (X).

σ∈NC (k) σ∧τ =π

Proof. Statement 1) is based on a purely combinatorial observation that X Prπ (X, S) = Stσ (X, S) σ∈P(k) σ≥π

and the fact that Stσ (X) = 0 for σ 6∈ NC (k); see Corollary 1 of [A]. Statement 2) is based on a purely combinatorial observation that n Y i=1

and the same fact.

Stπi ((Ci ; X), S) =

X

Stσ (X, S)

σ∈P(k) σ∧τ =π

Lemma 7. The limit defining Prπ (X) exists in norm if the corresponding limit exists for the free Poisson stochastic measure. Proof. Let S = (I1 , I2 , . . . , IN ) be a subdivision of [0, 1). Let T be another such subdivision, and let S ∧ T = (J1 , J2 , . . . , JM ) be their common refinement. Temporarily denote by p(s) the index i such that Js ⊂ Ii . Denote A(S) = Prπ (X, S), and similarly for S ∧ T . X X X A(S) − A(S ∧ T ) = Xs(1) (S ∧ T ) · · · Xs(k) (S ∧ T ) 1 k −1 (s )=v −1 (s )=v k p p v ∈[N ]≥π 1 1 k k X − Xu (S ∧ T ) k u∈[M ]≥π X = Xs (S ∧ T ). p(s)∈[N ]k≥π , s6∈[M ]k≥π The above expression A(S) − A(S ∧ T ) is a sum with positive coefficients. Hence so is ((A(S) − A(S ∧ T ))(A(S) − A(S ∧ T ))∗ )n . Therefore its expectation is a sum over a collection of indices, with weights given by products of |Js |, all of which are independent of the distribution of X, of free cumulants of X of order 2kn. Each of those

free

cumulants is bounded in norm by 2nk (i)

(16 kXk) , where kXk = maxi X . Since for the free Poisson process all such cumulants are equal to 1, the result is at most (16 kXk)2nk times

24

M. ANSHELEVICH

the corresponding quantity for the free Poisson process, for which we denote Prπ (X, S) by a(S). That is, ϕ[((A(S) − A(S ∧ T ))(A(S) − A(S ∧ T ))∗ )n ] ≤ (16 kXk)2nk ϕ[((a(S) − a(S ∧ T ))(a(S) − a(S ∧ T ))∗ )n ], and so kA(S) − A(S ∧ T )k2n ≤ (16 kXk)k ka(S) − a(S ∧ T )k2n , which implies in particular that kA(S) − A(S ∧ T )k ≤ (16 kXk)k ka(S) − a(S ∧ T )k . By assumption, the net a(S) converges in norm, and kA(S) − A(T )k ≤ kA(S) − A(S ∧ T )k + kA(T ) − A(S ∧ T )k ≤ (16 kXk)k (ka(S) − a(S ∧ T )k + ka(T ) − a(S ∧ T )k). Therefore the net A(S) is a Cauchy net, and so converges.

Corollary 8. Prπ (X), and hence Stπ (X), is well-defined for all π, X. Proof. Let X be a free Poisson stochastic measure. By Lemma 4, Stπ (X) is well-defined for π ∈ NC (k) with a single outer class. Since for such π and σ ∈ NC (k), σ ≥ π, σ also contains only one outer class, by Lemma 6 Part (1) we conclude that for such π, Prπ (X) is well-defined as well. By Lemma 5, Prπ (X) is then well-defined for an arbitrary π ∈ NC (k), and applying Lemma 6 Part (1) again implies that Stπ (X) is well-defined for an arbitrary π as well. Finally, by Lemma 7 the same is true for an arbitrary consistent k-tuple X of free stochastic measures. Corollary 9. Let X be a consistent k-tuple of free stochastic measures. For an interval I, define ∆(X)(I) = limδ(S)→0 Stˆ1 (X, S), where S is a subdivision of I in place of [0, 1). With this notation, ∆(X) is a free stochastic measure. Lemma 10. Let X be a consistent k-tuple of free stochastic measures. Let G1 , G2 , . . . , Gm ⊂ [k], and denote XG = (X (u1 ) , X (u2 ) , . . . X (u|G| ) ) for G = (u1 < u2 < · · · < u|G| ). Then the m-tuple (∆(XG1 ), ∆(XG2 ), . . . , ∆(XGm )) is also consistent. Proof. The free increments property and stationarity follow immediately from the corresponding properties of X. For a general n-tuple Y of free stochastic measures that has these two properties, by stationarity the continuity property is equivalent to the continuity of the function t 7→ ϕ[Y (v1 ) (t) . . . Y (vl ) (t)]


25

for all t, v. By Möbius inversion, this is equivalent to the continuity of t 7→ R(Y (v1 ) (t), . . . , Y (vl ) (t)) for all t, v. By additivity and the free increments property, this is equivalent to the continuity of this function, for all v, at t = 0, and so to the same property for M . Thus finally, for the m-tuple in the hypothesis, it suffices to prove that ϕ[∆(XG1 , t)∆(XG2 , t) . . . ∆(XGm , t)] → 0 as t → 0. Note that we do not need to put in a multi-index v since {Gi }m i=1 is already an arbitrary collection of subsets of [k]. Denote by σ ∈ NC (l) the partition (B1 , B2 , . . . , Bm ) with interval classes ( j−1 ! ) j X X Bj = |Gs | + 1, . . . , |Gs | , s=1

s=1

and let Y = (XG1 , XG2 , . . . , XGm ). Clearly Y is a consistent l-tuple. Then ϕ[∆(XG1 , t)∆(XG2 , t) . . . ∆(XGm , t)] X = ϕ[Prσ (Y(t))] = ϕ[Stτ (Y(t))] τ ≥σ

=

X

Rτ (Y(t)) =

τ ≥σ

X

t|τ | Rτ (Y)

τ ≥σ

by Lemma 1, and so goes to 0 as t → 0.

Lemma 11. Let σ = (B1 , B2 , . . . , Bn ) be an interval partition of [k]. Then ∆(∆(B1 ; X), . . . , ∆(Bn ; X)) = ∆(X). Proof. Let S = (I1 , I2 , . . . , IN ) be a subdivision of [0, 1). For each i, let Si = (Ii,1 , Ii,2 , . . . , Ii,Mi ) be a subdivision of Ii , and T be the subdivision of [0, 1) obtained by combining {Si }N i=1 . Then as δ(S1 ), . . . , δ(SN ) → 0, also δ(T ) → 0. Therefore ∆(X, T ) = ∆(X),

lim δ(S1 ),...,δ(SN )→0

and so ∆(X) is also the limit of the left-hand-side if in addition δ(S) → 0. Here ∆(X, T ) =

Mi Y N X k X

X (t) (Ii,s ).

i=1 s=1 t=1

On the other hand, Mi Y N Y n X X i=1 j=1 s=1 t∈Bj

X (t) (Ii,s ) =

N Y n X i=1 j=1

∆((Bj ; X), Si )

26

M. ANSHELEVICH

and lim

N Y n X

lim

δ(S)→0 δ(S1 ),...,δ(SN )→0

∆((Bj ; X), Si ) = lim

N Y n X

δ(S)→0

i=1 j=1

∆(Bj ; X)

i=1 j=1

= ∆(∆(B1 ; X), . . . , ∆(Bn ; X)). Therefore the difference ∆(∆(B1 ; X), . . . , ∆(Bn ; X)) − ∆(X) is the limit, as δ(S1 ), . . . , δ(SN ) → 0 and then as δ(S) → 0, of Mi Y N Y n X X

X (t) (Ii,s ) −

i=1 j=1 s=1 t∈Bj

Mi Y N X k X

X (t) (Ii,s ).

i=1 s=1 t=1

This expression is a sum with positive coefficients. Also, for the free Poisson process, ∆(∆(B1 ; X), . . . , ∆(Bn ; X)) − ∆(X) = ∆(X, . . . , X) − X = 0. By the same estimates as in Lemma 7, the result follows.

Lemma 12. For π ∈ NC (k), 

o(π)

Stπ (X) = RI(π) (C(π); X) · St(B1 (π),B2 (π),...,Bo(π) (π)) 

[

 Bi (π); X .

i=1

Proof. Let C be an inner class of π, and let π 0 ∈ NC (k − |C|) be the restriction of π to [k]\C. Then it suffices to prove that Stπ (X) = R(C; X) · Stπ0 (([k]\C); X). Denote A = Stπ (X) − R(C; X) · Stπ0 (([k]\C); X). X ϕ[(AA∗ )n ] = (−R(C; X))|G| ϕ[Stπ1 (X1 )Stπ2 (X2 ) . . . Stπ2n (X2n )], G⊂[2n]

where: • • • •

If If If If

j j j j

6∈ G, 6∈ G, ∈ G, ∈ G,

j j j j

odd, then πj = π, Xj = X. even, then πj = π op , Xj = Xop . odd, then πj = π 0 , Xj = (([k]\C); X). even, then πj = (π 0 )op , Xj = (([k]\C); X)op .

Denote πG = π1 + π2 + · · · + π2n . Let Ci (G) be the support of πi identified as a sub-partition of πG , and let τG = (C1 (G), C2 (G), . . . , C2n (G)). Then


27

by Part (2) of Lemma 6, kAk2n 2n =

X

X

ϕ[Stσ (X1 , X2 , . . . , X2n )]

σ∈NC (2nk−|G|·|C|) σ∧τG =πG

G⊂[2n]

=

X

(−1)|G| R(C; X)|G| X

(−1)|G|

G⊂[2n]

R(C; X)|G| Rσ (X1 , X2 , . . . , X2n ).

σ∈NC (2nk−|G|·|C|) σ∧τG =πG

Fix G ⊂ [2n]. Let σ ∈ NC (2nk), σ ∧ τ∅ = π∅ , where τ∅ = ˆ1k + · · · + ˆ1k and π∅ = π + π op + π + · · · + π op . Denote C op = (k + 1 − C) the class of π op corresponding to C. Since C is an inner class of π, the condition σ ∧ τ∅ = π∅ implies that (2jk + C) and ((2j + 1)k + C op ) are classes of σ for 0 ≤ j < n. Let gG map such a σ to the partition in NC (2nk − |G| · |C|) obtained by removing from σ the classes (2jk + C) for (2j + 1) ∈ G and ((2j + 1)k + C op ) for (2j + 2) ∈ G. It is easy to see that gG is a bijection onto {σ ∈ NC (2nk − |G| · |C|)|σ ∧ τG = πG }, and that R(C; X)|G| RgG (σ) (X1 , X2 , . . . , X2n ) = Rσ (X, X, . . . , X). Therefore kAk2n 2n =

X

(−1)|G|

G⊂[2n]

X

Rσ (X, X, . . . , X) = 0

σ∈NC (2nk) σ∧τ∅ =π∅

since the first sum equals to 0.

Proof of the Main Theorem. The statement of the theorem holds for π = ˆ0k . From now on, assume π > ˆ0k . The proof will proceed by induction on k. The statement of the theorem is vacuous for k = 1; assume that it holds for all tuples of less than k elements. By Lemma 12, StIi0 (π) (Bi (π); X) = RIi (π) (Bi (π)\Bi (π); X) · ∆(Bi (π); X). Therefore PrIi0 (π) (Bi (π); X) =

X

Stσi (Bi (π); X)

σi ≥Ii0 (π)

=

X σi ≥Ii0 (π)

(RI(σi ) (B(σi )\B(σi ); X) · ∆(B(σi ); X)).

28

M. ANSHELEVICH

Then by Lemma 5, o(π)

Prπ (X) =

Y

PrIi0 (π) (Bi (π); X)

i=1 o(π)

=

Y

X

i=1

σi ≥Ii0 (π)

(RI(σi ) (B(σi )\B(σi ); X) · ∆(B(σi ); X)) o(π)

=

X

RI(σ) (C(σ); X)

X

∆(Bj (σ); X)

j=1

σ≥π ∀i:Bi (σ)=Bi (π)

=

Y

RI(σ) (C(σ); X)

σ≥π ∀i:Bi (σ)=Bi (π)

× Prˆ0o(π) (∆(B1 (σ); X), . . . , ∆(Bo(π) (σ); X)). In its turn, Prˆ0o(π) (∆(B1 (σ); X), . . . , ∆(Bo(π) (σ); X)) X = Stρ (∆(B1 (σ); X), . . . , ∆(Bo(π) (σ); X)). ρ∈NC (o(π))

Since π > ˆ0k , σ has at most k −1 classes, so the induction hypothesis applies to Y = (∆(B1 (σ); X), . . . , ∆(Bo(σ) (σ); X)). Thus Stρ (Y) = RI(ρ) (C(ρ); Y) · ψ(∆(B1 (ρ); Y), . . . , ∆(Bo(ρ) (ρ); Y)). n o Define the map f : NC (o(π)) × σ ∈ NC (k)|σ ≥ π, ∀i : Bi (σ) = Bi (π) → (2)

f (ρ,σ)

σ

ρ

NC (k) by i ∼ j ⇔ ((i ∼ j) or (i ∈ Bs (σ), j ∈ Bt (σ), s ∼ t)). Note that the outer classes of f (ρ, σ) are in one-to-one correspondence with the outer classes of ρ, and each inner class of f (ρ, σ) corresponds to a unique inner class of either ρ or σ. It is easy to see that f is in fact a bijection onto {τ ∈ NC (k)|τ ≥ π}. Combining Equation (2) with Lemma 11, we see that RI(σ) (C(σ); X) · Stρ (∆(B1 (σ); X), . . . , ∆(Bo(π) (σ); X)) = RI(τ ) (C(τ ); X) · ψ(∆(B1 (τ ), X), . . . , ∆(Bo(τ ) (τ ); X)), with τ = f (ρ, σ). Therefore X Prπ (X) = RI(τ ) (C(τ ); X) τ ≥π

× ψ(∆(B1 (τ ), X), ∆(B2 (τ ); X), . . . , ∆(Bo(τ ) (τ ); X)).


29

P On the other hand, for all π, Prπ (X) = τ ≥π Stπ (X). Note that the Möbius inversion formula for π > ˆ0k involves only σ > ˆ0k . Therefore, applying this formula, Stπ (X) = RI(π) (C(π); X) × ψ(∆(B1 (π), X), ∆(B2 (π); X), . . . , ∆(Bo(π) (π); X)) i(π)

=

Y

R(Ci ; X) · ψ(∆(B1 ; X), ∆(B2 ; X), . . . , ∆(Bo(π) ; X)).

i=1

Remark 5 (Higher-dimensional analogs). The Main Theorem gives a complete description of the higher stochastic measures Stπ as given in Definition 3. However, under the original definitions of [RW] (modified for processes with freely independent increments) these only correspond to values on cubes, hence their dependence on only 1 and not k parameters. In this remark we briefly describe how one could extend the definition to more general rectangles of the form I = [a1 , b1 ) × [a2 , b2 ) × · · · × [ak , bk ). It is clear that it suffices to give the definition only for the case when for 1 ≤ i, j ≤ k the intervals [ai , bi ) and [aj , bj ) are either disjoint or the same (one then needs to show that the resulting definition is consistent). Assume that the rectangle I is of this form. Then we can define a parπ(I)

tition π(I) ∈ P(k) by i ∼ j ⇔ [ai , bi ) = [aj , bj ). Let π(I) have classes B1 , B2 , . . . , Bl . Let c(i) be the index such that i ∈ Bc(i) , 1 ≤ i ≤ k. Let X be a free stochastic measure, and X a k-tuple of free stochastic measures given by X (j) ([a, b)) = X([a − aj , b − aj )). The conditions on ai , bi imply that this k-tuple is consistent. Let S = {Sj } be subdivisions of [ac−1 (j) , bc−1 (j) ), 1 ≤ j ≤ l into intervals Ij,s . For σ ∈ NC (k), denote o n Q Sσ = v ∈ Nk : ( k Ic(i),v ) ∩ Rkσ 6= ∅ . Note that if π(I) = 1ˆ and S is a i=1

i

single subdivision with N classes, Sσ = [N ]kσ . Define Stσ (X, S) =

k X Y

X(Ii,vi )

v ∈Sσ i=1 and St(I) = limδ(S)→0 Stσ (X, S). It follows immediately that St(I) = 0 unless σ ≤ π(I). Indeed, if σ 6≤ π(I) then for any subdivision S, Sσ = ∅. Acknowledgments. This paper stems from the work started during the special semester “Free probability and operator spaces” at the Institut Henri Poincaré – Centre Emile Borel, with the support from the Fannie and John Hertz Foundation; it was completed with the support from an NSF postdoctoral fellowship. The author is grateful to the referee for numerous comments and suggestions.

30

M. ANSHELEVICH

References [A]

M. Anshelevich, Free stochastic measures via noncrossing partitions, Adv. Math., 155(1) (2000), 154-179, MR 2001k:46102, Zbl 0978.46041.

[BLS] M. Bo˙zejko, M. Leinert and R. Speicher, Convolution and limit theorems for conditionally free random variables, Pacific J. Math., 175(2) (1996), 357-388, MR 98j:46069, Zbl 0874.60010. [M]

W. Mlotkowski, Free probability on algebras with infinitely many states, Probab. Theory Related Fields, 115(4) (1999), 579-596, MR 2001b:46096, Zbl 0941.60020.

[RW] G.-C. Rota and T.C. Wallstrom, Stochastic integrals: A combinatorial approach, Ann. Probab., 25(3) (1997), 1257-1283, MR 98m:60081, Zbl 0886.60046. Received March 1, 2001 and revised July 13, 2001. Department of Mathematics University at California Berkeley, CA 94720 E-mail address: [email protected]


GROUPS ACTING ON CANTOR SETS AND THE END STRUCTURE OF GRAPHS Brian H. Bowditch We describe a variation of the Bergman norm for the algebra of cuts of a connected graph admitting a cofinite group action. By a construction of Dunwoody, this enables us to obtain nested generating sets for invariant subalgebras. We describe a few applications, in particular, to convergence groups acting on Cantor sets. Under certain finiteness assumptions one can deduce that such actions are necessarily geometrically finite, and hence arise as the boundaries of relatively hyperbolic groups. Similar results have already been obtained by Gerasimov by other methods. One can also use these techniques to give an alternative approach to the Almost Stability Theorem of Dicks and Dunwoody.

0. Introduction. In this paper, we describe some of the interconnections between the end structure of graphs, groups acting on protrees, and convergence actions on Cantor sets. Our work ties in with recent work of Gerasimov and Dunwoody and earlier work of Bergman. It was shown by Hopf in the 1930s that the space of ends of (any Cayley graph of) a finitely generated group, Γ, either consists of 0, 1 or 2 points, or else is a Cantor set. In the late 1960s, Stallings used topological methods to show that in the last case Γ splits nontrivially over a finite subgroup. Shortly afterwards, Bergman [Ber] gave a different proof of the same result. One can interpret his construction as giving us a Γ-invariant nested subset of the Boolean algebra of clopen sets of the space of ends, and hence a splitting of the group via Bass-Serre theory. In fact, one can think of this algebra combinatorially in terms of cuts of the Cayley graph, i.e., finite sets of edges which separate the graph into more than one infinite subset (cf. [Du1]). Bergman’s proof uses a certain “norm” defined on the set of such cuts. Recently Dunwoody and Swenson showed how one can use Bergman’s norm to construct nested generating sets of arbitrary invariant subalgebras of the algebra of cuts of a locally finite graph (see [DuS] and Section 8). In this paper we generalise Bergman’s norm, and some its consequences, to certain non-locally finite graphs (see Section 9). One of the main applications we 31

32

BRIAN H. BOWDITCH

describe here will be to convergence group actions on Cantor sets. Similar results have already been obtained by Gerasimov [Ger], by other methods. One can also use these ideas to give a simplified proof of the Almost Stability Theorem of [DiD] (see Section 14). The notion of a convergence group was defined by Gehring and Martin [GehM]. It extracts the essential dynamical features of a kleinian group acting on the boundary of (classical) hyperbolic space. This generalises to boundaries of proper hyperbolic spaces in the sense of Gromov [Gr], see for example, [T2, F, Bo3]. The celebrated result of [T1, Ga, CJ] tells us that every convergence action on the circle arises from a fuchsian group. (In particular, the work of Tukia [T1] deals with the case of cyclically ordered Cantor sets.) There has also been a study of convergence actions on more general continua and their applications to hyperbolic and relatively hyperbolic groups (see for example [Bo2] and the references therein). Here we turn our attention to the opposite extreme, namely convergence actions on Cantor sets. An extensive study of such actions in relation to median algebras can be found in [Ger]. An example of such an action is that of a finitely generated group acting on its space of ends, which is a Cantor set if the group is not virtually cyclic and splits nontrivially over a finite subgroup. We shall see that, under some finiteness assumptions on the group, this type of example is typical. Via Stone duality, a convergence action of a group, Γ, on a Cantor set is equivalent to a certain kind of action on a Boolean algebra. To obtain a splitting of Γ from this action, there are two issues to be addressed. The first is to find a Γ-invariant nested generating set for the Boolean algebra, and the second is to arrange that this set is cofinite (in other words to show that the algebra is finitely generated as a Γ-Boolean algebra). To get a handle on these issues, we assume that Γ is (relatively) finitely generated, and work with the algebra of cuts of a connected Γ-graph given by this hypothesis. The nested generating set is obtained using the Bergman norm (see Sections 8 and 9). In [Ger], a similar result is obtained via the theory of median algebras. For the second part, we need to assume that Γ is (relatively) almost finitely presented. We use a result of [DiD] or of [BesF] to obtain a cofinite generating set (Section 11). This is really an issue of accessibility. (We remark that it is shown in [DiD] that a finitely generated group, Γ, is accessible if and only if the Boolean algebra of almost invariant subsets is finitely generated as a Γ-Boolean algebra.) I am indebted to Victor Gerasimov for explaining to me some of his work in this area. I have benefited greatly from discussions with Martin Dunwoody, in particular for introducing me to the work of Bergman, as well as explaining to me how accessibility results could be used to obtain finitely generated algebras (see Sections 10 and 11). Some of the work for the present article was carried out while visiting the Centre de Recerca Matemàtica in

GROUPS ACTING ON CANTOR SETS

33

Barcelona. I am grateful for the support and hospitality of this institution, and for helpful discussions with Warren Dicks while there. 1. Summary of results. We shall introduce some of the results of this paper by giving a series of results, which are, in some sense, increasingly general, but which require greater elaboration in their formulation. They will be proven is Section 12. As mentioned in the introduction, most of these results have been obtained, in some form, by Gerasimov using median algebras [Ger]. The results will be made more precise in later sections. We shall assume throughout that the convergence actions are minimal, i.e., that there is no discontinuity domain. For definitions regarding convergence actions, see Section 4. The simplest example of a convergence action on a Cantor set is that of a (virtually) free group acting on its space of ends (which is the same as its boundary as a hyperbolic group). Such an action has no parabolic points. One result says that this is typical: Theorem 1.1. If Γ is an almost finitely presented group acting as a minimal convergence group on a Cantor set, M , without parabolic points, then Γ is virtually free, and there is a Γ-equivariant homeomorphism from M to the space of ends of Γ. By almost finitely presented we mean that Γ admits a cocompact properly discontinuous action on a locally finite connected 2-complex, Σ, with H1 (Σ; Z2 ) = 0. (Without loss of generality, one can assume that the action on Σ is also free.) Clearly, finite presentability in the usual sense implies almost finite presentability. More generally, it is well-known that any almost finitely generated group acts as a convergence group on its space of ends (see Section 5). If Γ is finitely presented, then Dunwoody’s accessibility theorem [Du2] tells us that we can represent Γ as a finite graph of groups with finite edge groups and finite or one-ended vertex groups. The one-ended vertex groups are uniquely determined (up to conjugacy) as the maximal one-ended subgroups of Γ, and are precisely the maximal parabolic subgroups with respect to this convergence action. We have a converse: Theorem 1.2. Suppose that Γ is an almost finitely presented group and acts as a minimal convergence group on a Cantor set, M , with all maximal parabolic subgroups finitely generated and one-ended. Then, M is equivariantly homeomorphic to the space of ends of Γ. The hypothesis on parabolic subgroups in the above result is somewhat unnatural. Note that the splitting given by Dunwoody’s accessiblity result gives us, via Bass-Serre theory, an action on a simplicial tree with finite edge stabilisers and finite quotient. To such a tree, we can associate a

34

BRIAN H. BOWDITCH

“boundary” as described in Section 6. The group acts as a convergence group on this boundary, with the infinite vertex stabilisers as the maximal parabolic subgroups. If these happen to be one-ended, then we recover the space of ends of Γ. We also have a converse: Theorem 1.3. Suppose that Γ is an almost finitely presented group and acts as a minimal convergence group on a Cantor set, M . Then, Γ has a representation as a finite graph of groups with finite edge groups such that M is equivariantly homeomorphic to the “boundary” of the associated BassSerre tree. In fact, we only really require that Γ be almost finitely presented relative to a set of parabolic subgroups (see Section 2 for a definition). More precisely: Theorem 1.4. Suppose that a group, Γ, acts as a minimal convergence group on a Cantor set, M . Suppose that G is a finite collection of parabolic subgroups of Γ with respect to this action. Suppose that Γ is almost finitely presented relative to G. Then, the conclusion of Theorem 1.3 holds. Note that, from the conclusion of Theorem 1.3, one may deduce that there are only finitely many conjugacy classes of maximal parabolic subgroups. In fact, it follows that the action of Γ on M is geometrically finite, and that Γ is hyperbolic relative to the collection of maximal parabolic subgroups (see Section 4 for definitions). Moreover, the boundary of Γ as a relatively hyperbolic group may be identified with the boundary of the Bass-Serre tree. Note that in Theorem 1.4, the splitting obtained is “relative to” G, i.e., each element of G is conjugate into a vertex group. We can further weaken the hypotheses, and assume only that Γ is finitely generated (relative to a class of parabolic subgroups). However, in this case we can only be assured of an action of Γ on a protree, as opposed to a simplicial tree. From this, one can deduce that the original action on M is an inverse limit of geometrically finite actions of the above type. This is elaborated on in Section 13. 2. Finiteness conditions. In this section, we elaborate on the finiteness conditions featured in Section 1. These are most conveniently expressed in terms of group actions on sets (cf. [Bo5]). Suppose Γ is a group. A Γ-set, V , is a set on which the group Γ acts. Given x ∈ V , we write Γ(x) = {g ∈ Γ | gx = x}. We write V∞ = {x ∈ V | |Γ(x)| = ∞}. We can interpret a property of V as a property of the group, Γ “relative to” the set of infinite point stabilisers, {Γ(x) | x ∈ V∞ }. We shall also speak of “Γ-graphs”, “Γ-trees” and “Γ-Boolean algebras” etc. for such objects admitting Γ-actions.


35

We shall say that the Γ-set, V , is cofinite if |V /Γ| is finite. A pair stabiliser is a subgroup of the form Γ(x) ∩ Γ(y) for x 6= y. We shall say that V is “0-connected” if it can be identified with the vertex set, V (K), of a connected cofinite Γ-graph, K (or equivalently, as a Γ-invariant subset of V (K) containing V∞ (K)). We say that V is 1-connected if it is the vertex set of a cofinite simply connected CW-complex (or equivalently, a Γ-invariant subset containing V∞ (Σ) of the vertex set of a cofinite simplicial complex, Σ). Note that 1-connectedness is equivalent to the assertion that for some (or equivalently any) cofinite connected Γ-graph with vertex set V , there is some n ≥ 2 such that Ωn (K) is simply connected. Here Ωn (K) is the 2-complex obtained by attaching a 2-cell along every circuit of length at most n in K. More generally, we say that V is Z2 -homologically 1-connected if Ωn (K) is Z2 -acyclic, i.e., H1 (Ωn (K); Z2 ) = 0. We say that a graph, K, is fine if, for each n, there are only finitely many circuits of length n containing any given edge. Clearly, this implies that the complex Ωn (K) described above is locally finite away from V = V (K) (and can thus be subdivided to give a simplicial complex that is locally finite away from V ). We say that a Γ-set is fine if some (hence any) cofinite connected Γ-graph, K, with vertex set V , is fine. Here, the fineness of K is equivalent to saying that there are finitely many circuits of any given length modulo Γ. Note that if V is fine and (Z2 -homologically) 1-connected we can embed V equivariantly in the vertex set, V (Σ), of a cocompact simply connected (Z2 -acyclic) 2-dimensional simplicial complex, Σ, which is locally finite away from V (Σ), and such that the stabiliser of each element of V (Σ) \ V is finite. Suppose that G is a nonempty collection of self-normalising subgroups of Γ, which is a finite union of conjugacy classes, and such that the intersection of any two distinct elements of G is finite. We may view G as a Γ-set with Γ acting by conjugation. We say that Γ is finitely generated relative to G if G is 0-connected. We say that Γ is (almost) finitely presented relative to G if G is (Z2 -homologically) 1-connected. In the case of interest to us, G will always be fine. This ties in with the usual group theoretical terminology. A group, Γ, is finitely generated if and only if it admits a cofinite action on a locally finite simplicial 1-complex. It is (almost) finitely presented if and only if it admits a cofinite action on a (Z2 -homologically) 1-connected locally finite simplicial 2-complex. Here the actions can in fact be taken to be free. If G ⊆ Γ is self-normalising with presentation hA; Ri, then Γ is finitely generated relative to G if and only if there is a finite set B ⊆ Γ, such that hA ∪ Bi. It is finitely presented relative to G, if and only if there is a finite set of words, S, such that hA ∪ B; R ∪ Si is a presentation of Γ. One can generalise this to a finite collection of self-normalising subgroups using the language of groupoids, though we shall not make that explicit here.

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3. Boolean algebras. Let B be a Boolean ring, i.e., a commutative ring with a unity element, 1, satisfying x2 = x for all x ∈ B. We write x∗ = 1 + x, x ∧ y = xy and x ∨ y = x + y + xy. Thus, (B, ∧, ∨, ∗) is a Boolean algebra. We write x ≤ y to mean that xy = x. Thus ≤ is a partial order on B, and [x 7→ x∗ ] is an order reversing involution. Given any set, X, its power set, P(X) is a Boolean algebra with P ∗ = X \ P , P ∧ Q = P ∩ Q and P ∨ Q = P ∪ Q, for P, Q ∈ P(X). Suppose M is a compact totally disconnected topological space. The set, B(M ) of clopen subsets of M is a Boolean subalgebra of P(M ). The Stone duality theorem [St] tells us that every Boolean algebra arises in this way (see for example [Si]). Suppose that B is a Boolean algebra. We associate to B a compact totally disconnected space Ξ = Ξ(B), called the Stone dual, such that B(Ξ) ∼ = B. This can be described in a number of equivalent ways. For example we can define Ξ as the set of Boolean ring homomorphisms from B to Z2 . This is a closed subset of the Tychonoff cube ZB 2 , and we topologise Ξ accordingly. Alternatively, we define Ξ as the maximal ideal spectrum of the ring B with the Zariski topology. Note that the complement of a maximal ideal in B is an ultrafilter. We therefore get the same thing by taking the set of ultrafilters on B. From this point of view, we can define a basis for the closed sets by taking a typical basis element to be the set of all ultrafilters that contain a given element of B. Suppose that B is a subalgebra of P(X) for some set X. If x ∈ X, then the ultrafilter O(x) = {P ∈ B | x ∈ P } determines a point of Ξ(B), so we get natural map from X to Ξ(B). If M is a compact and totally disconnected topological space, and B = B(M ), then this map gives us the Stone isomorphism from M to Ξ(B(M )). Note that if f : B −→ B 0 is a homomorphism, we get a continuous dual map, f∗ : Ξ(B 0 ) −→ Ξ(B). If f is surjective, then f∗ is injective, so we can identify Ξ(B 0 ) as a closed subset of Ξ(B). We say that two nonzero elements x, y ∈ B are nested if xy is equal to 0, x, y or 1 + x + y. This is equivalent to saying that one of xy, xy ∗ , x∗ y or x∗ y ∗ equals 0. We say that a subset, E ⊆ B is nested if 0, 1 ∈ / E and every pair of elements of E are nested. Note that if E is nested, then so is the ∗-invariant set E ∪ E ∗ . 4. Convergence groups. The notion of a convergence group was defined in [GehM]. For further discussion, see [T2, Bo3, T3]. Suppose that M is a compact metrisable space and that Γ is a group acting by homeomorphism on M . We say that this is a convergence action


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(or that Γ is a convergence group) if the induced action on the space of distinct triples of M (i.e., M × M × M minus the large diagonal) is properly discontinuous. This is equivalent to the statement that if (gn )n∈N is any infinite sequence of distinct elements of Γ, then there are points, a, b ∈ M , and a subsequence (gni )i such that gni |M \ {a} converges locally uniformly to b. We refer to the latter statement as the “convergence property” of Γ. A subgroup, G, of Γ is parabolic if it is infinite and fixes a unique point. Such a fixed point, x, is a called a parabolic point, and its stabiliser, Γ(x), is a maximal parabolic subgroup of Γ. We say that x is a bounded parabolic point if (M \ {x})/Γ(x) is compact. (We allow for the possibility of a parabolic group being an infinite torsion group.) A point x ∈ M is a conical limit point if there is a sequence of elements gn ∈ Γ, and distinct points, a, b ∈ M such that gn x → a and gn y → b for all y ∈ M \ {x}. It is shown in [T3] that a conical limit point cannot be a parabolic point. We say that the action of Γ on M is geometrically finite if every point of M is either a conical limit point or a bounded parabolic point. Such actions have been studied by Tukia [T3]. By the space of distinct pairs of M , we mean M × M minus the diagonal. Lemma 4.1. Suppose that Γ acts on M as a convergence group, and that x, y ∈ M are distinct and not conical limit points. Then, Γ(x) ∩ Γ(y) is finite, and the Γ-orbit of (x, y) is a discrete subset of the space of distinct pairs. Proof. Suppose, for contradiction, that gn ∈ Γ is a sequence of distinct elements of Γ with gn x → a and gn y → b with a, b ∈ M distinct. The convergence property tells us that after passing to a subsequence, either gn |M \ {x} converges (locally uniformly) to b, or else gn |M \ {y} converges (locally uniformly) to a. Thus, either x or y is a conical limit point. Recall the terms “connected” and “fine” as defined in Section 2. Lemma 4.2. Suppose that Γ acts on M as a convergence group, and that Π ⊆ M is a Γ-invariant subset. Suppose that no point of Π is a conical limit point. If Π is connected (as a Γ-set) then it is fine. Proof. We show that if K is any cofinite Γ-graph with vertex set V (K) = Π, then modulo Γ, there are only finitely many circuits of length n for any given n. Since Π is connected, we can take K to be connected, and we see that K is fine. Together with the first part of Lemma 4.1, this implies that Π is fine as claimed. Suppose, for contradiction, that (β k )k∈N is an infinite sequence of circuits of length n in K, each lying in a different Γ-orbit. We write β k = xk1 . . . xkn , taking subscripts mod n. Passing to a subsequence, we can suppose that for all i each of the edges {xki xki+1 }k lie in the same Γ-orbit. Thus, modulo Γ, we can suppose that xk0 = x0 and xk1 = x1 are independent of k. Now

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by Lemma 4.1, the set of pairs {(x1 , xk2 ) | k ∈ N} is discrete in the space of distinct pairs. Thus, again after passing to a subsequence, we can suppose that either xk2 = x2 is constant, or that xk2 → x1 . In the latter case, we can suppose that the xk2 are all distinct, so since the set of pairs (xk2 , xk3 )k is discrete, we must also have xk3 → x1 . It follows inductively that xki → x1 for all i, giving the contradiction that x0 = xkn → x1 . We can thus assume that xk2 = x2 is constant. But now, the same argument tells us that xk3 is constant, so by induction, xki is constant for all i. We derive the contradiction that β k is constant.

Note that, by the result of Tukia [T3], Lemma 4.2 applies to a set of parabolic points. A standard example of a convergence group is the action induced on the boundary, ∂X, by any properly discontinuous action of a group, Γ, on a proper (complete locally compact) hyperbolic space, X. If the action on ∂X is geometrically finite, we say that Γ is hyperbolic relative to the set, G, of maximal parabolic subgroups of Γ. (In [Bo5], we impose the additional requirement that each element of G be finitely generated, but this need not concern us here.) It is necessarily the case that G is cofinite and connected (and hence fine) as a Γ-set. Relatively hyperbolic groups were introduced by Gromov [Gr]. It turns out that they can be characterised dynamically as geometrically finite convergence groups [Y]. We note that, in the case where M is totally disconnected, we can express the convergence property in terms of the Boolean algebra, B = B(M ) of clopen subsets of M . By a ternary partition of B, we mean a triple of pairwise disjoint nonzero elements, A, B, C ∈ B such that A + B + C = 1. (In other words, M = A t B t C.) Note that A × B × C is a compact subset of the space of distinct triples of M . Suppose that Γ acts by isomorphism on a Boolean algebra B. Definition. We say that the action is a convergence action if, for any two ternary partitions, (a1 , a2 , a3 ) and (b1 , b2 , b3 ) of B, the set {g ∈ Γ | b1 ∧ga1 6= 0, b2 ∧ ga2 6= 0, b3 ∧ ga3 6= 0} is finite. To see that this agrees with the notion already defined for M , note that compact subset of the set of distinct triples of M can be finitely covered by sets of the form A × B × C, where A, B, C is a ternary partition of M . We finally note that if M is an inverse limit of compact spaces, (Mn )n∈N , each admitting a Γ-action that commutes with the inverse limit system, then the action of Γ on M is a convergence action if and only if the action on each Mn is a convergence action.


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5. Ends of graphs. In this section, we explain how to associate a “space of ends”, Ξ0 (K), to a connected graph, K. Let K be a connected graph, with vertex set V = V (K), and edge set E(K). Let V0 = V0 (K) be the set of vertices of finite degree. If W ⊆ V (K) we write I(W ) for the set of edges with precisely one endpoint in W . Definition. A subset W ⊆ V (K) is a K-break if I(W ) is finite. We write B = B(K) for the Boolean algebra of K-breaks. We write Ξ(K) = Ξ(B) for the Stone dual of B. There is a natural map, ξ : V −→ Ξ(K), defined by sending x ∈ V to the ultrafilter of elements of B containing x. Note that ξ|V0 is injective, and every point of ξ(V0 ) is isolated in Ξ(K). We write Ξ0 (K) = Ξ(K) \ ξ(V0 ). This is a closed subset of Ξ(K). Note that if K is locally finite, then Ξ0 (K) is the precisely the space of ends of K in the usual sense. Another way to define Ξ0 (K) is as follows. Let I be the ideal of B consisting of finite subsets of V , and let C(K) be the quotient B/I. There is an inclusion of Ξ(C) in Ξ(B) whose image is precisely Ξ0 (K). We shall say that K is one-ended if Ξ0 (K) consists of a single point, i.e., no finite set of edges separates K into two or more infinite subgraphs. Lemma 5.1. If a group Γ acts on a connected graph, K, with finite edge stabilisers, then the induced action of Γ on Ξ(K) is a convergence action. Proof. Note that if F ⊆ E(K) is any finite set of edges, then {g ∈ Γ | F ∩ gF 6= ∅} is finite. Suppose that (A1 , A2 , A3 ) and (B1 , B2 , B3 ) are ternary partitions of B. Let I = I(A1 ) ∪ I(A2 ) ∪ I(A3 ) and J = I(B1 ) ∪ I(B2 ) ∪ I(B3 ). Let X and Y be any connected finite subgraphs of K containing I and J respectively. Suppose that E(X) ∩ E(Y ) = ∅. Now, I ∩ E(Y ) = ∅ and so all the vertices of Y must lie in the same element of {A1 , A2 , A3 }, say V (Y ) ⊆ Ai . Similarly, V (X) ⊆ Bj for some j. We claim that V (K) ⊆ Ai ∪ Bj . For suppose x ∈ V (K) \ (Ai ∪ Bj ). Let α be a shortest path connecting x to Ai ∪ Bj . Let y, z be, respectively, the last and last but one vertices of α. Without loss of generality, y ∈ Ai . Now z ∈ / Ai , and so the edge connecting y and z must lie in I ⊆ E(X). It follows that z ∈ V (X) ⊆ Bj , giving a contradiction, and hence proving the claim. Note that if k 6= i, j, then Ak ∩ Ai = Bk ∩ Bj = ∅, so that Ak ∩ Bk = ∅. Now if g ∈ Γ with Bi ∩ gAi 6= ∅ for each i = 1, 2, 3, then, by the previous paragraph, we see that E(Y ) ∩ gE(X) 6= ∅. But the set of such g for given (A1 , A2 , A3 ) and (B1 , B2 , B3 ) is finite. This shows that Γ is a convergence group as claimed.

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In particular, we deduce the well-known fact that any finitely generated group acts a convergence group on its space of ends. (Take K to be any Cayley graph of Γ.) Suppose that f : K −→ L is a contraction onto a connected graph, L (i.e., a map such that the preimage of each edge of L is an edge of K, and the preimage of every vertex of L is a connected subgraph of K). There is a natural inclusion of B(L) into B(K). If the preimage of every finitedegree vertex of L is finite, then this descends to an injection from C(L) to C(K). If, in addition, the preimage of every infinite degree vertex of L is one-ended, then this is an isomorphism, so that Ξ0 (K) and Ξ0 (L) are canonically homeomorphic. 6. Simplicial trees. Let T be a simplicial tree with vertex set V = V (T ), edge set E(T ), and ~ ). Given ~e ∈ E(T ~ ), we write e for the underlying directed edge set E(T undirected edge, and −~e for the same edge pointing in the opposite direction. Let V (~e) ⊆ V be the set of vertices, v, such that ~e points towards v. If ~ ), we write ~e < f~ to mean that f~ points towards ~e and ~e points ~e, f~ ∈ E(T away from f~. (In some papers the opposite convention is used.) Note that this is equivalent to saying that V (~e) is strictly contained in V (f~). Clearly ~ ) with order reversing involution [~e 7→ −~e]. ≤ is a partial order on E(T ~ ) is a transversal if, for all ~e ∈ E(T ~ ), precisely one of A subset, F ⊆ E(T ~e or −~e lies in F . A transversal, F , is a flow on T if no two elements of F point away from each other (i.e., there do not exist ~e, f~ ∈ F with ~e ≤ −f~). A flow must be of one of two types. Either there is some (unique) v ∈ V (T ) such that each element of F points towards v, or else there is some infinite ray, α ⊆ T such that all edges of F ∩ E(α) point away from its basepoint, and all other elements of F point towards α. We can identify the set of flows of the second kind with the boundary, ∂T , of T , thought of as a hyperbolic space in the sense of Gromov [Gr]. Recall the definitions of B = B(T ) and Ξ(T ) from Section 5. Note that ~ )} is a nested set of generators for B. (The E = E(T ) = {V (~e) | ~e ∈ E(T ~ ) agrees with that on E ⊆ B, and the involution partial order, ≤, on E(T [~e 7→ −~e] corresponds to the involution [W 7→ W ∗ ].) We can think of an element of Ξ(T ) as an ultrafilter on B(T ). The elements of E lying in that ultrafilter define a flow on T . In this way we may identify Ξ(T ) with the set of flows on T , and hence with V (T ) t ∂T . Now, T is fine (as defined in Section 2) and hyperbolic (in the sense of Gromov). It thus has associated with it a compact space ∆T , as in [Bo5]. As a set, ∆T may be defined as V (T ) t ∂T . Note that any two elements, x, y ∈ ∆T can be connected by a unique arc, [x, y], which may be compact,


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a ray, or a bi-infinite geodesic depending on whether or not x, y ∈ V (T ). Given x ∈ ∆T and I ⊆ E(T ), let B(x, I) = {y ∈ ∆T | I ∩ E([x, y]) = ∅}. We define a topology on ∆T by taking a neighbourhood base of x ∈ ∆(T ) to be the collection {B(x, I)}I as I runs over all finite subsets E(T ). In this topology, ∆T is compact and Hausdorff. We see that, as sets, ∆T can be identified with Ξ(T ), and one can readily verify that the two topologies agree. As with more general graphs (Section 5), the isolated points of ∆T ≡ Ξ(T ) are precisely the vertices of T of finite degree. We shall define the ideal boundary ∆0 T , of the tree, T , as the space ∆T minus the vertices of T of finite degree. Again, this can be identified with with Ξ0 (T ) as defined in Section 5. If a group Γ acts on a simplicial tree, T , with finite edge stabilisers, then the induced action on Ξ(T ) ≡ ∆T is a convergence action. In fact, if the action of Γ on T is cofinite, then the action on ∆T is geometrically finite, with the infinite vertex groups precisely the maximal parabolic subgroups. Thus, Γ is hyperbolic relative to the infinite vertex groups, and its boundary can be identified with Ξ0 (T ) ≡ ∆0 T (see [Bo5]). Moreover, if Γ is finitely generated, and each infinite vertex group is finitely generated and one-ended, then from the last paragraph of Section 5, we see that ∆0 T is canonically homeomorphic to the space of ends of Γ. 7. Protrees and nested sets. A protree is a set Θ, with an involution [x 7→ x∗ ] and a partial order, ≤, with the property that for any x, y ∈ Θ, precisely one of the six relations x < y, x∗ < y, x < y ∗ , x∗ < y ∗ , x = y, x = y ∗ holds. This notion is due to Dunwoody (see for example [Du4]). We refer to a ∗-invariant subset of Θ as a subprotree. An example of a protree is the directed edge set of any simplicial tree, as described in the last section. In fact, any finite protree can be realised as the directed edge set of a finite tree (a property which could serve as an equivalent definition). More generally any “discrete” protree can be realised as the directed edge set of a simplicial tree. A protree is said to be discrete if, for all x, y ∈ Θ, the set {z ∈ Θ | x ≤ z ≤ y} is finite. We can define transversals and flows on a protree, exactly as for simplicial trees. (However, we cannot in general classify flows in the same way.) Suppose Θ is a protree. Let F = F(Θ) be the Boolean ring with generating set Θ, and with relations x + x∗ = 1 for all x ∈ Θ and xy = 0 for all x, y ∈ Θ satisfying x < y ∗ in Θ. (It follows that x < y ∗ also in F(Θ) in the sense defined in Section 3.) Note that it follows that if x, y ∈ Θ are distinct, then xy must be equal to 0, x, y or 1 + x + y. In particular, it follows that any element of F can be written as a sum of finitely many elements of Θ.

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Note that if x ∈ Θ, then we can define an epimorphism, θ : F −→ Z2 by θ(x) = 1 and θ(y) = 0 for all y ∈ Θ \ {x}. It therefore follows that x 6= 0 for all x ∈ Θ. Suppose that F ⊆ Θ is any transversal. We may identify F as a subset of F(Θ), and as such, it generates F(Θ) as a ring with unity (indeed as an additive group if we include also 1). If x, y ∈ F , then precisely one of the relations xy = 0, x∗ y = 0, xy ∗ = 0, x∗ y ∗ = 0 holds. Thus, Θ is a nested set of generators P for F(Θ). Any element of F(Θ) can be written in a standard form + ni=1 xi where ∈ {0, 1} and x1 , . . . , xn are distinct elements of F . If Θ is a discrete protree, and T is the corresponding simplicial tree, then there is an epimorphism φ : F(Θ) −→ B(T ) defined by φ(x) = V (~e), where ~ ) is the directed edge corresponding to x ∈ Θ. Let F ⊆ Θ be any ~e ∈ E(T transversal. If x1 , . . . , xn ∈ F are distinct, then the corresponding elements of B(T ) are distinct. If n 6= 0, it follows easily that their symmetric difference can be neither ∅ nor V (T ). In other words, we see that any standard form of any element in the kernel of φ must be trivial. Hence, the kernel is trivial, so φ is, in fact, an isomorphism. We now return to the case of a general protree, Θ, with transversal F . Suppose that Φ ⊆ Θ is any subprotree. Then, there is a natural epimorphism, θ : F(Θ) −→ F(Φ) defined by θ(x) = x if x ∈ Φ, and θ(x) = 0 if x ∈ Θ \ Φ. In particular, suppose x1 , . . . , xSn are distinct. We get an epimorphism from F(Θ) to F(Φ), where Φ = ni=1 {xi , x∗i }. Now, F(Φ) is isomorphic to the Boolean algebra on a finite tree, as above, and so it follows that x1 + · · · + xn ∈ / {0, 1}. We see that the standard form of an element of Θ (with respect to a given transversal, F ) is unique. We therefore have an explicit description of the ring F(Θ). If Φ ⊆ Θ is again any subprotree, we also get a homomorphism from F(Φ) to F(Θ) which extends the inclusion of Φ in Θ. From the above description, it is clear that this is injective. We therefore get a surjective map from Ξ(Θ) to Ξ(Φ). If Θ is an increasing union of subprotrees, (Θn )n∈N , then it is easily seen that Ξ(Θ) in an inverse limit of the system (Ξ(Θn ))n of topological spaces. Let Ξ(Θ) = Ξ(F(Θ)) be the Stone dual. If we think of an element of Ξ(Θ) as an ultrafilter on F(Θ), then its intersection with Θ is a flow on Θ. We may therefore identify Ξ(Θ) with the set of flows on Θ. For non-discrete protrees, however, we do not get a clear distinction between vertices and boundary points, as in the simplicial case. Suppose that B is any Boolean algebra with a nested set of generators, E ⊆ B. Now, E has the structure of a protree, with involution and partial order induced from B. We can therefore construct the Boolean algebra F = F(E) as above, and identify B as a quotient of F. Note that Ξ(B) can thus be identified as a closed subset of the space Ξ(E).


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In fact, we can say more than this. We can formally identify E as a subset of F. When composed with the canonical epimorphism from F to B, this gives the inclusion of E into B. Let I be the kernel of the canonical epimorphism from F to B. Note that if x, y ∈ E, then x, y, x + y ∈ / I (since, by the definition of a nested set, they correspond to distinct nonzero elements of B). Now a combinatorial argument shows that I is generated P by elements of the form 1 + ni=1 xi , where x1 , . . . , xn ∈ E have the property that xi xj = 0 for i 6= j, and if y ∈ E then for some i ∈ {1, . . . , n}, we have yxi ∈ {xi , 1+xi +y}. In other words, we can think of the elements x1 , . . . , xn as a set of edges whose tails all meet at a “vertex” of degree n of the protree E. (This statement is precise if the protree E happens to be discrete, and hence the edge set of a simplicial tree.) Suppose that a ∈ Ξ(E) \ Ξ(B). Now a corresponds to a flow, F , on E. This cannot be identically zero on I and so must be nonzero on some generator of I of the above form. From this, it is easy to see that F converges to some vertex of finite degree. But such a point is easily seen to be isolated in Ξ(E). We have shown that every point of Ξ(E) \ Ξ(B) is isolated. Finally, suppose that B is a subalgebra of P(V ) for some set, V . There is a natural map from V to Ξ(B) as defined in Section 3. We therefore get a map from V to Ξ(E). The image of a point x ∈ V in Ξ(E) is defined by the flow {A ∈ E | x ∈ A} on E. 8. Construction of nested generating sets. It was shown in [DuS] how the Bergman norm can be used to construct invariant nested subsets of a Boolean algebra. Dunwoody has observed how an elaboration of this argument in fact gives us nested generating sets. The central idea may be formulated in a general fashion as follows. Let B be a Boolean ring. We say that two elements, x, y ∈ B are disjoint if xy = 0. Suppose that S is an ordered abelian group (or cancellative semigroup). Suppose that to each disjoint pair, x, y ∈ B, we have associated an element σ(x, y) ∈ S. We suppose that σ(x, y) = σ(y, x) ≥ 0, and that σ(x, y) > 0 if x, y 6= 0. Moreover, if x, y, z ∈ B are pairwise disjoint, then σ(x, y + z) = σ(x, y) + σ(x, z). Given any x, y ∈ B, we write µ(x) = σ(x, x∗ ) and µ(x|y) = σ(xy, x∗ y). Clearly µ(x∗ ) = µ(x) and µ(x∗ |y) = µ(x|y). Suppose now that x, y ∈ B are non-nested, i.e., that xy, x∗ y, xy ∗ , x∗ y ∗ are all nonzero. If µ(y|x) ≤ µ(x|y ∗ ), then µ(xy) = σ(xy, (xy)∗ ) = σ(xy, x∗ + xy ∗ ) = σ(xy, x∗ ) + σ(xy, xy ∗ )

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≤ σ(xy, x∗ ) + σ(xy ∗ , x∗ y ∗ ) < σ(xy, x∗ ) + σ(xy ∗ , x∗ y ∗ ) + σ(xy ∗ , x∗ y) = σ(xy, x∗ ) + σ(xy ∗ , x∗ y ∗ + x∗ y) = σ(xy, x∗ ) + σ(xy ∗ , x∗ ) = σ(x, x∗ ) = µ(x). Similarly, if µ(y|x) ≤ µ(x|y), then µ(xy ∗ ) < µ(x). Now, if we allow ourselves to permute x, y, x∗ , y ∗ , then we can always arrange that µ(y|x) is minimal among {µ(x|y), µ(y|x), µ(x|y ∗ ), µ(y|x∗ )}, so that max{µ(xy), µ(xy ∗ )} < µ(x). Lemma 8.1. If µ(B \ {0, 1}) is well-ordered (as a subset of S), then B has a nested set of generators. Proof. Let E ⊆ B \ {0, 1} be the set of x ∈ B such that x does not lie in the ring generated by {z ∈ B \ {0, 1} | µ(z) < µ(x)}. Clearly, E generates B. Moreover, E is nested. For if x, y ∈ E were not nested, then, without loss of generality, max{µ(xy), µ(xy ∗ )} < µ(x). But x = xy + xy ∗ , giving a contradiction. Note that the generating set we obtain is ∗-invariant. 9. A variation on the Bergman norm. In this section, we give a generalisation of Bergman’s result [Ber]. Let V be a set, and let P(V ) be its power set. Let R = R(V ) of binary partitions of V , i.e., pairs {A, B} ⊆ P(V ) such that V = A t B. We say that {A, B} is nontrivial if A, B 6= ∅. Note that R has the structure of an abelian group, with addition defined by {A, A∗ } + {B, B ∗ } = {(A ∩ B) ∪ (A∗ ∩ B ∗ ), (A ∩ B ∗ ) ∪ (A∗ ∩ B)}. The same structure can be obtained by quotienting the additive group of the Boolean ring P(V ) by the subgroup {0, 1}. Let Ψ be the set of unordered pairs (i.e., 2-element subsets) of V . If π ∈ Ψ and P = {A, A∗ }, we say that π crosses P if π ∩ A 6= ∅ and π ∩ A∗ 6= ∅. Let Ψ(P ) = Ψ(A) be the set of π ∈ Ψ such that π crosses P . Suppose that a group Γ acts on V . Definition. A (directed ) slice of V is a subset A ⊆ V , such that Φ ∩ Ψ(A) is finite for every Γ-orbit, Φ, in Π. We refer to P = {A, A∗ } as an undirected slice. We write S(V ) for ˆ ) for the additive group of the Boolean algebra of directed slices, and S(V undirected slices.


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Suppose (Ψn )n∈N is a collection of cofinite Γ-invariant subsets of Ψ with S Ψ = Ψn . Given n ∈ N and P ∈ R, write Ψn (P ) = Ψn ∩ Ψ(P ). Thus, P ∈ S if and only if Ψn (P ) if finite for all P . We write Kn for the graph with vertex set V and edge set Ψn . Let NN be the set of infinite sequences of natural numbers. This has the structure of an ordered abelian group with the lexicographic order. We define a map, µ : Sˆ −→ NN by setting µ(P ) = (µn (P ))n , where µn (P ) = |Ψn (P )|. Recall that V is connected if it is the vertex set of some connected Γgraph with finite quotient (i.e., some finite union of the Kn is connected). We show: ˆ ⊆ NN is Theorem 9.1. Let V be a connected cofinite Γ-set. Then µ(S) well-ordered, and the map µ is finite-to-one modulo the action of Γ. It will be convenient for the proof to assume that the sets Ψn are increasing, i.e., Ψm ⊆ Ψn whenever S m ≤ n. There is no loss of generality in doing this, for if we set Ψ0n = m≤n Ψm , then if (P α )α∈N is an infinite sequence of undirected slices which is non-increasing with respect to the order determined by (Ψn )n , then some subsequence will be non-increasing with respect to the order determined by (Ψ0n )n . We may as well also assume that K0 , and hence every Kn , is connected. It is easily verified that if n ∈ N, then µn (P + Q) ≤ µn (P ) + µn (Q), with equality precisely if no element of Ψn crosses both P and Q. We now set about the proof of Theorem 9.1. For the moment, we can forget about the group, Γ. Let K be a connected graph, with vertex set V (K) = V and edge set E(K). A finite subset, I ⊆ E(K) is separating if K \ I is disconnected. (Here, K \ I denotes the graph with vertex set V (K) and edge set E(K) \ I.) Definition. A cut is a finite nonempty subset, I ⊆ E(K), such that every circuit (or equivalently, cycle) in K contains an even number of edges of I. Thus, every cut is separating. A cut is minimal if it contains no proper subcut. We similarly define minimality for separating sets. It is easily verified that if I ⊆ E(K) is finite, then the following three conditions are equivalent: (1) I is a minimal cut, (2) I is a minimal separating set, or (3) K \ I has precisely two connected components. Note that each cut I ⊆ K determines a nontrivial partition, P (K, I) = {A, A∗ }, where each path from A to A∗ contains an odd number of edges of I. Conversely, each nontrivial partition P = {A, A∗ } determines the subset, I = I(K, P ), of edges which cross from A to A∗ . If I is finite, then it is a

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cut, and we say that P is an (undirected) K-break. There is thus a natural bijection between cuts and K-breaks. Given two cuts I and J, we write I + J for their symmetric difference. This is also a cut, and the operation agrees with that already defined on the set of partitions. If J ⊆ I is a subcut, then I \ J = I + J is also a subcut, and I = J + (I \ J). We shall measure the “size” of a cut by its cardinality. We note: Lemma 9.2. Suppose e ∈ E(K) and n ∈ N. There are finitely many minimal cuts of size n containing e. Proof. Suppose, for contradiction, that the set, I, of such minimal cuts is infinite. Choose I ⊆ E(K) maximal such that I is contained in infinitely many elements of I. Let J = {J ∈ I | I ⊆ J}. Now, I cannot separate K (otherwise I would be a minimal cut, and J = {I}). Let γ be a path in K \ I connecting the endpoints of e. Now each element of J contains some edge of γ, and so some infinite subset of elements of J all contain the same edge, say f , of γ. But now I ∪ {f } is contained in infinitely many elements of J and hence of I, contradicting the maximality of I. Definition. We say that a cut, I, is blocklike if every pair of elements of I lie in a minimal subcut of I. Clearly, in Lemma 9.2, one can replace “minimal cut” by “blocklike cut”. Every cut can be uniquely decomposed into maximal blocklike cuts. One way to see this is as follows. Suppose Υ is a finite connected graph. A block of Υ is a maximal 2-vertexconnected subgraph. Two blocks intersect, if at all, in a single vertex. If e, f ∈ E(Υ) are distinct, then e, f lie in the same block if and only if they lie in some circuit in Υ, and if and only if they lie in some minimal separating set. Note that Υ is bipartite (i.e., E(Υ) is a cut) if and only if each of its blocks is bipartite. Suppose that K is any connected graph, and I ⊆ E(K) is a cut. Let Υ = Υ(K, I) be the graph obtained by collapsing each connected component of K \ I to a point. Thus, Υ is a finite connected bipartite graph. There is a canonical surjective map φ : K −→ Υ. The preimage of every connected subgraph is connected. Now it is easily checked that the preimage of every cut in Υ is a subcut of I. Moreover, every subcut, J, arises in this way: J = φ−1 (φJ). We also see easily that J is minimal if and only if φJ is minimal (since K \ J has the same number of components as φ(K \ J) = K \ φJ). Thus if follows that J is blocklike if and only if φJ is blocklike. We can thus decompose I canonically by taking the preimages of (the edge sets of) blocks of Υ. If P is a K-break, then we shall write Υ(K, P ) = Υ(K, I(K, P )). The canonical decomposition of I(K, P ) gives us a canonical decomposition of P (depending on K).


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Now suppose that K 0 is another graph on the same vertex set, V , with K ⊆ K 0 . Let φ : K −→ Υ = Υ(K, P ) and φ0 : K 0 −→ Υ0 = Υ(K 0 , P ) be collapsing maps described above. We can obtain Υ0 from Υ by identifying certain vertices of the same colour and/or adding edges between vertices of different colours. There is thus a natural map, ψ : Υ −→ Υ0 , such that ψ ◦ φ = φ0 ◦ ι, where ι is the inclusion of K in K 0 . If we measure the complexity, c(Υ), of a finite graph, Υ, by the number of edges in the complementary graph, i.e., c(Υ) = 21 |V (Υ)|(|V (Υ)| − 1) − E(Υ), then we see that the map ψ cannot increase complexity. We have c(Υ0 ) = c(Υ) if and only if ψ is an isomorphism. In this case, the canonical decomposition of P with respect to K is identical to its canonical decomposition with respect to K 0 . Note also that in general, |V (Υ0 )| ≤ |V (Υ)| ≤ |I(K, P )| + 1. Suppose now that V is a set with (Ψn )n an increasing collection of sets of pairs of V as described above. We thus get an increasing collection of graphs, (Kn )n . We suppose that K0 (and hence each of these graphs) is connected. If P is any undirected slice, then the sequence of complexities, c(Υ(Kn , P )), is non-increasing in n. Moreover, the cut I(Kn , P ) will be minimal for all sufficiently large n (depending on P ). In this case, Υ(Kn , P ) consists of a single edge (so its complexity is 0). Suppose that T ⊆ Sˆ is an infinite set of undirected slices with µ0 (P ) = |I(K0 , P )| bounded, by ν, say for P ∈ T . Thus, for each P ∈ T and n ∈ N, we have |V (Υ(Kn , P ))| ≤ |V (Υ(K0 , P ))| ≤ µ0 (P ) + 1 ≤ ν + 1, so there are only finitely many possibilities for the graph Υ(Kn , P ). Given a graph Υ, and n ∈ N, let T (n, Υ) = {P ∈ T | Υ(Kn , P ) = Υ}. For any fixed n, this partitions T into finitely many subsets. We can therefore choose Υ = Υ0 of minimal complexity, c(Υ0 ), with the property that for some n = n0 , say, the set T0 = T (n0 , Υ0 ) is infinite. For any n ≥ n0 , Υ(Kn , P ) = Υ0 for all but finitely many P ∈ T0 . (Since c(Υ(Kn , P )) ≤ c(Υ(Kn0 , P )) = c(Υ0 ), and if we had strict inequality for infinitely many P , then we would contradict the minimality of c(Υ0 ).) We now introduce the action of Γ. Note that by Lemma 9.2, Kn has only finitely many blocklike cuts of size s modulo Γ, for any n, s ∈ N. Suppose that (P α )α∈N is a sequence of undirected slices all lying in distinct Γ-orbits. Suppose that µ(P α ) is non-increasing. We want to derive a contradiction. First note that applying the construction above, with T = {P α | α ∈ N}, we can assume that, after passing to a subsequence, (P α )α has the following property. There exist n0 ∈ N and a finite bipartite graph, Υ0 , such that if n0 ≤ n ≤ α, then Υ(Kn , P α ) = Υ0 . For notational convenience (replacing Ψ0 by Ψn0 ), we can assume that n0 = 0. Now for each α, we decompose the cut I(K0 , P α ) into its maximal blocklike subcuts, I(K0 , P α ) = I1α + · · · + Ipα . Note that p is the number of blocks

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of Υ0 , and therefore constant. After passing to a subsequence, we can assume that each cut, Iiα , is the Γ-image of some fixed cut Ji , independent of α. Let Piα = P (K0 , Iiα ) and Qi = P (K0 , Ji ). Thus, P α = P1α + · · · + Ppα . Note that since the P α all lie in different Γ-orbits, we have p ≥ 2. Now for each α, I(Kn , P α ) is a minimal cut for all sufficiently large n. (To α see S this, note that K0 \ I(K0 , P ) has finitely many components. Since Ψ = n Ψn , we can find n so that any pair of such components are connected by an edge in Kn . It follows that Kn \I(Kn , P α ) has precisely two components.) Thus, there is some n (depending on α) such that the cuts {I(Kn , Piα )}pi=1 are not disjoint. Let n(α) be the smallest such n, and let m = min{n(α) | α ∈ N}. (Thus m > 0.) If n < m, then for all β ∈ N, the cuts {I(Kn , Piβ )}pi=1 are disjoint. Thus, for n < m, we have µn (P β ) = µn (P1β ) + · · · + µn (Ppβ ) = µn (Q1 ) + · · · + µn (Qp ), which is independent of β. Now for some α, the cuts {I(Km , Piα )}pi=1 are no longer disjoint, so this time we get strict inequality: µm (P α ) < µm (Q1 ) + · · · + µm (Qp ). However, for β ≥ m, we have Υ(Km , P β ) = Υ0 = Υ(K0 , P β ), and the natural map from Υ(K0 , P β ) to Υ(Km , P β ) is an isomorphism. Thus, the canonical decompositions of P β with respect to K0 and Km are identical. In particular, the cuts {I(Km , Piβ )}pi=1 are disjoint, and again, we have equality: µm (P β ) = µm (Q1 ) + · · · + µm (Qp ). Thus µm (P β ) > µm (P α ). Taking β > α, we derive a contradiction to the assumption that µ is non-increasing. This proves Theorem 9.1. We note the following corollary of this result. Suppose that V is countable. Let (Ψn )n be an enumeration of the Γ-orbits of the set of pairs, Ψ. Suppose A, B ∈ S are disjoint. Let σn (A, B) be the number of pairs in Ψn with one element in A and one element in B. Let σ(A, B) = (σn (A, B))n ∈ NN . Thus, µ(A) = σ(A, A∗ ). We are thus in the situation described in Section 8. Applying Lemma 8.1, we deduce: Theorem 9.3. Suppose Γ is a countable group, and V is a connected cofinite Γ-set. Let S be the Boolean algebra of slices of V . Then, any Γ-invariant subalgebra of S has a Γ-invariant nested set of generators. Proof. The construction of Section 8 was canonical, and hence Γ-invariant. We finish with the following observation with regard to slices. Lemma 9.4. Suppose that V is a cofinite fine Γ-set with finite pair stablisers and W ⊆ V is a Γ-invariant subset. Suppose that each element of V \ W has finite stabliser. Then the map [A −→ A ∩ W ] : S(V ) −→ S(W ) is an epimorphism of Γ Boolean algebras.


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Proof. We only really need to note that the map is surjective. To see this, suppose B ∈ S(W ). Choose any b ∈ B and any orbit transversal, {x1 , . . . , xn }, of V \ W . Let A = B ∪ {gxi | 1 ≤ i ≤ n, gb ∈ B}. To verify that A ∈ S(V ), suppose that x, y ∈ V are distinct. We connect x and y by an edge e. Let E0 be the set of edges connecting bSto each of the xi . Let K be the graph with vertex set V and edge set Γe ∪ ΓE0 . We see that K is fine. Now if some Γ-image, e0 , of e connects A to A∗ , then it lies in an arc of length at most 3 connecting B to B ∗ , and whose interior vertices are at Γ-images of x or y. Since x and y have finite degree in K, then modulo Γ, there are only finitely many possibilities for such arcs, and hence for its pair of endpoints. But since B ∈ S(Γ), there can only be finitely many such arcs in total. Thus, there are only finitely many such edges e0 . Since x, y were arbitrary, we have shown that A ∈ S(V ). 10. A finiteness result for Boolean algebras related to simplicial trees. In this section, using a result of [DiD], we give a proof of the following: Lemma 10.1. Suppose that Γ is a countable group, and that T is a cofinite simplicial Γ-tree with finite edge stabilisers. Suppose that A is any Γ-invariant Boolean subalgebra of the Boolean algebra, B(T ). Then A is finitely generated as a Γ-Boolean algebra. Here, B(T ) is the Boolean algebra of T -breaks, as defined in Section 5. To say that A is finitely generated as a Γ-Boolean algebra means that it has a generating set which is a finite union of Γ-orbits. Now, V = V (T ) has finite pair stabilisers, and so Theorem 9.3 gives us a Γ-invariant nested set, E, of generators of A. If we know already that A is finitely generated, then we can assume that E is cofinite. It follows that if e ∈ E(T ), the set {A ∈ E | e ∈ I(A)} is finite. We see that if x, y ∈ V , then {A ∈ E | x ∈ A, y ∈ / A} is finite. In particular, if A, B ∈ E, then {C ∈ E | A ⊆ C ⊆ B} is finite. In other words, E satisfies the finite interval ~ condition. We can thus identify E with the directed edge set, E(S), of a simplicial tree, S. Note that Γ acts on S with finite edge stabilisers. If x ∈ V (T ), then {A ∈ E | x ∈ A} is a flow on E, and hence determines a flow on S. Moreover, from the observation of the previous paragraph, there can be no infinite decreasing sequence in the flow (i.e., any strictly decreasing sequence, A1 ⊃ A2 ⊃ A3 ⊃ . . . with Ai ∈ E must terminate). The flow thus arises from a unique vertex of S. This therefore defines a Γ-equivariant map φ : V (T ) −→ V (S). Suppose y ∈ V (S). Let E(y) ⊆ E be the set of elements of E which ∗ ∗ correspond T ∗to directed edges with tail at y. Let E (y) = {A | A ∈ E(y)}. If x ∈ E (y) ⊆ V (T ), then each edge corresponding to an element of

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E ∗ (y) must point towards φ(x) ∈ V (S). It follows that T ∗ φ(x) = y. From this / φ(V (T )), then E S (y) = ∅, so that S observation, we see that if y ∈ E(y) = V (S). If it also happens that E(y) is finite, then E(y) is the sum (i.e., symmetric difference) of the elements of E(y). We deduce: P Lemma 10.2. If y ∈ V (S)\φ(V (T )) has finite degree in S, then ni=1 Ai = 1 in B(T ), where A1 . . . An are the elements of E which correspond to those edges with tails at y. This result will be used in the proof of Lemma 10.1. To prove Lemma 10.1, one can use an accessibility result of Dicks and Dunwoody. Following [DiD], we say that an edge of a Γ-tree is compressible if its endpoints lie in distinct Γ-orbits and if its stabiliser is equal to an incident vertex stabiliser. (Such an edge can be “compressed” in the corresponding graph of groups to give a smaller graph.) A Γ-tree is incompressible if it has no compressible edges. The following is shown in [DiD] (III 7.5, p. 92): Proposition 10.3. Let Γ be a group. Suppose that S, T are cofinite simplicial Γ-trees with finite edge stabilisers, and that S is incompressible. If there is a Γ-equivariant map from V (T ) to V (S), then |V (S)/Γ| ≤ |V (T )/Γ| + |E(T )/Γ|. (In [DiD] is assumed also that T is incompressible. However, it is clear that one can always collapse T to give another tree, T 0 , with this property, and with V (T 0 ) isomorphic as a Γ-set to a Γ-invariant subset of V (T ). This process can only decrease |V (T )/Γ| and |E(T )/Γ|.) Alternatively, we can use (the argument of) the “elliptic” case of the accessibility result of Bestvina and Feighn [BesF]. This gives a slightly different result: Proposition 10.4. Suppose that Γ is a group and that S, T are cofinite simplicial Γ-trees without edge inversions. Suppose that S is incompressible, and that every edge stabiliser of S fixes a vertex of T . If there is a Γequivariant map from V (T ) to V (S), then |V (S)/Γ| ≤ max{1, 5|E(T )/Γ|}. In particular, this applies to the case of finite edge-stabilisers. We shall sketch a proof below, which is condensed out of the relevant part of [BesF]. Our direct use of Grushko’s Theorem bypasses the use of folding sequences. We begin with some preliminary remarks. Let t be a graph of groups, and let Γ = π1 (t) be its fundamental group. Thus Γ acts on the corresponding Bass-Serre tree, T , with quotient the underlying graph |t| = T /Γ. Given v ∈ V (t) or e ∈ E(t), we write Γ(v) = Γt (v) or Γ(e) = Γt (e) for the corresponding vertex or edge groups. A subgroup of Γ is elliptic if it is conjugate into a vertex group. Let t0 be the graph of groups with the same underlying graph, and with all vertex and edge groups


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trivial. Thus, π1 (t0 ) = π1 (|t|) is free of rank β(t) = |E(t)|−|V (t)|+1. Moreover, S there is a natural epimorphism from Γ to π1 (t0 ) whose kernel contains hh v∈V (t) Γ(v)ii, where hh.ii denotes normal closure. In particular, if Γ is the normal closure of some vertex group, Γ(v), then |t| is a tree. We see easily that if every other vertex group is Γ-conjugate into Γ(v), then Γ = Γ(v). Indeed, if every vertex group is conjugate into a subgroup, G ≤ Γ(v), then Γ = G = Γ(v). Proof of Proposition 10.4. After subdividing the edges of T , we obtain a Γtree, T 0 , with V (T ) ⊆ V (T 0 ), and an equivariant morphism φ : T 0 −→ S (which sends each edge of T 0 to an edge or vertex of S). This descends to a graph-of-groups morphism φ : t0 −→ s (where |t0 | = T 0 /Γ and |s| = S/Γ) which induces the identity map on Γ = π1 (t0 ) = π1 (s). We make a series of observations. Claim 1. If v ∈ V (s) \ φ(V (t)) and G ⊆ Γ(v) is T -elliptic, then G is Γ(v)conjugate into an incident edge group. This is easily seen by considering the arc connecting a lift of v to V (S) (fixed by Γ(v)) to the φ-image of a fixed point of G in V (T ). In fact, the same argument shows that if v, w ∈ V (s) \ φ(V (t)) are the endpoints of an edge e ∈ E(s), then any T -elliptic subgroup, G, of hΓ(v) ∪ Γ(w)i is conjugate into an incident edge group adjacent to (but different from) e. In particular, from the T -ellipticity hypothesis of the proposition, this applies to G = Γ(e) = Γ(v) ∩ Γ(w). We say that a vertex, v ∈ V (s) is dead if v ∈ / φ(V (t)) and if Γ(v) is the normal closure of the incident edge groups. Otherwise it is live. We thus decompose V (s) = VD (s) t VL (s) into dead and live vertices. Claim 2. |VL (s) \ φ(V (t))| ≤ β(t) − β(s). To see this, let s be the graph of groups with underlying graph |s| obtained by collapsing each edge group in E(s) and each vertex group in φ(V (t)) to the trivial group, and by collapsing each remaining vertex group, Γs (v), to the quotient of Γs (v) by the normal closure of its incident edge groups. Thus if v ∈ V (s), then Γs (v) is nontrivial if and only if v ∈ VL (s) \ φ(V (t)). It follows that π1 (s) has at least |VL (s) \ φ(V (t))| + β(s) nontrivial free factors. Now there is a natural epimorphism from π1 (t0 ) to π1 (s). The former group is free of rank β(t), and so Claim 2 follows by Grushko’s Theorem. Clearly, |φ(V (t))| ≤ |V (t)| and so |VL (s)| ≤ |E(t)| − β(s) + 1. Claim 3. Suppose v ∈ VD (s) and G is an incident edge group. Suppose that every other incident edge group is Γ(v)-conjugate into G. Then Γ(v) = G. To see this, consider the action of Γ(v) on a minimal Γ(v)-invariant subtree of T . Let r be the corresponding graph of groups. Now by the T -ellipticity hypothesis, G is conjugate into a vertex group, Γr (w), where w ∈ V (r).

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Moreover, if H is any other vertex group of r, then again H is T -elliptic, and hence, by Claim 1, is Γ(v)-conjugate into an incident edge group to v in s. Thus H is Γ(v)-conjugate into G. But now Γ(v) = hhGii = hhΓ(w)ii, and so it follows by the discussion before the proof that Γ(v) = G. The claim follows. As an immediate corollary, we see (by the incompressibility of s) that any such vertex must have degree at least 3 in s. In particular, all terminal vertices of s are live. Claim 4. We cannot have two adjacent dead vertices of degree 2 in s. For suppose to the contrary that e ∈ E(s) has endpoints v, w ∈ VD (s), both of degree 2. From the remark after Claim 1, we see that, without loss of generality, Γ(e) is conjugate into Γ(f ), where f ∈ E(s) is the other edge incident on v. But now, from Claim 3, we derive the contradiction that v has degree at least 3. We now have enough information to bound |V (s)| in terms of the complexity of |t|. First note that since every terminal vertex of s is live (Claim 3), the number of such vertices is bounded by Claim 2. Moreover (Claim 2), we have β(s) ≤ β(t). This places a bound on the number of vertices of s of degree at least 3. Finally, Claim 4 together with the bound on live vertices places a bound on the number of vertices of degree 2. More careful bookkeeping shows that if |s| is not a point, then |V (s)| ≤ 5|E(t)| − |V (t)|. Proposition 10.4 now follows. Now since, β(s) ≤ β(t), we also get a bound on |E(s)| = |E(S)/Γ|. In fact, to obtain such a bound, we can weaken the hypotheses slightly: Corollary 10.5. Let Γ be a group and that S, T be cofinite Γ-trees with finite edge stablisers. Suppose that φ : V (T ) −→ V (S) is a Γ-equivariant map, and that each compressible edge of S is incident on some element of φ(V (T )). Then there is a bound on |E(S)/Γ| in terms of |E(T )/Γ|. Proof. To see this, note that we can obtain a Γ-tree, S 0 , by collapsing down a union of trees, F , in S/Γ, consisting of a union of compressible edges. Moreover, we can assume that if x ∈ φ(V (T )), then the image of x in S/Γ is incident to at most one edge of F that is terminal S/Γ. (Since collapsing such an edge will be sufficient to render all the other edges incident on x incompressible.) Now, Proposition 10.3 or 10.4 gives a bound on the complexity of S 0 /Γ. This, in turn, gives a bound on the number of edges of F , and hence a bound on the complexity of S/Γ as claimed. We can now set about the proof of Lemma 10.1. Suppose that Γ, T and A are as in the hypotheses. By Theorem 9.3, there is a Γ-invariant nested set, E, of generators of A. Now, if A is not finitely generated as a Γ-Boolean algebra, then we can find an infinite sequence, (An )n∈N , of elements of E such


53

that An does not lie in the Γ-Boolean algebra generated by {Ai | i < n}. Let Γ(An ) be the stabiliser of An . After passing to a subsequence, we can assume that |Γ(An )| is non-decreasing in n. Let En be the union of the Γorbits of {Ai , A∗i | i ≤ n}. Given A ∈ En , we write m(A) = i to mean that A or A∗ lies in the Γ-orbit of Ai . Now fix some n. As discussed after the statement of Lemma 10.1, we can identify En with the directed edge set of a simplicial tree, Sn , and there is an equivariant map, φ : V (T ) −→ V (Sn ). Note that |E(Sn )/Γ| = n. We claim that Sn satisfies the weakened hypotheses of Corollary 10.5. In fact, we show that any vertex whose stabiliser fixes an incident edge must lie in φ(V (T )). Suppose, to the contrary, that y ∈ V (Sn ) \ φ(V (T )) is incident on an edge ~ n ) with tail at y and with Γ(e) = Γ(y). Note that Γ(y) is finite, so ~e ∈ E(S that y has finite degree. Let A ∈ En be the element corresponding to ~e, so that Γ(A) = Γ(y). Let En (y) be the set of elements of En which correspond to edges with tails at y. We can assume that ~e is chosen so that m(A) is maximal among those elements of En (y) with stabilisers equal to Γ(y). Write P En (y) = {A, B1 , . . . , Bk }. By Lemma 10.2, we have that A = 1 + kj=1 Bj in the Boolean algebra B(T ). In particular, A lies in the Boolean algebra generated by {B1 , . . . , Bk }. Now, for each j, Γ(Bj ) ≤ Γ(A). Either Γ(Bj ) = Γ(A) so that, by the maximality of m(A), we have m(Bj ) < m(A), or else |Γ(Bj )| < |Γ(A)| so that, by the construction of the sequence (Ai )i , we again have m(Bj ) < m(A). We therefore deduce that A lies in the Γ-Boolean algebra generated by {Ai | i < m(A)}, contrary to the construction of (Ai )i . This proves the claim. Now applying Corollary 10.5, we get a bound on the complexity n = |E(Sn )/Γ|. But we could have chosen n arbitrarily large, thereby giving a contradiction. This proves Lemma 10.1. 11. An application to 1-connected Γ-sets. In this section, we apply Lemma 10.1 to show: Lemma 11.1. Suppose Σ is a countable Z2 -acyclic simplicial 2-complex which is locally finite away from the vertex set V (Σ). Suppose a group Γ acts on Σ with finite quotient and such that Γ(x) ∩ Γ(y) is finite for all distinct x, y ∈ V (Σ). Let S(V (Σ)) be the Boolean algebra of slices of V (Σ). Then, any Γ-invariant subalgebra, A, of S(V ) has a cofinite nested set of generators which is discrete as a protree. We can reduce Lemma 11.1 to Lemma 10.1 using the machinery of patterns and tracks as in [Du2]. (The overall strategy of the proof is thus

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similar to that of the accessibility result of [BesF].) Let Σ, Γ, V , A be as in the hypotheses, and let K be the 1-skeleton of Σ. Recall that a pattern, t, on Σ is a compact subset of Σ \ V (Σ) which meets each 1-simplex either in the empty set or a single point, and which meets each 2-simplex, σ, either in the empty set or in a single arc connecting two distinct faces of σ. It represents a subset, A ⊆ V (Σ) if it meets precisely those edges of Σ which connect A to A∗ . Every Σ-slice is represented by a pattern. A track is a connected pattern. Two disjoint tracks, s, t, are parallel if there is a closed subset of Σ \ V (Σ) homeomorphic to s × [0, 1] ∼ = t × [0, 1] with boundary s t t. If T is a Γ-equivariant set of disjoint pairwise non-parallel tracks, then there is a bound on |T /Γ|, (see [Du2]). By Theorem 9.3 there is a Γ-invariant nested set of generators, E, for A. By a standard construction (cf. [Du2]), we can find a set of patterns (t(A))A∈E such that t(A) represents A, t(A) = t(A∗ ), t(A) ∩ t(B) = ∅ if B 6= A, A∗ and t(gA) = gt(A) for all g ∈ Γ. By the observation of the previous paragraph, we can find a cofinite Γ-invariant set, T , of tracks such that if A ∈ E, then each connected component of the pattern t(A) is parallel to some element of T . Now T determines a simplicial tree, T , whose edges are in bijective correspondence with T , and whose vertices are in bijective S correspondence with the connected components of Σ \ T . (It is here that we use the fact that Σ is Z2 -acyclic, so that every track separates Σ.) There is a canonical map φ : V (Σ) −→ V (T ), where two vertices of Σ get mapped to the same vertex of T if and only if they are not separated by any element of A. Note that Γ acts with finite edge stabilisers on T . Suppose that A ∈ E. Let IA ⊆ E(T ) be the set of edges of T that correspond to the connected components of t(A). Now IA in turn determines an element, B(A) ∈ B(T ), with the property that I(B(A)) = IA and such that φ(A) ⊆ B(A). It follows that A = φ−1 (B(A)). Let F = {B(A) | A ∈ E}. Thus, F is a nested subset of B(T ). By Lemma 10.1, some cofinite Γinvariant subset of F is sufficient to generate the Boolean algebra generated by F. The corresponding elements of E now generate A. (This follows because A = φ−1 (B(A)) for all A ∈ E, so that any relation between the elements of F also holds between the corresponding elements of E.) We see that A has a cofinite generating set, as required. This proves Lemma 11.1. Corollary 11.2. Let V be a fine Z2 -homologically 1-connected Γ-set with finite pair stabilisers. Then any Γ-invariant subalgebra of S(V ) has a cofinite nested set of generators which is discrete as a protree. Proof. As described in Section 2, V can be embedded in a Z2 -acyclic 2complex which is locally finite away from V . Let A0 = {A ∈ S(V (Σ)) | V ∩ A ∈ A}. By Lemma 9.4, the map [A −→ V ∩ A] is an epimorphism from A0 to A. A nested set of generators for A0 as given by Lemma 11.1 then gives us the required generating set for V .


55

12. Convergence actions on Cantor sets in the finitely presented case. In this section, we shall give proofs of the main results stated in Section 1. Suppose the group Γ acts as a convergence group on the Cantor set, M . We write Π ⊆ M for the set of non-conical limit points, and B(M ) for the Boolean algebra of clopen sets of M . We suppose that there is a cofinite Γ-invariant collection, G, of parabolic subgroups of Γ such that Γ is almost finitely presented relative to G. As a Γ-set, G is isomorphic to a Γ-invariant subset, Π0 , of Π. By Lemma 4.2, Π0 is fine. Firstly consider the case where Π0 6= ∅. By hypothesis, Π0 is Z2 -homologically 1-connected. Let A = {A ∩ Π0 | A ∈ B(M )}. Since Π0 is dense in M , A is isomorphic to B(M ) as a Γ-Boolean algebra, and so Ξ(A) is Γ-equivariantly homeomorphic to Ξ(B(M )) ∼ = M . Moreover, applying Lemma 4.1, we see that A is a subalgebra of S(Π0 ). By Corollary 11.2, A has a cofinite Γ-invariant nested set of generators, E, which is discrete as a protree. It can thus be identified with the directed edge set of a cofinite simplicial tree, T , with finite edge stabilisers. We can construct the space Ξ(E) as in Section 7, where E is viewed as a protree. As in Section 6, we see that Ξ(E) be indentified with ∆T , and so ∆0 T is precisely Ξ(E) with its isolated points removed. Moreover, in Section 7, we saw that Ξ(A) can be canonically embedded in Ξ(E) so that every point of Ξ(E) \ Ξ(A) is isolated. Since Ξ(A) ≡ M is a Cantor set, we see that M can also be identified with Ξ(E) with its isolated points removed, and hence with ∆0 T . Now, Γ is hyperbolic relative to the infinite vertex stabilisers and its boundary as a relatively hyperbolic group is precisely ∆0 T (see Section 6). We have thus shown that the boundary is Γ-equivariantly homeomorphic to M . (In retrospect, we see that Π is precisely the set of parabolic points.) In the case where Π0 = ∅, we can find a Z2 -acyclic complex on which Γ acts freely. If Π 6= ∅, we take a Γ-equivariant map φ : V (Σ) −→ Π. Again, the image of φ is a fine Γ-set. Let A be the Boolean algebra {φ−1 A | A ∈ B(M )}. Again, A is a subalgebra of S(V ) isomorphic to B(M ) and the argument proceeds as before. It remains to deal with the case where Π = ∅, in other words, every point of M is a conical limit point. By [Bo1] it follows that Γ is hyperbolic with boundary M . It is now an easy consequence of Dunwoody’s accessibility theorem [Du2] that Γ is virtually free and that M ≡ ∂Γ may be identified with the space of ends of Γ. This concludes the proofs of Theorems 1.4 and 1.1. We immediately deduce Theorem 1.3. Theorem 1.2 follows from Theorem 1.4 and the discussion at the end of Section 5.

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13. Convergence actions arising from protrees. Suppose Θ is a Γ-protree such that the stabiliser of each element is finite. We say that Θ is locally discrete if each cofinite subprotree is discrete. Note that Γ acts on Ξ(Θ) by homeomorphism. Proposition 13.1. If Θ is locally discrete, then Γ acts on Ξ(Θ) as a convergence group. Proof. By definition, Ξ(Θ) is dual to the Boolean ring F(Θ) defined in Section 7. It is easily verified that the action of Γ on F(Θ) is a convergence action, as defined at the end of Section 4. If Θ is countable, then we can write it as an increasing union, Θ = S∞ n=1 Θn , of cofinite discrete Γ-protrees, Θn . We can identify Θn as the directed edge set of a simplicial Γ-tree, Tn . We see that Ξ(Θ) is an equivariant inverse limit of the spaces Ξ(Θn ) ∼ = ∆Tn , and that Ξ0 (Θ) is an equivariant inverse limit of the spaces ∆0 Tn . (This gives another proof of the fact that Γ acts as a convergence group on Ξ(Θ).) We see that the action of Γ on Ξ(Θ) is an inverse limit of geometrically finite actions. Note that an inaccessible group admits a locally discrete action on a nondiscrete protree. Dunwoody’s example of a finitely generated inaccessible group [Du3] thus gives an example of a non-geometrically finite action of such a group on a Cantor set. We show that examples of this type are typical of convergence actions of (relatively) finitely generated groups on Cantor sets: Theorem 13.2. Suppose that Γ acts as a minimal convergence group on a Cantor set, M , and that G is a finite collection of parabolic subgroups. Suppose that Γ is finitely generated relative to G. Then, Γ admits a locally discrete action on a countable protree, Θ, such that M is equivariantly homeomorphic to Ξ(Θ). In particular, the action of Γ on M is an inverse limit of geometrically finite actions. The proof of Theorem 13.2 proceeds exactly as with that of Theorem 1.4 (and 1.1) as described in Section 12, except that, in this case, we have to make do with a (fine) connected cofinite Γ-graph, K, instead of the 2complex, Σ. As before, M is equivariantly homeomorphic to Ξ(A) where A is a Γ-subalgebra of the Boolean algebra of K-breaks. Theorem 9.3 gives us an invariant nested generating set, E of A, which has the structure of a protree, Θ, as described in Section 7. We can canonically identify Ξ(Θ) as a closed subset of Ξ(A) ∼ = M , whose complement consists of isolated points and is thus empty in this case. We have equivariantly identified Ξ(Θ) with M as required.


57

14. Other applications. In this section, we sketch two further applications of Theorem 9.3. One concerns constructions of group splittings, and the other relates to the Almost Stability Theorem of [DiD]. Suppose that Γ is a one-ended finitely generated group, and that G ≤ Γ is any subgroup. Let X be a Cayley graph of Γ (or any cofinite locally finite Γ-graph). As in [Bo4] (cf. [DuS]) we say that G has codimension-one in Γ if there is a connected G-invariant subset, Y ⊆ X, such that Y /G is compact, and such that X \ Y has at least two distinct components neither of which is contained in a uniform neighbourhood of Y . (This is independent of the choice of X.) The following result also follows directly from a result of Niblo [N]: Proposition 14.1. Suppose that Γ is finitely generated and that G ≤ Γ is a codimension-one subgroup such that Γ is the commensurator of G. Then, Γ splits nontrivially as a graph of groups with G conjugate into one of the vertex groups. The “commensurator” condition means that G ∩ gGg −1 has finite index in G for all g ∈ Γ. If we assume that no vertex group is equal to an incident edge group, then it follows that all the vertex and edge groups will be commensurate with G. To prove Proposition 14.1, let K be a coset graph of G in Γ. In other words, K is a connected cofinite Γ-graph, with V (K) isomorphic as a Γ-set to the set of left translates of G, with Γ acting by left multiplication. Let x ∈ V (K) be a vertex stabilised by G. Let Y ⊆ X be as in the hypotheses. Thus, we can write X \ Y = P t Q where neither P nor Q is contained in any uniform neighbourhood of Y . Now all but finitely many Γ-images of Y are disjoint from Y , and hence contained in either P or Q. Each Γ-image of Y corresponds to a vertex of K. This therefore assigns all but finitely many elements of V (K) to one of two disjoint infinite subsets, A, B ⊆ V (K), corresponding to P and Q respectively. Assigning the remaining vertices arbitrarily, we can suppose that B = A∗ . If y ∈ A and z ∈ A∗ , then any path connecting the corresponding images of Y in X must intersect Y . Since Y /G is finite, it follows that only finitely many Γ-images of any pair {y, z} ⊆ V (K) can meet both A and A∗ . In other words, A is a slice. We have shown that the algebra of slices, B(K), of K is nontrivial, Theorem 9.3 now applies to give us a nested set of generators for B(K). Since the set of Γ-images of Y is locally finite in X, it follows that this generating set is locally discrete. We thus get an action of Γ on a simplicial tree, T . Moreover, there is an equivariant map from V (K) to V (T ). This proves Proposition 14.1.

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This result is, in some sense, “orthogonal” to the constructions of [Bo5]. Note that we cannot expect the splittings we obtain in this case to be canonical. Another application of Theorem 9.3 (as observed by Dunwoody) is to give an alternative proof of a version of the Almost Stability Theorem of [DiD]. This can be interpreted as giving a criterion for a Γ-set to be enbedded in the vertex set of a simplicial Γ-tree with finite edge stabilisers. Let Γ be a group, and X be a cofinite Γ-set. Let P(X) be the power set of X, thought of as a Γ-Boolean algebra, and let I be the ideal of finite subsets. Proposition 14.2. Suppose that Γ is a finitely generated group, and that X is a Γ-set with finite point stabilisers. Suppose that V ⊆ P(X) is a Γinvariant subset of P(X) with the property that if A, B ∈ V then A + B ∈ I. Then V can be equivariantly embedded in the vertex set of a simplicial Γ-tree. Suppose we already know that V embeds in a simplicial Γ-tree, T . We can take X to be the directed edge set of T . To each x in V , we associate the set of directed edges which point towards x. This gives a subset of P(X) satisfying the hypotheses of Proposition 13.2. Of course, the situation described by the hypotheses may be more general that this. Proposition 14.2 can be proven as follows. Given any x ∈ X, let A(x) ∈ P(V ) be the subset of V consisting of those elements of V that contain x. One verifies that A(x) is a slice of V . Let A be the subalgebra of slices of V generated by {A(x) | x ∈ V }. (A typical element of A has the form B or B ∗ , where B is either finite or is equal to A(x) for some x ∈ X.) We now apply Theorem 9.3. It is easily verified that the resulting generating set is locally discrete.

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[T1]

P. Tukia, Homeomorphic conjugates of fuchsian groups, J. Reine Angew. Math., 391 (1988), 1-54, MR 89m:30047, Zbl 0644.30027.

[T2]

, Convergence groups and Gromov’s metric hyperbolic spaces, New Zealand J. Math., 23 (1994), 157-187, MR 96c:30042, Zbl 0855.30036.

[T3]

, Conical limit points and uniform convergence groups, J. Reine. Angew. Math., 501 (1998), 71-98, MR 2000b:30067, Zbl 0909.30034.

[Y]

A.Yaman, A topological characterisation of relatively hyperbolic groups. Preprint, Southampton, 2001.

Received February 10, 2001 and revised September 20, 2001.

60

BRIAN H. BOWDITCH

Faculty of Mathematical Studies University of Southampton Highfield, Southampton SO17 1BJ Great Britain E-mail address: [email protected]


ON THE DIOPHANTINE EQUATION

xm − 1 x−1

=

yn − 1 y−1

Yann Bugeaud and T.N. Shorey m

n

−1 −1 We study the Diophantine equation xx−1 = yy−1 in integers x > 1, y > 1, m > 1, n > 1 with x 6= y. We show that, for given x and y, this equation has at most two solutions. Further, we prove that it has finitely many solutions (x, y, m, n) with m > 2 and n > 2 such that gcd(m − 1, n − 1) > 1 and (m − 1)/(n − 1) is bounded.

1. Introduction. Goormaghtigh [7] observed in 1917 that 53 − 1 213 − 1 903 − 1 25 − 1 = and 8191 = = 2−1 5−1 2−1 90 − 1 are two solutions of the Diophantine equation (1) xm − 1 yn − 1 = in integers x > 1, y > 1, m > 2, n > 2 with x 6= y. x−1 y−1 31 =

There is no restriction in assuming that y > x in (1) and thus we have m > n. This equation asks for integers having all their digits equal to one with respect to two distinct bases and we still do not know whether or not it has finitely many solutions. Even if we fix one of the four variables, it remains an open question to prove that (1) has finitely many solutions. However, when either the bases x and y, or the base x and the exponent n, or the exponents m and n are fixed, then it is proved that (1) has finitely many solutions (see [3] for references). In the first two cases, thanks to Baker’s theory of linear forms in logarithms, we can compute explicit (huge) upper bounds for the size of the solutions. As for the number of solutions, Shorey [14] proved that for two integers y > x, the Diophantine equation yn − 1 xm − 1 = in integers m > 1, n > 1, x−1 y−1 has at most 17 solutions, independently of x and y. One of the purposes of the present work is to considerably improve this estimate by showing that (2) has at most one solution provided that y is large enough and that otherwise (2) has at most two solutions.

(2)

61

62

Y. BUGEAUD AND T.N. SHOREY

When the exponents m and n are fixed, Davenport, Lewis and Schinzel [6] proved that (1) has finitely many solutions, but their proof rests on a theorem of Siegel and it is ineffective. However, when gcd(m − 1, n − 1) > 1, they are able to replace Siegel’s result by an effective argument due to Runge. Theorem DLS. Equation (1) with gcd(m − 1, n − 1) > 1 implies that max(x, y) is bounded by an effectively computable number depending only on m and n. Recently, this has been improved by Nesterenko & Shorey [13] as follows. Theorem NS. Let d ≥ 2, r ≥ 1 and s ≥ 1 be integers with gcd(r, s) = 1. Assume that m − 1 = dr and n − 1 = ds. If (x, y, m, n) satisfy (1), then max{x, y, m, n} is bounded by an effectively computable number depending only on r and s. This is the first result of the type where there is no restriction on the bases x and y and the exponents m and n extend over an infinite set. In the present work, we show that the assertion of Theorem NS continues to be valid when the ratio (m − 1)/(n − 1) is bounded. 2. Statement of the results. Our first result deals with the number of solutions of Equation (2) and improves a previous estimate of Shorey [14]. Theorem 1. Let y > x > 1 be integers. If gcd(x, y) > 1 or if y ≥ 1011 , then (2) has at most one solution. Further, if y ≥ 7, then (2) has at most two solutions. Finally, the only solutions of (2) with y ≤ 6 are given by (x, y, m, n) = (2, 5, 5, 3) or (2, 6, 3, 2). M. M¸akowski and A. Schinzel [12] proved that (2) with y ≤ 10 and m > n > 2 has only the solution (x, y, m, n) = (2, 5, 5, 3), however, for sake of completeness, we give a proof of the last statement of Theorem 1. Our proof is based on an idea of Le [10] and it combines the theory of linear forms in logarithms together with a strong gap principle proved by elementary means. As is apparent from the proof, one can derive several other interesting statements including the following. Theorem 2. Let y > x > 1 be coprime integers and assume that the smallest integer s ≥ 1 such that xs ≡ 1 (mod y) satisfies (3)

s > 4811 (log y)2 (log log y).

Then Equation (2) has no solution. Remark 1. We point out that for given y sufficiently large, Theorem 2 solves (2) for a wide set of integers x, indeed for at least ϕ(y) − ϕ(y)2/3 integers x, with 1 ≤ x ≤ y and gcd (x, y) = 1. Here, ϕ denote the Euler


63

totient function. A proof of this assertion is given just after the proofs of Theorems 1 & 2. Remark 2. Theorem 1 implies that the equations 2m − 1 5n − 1 2m − 1 90n − 1 = and = 2−1 5−1 2−1 90 − 1 have exactly one solution in integers m > 1 and n > 1, namely (m, n) = (5, 3) and (13, 3), respectively. Remark 3. M. M¸akowski and A. Schinzel [11] proved that (2) with y ≤ 10 and m > n > 2 has only the solution (x, y, m, n) = (2, 5, 5, 3), however, for sake of completeness, we give a proof of the last statement of Theorem 1. In Theorem NS quoted in the Introduction, the exponents m and n are allowed to vary such that the ratio (m−1)/(n−1) is constant. This condition also implies that y cannot be too large compared with x. We now present two new results under a similar hypothesis. The first one can be seen as an improvement of Theorem NS, although it does not imply the latter. The second one is of a different nature, namely, there is no restriction on the exponents m and n and the bases x and y extend to an infinite set. Theorem 3. Let α > 1. Equation (1) with gcd (m − 1, n − 1) ≥ 4α + 6 + α1 and (m − 1)/(n − 1) ≤ α implies that max (x, y, m, n) is bounded by an effectively computable number depending only on α. Theorem 3 does not contain Theorem NS because of the condition imposed on gcd(m − 1, n − 1). If r and s are fixed, we may remove this condition by Theorem DLS and then Theorem NS follows from Theorem 3. In the course of the proof of Theorem 3, we need an auxiliary result (namely, Lemma 4 below) which enables us to considerably improve Theorem 2 of [13]. Theorem 4. Let (x, y, m, n) be a solution of (1) with y > x. Then we have gcd(m − 1, n − 1) ≤ 33.4 m1/2 . Remark 4. Theorem 2 of [13] only asserts that there exist an effectively computable absolute constant C such that gcd(m − 1, n − 1) ≤ Cm4/5 (log m)3/5 . Further, its proof combines the theory of linear forms in logarithms together with sharp upper bounds for the size of the solutions of (1) obtained by Runge’s method, whereas the proof of Theorem 4 depends only on estimates for linear forms in two logarithms. Theorem 5. Let a > 1. Let y > x > 1 be integers such that x divides y − 1 and y ≤ xa . If (x, y, m, n) satisfies (1), then n < m ≤ 14000 a2 (log 3a)2

64


and x < n,

y < na .

Remark 5. It follows from Theorem 5 that for a given a > 1, Equation (1) has only finitely many solutions (x, y, m, n) with y ≤ xa and x | (y − 1). 3. Auxiliary lemmas. We begin with the following theorem of Baker and W¨ ustholz [2] on linear forms in logarithms. Lemma 1. Let α1 , . . . , αd be positive rational numbers of heights not exceeding A1 , . . . , Ad , respectively, where Aj ≥ e for 1 ≤ j ≤ d. Put Ω=

d Y

log Aj .

j=1

Then the inequalities 0 < |b1 log α1 + · · · + bd log αd | < exp −(16d)2(d+2) Ω log B

have no solution in integers b1 , . . . , bd of absolute values not exceeding B, where B ≥ e. In order to prove Theorems 1 & 2, we need an explicit estimate for the size of the solutions of (2). Lemma 2. Let y > x > 1 be integers and let (m, n) be a solution of (2). Then we have (4)

m ≤ 2 × 4810 (log y)2 (log m) + 1.

Proof. We rewrite (2) as yn 1 1 xm − = − , x−1 y−1 x−1 y−1 thus we get n −m x − 1 0 exp −(48)10 (log y)2 log(x + 1) log m y−1 which, combined with (5), yields the estimate stated in the lemma.


65

Apart from Lemma 1, we shall also need the following refinement, due to Mignotte [12], of a theorem of Laurent, Mignotte & Nesterenko [9] on linear forms in two logarithms. Lemma 3. Consider the linear form Λ = b2 log α2 − b1 log α1 , where b1 and b2 are positive integers. Suppose that α1 and α2 are multiplicatively independent. Put D = [Q(α1 , α2 ) : Q] / [R(α1 , α2 ) : R]. Let a1 , a2 , h, k be real positive numbers, and ρ a real number > 1. Put λ = log ρ, χ = h/λ and suppose that χ ≥ χ0 for some number χ0 ≥ 0 and that b2 b1 + + log λ + f (dK0 e) + 0.023, h ≥ D log a2 a1 ai ≥ max 1, ρ | log αi | − log |αi | + 2Dh(αi ) , (i = 1, 2), a1 a2 ≥ λ2 where f (x) = log

1+

√

√ x x−1 x log x 3 3 log x−1 + + + log + , x−1 6x(x − 1) 2 4 x−1

and 1 K0 = λ

√

2 + 2χ0 + 3

s

!2 √ 2(1 + χ0 ) 2λ 1 1 4λ 2 + χ0 + + + a1 a2 . √ 9 3 a1 a2 3 a1 a2

Put v = 4χ + 4 + 1/χ

and

m = max 25/2 (1 + χ)3/2 , (1 + 2χ)5/2 /χ .

Then we have the lower bound s !2 1 v 1 v 2 4λv 1 1 8λm log |Λ| ≥ − + + + + √ a1 a2 λ 6 2 9 3 a1 a2 3 a1 a2 n − max λ(1.5 + 2χ) + log (2 + 2χ)3/2 o √ + (2 + 2χ)2 k ∗ A + (2 + 2χ) , D log 2 where A = max{a1 , a2 }

and

1 k∗ = 2 λ

1 + 2χ 3χ

2

1 + λ

2 2 (1 + 2χ)1/2 + 3χ 3 χ

Proof. This is Theorem 2 of [12]. We apply Lemma 3 for deriving the following result:

! .

66


Lemma 4. Let α > 1 and d > 1 be an integer. Suppose that (x, y, m, n) with y > x is a solution of (1). Assume that m−1 gcd(m − 1, n − 1) = d, ≤ α. n−1 Then we have 1 . d ≤ 743 α + 2 Proof. Let α > 1 and d > 1 be an integer. We suppose that (1) with gcd(m − 1, n − 1) = d and (m − 1)/(n − 1) ≤ α is satisfied and we put m − 1 = dr,

n − 1 = ds

where r and s are positive integers. In view of the last assertion of Theorem 1, whose proof is independent of Lemma 3, we may assume that y ≥ 7. We write (1) as x y 1 1 xrd − y sd = − x−1 y−1 x−1 y−1 which implies that x(y − 1) ys (6) 0 < log − d log r < y −sd . y(x − 1) x Now we apply Lemma 3 with b1 = d, b2 = 1, α1 = y s /xr and α2 = (x(y − 1))/(y(x − 1)) in order to get a lower bound for Λ = b2 log α2 − b1 log α1 . We observe that h(α1 ) ≤ s log y, h(α2 ) ≤ 2 log y and that α1 and α2 are multiplicatively independent. Further, we put 3 log y ρ=1+ . 1 4 log(1 + x−1 ) Then we may take a1 = (2s + 3/4) log y

and a2 = (19 log y)/4.

Thus, we check that

(7)

3 log x ≤ log 1 + (x − 1) log y 4

≤ λ := log ρ ≤

4 log y 3

and (8)

a1 a2 ≥ 7λ2 , y n−1

a1 a2 /λ ≥ 19.

2xm−1

≤ which implies that log y 1 (9) ≤α+ . log x 2 We may assume that d ≥ 30. By (7) and since dK0 e ≥ 16, in order to apply Lemma 3, we have to choose the parameter h such that By (2), we observe that

h ≥ log d + 0.3.


67

Assume first that λ ≥ log d + 0.3 and apply Lemma 3 with h = λ and χ = 1. Thus v = 9, m = 16 and, using that y ≥ 7 in the definition of ρ, we infer that k ∗ ≤ 2.4. Now, we notice that λ/a1 + λ/a2 ≤ 0.77 and use (8), to obtain that 19.65 (10) log Λ > − a1 a2 − 3.5λ − log(56.7a1 + 4). λ Finally, using y ≥ 7, we conclude from (7), (9), (10) and (6) that 1 3 α+ . d ≤ 96 2 + 4s 2 We observe that the right hand side of the preceding inequality does not exceed 264(α + 1/2). Therefore, we may suppose that λ < log d + 0.3, that is, in view of (7) ! 3 log y (11) log 1 + < log d + 0.3. 1 4 log(1 + x−1 ) Next, we show that d ≤ 135 log y.

(12)

For the proof of (12), we assume that d > 135 log y and we shall arrive at a contradiction. We apply Lemma 3 once again, but with another choice of the radius ρ. Namely, we take ρ = e4 . Then λ = 4. We set b1 b2 57 h = log + + a2 a1 20 and we see that h ≥ 6, thus we may choose χ0 = 32 . We observe that 5/2 (1 + χ)3/2 a1 a2 ≥ 49 and dK0 e ≥ 27. Further, we have v ≤ h + 14 3 ,m ≤ 2 and we put H = h + 14 3 . Now, Lemma 3 yields 3 2 H a1 a2 − 6 − 2h − log(2h2 a1 a2 ) 40 17 2 ≥− H a1 a2 , 200

log Λ ≥ −

since H ≥ we get

32 3

and a1 a2 ≥ 49. Combining the preceding estimate with (6),

d 57 4d 1 451 2 ≤ log + + . log y 50 19 log y 5 60 This is not possible and the proof of (12) is complete. We combine (7), (11) and (12) to conclude that 3 (x − 1) log y < 183 log y. 4

68


Thus x ≤ 244. Finally, we apply (9) and (12) to conclude that d ≤ 743(α + 1/2). In addition to Lemma 4, the proof of Theorem 3 uses an irrationality measure [15] of certain algebraic numbers derived from a Theorem of Baker [1]. Lemma 5. Let A, B, K and n be positive integers such that A > B, K < n, n ≥ 3 and ω = (B/A)1/n is not a rational number. For 0 < φ < 1, put δ =1+

2−φ , K

σ=

δ 1−φ

and u1 = 40n(K+1)(σ+1)/(Kσ−1) ,

K+σ+1 n(K+1) u−1 40 . 2 = K2

Assume that A(A − B)−δ u−1 1 > 1. Then p u2 > q Aq K(σ+1) for all integers p and q with q > 0. ω −

Proof. This is Lemma 1 of Shorey & Nesterenko [15]. We notice that this has been refined by Hirata-Kohno in [8] but the statement of [15] is sufficient for our purpose. Our last auxiliary result is an estimate from the theory of p-adic linear forms in (two) logarithms. For this, we need some notation. Let m > 1 be an integer and write m = pu1 1 . . . puww , where p1 < · · · < pw are distinct prime numbers and the ui ’s are positive integers. Let x be a nonzero integer and let p be a prime. We recall that the p-adic valuation of x, denoted by vp (x), is the greatest nonnegative integer v such that pv divides x. Analogously, we define the m-adic valuation of x, which we denote by vm (x), to be the greatest nonnegative integer v such that mv divides x. We observe that vpi (x) , vm (x) = min 1≤i≤w ui where [·] denotes the integer part. Further, if a/b is a nonzero rational number with a and b coprime, we set vm (a/b) = vm (a) − vm (b). We let x1 /y1 and x2 /y2 be two nonzero rational numbers with x1 /y1 6= ±1. Lemma 6 provides an upper bound for the m-adic valuation of b1 b2 x1 x2 Λ= − , y1 y2


69

where b1 and b2 are positive integers, assuming that x1 x2 vpi − 1 ≥ ui , vpi − 1 ≥ 1 for all prime pi , 1 ≤ i ≤ w y1 y2 and that either m is odd or 4 divides m. Further, let A1 > 1, A2 > 1 be real numbers such that log Ai ≥ max{log |xi |, log |yi |, log m}, (i = 1, 2), and put b0 =

b2 b1 + . log A2 log A1

Lemma 6. With the previous notation and under the above hypothesis, if moreover m, b1 and b2 are relatively prime, then we have the upper estimate 2 66.8 0 vm (Λ) ≤ max{log b + log log m + 0.64, 4 log m} log A1 log A2 . (log m)4 Proof. This is Theorem 2 of [4].

4. Proofs. Proofs of Theorems 1 and 2. Let y > x ≥ 2 be integers and denote by d their greatest common divisor. Set x0 = x/d and y0 = y/d. Assume that (2) has two distinct solutions (m2 , n2 ) and (m1 , n1 ) with m2 > m1 > 1. Then we get (13)

(dy0 − 1)(dx0 )m2 − (dx0 − 1)(dy0 )n2 = dy0 − dx0

and (14)

(dy0 − 1)(dx0 )m1 − (dx0 − 1)(dy0 )n1 = dy0 − dx0 .

We first observe that d and x0 are coprime. Indeed, let p be a prime such that pa kd and p|x0 . Thus p2a must divide the left-hand side of (13) since m2 ≥ 2 and n2 ≥ 2. But pa+1 divides dx0 , hence p must divide y0 contradicting gcd(x0 , y0 ) = 1. Similarly, we prove that d and y0 are coprime. Using (14) to replace dy0 − dx0 in (13) and dividing (13) by (dx0 − 1)(dy0 )n1 , we get (dx0 )m1 (dy0 − 1)(1 − (dx0 )m2 −m1 ) + (dy0 )n2 −n1 . (dy0 )n1 (dx0 − 1) Since d and x0 y0 are coprime, this implies that d divides the right-hand side of (15), hence d = 1. Consequently, if x and y have a common factor, then (2) has at most one solution. In the sequel of the Proof of Theorem 1, we always assume that x and y are coprime. Further, we assume that there exist pairs of positive integers (m3 , n3 ), (m2 , n2 ) and (m1 , n1 ), with m3 > m2 > m1 ≥ 1 and xmj − 1 y nj − 1 = , j = 1, 2, 3. x−1 y−1 (15)

1=

70


Since y > x, we clearly have mj+1 − mj ≥ 2 for j = 1, 2. Further (16)

nj+1 − nj ≥ 2,

j = 1, 2.

Indeed, if for example n2 = n1 + 1, then we get xm1 − 1 xm2 − 1 = + y n1 , x−1 x−1 and x divides y, which is a contradiction. This proves (16). Let δ = gcd(x−1, y−1) and put a = (y−1)/δ, b = (x−1)/δ, c = (y−x)/δ. Let s denote the smallest integer ≥ 1 such that xs ≡ 1(mod by n1 ). Then, for j = 1, 2, we have axmj+1 − by nj+1 = c

(17) and

axmj − by nj = c.

(18) Therefore

xmj+1 −mj ≡ 1(mod by nj ),

(19)

thus m3 − m2 and m2 − m1 are multiples of s. Further, (17) and (18) with j = 1 yield 1=

axm1 (1 − xm2 −m1 ) + y n2 −n1 , by n1

from which we deduce that y and (1 − xm2 −m1 )/(by n1 ) are coprime. By the definition of s, we deduce that xs − 1 (20) gcd y, = 1. by n1 Write m3 − m2 = su, with u ≥ 1 integer. By (19), we see that by n2 divides xsu − 1, thus y n2 −n1 divides xs − 1 (21) (1 + xs + · · · + xs(u−1) ) n1 . by From (20) and (21), we then obtain that (22)

1 + xs + · · · + xs(u−1) ≡ 0 (mod y n2 −n1 ).

Let p be a prime and assume that pα ky. By the definition of s, there exists an integer λ ≥ 1 such that xs = 1 + λpαn1 . We infer from (22) that pα(n2 −n1 ) divides (1 + λpαn1 )u − 1 /(λpαn1 ) = u + u(u − 1)λpαn1 /2 + . . . If p is odd or if αn1 ≥ 2, we get that pα(n2 −n1 ) divides u. If p = 2, then x is odd and b must be even, thus λ is also even and, arguing as above, we see that 2α(n2 −n1 ) divides u. Consequently, y n2 −n1 divides u and we get (23)

m3 − m2 ≥ y n2 −n1 .


71

By (17) and (18), we have also y nj+1 − y nj ≡ 1 (mod axmj ) for j = 1, 2 and we argue as above to conclude that (24)

n3 − n2 ≥ xm2 −m1 .

Assume that (2) has two solutions (m3 , n3 ) and (m2 , n2 ) with m3 > m2 > 1. We apply our preceding results with m1 = n1 = 1 and (23) yields m3 > y n2 −1 . By (16), we have n2 ≥ 3, thus (25)

m3 > y 2 .

Combined with Lemma 2, (25) yields y < 1011 , as claimed. Assume now that (2) has three solutions (m4 , n4 ), (m3 , n3 ) and (m2 , n2 ) with m4 > m3 > m2 > 1. By (23) we get m4 > y n3 −n2 , and by (24), (2) and (16) we see that n3 − n2 > xm2 −1 ≥ y n2 −1 /2 ≥ y 2 /2. Finally, we have 2

m4 > y y /2 , which, combined with Lemma 2, yields y ≤ 6. Now, we solve (2) for any pair (x, y) with 2 ≤ x < y ≤ 6. For (2, 3) and (3, 4), we observe that the equation |3u − 2v | = 1 has the only solution (2, 3) in integers u > 1 and v > 1. For (x, y) = (2, 4), (2, 6), (3, 6) and (4, 6) we conclude by arguing, respectively, modulo 4, 8, 9 and 4 that (2) has no solution other than the one given by x = 2, y = 6, m = 3, n = 2. For (x, y) = (4, 5), we have to solve the equation 4u − 3 · 5v = 1. Arguing modulo 5, we see that u is even, hence (4u/2 − 1)(4u/2 + 1) = 3 · 5v and u = 2, v = 1. We deal with (x, y) = (5, 6) in a similar manner. Now, for (x, y) = (2, 5), we are left with the equation 5u +3 = 2v . Modulo 8, we see that u is odd, hence (5(u−1)/2 , v) is a solution of 5X 2 + 3 = 2k . By Theorem 1 of [5], this equation has only two solutions, namely (X, k) = (1, 3) and (5, 7). Finally, it remains us to treat the pair (3, 5), hence the equation 2 · 3u − v 5 = 1. Modulo 3, we see that v is odd, thus (5(v−1)/2 , u) is a solution of 1 + 5X 2 = 2 · 3k . By Theorem 2 of [5], this equation has only the solution (X, k) = (1, 1). This completes the proof of Theorem 1. The Proof of Theorem 2 is now easy. Indeed, set m1 = n1 = 1 and let (m2 , n2 ) be a solution of (2). As noticed just below (19), s divides m2 − 1. Thus s < m2 which, together with (4), contradicts (3). Theorem 2 is then a straightforward consequence of Lemma 1.

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Now, we give a proof of Remark 1. Let y be a given positive integer. For an integer 1 ≤ x ≤ y coprime with y, we denote by ordy (x) the smallest integer s ≥ 1 such that xs ≡ 1 (mod y). Further, we write X X ϕ(d) + ϕ(d) = ϕ(y), √ √ d|ϕ(y),d
1. We denote by C1 , C2 and C3 effectively computable positive numbers depending only on α. Let (x, y, m, n) be a solution of (1) such that gcd (m − 1, n − 1) = d ≥ 3 and (m − 1)/(n − 1) ≤ α. We write m − 1 = dr,

n − 1 = ds

where r and s are positive integers. Now we infer from (1) that x ≤ 2y s/r ,

y ≤ 2xr/s .

By Lemma 4, we know that d ≤ C1 . By Theorem 1, we may also suppose that y ≥ C2 with C2 sufficiently large. Further, we rewrite (1) as x

y ds 1 1 xdr −y = − , x−1 y−1 x−1 y−1

which implies that (26)

y(x − 1) 1/d xr 1 − s < ds . x(y − 1) y y

Now we apply Lemma 5 with A = x(y − 1), B = y(x − 1), d = n, σ = s and K = [2α] + 1. We may suppose that K < n. We choose φ, depending only on α, suitably such that 1 1 2−φ 1 1 + + sK 1 + + < 4α + 6 + s. α 1 − φ K(1 − φ) α


73

Finally, setting δ = 1 + (2 − φ)/K and u1 = 40d(K+1)(δ+1−φ)/(Kδ+1−φ) , we observe that 1

1

1+ α A(A − B)−δ u−1 (y − x)−δ > C3 y 1+ α −δ > 1 1 > C3 y

if C2 is sufficiently large. Hence, we conclude from Lemma 5 that the left hand side of (26) exceeds y −s(4α+6+1/α) , hence 1 d < 4α + 6 + , α as claimed. Proof of Theorem 4. Set d = gcd(m − 1, n − 1). We apply Lemma 4 with α = (m − 1)/(n − 1). We conclude that d2 ≤ d(n − 1) ≤ 1114.5 m, whence d ≤ 33.4 m1/2 . Proof of Theorem 5. Let (x, y, m, n) be a solution of (1) with x < y ≤ xa and y ≡ 1 (mod x). Regarding (1) modulo x, we deduce that n ≡ 1 (mod x), thus n > x. We set Λ := (y − 1)xm = (x − 1)y n − (x − y) and we observe that vx (Λ) ≥ m + 1. Further, if x 6= 2, then according as 2kx or not we have y−1 vx (Λ) ≤ vx/2 y n − 1 − x−1 or y−1 n vx (Λ) = vx y − 1 − . x−1 We check that the hypotheses of Lemma 6 are satisfied, and, thanks to that Lemma, we obtain for x > 2 that 2 66.8 n+1 x x m+1≤ max log + log log + 0.64, 4 log (log y)2 . (log x2 )4 log y 2 2 Further, log y ≤ a log x and log x/ log x2 ≤ log 6/ log 3 ≤ 1.631, hence we get m ≤ max{2843a2 , 148(log2 m + 0.64)a2 }. It follows that m ≤ 14000a2 (log 3a)2 , as claimed.

Acknowledgements. This work was done when the first author was visiting the Tata Institute of Fundamental Research, Mumbai. He would like to thank the mathematical department for its hospitality and the CNRS for its financial support.

74


References [1] A. Baker, Simultaneous rational approximations to certain algebraic numbers, Proc. Cambridge Phil. Soc., 63 (1967), 693-702, MR 35 #4167, Zbl 0166.05503. [2] A. Baker and G. W¨ ustholz, Logarithmic forms and group varieties, J. Reine Angew. Math., 442 (1993), 19-62, MR 94i:11050, Zbl 0788.11026. [3] R. Balasubramanian and T.N. Shorey, On the equation a(xm − 1)/(x − 1) = b(y n − 1)/(y − 1), Math. Scand., 46 (1980), 177-182, MR 81m:10022, Zbl 0434.10013. [4] Y. Bugeaud, Linear forms in two m-adic logarithms and applications to Diophantine problems, Compositio Math., 132 (2002), 137-158. [5] Y. Bugeaud and T.N. Shorey, On the number of solutions of the generalized Ramanujan–Nagell equation, J. Reine Angew. Math., 539 (2001), 55-74, CMP 1 863 854. [6] H. Davenport, D.J. Lewis and A. Schinzel, Equations of the form f (x) = g(y), Quart. Jour. Math. Oxford, 2 (1961), 304-312, MR 25 #1152, Zbl 0121.28403. [7] R. Goormaghtigh, L’Intermédiaire des Mathématiciens, 24 (1917), 88. [8] N. Hirata-Kohno and T.N. Shorey, On the equation (xm − 1)/(x − 1) = y q with x power, in ‘Analytic Number Theory’ (Kyoto, 1996), 119-125, London Math. Soc. Lecture Note Ser., 247, Cambridge Univ. Press, Cambridge, 1997, MR 2000c:11051, Zbl 0904.11009. [9] M. Laurent, Yu. Nesterenko and M. Mignotte, Formes linéaires en deux logarithmes et déterminants d’interpolation, J. Number Th., 55 (1995), 285-321, MR 96h:11073, Zbl 0843.11036. [10] Maohua Le, A note on the Diophantine equation axm − by n = k, Indag. Math. (N.S.), 3 (1992), 185-191, MR 93c:11016, Zbl 0762.11012. [11] M. M¸akowski and A. Schinzel, Sur l’équation indéterminée de R. Goormaghtigh, Mathesis, 68 (1959), 128-142, MR 22 #9472, Zbl 0085.02902. [12] M. Mignotte, A corollary to a theorem of Laurent–Mignotte–Nesterenko, Acta Arith., 86 (1998), 215-225, MR 99i:11060, Zbl 0919.11051. [13] Yu. Nesterenko and T.N. Shorey, On an equation of Goormaghtigh, Acta Arith., 83 (1998), 381-389, MR 98m:11022, Zbl 0896.11010. [14] T.N. Shorey, On the equation axm − by n = k, Indag. Math., 48 (1986), 353-358, MR 88a:11030, Zbl 0603.10019. [15] T.N. Shorey and Yu. Nesterenko, Perfect powers in product of integers from a block of consecutive integers (II), Acta Arith., 76 (1996), 191-198, MR 97d:11005, Zbl 0859.11025. Received January 22, 2001.

´ Louis Pasteur Universite ´matiques U. F. R. de mathe ´ Descartes 7, rue Rene 67084 Strasbourg France


E-mail address: [email protected] School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road Mumbai 400 005 India E-mail address: [email protected]

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TODA LATTICE AND TORIC VARIETIES FOR REAL SPLIT SEMISIMPLE LIE ALGEBRAS Luis G. Casian and Yuji Kodama The paper concerns the topology of an isospectral real smooth manifold for certain Jacobi element associated with real split semisimple Lie algebra. The manifold is identified as a compact, connected completion of the disconnected Cartan e which is a disjoint subgroup of the corresponding Lie group G union of the split Cartan subgroups associated to semisimple portions of Levi factors of all standard parabolic subgroups of e The manifold is also related to the compactified level sets G. of a generalized Toda lattice equation defined on the semisimple Lie algebra, which is diffeomorphic to a toric variety in the e e We then give flag manifold G/B with Borel subgroup B of G. a cellular decomposition and the associated chain complex of the manifold by introducing colored-signed Dynkin diagrams which parametrize the cells in the decomposition.

1. Introduction. In this paper, we study the topological structure of certain manifolds that are interesting in two different ways. First they are isospectral manifolds for a signed Toda lattice flow [14]; an integrable system that arises in several physical contexts and has been studied extensively. Secondly they are shown in §8 to be the closures of generic orbits of a split Cartan subgroup on a real flag manifold. These are certain smooth toric varieties that glue together the disconnected pieces of a Cartan subgroup of a semisimple Lie group of the e In this paper we start from the Toda lattice aspect of this object and form G. end the paper inside a real flag manifold. We thus start by motivating and describing our main constructions from the point of view of the Toda lattice; then we trace a path that starts with this Toda lattice and that naturally leads to the (disconnected) Cartan subgroup of a split semisimple Lie group and a toric orbit. Another path that also leads to a Cartan subgroup starts with Kostant’s paper [16]; this approach is described in §8. From the Toda lattice end of this story, these manifolds are related to the compactified level set of a generalized (nonperiodic) Toda lattice equation defined on the semisimple Lie algebra (see for example [16]) and, although they share some features with the Tomei manifolds in [21], they are different 77

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L.G. CASIAN AND Y. KODAMA

from those (e.g., nonorientable). As background information, we start with a definition of the generalized Toda lattice equation which led us to our present study of the manifolds. Let g denote a real split semisimple Lie algebra of rank l. We fix a split Cartan subalgebra h with root system ∆ = ∆(g, h), real root vectors eαi associated with simple roots {αi : i = 1, . . . , l} = Π. We also denote {hαi , e±αi } the Cartan-Chevalley basis of g which satisfies the relations, [hαi , hαj ] = 0, [hαi , e±αj ] = ±Cj,i e±αj , [eαi , e−αj ] = δi,j hαj where the l × l matrix (Ci,j ) is the Cartan matrix corresponding to g, and Ci,j = αi (hαj ). Then the generalized Toda lattice equation related to real split semisimple Lie algebra is defined by the following system of 2nd order differential equations for the real variables fi (t) for i = 1, . . . , l,   l X d2 fi (1) = i exp − Ci,j fj  dt2 j=1

where i ∈ {±1} which correspond to the signs in the indefinite Toda lattices introduced in [5, 14]. The main feature of the indefinite Toda equation having at least one of i being −1 is that the solution blows up to infinity in finite time [14, 10]. Having introduced the signs, the group corresponding to e with Lie algebra g which is defined the Toda lattice is a real split Lie group G e = SL(n, R), in §3. For example, in the case of g = sl(n, R), if n is odd, G ± e and if n is even, G = Ad(SL(n, R) ). The original Toda lattice equation in [20] describing a system of l particles on a line interacting pairwise with exponential forces corresponds to the case with g = sl(l + 1, R) and i = 1 for all i, and it is given by d2 qi = exp(qi−1 − qi ) − exp(qi − qi+1 ), i = 1, . . . , l, dt2 where the physical variable qi , the position of the i-th particle, is given by qi = fi − fi+1 ,

i = 1, . . . , l,

with fl+1 = 0 and f0 = fl+2 = −∞ indicating q0 = −∞ and ql+1 = ∞. The system (1) can be written in the so-called Lax equation which describes an iso-spectral deformation of a Jacobi element of the algebra g. This is formulated by defining the set of real functions {(ai (t), bi (t)) : i = 1, . . . , l} with   l X dfi (t) (2) ai (t) := , bi (t) := i exp − Ci,j fj (t) dt j=1

TODA LATTTICE AND TORIC VARIETIES

79

from which the system (1) reads   l X dbi = −bi  Ci,j aj . dt

dai = bi , dt

(3)

j=1

This is then equivalent to the Lax equation defined on g (see [7] for a nice review of the Toda equation). dX(t) = [P (t), X(t)], dt where the Lax pair (X(t), P (t)) in g is defined by  l l X X    X(t) = a (t)h + (bi (t)e−αi + eαi )  i αi  i=1 i=1 (5) l  X    bi (t)e−αi .  P (t) = − (4)

i=1

Although a case with some bi = 0 is not defined in (2) the corresponding system (3) is well-defined and is reduced to several noninteracting subsystems separated by bi = 0. The constant solution ai for bi = 0 corresponds to an eigenvalue of the Jacobi (tridiagonal) matrix for X(t) in the adjoint representaion of g. We denote by S(F ) the ad diagonalizable elements in g⊗R F with eigenvalues in F , where F is R or C. Then the purpose of this paper is to study the disconnected manifold ZR of the set of Jacobi elements in g associated to the generalized Toda lattices, ZR =

X =x+

Xl i=1

(bi e−αi

+ eαi ) : x ∈ h, bi ∈ R \ {0}, X ∈ S(R) ,

its iso-spectral leaves Z(γ)R , γ ∈ Rl and the construction of a smooth ˆ R of each Z(γ)R . The construction of connected compactification, Z(γ) ˆ R generalizes the construction of such a smooth compact manifold which Z(γ) was carried out in [15] in the important case of g = sl(l+1, R). The construction there is based on the explicit solution structure in terms of the so-called τ -functions, which provide a local coordinate system for the blow-up points. Then by tracing the solution orbit of the indefinite Toda equation, the disconnected components in Z(γ)R are all glued together to make a smooth compact manifold. The result is maybe well explained in Figure 1 for the case of sl(3, R). In the figure, the Toda orbits are shown as the dotted lines, and each region labeled by the same signs in (1 , 2 ) with i ∈ {±} are glued together through the boundary (the wavy-lines) of the hexagon. At a point of the boundary the Toda orbit blows up in finite time, but the orbit can be uniquely traced to the one in the next region (marked by the same letter

80


ˆ R in this case is shown A, B or C). Then the compact smooth manifold Z(γ) to be isomorphic to the connected sum of two Klein bottles. In the case of ˆ R is shown to be nonorientable and the symmetry sl(l + 1, R) for l ≥ 2, Z(γ) group is the semi-direct product of (Z2 )l and the Weyl group W = Sl+1 , the permutation group. One should also compare this with the result in [21] where the compact manifolds are associated with the definite (original) Toda lattice equation and the compactificationis done by adding only the subsystems. (Also see [4] for some topological aspects of the manifolds.)

A

B (_ ,+)

(+ ,_)

(_ ,_)

(_ ,_)

B (_ ,+)

(+ ,_)

(_ ,+)

(+ ,+)

B

(+ ,_)

(_ ,_) A

C A

C

(_ ,+)

(_ ,_)

B

A (+ ,_) C

C

Figure 1. The manifold Z(γ)R and the Toda orbits for sl(3, R). ˆ R , can be physiThe study of Z(γ)R and of the compact manifolds Z(γ) cally motivated by the appearance of the indefinite Toda lattices in the context of symmetry reduction of the Wess-Zumino-Novikov-Witten (WZNW) model. For example, the reduced system is shown in [6] to contain the inˆ R can then be viewed as definite Toda lattices. The compactification Z(γ) a concrete description of (an expected) regularization of the integral manifolds of these indefinite Toda lattices, where infinities (i.e., blow up points) of the solutions of these Toda systems glue everything into a smooth compact manifold. In addition our work is mathematically motivated by: a) The work of Kostant in [16] where he considered the real case with all bi > 0,


81

b) the construction of the Toda lattice in [17]. In [17], the solution {bi (t) : i = 1, . . . , l} in (2) of the generalized Toda lattice equation is shown to be expressed as an orbit on a connected component H labeled by = (1 , . . . , l ) with i = ±1, of the Cartan subgroup HR defined in §3, [ H . HR = ∈{±1}l

This can be seen as follows: Let g be an element of H given by ! l X g = h exp (6) fi hαi , i=1

which gives a map from ZR into HR . Here the element h ∈ H satisfies χαi (h ) = i for the group character χφ determined by φ ∈ ∆. The solution {bi (t) : i = 1, . . . , l} in (2) is then directly connected to the group character χ−αi evaluated at g , i.e., bi (t) = χ−αi (g ). The Toda lattice equation is now written as an evolution of g , d −1 d g g = g−1 e+ g , e− , dt dt where e± are fixed elements in the simple root spaces g±Π so that all the elements in g±Π can be generated by e± , i.e., g±Π = {Adh (e± ) : h ∈ H}. In particular, we take e± =

l X

e±αi .

i=1

Thus the Cartan subgroup HR can be identified as the position space (e.g., fi = qi + · · · + ql for sl(l + 1, R)-Toda lattice) of the generalized Toda lattice equation, whose phase space is given by the tangent bundle of HR . One should also note that the boundary of each connected component H is given by either bi = 0 (corresponding to a subsystem) or |bi | = ∞ (to a blow-up). We are then led to our main construction of the compact smooth manifold of HR in attempting to generalize [16] Theorem 2.4 to our indefinite Toda case including bi < 0. ˆ R containing the Cartan subgroup HR by adding pieces We define a set H corresponding to the blow-up points and the subsystems (Definition 6.1). ˆ R is defined as a disjoint union of split Cartan subgroups H A Thus the set H R associated to semisimple portions of Levi factors of all standard parabolic subgroups determined by A ⊂ Π, (Definition 3.10), [ [ ˆR = H w HRA × {[w]A }. A⊂Π w∈W/WΠ\A

82


ˆ R then constitutes a kind of compact, connected completion The space H of the disconnected Cartan subgroup HR . Figure 1 also describes how to connect the connected components of HR to produce the connected maniˆ R in the case of sl(3, R). The signs must now be interpreted as the fold H signs of simple root characters on the various connected components. The pair of signs (+, +) corresponds to the connected component of the identity H and regions with a particular sign represent one single connected component of HR . Boundaries between regions with a fixed sign correspond to connected components of Cartan subgroups arising from Levi factors of ˆ R. parabolic subgroups. In addition the Weyl group W acts on H Most of this paper is then devoted to describing the detailed structure ˆ R , and in §8 we conclude with a diffeomorphism defined of the manifold H ˆ ˆ R as identified with a toric between HR and the isospectral manifold Z(γ) e variety (HR gB) in the flag manifold G/B. ˆ R , we intro1.1. Main Theorems. In connection with the construction H duce in Definition 4.1 the set of colored Dynkin diagrams. The colored Dynkin diagrams are simply Dynkin diagrams D where some of the vertices have been colored red or blue. For example in the case g = sl(3, R):◦R − ◦ (the sub-index R indicates that ◦ is colored red). The full set of colored Dynkin diagrams consists of pairs: (D, [w]Π\S ) where D is a colored Dynkin diagram, S ⊂ Π denotes the set of vertices that are colored in D and [w]Π\S is the coset of w in W/WS . To each pair (D, [w]Π\S ) one can associate a set ˆ R also which is actually a cell. First Notation 6.2 associates a subset of H Π\S denoted (D, [w] ).

sα ( (

R,

[e])

, Sα

1

1

(

) (

B,

[e])

,e

) (

R,

[e])

ˆ R parametrized by colored Dynkin Figure 2. The manifold H diagrams for sl(2, R). This turns out to be a cell of codimension k with k = |S|. We illustrate the example of sl(2, R) in Figure 2. We consider the set Dk of colored Dynkin diagrams (D, [w]Π\S ) with |S| = k. In Figure 2, we have D0 = {(◦, e), (◦, sα1 )}, D1 = {(◦B , [e]), (◦R , [e])}. We then obtain the following theorem:


83

Theorem 1.1. The collection of the sets {Dk : k = 0, 1, . . . , l} gives a cell ˆ R. decomposition of H The chain complex M∗ is introduced in §4. The construction, in the case of sl(2; R), is as follows: ∂1 M1 = Z D0 −→ M0 = Z D1 . Here the boundary map ∂1 is given by ∂1 (◦, e) = (◦B , [e]) − (◦R , [e]), ∂1 (◦, sα1 ) = (◦B , [sα1 ]) − (◦R , [sα1 ]). P In particular, (◦) := w∈W (−1)`(w) (◦, w) = (◦, e)−(◦, sα1 ) is a cycle, where ˆ R . The following theorem `(w) is the length of w, and (◦) represents the H ˆ gives a topological description of HR : ˆ R is compact, nonorientable (except if g is of Theorem 1.2. The manifold H type A1 ), and it has an action of the Weyl group W . The integral homology ˆ R can be computed as a Z[W ]-module as the homology of the chain of H complex M∗ in (13). Theorem 1.2 is completed in Proposition 7.7. The W -action is (abstractly) introduced in Definition 4.4 in terms of a representation-theoretic induction process from smaller parabolic subgroups of W . The proof that the W action is well-defined is given in Proposition 4.5 and Proposition 5.3. The in Definition 7.6 is defined so that it computes inchain complex MCW ∗ tegral homology. Since each Xr \ Xr−1 in Definition 7.6 is a union of cells (D, [w]Π\S ) ∈ Dl−r , we obtain an identification between MCW and the chain ∗ complex M∗. ˆR Then using the Kostant map which can be described as a map from H e into the flag manifold G/B (in Definition 8.8), that is, a torus imbedding, we obtain the following theorem: ˆ R is a smooth compact manifold which Theorem 1.3. The toric variety Z(γ) ˆ R. is diffeomorphic to H The complex version of this theorem has been proven in [9], and our proof is essentially given in the same manner. 1.2. Outline of the paper. The paper is organized as follows: In §2, we begin with two fundamental examples, g = sl(l + 1, R) for l = 1, 2, which summarize the main results in the paper. In §3, we present the basic notations necessary for our discussions. We e of rank l whose split Cartan subgroup HR conthen define a real group G l e cortains 2 connected components. We also define the Lie subgroups of G responding to the subsystems and blow-ups of the generalized Toda lattice

84


e and the Cartan subgroup equations. The reason for the introduction of G HR can be appreciated in Remark 8.10 and Corollary 8.11. In §4, we introduce colored Dynkin diagrams which will be shown to paˆ R . We then rametrize the cells in a cellular decomposition of the manifold H construct a chain complex M∗ of the Z[W ]-modules Ml−k (Definition 4.9). The parameters involved in the statement of Theorem 1.1 are given here. In §5 we show that Weyl group representations introduced in §4 are welldefined. In addition we define HR◦ in §5 by adding some Cartan subgroups associated to semisimple Levi factors of parabolic subgroups to HR . ˆ R as a union of the Cartan subgroup associated to In §6, we define H semisimple Levi factors of certain parabolic subgroups (Definition 6.1) using ˆ R to translation by Weyl group elements. We also associate subsets of H colored Dynkin diagrams. ˆ R expressing H ˆ R as the In §7, we discuss the toplogical structure of H union of the subsets determined by the colored Dynkin diagrams. We then ˆ R is a smooth compact manifold and those subsets naturally show that H determine a cell decomposition (Theorem 1.2). In §8, we consider a Kostant map between the isotropy subgroup GzC of GC with Ad(g)z = z and the isospectral manifold J(γ)C for some γ ∈ Rl , e which can be also described as a map into the flag manifold G/B. Then we show that the toric variety (HR nB) is a smooth manifold and obtain Theorem 1.3. 2. Examples of sl(l + 1, R), l = 1, 2. This section contains two examples which are the source of insight for the main theorems in this paper. Most of the notation and constructions used later on in the paper can be anticipated through these examples. ˆ R , has nothing to do, in its Our main object of study in this paper, H construction, with the moment map and the Convexity Theorem of [1]. However, because of [1] it is expected to contain a polytope with vertices given by the action of the Weyl group which, gives rise to it through some gluings along its faces. We will identify a convenient polytope of this kind ˆ R would in h0 (dual of Cartan subalgebra) and show how all the pieces of H fit inside it. The polytope in the example of sl(3, R) is a hexagon. Since we are not dealing with the moment map in this paper, this is just done for the purposes of motivation and illustration. The polytope of the Convexity Theorem in [1], strictly speaking, is the convex hull of the orbit of ρ. This ˆ R. sits inside the convex hull of the orbit of 2ρ which is all a part of H The terms dominant and antidominant being relative to a choice of a Borel subalgebra, we refer to the chamber in h0 containing ρ as antidominant for no other reason than the fact that we will make it correspond to what we call the antidominant chamber of the Cartan subgroup HR . This odd


85

convention is applicable throughout this section always in connection with the 2ρ polytope and our figures. At this stage it is useful to keep in mind that: ˆ R is made up of a split Cartan sub1) The bulk — interior — of H l group HR which has 2 connected components HS parametrized by signs = (±, . . . , ±) for g with rank l, i.e., HR = ∈{±1}l H . These disconnected pieces are glued together into a connected manifold by using Cartan subgroups HRA associated to Levi factors of parabolic subgroups, which are determined by the set of simple roots Π \ A for each A ⊂ Π. 2) The language of colored Dynkin diagrams and signed colored Dynkin diagrams is introduced in this paper in order to parametrize the pieces ˆ R . However the motivation for their introduction is that these diof H agrams parametrize the pieces of a convex polytope (hexagon) which lives in h0 , including its external faces and internal walls. The parameters for the pieces are colored and signed-colored Dynkin diagrams. 3) If one just looks at the antidominant chamber intersected with the 2ρ polytope, then it is easy to see that it forms a box. This box is the closure of the antidominant chamber HR< of the Cartan subgroup S < ˆ ˆ w H . The box HR (Definition 3.10) inside HR , i.e., HR = w∈W

HR
0, b > 0, abc = 1}, ∈{±1}2

where h = h(1 ,2 ) = diag(2 , 1 2 , 1 ) satisfying χαi (h ) = i . Since A = ∅, ˆ R construction. WΠ\A = W , and there is only one coset [e]A = [e] in the H We now consider the signed-colored Dynkin diagrams setting S = ∅, that is, all the vertices are uncolored and they give a subdivision of the antidominant chamber inside the 2ρ hexagon. In order to move around this chamber and its subdivisions using the Weyl group we must also consider six elements [w]Π = w ∈ W . We then have, for S = ∅ and A = ∅, and (1 , 2 ) = (±1, ±1), (◦1 − ◦2 , e) = h diag(a, b, (ab)−1 ) : ab−1 < 1, ab2 < 1 × {[e]} Note here that the inequalities |χαi | < 1 guarantee that the set in question is contained in the chamber associated to e. Here the local coordinate φe is given by (χα1 , χα2 ) which equals (ab−1 , ab2 ). For A = {α1 }, we have h = diag(2 , 2 , 1) ∈ HA with = (1, 2 ); and we multiply this element with diag(1, a, a−1 ). This is a typical element in

90


the connected component of the Cartan associated to the Levi factor, in accordance with the definition of HRA in Definition 3.10. This gives: (◦0 − ◦2 , e) = diag(2 , 2 a, a−1 ) : 0 < a < 1 × {[e]A } where the local coordinate φe is given by (0, 2 a2 ). For A = {α2 }, we have in a similar way with h(1 ,1) = diag(1, 1 , 1 ): (◦1 − ◦0 , e) = diag(a, 1 a−1 , 1 ) : 0 < a < 1 × {[e]A } where φe now equals (1 a2 , 0). We have an open square associated with the interior of the antidominant chamber, [ (◦ − ◦, e) = (◦ν1 − ◦ν2 , e). (9) (ν1 ,ν2 )∈{±1,0}2

The image of this set under the map φe is an open square (−1, 1) × (−1, 1). We now write down the boundary of this square: We here give an explicit form of (◦R − ◦, [e]Π\S ), and the others can be obtained in the similar way. We first have, for S = {α1 }, A = ∅, so that [e]A = [e] = [sαi ] for i = 1, 2, ◦R − ◦2 , [e]{α2 } = diag 2 a, −2 a, −a−2 : 0 < a < 1 × {[e]}. Here φe equals (χα1 , χα2 ), and is given by (−1, 2 a3 ). With A = {α2 }, we have: o n ◦R − ◦0 , [e]{α2 } = {diag(1, −1, −1)} × [e]{α2 } . A

The map φe is (χ∆ α1 , 0) and equals (−1, 0). We then have [ ◦R − ◦, [e]{α2 } = ◦R − ◦ν , [e]{α2 } . ν∈{±1,0}

The image of this set under φe is thus {−1} × (−1, 1). We now consider the parts of the Cartan subgroup of Levi factors corresponding to other chambers inside the hexagon. Note that if we apply sα1 = w to (◦1 − ◦2 , e), we obtain (◦1 − ◦1 2 , sα1 ) = diag 1 2 b, 2 a, 1 (ab)−1 : ab−1 < 1, ab2 < 1 × {[e]}. Since sign(χα1 ) = 1 and sign(χα2 ) = 1 2 , this set is no longer contained in H(1 ,2 ) but rather in H(1 ,1 2 ) . This justifies the notation (◦1 − ◦1 2 ). One should note that for the component H in HR did not change when one uses the simple roots associated to the new positive system sα1 ∆+ .


sα

91

2

(-,+) (-,-) (+,-) (+,+) (+,+)

sα

1

(+,-) (-,-) (-,+)

Figure 5. Gluing creates a Mobius band out of two contiguous chambers. Also notice that for S = {α1 } we have ◦R − ◦+ , [sα1 ]Π\S = diag −a, a, −a−2 : a > 0 × {[e]} and we have an identification of (◦R − ◦+ , [sα1 ]Π\S ) and (◦R − ◦− , [e]Π\S ). They are the same set. In fact, now S = {α1 } and e and sα1 give the same coset in W/WS so the corresponding signed-colored Dynkin diagrams agree too. Similarly (◦R − ◦− , [sα1 ]Π\S ) is the same as (◦R − ◦+ , [e]Π\S ). In our hexagon this means that two of the outer walls must be glued. The fact that a segment with a sign + is glued to one with − corresponds to the fact that the two contiguous chambers will form a Mobius band after the gluing (see Figure 5). What this means is that in our geometric picture consisting of the inside of the 2ρ hexagon, some portions of the boundary need to be identified. Such identifications take place on all the chambers. This identification provides the gluing rule given in Lemma 4.2 in [15] for the case of sl(n, R). 2.3. The chain complex M∗ . We describe M∗ in terms of colored Dynkin diagrams. The Z modules of chains Mk are then given by (see Figure 6): • M2 = Z [(◦ − ◦, w) : w ∈ W ]. The cells are the chambers of the 2ρ hexagon, and dimM2 = 6. • M1 consists of the cells parametrized by the colored Dynkin diagrams (◦R − ◦, [w]{α2 } ), (◦B − ◦, [w]{α2 } ) with w ∈ {e, sα2 , sα1 sα2 } and (◦ − ◦R , [w]{α1 } ), (◦ − ◦B , [w]{α1 } ) with w ∈ {e, sα1 , sα2 sα1 }. These are the sides of the different chambers of the hexagon. The blue (B) stands for internal chamber wall and the red (R) for external face of the hexagon. The dimension is then given by dim M1 = 12.

92


S2 ( R _ B ,[e])

(

_

( R_

,[e]

(

,

B ,[e])

_

( B_

2

)

( R _ R ,[e]) (

e)

_

( B _ R ,[e])

,[e])

( ( B _ B ,[e])

( (

_

_

S1

1

R ,[e] )

,S1

)

1

B ,[S1] )

( R _ R ,[e])

_

1

R ,[S1] )

( R _ B ,[e]) ( R_

,[e]

2

)

S1S2 S1

ˆ R parametrized by colored Dynkin Figure 6. The manifold H diagrams for sl(3, R). • M0 = Z [(◦s − ◦t , [e]) : s, t ∈ {R, B}]. These are the four vertices of a chamber. Because of identifications, there are only four different points coming from all the six chambers, that is, dimM0 = 4. ∂

∂

2 1 The chain complex M∗ : M2 −→ M1 −→ M0 leads to the following 3 integral homology: H2 = 0 , H1 = Z ⊕ Z/2Z, H0 = Z. This implies ˆ R is nonorientable and is equivalent to the connected sum of two that H Klein bottles. Also note that the Euler character is 6 − 12 + 4 = −2. ˆ R , Z) of the According to Proposition 7.7 this computes the homology H∗ (H ˆ compact smooth manifold HR . The torsion Z/2Z in H1 has the following representative: X c1 = (−1)`(w) ◦R − ◦, [w]{sα2 }

w∈W/W{sα

−

2}

(−1)`(w) ◦R − ◦, [w]{sα1 } .

X w∈W/W{sα

1}

Here `(w) denotes the length of w. If we let X c2 = (−1)`(w) (◦ − ◦, w), w∈W

then ∂2 (c2 ) = 2c1 . From the chain complex one can compute the three dimensional vector ˆ R , Q) as a W -module. This is a direct sum of a one dimensional space H1 (H nontrivial (sign) representation and the two dimensional reflection represenˆ R , Q) is trivial. tation. The representation H0 (H


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e 3. Notation, the group G. 3.1. Basic Notation. The important Lie group for the purposes of this e that will be technically introduced in Subsection 3.2. paper is a group G This group is R split and has a split Cartan subgroup with 2l components with l = rank(G). For example SL(3, R) is of this kind but SL(4, R) is not. We need to introduce the following standard objects before it is possible to e define and study G. Notation 3.1 (Standard Lie theoretic notation). We adhere to standard notation and to the following conventions: . . .C denotes a complexification; for any set S ⊂ Π, . . .S is an object related to a parabolic subgroup or subalgebra determined by S; for any subset A ⊂ Π , . . .A is associated to the parabolic determined by Π \ A. However we will not simultaneously employ . . .A and . . .S for the same object . . .. As in the introduction g denotes a real split semisimple Lie algebra of rank l with complexification gC = g ⊗ C. We also have GC the connected adjoint Lie group with Lie algeR

bra gC (GC = Ad(GsC )) and G the connected real semisimple Lie subgroup of GC with Lie algebra g. Denote GsC the simply connected complex Lie group associated to gC . We list some additional very standard Lie theoretic notation: • g0 = HomR (g, R) and g0C = HomC (gC , C). • Given λ ∈ g0 and x ∈ g, hλ, xi is λ evaluated in x, hλ, xi = λ(x). • (, ) the bilinear form on g or gC given by the Killing form (the same notation applies to the Killing form on g0 and g0C ). • θ a Cartan involution on g. • g = k + p the Cartan decomposition of g associated to θ, where k is the Lie algebra of a maximal compact subgroup K of the adjoint group G and p is the orthogonal complement to k with respect to the Killing form. • eφ ∈ g, h root vectors chosen so that (eφ , e−φ ) = 1. • ∆+ ⊂ ∆ be P a fixed system of positive roots. • b=h+ Reφ (Borel subalgebra). φ∈∆+ P P Re−φ . • n = [b, b] = Reφ and n = φ∈∆+

φ∈∆+

• nC = n ⊗ C, nC = n ⊗ C. R

R

• HC Cartan subgroup of GC with Lie algebra hC . • H = H(∆) the connected Lie subgroup of G with Lie algebra h, H = exp(h). • HR1 Cartan subgroup of G with Lie algebra h. • W the Weyl group of g with respect to h or the Weyl group of HR1 .

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• WS ⊂ W , the group generated by the simple reflections corresponding to the elements in S ⊂ Π. Remark 3.2. The group HR1 [22] p. 59, 2.3.6 consists of all g ∈ G such that Ad(g) restricted to h is the identity. This Cartan subgroup will be usually disconnected and H is the connected component of the identity e. The Weyl group W of the Cartan subgroup HR1 is isomorphic to the group which is generated by the simple reflections sαi with αi ∈ Π and which agrees with the Weyl group associated to the pair (gC , hC ). This is because our group is assumed to be R split. Example 3.3. The reader may wish to read the whole paper with the following well-known example in mind. Let GsC = SL(n, C) and GC = Ad(SL(n, C)). The second group is obtained by dividing SL(n, C) by the finite abelian group consisting of the n roots of unity times the identity matrix. We can set G = Ad(SL(n, R)). Note that if n is odd then SL(n, R) = Ad(SL(n, R)) = G. If n is even then Ad(SL(n, R)) is obtained by dividing by {±I} (I the identity matrix). In this example, g consists of traceless n×n real matrices. The Cartan subalgebra h in the Ad(SL(n, R)) case can be taken to be the space of traceless n × n real diagonal matrices. The root vectors eφ are the various matrices with all entries ai,j = 0 if (i, j) 6= (io , jo ) and aio ,jo = 1 where io 6= jo are fixed integers. In this case of G = Ad(SL(n, R)), with h chosen as above, the Cartan subgroup HR1 of G consists of Ad applied to the group of real diagonal matrices of determinant one. The group H consists of Ad applied to all diagonal matrices diag(r1 , . . . , rn ) with ri > 0 the connected component of the identity of HR1 . Notation 3.4. The following elements in h and its dual h0 will appear often in this paper: i the coroots. • α ˇ i = (α2α i ,αi ) • mαi i = 1, . . . , l, the fundamental weights, (mαi , α ˇ j ) = δi,j . • m◦αi the unique element in h defined by hmαi , xi = (m◦αi , x). • hαi is the unique element in h such that (hαi , x) = hˇ αi , xi. 2πm◦

• yi = (αi ,ααii) . • h< = {x ∈ h : hαi , xi < 0 for all αi ∈ Π}. Example 3.5. Consider GC = Ad(SL(n, C)) ⊂ Ad(GL(n, C)) and n even. We view gC inside the Lie algebra of GL(n, C) (as n × n complex traceless matrices). Then m◦αi = diag(t1 , . . . , tn ) + z where tj = 1 for j ≤ i and tj = 0 for j > i. The element z is in the center of the Lie algebra of GL(n, C) and thus ad(z) = 0. For instance if n = 2 then m◦α1 = diag( 21 , − 12 ) = diag(1, 0) − diag( 21 , 21 ). e We now define, following [16], 3.4.4 in p. 241 an en3.2. The group G. e largement G of the split group G. The purpose of this is to “complete”


95

the Cartan subgroup HR1 forcing it to have 2l connected components where e =G l is the rank of G. In the case G = SL(l + 1, R) where l is even, G already. We thus consider √ another real split group, slightly bigger than G. linear automorLet x, y ∈ h so that x+ −1y ∈√hC . We recall the conjugate √ phism of gC given by z c = x − −1y where z = x + −1y and x, y ∈ g. We recall that this automorphism induces an automorphism GC → GC , g 7→ g c . We thus let e = {g ∈ GC : g c = g}. G By [16] Proposition 3.4, we also have: e = {g ∈ GC : Ad(g)g ⊂ g}. G Then we have the following Proposition whose proof will be given later (right after Proposition 3.15): e ∼ Proposition 3.6. Let G = Ad(SL(n, R)), then G = SL(n, R) for n odd ± ∼ e e and G = Ad(SL(n, R) ) if n is even. Thus G is disconnected whenever n is even. e First we We now describe an element hi in the Cartan subgroup of G. have: Lemma 3.7. Let x, y ∈ h and αi ∈ Π. Then the numbers exp(hαi , x + √ P −1yi) are real if and only if y has the form: y = li=1 ki yi √ where ki is an integer and yi is as in Notation 3.4. The elements hi = exp( −1ki yi ) with e and satisfy h2 = e with hi 6= e. ki odd are in G i Proof. It is enough to consider the case when x √ = 0. Suppose √ first that Pl −1hα ,yi i y = i=1 ki yi where each ki is an integer. Then e √ = e −1πki takes either +1 or −1. Conversely, suppose that all the ehαi , −1yi with i = 1, . . . , l √ hα , −1yi i are real. Then e equals ±1 and hαi , yi = ki π for all i = 1, . . . , l. iπ This implies that, for each i, hˇ αi , yi = (α2ki ,α . Since g is semisimple and i) P ◦ the {mαi : i = 1, . . . , l} forms a basis of h, necessarily y = li=1 ci m◦αi with iπ hˇ αi , yi = ci and hˇ αi , yi = (α2ki ,α . This proves the first part of the statement i) √

in Lemma 3.7. We note that since all the ehαj ,yi i are real then ehφ, −1yi i is also real for any root φ which is not necessarily simple. Therefore each Ad(hi ) stabilizes all the root spaces of h in g. Since h is also stabilized, √ e Clearly hi = exp( −1yi ) we obtain that Ad(hi )(g) ⊂ g and thus hi ∈ G. satisfies h2i = 1 where hi 6= e. This is because hi is representable by a diagonal matrix with entries of the form ±1. Moreover, at least one diagonal entry must be equal to −1. √ Example 3.8. In the case of Ad(SL(n, R)± ) with n even, hi = exp( −1yi ) is just Ad(diag(r1 , . . . , rn )) where rj = 1 if j ≤ i and rj = −1 if j > i. When

96


n is odd then we have hi = (−1)n−i diag(r1 , . . . , rn ) with the same notation as above for the ri . In this case SL(n, R) = Ad(SL(n, R)). e and other items related We now describe a split Cartan subgroup HR of G e to it. The HR is the real part of HC on G, e HR = HC ∩ G, which has 2l components (see Proposition 3.15 below). We denote by B a e or G as will be clear Borel subgroup with Lie algebra h + n contained in G e this is HR N , N = exp(n). From the Bruhat from the context. Thus in G e decomposition applied to G, we have [ e= ˆ R N, G N wH w∈W

ˆ stands for any representative of the Weyl group with N = exp(n). Here w element w ∈ W in the normalizer of the Cartan subgroup. Keeping this in mind we will harmlessly drop theˆfrom the notation. In addition let χφ denote the group character determined by φ ∈ ∆; on HR each χφ is real and cannot take the value zero. Thus a group character has a fixed sign on each connected component. We denote by sign(χφ (g)) the sign of this character on a specified element g of HR . We let E = E(∆) = {±1}l = {(1 , . . . , l ) : i ∈ {±1}, i = 1, . . . , l}. Then the set E of all 2l elements (or functions : Π → {±1}) will parametrize connected components of the e and corresponds to the signs k = sk sk+1 in the Cartan subgroup HR of G indefinite Toda lattice √ with the connected Q in p. 323 of [14]. In connection components let h = i 6=1 hi for ∈ E and hi = exp( −1yi ) in Lemma 3.7. Now H = h H = {h ∈ HR : sign(χαi (h)) = i for all αi ∈ Π}, and we have: [ HR = H . ∈E

Notation 3.9. We need notation to parametrize connected the components of a split Cartan subgroup, roots and root characters. Unfortunately we need such notation for all the parabolic subgroups associated to arbitrary subsets of Π. Recall (Notation 3.1) that notation associated to a parabolic subgroup determined by a subset Π \ A, A ⊂ Π is usually indicated by changing the standard notation with the use of a superscript . . .A . Thus we have: ∆A ⊂ ∆, root system giving rise to a semisimple Lie algebra lA ⊂ g. Also there are corresponding connected semisimple Lie subgroups LA C ⊂ GC , A ) = Ad(LA ) and it has a LA ⊂ LA . The adjoint group is denoted by L (∆ C C C e A ) be defined in the same way real connected Lie subgroup L(∆A ). Let L(∆ e but relative to the root system ∆A . We let hA be the real span of the hα as G i with αi 6∈ A. This is a (split) Cartan subalgebra of lA denoted as HRA . The


97

corresponding connected Lie subgroup is H A = exp(hA ) (exponentiation taking place inside GC ). e corresponding to the subsystems We also consider Lie subgroups of G of the Toda lattice. In accordance to our convention for Levi subgroups associated with A ∈ Π, we have E A = { = (1 , . . . , l ) : i = 1 if αi ∈ A}. Thus H A is by definition a subgroup of H. This Lie subgroups of H is isomorphic to the connected component of the identity of a Cartan subgroup of a real semisimple Lie group that corresponds to lA . There is a bijection (10) E A ∼ = E(∆A ) = {(j , . . . , j ) : αj 6∈ A for i = 1, . . . , m = |Π \ A|} 1

m

i

which is given in the obvious way by restricting a function : Π → {±1} such that i = (αi ) = 1 for αi ∈ A to a new function (∆A ) with domain Π \ A. We now consider the Cartan subalgebras and the Cartan subgroups for Levi factors of parabolic subalgebras and subgroups dtermined by A ⊂ Π: Definition 3.10. For A ⊂ Π, we denote: S • HRA = h H A (if ∆A = ∅ (i.e., A = Π) then HRA = {e}). ∈E A

e A ), defined in the same way as HR • HR (∆A ) a Cartan subgroup of L(∆ and having Lie algebra hA (HR (∆A ) = {e} if ∆A = ∅). • HA,≤ = {h ∈ h H A : for all αi ∈ Π \ A : |χαi (h)| ≤ 1} the antidominant chamber of HA = h H A . Similarly we consider HA,< using strict inequalities. A ∆A • H(∆A )≤ = {h ∈ H(∆ ) : for all αi ∈ Π \ A : |χαi (h)| ≤ 1}. Similarly we consider a version with strict inequalities. A A • χ∆ αi the root character associated to αi on the Cartan HR (∆ ). We use notation . . .≤ to indicate the antidominant chamber on Cartan subalgebras and subgroups and . . .< for strictly antidominant chambers. e = SL(3, R), HR is the group Example 3.11. In the case of G HR = {diag (a, b, c) : a 6= 0, b 6= 0, abc = 1} . For A = {α1 }, H A is the group {diag(1, a, a−1 ) : a > 0}. The set E A consists of (1, 1) and (1, −1). The element h(1,−1) = diag(−1, −1, 1) and thus h(1,−1) H A = {diag(−1, −a, a−1 ) : a > 0}. These two components form HRA . Note that HR LA is the Lie subgroup of SL(3, R) consisting of all real matrices of the form,   a 0 0 HR L{α1 } = 0 b c  0 d e

98


having determinant one. The group LA is obtained by setting a = 1 and the determinant equal to one. The group L(∆A ) is isomorphic to e A ) is isomorAd(SL(2, R)) the adjoint group obtained from LA and L(∆ e A ) are phic to Ad(SL(2, R)± ). Thus these three groups LA , L(∆A ) and L(∆ all different in this case. Definition 3.12. Recall that the S is a subset of Π indicating colored vertices in a Dynkin diagram. Let η : S → {±1} be any function. We let η ∈ E Π\S defined by η (αi ) = η(αi ) if αi ∈ S, η (αi ) = 1 if αi 6∈ S. Thus there is a bijective correspondence between the set of all functions η and the set E Π\S and (by Equation (10)) a second bijection with the set E(∆Π\S ): η 7→ η ∈ E Π\S , and η 7→ η (∆Π\S ) ∈ E(∆Π\S ). The exponential map h exp : h → h H allows us to define chamber walls in H and therefore in any of the connected components of HR . For any root φ ∈ ∆ the set {h ∈ h H : |χφ (h)| = 1} defines the φ-wall of H = h H. The intersection of all the αi -walls of H is the set {h }. This is also the intersection of all the φ-walls of H with φ ∈ ∆. We also define the following subsets of H : Definition 3.13. We denote: • D = D(∆) = {h : ∈ E}. • D(∆Π\S ) = {h : ∈ E(∆Π\S )}. The set D has two structures: It is a finite group and also a set with an action of W . In Proposition 3.15 it is the first structure that is emphasized but in the Proof of Proposition 4.5 it is the second structure which is relevant. We now look at the W -action. Since W acts on HR and w ∈ W sends a φ-wall of H to the w(φ)-wall of some other H0 , this set D is preserved by W and thus acquires a W -action. Given S ⊂ Π we similarly obtain that D(∆Π\S ) has a WS -action. The map (11)

D(∆Π\S ) → E(∆Π\S )

sending h 7→ also defines an action of WS on the set of signs E(∆Π\S ). Recall (Notation 3.9) that E Π\S ⊂ E denotes those for which i = 1 whenever αi 6∈ S. We have a bijection (by (11) together with(10)) E Π\S ∼ = D(∆Π\S ). The WS -action on the set D(∆Π\S ) thus gives a WS -action on E Π\S such that for any ∈ E Π\S only the j with αj ∈ S may change in sign under


99

the action. Note that this construction requires looking at the h with ∈ E(∆Π\S ) in the adjoint representation of lA . Remark 3.14. The root characters can be expressed as a product of the P simple root characters raised to certain integral powers φ = li=1 ci αi with Q ci ∈ Z, eφ = li=1 χcαii . Therefore if h ∈ HC the scalars χαi (h) determine all the√scalars eφ (h) = χφ (h) and thus h is uniquely determined. Moreover √ hφ,x+ −1yi hα ,x+ −1yi is real for all φ ∈ ∆ if and only if e i is real for all e αi ∈ Π. e has 2l components. We Proposition 3.15. The Cartan subgroup HR of G have HR = DH where D (Definition 3.13) is the finite group of all the h , ∈ E. √ Proof. It was shown in Lemma 3.7 that the elements exp(x + −1y) with √ x, y ∈ h such that ehαi ;x+ −1yi is real for all αi ∈ Π are those for which y P and ki are integers. has the form y = li=1 ki yi with yi as in Notation 3.4, √ hφ,x+ −1yi Moreover as in Remark 3.14 we also have that e is real for any φ ∈ P ∆ exactly when y = li=1 ki yi for some integers ki . Therefore √ all the root spaces of g are stabilized under the adjoint √ action of exp(x + −1y). Since clearly h is also stabilized, then exp(x + −1y) stabilizes all of g and this √ √ e In fact this shows that exp(x+ −1y) ∈ G e implies that exp(x+ −1y) ∈ G. Pl if and only if y has the form y = i=1 ki yi for certain ki integers. Thus e ∩ HC . From here Lemma 3.7 and these remarks compute the intersection G it is easy to conclude. Note that Proposition 3.15 implies that D(∆) in Definition 3.13 is isomorphic to HR /H as a set with a W -action. That the W -actions agree is verified in Corollary 4.6. We now give the Proof of Proposition 3.6: Proof. Let Ad denote the representation of GL(n, C) on sl(n, C). Then we have Ad(GL(n, C)) = Ad(SL(n, C)). If n is odd, Ad(SL(n, C)) is isomorphic to SL(n, C). Denote Di = diag(r1 , . . . , rn ) with ri as in Example 3.8. We set hi = Di when n is even and hi = (−1)n−i Di when n is odd. Let √ e χα (hi ) = eπ −1δi,j for αi ∈ Π. hi = Ad(hi ). We have that for each hi ∈ G, i The hi generate the group D with 2l elements where l = n − 1 and now e and it is the group G1 = hAd(SL(n, R)), hi , i = 1, . . . , li is a subgroup of G ± isomorphic to Ad(SL(n, R) ) for n even and to Ad(SL(n, R)) ∼ = SL(n, R) e for n odd. What remains is to verify that G ⊂ G1 . e = {Ad(g) : g ∈ U (n)} ∩ G. e By the Iwasawa decomposition of G e Let K e and G1 , it suffices to show that K = KD, K = SO(n). The right side, KD is either O(n) or SO(n) according to the parity of n. Recall that the

100


e of G e acts transitively on the set X of all maximal compact Lie subgroup K maximal abelian Lie subalgebras a which are contained in the vector space p. The action of K = Ad(SO(n)) is also transitive on X (by (2.1.9) of [22]) e there is k ∈ K such that g = kD where D is the and thus for any g ∈ K e isotropy group (in K) of an element in X, for instance of the element h ∈ X. However this isotropy group D has been computed implicitly in the Proof e = KD. of Proposition 3.15 and D = D. Therefore K Proposition 3.16. Let αi ∈ Π and assume that = (1 , . . . , l ) ∈ E. Then −C sαi h = h0 where 0j = j i j,i with (Ci,j ) the Cartan matrix. Proof. This follows from the expression of the Weyl group action on elements αi , x)αi . If this expression is applied to x = αj in h0C given by: sαi x = x − (ˇ it gives sαi αj = αj − Cj,i αi . On the level of root characters this becomes, by exponentiation of the previous identity, −Cj,i

sαi χαj = χαj χαi

.

Now recall that j is just χαj evaluated at h . Also 0j will be χαj evaluated at sαi h . When we evaluate χαj on sαi h in order to compute the corresponding j-th sign, we obtain χsαi αj (h ). Therefore the sign of χαj on the sαi h is −C

given by the product j i j,i . Finally we use the fact that the set of all scalars χαi (h) determines h. Thus 0 determines the element h0 giving rise to the equation sαi h = h0 . The sign change j → 0j in Proposition 3.16 is precisely the gluing rule in Lemma 4.2 for the indefinite Toda lattice in [15]. Then the gluing pattern using the Toda dynamics is just to identify each piece of the connected component H (see Figure 1). The sign change on subsystem corresponding to HRA with A ⊂ Π can be also formulated as: Proposition 3.17. Let αi ∈ Π \ A and assume that = (1 , . . . , l ) ∈ E A . Then: −C a) If i = 1, sαi h = h . If i = −1 then sαi h = h0 where 0j = j i j,i . Q In addition h0 factors as a product ( αj ∈A,Cj,i is odd hj )hA where A ∈ E A . b) If i = 1, sαi HA ⊂ HA . If i = −1 then sαi HA ⊂ h0 H A where 0j = Q −C j i j,i . In addition h0 factors as a product ( αj ∈A,Cj,i is odd hj )hA where A ∈ E A . c) The sign k = sign(χαk (h)) for any h ∈ H agrees with sign(χsαi αk (h0 )) for any h0 ∈ sαi (H ). Proof. Part a) follows from Proposition 3.16 but with the observation that in this case, 0 may fail to be in E A even if ∈ E A . This happens exactly


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when i = −1 and αj ∈ A with Cj,i odd (either −3 or −1). Under these circumstances 0j = −1 (rather than one as required in the definition of E A ). Q We can fix this problem by factoring h0 as a product ( αj ∈A,Cj,i is odd hj )hA where A ∈ E A . Part b) follows from Part a) and the fact that each h H is a connected component of HR in Proposition 3.15. Part c) follows easily from χsαi αk (sαi h) = χαk (sαi (sαi h)) = χαk (h). Also by the fact that the sign of a root character is constant along a connected component and that we have h ∈ H and sαi h is in the new component sαi H . The two desired signs have thus been computed in the two connected components and they agree. Remark 3.18. From Proposition 3.17 it follows that anyconnected compoA,≤ , with w ∈ WΠ\A , nent HA , is the union of chambers of the form w H(w) (w) ∈ E, [ A,≤ HA = w H(w) . w∈WΠ\A

4. Colored Dynkin diagrams. We now introduce some notation that will ultimately parametrize the cells in ˆ R to be defined a cellular decomposition of the smooth compact manifold H in §6. 4.1. Colored Dynkin Diagrams. Let us first define: Definition 4.1 (Colored Dynkin diagrams D(S)). A colored Dynkin diagram is a Dynkin diagram where all the vertices in a set S ⊂ Π have been colored either red R or blue B. For example, in sl(4, R), ◦R − ◦ − ◦B is a colored Dynkin diagram with S = {α1 , α3 }. Thus a colored Dynkin diagram where S 6= ∅, corresponds to a pair(S, η ) with S ⊂ Π and η : S → {±1} any function. Here η(αi ) = −1 if αi is colored R and η(αi ) = 1 if αi is colored B. If S = ∅ then o with o (αi ) = 1 for all αi ∈ Π replaces η . We denote: • D = (S, η ) or (S, η (∆Π\S )) with η ∈ E Π\S ; • D(S) = {D = (S, η ) : the vertices in S are colored}. We also introduce an oriented colored Dynkin diagram which is defined as a pair (D, o) with o ∈ {±1} and D a colored Dynkin diagram. 4.2. Boundary of a colored Dynkin diagram. We now define the boundaries of a cell parametrized by a colored Dynkin diagram D. Definition 4.2 (The boundary ∂j,c D). For each (j, c) with c = 1, 2 and j = 1, . . . , m we define a new colored Dynkin diagram ∂j,c D, the (j, c)boundary of the D by considering {αij : 1 ≤ i1 < · · · < im ≤ l} the set

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s α2

oR -o

j=1, c=1

j=2

,c =2

j =1 ,c =1

o

oB -

sα1

=1

=2 ,c 1 j=

,c j= 2

o -oB

oR 1 =

j=2, c=2

o-

,c j= 2

o-o

Figure 7. The boundary ∂j,c in the case of sl(3, R). Π \ S of uncolored vertices and m = |Π \ S|. The ∂j,c D is then a new colored Dynkin diagram obtained by coloring the ij -th vertex with R if c = 1 and with B if c = 2. The boundary of an oriented colored Dynkin diagram (D, o), o ∈ {±1} is, in addition, given an orientation defined to be the sign (−1)j+c+1 o. Recall that a colored Dynkin diagram D corresponds to a pair (S, η ) with S ⊂ Π and η : S → {±1} (Definition 3.12). Thus the boundary ∂j,c determines a new pair (S ∪ {αij }, η0 ) associated to ∂j,c D. We can then define the following boundary maps, (12) (−1)j+c+1 ∂j,c : Z [D(S)] → Z D(S ∪ {αij }) . Example 4.3. The boundary of ◦ − ◦ (which we can picture as a box) consists of segments (one dimensional boxes) given by ∂1,1 (◦ − ◦) = ◦R − ◦, ∂2,1 (◦ − ◦) = ◦ − ◦R , ∂1,2 (◦ − ◦) = ◦B − ◦ and ∂2,2 (◦ − ◦) = ◦ − ◦B . The orientation sign associated to ◦R − ◦ is the following: With c = 1 and j = 1 one has (−1)j+c+1 = (−1)3 = −1. We illustrate the example in Figure 7. 4.3. WS -action on colored Dynkin diagrams. We now move these colored Dynkin diagrams around with elements in W . A WS -action on the diagram D ∈ D(S), WS : D(S) → D(S), is defined as follows: Definition 4.4. For any αi ∈ S (the αi vertex is colored), sαi D = D0 is a new colored Dynkin diagram having the colors according to the sign change −C 0j = j i j,i in Proposition 3.17 with the identification that R if the sign is −1, and B if it is +1. For example, in the case of sl(3, R),sα1 (◦R − ◦B ) = ◦R − ◦R , sα1 (◦B − ◦R ) = ◦B − ◦R .


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We also define a WS -action on the set D(S) × {±1} of oriented colored Dynkin diagrams. If αi ∈ S, the action of sαi on the pair (D, o) is given by sαi (D, o) = (sαi D, (i )rαi o) where rαi is the number of elements in the set {αj ∈ Π \ S : Cj,i is odd} and i = ±1 depending on the color of αi . We confirm the Definition: Proposition 4.5. Definition 4.4 above gives a well-defined action of WS on the set D(S) of colored Dynkin diagrams with set of colored vertices S. Proof. This follows from Proposition 3.16 and the correspondence (S, η ) → hη giving a bijection between D(S) and D(∆Π\S ). Corollary 4.6. There is a bijection of D(∆) → HR /H intertwining the W -actions on both sets. Proof. The two sets D(∆) and HR /H are clearly in bijective correspondence and an element h corresponds to the coset h H. By Proposition 3.17, Notation 4.4 and Proposition 4.5, the actions on these two sets agree. They are both given by Part a) in Proposition 3.17. Notation 4.7. Motivated by Example 2.2 we consider the set W × D(S) WS

the W translations of the set D(S). We write the elements in this set as pairs (w, D) and introduce an equivalence relation ∼ on these pairs, where for any x ∈ WS , (wx, D) ∼ (w, xD). Denote the equivalence classes [(w, D)]. This set of equivalence classes is then in bijective correspondence with the set D(S) × W/WS of pairs (D, [w]) with [w] = [w]Π\S ∈ W/WS and D in D(S). The correspondence is such that [(w• , D)] corresponds to (D, [w• ]) with w• ∈ [w]Π\S a minimal length representative of a coset in W/WS . We denote Dk = {(D, [w]Π\S ) : S ⊂ Π, |S| = k, w ∈ W }, which also parametrizes all the connected components of the Cartan subgroups of the form HR (w(∆Π\S )). Remark 4.8. For the reader who is not interested in the W -action or torsion, the object defined in Definition 4.9 becomes over Q the vector space with basis given by the full set of colored Dynkin diagrams having a fixed number of uncolored vertices. The boundary maps are obtained by translating around the ∂j,c defined for the antidominant chamber (see (14) below). Perhaps this boundary construction can be appreciated in Example 2.2 in Section 2 where portions of the boundary of a chamber are translated and some identifications take place. The tensor product notation below accomplishes the required boundary identifications algebraically. Definition 4.9 (The Z[W ]-modules M(S)). The full set of colored Dynkin diagrams is the set D(S) × W/WS of all pairs (D, [w]Π\S ).

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We can also define the full set of oriented colored Dynkin diagrams by considering D(S) × {±1} × W/WS for different subsets S ⊂ Π. As W sets these correspond to W × D(S) × {±1}. WS

If we consider D(S) × {±1} as imbedded in Z[D(S)] by sending (D, o) to oD, o ∈ {±1} we may consider a WS -action on ±D(S), namely the action on oriented colored Dynkin diagrams. This produces a Z[W ]-module Z[W ] ⊗ Z[D(S)], and we denote this module by Z[WS ]

M(S) = Z[W ] ⊗ Z[D(S)]. Z[WS ]

Also denote by Ml−k the direct sum of all these modules over all sets S with exactly k elements, M Ml−k = M(S). |S|=k

Remark 4.10. Assume that A is a Z[WS ]-module and B is a Z[WS 0 ]module with S ⊂ S 0 . In particular B can be regarded as a Z[WS ]-module by restriction. Let f : A → B be a map intertwining these two Z[WS ]module structures involved. Then, tensoring with Z[W ] we obtain a map of Z[W ]-modules: F (f ) : Z[W ] ⊗ A → Z[W ] ⊗ B. Z[WS ]

Z[WS ]

Since, in addition, B is a Z[WS 0 ]-module where S ⊂ S 0 then there is a second map: g : Z[W ] ⊗ B → Z[W ] Z[WS ]

⊗

B.

Z[WS 0 ]

Let g ◦ F (f ) = T (f ). This is then a map of Z[W ] modules T (f ) : Z[W ] ⊗ A → Z[W ] Z[WS ]

⊗

B.

Z[WS 0 ]

We now give the boundary maps of Ml−k . We regard D(S) × {±1} as a Z[WS ]-module using Definition 4.4 the oriented case. We can view a pair (D, o) instead as ±D ∈ Z[D(S)]. This gives a Z[WS ]-module structure to each of the Z modules involved on the domain of the map in (12) and a WS∪{αij } to those on the co-domain. The map in (12) now has the form described in Remark 4.10. We apply the construction T (∂j,c ) in Remark 4.10 to (12) and add over all possible subsets S ⊂ Π with |S| = k on the left side and with |S| = k + 1 on the right side. We then obtain the boundary maps, (13)

∂l−k : Ml−k → Ml−(k+1) ,


105

which are all given by ∂l−k (w• ⊗ X) =

(14)

l−k X 2 X

(−1)j+c+1 w• ⊗ T (∂j,c )X,

j=1 c=1

w•

where X ∈ D(S) and ∈ [w]Π\S is the minimum length representative in W/WS (see Notation 4.7). We thus have: Proposition 4.11. The maps ∂l−k of (14) define a chain complex M∗ of Z[W ]-modules. 5. Cartan subgroups and Weyl group actions. We here discuss relations between some Cartan subgroups of Levi factors, and verify the WS -action on oriented colored Dynkin diagrams. A

5.1. Cartan subgroups of Levi factors. Let Ad∆ denote the adjoint representation of the Lie subgroup HR LA of G on the Lie algebra lA . Note that we are deviating slightly from the standard convention and denoting by A Ad∆ the representation of HR LA acting on the semisimple part of the Levi factor lA , seen as a quotient of the group action on hA + lA (thus dividing by the center of this Lie algebra which corresponds to a trivial representation A summand). We will use this same notation Ad∆ when restricting to various A Lie subgroups of HR LA containing LA . The notation Adw(∆ ) with w ∈ W refers to the similar construction with respect to w(∆A ). When A = A Π and ∆A = ∅ then Adw(∆ ) will refer to the (trivial) one dimensional representation of {e}. A be defined by: Definition 5.1. Let Hfund   X  ◦ A  (15) . exp ci mαi : ci ∈ R  = Hfund   αi 6∈A

A . Note that usually H A 6= Hfund

Then we have: A

Proposition 5.2. The images of Ad∆ on subsets of Cartan subgroups satisfy A

1) Ad∆ (D(∆)) = D(∆A ), A 2) Ad∆ (HRA ) = HR (∆A ), A A ) = Ad∆A (H A ). 3) Ad∆ (Hfund Proof. We first point out that the exponential map in the Lie groups LA R, ∆A A Ad gives a diffeomorphism between h and the corresponding connected

106

L.G. CASIAN AND Y. KODAMA A

Lie group. Thus the Lie groups H A , Ad∆ (H A ) are isomorphic. This takes care of Part 2) on the level of the connected component of the identity. A Also the image under ad∆ of the set {m◦αi : αi 6∈ A} gives rise to a A basis of the Cartan subalgebra of ad∆ (lA ). Thus exponentiating we obtain Part 3). √ A Recall that χαi (hj ) = exp(π −1δi,j ) and Ad∆ (h) = I if and only if A χαi (h) = 1 for all αi ∈ Π \ A. We have thus Ad∆ (hi ) = I for αi ∈ A and A A {Ad∆ (h ) : ∈ E} = {Ad∆ (h ) : ∈ E A } = D(∆A ). The elements hi with αi ∈ A are in the center of HR LA . This proves Part 1). For Part 2), we proceed by noting that by definition of HR (∆A ) it must be A generated by the Lie group Ad∆ (H A ) with Lie algebra hA and the elements Ad(hi ) with αi ∈ Π \ A (playing the role that the hi play in the definition A of HR ). This is the same as Ad∆ (HRA ). 5.2. Action of the Weyl group on oriented colored Dynkin diagrams. Proposition 5.3. The action of WS on D(S) × {±1} in Definition 4.4 is well-defined. Proof. Recall that W is a Coxeter group, (Proposition 3.13 [12]) and it thus has defining relations s2αi = e and (sαi sαj )mij = e where mij is 2, 3, 4, 6 depending on the number of lines joining αi and αj in the Dynkin diagram. The case mij = 2 occurring when αi and αj are not connected in the Dynkin diagram. The only relevant cases then are when αi , αj are both colored and connected in the Dynkin diagram. The only relevant vertices in the Dynkin diagram are those connected with these two and which are uncolored. We are thus reduced to very few nontrivial possibilities: D5 , A4 , F4 , B4 , C4 ; smaller rank cases being very easy cases. We verify only one of these cases, the others being almost identical with no additional difficulties. Consider the case of D5 where αi = α, αj = β are the two simple roots which are not “endpoints” in the Dynkin diagram and all the others are uncolored. Then rα = 1 but rβ = 2. For book-keeping purposes it is convenient to temporarily represent red as −1 and blue as 1 and simply follow what happens to these two roots and an orientation o = 1. Hence all the information can be encoded in a triple (1 , 2 , o) representing the colors of these two roots and the orientation o. Now we apply (sα sβ )3 to (1 , 2 , o): sβ

s

α (1 , 2 , o = 1) −→ (1 2 , 2 , (2 )2 o = 1) −→ sβ sα (1 2 , 1 , (1 2 )1 o) −→ (2 , 1 , (1 )2 1 2 o = 1 2 o) −→ sβ sα (2 , 1 2 , (2 )1 1 2 o = 1 o) −→ (1 , 1 2 , (1 2 )2 1 o = 1 o) −→ (1 , 2 , o).


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A

5.3. The set HR◦ . By Proposition 5.2, if χ∆ αi , αi ∈ Π\A is a root character A ∆A A A of HR (∆ ) acting on l , then it is related to χαi by χ∆ = χαi . The αi ◦ Ad A ∆ ◦ map χαi , αi ∈ Π \ A provides the local coordinates on HR which consists of the Cartan subgroups HR (∆A ). Definition 5.4. Let HR◦ be defined as [ HR◦ = HR (∆A ) × {[e]A }. A⊂Π

We then define a map φe :

HR◦

→ Rl as follows:

φe (h, [e]A ) = (φe,1 (h), . . . , φe,l (h)), A

where φe,i (h) = χ∆ αi (h) whenever αi 6∈ A and φe,i (h) = 0 if αi ∈ A. Denote φA the restriction of φe to HR (∆A ) × {[e]A }. e A

By Proposition 5.2 Part 2) we can compose with Ad∆ × 1 and re-write S A A HR × {[e]A } . We also define the domain of φe ◦ (Ad∆ × 1) as A⊂Π

φw = (φw,1 , . . . , φw,l ) : w(HR◦ ) −→ Rl , with w(HR◦ ) :=

[

HR (w(∆A )) × {[w]A },

A⊂Π

by setting A

A

) φw,i (x, [w]A ) = χw(∆ (Adw(∆ ) (wh)) = χwαi (wh) = χαi (h), wαi A

where x = Adw(∆ ) (wh) ∈ HR (w(∆A )) with h ∈ HR , if αi 6∈ A. For the case when αi ∈ A, we set φw,i (x, [w]A ) = 0. 5.4. The sets HRA . We here note an isomorphism between several presentations of a split Cartan subgroup of a Levi factor. Proposition 5.5. All the following are isomorphic as Lie groups: 1) HRA , A 2) Ad∆ (HR ), A 3) Ad∆ (HRA ), 4) HR (∆A ). Proof. The groups in 3) and 4) are isomorphic by Proposition 5.2. We can write hC = z + hA C where z is defined by z ∈ z if and only if hαi , zi = 0 for all αi ∈ Π \ A.

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e Now the group exp(z) is the center of HC LA C . Intersecting with G we A e obtain √ the center of HR L . To find √ this intersection with G, we must find −1yi hα ,x+ j all x + −1y ∈ z such that e is real for all αi ∈ Π (see the Proof of Lemma 3.7). We find that x ∈ z ∩ h and that y is an integral linear combination of the yi . As in the Proof of Proposition 5.2 the elements √ exp( −1yi ) which are in the center of HR LA are those for which αi ∈ A. The center of HR LA is then the group generated by hi with αi ∈ A and exp(z∩h). Since exp(z∩h)∩H A = {e}, we have that HR divided by exp(z∩h) is isomorphic to DHRA . The groups HRA , DHRA and those involved in 2), 3) or 4) all differ by a subgroup of the D ⊂ exp(z) which is annihilated by A Ad∆ . From here and Proposition 5.2 Part 1), the isomorphism between 2), 3) and 4) follows. ˆ R. 6. The set H ˆ R as a union of Cartan subgroups of Levi We here define our main object H factors and their W -translations. ˆ be the disjoint union of all the quotients W/WΠ\A over A ⊂ Π. Let W Each of the elements [w]A ∈ W/WΠ\A parametrizes a parabolic subgroup. First [e]A corresponds to a standard parabolic subgroup of GC (Proposition 7.76 or Proposition 5.90 of [13]). This is just the parabolic subgroup determined by the subset Π\A of Π. Then we translate such a parabolic subgroup with w ∈ W . The resulting parabolic subgroup corresponds to [w]A . Thus ˆ is the set of all the W -translations of standard parabolic subgroups. W Definition 6.1. We define [ [ ˆR = (16) H HR (w(∆A )) × {[w]A }. A⊂Π w∈W A

From Proposition 5.5, HRA can be replaced by HR (∆A ) or Adw(∆ ) (HR ). ˆ R as a subset of HR × W ˆ . We recall w•,A Thus we can alternatively write H or just w• as in Notation 4.7. We then have: [ [ ˆR = (17) H w•,A (HRA ) × {[w]A }. A⊂Π w•,A ∈W/WΠ\A

Also fixing an isomorphism ξw : HR (∆A ) → HR (w(∆A )) inducing a set ∗ : w(Π) → Π, we have bijection (by composition) ξw [ ˆR ∼ H W × HR (∆A ), = A⊂Π

WΠ\A

where an element (w•,A , h) on the right-hand side is sent to (ξw (h), [w]A ). ˆ R with a W -action. This endows H


109

In what follows it is useful to think of a colored Dynkin diagram (e.g., ◦R −◦) as parametrizing a “box” (e.g., [−1, 1]). We need to further subdivide this box into 2l smaller boxes by dividing it into regions according to the sign of each of the coordinates (e.g., [−1, 0] [0, 1]). We also need to consider the boundary between these 2l regions (e.g., {0}). We will introduce an additional sign or a zero to keep track of such subdivisions (e.g., ◦R − ◦+ , ◦R − ◦− , ◦R − ◦0 for [−1, 0], [0, 1], {0}) (see also Example in Section 2). We will then do the same thing with a colored Dynkin diagram of the form (D, [w]Π\S ) by assigning labels in {±1, 0} to the vertices in Π \ S. ˆ R to The main purpose of the following is to associate certain sets in H the colored Dynkin diagrams, the signed-colored Dynkin diagrams and the sets associated to them only play an auxiliary role. We introduce the following notation. This notation is illustrated in Example in Section 2 and Figure 3. ˇ is a Dynkin diagram Notation 6.2. A signed-colored Dynkin diagram D with some vertices colored (R or B) and the remaining vertices labled +, − or 0. The followings are auxiliary objects to keep track of signs, zeros and colors. • ηˇ : Π → {±1, 0} function which agrees with η on S and determines the ˇ sign labels in D. ˇ = A(ˇ • A = A(D) η ) = {αi ∈ Π : ηˇ(αi ) = 0}. • K(η) the set of all ηˇ : Π → {±1, 0} which agree with η in S. ˇ • ηˇ ∈ E A the element which agrees with ηˇ on Π \ A (A = A(D)). Π\S ˇ • D(S, A, ηˇ, [w] ) the unique signed-colored Dynkin diagram attached to a colored Dynkin diagram (D, [w]Π\S ) or to (S, η, [w]Π\S ), where the vertices in Π \ S are given a label in {±1, 0}. ˆ R to a signed-colored Dynkin diagram with We now associate a subset of H S = ∅. Notice that in this case ηˇ can be any element of E A . Recall that ηˇ(∆A ) is then the element of E(∆A ) that corresponds. We associate to a signed colored Dynkin diagram (∅, A, ηˇ, [e]Π\S ) two sets, one includes walls and the other doesn’t (in order to avoid duplicate notation we denote the signed-colored Dynkin diagram and the set with the same notation): ≤

(∅, A, ηˇ, [e]Π\S ) = H(∆A )≤ × {[e]A }. ηˇ (∆A ) When A = ∅ (no zeros), these are the 2l boxes in the antidominant chamber. We define the second related set as: (∅, A, ηˇ, [e]Π\S ) = H(∆A )< × {[e]A }. ηˇ (∆A ) The chamber walls of the antidominant chamber of the Cartan subgroup are defined as:

110


• D(αi , A, )≤ = {h ∈ H(∆A )≤ : |χαi (h)| = 1} (the αi -wall), (∆A ) • D(αi , A, )< = D(αi , A, )≤ ∩ {h ∈ H(∆A )≤ : |χαj (h)| < 1 if j 6= (∆A ) i, αj ∈ Π \ A}. We next consider the case of S = {αi } 6∈ A and then the general case of any S ⊂ Π with A ⊂ Π \ S and any ηˇ ∈ E A . This defines the walls for Levi factor pieces corresponding to subsystems of the Toda lattice (we here list open walls): < A • ({αi }, A, ηˇ) = TD(αi , A, ηˇ) × {[e] }. • (S, A, ηˇ) = ({ αi }, A, ηˇ). αi ∈S

ˆ R to a colored Dynkin diagram. We define a set We now associate a set in H denoted D = (S, η ) as follows: For S 6= ∅ (so that there is an η : S → {±1}), S S (S, A(ˇ η ), ηˇ), and if S = ∅, (∅, o ) = (∅, ηˇ). • (S, η ) = ηˇ∈K(η)

ηˇ∈{±1,0}l

Here o = (1, . . . , 1), and see (7) and (9) in Section 2. We consider the w-translations of the colored Dynkin diagrams: For this write w uniquely as w = w• w• with w• ∈ WS . • (w• D, [w]Π\S ) = w((D, [e]Π\S )). In order to define the W -translations of sets associated to signed-colored Dynkin diagrams we have to extend the definition of the WS -action to the signed-colored Dynkin diagrams. The definition is exactly the same if we treat the label − as if it were an R and the label + as if it were a B as in ˇ [w]Π\S ) and let: Definition 4.4. Then consider (w• D, ˇ [w]Π\S ) = w((D, ˇ [e]Π\S )). • (w• D, ˆ R as the antidominant chamber Remark 6.3. We refer to the set (∅, o ) ⊂ H •,A A ˆ of HR . Recall Notation 4.7 w ∈ [w] . The following justifies our definition ˆ R using (17) in the definition of H ˆ R. of the “antidominant chamber” of H Proposition 6.4. We have [ [ ˆR = H (18)

A,≤ w•,A σ H(σ) × {[w]A }.

A⊂Π w•,A ∈W σ∈WΠ\A

Proof. We set w• = w•,A for simplicity. By Proposition 3.17 Part b), we have that each HRA can be written as a union over σ ∈ WΠ\A of sets of A,≤ ˆ R , we conclude the the form σ H(σ) and thus by the definition (17) of H statement.


111

7. Colored Dynkin diagrams and the corresponding cells. ˆ R as the union Here we consider the manifold structure and the topology of H of cells parametrized by colored Dynkin diagrams. 7.1. Action of the Weyl group on the sets (S, η ). Let Mgeo be the complex with the sets (D, [w]Π\S ). We then consider the action of WS on the union of all the sets D = (S, η ) having a fixed nonempty set S of colored vertices and endowed with an orientation o. Similarly we can endow each of the terms in the chain complex Mgeo with a action of W by translating the sets corresponding to colored Dynkin diagrams with the W -action and taking into account changes of orientation induced on the oriented boxes. This new action of W on Mgeo could in principle be different from the W -action on the chain complex M∗ (see Proposition 4.11). We now become more explicit about the W -action that was just introduced on Mgeo ∗ : Note that since χαji (h) = ±1 for any αji ∈ S and h ∈ (S, η ) (Notation 6.2), if h ∈ (S, η ) then sαji χαj (h) = χαj (h)χαji (h)−Cj,ji = ±χαj (h). Hence, in terms of the coordinates given by φe , the action of sαji is given by a diagonal matrix whose nonzero entries are ±1. This matrix has some entries corresponding to the set S and other entries corresponding to Π \ S. The entries corresponding to the set S change by a sign as described in Proposition 3.16. The statement in Proposition 3.16 just means that the set (S, η ) is sent to (S, η0 ) with η 0 = (ηj1 , . . . , ηjs ), and ηj0 i = ηji (−1)Cj,ji . The determinant of the diagonal submatrix corresponding to elements in Π \ S is the sign (ηi )r with (see Definition 4.4) r = {αj ∈ Π \ S : Cj,i is odd} . Consider the set (S, η ) endowed with a fixed orientation ω corresponding to o = 1. Then sαi with αi ∈ S sends (S, η ) to (S, η0 ) endowed with the orientation (ηi )r ω. We now consider the Z module of formal integral combinations of the sets (S, η ), h i Z (S, η ) : S ⊂ Π, |S| = k, η ∈ E Π\S . We keep track of the orientation by putting a sign ± in front of (S, η ). This Z module acquires a Z[WS ]-action that corresponds to the abstract construction given in Definition 4.9 with colored Dynkin diagrams. By considering all the W -translations of the (S, η ) we generate the module denoted above by M(S). The direct sum of all these M(S) over |S| = k is denoted Ml−k (see Definition 4.9). Hence there is no difference as W -modules between Mgeo and M∗ . ∗ We can now summarize this discussion in the following: Proposition 7.1. The action of W on M∗ (Subsection 4.3) and the action of W on Mgeo are isomorphic.

112


We will then drop the superscript . . .geo from the notation in view of Proposition 7.1. ˆ R . Recall the map φe in Definition 5.4 7.2. Manifold structure on H ◦ S whose domain is H R = HR (∆A ) × {[e]A } and co-domain is Rl . We also A⊂Π S have defined φw , with domain HR (w(∆A ) × {[w]A }). We will use these A⊂Π

ˆ R coordinate charts leading to a manifold structure. We then maps to give H have the following three Propositions: Proposition 7.2. The image φe (HR (∆A )) × {[e]A } consists of all (t1 , ..., tl ) ∈ Rl such that ti 6= 0 if and only if αi 6∈ A. The map φe is a bijection ◦ S between H R = (HR (∆A )) × {[e]A } and Rl . A⊂Π

Proof. We start with the last statement, that φe is a bijection; we have that A ∆A φe is injective because the scalars χ∆ αj (h), . . . , χαjm (h) determine all the root A

1

A A characters χ∆ φ (h), φ ∈ ∆ for h ∈ HR (∆ ) and these scalars determine h in the adjoint group (Remark 3.14). From Proposition 5.2 Part 2) it follows A that we can regard HR (∆A ) as Ad∆ (HR ) (see (15)). We prove surjectivity A by proving first the statement concerning the image φe ◦ (Ad∆ × 1)(HA × l {[e]A }) . When all the sets A are considered then all of R willbe seen to be P in the image of φe . First consider h h with h = exp ci m◦αi and ∈ E A . αi 6∈A P ∆A ◦ cj mαj = ci (αi2,αi ) we obtain, We now apply φe ◦(Ad ×1). Since αi , αj 6∈A

ci

(αi ,αi ) 2

by exponentiating, χαi (h h) = i e . The set φe (HR (∆A ) × {[e]A }) becomes the image of the map: Rl → Rl given by first defining a map that (αi ,αi )

sends (1 t1 , . . . , l tl ) → (f1 , . . . , fl ) with fi = i ti 2 for ti > 0. This map is modified so that whenever αi ∈ A then the i-th coordinate is replaced with 0. We denote this modified map by F A . The domain and the image of F A therefore consists of the set {(s1 , . . . , sl ) : si = 0 if αi ∈ A}. Together all these F A give rise to one single map F : Rl → Rl which is surjective. Proposition 7.3. The image φw (HR (w(∆A )) × {[w]A }) consists of all (t1 , . . . , tl ) ∈ Rl such that ti 6= 0 if and only if αi 6∈ A. Proof. This follows from Proposition 6.4 and the fact that φe,i (w(h), A) = χwαi (w(h)) = χαi (h) for αi 6∈ A. Proposition 7.4. The image φw ((∅, o )) consists of all (t1 , . . . , tl ) ∈ Rl such that −1 < ti < 1. The sets φw ((S, η , [w]Π\S )) as S ⊂ Π, S 6= ∅ varies,


113

give a cell decomposition of the boundary of the box [−1, 1]l . In particular, ˆ R. the sets (S, η , [w]Π\S ) give a cell decomposition of the smooth manifold H Proof. This follows from Proposition 7.3 but is better understood in Example 2.2. We omit details. ˆ R for the Remark 7.5. There is a more convenient cell decomposition of H purpose of calculating homology explicitly. The only change is that the l dimensional cell becomes the union of all the l-cells together with all the (internal) boundaries corresponding to colored Dynkin diagrams where all the colored vertices are colored B. This is the set: [ ˆR \ H (S, η , [w]Π\S ). S⊂Π, w∈W η such that η(αi )=−1 for some αi ∈S

This set can be seen to be homeomorphic to Rl . With this cell decomposition there is exactly one l cell; and the other lower dimensional cells correspond to colored Dynkin diagrams which are parametrized by pairs (D, [w]Π\S ), such that, at least one vertex of D has been colored R. In terms of the (S, η , [w]Π\S ), the top cell would instead be defined to consist of HR◦ . (See Figure 6 for this remark.) The big cell in thisPdecomposition with a fixed orientation corresponds to the element cl = (−1)l(w) (D, o , w) (S = ∅). This element satisfies w∈W

∂l (cl ) = 2(cl−1 ) for some cl−1 6= 0 (except in the case of type A1 ). Note that (D, w) and (D, wsαi ), with S = ∅ and l(wsαi ) = l(w) + 1, appear with opposite signs in cl . When ∂l is applied and the αi is colored R the sign (−1)rαi in Definition 4.4 makes these two terms contribute as 2(D0 , [w]Π\{αi } ) with D0 the new colored Dynkin diagram obtained. The terms from the boundary ∂l obtained by coloring αi with B will cancel since the action of sαi on colored Dynkin diagrams is trivial when αi is colored B. The case of A1 is an exception because, in that case, once α1 is colored R no more uncolored vertices remain. The set {αj ∈ Π \ αi : Cj,i is odd} is empty and rαi = 0. Thus there is cancellation in this case. ˆ R , coordinate charts, integral homology. We de7.3. Topology on H ˆ R in which U ⊂ S H(w(∆A )) × {[w]A } is open if and fine a topology on H A⊂Π

only if φw (U ) is open in the usual topology of Rl . The maps φw become ∞ on their docoordinate charts and since the compositions φw ◦ φ−1 σ are C ˆ R acquires the structure of a smooth manifold. The W -action main, then H becomes a smooth action. ˆ R and the chain complex MCW Definition 7.6 (Filtration of H ). We con∗ ˆ R in the sense of [18] p. 222. struct a filtration of the topological space H

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Let Xl−k denote the union of all the sets of the form (D, [w]Π\S ) over all w ∈ W and S ∈ P (Π) such that |S| ≥ k. This is a closed set and Xr \ Xr−1 is a union of sets of the form (D, [w]Π\S ) with |S| = l − r. The filtration Xr r = 0, 1, . . . , l satisfies the conditions of Theorem 39.4 in [18]. We define a chain complex MCW with boundary operators as in [18] ∗ ∂r : Hr (Xr , Xr−1 , Z) → Hr−1 (Xr−1 , Xr−2 , Z). ˆ R is compact, nonorientable Proposition 7.7. The smooth manifold H (except if g is of tupe A1 ). The homology of the chain complex M∗ , H k (M∗ ) ˆ R , Z). = Ker ∂k /image(∂k−1 ) is isomorphic as a Z[W ] module to Hk (H ˆ R is the finite union of the chambers as in ProposiProof. The manifold H ˆ R is by continuous transformations, it then tion 6.4. Since the W -action on H suffices to observe that the antidominant chamber is compact. The antidominant chamber (∅, o ) of Definition 6.2 can be seen to be compact by describing explicitly its image under φe . This image is a “box” inside Rl , as can be seen in Propositions 7.2, 7.4 namely the set {(t1 , . . . , tl ) : −1 ≤ ti ≤ 1}. ˆ R is now the finite union of the W -translates of this compact The space H set. That the boundary operators of MCW agree with the boundary operators of M∗ will follow from the fact that in the φw coordinates the (D, [w]) is a “box” which is itself part of the boundary of a bigger “box” (Proposition 7.4). We start with the set {(t1 , . . . , tl ) : −1 ≤ ti ≤ 1} and note that its boundary is combinatorially described by (12) or (13). Note that (D, w) (with S = ∅, [w]Π = w) represents the open box. The faces are parametrized by coloring each of the l vertices R or B which then represent opposite faces in the boundary. The signs are just chosen so that ∂k−1 ◦∂k = 0 for k = 1, . . . , l. This description may be best understood by working out Example 2.2. All the cells thus appear by taking the faces of a box [−1, 1]l and then faces of faces etc. By the same process of coloring uncolored vertices R or B which give rise, each time, to a pair of opposite faces in a box. In each case (12) or (13) correctly describe the process of taking the boundary of a box. Note that it is enough to study what happens when w = e and then consider the W -translates. We now use Theorem 39.4 of [18] to conclude that MCW computes in∗ tegral homology. However each Z[W ]-module appearing in MCW in a fixed ∗ degree, can easily be seen to be identical with the corresponding term in M∗ . By Proposition 7.1 and the agreement of the boundary operators, we obtain that M∗ computes integral homology. The nonorientability follows if we use the second cell decomposition described in Remark 7.5. The unique top cell then has a nonzero boundary (except in the case of g = sl(2, R)).


115

ˆ R. 8. Toda lattice and the manifold H ˆ R with the Toda lattice by extending the We now associate the manifold H results of Kostant in [16]. We start with the definition of the variety ZR of Jacobi elements on g where the Toda lattice is defined. ez. 8.1. The variety ZR and isotropy group G Definition 8.1 (Varieties of Jacobi elements). Let S(g) be the symmetric algebra of g. We may regard S(g) as the algebra of polynomial functions on the dual g0 . If we consider the algebra of G-invariants of S(g), then by Chevalley’s theorem there are homogeneous polynomials I1 , . . . , Il in S(g)G which are algebraically independent and which generate S(g)G . Thus S(g)G can be expressed as R[I1 , . . . , In ]. For F = C or F = R, we consider the variety ZF of normalized Jacobi elements of gF . Our notation, however, is slightly different from the notation of [16] in the roles of eαi and e−αi . Thus we let ( JF = ZF =

X =x+ ( X =x+

l X

) (bi e−αi + eαi ) : x ∈ h, bi ∈ F \ {0} ,

i=1 l X

) (bi e−αi + eαi ) : x ∈ h, bi ∈ F \ {0}, X ∈ S(F ) .

i=1

We also allow subsystems which correspond to the cases having some bi = 0: ( ) l X ◦ JF = X = x + (bi e−αi + eαi ) : x ∈ h, bi ∈ F , i=1 ( ) l X ◦ ZF = X = x + (bi e−αi + eαi ) : x ∈ h, bi ∈ F, X ∈ S(F ) . i=1

Kostant defines in [16] p. 218 a real manifold Z by considering all elements l X x+ (bi e−αi +eαi ) in ZR which in addition satisfy bi > 0. We are departing i=1

in a crucial way from [16] by allowing the bi to be negative or even zero when A 6= ∅. This extension gives the indefinite Toda lattices introduced in [14]. We let for any ∈ E, ( ) l X Z = X = x + (bi e−αi + eαi ) : i bi > 0, X ∈ S(R) . i=1

116


This is a set of real normalized Jacobi elements, that is, the elements of ZR . Thus the union of all the Z is all of ZR , [ ZR = Z . ∈E

The elements in ZR are thus the signed normalized Jacobi elements (in S(R) and Z simply denotes Zo where o = (1, . . . , 1)). Definition 8.2 (Chevalley invariants and isospectral manifold). If x ∈ g and gx ∈ g0 is defined by hgx , yi = (x, y) for any y ∈ g then the map g → g0 sending x to gx defines an isomorphism. We can then regard S(g) as the algebra of polynomial functions on g itself by setting for f ∈ S(g) and x ∈ g, f (x) = f (gx ). The functions I1 , . . . , Il now on g and then restricted to JR , ZR or to ZC are called the Chevalley invariants which are the polynomial functions Pl of {a1 , . . . , al , b1 , . . . , bl } for X = i=1 (ai hαi + bi e−αi + eαi ). The map I = (I1 , . . . , Il ) then defines by restriction a map I = IF : ZF → F l . Fix γ ∈ F l in the image of the map I, and denote \ Z(γ)F = IF−1 (γ) = I−1 (γ) ZF , which defines the isospectral manifold of Jacobi elements of g. Note that in the real (isospectral) manifold Z(γ) studied in [16] will just be one connected component of Z(γ)R . e y on G). e Let Gy be the isotropy Definition 8.3 (The isotropy subgroup G C subgroup of GC for an element y ∈ gC . The group GyC is an abelian connected algebraic group of complex dimension l (Proposition 2.4 of [16]). If y ∈ g e y the intersection we denote by G e y = Gy ∩ G. e G C If x ∈ g, the centralizer of x is denoted gx and dim gx ≥ l. We say that x is regular if dim gx = l. We consider an open subset of GC given by the biggest piece in the Bruhat decomposition: (19)

(GC )∗ = N C HC NC = N C BC ,

e∗ = G e∩ where NC = exp(nC ), N C = exp(nC ) and BC = HC NC . We let G (GC )∗ and, as in Notation 3.9, B = HR exp(n). We then have a map N C × BC → (GC )∗ , given by (n, b) 7→ nb which is an isomorphism of algebraic varieties. Given d ∈ (GC )∗ , d has a unique decomposition as d = nd bd as in (2.4.6) of [16].


117

e y∗ is not an intersection with exp(n)H exp(n) We caution the reader that G but rather with exp(n)HR exp(n). This is the object that appears, for example, in (3.4.10) of [16] and properly contains (3.2.9) in Lemma 3.2 of [16]. e y and HR : The following Proposition gives the relation between G e y is G e conjugate to the Cartan Proposition 8.4. Let y ∈ Zo . Then G e subgroup HR of G. Proof. By Lemma 2.1.1 in [16], y is regular. Then as in Lemma 3.2 of [16], y must be conjugate, under an element in H, to an element x in p. Using Proposition 2.4 of [16] GxC is connected. Since x is also a regular element it follows that gxC is a Cartan subalgebra and GxC is a Cartan subgroup of GC . Using conjugation by an element in K we may conjugate this Cartan subgroup if necessary to HC (Proposition 6.61 or Lemma 6.62 of [13]). We y e y = G∩G e e can thus assume that GxC = HC . We obtain that G C is G conjugate e ∩ HC = HR (see Notation 3.9). to G 8.2. Kostant’s map βCy . Fix y ∈ J(γ)R . Kostant defines a map βCy :

(20)

(GyC )∗ −→ J(γ)C d 7→ Ad(n−1 d )(y)

with d = nd bd , nd ∈ N and bd ∈ B. Note that we have deviated from the convention in [16] by exchanging the roles of N C and NC . We did not exchange the roles of these two groups in (19) but this is compensated by our use of an inverse in the definition of the Kostant map. Theorem 2.4 of [16] then implies that βCy is an isomorphism of algebraic varieties. Denote β y the e Thus we have β y : G e y∗ → Z(γ)R restriction of βCy to the intersection with G. e y∗ = (Gy )∗ ∩ G. e and G C

Proposition 8.5. Let y ∈ J(γ)R . The map β y is an isomorphism of smooth e y∗ → J(γ)R . manifolds G e y of the diffeomorProof. The map β y is the restriction to the Lie group G y phism of complex analytic manifolds βC . We obtain that β y must be an injective map. We show surjectivity. If z ∈ J(γ)R then by surjectivity of βCy there is gC ∈ (GyC )∗ such that βCy (gC ) = z and gC = nC bC . Thus gCc = ncC bcC with ncC ∈ N C and bcC ∈ BC . Therefore β y (gCc ) = Ad(ncC )−1 y. Since y c = y we obtain that (Ad(nC )−1 y)c = z c . But our assumption is that z c = z. Hence we have obtained that βCy (gCc ) = z. By the injectivity of βCy we obe and thus gC = g ∈ G e y . This proves β y tain that gCc = gC . Therefore gC ∈ G is a bijection. By Proposition 2.3.1 of [16], J(γ)R is a submanifold of real dimension l of J(γ)C . The diffeomorphism βCy restricts to the smooth nonsingular map β y . Since we have shown that β y is a bijection, then it is a diffeomorphism.

118


e y is a Cartan subgroup conjugate to HR Remark 8.6. If y ∈ Zo then G (see Proposition 8.4). However, in general it may happen that y ∈ J(γ)R e y is a Cartan subgroup not is not semisimple or that y is semisimple but G conjugate to HR . For example, take y = a1 hα1 + b1 e−α1 + eα1 in the case e = Ad(SL(2, R)± )). We get a nilpotent matrix if a2 + b1 = 0. of sl(2, R) (G 1 e y is a compact Cartan subgroup When a21 +b1 < 0 then y is semisimple but G and thus it is not conjugate to HR . In the cases a21 + b > 0 one obtains that e y is conjugate to HR . G e y is conjugate to HR (for example Assume that we are in the case when G y ∈ Zo ). Combining Proposition 8.4 and Proposition 8.5, Kostant’s map ˆ R as gives an imbedding of Z(γ)R , where γ ∈ Rl is in the image of I, into H an open dense subset. 8.3. Kostant’s map and toric varieties. We first remark that for a fixed y ∈ Z(γ)o Kostant’s map β y in (20) is just the map d → n−1 d with d = y e ∗ and nd ∈ N , so that it can be described as a map into the flag nd bd ∈ G e e y∗ . manifold: d → gB in G/B, restricted to G By Proposition 8.5 the map into the flag manifold is a diffeomorphism e y∗ . Hence, since the map is given by the onto its image when restricted to G action of a Cartan subgroup on the flag manifold; this action has a trivial e y diffeomorphically isotropy group and the map to the flag manifold sends G to its image. e y is as good as its conjugate HR but, for conveThe Cartan subgroup G nience, we prefer to deal with HR for which we have established notation. e y , x−1 HR x = G e y . Then the We let x be an element that conjugates HR to G e y -orbit of B in G/B e G is x−1 HR xB. Since we can translate this set using multiplication by the fixed element x, we can just study the HR -orbit of xB e in G/B. We denote: e y → G/B. e • The map q : G ˆ R = (q ◦ (β y )−1 Z(γ)R ) = x−1 ((HR xB)). • Z(γ) ˆ To study Z(γ), it is enough to describe in detail the toric variety (HR xB). Thus we focus our attention on objects that have this general form: e such that nB ∩ HR = {e}, the toric variety Definition 8.7. For any n ∈ G, T (HR nB) is called generic in the sense of [8], if n ∈ w(N B)w−1 . w∈W

We then assume n ∈

T

w(N B)w−1

w∈W

and nB ∩ HR = {e}. With these

hypotheses we have: nB =

Y φj ∈−∆+

exp(tej eφj )B


119

for tej ∈ R, and Y

nB = n(w(∆))wB =

exp(tw j ew(φj ) )wB

φj ∈−∆+

for w ∈ W and tw j ∈ R. We now define: T

Definition 8.8. Assume n ∈

w∈W

w(N B)w−1 and nB ∩ HR = {e}. Denote

for any A ⊂ Π, Y

n(w(∆A ))wB =

exp(tw j ew(φj ) )wB.

φj ∈−∆A +

We define a map, ˇ : B

ˆR H

w(∆A )

(Ad

g, [w]A )

−→ 7 →

(HR nB) gn(w(∆A ))wB

ˆ R ). Note here that where g ∈ HR (see Definition 6.1 for H Y (21) gn(w(∆A ))wB = exp tw χ (g)e w(φj ) wB. j w(φj ) φj ∈−∆A +

ˇ can be interpreted as a version of Kostant’s map β y for the The map B subsystem determined by the set A ∈ Π and its w-translation. A detailed ◦

correspondence with the set Z R (γ) could be made but it requires additional w(∆A )

A

notation. Note also that we have χw(φj ) (g) = χw(φj ) ◦ Adw(∆ ) (g) for φj ∈ ∆A . Then we obtain: Theorem 8.9. Assume n ∈

T w∈W

w(N B)w−1 and nB ∩ HR = {e}. The

ˇ is a homeomorphism of topological spaces. The toric variety function B ˇ is a diffeomorphism. (HR nB) is a smooth manifold and the map B ˆ R and Proof. First we point T out that we already have a smooth manifold H −1 the assumption n ∈ w(N B)w will simply ensure that we can define w∈W

the map to the flag manifold. ˇ We use the local coordinates, {φw : w ∈ We first show the continuity of B: ˆ R given in Definition 5.4. In these local coordinates, we assume W } for H wo (A) ) that for each i = 1, . . . , l, φwwo ,i (Adwwo (∆ (g), [wwo ]wo (A) ) (which equals w(∆A )

χw(−αi ) (g) or zero) converges to a scalar χow(−αi ) . We note that if φj =

120


l X − cij αi with each cij a nonnegative integer, then i=1

χw(φj )

l l Y Y wo (A) ) cij cij = χ−w(αi ) = φww (Adwwo (∆ , [wwo ]wo (A) ). o ,i i=1

i=1

We thus let χow(φj ) =

l Y

χo−w(αi )

cij

.

i=1

Let A0 = {αi ∈ Π : χo−w(αi ) = 0}. Then note that χow(φj ) = 0 if and only if cij 6= 0 for some αi ∈ A0 . Thus the 0

only χow(φj ) which are nonzero correspond to roots in ∆A +. By the assumption made, we have Y gn(w(∆A ))wB = exp tw χ (g)e w(φj ) wB, j w(φj ) φj ∈−∆+

which can be written in terms of the coordinate functions φwwo as ! l Y Y w (A) o cij ) exp tw φww (Adwwo (∆ , [wwo ]wo (A) )ew(φj ) wB. j o ,i φj ∈−∆+

i=1

This then converges (by continuity of φwwo ) to Y o exp tw χ (g)e w(φj ) wB, j w(φj ) φj ∈−∆+ 0

0

A which only involves roots in ∆A + , and can be written as gn(w(∆ ))wB. A ˆ R is completely determined by the coorSince any (Adw(∆ ) (g), [w]A ) ∈ H A0

A

0

dinates φw (Adw(∆ ) (g), [w]A ) and some (Adw(∆ ) (g), [w]A ) uniquely corresponds to the coordinates (χow(α1 ) (g), . . . , χow(αl ) (g)), then we can con0

0 w(∆A ) w(∆A ) ˇ ˇ clude that B(Ad (g), [w]A ) converges to B(Ad (g), [w]A ) whenA

A0

0

ever the argument (Adw(∆ ) (g), [w]A ) approaches to (Adw(∆ ) (g), [w]A ) in ˆ R . This proves the continuity of the map B. ˇ H ˆ R is compact (Proposition 7.7) and B( ˇ H ˆ R ) contains the orbit Since H ˇ ˆ ˇ it HR nB, then (HR nB) ⊂ B(HR ). From the construction of the map B ˇ H ˆ R) = is easy to see that its image is contained in (HR nB) and thus B( (HR nB).


121

The smoothness of the map and that it gives a diffeomorphism follows |∆+ | Y from the fact that (s1 , . . . , s|∆+ | ) → exp(sj ew(φj ) )wB constitutes a coj=1

ordinate system in the flag manifold.

Remark 8.10. We caution the reader that if one replaces the Cartan subgroup HR1 of G instead of HR in the statement of Theorem 8.9 then the closure of the orbit HR1 nB may not be smooth. In fact the structure of HR1 nB can be explicitly described too. Consider only the connected components of each HR (w(∆A )) associated to such that Ad(h ) ∈ Ad(HR1 ) in ˆ R that will correspond to H 1 nB. Definition 5.4. This gives a subspace of H R In terms of the coordinate charts φw one gets locally Rl but now some of its 2l quadrants may be missing (since Ad(HR1 ) may have fewer than 2l connected components). Thus smoothness is obtained exactly when Ad(HR1 ) contains 2l connected components. Examples that lead to nonsmooth closures if one uses the Cartan subgroup HR1 are all the G = SL(n, R) with n even. In terms of the Toda lattice this corresponds to considering the indefinite Toda lattice in (1) in the Introduction but leaving out some of the signs i . When n = 2, for example, one obtains a closed interval inside G/B (which is a e which leads to the Cartan subgroup circle). The disconnected Lie group G HR is then a requirement in all our constructions and main results. We thus have: T w(N B)w−1 and nB ∩ HR1 = {e}. Then Corollary 8.11. Assume n ∈ w∈W

HR1 nB

is smooth if and only if Ad(HR1 ) has 2l connected components.

We also have: Corollary 8.12. If n ∈

T w∈W

w(N B)w−1 and nB ∩ HR = {e}, then the toric

variety X = (HR nB) satisfies: X HR = (G/B)HR , the HR -fixed points. ˆ R has an HR -action and the only Proof. We use Theorem 8.9. The manifold H fixed points are the (e, w) with w ∈ W . These get mapped to the fixed points of the HR -action in G/B, the cosets wB, w ∈ W . Thus X HR = (G/B)HR . This also follows directly if we consider (21) with A = ∅ and we just let each χw(φj ) go to zero and obtain wB.

122


Remark 8.13. 1) By Theorem 3.6 in [9] and Remark 3 in p. 257 of [8] we may assume that the map into the flag manifold which appears as a consequence of Kostant’s map in Subsection that one in fact obtains the T 8.3 is such−1 w(N B)w . HR orbit of an element n ∈ w∈W

2) In §7 of [2] and in [3] a more restrictive definition of the notion of genericity is given. This discussion is only relevant for the case of toric varieties in G/P where P is a parabolic subgroup rather than just a Borel subgroup as is the case in the present work. In these more general situations sometimes generic varieties as defined in [8] are not normal. In T any case, note that if P = B our Corollary 8.12 implies w(N B)w−1 and nB ∩HR = {e} then the corresponding that if n ∈ w∈W

toric variety is also generic in the sense discussed in [2] or [3]. Finally we observe that normality is not used in any of our results. Here we rely instead on explicit coordinate charts to obtain the smoothness of our toric varieties and describe their topological structure. (We would like to thank H. Flaschka for sending the papers [2, 3] to us.) We conclude: ˆ R is a smooth compact manifold Theorem 8.14. Let γ ∈ Rl , then Z(γ) ˆ diffeomorphic to HR . ˆ Proof. This is just Theorem 8.9 and the definition of Z(γ). The two conditions in Theorem 8.9 are satisfied by Proposition 8.5 (Subsection 8.3) and by Theorem 3.6 in [9] and Remark 3 in p. 257 of [8] as noted above in Remark 8.13.

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[16] B. Kostant, The solution to a generalized Toda lattice and representation theory, Advances in Math., 34 (1979), 195-338, MR 82f:58045, Zbl 0433.22008. [17] A.N. Leznov and M.V. Saveliev, Group-Theoretical Methods for Integration of Nonlinear Dynamical systems, Birkhauser Verlag, Boston-Basel-Berlin, 1992, MR 93d:58194, Zbl 0746.58004. [18] J. Munkres, Elements of Algebraic Topology, Addison Wesley Publishing Company, 1984, MR 85m:55001, Zbl 0673.55001. [19] I.R. Shafarevich, Basic Algebraic Geometry 1, Varieties in Projective Space, Springer Verlag, 1994, MR 95m:14001, Zbl 0797.14001. [20] M. Toda, Theory of Nonlinear Lattices, Springer Verlag, 1981, MR 82k:58052b, Zbl 0465.70014. [21] C. Tomei, The topology of isospectral manifolds of tridiagonal matrices, Duke Math. Journal, 51 (1984), 981-996, MR 86d:58091, Zbl 0558.57006. [22] N. Wallach, Real Reductive Groups I, Pure and Applied Mathematics, 132, Academic Press, 1988, MR 89i:22029, Zbl 0666.22002. Received January 19, 2000 and revised April 13, 2001. This research was supported by NSF grant DMS-0071523.

Department of Mathematics The Ohio State University Columbus, OH 43210

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E-mail address: [email protected] Department of Mathematics The Ohio State University Columbus, OH 43210 E-mail address: [email protected]


ON DIOPHANTINE MONOIDS AND THEIR CLASS GROUPS Scott T. Chapman, Ulrich Krause, and Eberhard Oeljeklaus A Diophantine monoid S is a monoid which consists of the set of solutions in nonnegative integers to a system of linear Diophantine equations. Given a Diophantine monoid S, we explore its algebraic properties in terms of its defining integer matrix A. If dr (S) and dc (S) denote respectively the minimal number of rows and minimal number of columns of a defining matrix A for S, then we prove in Section 3 that dr (S) = rank Cl(S) and dc (S) = rank Cl(S)+rank Q(S) where Cl(S) represents the divisor class group of S and Q(S) the quotient group of S. The proof relies on the characteristic properties of the so-called essential states of S, which are developed in Section 2. We close in Section 4 by offering a characterization of factorial Diophantine monoids and an algorithm which determines if a Diophantine monoid is halffactorial.

1. Introduction. Because of their applications in commutative algebra, algebraic geometry, combinatorics, number theory, and computational algebra, the study of commutative cancellative monoids has recently increased in popularity (see for example [14]). Let Z and N represent the integers and the nonnegative integers respectively. For 1 ≤ m, n in N and A ∈ Zm×n set (1)

MA = Nn ∩ {x ∈ Zn | Ax = 0}.

We will refer to MA as a Diophantine monoid and to A as a matrix which determines MA . Admitting the possibility that m = 0, we set MA = Nn if m = 0. The special case of a Diophantine monoid MA for A ∈ Z1×n (i.e., one single homogeneous linear Diophantine equation) has been studied in [2], where it was shown that the divisor class group of MA (denoted Cl (MA )) in this case must be cyclic [2, Theorem 1.3]. It is natural to consider the question of whether a Diophantine monoid MA given by a matrix A ∈ Zm×n can be described (up to isomorphy) by another matrix with less rows or less columns. For a Diophantine monoid S, let dr (S) and dc (S) denote respectively the minimal number of rows and minimal number of columns of a matrix A with S ' MA . One has that 125

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S.T. CHAPMAN, U. KRAUSE, AND E. OELJEKLAUS

dc (S) = dr (S) + rank Q(S) where Q(S) is the quotient group of S. In this paper, we prove for a monoid which is root-closed and finitely generated that dr (S) = rank Cl (S) and hence dc (S) = rank Cl (S) + rank Q(S) (Theorem 3.8). Furthermore, this theorem also shows that for such an S there exists an A ∈ Zm×n with m = dr (S), n = dc (S) and S ' MA . More precisely, we divide our work into three additional sections. After a very brief review of Krull monoids, we focus our considerations in Section 2 on the so-called essential states of these monoids and provide in Lemma 2.3 and Proposition 2.4 their characteristic properties. Using these properties, we obtain the above mentioned result (Theorem 3.8) from our crucial Theorem 3.1 together with Lemma 3.5. Theorem 3.1 demonstrates that any Krull monoid with finitely many essential states must be isomorphic to a Diophantine monoid and describes the structure of the representing matrix as well as the structure of the class group. As a consequence, we obtain in Corollary 3.3 various equivalent characterizations of the above Krull monoids or, equivalently, Krull monoids with finitely generated divisor class group and finitely many prime divisors. For some results related to Corollary 3.3, the interested reader can consult [7, Theorem 5], [8, Lemma 3], [10, Proposition 2], and [13, Corollary 1]. We note that Lemma 3.5 yields a description of the class group of a Diophantine monoid S purely in terms of linear algebra. In Section 4, we present some examples to illustrate the results of Sections 2 and 3 and also obtain in Proposition 4.1 a characterization of when MA is factorial (recall that MA is factorial if any nonzero element has a representation α1 + · · · + αk by irreducible elements αi which is unique up to ordering). We close by presenting an algorithm, which when teamed with known algorithms for computing the set of minimal nonnegative solutions to MA , will compute the class group of MA and determine if MA is half-factorial (recall that MA is half-factorial if whenever α1 , α2 , . . . , αk and β1 , β2 , . . . , βl are irreducible elements of MA with α1 + · · · + αk = β1 + · · · + βl , then k = l). Although the literature concerning the algebraic structure of the monoids MA is not extensive, we take interest in this topic partly because of the rich mathematical history behind the study of Diophantine equations. It is easy to determine the solution set of a system of linear Diophantine equations over the integers, but this is not the case for determining the set of solutions over the nonnegative integers. While a modern treatment of the combinatorial aspects of this subject can be found in the works of Stanley (see [15] and [16] for example), attempts to determine the set of “irreducible solutions” of MA can be traced back almost 100 years to a paper of Elliott [4], where the author produces generating functions to determine these solutions. Another early attempt at producing this set of minimal solutions can be found in [5]. The development of modern algorithms connected with these solutions has become a popular topic of research in computational algebra (see [3] and [14]).

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2. Essential states. Let S be a commutative cancellative monoid (i.e., a subsemigroup of an abelian group written additively with 0 ∈ S). Throughout this paper we assume that S 6= {0}. If {0} is the only subgroup of S, then S is called reduced. Let Q(S) := {x − y | x, y ∈ S} be the group generated by S. A homomorphism π : Q(S) → Z is called a state of S if π(S) ⊆ N. The monoid S is a Krull monoid if there exists, for some set K, a monomorphism ϕ : Q(S) → Z(K)

(direct sum)

such that ϕ(S) = N(K) ∩ ϕ(Q(S)) = N(K) ∩ Q(ϕ(S)). As a consequence of our definition, a Krull monoid is always reduced. For j ∈ K let pj : Z(K) → Z be the surjection onto the j-th component, pj ((xk )k∈K ) = xj , and πj : Q(S) → Z,

πj = pj ◦ ϕ.

Then (∗)

S = {x ∈ Q(S) | πj (x) ≥ 0 for all j ∈ K}

where (πj )j∈K is a family of states of S with πj (x) = 0 for almost all j ∈ K and any fixed x ∈ Q(S). We may assume that πj 6= 0 for all j ∈ K. Example 2.1. For every A ∈ Zm×n the Diophantine monoid MA = {x ∈ Nn | Ax = 0} ⊆ Zn is a Krull monoid. With K = {1, 2, . . . , n} and ϕ : Q(MA ) → Zn the canonical embedding, we have MA = {(x1 , . . . , xn ) ∈ Q(MA ) | πj (x) = xj ≥ 0 for j ∈ K}. When a Krull monoid S is given in the form (∗), it is natural to ask for minimal subsets E ⊆ K with the property that S = {x ∈ Q(S) | πi (x) ≥ 0 for all i ∈ E}. Such subsets exist and can be described by the so-called essential states of S. Definition 2.2. A nonzero state π of S is called essential, if for every x, y ∈ Q(S) with π(x) ≥ π(y) there exists z ∈ Q(S) with π(z) = π(x), z − x ∈ S and z − y ∈ S. The essential states can be characterized as follows:

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Lemma 2.3. Let S be a Krull monoid and (πj )j∈J the family of all nonzero states of S. For every i ∈ J the following statements are equivalent: i) πi is essential. ii) For every j ∈ J with πj 6∈ Qπi there exists x ∈ S such that πi (x) = 0 and πj (x) > 0. iii) S ∩ Ker πi is a maximal element in the set {S ∩ Ker πj | j ∈ J} with respect to set inclusion. iv) If j ∈ J and (S ∩ Ker πi ) ⊆ (S ∩ Ker πj ) then πj = απi for some α ∈ Q, α > 0. In particular, the family of essential states of S is not empty. Proof. Note that for any finite subset I ⊆ J, there exists x ∈ S with πi (x) > 0 for all i ∈ I. Statements ii) and iv) are obviously equivalent. We begin by showing that i) ⇒ iii). Assume that i) holds and that, for some j ∈ J, (S ∩ Ker πi ) $ (S ∩ Ker πj ). Choose y ∈ (S ∩ Ker πj )\ Ker πi and x ∈ S with πj (x) > 0. Then u := πi (y)x − πi (x)y ∈ Ker πi . Condition i) yields an element z ∈ Q(S) with πi (z) = πi (πi (y)x) = πi (πi (x)y), z − πi (y)x ∈ S ∩ Ker πi ⊆ Ker πj and z − πi (x)y ∈ S ∩ Ker πi ⊆ Ker πj . In particular, πj (z − πi (y)x) = πj (z − πi (x)y) = 0. But this gives 0 = πj (z − πi (x)y) = πj (z) − πi (x)πj (y) = πj (z), and the contradiction 0 = πj (z − πi (y)x) = πj (z) − πi (y)πj (x) = −πi (y) · πj (x) < 0. For the rest of the proof, let K ⊆ J be a subset of J with the property that S = {x ∈ Q(S) | πj (x) ≥ 0 for all j ∈ K} where πj (x) = 0 for almost all j ∈ K and any fixed x ∈ Q(S). The representation (∗) shows that such subsets K exist. For iii) ⇒ iv), we fix i ∈ J and assume that iii) is true for πi . Moreover we fix j ∈ J and assume that S ∩ Ker πi ⊆ S ∩ Ker πj . Define L := K ∪ {i, j}.

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From iii) we know that L0 : = {k ∈ L | (S ∩ Ker πi ) ⊆ Ker πk } = {k ∈ L | S ∩ Ker πi = S ∩ Ker πk }. We choose some v ∈ S with πi (v) > 0. Then πj (v) > 0 by our assumption π (v) and α = πji (v) > 0. Let w ∈ S be arbitrary. If πi (w) = 0, then πj (w) = 0 and πi (w) = απj (w). Assume that πi (w) > 0. Then i ∈ L00 := {k ∈ L0 | πk (w) 6= 0}. Q Since L00 is finite, we can define m = k∈L0 πk (w) and 0

λ = max {r ∈ N | r ≥ 1, πk (mv − rw) ≥ 0 for all k ∈ L0 }. For some k1 ∈ L00 the equality πk1 (mv − λw) = 0 holds. For every k ∈ L2 := {k ∈ L\L0 | πk (mv − λw) 6= 0} we choose xk ∈ S∩ Ker πi with πk (xk ) > max{0, πk (λw − mv)}. P Since L2 is finite (and possibly empty), the element x := µ∈L2 xµ is welldefined and x ∈ S ∩ Ker πi ⊆ Ker πj . In the next step, we show that u = x + mv − λw ∈ S (i.e., that πk (u) ≥ 0 for all k ∈ L). We have already seen that for k ∈ L0 we have πk (u) = πk (x) + πk (mv − λw) ≥ πk (mv − λw) ≥ 0. For k ∈ L2 we get πk (u) = πk (x) + πk (mv − λw) ≥ πk (xk ) + πk (mv − λw) ≥ 0, and finally for k ∈ L\(L0 ∪ L2 ) = (L\L0 )\L2 we have that πk (u) = πk (x) ≥ 0. Therefore u ∈ S, and it follows for the above chosen k1 ∈ L00 with πk1 (mv − λw) = 0 that πk1 (u) = πk1 (x) + πk1 (mv − λw) = πk1 (x) = 0, since x ∈ S∩ Ker πi = S∩ Ker πk1 . In particular, u ∈ S∩ Ker πi ⊆ Ker πj . This gives πj (u) = 0 = πi (u) = πi (mv − λw) = πj (mv − λw). Hence λπj (w) = πj (mv), λπi (w) = πi (mv) and πj (w) =

πj (mv) πj (v) 1 πj (mv) = πi (w) = πi (w) = απi (w). λ πi (mv) πi (v)

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Since w ∈ S was chosen arbitrarily, it follows that πj = απi . For iv) ⇒ i), assume that x, y ∈ Q(S) are given with πi (x) ≥ πi (y). As above, we define L := K ∪ {i, j}, L0 := {k ∈ L | (S ∩ Ker πi ) ⊆ Ker πk }, L1 := L\L0 , and L2 := {k ∈ L1 | πk (x − y) 6= 0}. Note that L2 is a finite set. By iv) we know that πk = αk πi , for all k ∈ L0 where 0 < αk ∈ Q. For every k ∈ L1 there exists xk ∈ S ∩ Ker πi with πk (x − y) + πk (xk ) ≥ 0. P

Defining z = x +

µ∈L2 xµ ,

we get X

z−x=

xµ ∈ S.

µ∈L2

We prove that z − y ∈ S by showing that πk (z − y) ≥ 0 for all k ∈ L. If k ∈ L2 , then  πk (z − y) = πk x − y +

 X

xµ  = πk (x − y) +

µ∈L2

X

πk (xµ )

µ∈L2

X

= πk (x − y) + πk (xk ) +

πk (xµ ) ≥ 0.

µ∈L2 \{k}

For k ∈ L0 we get πk (z − y) = πk (x − y) = αk πi (x − y) ≥ 0. Finally, for k ∈ L1 \L2 we have  πk (z − y) = πk 

 X

xµ  ≥ 0.

µ∈L2

Again, let the Krull monoid S be given in the form (∗). Thus, S = {x ∈ Q(S) | πj (x) ≥ 0 for all j ∈ K} where (πj )j∈K is a family of nonzero states of S such that πj (x) = 0 for almost all j ∈ K and any fixed x ∈ Q(S). Proposition 2.4. For every ∅ = 6 I ⊆ K the following statements are equivalent: i) S = {x ∈ Q(S) | πi (x) ≥ 0 for all i ∈ I}.

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ii) If π is an essential state of S, then there exists i ∈ I and α ∈ Q, α > 0, such that π = απi . Proof. i) =⇒ ii) Let π be an essential state of S. It suffices to show that π ∈ Qπi for some i ∈ I. Let i0 ∈ I and v, w ∈ S with π(v) > 0, πi0 (w) > 0 and define u := v + w. Then π(u) > 0 and i0 ∈ I1 := {i ∈ I | πi (u) > 0},

|I1 | < ∞.

We claim that π ∈ Qπi for some i ∈ I1 . Assume the contrary. Then πi ∈ / Qπ and we apply Lemma 2.3 ii) to get forPevery i ∈ I1 an element xi ∈ S with π(xi ) = 0 and πi (xi ) > 0. With x := i∈I1 xi ∈ S it follows that π(x) = 0 and for every i ∈ I1 X πi (x) = πi (xi ) + πi (xj ) > 0. j∈I1 \{i}

Let i1 ∈ I1 with πi1 (x) = min πi1 (u)

πi (x) i ∈ I1 πi (u)

and z := πi1 (u)x − πi1 (x)u. From i) we get z ∈ S, since πi (z) = πi1 (u)πi (x) − πi1 (x)πi (u) ≥ 0 for i ∈ I1 , and πi (z) = πi1 (u)πi (x) ≥ 0 for i ∈ I \ I1 . But π(z) = −πi1 (x)π(u) < 0, a contradiction to the fact that π(S) ⊆ N. ii)=⇒ i) Let N := {w ∈ Q(S) | πi (w) ≥ 0 for all i ∈ I}. For y ∈ N we consider the finite set K(y) := {j ∈ K | πj (y) < 0} and prove that |K(y)| = 0 (i.e., that A := {y ∈ N | |K(y)| ≥ 1} = ∅). Assume the contrary and choose y ∈ A such that |K(y)| is minimal. Let j0 ∈ K such that πj0 is an essential state of S. From ii) we know that j0 6∈ K(y). For every P j ∈ {j0 } ∪ K(y) there exists xj ∈ S with πj (xj ) > 0. Thus x := j∈{j0 }∪K(y) xj ∈ S and πj (x) > 0 for every j ∈ {j0 } ∪ K(y). Let j1 ∈ K(y) such that πj (y) πj1 (y) = max 0> j ∈ K(y) . πj1 (x) πj (x) We define z := πj1 (x)y − πj1 (y)x. Since πj1 (y) < 0 it follows for every j ∈ K \ K(y) that πj (z) = πj1 (x)πj (y) − πj1 (y)πj (x) ≥ −πj1 (y)πj (x) ≥ 0.

132


In particular, z ∈ N since I ⊆ K\K(y), and K(z) ⊆ K(y). Moreover |K(z)| < |K(y)| because j1 6∈ K(z). From the minimality of |K(y)| we conclude |K(z)| = 0 (i.e., z ∈ S). Since πj0 (z) = πj1 (x)πj0 (y) − πj1 (y)πj0 (x) ≥ −πj1 (y)πj0 (x) > 0 it follows that z ∈ (S ∩ Ker πj1 ) 6⊆ (S ∩ Ker πj0 ). But, by Lemma 2.3 iii), there has to be an essential state π of S with (S ∩ Ker πj1 ) ⊆ (S ∩ Ker π). Since πj0 was already arbitrarily chosen, this contradiction finishes the proof. We call a state π of S a normal state, if π(Q(S)) = Z. For every nonzero state π : Q(S) → Z, the image π(Q(S)) is a nonzero ideal dZ of Z where d ∈ N, d ≥ 1, and πnor := d1 π is normal. Let KN denote the set of all normal states of a given Krull monoid S. From Proposition 2.4 we obtain the following. Corollary 2.5. There is a unique minimal subset I(S) ⊆ KN such that S = {x ∈ Q(S) | π(x) ≥ 0 for all π ∈ I(S)}. The set I(S) consists exactly of the essential normal states of S. Moreover π(x) = 0 for almost all π ∈ I(S) and any fixed x ∈ Q(S). Proof. Let L be a set of nonzero states of S such S = {x ∈ Q(S) | π(x) ≥ 0 for all π ∈ L} e := and π(x) = 0 for almost all π ∈ L and any fixed x ∈ Q(S). Define L {πnor | π ∈ L}. Obviously S = {x ∈ Q(S) | πnor (x) ≥ 0 for all π ∈ L}, and e by Proposition 2.4. the set I(S) of normal essential states of S is a subset of L In particular π(x) = 0 for almost all π ∈ I(S) and any fixed x ∈ Q(S). Now we may again apply Proposition 2.4 and conclude that S = {x ∈ Q(S) | π(x) ≥ 0 for all π ∈ I(S)}. We consider the form S = {x ∈ Q(S)|π(x) ≥ 0 for all π ∈ I(S)} as the (uniquely determined) normal representation of the Krull monoid S. It gives rise to a divisor theory ϕ : Q(S) → Z(I(S)) , ϕ(x) := (π(x))π∈I(S) ,

(direct sum)

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with Ker ϕ = S ∩ (−S) = {0}, which maps Q(S) onto a free subgroup of Z(I(S)) such that ϕ(S) = N(I(S)) ∩ ϕ(Q(S)) = N(I(S)) ∩ Q(ϕ(S)). The quotient group Cl (S) := Z(I(S)) /ϕ(Q(S)) is the divisor class group of S. It is a direct consequence of the construction that isomorphic Krull monoids have isomorphic divisor class groups. Now suppose that A ∈ Zm×n and S is isomorphic to the Diophantine monoid MA = {x ∈ Nn | Ax = 0}. For 1 ≤ i ≤ n let pi : Zn → Z, with pi (x1 , . . . , xn ) = xi , denote the natural surjection. The restriction maps πi = pi |Q(MA ) from Q(MA ) into Z are states of MA . We call them the canonical projections of Q(MA ). Note that a relation πi = πj does not imply i = j. Moreover, for 1 ≤ i ≤ n we define ci = gcd (xi | x = (x1 , . . . , xn ) ∈ MA ) ∈ N where ci = 0 if and only n Y if xi = πi (x) = 0 for all x ∈ MA . We call the product w(MA ) = ci the i=1

weight of the monoid MA . If n ≥ 2 and ci = 0, then MA is canonically e ∈ Zm×(n−1) is given by isomorphic to the monoid MAe ⊆ Nn−1 , where A canceling the i-th column of A. Therefore it suffices to study the situation w(MA ) 6= 0. All canonical projections πi , 1 ≤ i ≤ n, are normal if and only if w(MA ) = 1. In this case, every normal essential state of MA is a canonical projection (as follows immediately from Corollary 2.5), hence there are at most n essential states of MA . In general, not all normal canonical projections of Q(MA ) are essential states of MA . Let I(MA ) := {i | 1 ≤ i ≤ n, πi is an essential state of MA } and note that the following statements are equivalent: i) w(MA ) 6= 0, ii) there exist elements x = (x1 , . . . , xn ) ∈ MA with xi > 0 for 1 ≤ i ≤ n. Lemma 2.6. Let MA be a Diophantine monoid with A ∈ Zm×n and w(MA ) 6= 0. The following statements are true: i) Q(MA ) = {x ∈ Zn |Ax = 0}, and there exists a linearly independent system of vectors v1 , v2 , . . . vn−r ∈ MA , where r = rank A. ii) Let ci be defined as above and D := diag (c1 , . . . , cn ) ∈ Zn×n . The map MAD → MA , defined by y 7→ Dy, is an isomorphism of monoids and w(MAD ) = 1. iii) There are at most n normal essential states of MA .

134


Proof. i): The assumption w(MA ) 6= 0 yields an element x = (x1 , . . . , xn ) ∈ MA with xi > 0 for 1 ≤ i ≤ n. Obviously Q(MA ) ⊆ G = {u ∈ Zn |Au = 0}. If y ∈ G then z = kx − y ∈ Nn ∩ G = MA for k ∈ N sufficiently big, hence y = kx − z ∈ Q(MA ) and Q(MA ) = G. Let r = rank A and w1 , . . . , wn−r be a basis of the Q-vector space W = {z ∈ Qn |Az = 0}. We may assume that wj ∈ Zn for 1 ≤ j ≤ n − r. Let m0 ∈ N with m0 x + wj ∈ MA , 1 ≤ j ≤ n − r. Since the family (kx + wj |1 ≤ j ≤ n − r) is linearly independent for all but at most one k ∈ N, we are done. We leave the verification of statements ii) and iii) to the reader. 3. The representation of Krull monoids by matrices. In this section, we show that any Krull monoid S with a finite number e of essential states must be isomorphic to a Diophantine monoid. We describe the structure of the corresponding matrix and show in principle its computation, as well as the computation of the divisor class group of S. For the class group Cl (S) defined in the previous section, we have the short exact sequence ϕ

ρ

0 −→ Q(S) −→ Z(I) −→ Cl (S) −→ 0 where I = I(S) indexes the set of all normalized essential states of S and where ρ denotes the canonical epimorphism. To obtain the desired description of S by a matrix, we will use a particular basis for the Z-module Z(I) . For an arbitrary Z-module M , let rank M denote the minimal number of generators of M and free rank M the maximal length of a free family in M . Of course, if M is a free module, then the rank and the free rank of M coincide and are equal to the cardinality of a Z-basis of M (since M = {0} is generated by the empty set, one has rank {0} = free rank {0} = 0). Theorem 3.1. For a (reduced) Krull monoid S, the number e of essential states is finite if and only if the rank r of Q(S), the free rank h of Cl (S), and the rank k of the torsion group of Cl (S) are all finite. In this case, the following statements apply. i) There exist natural numbers α1 , . . . , αk ≥ 2 with αi+1 | αi for 1 ≤ i ≤ k − 1 such that Cl (S) ' Zα1 ⊕ · · · ⊕ Zαk ⊕ Zh and h + r = e. ii) S is isomorphic to a Diophantine monoid MA with A ∈ Zm×n for m = k + h and n = k + e (if h = k = 0, then m = 0 and MA = Ne ). iii) The matrix A has a block structure

DIOPHANTINE MONOIDS

A11

A12

A21

0

135

A=

where A11 ∈ Nk×e with entries of the i-th row in {0, 1, . . . , αi − 1}, A12 = −diag(α1 , . . . , αk ) ∈ Nk×k and A21 ∈ Nh×e contains in each row at least two strictly positive and two strictly negative entries. Proof. M = ϕ(Q(S)) is a submodule of the free module Z(I) and therefore free. If e = |I| is finite then rank M is finite and, hence, r is finite. By its definition as the quotient of Z(I) and M , the class group is finitely generated and h and k are finite. Conversely, if r, h and k are finite then k + h = rank Cl (S) ≥ rank Z(I) − rank ϕ(Q(S)) and hence e ≤ k + h + r is finite. Step I. In a first step, we analyze certain submodules of Z(I) for an arbitrary index set I. Let (bi )i∈I be a basis of the free module Z(I) . For a given set {αj | j ∈ J} ⊆ N\{0}, ∅ 6= J ⊆ I, let M be the free submodule with L L (I\J) Z Zαi basis (αj bj )j∈J . Consider the epimorphism ψ : Z(I) −→ for J 0 = {j ∈ J | αj 6= 1} defined by ! X X ψ ri bi = (ri

i∈J 0

mod αi )bi +

i∈J 0

i∈I

X

ri bi .

i∈I\J

Obviously, Ker ψ = M and, hence, M M (2) Z(I) M ' Zαi Z(I\J) . i∈J 0

To describe M by a matrix, let (ui )i∈I be the standard basis of Z(I) and P uj = cij bi , cij ∈ Z for i, j ∈ I. Define a block matrix i∈I

(3)

I

J0

J0

A11

A12

I\J

A21

0

A=

where the blocks in the above diagram are given as follows:

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A11 : A12 : A21 :

For i ∈ J 0 , j ∈ I, let aij ∈ {0, 1, . . . , αi − 1} be the residue of cij mod αi . For i, j ∈ J 0 , let aij = −αi δij . For i ∈ I\J, j ∈ I, let aij = cij .

Note that A21 cannot have P a zero-row. Namely, ci0 j = 0 for i0 ∈ I\J and all cij bi for all j ∈ I, which contradicts the freeness j ∈ I would imply uj = i6=i0

of (bi )i∈I .

0 x For NA : = (x, y) ∈ × |A = 0 and π : Z(I) × Z(J ) −→ y Z(I) , defined by π(x, y) = x, we shall show that π : NA −→ M is an isomorphism of Z-modules. For x ∈ Z(I) we have that   X X X X X  cij xj  bi . x= xj uj = xj cij bi = 0 Z(J )

Z(I)

j

j

i

i

j

Therefore, x ∈ M if and only if X X αi cij xj for i ∈ J 0 and cij xj = 0 for i ∈ I\J, j

j

which is equivalent to X X (4) aij xj = αi yi with yi ∈ Z for i ∈ J 0 and aij xj = 0 for i ∈ I\J. j∈I

j∈I

According to the definition of A, this means that x ∈ M if and only if 0 (x, y) ∈ NA for some y ∈ Z(J ) . Thus, π : NA −→ M is a well-defined epimorphism. π is injective because by (4) y = (yi )i∈J 0 is uniquely determined by x = (xj )j∈I . This shows that π : NA −→ M is an isomorphism. Step II. Consider now a (reduced) Krull monoid S for which I = I(S) = {1, . . . , e} is finite and let M = ϕ(Q(S)) ⊆ Z(I) . Since Z(I) is finitely generated, by the elementary divisor theorem (see [9]) there exist a basis (bi )i∈I of Z(I) and numbers αj ∈ N\{0} for 1 ≤ j ≤ r ≤ e with αj 6= 1 for 1 ≤ j ≤ k ≤ r and αj+1 | αj for 1 ≤ j ≤ k − 1 such that (αj bj )1≤j≤r is a basis of M (k = 0 admitted). Obviously, r = rank ϕ(Q(S)) = rank Q(S). Setting J = {1, . . . , r} and J 0 = {1, . . . , k}, from (2) in Step I, we obtain for k L L I\J M = ϕ(Q(S)) that Cl (S) = Z(I) M ' Zαi Z . i=1

It follows that k is the rank of the torsion group of Cl (S) and that e − r = |I\J| is the free rank of Cl (S). This proves Part i) of Theorem 3.1. Further, for the matrix A defined by (3) in Step I, we have that π : NA −→ M is an isomorphism. Hence the equations in (4) show that π(x, y) ∈ 0 NI implies (x, y) ∈ NI × NJ . Since S is a Krull monoid it follows that

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ϕ(S) = NI ∩ ϕ(Q(S)) = NI ∩ M . Combining these facts, we obtain for ϕ

0

π −1

MA : = (NI × NJ ) ∩ NA that S −→ NI ∩ M −−−→ MA . Since ϕ and π −1 are monoid isomorphisms, we have that S is isomorphic to the Diophantine monoid MA . Also, by (3) of Step I, the matrix A is a block matrix where A11 ∈ Zk×e with entries aij ∈ {0, 1, . . . , αi − 1}, A12 = −diag(α1 , . . . , αk ) ∈ Nk×k and A21 ∈ Zh×e . In particular, A ∈ Zm×n with m = k + h, n = k + e. This proves Part ii) and the first part of iii) of Theorem 3.1. It remains to show the statement about A21 in iii). Pick i ∈ I\J and let I+ = {j ∈ I | aij > 0} and I− = {k ∈ I | aik < 0}. Since by Step I the submatrix A21 cannot have a zero-row, we must have that I+ or I− is nonempty. We shall show in fact that both I+ and I− must be nonempty. Suppose that j1 ∈ I+ . Since J 6= ∅ and i ∈ I\J imply |I| ≥ 2, there exists j2 ∈ I, j2 6= j1 . Let (πj )j∈I be the set of normalized essential states of I S. P By the isomorphism of S and MA proved above, one has x ∈ N with aij xj = 0 for xj = πj (s) with j ∈ I and s ∈ S. Since πj2 is essential, j∈I

there exists by Lemma 2.3 an s ∈ S with πj2 (s) = 0 and πj1 (s) > 0. Therefore, there must be some j ∈ I with aij < 0. Thus, I+ 6= ∅ implies I− 6= ∅. Similarly, I− 6= ∅ implies I+ 6= ∅. Therefore, there exist j1 ∈ I+ and k1 ∈ I− . Suppose now I+ = {j1 }. Since πj1 is essential, by Lemma 2.3 there exists s ∈ S with πj1 (s)P = 0 and πk1 (s) > 0. But then, for x = (πj (s))j , we obtain 0 = −aij1 xj1 = aij xj < 0, which is a contradiction. j6=j1

Thus we must have that |I+ | ≥ 2. Similarly, |I− | ≥ 2. This proves the statement about A21 in Part iii) of Theorem 3.1. Remarks 3.2. 1. The Diophantine monoid MA in Theorem 3.1 ii) consists of two different kinds of equations. The first k equations are of type 1 and the remaining h equations are of type 2 in the sense of [2, Theorem 1.3]. The latter type does not occur if and only if h = 0, or, equivalently, r = e which for e ≤ 3 must happen by Theorem 3.1 iii). 2. Concerning Krull monoids S with infinitely many essential states, Step I in the proof of Theorem 3.1 can be used provided the module M = ϕ(Q(S)) meets the assumptions made there. In general, a submodule M of Z(I) does not satisfy these assumptions as the following simple example shows. Let I = N, (ui )i∈N the standard basis of Z(N) , Q = {qi }i∈N and τ : Z(N) −→ Q be the Z-homomorphism defined by τ (ui ) = qi for all i ∈ N. If the submodule M = Ker τ of Z(N) would satisfy the assumptions of Step I, then by (2) Q ' Z(N) /M should be isomorphic to a direct sum of copies of Z and Zα (which is not the case). If, however, for I infinite Cl (S) is finitely generated, then one

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can argue that S is isomorphic to a monoid MA where the number of rows of A is rank Cl (S) and the “number” of columns of A is given by the cardinality of I augmented by a finite set. Similarly, if Cl (S) is free, then S is isomorphic to a monoid MA with |I\J| equations in |I| unknowns. From Theorem 3.1 we obtain the following characterization of Diophantine monoids. Corollary 3.3. For a reduced monoid S the following statements are equivalent: i) S is a Krull monoid with finitely many essential states. ii) S is isomorphic to a Diophantine monoid. iii) S is isomorphic to a monoid W ∩ Nn for a vector subspace W of Qn and some n ≥ 1. iv) S is isomorphic to a full and expanded submonoid T of Nn (i.e., x, y ∈ T, y − x ∈ Nn imply y − x ∈ T , and kz ∈ T for k ≥ 1, z ∈ Nn , implies z ∈ T ). v) S is root-closed and finitely generated. Proof. i) ⇒ ii) follows directly from Theorem 3.1. Obviously, ii) ⇒ iii). Since T = W ∩ Nn is full and expanded, iii) ⇒ iv). From iv) it follows by Dickson’s Lemma (see [14, Theorem 5.1]) that T , and hence S, is finitely generated. Since T is expanded, S must be root-closed (i.e., kz ∈ S for k ≥ 1, z ∈ Q(S) implies z ∈ S). This yields iv) ⇒ v). Finally, v) ⇒ i) follows from a well-known theorem of Halter-Koch [7, Theorem 5] and the main result in [13], by which a root-closed and finitely generated monoid is a Krull monoid described by finitely many states. Remarks 3.4. 1. In [13, Corollary 1] a description of the class group is given by employing terminology and methods from convex analysis. Using results from [13], in [10, Proposition 2] it is shown that a monoid S satisfying Q(S) = Zn is isomorphic to the monoid of nonnegative solutions of a homogeneous system of integral linear equations if and only if S is a Krull monoid which holds if any only if S is finitely generated and root-closed. The method used in the proof of Theorem 3.1 is direct and does not involve convex analysis. Moreover, Theorem 3.1 describes the representing matrix A ∈ Zm×n in terms of the class group and the number of essential states of S. This yields in particular that m = rank Cl (S). 2. Corollary 3.3 applies in particular to additive submonoids S of Nn . For this case it has been shown in [8, Lemma 3] that S is the set of solutions in the nonnegative integers of a system of homogeneous linear equations with rational coefficients if and only if S = W ∩ Nn

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for a vector subspace W of Qn or, equivalently, S is full and expanded. An additive submonoid S of Nn is called a full affine semigroup if S = M ∩ Nn for a subgroup M of Zn or, equivalently, S = Q(S) ∩ Nn (see [14, Chapter 7] for more about these semigroups). Obviously, such a semigroup is root-closed, but it need not be expanded as the example S = 2N shows. Furthermore, such a semigroup is a Krull monoid with finitely many essential states. From Corollary 3.3 one concludes that an arbitrary reduced monoid is a Krull monoid with finitely many essential states if and only if it is isomorphic to a full affine semigroup. By Theorem 3.1 Part ii), we know how to describe an arbitrary Krull monoid S having a finite number of essential states by a particular matrix adapted to S. The following Lemma addresses, conversely, a Krull monoid given by an arbitrary matrix. For a matrix A, let im A denote the image of the mapping induced by A. Lemma 3.5. Let S ' MA with A ∈ Zm×n , m ≥ 1, n ≥ 1 and weight w(MA ) = 1. Arrange A such that the first e canonical projections are exactly the normal essential states of MA and let A = [A0 A00 ] where A0 ∈ Zm×e , A00 ∈ Zm×(n−e) . The following statements hold: i) Cl (S) ' im A0 /(im A0 ∩ im A00 ). ii) n = rank A + rank Q(S) = rank A00 + e. iii) rank A0 − rank (im A0 ∩ im A00 ) = free rank Cl (S) ≤ rank Cl (S) ≤ rank A0 . iv) rank Cl (S) ≤ m and rank Cl (S) + rank Q(S) ≤ n. In particular, if A0 = A, then one has Cl (S) ' Zk , k = rank A and rank Cl (S) + rank Q(S) = n. Proof. It can be assumed that S = MA . By assumption the divisor theory ϕ : Q(S) −→ Z = EA is given by ϕ(y1 , . . . , yn ) = (y1 , . . . , ye ). From Lemma 2.6 it follows that Q(S) = KerA. i) Consider f : im A0 −→ Cl (S) defined by A0 x 7−→ x + ϕ Q(S) for x ∈ x Ze . The mapping f is well-defined. For, if A0 x = 0, then y = ∈ 0 Q(S) and, hence, x = ϕ(y) ∈ ϕ Q(S) . Obviously, f is surjective. Furthermore, if A0 x ∈ Kerf then x ∈ ϕ Q(S) and A0 x + A00 y = 0 for some y ∈ Zn−e . This shows Kerf ⊆ im A0 ∩ im A00 . Conversely, if A0 x ∈ im A00 , then A0 x + A00 y = 0 for some y ∈ Zn−e and x = x ϕ ∈ ϕ Q(S) . This shows im A0 ∩ im A00 ⊆ Kerf . Thus, y Kerf = im A0 ∩ im A00 and we have the short exact sequence i

f

0 −→ im A0 ∩ im A00 − → im A0 − → Cl (S) −→ 0,

140


where i denotes the embedding. Therefore, Cl (S) ' im A0 /(im A0 ∩ im A00 ). ii) Obviously, n = rank im A + rank KerA. Since KerA = Q(S) it follows that n = rank A+ rank Q(S). Also, n−e = rank im A00 + rank KerA00 . 00 00 We show that KerA = 0 which proves ii). If z ∈ KerA , then y, −y ∈ 0 KerA for y = . It follows that y, −y ∈ Q(MA ) and πi (−y) = z πi (y) ∈ N for every i ∈ I = {1, 2, . . . , e}. Since πi , i ∈ I are the essential projections of MA , we obtain that y, −y ∈ MA . Since MA ⊆ Nn we must have that y = 0 and, hence, z = 0. iii) Follows directly from i). iv) From iii) we have that rank Cl (S) ≤ rank A0 ≤ min{m, rank A} and using ii) we obtain rank Cl (S) + rank Q(S) ≤ rank A + rank Q(S) = n. The result for A0 = A follows immediately since in this case the proof for i) shows that Kerf = {0}. Concerning the representation of a monoid by a matrix, it is a natural question to ask for a matrix with a minimal number of rows and columns, respectively. Definition 3.6. For a monoid S the row degree of S is defined by dr (S) = min{m ∈ N | S ' MA for A ∈ Zm×n } and the column degree of S is defined by dc (S) = min{n ∈ N\{0} | S ' MA for A ∈ Zm×n }. Remark 3.7. Since we defined for A ∈ Zm×n that MA = Nn if m = 0, it follows that dr (S) = 0 if and only if S ' Nn for some n. Also, for S ' Nn it follows from Lemma 3.5 ii) that dc (S) = n. From Theorem 3.1 together with Lemma 3.5, we obtain the theorem announced in the Introduction. Theorem 3.8. For a reduced monoid S which is root-closed and finitely generated the following holds: i) Row degree and column degree of S are finite and dr (S) = rank Cl (S), dc (S) = rank Cl (S) + rank Q(S). ii) There exists a matrix A ∈ Zm×n with m = dr (S) and n = dc (S) such that S is isomorphic to the Diophantine monoid MA . Proof. By Corollary 3.3, S is a Krull monoid with finitely many essential states.

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i) By Theorem 3.1, dr (S) ≤ rank Cl (S) < ∞ and dc (S) ≤ rank Cl (S) + rank Q(S) < ∞. Suppose S ' MA for A ∈ Zm×n . We may assume that w(MA ) 6= 0 and, by Lemma 2.6, that S ' MAD with w(MAD ) = 1. From Lemma 3.5 iv) we obtain rank Cl (S) ≤ m and rank Cl (S) + rank Q(S) ≤ n. This proves i). ii) Follows immediately from i) and Theorem 3.1. 4. Some consequences and examples. The results of the previous section can be used to check if a Diophantine monoid is factorial or half-factorial. Proposition 4.1. Let S ' MA and A = [A0 A00 ] be given as in Lemma 3.5. i) S is factorial if and only if each column of A0 is in the column space of A00 . ii) If any two columns of A0 not in the column space of A00 have their difference in the column space of A00 , then S is half-factorial. Proof. It is well-known that any Krull monoid S is factorial if and only if Cl (S) = {0} and that S is half-factorial if Cl (S) ' Z2 (cf. [12, Proposition 2]). i) By Lemma 3.5, Cl (S) = {0} if and only if im A0 ⊆ im A00 . Therefore, S is factorial if and only if each column of A0 is in the Z-span of the columns of A00 . ii) If all columns of A0 are in im A00 then by i) S is factorial and, a fortiori, half-factorial. Suppose there is a column A0i0 of A0 which is not in im A00 . By assumption, for any column A0i 6∈ im A00 it holds that A0i − A0i0 ∈ im A00 . Therefore, for each column A0i of A0 either A0i ∈ im A00 or A0i ∈ A0i0 + im A00 . Lemma 3.5 implies that Cl (S) ' Z2 and, hence, S is half-factorial. Remark 4.2. Neither the condition given in ii) nor the condition Cl (S) ' Z2 are necessary for half-factoriality. This is so even for m = 1; see [2] for various sufficient or necessary conditions of half-factoriality in this particular case.

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In [2, Theorem 1.3], the current authors found a formula for computing the class group of a Diophantine monoid given by just one equation. The results of Section 3 yield a simpler proof of this formula, as well as an additional characterization of such monoids. Proposition 4.3. i) A monoid S is isomorphic to a nonfactorial Diophantine monoid given by just one equation if and only if S is reduced, root-closed, and finitely generated with rank Cl (S) = 1. ii) Let S = MA with A = [a1 a2 . . . an ] ∈ Z1×n and, without restriction, ai 6= 0 for all i, not all ai of equal sign and gcd (a1 , . . . , an ) = 1. There are exactly two possible cases: a) Cl (S) ' Zα with α ∈ N. This case occurs if and only if all ai except one, say an , are of equal sign. Thus we have α = |acn | , where n−1 Q c= ci and 0 < ci = gcd (|aj | | j 6= i) for 1 ≤ i ≤ n. i=1

b) Cl (S) ' Z. This case occurs if and only if there are at least two ai with positive sign and at least two ai with negative sign. Proof. The proof of i) follows directly from Theorem 3.8. For ii), let aej = Q aj e = [ae1 . . . a ci . For A fn ] the mapping ψ : MA −→ MAe, dj with dj = i6=j

ψi (x1 , . . . , xn ) = xcii is a monoid isomorphism. Hence Cl (MA ) ' Cl (MAe). Furthermore, w(MAe) = 1. Thus, we can assume that ci = 1 for all i, c = 1 and w(MA ) = 1. a) Let ai > 0 for 1 ≤ i ≤ n − 1 and an < 0. Then A = [A0 A00 ] with A0 = [a1 . . . an−1 ] and A00 = an = −α. Since cn = 1, we have that 1 ∈ im A0 and, hence, im A0 = Z. Obviously, im A00 = αZ and from Lemma 3.5 i) we obtain Cl (MA ) ' Zα . b) From the assumption in b), all projections must be essential and hence, A = A0 . From gcd(a1 , . . . , an ) = 1 it follows that im A0 = Z and Lemma 3.5 yields Cl (MA ) ' Z. The following example illustrates the concept of row degree and column degree, respectively and the statements made in Theorem 3.8. Example 4.4. Consider the Diophantine monoid S = {x ∈ N5 | x1 + x2 = x4 + x5 , x2 + x5 = x3 + x4 }.

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1 1 0 −1 −1 Obviously, S = MA for A = ∈ Z2×5 . All canon0 1 −1 −1 1 ical projections are normal and by Lemma 2.3 the projections π1 to π4 are essential while π5 is not. We havew(MA) = 1 and A = [A0 A00 ] with 1 1 0 −1 −1 A0 = and A00 = . It follows that im A0 = 0 1 −1 −1 1 −1 Z ⊕ Z, im A00 = Z , and, by Lemma 3.5 i), Cl (MA ) ' Z. Since +1 rank Q(MA ) = 3, it follows from Theorem 3.8 i) that dr (S) = 1 and dc (S) = 1 + 3 = 4. By Theorem 3.8 ii) there exists a matrix B ∈ Z1×4 such that MA is isomorphic to the Diophantine monoid MB given by the smaller matrix B. Indeed, one can reduce the given system of two equations to just one equation as follows. Eliminating x5 = x1 + x2 − x4 yields x2 + (x1 + x2 − x4 ) = x3 + x4 . That is x1 + 2x2 − x3 − 2x4 = 0. One has to make sure, however, that for any solution in N of the latter equation, it automatically holds that x5 = x1 + x2 − x4 ≥ 0. This is obvious in case of x2 ≥ x4 . If x2 ≤ x4 , then one obtains x1 +x2 −x4 ≥ x1 +2(x2 −x4 ) = x3 ≥ 0. By Theorem 3.8, one can easily check if a given Diophantine monoid can be described by a smaller matrix without actually carrying out the elimination procedure as in the above example. This latter process might be quite difficult in general. The following example also illustrates Theorem 3.8, and essentially covers all class group possibilities of rank 2. In contrast to Example 4.4, here there is no possible smaller description of the given monoids. Example 4.5. Let G be a finitely generated abelian group of rank 2. We show how to construct a Diophantine monoid S defined by 2 equations such that Cl (S) ' G. There are three cases to consider. (a) Suppose G ' Zn ⊕ Zkn for positive integers n > 1 and k ≥ 1. Let S = {x ∈ N5 | x1 + x3 = nx4 , x2 + x3 = knx5 }.

1 0 1 −n 0 Obviously, S = MA for A = ∈ Z2×5 . It is easy to 0 1 1 0 −kn check that all canonical projections are normal and, using Lemma 2.3, that the projections π1 , π2 , π3 areessential while π4 , π5 are not. Thus,w(MA ) = 1 1 0 1 −n 0 and A = [A0 A00 ] with A0 = and A00 = . It follows 0 1 1 0 −kn

144


that im A0 = Z ⊕ Z and im A00 = nZ ⊕ knZ. Therefore, by Lemma 3.5 i), Cl (S) ' (Z ⊕ Z)/(nZ ⊕ knZ) ' Zn ⊕ Zkn . Since rank Q(S) = 3, by Theorem 3.8 we have that dr (S) = 2 and dc (M ) = 2 + 3 = 5. Therefore, the Diophantine monoid S can neither be described by one equation alone, nor by less than five variables. (b) Suppose G ' Z ⊕ Zn for some positive integer n > 1. Let S = {x ∈ N6 | x1 + x2 − x3 − x4 = 0, x1 + x3 + x5 = nx6 }. 1 1 −1 −1 0 0 Clearly, S = MA for A = ∈ Z2×6 . As in (a), 1 0 1 0 1 −n w(MA ) = 1, the projections π1 , π2 , π3 , π4 and π5are essential and π6 is not 1 1 −1 −1 0 0 essential. Hence, if A0 = and A00 = , then 1 0 1 0 1 −n im A0 = Z ⊕ Z and im A00 = nZ. Thus, Cl (S) ' (Z ⊕ Z)/nZ ' Z ⊕ Zn . Since rank Q(S) = 4, we have that dr (S) = 2 and dc (M ) = 2 + 4 = 6. (c) Suppose that G ' Z ⊕ Z. Let S = {x ∈ N6 | x1 + x3 − x4 − x6 = 0, x2 + x3 − x5 − x6 = 0}. 1 0 1 −1 0 −1 Clearly, S = MA for A = ∈ Z2×6 . As above, 0 1 1 0 −1 −1 w(MA ) = 1, but here all projections are essential. Thus, A0 = A and Lemma 3.5 yields Cl (S) ' Z2 . Since rank Q(S) = 4, we have that dr (S) = 2 and dc (M ) = 2 + 4 = 6. To close Section 4, we present an algorithm which reflects the techniques developed in Sections 2 and 3. The method is based on having the complete set of minimal solutions over the nonnegative integers of the system Ax = 0. An algorithm for that computation, by Garc´ıa-Sánchez and Rosales, can be found in [14, Section 3, p. 80]. Before proceeding, we will require one additional result. Let M be a finitely generated submonoid of (Nn , +) and u1 , . . . , ut the irreducible elements of M . Let V and W be the Q-vector spaces generated by u1 , . . . , ut and u2 − u1 , . . . , ut − u1 respectively. Then dim V − dimW ≤ 1 and V = W if and only if u1 ∈ W . Lemma 4.6. Let M , V and W be as above. The following statements are equivalent: i) M is half factorial ii) dim V = 1 + dim W . Pt Proof. For i) ⇒ ii), assume that u1 = j=2 αj (uj − u1 ) ∈ W with each Pt αj ∈ Q. Then k1 u1 = j=2 kj (uj − u1 ) for suitable k1 , . . . , kt ∈ Z, k1 6= 0. P P Since (k1 + tj=2 kj )u1 − tj=2 kj uj = 0, the monoid M is not half factorial.

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For ii) ⇒ i), let r1 , . . . , rt ∈ Z with t X

rj u j = 0 =

j=1

Since u1 6∈ W it follows that

t X

  t X rj (uj − u1 ) +  rj  u 1 .

j=2

j=1

Pt

j=1 rj

= 0.

Algorithm 4.7. Assume that A ∈ Zm×n . The following algorithm calculates the class group of MA and determines whether or not MA is halffactorial. Suppose that (uτ |1 ≤ τ ≤ t) 6= ∅ is the family of all irreducible elements of the monoid MA . Let C := (uiτ ) ∈ Nn×t be the matrix with column vectors uτ and row vectors vi := (uiτ ) ∈ N1×t , 1 ≤ i ≤ n. Let C1 := (uiτ − ui1 ) ∈ Zn×t . By Lemma 4.6, MA is not half factorial if and only if im Z C = im Z C1 . The calculation of the class group proceeds as follows. I) Reduction of the system. Let vi := (uiτ ) ∈ N1×t , 1 ≤ i ≤ n, be the i-th row vector of the matrix C. Step 1): For all i ∈ {1, . . . , n}, if vi = 0, then cancel the i-th row of C. Step 2): For all i, j ∈ {1, . . . , n} with i < j, if λvi = vj for some λ ∈ Q, then cancel the j-th row of C. After these canceling steps, we denote the new matrix again with C. Then C ∈ Zne×t for some n e ≤ n. Step 3): For all j ∈ {1, . . . , n e}, calculate cj := gcd (ujτ |1 ≤ τ ≤ t) ∈ N and replace the row vector vj of C by c1j vj . II) Determining the essential states. Step 4): For every i ∈ {1, . . . , n e}, define Ji := {τ ∈ {1, . . . , t}|uiτ = 0}. If Ji 6= ∅, calculate the sums X (i) vj := ujτ τ ∈Ji

for all j ∈ {1, . . . , n e}, j 6= i. Define (i)

I := {i ∈ {1, . . . , n e}| Ji 6= ∅, vj 6= 0 for all j ∈ {1, . . . , n e}, j 6= i} and e := |I|. The projections πi , i ∈ I, are exactly the essential states of M . III) The final calculation. e ∈ Ne×t be the matrix with the row vectors vi , i ∈ I. TransStep 5): Let C e form C into its Smith normal form, using e.g., the algorithm given in [14].

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e and r ≥ 0 elementary From the Smith normal form one gets k := rank C e divisors αi ≥ 2 of C with αi+1 |αi for 1 ≤ i ≤ r − 1 if r ≥ 2. Then Cl (M ) = Ze−k ⊕ Z/α1 Z ⊕ ... ⊕ Z/αr Z. Acknowledgment. The authors would like to acknowledge many useful suggestions made by an anonymous referee on an earlier draft of the paper. References [1] S. Chapman and A. Geroldinger, Krull domains and monoids, their sets of lengths, and associated combinatorial problems, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, 189 (1997), 73-112, MR 98g:13017, Zbl 0897.13001. [2] S. Chapman, U. Krause and E. Oeljeklaus, Monoids determined by a homogeneous linear Diophantine equation and the half-factorial property, J. Pure Appl. Algebra, 151 (2000), 107-133, MR 2001g:11034. [3] E. Contejean and H. Devie, An efficient incremental algorithm for solving systems of linear Diophantine equations, Inform. and Comput., 113 (1994), 143-172, MR 96a:11151, Zbl 0809.11015. [4] E.B. Elliott, On linear homogenous Diophantine equations, Quart. J. Pure Appl. Math., 34 (1903), 348-377. [5] J.H. Grace and A. Young, The Algebra of Invariants, Cambridge University Press, Cambridge, 1903. [6] F. Halter-Koch, Halbgruppen mit divisorentheorie, Exposition Math., 8 (1990), 27-66, MR 91c:20091, Zbl 0698.20054. [7]

, The integral closure of a finitely generated monoid and the Frobenius problem in higher dimensions, in ‘Semigroups: Algebraic Theory and Applications to Formal Languages and Codes’ (eds. C. Bonzini et al.), World Scientific 1993, 86-93, CMP 1 647 172, Zbl 0818.20079.

[8] M. Hochster, Ring of invariants of tori, Cohen-Macauley rings generated by monomials, and polytopes, Ann. of Math., II Ser., 96 (1972), 318-337, MR 46 #3511, Zbl 0237.14019. [9] N. Jacobson, The Theory of Rings, Mathematical Surveys II, AMS, Providence, 1943, MR 5,31f, Zbl 0060.07302. [10] F. Kainrath and G. Lettl, Geometric notes on monoids, Semigroup Forum, 61 (2000), 298-302, CMP 1 832 189, Zbl 0964.20037. [11] U. Krause, On monoids of finite real character, Proc. Amer. Math. Soc., 105 (1989), 546-554, MR 89i:20102, Zbl 0692.20058. [12] U. Krause and C. Zahlten, Arithmetic in Krull monoids and the cross number of divisor class groups, Mitt. Math. Ges. Hamburg, 12 (1991), 681-696, MR 93b:11142, Zbl 0756.20010. [13] G. Lettl, Subsemigroups of finitely generated groups with divisor-theory, Monatsh. Math., 106 (1988), 205-210, MR 89m:20078, Zbl 0671.20059. [14] J.C. Rosales and P.A. Garc´ıa-S´ anchez, Finitely Generated Commutative Monoids, Nova Scientific Publishers, 1999, MR 2000d:20074, Zbl 0966.20028.

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[15] R. Stanley, Linear Diophantine equations and local cohomology, Inv. Math., 68 (1982), 175-193, MR 83m:10017, Zbl 0516.10009. [16]

, Combinatorics and Commutative Algebra, Birkh¨ auser, Boston, 1983, MR 85b:05002, Zbl 0537.13009.

Received September 25, 2000 and revised November 9, 2001. Part of this work was completed while the first author was on an Academic Leave granted by the Trinity University Faculty Development Committee. Part of this work was completed while the second author was visiting Trinity University during the fall of 1999 as the Eva and O.R. Mitchell Visiting Distinguished Professor of Mathematics. Trinity University Department of Mathematics 715 Stadium Drive San Antonio, Texas 78212-7200 E-mail address: [email protected] ¨ t Bremen Universita Fachberich Mathematik/Informatik DW-2800 Bremen, Germany E-mail address: [email protected] ¨ t Bremen Universita Fachberich Mathematik/Informatik DW-2800 Bremen, Germany E-mail address: [email protected]


THE GROUP OF ISOMETRIES OF A FINSLER SPACE Shaoqiang Deng and Zixin Hou

We prove that the group of isometries of a Finsler space is a Lie transformation group on the original manifold. This generalizes the famous result of Myers and Steenrod on a Riemannian manifold and makes it possible to use Lie theory on the study of Finsler spaces.

Introduction. Let (M, F ) be a Finsler space, where F is positively homogeneous but not necessary absolutely homogeneous. As in the Riemannian case, we have two kinds of definitions of isometry on (M, F ). On one hand, we can define an isometry to be a diffeomorphism of M onto itself which preserves the Finsler function. On the other hand, since on M we still have the definition of distance function (although generically it is not a real distance), we can define an ismotry of (M, F ) to be a mapping of M onto M which keeps the distance of each pair of points of M . The equivalence of the two definitions of isometry in the Riemannian case is a famous result of Myers and Steenrod. They used this result to prove that the group of isometries of a Riemannian manifold is a Lie transformation groups on the original manifold [5]. This result plays a fundamental role on the theory of homogeneous Riemannian manifolds. Since then, many different proofs were provided, cf., e.g., Palais [6], S. Kobayashi [4]. In this paper we prove that the two definitions of isometry are equivalent for a Finsler space. Then we prove that the group of isometries has a diferentiable structure which turns it into a Lie transformation on the manifold. This result makes it possible to use Lie theory on the study of Finsler spaces. In this paper, Finsler structure F is only assumed to be positively homogeneous but not necessary absolutely homogeneous. We will not point out this each time. For a mapping φ of a manifold M , we use dφ to denote its differential. If p ∈ M , dφ|p will denote the differential of φ at p. The notations of forward and backward metric ball in a Finsler spaces comes from the newly published book by D. Bao, S.S. Chern and Z. Shen [1]. 149

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1. A result on distance function. Let (M, F ) be a Finsler space, d be the distance function of (M, F ). We first need to prove a result on the distance function. Lemma 1.1. Let x ∈ M . Then for any > 0, there exists a neighborhood U of the original of Tx (M ) such that expx is a C 1 -diffeomorphism from U onto its image and for any A, B ∈ U , A 6= B, and any C 1 curve σ0 (s), 0 ≤ s ≤ 1, connecting A and B which satisfies σ0 (s) ∈ U and σ˙ 0 (s) 6= 0, s ∈ [0, 1], we have L(σ) L(σ0 ) − 1 ≤ , where L(·) denotes the arc length of a curve and σ(s) = expx σ0 (s). Proof. Let Bx (r) = {A ∈ Tx (M )| F (x, A) < r} be a tangent ball in Tx (M ) such that exp = expx is a C 1 -diffeomorphism from Bx (r) onto Bx+ (x) = {w ∈ M |d(x, w) ≤ r} (cf. [1]). Assume A, B ∈ Bx (r), A 6= B. Let σ0 (s), 0 ≤ s ≤ 1 be a C 1 curve connecting A and B and ∀s, σ0 (s) ∈ Bx (r) and σ˙ 0 (s) 6= 0. Then we can write the velocity vector of σ0 (s) as σ˙ 0 (s) = t(s)X(s), where X(s) satisfies F (x, X(s)) = 2r , ∀s, and t(s) ≥ 0 is a continuous function on [0, 1]. Therefore the arc length of σ0 is Z 1 t(s)F (x, X(s))ds. L(σ0 ) = 0

Denote X1 (s) = d(expx )|σ0 (s) X(s). Then the velocity vector of the curve σ(s) = expx (σ0 (s)), 0 ≤ s ≤ 1 is σ(s) ˙ = d(expx )|σ0 (s) (t(s)X(s)) = t(s)d(expx )|σ0 (s) (X(s)) = t(s)X1 (s). Therefore, the arc length of σ is Z 1 L(σ) = t(s)F (σ(s), X1 (s))ds. 0

Now we select a neighborhood V1 of x in M with compact closure which is contained in Bx+ (r) and fix a coordinate system (x1 , x2 , . . . , xn ) in V1 . Let U1 = exp−1 V1 . Suppose σ0 ⊂ U1 . Denote by M (s) the matrix of d(expx )|σ0 (s) under the base ∂x∂ 1 , ∂x∂ 2 , . . . , ∂x∂n . Given any positive number δ < 2r . Since d(expx )|0 = In and exp is C 1 smooth, there exists a neighborhood U2 ⊂ U1 of the original of Tx (M ) such that for any C 1 curve σ0 satisfying σ0 (s) ∈ U2 , ∀s, we have kM (s) − Ik
0. Now write the n P Finsler function F (w, y) as F (w, y1 , y2 , . . . , yn ) for y = yj ∂ . Consider y ∈ TW (T x(M )) = T x(M ), F (x, y) =

j=1

∂xj w

the closure D1 of the set D0 = {(w, y) ∈ T M |w ∈ V1 , F (x, y) ≤ 2r + r1 }. Since F is continuous and D1 is compact, F is uniformly continuous on D1 . Therefore for the given > 0, there exists δ1 > 0 and a neighborhood V2 ⊂ V1 of x such that for any w ∈ V2 , |yj − yj0 | < δ1 , j = 1, 2, . . . , n, F (x, y1 , y2 , . . . , yn ) < 2r + r1 , F (w, y10 , . . . , yn0 ) < 2r + r1 , we have r |F (x, y1 , y2 , . . . , yn ) − F (w, y10 , y 0 2, . . . , yn0 )| < . 2 Therefore if we select the above δ such that δ < δ1 . Then for the corresponding U2 and any C 1 curve σ0 , σ0 ⊂ U2 ∩ (exp)−1 V2 , we have R 1 t(s)(F (x, X(s)) − F (σ(s), X (s)))ds 1 L(σ) 0 R L(σ0 ) − 1 = 1 0 t(s)F (x, X(s))ds R1 t(s)|F (x, X(s)) − F (σ(s), X1 (s))|ds ≤ 0 R1 r 0 t(s)ds R1 r t(s)ds ≤ 2r R01 = . 2 0 t(s)ds

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Theorem 1.2. Let x ∈ M and Bx (r) be a tangent ball of Tx (M ) such that expx is a C 1 diffeomorphism from Bx (r) onto Bx+ (r). For A, B ∈ Bx (r), A 6= B, let a = expx A, b = expx B. Then we have F (x, A − B) →1 d(a, b) as (A, B) → (0, 0). Proof. Let Bx− (r) = {w ∈ M | d(w, x) < r}. Suppose r is so small that each pair of points in Bx+ ( 2r ) ∩ Bx− ( 2r ) can be joint by a unique minimal geodesic contained in Bx+ (r)(cf. [1]). Let Γ0 (s), 0 ≤ s ≤ 1 be the line segment connecting A and B, and Γ(s) = expx Γ0 (s). By Lemma 1.1, we have F (x, A − B) L(Γ0 ) = →1 L(Γ) L(Γ) as (A, B) → (0, 0). Now let a = expx A, b = expx B. Suppose a, b ∈ Bx+ ( 2r ) ∩ Bx− ( 2r ). Let γab (s), 0 ≤ s ≤ 1 be the unique minimal geodesic of constant speed connecting a and b. Let γ0 (s), 0 ≤ s ≤ 1 be the unique curve in Bx (r) which satisfies γab (s) = expx γ0 (s). Then by Lemma 1.1, we also have L(γ0 ) →1 L(γab ) as (A, B) → (0, 0). Since d(a, b) ≤ L(Γ), L(γ0 ) ≥ F (x, A − B), we have

F (x, A − B) F (x, A − B) L(γ0 ) ≤ ≤ . L(Γ) d(a, b) L(γab ) Theorem 1.2 follows.

2. Differentiability of isometries. First we have: Proposition 2.1. Let k · k1 , k · k2 be two Minkowski norms on Rn . Let φ be a mapping of Rn into itself such that kφ(A) − φ(B)k2 = kA − Bk1 , ∀A, B ∈ Rn . Then φ is a diffeomorphism. Proof. Consider Rn endowed with k · kj , j = 1, 2 as two Finsler spaces, denoted by (M1 , F1 ), (M2 , F2 ). Then geodesics in Mj , j = 1, 2 are straight lines (cf. [1]). And the distance function of Mj are dj (A, B) = kA − Bkj , j = 1, 2. Consider φ as a mapping from the Finsler space (M1 , F1 ) to (M2 , F2 ). Then φ preserves the distance function. Since in a Finsler space short geodesics minimize distance between its start and end points (cf. [1]), we can prove (similarly as in the Riemannian case) that φ transforms geodesics to geodesics. First suppose φ(0) = 0. For A ∈ Rn , A 6= 0, the curve φ(tA),

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t ≥ 0 is a ray which coincides with the ray tφ(A) for t = 0 and t = 1. Therefore they coincide as point sets. Thus φ(tA) = µ(t)φ(A) for some nonnegative function µ(t). Since kφ(tA) − 0k2 = ktA − 0k1 = tkAk1 = kµ(t)φ(A) − 0k2 = µ(t)kφ(A)k2 = µ(t)kAk1 , t ≥ 0, we have µ(t) = t. Thus φ(tA) = tφ(A), for t ≥ 0. Suppose A 6= B, a similar argument as the above shows that there exists a nonnegative function λ(t) such that φ(tA + (1 − t)B) = λ(t)φ(A) + (1 − λ(t))φ(B), t ≥ 0. And we can similarly show that λ(t) = t. In particular, for t = 21 we have, 1 1 1 1 φ(A + B) = φ (A + B) = φ(A) + φ(B). 2 2 2 2 Thus φ(A+B) = φ(A)+φ(B). Taking A = −B in the above equality we have φ(−A) = −φ(A). Therefore φ is a linear transformation. Since Ker(φ) = {0}, it is a diffeomorphism. If A1 = φ(0) 6= 0, consider the composition mapping φ1 = πA1 ◦ φ, where πA1 (A) = A − A1 is the parallel translation, which is a diffeomorphism. Since φ1 (0) = 0 and kφ1 (A)−φ(B)k2 = kA−Bk1 , φ1 is a diffeomorphism. Hence φ is a diffeomorphism. Remark. The proposition is an interesting application of Finsler geometry to Functional Analysis. Now we can prove the main result of this paper. Theorem 2.2. Let (M, F ) be a Finsler space and φ be a distance-preserving mapping of M onto itself. Then φ is a diffeomorphism. Proof. Let p ∈ M and put q = φ(p). Let r > 0, > 0 be so small that both expp and expq are C 1 diffeomorphisms on the tangent ball Bp (r+), Bq (r+) of Tp (M ) and Tq (M ), respectively. For any nonzero X ∈ Tp (M ), consider the r . The image γ(t) = φ(expp (tX)) radial geodesic expp (tX), 0 ≤ t ≤ 2F (p,X) is a geodesic since φ is distance-preserving. Let X 0 denote the tangent vector of γ at the point q. We have obtained a mapping X → X 0 of Tp (M ) into Tq (M ). Denoting this mapping by φ0 we have φ0 (λX) = λφ0 (X), for X ∈ Tp (M ) and λ ≥ 0. Let A, B ∈ Tp (M ) , A 6= B and t is so small that both tA and tB lie in Bp (r). Let at = expp (tA), bt = expp (tB). Then by Theorem 1.2 we have lim

t→0+

F (p, tA − tB) = 1. d(at , bt )

On the other hand, by the definition of φ0 we have expq (φ0 (tX)) = φ(exp tX),

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for any X and t small enough. Thus by Theorem 1.2 we also have lim

t→0+

F (q, φ0 (tA) − φ0 (tB)) = 1. d(φ(at ), φ(bt ))

Since d(φ(at ), φ(bt )) = d(at , bt ), we get F (p, tA − tB) F (q, φ0 (tA) − φ0 (tB)) tF (p, A − B) F (p, A − B) = lim = . 0 0 F (q, φ0 (A) − φ0 (B)) t→0+ tF (q, φ (A) − φ (B))

1 = lim

t→0+

Therefore F (q, φ0 (A) − φ0 (B)) = F (p, A − B). By Proposition 2.1, φ0 is a diffeomorphism of Tp (M ) onto Tq (M ). Although on Bp+ (r) = expp Br (p) we have φ = expq ◦ φ0 ◦ (expp )−1 , we still cannot conclude that φ is smooth on Bp+ (r), since in a Finsler space the exponential mapping is only C 1 at the zero section. That is, we can only conclude that φ is smooth in Bp (r) − {p}. To finish the proof, we proceed to take r so small so that every pair of points in Bp+ (r) ∩ Bp− (r) can be joint by a unique minimizing geodesic. Select p1 ∈ Bp+ ( 2r ) ∩ Bp− ( 2r ), p1 6= p. Consider the tangent ball Bp1 ( 2r ) of Tp1 (M ). The exponential mapping is a C 1 diffeomorphism from Bp1 ( 2r ) onto Bp+1 ( 2r ). The above argument shows that φ is smooth in Bp+1 ( 2r ) − {p1 }, which is a neighborhood of p. This completes the proof. 3. Group of isometries. Theorem 2.2 justifies the following definition of isometry for a Finsler space. Definition 3.1. Let (M, F ) be a Finsler space. A mapping φ of M onto itself is called an isometry if φ is a diffeomorphism and for any x ∈ M, X ∈ Tx (M ), F (φ(x), dφx (X)) = F (x, X). In the following we denote the group of isometries of (M, F ) by I(M ). Let N be a connected, locally compact metric space and I(N ) be the group of isometries of N , for each point x of N , let Ix (N ) denote the isotropy subgroup of I(N ) at x. Van Danzig and van der Waerden [7] proved that I(N ) is a locally compact topological transformation group on N with respect to the compact-open topology and Ix (N ) is compact. Now on M we have a distance function d defined by the Finsler function F . By Theorem 2.2, the group I(M ) coincides with the group of isometries I(M ) of (M, d). Although generically d is not a distance (d is not symmetric unless F is absolutely homogeneous), we still have: Theorem 3.2. Let (M, F ) be a connected Finsler space. The compact-open topology turns I(M ) into a locally compact transformation group of M . Let

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x ∈ M and Ix (M ) denote the subgroup of I(M ) which leaves x fixed. Then Ix (M ) is compact. Proof. A proof of this result for the Riemannian case was given in Helgason [3] (cf. Helgason [3], pp. 201-204), which is valid in general cases after some minor changes. Just note that on a Finsler manifold the topology generated by the forward metric balls Bp+ (r) = {x ∈ M |d(p, x) < r}, p ∈ M, r > 0 is precisely the underlying manifold topology and this is true for the topology generated by the backward metric balls Bp− (r) = {x ∈ M |d(x, p) < r}, p ∈ M, r > 0 (cf. [1]). Bochner-Montgomery [2] proved that a locally compact group of differentiable transformations of a manifold is a Lie transformation group. Therefore we have the following theorem. Theorem 3.3. Let (M, F ) be a Finsler space. Then the group of isometries I(M ) of M is a Lie transformation group of M . Let x ∈ M and Ix (M ) be the isotropy subgroup of I(M ) at x. Then Ix (M ) is compact. References [1] D. Bao, S.S. Chern and Z. Shen, An Introduction to Riemannian-Finsler Geometry, Springer-Verlag, Berlin, 1999, MR 2001g:53130, Zbl 0954.53001. [2] S. Bochner and D. Montgomery, Locally compact groups of differentiable transformations, Ann. of Math., 47 (1946), 639-653, MR 8,253c, Zbl 0061.04407. [3] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, 2nd ed., Academic Press, New York, 1978, MR 80k:53081, Zbl 0451.53038. [4] S. Kobayashi, Transformation Groups in Differential Geometry, Springer-Verlag, Berlin, 1972, MR 50 #8360, Zbl 0246.53031. [5] S.B. Myers and N. Steenrod, The group of isometries of a Riemannian manifold, Ann. of Math., 40 (1939), 400-416, Zbl 0021.06303. [6] R. Palais, On the differentiability of isometries, Proc. Math. AMS., 8 (1957), 805-807, MR 19,451a, Zbl 0084.37405. ¨ [7] D. van Dantzig and B.L. van der Waerden, Uber metriseh homogene r¨ aume, Abh. Math. Sem. Univ. Hamburg, 6 (1928), 374-376. Received March 15, 2001. This work is Project 19901015 and 19731004 supported by NSFC. Department of Mathematics Nankai University Tianjin, P. R. China E-mail address: [email protected] Department of Mathematics Nankai University Tianjin, P. R. China E-mail address: [email protected]


ON SPACES OF MATRICES CONTAINING A NONZERO MATRIX OF BOUNDED RANK Dmitry Falikman, Shmuel Friedland, and Raphael Loewy Let Mn (R) and Sn (R) be the spaces of n × n real matrices and real symmetric matrices respectively. We continue to study d(n, n − 2, R): The minimal number ` such that every `-dimensional subspace of Sn (R) contains a nonzero matrix of rank n−2 or less. We show that d(4, 2, R) = 5 and obtain some upper bounds and monotonicity properties of d(n, n − 2, R). We give upper bounds for the dimensions of n − 1 subspaces (subspaces where every nonzero matrix has rank n − 1) of Mn (R) and Sn (R), which are sharp in many cases. We study the subspaces of Mn (R) and Sn (R) where each nonzero matrix has rank n or n − 1. For a fixed integer q > 1 we find an infinite sequence of n such that any q+1 dimensional sub2 space of Sn (R) contains a nonzero matrix with an eigenvalue of multiplicity at least q.

1. Introduction. Let F be a field, Mm,n (F) the space of all m×n matrices over F and Sn (F) the space of all n × n symmetric matrices over F. We write Mn (F) for Mn,n (F). Let V be either Mm,n (F) or Sn (F), and let k be a positive integer. In the last 80 years there has been interest in the following types of subspaces, all related to the rank function: (a) A subspace of V where each matrix has rank bounded above by k. (b) A subspace of V where each nonzero matrix has rank k. A subspace of this type is said to be a k-subspace. (c) A subspace of V where each nonzero matrix has rank bounded below by k. See the works [13], [11], [2], [8], [5], [3], [4], [7] and many others. Roughly speaking these problems are divided into two classes depending on whether F is algebraically closed or not. The classical case, which goes back to Radon-Hurwitz, discusses the maximal dimension ρ(n) of an n-subspace U of Mn (R) where each 0 6= A ∈ U is an orthogonal matrix times r ∈ R∗ . Write n = (2a + 1)2c+4d , where a and d are nonnegative integers, and c ∈ 157

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{0, 1, 2, 3}. Then the Radon-Hurwitz number ρ(n) is defined by (1.1)

ρ(n) = 2c + 8d.

In his famous work Adams [1] gave a nonlinear version of the Radon-Hurwitz number by showing that ρ(n) − 1 is the maximal number of linearly independent vector fields on the n − 1 dimensional sphere S n−1 . (S n−1 ⊂ Rn denotes the Eucledian sphere of radius one centered at the origin.) From this result Adams deduced that the maximal dimension of an n-subspace of Mn (R) is exactly ρ(n). Let ρ(x) = 0 if x is not a positive integer. Define now n (1.2) ρs (n) = ρ + 1. 2 Adams, Lax and Phillips [2] showed that the maximal dimension of an nsubspace of Sn (R) is exactly ρs (n). Friedland, Robbin and Sylvester [8] and Berger and Friedland [5] gave further nonlinear versions of the above results by considering odd maps φ from S p to matrices of rank n in Mn (R), Sn (R) and Mn,n+1 (R) respectively. In this paper (§4) we generalize these results to odd maps from S p to rank n − 1 matrices in Mn (R) and Sn (R). The main motivation of this paper is the following quantity studied in [7]. For an integer k, such that 1 ≤ k ≤ n−1, let d(n, k, F) be the smallest integer ` such that every ` dimensional subspace of Sn (F) contains a nonzero matrix whose rank is at most k. Note that it is clear that the maximal dimension of a subspace of Sn (F) of type (c) above is exactly d(n, k − 1, F) − 1. It is our purpose to continue the study of d(n, k, R) started in [7]. Note that any d(n, k, R) − 1 dimensional subspace of Sn (R) contains a nonzero matrix with an eigenvalue of multiplicity at least n − k. (One of the main results of [8] was that any subspace of Sn (R) of dimension σ(n) + 1, where σ(n) = 2 if n 6≡ 0, ±1 (mod 8), σ(n) = ρ(4b) if n = 8b, 8b ± 1, contains a nonzero matrix with a multiple eigenvalue.) The equality n−k+1 (1.3) d(n, k, C) = + 1, 2 established in [7] is derived straightforwardly from the following dimension computations. Let Vk,n (C) (Vk,n (R)) be the variety of all matrices in Sn (C) (Sn (R)) of rank k or less. Then in the projective space PSn (C) the projective variety PVk,n (C) is an irreducible variety of codimension d(n, k, C)−1, which yields (1.3). In particular d(n, n−1, C) = 2. The results of Adams, Lax and Phillips, cited above, yield that d(n, n−1, R) = ρs (n)+1. This shows that in general the computation of d(n, k, R) is much more difficult than the computation of d(n, k, C). In [7] we gave a simple condition on n when d(n, k, C) = d(n, k, R) for k ≤ n − 2, which trivially holds for

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k = n − 1. It was shown by Harris and Tu [10] that the degree of the variety PVk,n (C) is given by the formula n+j n−k−1 Y n−k−j (1.4) δk,n := deg PVk,n (C) = . 2j+1 j

j=0

Then d(n, k, C) = d(n, k, R) if δk,n is odd. (In this case the result that any d(n, k, R) − 1 dimensional subspace of Sn (R) contains a nonzero matrix with an eigenvalue of multiplicity at least n − k is best possible.) We show that δn−q,n is odd if n > q ≥ 1 and (1.5) n ≡ ±q mod 2dlog2 2qe . In particular, under the above conditions, (1.6)

d(n, n − q, C) = d(n, n − q, R) =

q+1 + 1. 2

If n ≥ q and n satisfies (1.5) then any q+1 subspace of Sn (R) contains a 2 nonzero matrix with an eigenvalue of multiplicity at least q. This statement for q = 2 yields the original Lax’s result [12] that any 3 dimensional subspace of Sn (R) contains a nonzero matrix with a multiple eigenvalue for n ≡ 2 (mod 4). (This result and its generalization in [8] is of importance in the study of singularities of hyperbolic systems.) An important part of the paper is devoted to the study of the numbers d(n, n − 2, R). Besides the cases given in (1.5-1.6) for q = 2 it is easy to see that d(3, 1, R) = 6 [7]. (Hence the inequality d(n, n − 2, R) ≥ σ(n) + 2 established in [7] is not sharp for some n.) Partial results regarding d(n, n − 2, R) were obtained in [7]. In particular, it was shown that for m ≥ 1 (1.7)

d(4m, 4m − 2, R) ≤ 4m + 1.

We obtain here additional results on d(n, n − 2, R). In particular, using numerical and symbolic computations we show that d(4, 2, R) = 5, which implies that the upper bound in (1.7) is sharp for m = 1. The paper is organized as follows: In Section 2 we show that d(4, 2, R) = 5. In Section 3 we obtain some upper bounds and some monotonicity properties of d(n, n − 2, R). In Section 4 we consider the existence of continuous and smooth odd maps from the sphere S p to matrices of rank n−1 in Mn (R) and Sn (R). As a consequence we obtain new upper bounds for the dimension of (n − 1)-subspaces of Mn (R) and Sn (R). These upper bounds are shown to be sharp in many cases. In Section 5 we discuss the existence of subspaces U ⊂ S2n−1 (R) of dimension 2n, which do not contain a nonzero matrix of rank less than 2n−2. The last section is devoted to remarks and conjectures.

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2. Computation of d(4, 2, R). As indicated in the introduction, the first unknown number in the sequence {d(n, n − 2, R)}∞ n=3 is d(4, 2, R). By (1.7) d(4, 2, R) ≤ 5. It is our purpose to prove: Theorem 2.1. d(4, 2, R) = 5. Proof. It suffices to exhibit a 4-dimensional subspace L of S4 (R) with the property that every 0 6= A ∈ L has rank 3 or 4. Let L be the subspace of S4 (R) spanned by the matrices 

 1 −1 −1 1  −1 −1 1 1  , B1 =   −1 1 1 1  1 1 1 −1 

1  1 B3 =   1 1

 1 1 1 1 1 1  , 1 1 −1  1 −1 −1



−1 1 −1  1 −1 1 B2 =   −1 1 1 −1 −1 −1

 −1 −1  , −1  −1



 −1 −1 −1 1  −1 1 1 −1  . B4 =   −1 1 −1 −1  1 −1 −1 1

It is straightforward to check that dim L = 4. LetPα1 , α2 , α3 , α4 be indeterminates and let α = (α1 , α2 , α3 , α4 ), and B(α) = 4i=1 αi Bi . We show that if 0 6= α ∈ R4 then rank B(α) ≥ 3. So let d(α) = det B(α), and for 1 ≤ j ≤ 4 let dj (α) denote the determinant of the principal submatrix of B(α) obtained by deleting row and column j. We let Maple compute the common zeros of the 5 polynomials d(α), d1 (α), d2 (α), d3 (α), d4 (α) in C4 . It turns out that there are no common zeros with α1 = 0 except the trivial solution α = 0. Now we can assume α1 = 1. We list the ten solutions found by Maple. They are: (i) The two solutions given by α = (1, 0, 1 + x, x), where x is a zero of the polynomial f (z) = z 2 + z + 1. (ii) The two solutions given by α = (1, x, 2x − 1, 3x − 2), where x is a zero of the polynomial f (z) = 7z 2 − 8z + 3. (iii) The two solutions given by α = (1, x, −x, −5x − 1), where x is a zero of the polynomial f (z) = 14z 2 + 4z + 1.

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(iv) The four solutions given by 1 2 1 1 2 1 α = 1, x, x + x + , − x − x − , 2 2 2 2 where x is a zero of the polynomial f (z) = z 4 + 4z 3 + 8z 2 + 4z + 3 = (z + 1)4 + 2z 2 + 2. It follows that for any 0 6= α ∈ R4 either d(α) 6= 0 or ∃ 1 ≤ j ≤ 4 such that dj (α) 6= 0. Thus, the rank of every nonzero matrix in L is at least 3. We explain now why Maple did indeed give us all the common zeros of d(α), d1 (α), d2 (α), d3 (α), d4 (α). Let L1 be the subspace of S4 (C) spanned by B1 , B2 , B3 , B4 . We have by (1.3) d(4, 2, C) = 4. Moreover, by (1.4) we get (2.1)

deg P(V2,4 (C)) =

4 2

5 1

3

= 10.

We also use the known fact that a rank r symmetric matrix with entries in a field has a nonsingular r × r principal submatrix. Thus, the ten distinct solutions found by Maple yield ten matrices in L1 whose rank is at most 2, and no two of those matrices lie on the same line. In fact, a computation showed that each of those ten matrices has rank 2. It remains to explain why Maple did not omit other solutions, assuring us it did not miss any real solutions in particular. It follows from (2.1) that if L0 is a four dimensional generic subspace of S4 (C) then P(V2,4 (C)) meets P(L0 ) in at most ten distinct points. Hence, we have to check that L1 is indeed generic. It suffices to show that P(V2,4 (C)) and P(L1 ) intersect transversally. So let A denote any of the ten matrices in L1 of rank 2 found by Maple. Let x, y ∈ C4 be linearly independent vectors such that Ax = Ay = 0. A computation showed that the rank of   t x B2 x xt B3 x xt B4 x  y t B2 y y t B3 y y t B4 y  ∈ M3 (C) xt B2 y xt B3 y xt B4 y is 3. This implies that the linear system of three equations xt Bx = 0 y t By = 0 xt By = 0, has no solution B in L1 such that A, B are linearly independent. This shows that L1 is generic.

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In addition to the computation using Maple, a numerical procedure (using Matlab) was performed to check that there are no nonzero matrices of rank 1 or 2 in L. 3. Upper bounds and monotonicity properties of d(n, n − 2, R). It follows from (1.5) and (1.6) that if n ≡ 2(mod 4) then d(n, n − 2, R) = d(n, n − 2, C) = 4. It was shown in [7] that (3.1)

d(n, n − 2, R) ≤ 7

for n ≡ 4(mod 8).

It is our purpose here to get some additional upper bounds and some monotonicity properties of d(n, n − 2, R). Proposition 3.1. Let n ≡ 3, 5(mod 8). Then d(n, n − 2, R) ≤ 7. Proof. It follows from (1.4) that for every n ≥ 3 n n+1 n+2 (n + 2)(n + 1)n2 (n − 1)(n − 2) 1 2 3 = , deg P(Vn−3,n (C)) = 1 · 3 · 10 23 · 32 · 5 and this is odd for n ≡ 3, 5(mod 8). So, as indicated in the introduction, and using (1.3) we have for n ≡ 3, 5(mod 8) 4 d(n, n − 3, R) = d(n, n − 3, C) = + 1 = 7, 2 and since d(n, n − 2, R) ≤ d(n, n − 3, R), the proposition follows.

It follows from [7] that d(n, n − 2, R) ≥ 4 for n ≡ 3, 4, 5(mod 8). Thus, we have the exact value for d(n, n − 2, R) whenever n ≡ 2, 6(mod 8), and good upper and lower bounds whenever n ≡ 3, 4, 5(mod 8). Proposition 3.2. Let k be a fixed integer such that k ∈ {3, 4, 5, 11, 12, 13}. Then the sequence {d(16m + k, 16m + k − 2, R)}∞ m=0 is a (weakly) monotone increasing sequence, bounded above by 7. There exists M = M (k) such that d(16m + k, 16m + k − 2, R) = d(16M + k, 16M + k − 2, R)

for all m ≥ M.

Proof. It suffices to prove the monotonicity of the given sequence, because the required boundedness follows from (3.1) and Proposition 3.1. Let L be any 8-dimensional 8-subspace of M8 (R). Such a subspace exists because ρ(8) = 8 by (1.1). Let 0 A L1 = , A ∈ L . At 0 Then L1 is an 8-dimensional subspace of S16 (R). Note that every nonzero matrix in L1 is nonsingular. Let {Bi }8i=1 be a basis of L1 . Let dm,k = d(16m + k, 16m + k − 2, R) and d0 = dm,k − 1. Let Wm,k be any d0 -dimensional subspace of S16m+k (R) such that the rank of any

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nonzero matrix in Wm,k is at least 16m + k − 1. Note that d0 ≤ 7 − 1 = 6. 0 fm,k = span{Ci ⊕ Bi : 1 ≤ i ≤ do }. Let {Ci }di=1 be a basis of Wm,k . Let W f Then Wm,k is a d0 -dimensional subspace of S16(m+1)+k (R), and clearly every nonzero matrix in it has rank ≥ 16 + 16m + k − 1 = 16(m + 1) + k − 1. This shows that d(16(m + 1) + k, 16(m + 1) + k − 2, R) ≥ d0 + 1 = dm,k = d(16m + k, 16m + k − 2, R). The following proposition justifies some claims we made in the introduction. Proposition 3.3. Let 1 ≤ k ≤ n − 2. Then any d(n, k, R) − 1 dimensional subspace of Sn (R) contains a nonzero matrix which has an eigenvalue of multiplicity at least n − k. If d(n, k, R) = d(n, k, C) then this result is sharp. Proof. Let V be a d(n, k, R) − 1 dimensional subspace of Sn (R). If In ∈ V then 1 is the eigenvalue of In of multiplicity n ≥ n−k. Suppose that In ∈ / V. Let Vˆ = span{V, In }. Then Vˆ contains 0 6= Aˆ such that rank Aˆ ≤ k. Aˆ = A + aI, 0 6= A ∈ V , so −a is an eigenvalue of A of multiplicity at least n − k. Let Σn−k ⊂ Sn (R) be the variety of all A ∈ Sn (R) such that A has an eigenvalue of multiplicity at least n − k. Clearly Σn−k = {B ∈ Sn (R) : B = A + aIn

for some A ∈ Vk,n (R) and a ∈ R}.

Since Vk,n (R) is an irreducible variety of codimension d(n, k, C)−1 it follows that Σn−k is an irreducible variety of codimension d(n, k, C) − 2. Hence there exists an d(n, k, C) − 2 dimensional subspace W ⊂ Sn (R) such that Σn−k ∩ W = {0}. 4. Upper bounds for the dimension of (n − 1)-subspaces of Mn (R) and Sn (R). In this section we obtain upper bounds for the dimension of (n−1)-subspaces of Mn (R) and Sn (R). This is done by considering certain odd, smooth maps from S p into Mn (R) and Sn (R). We also need several known results. (k) (k) Let Mn (R) Sn (R) denote the set of all rank k matrices in 0 (F) Mn (R)(Sn (R)). Given any field F and positive integers m, n, let Mm,n denote the set of all matrices in Mm,n (F) whose rank is equal to min{m, n}. 0 Lemma 4.1. Let C ∈ Mn,n+1 (F). For j = 1, 2, . . . , n + 1, let C (j) denote the n × n matrix obtained from C by deleting its j-th column. Then the

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solution of the homogeneous linear system Cy = 0 is a line spanned by t (4.1) y = −detC (1) , detC (2) , −detC (3) , . . . , (−1)n+1 detC (n+1) . Proof. This is an easy consequence of well-known properties of the determinant. The next theorem appears as a part of Theorems A and F in [8]. See there how it is related to classical results due to Radon, Hurwitz, Adams and Adams-Lax-Phillips. Theorem 4.1. Let ϕ : S p → Mn (R) be an odd continuous map, i.e., ϕ(−α) = −ϕ(α) ∀α ∈ S p . (n)

(i) Suppose that ϕ(S p ) ⊂ Mn (R) = GL(n, R). Then, p ≤ ρ(n) − 1. (n) (ii) Suppose that ϕ(S p ) ⊂ Sn (R). Then, we have p = 0 if n is odd, and n p ≤ ρ( 2 ) if n is even. All the inequalities in (i) and (ii) are sharp. The next theorem appears in [5]. 0 Theorem 4.2. Let ϕ : S p → Mn,n+1 (R) be an odd continuous map. Then p ≤ max{ρ(n) − 1, ρ(n + 1) − 1}, and this inequality is sharp. (n−1)

Theorem 4.3. Let n ≥ 2. Let ϕ : S p → Mn Then, if n 6= 3, 5, 9 (4.2)

(R) be n odd smooth map.

p ≤ max{ρ(n − 1) − 1, ρ(n) − 1, ρ(n + 1) − 1, 2}. (n−1)

Furthermore, if ϕ(S p ) ⊂ Sn (R) then p = 0 if n is even, and n−1 p ≤ max ρ (4.3) , ρ(n + 1) − 1 if n is odd. 2 Proof. We prove first (4.2). If p ≤ 1 (4.2) certainly holds, so we may assume p ≥ 2. Hence S p is simply connected. Therefore, we can choose x(α) ∈ S n−1 in a well-defined and continuous way such that (4.4)

xt (α)ϕ(α) = 0

∀α ∈ S p .

For α ∈ S p let B(α) be the matrix in Mn,n+1 (R) obtained from ϕ(α) by augmenting it with the column vector x(α), i.e., B(α) = (ϕ(α), x(α)) ∈ Mn,n+1 (R). 0 Then rank ϕ(α) = n − 1 and (4.4) imply that B(α) ∈ Mn,n+1 (R) for all p α∈S . Note that the smoothness of ϕ implies that x(α) is also smooth. It is also clear that for each α ∈ S p x(−α) = ±x(α). Hence, we must have x(−α) = −x(α) for all α ∈ S p , or x(−α) = x(α) for all α ∈ S p .

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Case 1. Suppose that x(−α) = −x(α) for all α ∈ S p . Then the map from 0 S p to Mn,n+1 (R) defined by α → B(α) is an odd continuous map, so by Theorem 4.2 we have p ≤ max{ρ(n) − 1, ρ(n + 1) − 1},

(4.5) and (4.2) holds.

Case 2. We now assume x(−α) = x(α) for all α ∈ S p . We apply Lemma 4.1 0 to B(α) ∈ Mn,n+1 (R), for each α ∈ S p , and denote the normalized solution 1 y by ψ(α). Let given by (4.1) that is, kyk ψ(α) = (ψ1 (α), ψ2 (α), . . . , ψn (α), ψn+1 (α)). There are two subcases now. Case 2a. Suppose that n is even. It follows from (4.1) that ψ(−α) = (−ψ1 (α), −ψ2 (α), . . . , −ψn (α), ψn+1 (α)). We define C(α) =

B(α) ψ(α)

∈ Mn+1 (R).

This matrix is nonsingular because ψ(α) is orthogonal to all the rows of B(α). Let η(α) = (ψ1 (α), ψ2 (α), . . . , ψn (α)) and let ϕ(α) D(α) = ∈ Mn+1,n (R), η(α) that is, D(α) is the matrix obtained from C(α) by deleting its last column. Its rank is n for each α ∈ S p . Since D(−α) = −D(α) for all α ∈ S p we apply Theorem 4.2 again and conclude that (4.5) holds, so (4.2) holds. Case 2b. Suppose that n is odd. So n ≥ 3. Suppose first that p < n − 1. Since x(α) is smooth, it follows from Sard’s theorem that {x(α) : α ∈ S p } has measure zero in S n−1 . In particular, ∃ ξ ∈ S n−1 such that x(α) 6= ±ξ for all α ∈ S p . For every α ∈ S p let Lα = span{x(α), ξ}. Let Q(α) be the unique n × n orthogonal matrix satisfying: Q(α) is the identity on L⊥ α and its restriction to Lα is the rotation in that plane by an angle < π that sends x(α) to ξ. Since x(−α) = x(α) we have Q(−α) = Q(α). Also, Q(α) is continuous in α (cf. the Proposition in [8]). It follows from (4.4) that (4.6)

ξ t (Qt (α)ϕ(α)Q(α)) = 0,

for all α ∈ S p .

Let ϕ1 (α) = Qt (α)ϕ(α)Q(α). Then ϕ1 (α) is an odd continuous function. Without loss of generality we may assume ξ t = (0, 0, . . . , 0, 1). It follows from (4.6) that for each α ∈ S p the last row of ϕ1 (α) is 0, so deleting it

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results in an (n − 1) × n matrix of rank n − 1. Applying Theorem 4.2 again we conclude that p ≤ max{ρ(n − 1) − 1, ρ(n) − 1} so (4.2) holds. Now suppose that p ≥ n − 1. We may consider S n−2 as contained in S p , and then consider the restriction of the given ϕ(α) and x(α) to S n−2 . We repeat the proof given for p < n − 1 and conclude in the same way that n − 2 ≤ max{ρ(n − 1) − 1, ρ(n) − 1} = ρ(n − 1) − 1. Since this cannot happen by (1.1) when n 6= 3, 5, 9 we have completed the proof of (4.2). (n−1) Suppose now that ϕ(S p ) ⊂ Sn (R). Recall that the inertia of A ∈ Sn (R) is the triple (π, ν, δ), where π is the number of positive eigenvalues of A, ν is the number of negative eigenvalues of A and δ is the number of zero eigenvalues of A. Let n be even and suppose that p > 0. Let α ∈ S p , and consider a path in S p from α to −α. Let J denote the image of this path under ϕ. Since every matrix in J has rank n − 1, it follows that the inertia of each matrix on this path is equal to inertia (ϕ(α)). In particular, inertia (ϕ(−α)) = inertia(ϕ(α)), but this is impossible since ϕ(−α) = −ϕ(α). Hence p = 0. Let n be odd. Suppose first that n = 3. We show that p ≤ 1 in this case, so (4.3) holds. Suppose to the contrary that p ≥ 2. It is clear that ϕ(α) has the same inertia for each α ∈ S p . In particular, given any α ∈ S p , ϕ(α) and ϕ(−α) = −ϕ(α) have the same inertia. Hence each ϕ(α) has one positive, one negative and one zero eigenvalue. So for each α ∈ S p , the eigenvalues of ϕ(α) are pairwise distinct, contradicting Theorem B of [8]. Note that while we have shown that there is no odd continuous map from (2) (2) S 2 to S3 (R), there is an odd continuous map from S 2 to M3 (R). For example, consider the map that sends (x, y, z) ∈ S 2 to   0 x y  −x 0 z . −y −z 0 We assume now n ≥ 5. Since max{ρ( n−1 2 ), ρ(n + 1) − 1} ≥ 2, (4.3) holds if p ≤ 1. Hence we may assume that p ≥ 2. We go along the proof of the nonsymmetric part of the theorem. If x(α) is odd, then (4.5) holds, and since ρ(n) = 1 for odd n, we get n−1 p ≤ ρ(n + 1) − 1 ≤ max ρ , ρ(n + 1) − 1 , 2 and (4.3) holds.

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So we may assume that x(α) is even. Suppose also that p < n − 1. Then we get (4.6) again, where ϕ1 (α) = Qt (α)ϕ(α)Q(α) is an odd continuous function, and we may assume ξ t = (0, 0, . . . , 0, 1). It follows that for each α ∈ S p the matrix obtained from ϕ1 (α) by deleting its last row and column (n−1) is in Sn−1 (R). Hence, by part (ii) of Theorem 4.1 we get n−1 , p≤ρ 2 so (4.3) holds. If p ≥ n − 1 we consider S n−2 as contained in S p , and then consider the restriction of ϕ(α), x(α) to S n−2 . A repetition of the argument for p < n − 1 yields n − 2 ≤ ρ( n−1 2 ), which is impossible for n odd, n ≥ 5. This completes the proof. Corollary 4.1. Let n ≥ 2 and let L be an (n − 1)-subspace of Mn (R). (a) If n 6= 3, 5, 9, then (4.7)

dim L ≤ max{ρ(n − 1), ρ(n), ρ(n + 1), 3}.

(b) If L ⊂ Sn (R) (that is, L is a subspace of Sn (R)), then ( 1 if n is even, n−1 (4.8) dim L ≤ if n is odd. max ρ 2 + 1, ρ(n + 1) Proof. (a) Let d = dim L and let A1 , A2 , . . . , Ad be a basis of V . For A ∈ V P we write A = di=1 αi Ai and let α = (α1 , α2 , . . . , αd ). Then we may view the map α → A, restricted to α such that kαk = 1, as an odd smooth map (n−1) from S d−1 to Mn (R). Then d − 1 is bounded by the right-hand side of (4.2), so (4.7) follows. (b) The proof is similar to the previous case, using now (4.3) instead of (4.2). Consider next the sharpness of the inequalities (4.2), (4.3), (4.7) and (4.8). For that purpose we have the following lemma. Lemma 4.2. (i) There exists a ρ(n − 1) dimensional (n − 1)-subspace V of Mn (R). (ii) There exists an (n − 1)-subspace V of Sn (R) such that ( 1 if n is even, dim V = n−1 if n is odd. ρ 2 + 1, Proof. (i) As indicated in the introduction, there exists an ρ(n − 1) dimensional (n − 1)-subspace V1 of Mn−1 (R). Then we let V be the subspace of Mn (R) obtained from V1 by appending a row and column of 0’s to each matrix in V1 .

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(ii) The claim is trivial if n is even, so suppose n is odd. As indicated in the introduction, there exists an (n−1)-subspace V1 of Sn−1 (R) of dimension ρ n−1 + 1. Now V is obtained from V1 as in Part (i). 2 Corollary 4.2. Let n ≡ 1 (mod 4). Then the inequalities (4.3) and (4.8) are sharp, and the inequalities (4.2) and (4.7) are sharp provided that n 6= 5, 9. Proof. For n ≡ 1 (mod 4) the maximum of the right-hand sides of (4.2) and (4.7) are ρ(n − 1) − 1 and ρ(n − 1), respectively. Let V be the ρ(n − 1) dimensional (n − 1)-subspace defined in the Proof of Lemma 4.2, Part (i). This shows the sharpness of (4.7) provided that n 6= 5, 9. The sharpness of (4.2) is obtained by considering an odd smooth map from S ρ(n−1)−1 to (n−1) Mn (R) as in the proof of Corollary 4.1 Part (a). +1 The sharpness of (4.3) and (4.8) is proved similarly, using the ρ n−1 2 dimensional (n − 1)-subspace V of Sn (R) in Part (ii) of Lemma 4.2. Remark. In Theorem 4.3 it is possible to replace the assumption that ϕ is an odd smooth map by the assumption that ϕ is an odd continuous map. The proof of the remark is achieved as follows: First approximate ϕ(α) (n−1) arbitrarily by a smooth odd ϕ(α) e (which is in Sn (R) if ϕ(S p ) ⊂ Sn (R)). We can assume that for each α ∈ S p , ϕ(α) e has a simple eigenvalue λ(α) which is the smallest in absolute value among all eigenvalues of ϕ(α) e (and is (n−1) p p also real if ϕ(S ) ⊂ Sn (R)). Clearly λ(−α) = −λ(α) ∀α ∈ S . Let ψ(α) be the projection of ϕ(α) e corresponding to λ(α). Then ψ(α) is a smooth odd function of rank at most 1, and it follows that ϕ(α) e − ψ(α) satisfies the conditions of Theorem 4.3, so (4.2) and (4.3) hold respectively. 5. Existence of certain 2n dimensional subspaces of S2n−1 (R). Let n ≥ 4 be an even integer. It follows from [7] and this paper that d(n, n − 2, R) ≤ n + 1, with equality for n = 4. Now suppose that n ≥ 5 is an odd integer. We do not know if a similar result holds, although it seems plausible. In this section we show that if there exists an n + 1 dimensional subspace L of Sn (R) such that each nonzero matrix in L has rank n − 1 or n, then there is a severe restriction on the possible inertias attained in L. We also derive here additional results on n + 1 dimensional subspaces of n × n matrices where each nonzero matrix has rank n − 1 or n. Let F be any field and let V be an n + 1 dimensional subspace of Mn (F) spanned by A1 , A2 , . . . , An+1 . Sometimes we find it convenient to identify e∈ PFn with Fn+1 and PV with PFn . That is, we identify α e ∈ PFn and A t e V := PV with lines spanned by 0 6= α = (α1 , α2 , . . . , αn+1 ) and 0 6= A in P ] F n+1 and V respectively. Let A(α) = n+1 e with A(α). i=1 αi Ai . We identify α

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Given any 0 6= y ∈ Fn , let Ly = {A ∈ V : Ay = 0}.

(5.1)

Since dim V ≥ n + 1 it is clear that dim Ly ≥ 1 for every y 6= 0. Let M (y) = [A1 y, A2 y, . . . , An+1 y] ∈ Mn,n+1 (F). P n+1 For α = (α1 , α2 , . . . , αn+1 )t it is clear that i=1 αi Ai y = 0 if and only if M (y)α = 0. It follows that (5.2)

dim Ly = 1 if and only if rank M (y) = n. Observe that if rank M (y) = n then, by Lemma 4.1, ker M (y) is spanned by t (5.3) α(y) = − det M (y)(1) , det M (y)(2) , . . . , (−1)n+1 det M (y)(n+1) . Let T = ye ∈ PFn−1 : α(y) = 0 .

(5.4)

Hence, T consists of those ye for which rank M (y) < n. Given α = (α1 , α2 , . . . , αn+1 ) we consider the homogeneous polynomial ! n+1 X ψ(α) = det αi Ai , i=1

and let (5.5)

e Z(ψ) = the zero set of ψ(α), considered as a subset of PFn .

Lemma 5.1. Let V be an n + 1 dimensional subspace of Mn (C), spanned by A1 , A2 , . . . , An+1 , and such that: (i) ∃A ∈ V such that det A 6= 0. (ii) ∃B ∈ V such that rank B = n − 1 and ker B is spanned by a vector 0 6= u ∈ Cn with dim Lu = 1. Then the following holds: e Let H ⊂ PCn be the irreducible component of the hypersurface Z(ψ) passe ing through B. Then there exists a birational map ϕ : H − − → PCn−1 ,

θ = ϕ−1 : PCn−1 − − → H

defined as follows: P (a) For each α e ∈ H with rank C = n−1, where C = n+1 α) = ye, i=1 αi Ai , ϕ(e where y ∈ Cn is the basis of ker C. (b) For each ye ∈ PCn−1 such that dim Ly = 1, θ(e y) = α e, where α = α(y).

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Proof. The maps ϕ and θ are rational. For y in the neighborhood of u, dim Ly = 1. Hence θ is holomorphic in the neighborhood of u e. Let P β A . The assumptions (ii) imply that ϕ is holomorphic in the B = n+1 i=1 i i e neighborhood of β and ϕ · θ is the identity map (as a rational map). Hence θ : PCn−1 − − → H. Remark. As θ : PCn−1 − − → H is a rational map, it is not holomorphic on a variety of codimension at least 2 (cf. [9, Chapter 4, Section 2]). Hence codim T ≥ 2. Lemma 5.2. Let V be an n + 1 dimensional subspace of Mn (R), such that each nonzero matrix in V has rank n − 1 or n. Then: (i) ∃A ∈ V such that det A 6= 0. (ii) ∃B ∈ V such that rank B = n − 1 and ker B is spanned by a vector 0 6= y ∈ Rn with dim Ly = 1. Proof. By Theorem 2 of [3], V cannot be an (n − 1)-subspace or an nsubspace. Hence V contains nonsingular matrices and singular matrices. Let e ∈ Ve : det A = 0}. (5.6) Ves = {A Let η : Ves → PRn−1 be defined as follows: Suppose that B ∈ V is any singular matrix, and suppose 0 6= y ∈ Rn e = ye. satisfies By = 0. Then let η(B) It is known that the set of singular points of Ves is a proper subvariety of e of Ves Ves . As η(Ves ) = PRn−1 , it follows that there exists a regular point B n−1 e such that Dη(B) has the rank n − 1. Let y ∈ S be such that By = 0. e ≥ 1. Suppose that dim Ly ≥ 2. Then it is clear that dim ker Dη(B) e in Ves has dimension It follows that the dimension of the tangent space to B e ≥ n, implying that dim Vs ≥ n, a contradiction. Lemma 5.3. Let n be odd, n ≥ 3, and let V be an n + 1 dimensional subspace of Sn (R) such that each nonzero matrix in V has rank n − 1 or n. Suppose that V ⊥ = {B ∈ Sn (R) : tr (AB) = 0 ∀A ∈ V } does not contain rank one matrices. Let Ves be as in (5.6). Then, Ves is a smooth connected variety in PRn . (k) n Proof. Let Ak = aij , k = 1, 2, . . . , n + 1 be a basis of V . Let ψ(α) = i,j=1 P P n+1 det e ∈ Ves we have ψ(α) = 0, so A(α) = n+1 i=1 αi Ai . For α i=1 αi Ai has

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rank n − 1. Hence the (n − 1)th compound of A(α), Cn−1 (A(α)), has rank 1. So there exists 0 6= x(α) = (x1 (α), x2 (α), . . . , xn (α))t and = ±1 such that Cn−1 (A(α)) = x(α)xt (α). Let Cn−1 (α) = (cij (α))ni,j=1 . Observe that n X ∂ψ(α) (k) = aij cij (α) = xt (α)Ak x(α), k = 1, 2, . . . , n + 1. ∂αk i,j=1

Since V ⊥ does not contain a rank one matrix, we deduce that Oψ(α) 6= 0. Thus each α e ∈ Ves is a smooth point and the local dimension of Ves at α e is n − 1. Let VC be the complexification of V . Let TC , TR be defined by (5.5) for F = C, R, respectively. By the remark following Lemma 5.1, codimC TC ≥ 2. Since TR = TC ∩ PRn−1 , we have codimR TR ≥ 2. Let Yr = PRn−1 \TR . Then Yr is connected. Using (ii) of Lemma 5.2 we can define H and θ as in Lemma 5.1. Let HR = H ∩ PRn . Then θ(Yr ) is connected in HR , and this implies that θ(Yr ) = HR is connected. To finish the proof it suffices to show that Ves = HR . Since Y r = PRn−1 it follows that for any ye ∈ PRn−1 there exists βe ∈ HR such that A(β)y = 0. Let γ e be an arbitrary point in Ves . Then ∃ 0 6= u ∈ Rn such that A(γ)u = 0. If dim Lu = 1, then γ e ∈ HR . So suppose dim Lu ≥ 2. Then ∃ βe ∈ HR eγ e u , and clearly βe and γ such that A(β) ∈ Lu . Hence β, e∈L e are connected e e e in Lu ⊂ Vs . As Vs is a smooth variety, it follows that γ e ∈ HR . Hence Ves = HR . Recall that the standard inner product in Sn (R) is given by hA, Bi = tr(AB), so with respect to this inner product a matrix A is normalized if and only if tr(A2 ) = 1. Corollary 5.1. Let n and V satisfy the assumptions of Lemma 5.3. Let Vν = {A ∈ V : tr (A2 ) = 1} and Then Vνs

Vνs = {A ∈ Vν : det A = 0}. is a smooth connected hypersurface in Vν .

Proof. Assume that the basis A1 , A2 , . . . , An+1 is orthonormal with respect to the standard inner product in Sn (R). Let Ves , TR and Yr be as in Lemma 5.3. Identify Ves with Vνs /{±I}. It follows from Lemma 5.3 that Vνs is smooth. To show that Vνs is connected, it suffices to show we can connect some A ∈ Vνs to −A in Vνs . Take y ∈ S n−1 such that ye ∈ Yr . Connect y to −y by a path J (in S n−1 ) such that Je ⊂ Yr . For z ∈ S n−1 such that ze ∈ / TR let 1 α ˆ (z) = α(z), kα(z)k2

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where α(z) is given by (5.3). Since α is odd, A(ˆ α(z)), z ∈ J , connects A(ˆ α(y)) to A(ˆ α(−y)) = −A(ˆ α(y)). Theorem 5.1. Let n be odd, n ≥ 3, and let V be an n + 1 dimensional subspace of Sn (R) such that each nonzero matrix in V has rank n − 1 or n. Then each nonzero matrix in V has at least n−1 2 positive and negative eigenvalues. Proof. Let A1 , A2 , . . . , An+1 be a basis of V . We distinguish two cases. Case 1. Suppose that V ⊥ does not contain a matrix of rank one. Let Vνs be as in Corollary 5.1. Then, each matrix in Vνs has rank n − 1, and by the connectedness of Vνs it follows that A and −A have the same inertia for each A ∈ Vνs . It follows that every singular matrix in V \{0} has n−1 2 positive eigenvalues, n−1 negative eigenvalues, and 1 zero eigenvalue. Continuity 2 n−1 yields immediately that each 0 6= A ∈ V has at least 2 positive and negative eigenvalues. Case 2. Suppose now that V ⊥ does contain a rank one matrix. Note that we have dim P(V ⊥ ) = n(n+1) − (n + 1) − 1, while dim PV1,n (R) = n − 1. 2 Hence, for any > 0 there exist A1, , A2, , . . . , An+1, ∈ Sn (R) which satisfy the following three conditions: (i) kAi, − Ai k < , i = 1, 2, . . . , n + 1. (ii) V = span{A1, , A2, , . . . , An+1, } is n + 1 dimensional. (iii) V⊥ does not contain a rank one matrix. For > 0 small enough, we can apply Case 1 and conclude that any nonzero matrix in V has at least n−1 2 positive and negative eigenvalues. Let → 0 and use continuity to finish the proof. 6. Concluding remarks and conjectures. Theorem 6.1. Let 1 ≤ q ∈ Z. Then deg PVn−q,n (C) is odd if n > q and (6.1) n ≡ ±q mod 2dlog2 2qe . Proof. Let 0 6= q ∈ Z. Let µ(q) = dlog2 2|q|e and 0 ≤ ν(q) ∈ Z be the largest integer such that 2ν(q) divides q. Then ν(q) + 1 ≤ µ(q). Clearly if ` ≡ m 6≡ 0 (mod 2µ ) for some 1 ≤ µ ∈ Z then ν(`) = ν(m). Let δk,n := deg PVk,n (C). We claim that for n satisfying (6.1)

(6.2)

q−1 X qˆ + j 2j + 1 ν(δn−q,n ) = ν −ν , q−j j j=0

qˆ = ±q.

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Here for any real x and 0 ≤ k ∈ Z we let x x(x − 1) . . . (x − k + 1) = . k! k In particular (6.3)

x+k−1 −x , = (−1)k k k

k = 1, . . . .

Observe that for j ≤ q − 1 n+j (n + j)(n + j − 1) . . . (n − q + 2j + 1) = . q−j 1 · 2 . . . (q − j) For n satisfying (6.1) it follows that n + ` ≡ ±q + ` 6≡ 0 (mod 2µ(q) ) if −q + 1 ≤ ` ≤ q − 1. Hence (6.2) holds. Our theorem is equivalent to the statement that ν(δn−q,n ) = 0 if (6.1) holds. We first consider the case n ≡ −q (mod 2µ(q) ). Then q−1 X −q + j 2j + 1 ν(δn−q,n ) = ν −ν q−j j j=0

=

q−1 X

q−j

ν (−1)

j=0

=

q−1 X j=0

X q−1 2(q − j) − 1 2j + 1 − ν q−j j j=0

ν

2(q − j) − 1 q−j−1

−

q−1 X j=0

ν

2j + 1 j

= 0.

We now consider the case n ≡ q (mod 2µ(q) ). In this case it is enough to show the identity q+j q−1 Y q−j (6.4) = 1, q = 1, . . . . 2j+1 j=0

j

We prove the above identity by induction on q. For q = 1 (6.4) trivially holds. Assume that (6.4) holds for q = m ≥ 1. Let q = m + 1. Use the identity m+1+j m+1+j m+j , j = 0, 1, . . . , m, = m+1−j m−j m+1−j to deduce (6.4) for q = m + 1 from q = m. Corollary 6.1. Let n > q and let (6.1) hold. Then q+1 d(n, n − q, R) = d(n, n − q, C) = + 1. 2

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Corollary 6.2. Let n ≥ q ≥ 2 and assume that (6.1) holds. Then any q+1 dimensional subspace U of Sn (R) contains a nonzero matrix with an 2 eigenvalue of multiplicity q at least. Furthermore, there exists an q+1 −1 2 dimensional subspace V ⊂ Sn (R) such that each eigenvalue of 0 6= A ∈ V has a multiplicity less than q. Proof. For n = q U = Sn (R) and the eigenvalue 1 of Iq has multiplicity q. Assume that V ⊂ Sq (R) does not contain Iq . Then each eigenvalue of 0 6= A ∈ V has multiplicity less than q. Assume that q < n. Then the claim of the corollary follows from Corollary 6.1 and Proposition 3.3. Clearly δn−1,n = n is odd if and only if n is odd. As d(n, n − 1, R) = ρs (n) + 1 it follows that d(n, n − 1, R) > d(n, n − 1, C) = 2 if n is even. Conjecture 6.1. Let n > q ≥ 2 and assume that (6.1) does not hold. Then δn−q,n is even. Furthermore d(n, n − q, R) > d(n, n − q, C). For q = 2, 3, 4, 5 it is straightforward to check that δn−q,n is odd if and only if (6.1) holds. Finally, we return to the sequence {d(n, n − 2, R)}∞ n=3 and state two questions. The first unknown number in this sequence is d(5, 3, R). As we have seen in Section 3, d(5, 2, R) = 7, so d(5, 3, R) ≤ 7. Question 1. Is d(5, 3, R) = 6? We think that the answer is yes. In [6] it is shown that d(5, 3, R) ≤ 6. Let L be the 5 dimensional subspace of S5 (R) spanned by the matrices     1 1 1 1 −1 1 1 1 −1 1  1  1 −1 −1 −1 −1  1 −1 −1 1        , 1 −1 −1 1 1 B1 =  1 −1 −1 −1 −1  , B2 =     1 −1 −1 −1   1 −1 −1 1 −1 1  −1 1 −1 1 1 1 −1 1 1 1     −1 1 1 −1 −1 1 1 −1 1 −1  1 −1  1 −1 −1 −1 1 1 1  1         1 −1 −1 1  , B4 =  −1 −1 −1 1 1  B3 =  1 ,  −1  1 −1 1 −1 1 −1  1 1 −1  −1 1 1 −1 1 −1 1 1 −1 −1   1 1 1 1 −1  1 1 −1 1 −1    1 1 1  B5 =  .  1 −1  1 1 1 −1 −1  −1 −1 1 −1 −1 A local minimization procedure using Matlab seems to indicate that every nonzero matrix in L has rank 4 or 5. This has yet to be confirmed by other means, but if it is correct, then d(5, 3, R) = 6.

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Note that for every field F we have (6.5)

d(n, 1, F) ≥ d(n, 2, F) ≥ · · · ≥ d(n, n − 2, F) ≥ d(n, n − 1, F).

Question 2. Is there strict inequality everywhere in (6.5) if F = R? Acknowledgement. We would like to thank H. Tracy Hall for his help in the computations using Maple associated with the example that is discussed in Theorem 2.1. References [1] J.F. Adams, Vector fields on spheres, Ann. of Math., 75 (1962), 603-632, MR 25 #2614, Zbl 0112.38102. [2] J.F. Adams, P.D. Lax and R.S. Phillips, On matrices whose real linear combinations are nonsingular, Proc. Amer. Math. Soc., 16 (1965), 318-322, MR 31 #3432, Zbl 0168.02404. [3] L.B. Beasley and T.J. Laffey, Linear operators on matrices: The invariance of rank-k matrices, Linear Algebra Appl., 133 (1990), 175-184, MR 91f:15008, Zbl 0721.15006. [4] L.B. Beasley and R. Loewy, Rank preservers on spaces of symmetric matrices, Linear and Multilinear Algebra, 43 (1997), 63-86, MR 98k:15002, Zbl 0890.15002. [5] M.A. Berger and S. Friedland, The generalized Radon-Hurwitz numbers, Compositio Mathematica, 59 (1986), 113-146, MR 88a:11035, Zbl 0599.10014. [6] S. Friedland and A. Libgober, Generalizations of the odd degree theorem and applications, to appear in Israel J. Math.. [7] S. Friedland and R. Loewy, Spaces of symmetric matrices containing a nonzero matrix of bounded rank, Linear Algebra Appl., 287 (1999), 161-170, MR 99j:15024, Zbl 0939.15009. [8] S. Friedland, J.W. Robbin and J.H. Sylvester, On the crossing rule, Comm. on Pure and Appl. Math., 37 (1984), 19-37, MR 85b:15010, Zbl 0523.15011. [9] P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley & Sons, 1978, MR 80b:14001, Zbl 0408.14001. [10] J. Harris and L.W. Tu, On symmetric and skew-symmetric determinantal varieties, Topology, 23 (1984), 71-84, MR 85c:14032, Zbl 0534.55010. ¨ [11] A. Hurwitz, Uber die komposition der quadratischen formen, Math. Ann., 88 (1923), 1-25. [12] P.D. Lax, The multiplicity of eigenvalues, Bull. Amer. Math. Soc., 6 (1982), 213-215, MR 83a:15009, Zbl 0483.15006. [13] J. Radon, Lineare Scharen orthogonaler matrizen, Abh. Math. Sem. Univ. Hamburg, 1 (1923), 1-14. Received March 5, 2001 and revised August 6, 2001 Department of Industrial Engineering Technion – Israel Institute of Technology E-mail address: [email protected] Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago Chicago, IL 60607-7045

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E-mail address: [email protected] Department of Mathematics Technion – Israel Institute of Technology E-mail address: [email protected] The second author was a Lady Davis Visiting Professor at the Technion in the Fall of 2000. The research of the third author was supported by the Fund for the Promotion of Research at the Technion.


ON THE COMMUTATOR FORMULA OF A SPLIT BN-PAIR ¨lle Genet Gwenae The purpose of this note is to prove in an elementary way and with geometric considerations that the Levi decomposition of a finite group with a split BN-pair of characteristic p (a prime integer) implies the commutator formula.

1. Introduction. Split BN-pairs are defined by a set of axioms devised to study finite reductive groups (see [CR, 69.1], [R, 3.1]). Such finite groups G are assumed to contain subgroups B, N and S ⊆ W := N/T , with T := N ∩ B, such that among other things, T / N , (W, S) is a Coxeter system, B can be written as a semi-direct B = U T where U is a Sylow p-subgroup of G and B ∩ B w0 = T (w0 the longest element of W ). Recalling the geometric representation of W in an euclidean space E, we denote by Φ the associated root system (a subset of the unit sphere), by ∆ ⊆ Φ+ the fundamental and positive systems of Φ, so that the fundamental reflections {sδ , δ ∈ ∆} correspond with the elements of S (see [CR, 64.28] or [B]). For δ ∈ ∆, one defines Xδ := U ∩ U w0 sδ . An elementary consequence of the axioms of BN-pairs ([CR, 69.2]) is: (R0) For arbitrary γ ∈ Φ, written as γ = w(δ) with δ ∈ ∆, w ∈ W , one may define Xγ :=wXδ . This only depends on γ and Xγ 6= {1}. A nice extra property, satisfied in all finite reductive groups, is the commutator formula:

(1) ∀α, β ∈ Φ, α 6= ±β, [Xα , Xβ ] ⊆ Xiα+jβ , i > 0, j > 0, iα + jβ ∈ Φ . This is useful to check the crucial property of Levi decompositions, which is that for all subsets I ⊆ ∆, U ∩ U wI is normal in U (wI the longest element of WI ). In particular, (2)

∀δ ∈ ∆, U ∩ U sδ / U.

N. Tinberg ([T]) has shown that (2) implies in an elementary way the Levi decomposition. Here, we show a little more, namely that (1) follows from (2), without using the classification of BN-pairs or the case of rank 2 ([FS]). Note that (2) 177

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is always satisfied when the root subgroups Xδ have order p. Our arguments are easy considerations in the reflection representation space. We recall two elementary properties of BN-pairs. Denote by lS the length in W relative to S. (R1) (See [CR, 69.2] or [R, 3.3] by iteration.) There is at least one sequence such that Φ+ = {γ1 , . . . , γN }, N = |Φ+ | and U = Xγ1 . . . XγN . If γ ∈ Φ+ , then X−γ ∩ U = {1} = U ∩ U w0 . (R2) ([R, 2.3]) If w ∈ W and s ∈ S is such that lS (ws) = lS (w) + 1, then U ∩ U ws ⊆ U ∩ U s . The author thanks M. Cabanes for his suggestions.

Notation. If E 0 is a subset of E, we denote X(E 0 ) := Xγ , γ ∈ E 0 ∩ Φ and if A is a nonempty subset of W , we denote VA := ∩w∈A U w . 2. First results. The following proposition shows how to express VA , A ⊆ W, A 6= ∅, with the root groups Xγ , γ ∈ Φ. For this, we need: Lemma 1. Let m ≥ 1, let γ1 , . . . , γm be pairwise distinct in Φ+ and for every i, 1 ≤ i ≤ m, let xi ∈ Xγi \{1}. Let w ∈ W, then x1 x2 . . . xm ∈ U w if and only if for every i, w(γi ) ∈ Φ+ . Proof. If for all i, w(γi ) ∈ Φ+ , then for all i, Xw(γi ) =w Xγi ⊆ U because of (R1). We prove the converse by induction on lS (w). If w = 1, this is obvious. If w = sδ , δ ∈ ∆, one must check δ is none of the γi ’s. Suppose on the contrary that δ = γi0 . Then sδ (γi ) ∈ Φ+ for any i 6= i0 . So x1 . . . xi0 −1 ∈ U sδ and xi0 +1 . . . xm ∈ U sδ by the ‘if’. Then xi0 ∈ U sδ ∩Xδ ⊆ U sδ ∩U w0 sδ = {1}, a contradiction. For an arbitrary w ∈ W with lS (w) ≥ 1, we can write w = w0 sδ for 0 w ∈ W, δ ∈ ∆ and lS (w0 ) = lS (w)−1. Then U ∩U w ⊆ U ∩U sδ by (R2). We have just seen that, for all i, sδ (γi ) ∈ Φ+ . Set γi0 := sδ (γi ), x0i := xi s˙δ ∈ Xγi0 where s˙δ ∈ N is a representative of sδ . The result comes by applying the 0 induction hypothesis to x01 x02 . . . x0m ∈ U w . Proposition 1. For all nonempty subset A ⊆ W, we have VA = X(ΨA ) where ΨA := {γ ∈ Φ | ∀w ∈ A, w(γ) ∈ Φ+ }. Moreover, ΨA = {γ ∈ Φ, Xγ ⊆ VA }. Proof. Suppose 1 ∈ A, so that ΨA ⊆ Φ+ . Take a sequence γ1 , . . . , γN ∈ Φ+ as in (R1), so that ΨA = {γi1 , . . . , γim } for 1 ≤ i1 < · · · < im ≤ N . Then any x ∈ VA ⊆ U, can be written as x = x1 . . . xN with xi ∈ Xγi . The above lemma implies that xi = 1 whenever γi 6∈ ΨA . Then VA = X(ΨA ). So it makes clear that ΨA ⊆ {γ ∈ Φ, Xγ ⊆ VA }. But if γ ∈ Φ is such that

ON THE COMMUTATOR FORMULA

179

Xγ ⊆ VA then γ ∈ Φ+ by (R0) and (R1). Lemma 1 tells us γ ∈ ΨA . The last equality is proved. Now, suppose 1 6∈ A. Take a ∈ A and define A0 := Aa−1 ⊆ W. The above applies to A0 and gives our claim since VA = (VA0 )a and ΨA = a−1 ΨA0 . The following is a slight adaptation to our needs of the standard theorem about separation of convex sets. Lemma 2. Let C be a closed convex cone of E such that C ∩−C = {0}. Let Ψ be a finite set of E \ {0} and let F be the subset of the dual E ∨ of E, F = {f ∈ E ∨ : f (C\{0}) ⊆ R∗+ and 0 6∈ f (Ψ)}. Then C = ∩f ∈F f −1 (R∗+ ) ∪ {0}. Proof. One inclusion is clear. For the other, let x 6∈ C, so x 6= 0. We will find a linear form f such that f (C\{0}) ⊆ R∗+ , f (x) < 0 and 0 6∈ f (Ψ). One may assume that x has norm 1. If r > 0, let Cr (resp. Dr ) be the open cone generated by the elements of the unit sphere at distance < r from the elements of C (resp. from x). For r sufficiently small, we clearly have Cr ∩ Dr = Cr ∩ −Cr = ∅. Taking a hyperplane separating the open convex sets Cr and Dr , it is clear that such a hyperplane must contain 0. This tells us that the set G := {f ∈ E ∨ , f (C\{0}) ⊆ R∗+ , f (x) < 0} is not empty. Now, it is easy to see that G is an open set of the dual E ∨ and that the set {f ∈ G, 0 ∈ f (Ψ)} 6= G since Ψ is finite. So the outcome is still nonempty. Thus our claim. The consequence of the above on “convex” subsets of Φ and corresponding subgroups of G is as follows. Proposition 2. Let C be a closed convex cone of E such that C∩−C = {0}. Denote A(C) := {w ∈ W | C ∩ Φ ⊆ w−1 (Φ+ )}. Then (i) C ∩ Φ = ΨA(C) with the notation of Proposition 1. (ii) X(C) = VA(C) . Proof. (i) The inclusion C ∩ Φ ⊆ ΨA(C) = ∩w∈A(C) w−1 (Φ+ ) is trivial. On the other hand, we have C ∩ Φ = ∩f ∈F f −1 (R∗+ ) ∩ Φ where the set F consists of the f ∈ E ∨ such that f (C\{0}) ⊆ R∗+ and 0 6∈ f (Φ) (Lemma 2). But for f ∈ F, f −1 (R∗+ ) ∩ Φ defines an order relation on Φ ([B]) and therefore some unique fundamental system. By transitivity of W on fundamental systems, there is wf ∈ W such that f −1 (R∗+ ) ∩ Φ = wf −1 (Φ+ ). Obviously, wf ∈ A(C). So we get the reverse inclusion we seek. (ii) By (i) above, we have X(C) = X(ΨA(C) ). But Proposition 1 tells us that the latter is indeed VA(C) . Corollary 1. Let C and D be two closed convex cones of E such that C ∩ −C = D ∩ −D = {0}. Then, X(C) ∩ X(D) = X(C ∩ D). Proof. With Proposition 2 (ii), we clearly have X(C) ∩ X(D) = VA(C) ∩ VA(D) = VA(C)∪A(D) , and the latter is X(ΨA(C)∪A(D) ) by Proposition 1. But

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ΨA(C)∪A(D) = ΨA(C) ∩ΨA(D) that is again equal to C∩D∩Φ by Proposition 2 (i). We get our claim. 3. The commutator formula. Theorem 1. If G is a finite group with a split BN-pair satisfying the hypothesis (2), then it satisfies the commutator formula (1). Proof. For all finite subset Y ⊆ E, we denote by C(Y ) the closed convex cone generated by Y . We use the abbreviation C(y, z) := C({y, z}). Suppose G satisfies (2). Denote Uδ = U ∩ U sδ when δ ∈ ∆. Let α ∈ ∆ and β ∈ Φ+ , α 6= β. We have Uα = V{1,sα } = X(Ψ{1,sα } ) and Xβ ⊆ Uα by Proposition 1 because Ψ{1,sα } = Φ+ \{α}. But Φ+ \{α} = Φ ∩ C(Φ+ \{α}) because any positive linear combination of positive roots that is again a root is a positive one and α is a fundamental root, so is a minimal positive linear combination of positive roots. Therefore, Uα = X(C(Φ+ \{α})). Besides, since Xα ⊆ U , [Xα , Xβ ] ⊆ [Xα , Uα ] ⊆ Uα by hypothesis (2). On the other hand, both Xα and Xβ are subgroups of X(C(α, β)), so [Xα , Xβ ] ⊆ X(C(α, β)). The closed convex cones C(Φ+ \{α}) and C(α, β) satisfy C ∩ −C = {0} by property of positive roots ([B]) then Corollary 1 implies [Xα , Xβ ] ⊆ Uα ∩ X(C(α, β)) = X(C(Φ+ \ {α}) ∩ C(α, β)) = X(C(α, β) \ {α}). So we get, for all α ∈ ∆, β ∈ Φ+ \{α}, (3)

[Xα , Xβ ] ⊆ X(C(α, β)\{α}).

If α ∈ ∆, β ∈ Φ− \{−α}, we also have (3) by applying the above to w0 sα (α) ∈ ∆, w0 sα (β) ∈ Φ+ and conjugating the corresponding subgroups of G by sα w0 ∈ W . So we have (3) for any α ∈ ∆, β ∈ Φ\{±α}. If α, β are any non-proportional arbitrary roots, there is w ∈ W such that w(α) ∈ ∆, then again we have (3). Now, exchanging the rôles of α and β, (3) becomes [Xα , Xβ ] ⊆ X(C(α, β)\{β}). It is clear that each set Φ ∩ C(α, β)\{α} , Φ ∩ C(α, β)\{β} , and Φ ∩ C(α, β)\{α, β} is of the type Φ ∩ C where C is a closed convex cone such that C ∩ −C = {0} (draw a picture in the plane generated by α and β). Then Corollary 1 gives X(C(α, β)\{α}) ∩ X(C(α, β)\{β}) = X(C(α, β)\{α, β}). We get our claim.

References [B]

N. Bourbaki, Groupes et Algèbres de Lie, Ch. 4, 5 et 6, Hermann, 1968, MR 39 #1590, Zbl 0186.33001.

[CR] C.W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Vol. 2, Wiley Interscience, 1988, MR 90g:16001, Zbl 0634.20001. [FS]

P. Fong and G. Seitz, Groups with a BN -pair of rank 2, I & II, Inventiones Math., 21 (1973), 1-57; 24 (1974), 191-239, MR 50 #7335, Zbl 0295.20048, Zbl 0295.20049.

ON THE COMMUTATOR FORMULA

181

[R]

F. Richen, Modular representations of split BN -pairs, Trans. Amer. Math. Soc., 140 (1969), 435-460, MR 39 #4299, Zbl 0181.03801.

[T]

N.B. Tinberg, The Levi decomposition of a split BN -pair, Pacific J. Math., 91(1) (1980), 233-238, MR 82m:20009, Zbl 0452.20047.

Received January 8, 2001 and revised May 4, 2001. ´matiques - Case 7012 UFR de Mathe ´ Denis Diderot - Paris 7 Universite 2, Place Jussieu 75251 Paris Cedex 05 France E-mail address: [email protected]


` ¨ THEOREM FOR MINIMAL PHRAGMEN–LINDEL OF SURFACE EQUATIONS IN HIGHER DIMENSIONS Chun-Chung Hsieh, Jenn-Fang Hwang, and Fei-Tsen Liang Here we prove that if u satisfies the minimal surface equation in an unbounded domain which is properly contained in a half space of Rn , with n ≥ 2, then the growth rate of u is of the same order as that of the shape of Ω and the boundary value of u.

1. Introduction. Consider the minimal surface equation div T u = 0, where Tu = p

5u 1 + | 5 u|2

and

5 u = (ux1 , . . . , uxn ).

In 1965, Nitsche [7] announced the following result: “Let Ωα ⊂ R2 be a sector with angle 0 < α < π. If u satisfies the minimal surface equation with vanishing boundary value in Ωα , then u ≡ 0”. Hwang extends this result in [4], [5], [6] and proves that, in an unbounded domain Ω properly contained in the half plane in R2 , if u satisfies the minimal surface equation, then, the growth property of u is determined completely by the shape of Ω and the boundary value of u. In this respect, the Phragmèn-Lindelöf theorem for the minimal surface equation is better than that for the Laplace equation. (Indeed, if u satisfies the Laplace equation in an unbounded domain Ω, the growth property of u cannot be determined completely by the shape of Ω and the boundary data of u alone (cf. [10]).) The purpose of this paper is to generalize the two-dimensional PhragmènLindelöf theorems in [4], [5] and [6], to higher dimensions. In §2, we review the statements of the Phragmèn-Lindelöf theorem of [4], [5] and [6]. The higher-dimensional version is similar in content, but proof is different. In §3, based on an argument of [2], we established the suitable comparison principle. In §4, we compute the mean curvature of our comparison function, and use it to finish the proof of our main theorems in §5. 183

184

C.-C. HSIEH, J.-F. HWANG, AND F.-T. LIANG

2. Preliminary. The main purpose of this paper is to generalize the two-dimensional Phragmèn-Lindelöf theorem in [4], [5], [6] to higher dimensions. We may, first of all, recall some results in these papers and consider functions f : [0, ∞) → [0, ∞), f ∈ C 2 ([0, ∞)), f 0 ≡

df (y) > 0, dy

from which we define

f f 00 . (f 0 )2 In particular, for f (y) = y m , m being a positive constant, we have 1 p(f ) = , m which is precisely the reciprocal of the order of f , while for f (y) = ey , we have p(f ) = 0; moreover, in case f grows faster than the exponential function, we can assume p(f ) ≥ − for some small positive constant , essentially (cf. [5, Remark 2.7]). Accordingly, we may proceed to solve the ordinary differential equation in [−1, 1] p(f ) = 1 −

(∗)

2

(1 − p(f ))(h − th0 )(1 + h0 ) + h00 (h2 + t2 ) = 0

with initial values π and h0 (−1) = tan (1 − p(f )) , 2 and then denote its solution, if exists, by hm if f (y) = y m (and hence 1 p(f ) = m ), and by h∞ if f (y) = ey (and hence p(f ) = 0). In general, (∗) and (∗∗) cannot be solved explicitly; but, for some specific m, its solution can be written out explicitly. For example, we have (∗∗)

h(−1) = 0

h2 =

1 − t2 , 2

and also p h ∞ = 1 − t2 . It is useful to know some interesting properties of hm , 0 < m ≤ ∞, in the following: Lemma 1 ([6]). For 1 < m, m0 ≤ ∞ and t ∈ (−1, 1), then we have (i)

hm (t) > hm0 (t),

whenever

m > m0 ,

hm (t) < hm (t0 ),

whenever

|t| > |t0 |.

and (ii)

` ¨ PHRAGMEN-LINDEL OF

185

The Phragmèn-Lindelöf theorems in [5], [6] can now be formulated as follows. Theorem 2. Let Ω ⊆ {(x, y) ∈ R2 | − ay m < x < ay m , y > 0} ⊆ R2 be an unbounded domain, where a and m are positive constants, m ≥ 1. Let u ∈ C 2 (Ω) ∩ C 0 (Ω) and suppose that ( div T u ≥ 0 in Ω u ≤ ay m hm ( ayxm ) on ∂Ω. p Then we have u ≤ ay m hm ( ayxm ) ≤ ay m h∞ ( ayxm ) = a2 y 2m − x2 in Ω. Theorem 2*. Let Ω ⊆ {(x, y) ∈ R2 | − aeby < x < aeby , y > 0} be an unbounded domain where a, b are positive constants. Let u ∈ C 2 (Ω) ∩ C 0 (Ω) and suppose that ( div T u ≥ 0 in Ω √ 2 2by 2 on ∂Ω. u≤ a e −x √ Then we have u ≤ a2 e2by − x2 in Ω. Theorem 3. Let f ∈ C 2 ([0, ∞)), f > 0, f 0 > 0 in (0, ∞) and p(f ) ≥ p0 , where p0 is a negative constant, and let f1 ∈ C 0 ([0, ∞)) and f1 > 0 in (0, ∞). For a given unbounded open domain Ω ⊂ {(x, y) ∈ R2 | − f1 (y) < x < f1 (y), y > 0}, and u ∈ C 2 (Ω) ∩ C 0 (Ω) with ( div T u ≥ 0 p u ≤ a f 2 − x2 where f 2 ≥

(a2 −1)(2−p0 ) 2 f (a2 −(1−p0 )) 1

in Ω on ∂Ω,

and a is a positive constant satisfying a2 − 1 + p0 > 0.

Then, we have p u ≤ a f 2 − x2

in

Ω.

Remark. In Theorem 3, since p0 < 0 and a > 0, we have (a2 − 1)(2 − p0 ) a2 − 1 = (2 − p0 ) > 2. (a2 − (1 − p0 )) a2 − (1 − p0 ) Thus, in case u ≤ 0 on ∂Ω, our estimates are not good enough since we use worse boundary conditions, whereas the best estimates remain unknown. These theorems will be generalized to higher dimensions in §5.

186


3. A comparison principle. To establish the higher-dimensional Phragmèn-Lindelöf theorem, we shall need the following comparison principle. Lemma 4. Let Ω be an unbounded domain in Rn , and let u, v ∈ C 2 (Ω) ∩ C 0 (Ω). Suppose that ( div T u − div T v ≥ C in Ω, u≤v on ∂Ω, where C is a positive constant. Then we have u ≤ v in Ω. Proof. The idea of proof is analogous to that of [2]. Suppose that this lemma fails to hold. There then exists a positive constant such that Ω0 = {x ∈ Ω | u(x) > v(x) + } is not empty; by Sard’s theorem, we may further assume that ∂Ω0 ∩ Ω is smooth. For every R > 0, set BR = {x ∈ Rn |

|x| < R},

0

ΩR = BR ∩ Ω , ΓR = ∂BR ∩ ∂ΩR , and |ΓR | = the Hausdorff (n − 1) − dimensional measure of ΓR . Also, let I (1)

tan−1 (u − v − )(T u − T v) · ν

g(R) = ∂ΩR

Z

tan−1 (u − v − )(T u − T v) · ν

= ΓR

where ν is the unit outward normal of ∂ΩR . Then we have ZZ (5u − 5v) · (T u − T v) (2) g(R) = 1 + (u − v − )2 Ω ZZR + tan−1 (u − v − )(div T u − div T v). ΩR

Since the integrand of the right-hand side of (1) is nonnegative, Fubini’s theorem tells us that g 0 (R) exists for almost all R > 0, and whenever it


187

exists, we have, by (2), Z (5u − 5v) · (T u − T v) 0 (3) g (R) = 1 + (u − v − )2 ΓR Z + tan−1 (u − v − )(div T u − div T v) Γ Z R ≥C tan−1 (u − v − ), (by assumption) ΓR Z C ≥ tan−1 (u − v − ) |T u − T v|, 2 ΓR (since |T u| < 1 and |T v| < 1) C ≥ g. 2 Since g is an increasing function of R and g ≥ 0, it is easy to see that Lemma 4 holds in the case that g ≡ 0. If, on the other hand, g 6≡ 0, there would exist a positive constant R0 such that g(R) > 0 for all R ≥ R0 , and hence, for every R > R0 , in virtue of (3) Z R 0 C g (r) dr ≥ (R − R0 ), g(r) 2 R0 i.e., R C log g(r) ≥ (R − R0 ), 2 R0 and therefore, c

g(R) ≥ g(R0 ) e 2 (R−R0 ) .

(4) However, we have, by (1)

Z g(R) ≤ ΓR

π · 2 ≤ π|ΓR |; 2

since ΓR ⊂ ∂BR , this yields a positive constant C1 completely determined by n such that g(R) ≤ C1 Rn−1 , which contradicts (4) and yields the truth of Lemma 4.

Remark. The above proof works well and so the lemma is valid if v = +∞ on some parts of ∂Ω.

188


4. An estimation of the growth of solutions. Henceforth, we will denote Ω as an unbounded domain in Rn , n ≥ 2 , such that, for some f ∈ C 2 ([0, ∞)), f > 0, f 0 > 0 and f 00 > 0 in (0, ∞), we have Ω ⊂ {(x, y) ∈ R2 | − f (y) < x < f (y), y > o} × Rn−2 ⊂ Rn . We shall extend the results in §2 to such a domain Ω. First, for every positive constant y0 , since f > 0, f 0 > 0 and f 00 > 0 in (0, ∞), it is easy to see that there exists a positive constant δ1 , depending on y0 , such that {(f (y), y) ∈ R2 | y > 0}∩{(x, y) ∈ R2 | y0 −y+ δ21 x2 = 0} has exactly one point. And also, {(f (y), y) ∈ R2 | y > 0} ∩ {(x, y) ∈ R2 | y0 − y + 2δ x2 = 0} has exactly two points, say (f (y1 ), y1 ) and (f (y2 ), y2 ) with 0 < y1 < y2 , for all δ with 0 < δ < δ1 . In general, we have y1 = y1 (y0 , δ), y2 = y2 (y0 , δ) and also limδ→0 y1 (y0 , δ) = y0 . From now on, we always assume that the positive constant δ is less than the above δ1 . To apply Lemma 4 to estimate the speed of growth of solutions in Ω, we may consider comparison functions of the following form 1

Fy0 ,δ

A(f 2 (y) − x2 ) 2 , = y0 − y + 2δ x2

which is defined on δ 2 n−2 2 , Ωy0 ,δ = Ω ∩ (x, y) ∈ R y0 − y + x > 0, 0 < y < y1 × R 2 where δ, y0 , and A are positive constants. We first proceed to calculate the mean curvature of Fy0 ,δ . For convenience of computation, we may set 1

F = A · P 2 Q−1 , where P = f 2 (y) − x2 and Q = y0 − y + 2δ x2 . We observe that (5)

div T F =

=

(1 + Fx2 )Fyy − 2Fx Fy Fxy + (1 + Fy2 )Fxx 3

(1 + Fx2 + Fy2 ) 2 ( F12 +

Fx2 Fyy ) F2 F ( F12

Fy Fxy 1 F F + (F2 3 F ( FFx )2 + ( Fy )2 ) 2

− 2 FFx +

+

Denoting I =

Fy2 Fxx Fx2 Fyy Fx Fy Fxy + −2 2 2 F F F F F F F

and II =

Fxx Fyy + 3, F3 F

Fy2 Fxx ) F2 F

.


189

we note that the numerator in (5) is the sum of these two expressions and we shall treat them seperately. For the first expression, we have 2 ! 2 ! Fy2 Fy Fy Fx Fx Fx2 + + 2 ∂x + I = 2 ∂y F F F F F F Fy F x Fy Fx Fy ∂x + −2 F F F F2 Fy2 Fy Fx Fy Fy Fx Fx2 = 2 ∂y + 2 ∂x −2 2 ∂x F F F F F F = I* + I** where ! Fy2 1 Py2 Q2y 1 Px2 Q2x − + + 2 − + 2 2 P 2 Q2 F 2 P2 Q Fx Fy Qx Qy 1 Px Py −2 2 − + , F 2 P2 Q2

F2 I* = x2 F

and Fy2 1 Pxx Qxx Qyy 1 Pyy − + 2 − 2 P Q F 2 P Q Fx Fy 1 Pxy Qxy −2 2 − . F 2 P Q

F2 I** = x2 F

By a direct computation, I* =

−1 1 (Py Qx − Px Qy )2 , 4 P 2 Q2

while Qyy 1 Px Qx 2 1 Pyy − − I** = 2P Q 2 P Q 2 Qy 1 Py 1 Pxx Qxx + − − 2P Q 2 P Q Qy Qxy 1 Px Qx 1 Py 1 Pxy −2 − − − . 2P Q 2P Q 2 P Q

Thus, in particular, we have I* ≤ 0.

(6) As for I** and II, we recall that P = f 2 (y) − x2

and

Q = y0 − y +

δ 2 x , 2

190


and hence Py = 2f 0 f,

Px = −2x,

Qx = δx,

Qy = −1;

moreover Pxx = −2,

Pxy = 0,

Pyy = 2(f 00 f + f 0 f 0 )

and Qxx = δ,

Qxy = 0,

Qyy = 0.

Thus, we have I** =

1 2 00 [x (f f + f 02 ) − f 2 f 02 ] P3 (−2) δ 2 02 2 00 02 0 + 2 −δx (f f + f ) + f f + f f P Q 2 1 + [δ 2 x2 (f 00 f + f 02 ) − 1 − 2δf f 0 ] − δQ−3 , P Q2

and also ! Fy2 Fy Fx2 ∂x + 2 + ∂y + 2 F F F " Qy Q2 1 Px Qx 1 Py = 2 ∂x − + ∂y − A P 2P Q 2P Q # Qy 2 1 Px Qx 2 1 Py + − + − 2P Q 2P Q 2 1 2 00 Q [f (f f − 1) − x2 (f f 00 + f 02 )] = 2 A P3 2 1 0 2 2 2 2 (δ x + 1) . + 2 (2f f − δf + 3δx ) + P Q P Q2 Thus the numerator of div T F is Q2 II = 2 A P

Fx F

(7) I + II = I* + I** + II 1 Q2 = 3 x2 (f f 00 + f 02 ) − f 2 f 02 + 2 ((f 2 (f f 00 − 1) − x2 (f f 00 + f 02 ))) P A 1 + 2 2δx2 (f f 00 + (f 0 )2 ) − 2f f 0 − δf 2 f 02 P Q Q2 2 0 2 + 2 (−δf + 2f f + 3δx ) A 1 Q2 2 2 2 2 00 0 2 0 + δ x (f f + (f ) ) − 1 − 2δf f + 2 2 (δ x + 1) P Q2 A 1 − δQ−3 − P −2 Q−2 (Px Qy − Qx Py )2 . 4


191

We want to choose δ and A to make the third bracket of the right-hand side of (7) negative. For this, substituting the expression for Q in the bracket and rewriting it as (8) III = −1 +

2 A2

2 δx2 y0 − y + (1 + δ 2 x2 ) + δ 2 x2 (f 00 f + (f 0 )2 ) − 2δf f 0 . 2

For any given λ, 0 < λ < 41 , if we take δ such that ) ( λ λy0 λ λ (9) , , , 0 < δ < inf min f 2 f f 2 f (f 00 f + (f 0 )2 ) 21 y∈(0,y1 ) √ and A = 4 2y0 , then we have 1 λy0 2 1 + λ2 2 2 III ≤ −1 + y + (1 + λ ) + λ ≤ −1 + + λ2 < 0. 0 4(2y0 )2 2 4 As of the second bracket of the right-hand side of (7), to make it negative, it clearly suffices to make the following expression negative, namely 2 3 2 Q 0 (10) f f + δx − f f 0 + δx2 (f f 00 + (f 0 )2 ). IV = A2 2 For this, we observe that, as x2 < f 2 in Ω and f > 0, f 0 > 0 and f 00 > 0 in (0, ∞), 3 2 3 2 3 f2 0 0 0 δx + f f ≤ δf + f f = f f 1 + δ 0 , 2 2 2 ff while −f f 0 + δx2 (f f 00 + (f 0 )2 ) ≤ −f f 0 + δf 2 (f f 00 + (f 0 )2 ) f 2 (f f 00 + (f 0 )2 = −f f 0 1 − δ ff0 and furthermore, if we require that λf f 0 λf f 0 (9*) δ < inf min , f 2 (f f 00 + (f 0 )2 ) 2f 2 y∈(0,y1 ) it follows from (9) that (11)

Q2 1 ≤ . 2 A 4

And also, we have 2 −1 0 0 Q 0 1 IV ≤ f f (1 + λ) + λ − 1 ≤ f f (1 + λ) + λ − 1 ≤ ff . 2 A 4 4 Thus, the condition that f > 0, f 0 > 0 in (0, ∞) ensures us of the negativity of (10). It remains to consider the first bracket of the right-hand side of (7).

192


To make it negative, it suffices to make negative the following expression V = x2 (f f 00 + (f 0 )2 ) − f 2 f 02 +

Q2 2 f (f f 00 − 1), A2

or, in view of (11), 1 V ≤ x2 (f f 00 + (f 0 )2 ) − f 2 f 02 + f 2 (f f 00 ). 4 Recall that for given function f as above, we define (12)

p(f ) = 1 −

f f 00 . (f 0 )2

For §5, and from now on, we assume that −1 ≤ p(f ) ≤ 1, following a remark concerning p(f ) for our interesting functions, [5, Remark 2.7]. And so, in 1 particular for f (y) = (y + z)m , p(f ) = m and for f (y) = aeby , p(f ) = 0 where z, m > 1, a, and b are positive constants, and also it is easy to see α that for f (y) = ey , with α > 1, then p(f ) −→ 0− as y −→ +∞. 1 Rewriting (12) in terms of p(f ), and noticing that ( 34 + p4 ) × ( 2−p ) ≥ 16 , we have ! p 3 1 2 0 2 2 0 2 2 4 + 4 2 V ≤ (2 − p)(f ) x − ≤ (2 − p)(f ) x − f , (13) f 2−p 6 and so if we assume furthermore that 1 1 2 (14) Ω ⊆ (x, y) ∈ R | − √ f (y) < x < √ f (y), y > 0 × Rn−2 ⊆ Rn , 6 6 then V ≤ 0 and get the following conclusion about √ the estimation of our p 4 2y0 comparison function: If F = f 2 (y) − x2 with δ as in our (y0 − y + 2δ x2 ) assumptions, (9), (9*), then div T F ≤ 0 in Ωy0 ,δ , where Ω is assumed as in (14). Now we state what we achieved as follows: Proposition 5. Let f1 : [0, ∞) −→ [0, ∞), and f1 ∈ C 2 ([0, ∞)) with f1 > 0, f1 0 > 0, f1 00 > 0 on [0, ∞), and −1 ≤ p(f1 ) ≤ 1. Suppose that Ω ⊆ {(x, y) ∈ R2 | − f1 (y) < x < f1 (y), y > 0} × Rn−2 ⊆ Rn and that u ∈ C 2 (Ω) ∩ C 0 (Ω) and for some constant β with 0 < β < 1 satisfying ( div T u ≥ 0 in Ω √ p 2 2 u ≤ 4 2β 6f1 (y) − x on ∂Ω. q √ Then u ≤ 4 2 6f1 2 (y) − x2 in Ω. √

1

√ (f 2 (y) − x2 ) 2 Proof. Set f (y) = 6f1 (y) and define F (x, y) = 4 2y0 as (y0 − y + 2δ x2 ) above, where y0 > 0 and δ > 0, small as in (9) and (9*) and we also require


193

(2 − 2β)y0 . Then following the computation as above, in particular β(f (y1 ))2 that of (7), and also noticing that the first three brackets of the right-hand side of (7) are negative in Ωy0 ,δ as shown above, it is easy to see that that δ ≤

div T F =

( F12 +

Fy2 Fxx ) F2 F

− 2 FFx

1 (1 F3

Fy Fxy F F

+ ( F12 +

Fx2 Fyy ) F2 F

3

+ | 5 F |2 ) 2

and ! Fy2 Fxx 1 Fx2 Fyy Fx Fy Fxy 1 δ 2 −3 + −2 + + < −δ y0 − y + x , F2 F2 F F F F F2 F2 F 2 when (x, y) is close to {(x, y) ∈ R2 | y0 − y + 2δ x2 = 0}. √ And so, noticing that P = f 2 (y) − x2 , Q = y0 − y + 2δ x2 and A = 4 2y0 , when (x, y) is close to {(x, y) ∈ R2 | y0 − y + 2δ x2 = 0}, we have ! 5F 2 −3 1 −3 2 div T F ≤ −δQ + F2 F 2 ! 1 5P −3 Q2 5Q −3 2 ≤ −δQ + − A2 P 2 P Q 2 ! 1 Q −3 Q4 2 ≤ −δ + 5 P − 5Q 2 A P 2P 4 −3 Q 1 Q2 Q 2 2 2 ≤ −δ + | 5 P | + | 5 Q| − 5 P · 5Q A2 P 4 P2 P −3 δ ≤ − (1 + δ 2 x2 ) 2 , 2 −Q 5 P · 5Q ≥ 0 and | 5 Q|2 = 1 + δ 2 x2 . since P But the bounded connected component of the closure of {(x, y) ∈ R2 | − f1 (y) < x < f1 (y), y > 0}∩{(x, y) ∈ R2 | y0 −y + 2δ x2 > 0}, which is denoted as Ω∗ , is compact. And we have Ωy0 ,δ ⊆ Ω∗ × Rn−2 , and so, there exists a positive constant c, such that   in Ωy0 ,δ , div T F ≤ −c F ≥u on ∂Ωy0 ,δ ∩ {(x, y) ∈ R2 | y0 − y + 2δ x2 > 0} × Rn−2 ,   F = +∞ on ∂Ωy0 ,δ ∩ {(x, y) ∈ R2 | y0 − y + 2δ x2 = 0} × Rn−2 . Now, by Lemma 4, we have u ≤ F in Ωy0 ,δ , which is √

1

(6f12 (y) − x2 ) 2 u(x, y, z1 , . . . , zn−2 ) ≤ 4 2y0 (y0 − y + 2δ x2 )

in Ωy0 ,δ .

194


Now, let δ −→ 0 and then let y0 −→ +∞, we get the conclusion of the proof. 5. Phragm` en-Lindel¨ of theorem in higher dimensions. First, let’s generalize Theorem 2 as follows: Theorem 6. Let Ω ⊆ {(x, y) ∈ R2 | −ay m < x < ay m , y > 0}×Rn−2 ⊆ Rn be an unbounded domain, where m ≥ 1 and a are positive constants. Let u ∈ C 2 (Ω) ∩ C 0 (Ω) and suppose that ( div T u ≥ 0 in Ω x m u ≤ ay hm ( aym ) on ∂Ω. p Then we have u ≤ ay m hm ( ayxm ) ≤ ay m h∞ ( ayxm ) = a2 y 2m − x2 in Ω. Proof. For every given positive constant > 0, we now set f (x, y) = x a(y + )m+ , F (x, y, z1 , z2 , . . . , zn−2 ) = a(y + )m+ hm+ ( a(y+) m+ ), where (x, y, z1 , z2 , . . . , zn−2 ) ∈ Ω. Since −ay m < x < ay m , y > 0, we have ym x ≤ a(y + )m+ (y + )m+ −→ 0 as y −→ +∞. x By Lemma 1, hm+ ( a(y+) m+ ) −→ hm+ (0) uniformly as y −→ +∞, and so it p is easy to see that there exists a large constant y3 such that F (x, y) ≥ 200a2 y 2m − x2 for y ≥ y3 . Next by [6, Theorem 2], setting f (y) = a(y+)m+ , t = fx(y) and recalling 1 that p(f ) = m+ , we have

div T F 0 2 3 (f ) = (1 + | 5 F |2 )− 2 f h00m+ 0 0 2 00 2 2 · (1 − p(f ))(hm+ − thm+ )((hm+ ) + 1) + hm+ (hm+ + t ) + 0 2 . (f )

Since hm+ (t) is the solution of (∗) and (∗∗) with p(f ) = 3

div T F = (1 + | 5 F |2 )− 2

1 m+ ,

we have

(f0 )2 h00m+ · 0 2 f (f )

and so obviously that div T F < 0 on Ω0 where Ω0 = Ω ∩ {(x, y, z1 , . . . , zn−2 ) ∈ Rn |0 < y < y3 }. And so, there exists a positive constant C1 > 0 such that div T F ≤ −C1

on Ω0 .


195

But, noticing that p √ p x ≤ a2 y 2m − x2 ≤ 4 2β 6a2 y 2m − x2 , u ≤ ay hm m ay for some constant β < 1 on ∂Ω, by Proposition 5, we also have p √ p u ≤ 4 2 6a2 y 2m − x2 ≤ 200 · a2 y 2m − x2 in Ω \ Ω0 . m

By Lemma 4, we have u ≤ F

in Ω0 .

In conclusion, we have u ≤ F and let −→ 0, the proof is done.

in Ω,

As a corollary of Theorem 6, we state a generalization of Nitsche’s theorem [7] as follows. Corollary. Let Ω = {(x, y) ∈ R2 | − ay < x < ay, y > 0} × Rn−2 be a wedge domain, where a is a positive constant. Let u ∈ C 2 (Ω) ∩ C 0 (Ω) and suppose that ( div T u = 0 in Ω, u=0 on ∂Ω. Then u ≡ 0 in Ω. Proof. Apply Theorem 6 to functions u and −u, we have u ≤ 0 in Ω and −u ≤ 0 in Ω, and so u ≡ 0 as claimed. Next, let’s generalize Theorem 2* as follows: Theorem 6*. Let Ω ⊆ {(x, y) ∈ R2 | − aeby < x < aeby , y > 0} × Rn−2 ⊆ Rn , where a, b are positive constants. Let u ∈ C 2 (Ω) ∩ C 0 (Ω) and suppose that ( div T u ≥ 0 in Ω √ 2 2by 2 u≤ a e −x on ∂Ω. √ Then we have u ≤ a2 e2by − x2 in Ω. Proof. The proof is similar to that of Theorem 6. For every > 0, we consider the following function x p F (x, y, z1 , z2 , . . . , zn−2 ) = ae(b+)y h∞ = a2 e2(b+)y − x2 ae(b+)y with (x, y, z1 , . . . , zn−2 ) ∈ Ω. Since −aeby < x < aeby , y > 0, we have x aeby (b+)y ≤ (b+)y −→ 0 as y −→ +∞ ae ae and notice that F = ae(b+)y (1 −

1 x2 )2 . a2 e2(b+)y

196


Hence, there exists a positive constant y3 > 0 such that p F ≥ 200a2 e2by − x2 for y ≥ y3 . Next, by [6, Theorem 2], and setting f (y) = ae(b+)y , t = noticing that p(f ) = 0, we have

x f (y)

and

0 2 3 (f ) div T F = (1 + | 5 F |2 )− 2 f h00∞ 0 0 2 00 2 2 · (h∞ − th∞ )(h∞ + 1) + h∞ (h∞ + t ) + 0 2 f 00 0 2 3 (f ) h∞ = (1 + | 5 F |2 )− 2 . f (f0 )2 So, we have div T F < 0 on Ω0 , where Ω0 = Ω ∩ {(x, y, z1 , . . . , zn−2 ) ∈ Rn | 0 < y < y3 }, and so there exists a positive constant C1 > 0 such that

div T F ≤ −C1

in Ω0 .

Finally, by Proposition 5, notice that p √ p u ≤ a2 e2by − x2 ≤ 4 2β 6a2 e2by − x2 , for some constant β < 1 on ∂Ω, we also have √ p u ≤ 4 2 6a2 e2by − x2 ≤ F

in Ω \ Ω0 .

So, by Lemma 4, we have u ≤ F

on Ω0 ,

and so obviously, we get u ≤ F and let −→ 0, the proof is finished.

in Ω,

Finally, let’s generalize Theorem 3 as follows: Theorem 7. Let f1 ∈ C 2 ([0, ∞)) with f1 ≥ 0, f10 > 0, and f100 ≥ 0 in (0, ∞) such that p(f1 ) ≥ p0 , where p0 is a constant with −1 ≤ p0 ≤ 0. Suppose that Ω ⊆ {(x, y) ∈ R2 | − f1 (y) < x < f1 (y), y > 0} × Rn−2 ⊆ Rn and u ∈ C 2 (Ω) ∩ C 0 (Ω) satisfying ( div T u ≥ 0 in Ω p 2 2 u ≤ a f (y) − x on ∂Ω, 1 2 0) 2 f1 and a is a positive constant with a2 −1+p0 > 0. where f = (a(a−1)(2−p 2 −1+p ) 0 p Then we have u ≤ a f 2 (y) − x2 in Ω.


197

Proof. Forpany given > 0, we define f (y) = ey f (y + ) and F (x, y, z1 , . . . , zn−2 ) = a f2 (y) − x2 , then there exists y3 > 0 such that 1

1

F ≥ a(e2y f 2 (y) − x2 ) 2 ≥ (200 · a2 f 2 (y) − x2 ) 2

for y > y3 .

Computing the mean curvature of F and using the definition, p(f ) = 1 − f f00 , we have (f 0 )2

3

−3

div T F = (1 + | 5 F |2 )− 2 (f2 − x2 ) 2 · a(f0 )2 [(a2 − 1)(2 − p(f ))x2 − f2 (a2 − 1 + p(f ))] − af2 . Obviously, we have f2 (y) ≥ f 2 (y) ≥

(a2 − 1)(2 − p0 ) 2 (a2 − 1)(2 − p(f )) 2 f f (y), (y) ≥ 1 (a2 − 1 + p0 ) (a2 − 1 + p(f )) 1

and so, we have 3

div T F ≤ −a(1 + | 5 F |2 )− 2 f2 < 0

in Ω,

and by compactness, there exists a positive constant C1 > 0 such that div T F ≤ −C1

in Ω1 = {(x, y, z1 , z2 , . . . , zn−2 ) ∈ Ω|y < y3 }.

But by Proposition 5, notice that p √ p u ≤ a f 2 (y) − x2 ≤ 4 2β 6a2 f 2 − x2 , for some constant β < 1 on ∂Ω, we also have p u ≤ 200a2 f 2 (y) − x2 ≤ F

in Ω \ Ω1 .

By Lemma 4, we have u ≤ F in Ω1 . In conclusion, we have u ≤ F in Ω, and then let −→ ∞. We thus finish the proof.

References [1] R. Finn, Equilibrium Capillary Surfaces, Springer-Verlag, New York-Berlin-Heidelberg-Tokyo, 1986, MR 88f:49001, Zbl 0583.35002. [2] R. Finn and J. Hwang, On the comparison principle for capillary surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA, Math., 36 (1989), 131-134, MR 90h:35099, Zbl 0684.35007. [3] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1983, MR 86c:35035, Zbl 0562.35001. [4] J. Hwang, Phragmèn-Lindel¨ of theorem for the minimal surface equation, Proc. Amer. Math. Soc., 104 (1988), 825-828, MR 89j:35016, Zbl 0787.35012. [5]

, Growth property for the minimal surface equation in unbounded domains, Proc. Amer. Math. Soc., 121 (1994), 1027-1037, MR 94j:35019, Zbl 0820.35010.

198

[6]


, Catenoid-like solutions for the minimal surface equation, Pacific J. Math., 183 (1998), 91-102, MR 99d:58041, Zbl 0905.35029.

[7] J.C.C. Nitsche, On new results in the theory of minimal surface, Bull. Amer. Math. Soc., 71 (1965), 195-270, MR 30 #4200, Zbl 0135.21701. [8]

, Vorlesungen u ¨ber Minimalfl¨ achen, Springer-Verlag, Berlin-Heidelbert-New York, 1975, MR 56 #6533, Zbl 0319.53003.

[9] R. Osserman, A Survey of Minimal Surfaces, Van Nostrand-Reinhold, New York, 1969, MR 41 #934, Zbl 0209.52901. [10] M.H. Protter and H.F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, N.J., 1967, MR 36 #2935, Zbl 0153.13602. Received February 27, 2001. Mathematics Department Academia Sinica Nan-kang Taipei, 115, Taiwan E-mail address: [email protected] Mathematics Department Academia Sinica Nan-kang Taipei, 115, Taiwan E-mail address: [email protected] Mathematics Department Academia Sinica Nan-kang Taipei, 115, Taiwan E-mail address: [email protected]


REMARK ON THE RATE OF DECAY OF SOLUTIONS TO LINEARIZED COMPRESSIBLE NAVIER–STOKES EQUATIONS Takayuki Kobayashi and Yoshihiro Shibata We consider the Lp −Lq estimates of solutions to the Cauchy problem of linearized compressible Navier–Stokes equation. Especially, we investigate the diffusion wave property of the compressible Navier–Stokes flows, which was studied by D. Hoff and K. Zumbrum and Tai-P. Liu and W. Wang.

1. Introduction. In this paper, we consider the Cauchy problem of the following linearized compressible Navier-Stokes equations: (1.1)

ρt + γdiv v = 0

in (0, ∞) × Rn ,

vt − α∆v − β∇div v + γ∇ρ = 0

in (0, ∞) × Rn ,

ρ|t=0 = ρ0 ,

in Rn ,

v|t=0 = v0

where v = v(t, x) = T (v1 (t, x), . . . , vn (t, x)) a vector valued unknown function, ρ = ρ(t, x) is a scalar valued unknown function; t is time variable; we denote the spatial point of n-dimensional Euclidian Space Rn by x = (x1 , . . . , xn ) (n = 2); ρt =

∂ρ , ∂t

vt =

n X ∂vj , div v = ∂xj j=1

n X ∂2v 2, ∂x j j=1 ∂ρ ∂ρ ∇ρ = ,..., ; ∂x1 ∂xn

∂v , ∂t

∆v =

ρ0 and v0 are given initial data; α and γ are positive constants and β a nonnegative constant. Concerning the decay property, asymptotically, the solution decomposed into sum of two parts under the influence of a hyperbolic aspect and a parabolic aspect. One of which dominates in Lp for 2 5 p 5 ∞, the other 1 5 p < 2. For p = 2, the time asymptotic behavior of solutions is similar to the solution of pure diffusion problem. Namely, the decay at the rate of the solution is similar to the solution of a linear, second order, strictly parabolic system with L1 initial data. Moreover, the decay order of the term that is given by the convolution of Green functions of diffusion equation and 199

200

T. KOBAYASHI AND Y. SHIBATA

wave equation is better than the solution to pure diffusion system. On the other hand, for p < 2, the asymptotically dominant term reflects the spreading effect of the solution operator for the standard multi-dimensional wave equation. As a result, the solution may grow without bound in Lp for p < 2. This result was investigated by D. Hoff and K. Zumbrun [2, 3] in the case of the Navier-Stokes system describing the compressible fluid flow, and Y. Shibata [6] in the case of the linear viscoelastic equation. D. Hoff and K. Zumbrun [2, 3] considered the linear effective artificial viscosity system as the first approximation of the compressible Navier-Stokes equation in several space dimension. The Green function of this system is written exactly by the convolution of the Green function of diffusion equation and wave equation. In view of this, they gave the pointwise estimate and Lp estimate of the Green function in [2, 3], and Lp estimate for the solutions to the nonlinear problem in [2]. But, the Green function of the system (1.1) and the linear viscoelastic equation is not written exactly. Tai-P. Liu and W. Wang [4] gave the pointwise estimate for the solutions to the system (1.1) and the nonlinear problem in odd multi-dimension case, and Y. Shibata [6] gave the Lp estimate for the solution to the linear viscoelastic equations by directly using Fourier transform method. The main difference of the structure to the solutions between (1.1) or effective artificial viscosity system and linear viscoelastic equation is the Riesz kernel Rj (x) = F −1 [ξj /|ξ|] (x), where F −1 denotes the Fourier inverse transform. The Green matrix of the system (1.1) and effective artificial viscosity system includes the Riesz kernel. Since the convolution operator u → Rj ∗ u is not bounded from L1 to L1 and from L∞ to L∞ , if we consider L1 or L∞ estimate, then these features will lead to a great deal of cancellation in the convolution operator of the Green function. D. Hoff and K. Zumbrun [2] overcame this difficulty by applying the weak version of the Paley-Wiener theorem to the general, symmetrizable, hyperbolic-strictly parabolic systems. In this paper, we shall estimate directly using Fourier transform method in [6]. In particular, we shall detect the cancellation in the Green function. 2. Main results. First of all, we shall introduce the solution operator of (1.1). Applying the Fourier transform with respect to x = (x1 , . . . , xn ), (1.1) is reduced to the following ordinary differential equation with parameter ξ = (ξ1 , . . . , ξn ) ∈ Rn :

(2.1)

 dˆ ρ   dt (t, ξ) + iγξ · vˆ(t, ξ) = 0, dˆ v 2 ˆ(t, ξ) + βξ (ξ · v ˆ(t, ξ)) + iγξ ρˆ(t, ξ) = 0, dt (t, ξ) + α|ξ| v   ρˆ(0, ξ) = ρˆ0 (ξ), vˆ(0, ξ) = vb0 (ξ),

LINEARIZED COMPRESSIBLE NAVIER–STOKES EQUATIONS

201

where Z

−ix·ξ

e

u ˆ(t, ξ) =

Z u(t, x)dx,

Rn

ubj (ξ) =

e−ix·ξ uj (x)dx.

Rn

By (2.1) we have ( 2 d ρˆ ρ 2 2 ˆ(t, ξ) = 0, (t, ξ) + (α + β)|ξ|2 dˆ dt (t, ξ) + γ |ξ| ρ dt2 (2.2) ρˆ(0, ξ) = ρb0 (ξ), ρˆt (0, ξ) = −iγξ · vb0 (ξ). The characteristic equation corresponding to the (2.2) is (2.3)

λ2 + (α + β)|ξ|2 λ + γ 2 |ξ|2 = 0.

The roots λ± (ξ) of (2.3) are given by the formula p (2.4) λ± (ξ) = −A |ξ|2 ± |ξ|4 − B 2 |ξ|2 , where A = (α + β)/2, B = 2γ/(α + β). When |ξ| 6= 0, B, the solution of (2.2) is given by the formula (2.5) ρˆ(t, ξ) =

λ+ (ξ)eλ− (ξ)t − λ− (ξ)eλ+ (ξ)t eλ+ (ξ)t − eλ− (ξ)t ρb0 (ξ) − iγ ξ · vb0 (ξ). λ+ (ξ) − λ− (ξ) λ+ (ξ) − λ− (ξ)

Since λ+ (ξ) = λ− (ξ) when |ξ| = B, as the solution of (2.2), when B/2 < |ξ| < 2B, we use the following formula I (z + |ξ|2 )ezt 1 (2.6) ρˆ(t, ξ) = dz ρb0 (ξ) 2πi Γ z 2 + (α + β)|ξ|2 z + γ 2 |ξ|2 I ezt iγ dz ξ · vb0 (ξ), + 2πi Γ z 2 + (α + β)|ξ|2 z + γ 2 |ξ|2 where Γ is a closed path containing λ± (ξ) and contained in {z ∈ C | Rez 5 −c0 } and c0 is a positive number such that max

(2.7)

B 5|ξ|52B 2

Reλ± (ξ) 5 −2c0 .

Also, by (2.1) we have ( (2.8)

+ α|ξ|2 vˆ(t, ξ) = fˆ(t, ξ), vˆ(0, ξ) = vb0 (ξ), dˆ v dt (t, ξ)

where ξ fˆ(t, ξ) = iγ

dˆ ρ 2 β (t, ξ) + γ ρˆ(t, ξ) . dt

Therefore, by (2.5) and (2.8), the solution of (2.6) given by the formula:

202


when |ξ| = 6 0, B,

(2.9)

−α|ξ|2 t

Z

t

e−α|ξ|

2 (t−s)

fˆ(s, ξ) ds ! eλ+ (ξ)t − eλ− (ξ)t −α|ξ|2 t =e vb0 (ξ) − iγξ ρb0 (ξ) λ+ (ξ) − λ− (ξ) ! λ+ (ξ)eλ+ (ξ)t − λ− (ξ)eλ− (ξ)t ξ (ξ · vb0 (ξ)) −α|ξ|2 t + , −e λ+ (ξ) − λ− (ξ) |ξ|2

vˆ(t, ξ) = e

vb0 (ξ) +

0

and when B/2 < |ξ| < 2B, I ezt iγξ 2 dz ρb0 (ξ) vˆ(t, ξ) = e−α|ξ| t vb0 (ξ) − 2πi Γ z 2 + (α + β)|ξ|2 z + γ 2 |ξ|2 I (z + |ξ|2 )ezt ˆ0 (ξ)) 1 −α|ξ|2 t ξ (ξ · v dz − e . + 2 2 2 2 2πi Γ z + (α + β)|ξ| z + γ |ξ| |ξ|2 Let ϕ0 (ξ), ϕM (ξ) and ϕ∞ (ξ) be functions in C ∞ (Rn ) such that

(2.10)

( 1 ϕ0 (ξ) = 0

|ξ| 5 B/2, √ |ξ| = B/ 2,

( 1 ϕ∞ (ξ) = 0

|ξ| = 2B, √ |ξ| 5 2B,

ϕM (ξ) = 1 − ϕ0 (ξ) − ϕ∞ (ξ). Put (2.11)

E0 (t) = (E0,ρ (t), E0,v (t)), E∞ (t) = (E∞,ρ (t), E∞,v (t)), E0,ρ (t)(ρ0 , v0 )(x) = F −1 [ϕ0 (ξ)ˆ ρ(t, ξ)] (x), E0,v (t)(ρ0 , v0 )(x) = F −1 [ϕ0 (ξ)ˆ v (t, ξ)] (x), E∞,ρ (t)(ρ0 , v0 )(x) = F −1 [(ϕM (ξ) + ϕ∞ (ξ))ˆ ρ(t, ξ)] (x), E∞,v (t)(ρ0 , v0 )(x) = F −1 [(ϕM (ξ) + ϕ∞ (ξ))ˆ v (t, ξ)] (x).

√ √ Noting that ϕM (ξ) = 1 for B/ 2 5 |ξ| 5 2B and ϕM (ξ) = 0 for |ξ| = 2B or |ξ| 5 B/2, by (2.10) and (2.11) we see that (ρ(t, x), v(t, x)) = E0 (t)(ρ0 , v0 )(x) is a solution of (1.1). The main purpose of the paper is to show the following two theorems.


203

Theorem 2.1 (L1 − L∞ and L1 − L1 estimate of E0 (t)). (1) For any t > 0, we have k∂tj ∂xα E0,ρ (t)(ρ0 , v0 )kL∞ (Rn ) −

“

5 Cj,α,n (1 + t)

j+|α| 3n−1 + 2 4

”

kρ0 kL1 (Rn ) + kv0 kL1 (Rn ) ;

k∂tj ∂xα E0,v (t)(ρ0 , v0 )kL∞ (Rn ) −

“

5 Cj,α,n (1 + t)

j+|α| 3n−1 + 2 4

−

+ Cj,α,n (1 + t)

“

j+|α| n + 2 2

”

kρ0 kL1 (Rn ) + kv0 kL1 (Rn )

”

kv0 kL1 (Rn ) .

Here and hereafter, we write ∂j ∂ |α| α , ∂ = , α x ∂tj ∂x1 1 . . . ∂xαnn α = (α1 , . . . , αn ), |α| = α1 + · · · + αn , ∂tj =

CA,B,... means the constant depending on A, B, . . . . (2) For any t > 0, we have k∂tj ∂xα E0 (t)(ρ0 , v0 )kL1 (Rn ) j+|α| 5 Cj,α,n (1 + t)q(n)− 2 kρ0 kL1 (Rn ) + kv0 kL1 (Rn ) , where  n − 1 q(n) = n 4  4

if n = 3 and n is an odd number, if n = 2 and n is an even number.

Remark. The estimate (1) is better than [3, Theorem 1.2] when n = 2, j = 0 and |α| = 0. Theorem 2.2 (L1 − L1 and L∞ − L∞ estimate of E∞ (t)). Let p = 1 or ∞. For any t > 0, we have k∂tj ∂xα E∞,ρ (t)(ρ0 , v0 )kLp (Rn ) −ct −(j−k) 5 Cj,α,n e Ck t kρ0 k (2k+|α|−1)+ n + kρ0 kW |α| (Rn ) Wp (R ) p h i + Cj,α,n e−ct Ck t−(j−k) kv0 kW 2k+|α| (Rn ) + kv0 kW |α| (Rn ) ; p

p

k∂tj ∂xα E∞,v (t)(ρ0 , v0 )kLp (Rn ) h i 5 Cj,α,n e−ct Ck t−(j−k) kρ0 kW 2k+|α| (Rn ) + kρ0 kW |α| (Rn ) p p h i −(j−k) −ct − 12 + Cj,α,n e (1 + t ) Ck t kv0 kW 2k+|α| (Rn ) + kv0 kW |α| (Rn ) . p

p

204


Here and hereafter, we put K + = max(K, 0) and     X Wpk (Rn ) = u ∈ Lp (Rn ) | kukWpk (Rn ) = k∂xα ukLp (Rn ) < ∞ .   |α|5k

Y. Shibata [6] gave the Lp − Lq type estimates for the solution to the linear viscoelastic equation: ( vtt − ∆v − ∆vt = 0 in [0, ∞) × Rn , (2.12) v(0) = v0 , vt (0) = v1 in Rn . The solution of (2.12) are representated by the Fourier transform as follows: When |ξ| = 6 0, 2 vˆ(t, ξ) =

eλ+ (ξ)t − eλ− (ξ)t λ+ (ξ)eλ− (ξ)t − λ− (ξ)eλ+ (ξ)t vb0 (ξ) + vb1 (ξ), λ+ (ξ) − λ− (ξ) λ+ (ξ) − λ− (ξ)

where λ± (ξ) =

−|ξ|2 ±

p

|ξ|4 − |ξ|2 2

and when 1 < |ξ| < 4, I I 1 (z + |ξ|2 )ezt 1 ezt vˆ(t, ξ) = dz v b (ξ) + dzˆ v1 (ξ), 0 2πi γ z 2 + |ξ|2 z + |ξ|2 2πi γ z 2 + |ξ|2 z + |ξ|2 where γ is a closed path containing λ± (ξ) and contained in {z ∈ C | Rez 5 −c0 } and c0 is a positive number such that max Reλ± (ξ) 5 −2c0 .

15|ξ|54

The difference of the structure to the solutions between (1.1) and (2.12) is the Riesz kernel Rj (x) = F −1 (ξj /|ξ|)(x). The Green matrix of the solution of (1.1) includes the Riesz kernel (cf. (2.9)). Since the convolution operator u → Rj ∗ u is bounded from Lp to Lp for 1 < p < ∞, the following theorems directly follow from [6, Theorems 2.1 and 2.2]. Theorem 2.3 (Lp − Lq estimate of E0 (t)). (1) Let M be the positive number = 1 and let 1 5 p 5 q 5 ∞, (p, q) 6= (∞, ∞), (1, 1). Then, for any t ∈ [0, M ], we have k∂tj ∂xα E0 (t)(ρ0 , v0 )kLq (Rn ) 5 Cn,p,q,j,α,M kρ0 kLp (Rn ) + kv0 kLp (Rn ) . (2) Let 1 5 p 5 2 5 q 5 ∞. For any t > 0, we have k∂tj ∂xα E0 (t)(ρ0 , v0 )kLq (Rn ) −

5 Cn,p,q,j,α (1 + t)

“ “ n 2

1 − 1q p

” ” j+|α| + 2

kρ0 kLp (Rn ) + kv0 kLp (Rn ) .


205

Remark. (1) The estimate (1) in Theorem 2.1 is better than the estimate (2) in Theorem 2.3 with (p, q) = (1, ∞). (2) By Theorem 2.1 and Theorem 2.2, the estimate (1) in Theorem 2.3 also holds when (p, q) = (1, 1) or (∞, ∞). Theorem 2.4 (Lp − Lp estimate of E∞ (t)). Let 1 < p < ∞. For any t > 0, we have k∂tj ∂xα E∞,ρ (t)(ρ0 , v0 )kLp (Rn ) −N −ct 2 t kρ0 k (2j+|α|−N −2)+ n + kρ0 kW |α| (Rn ) 5 Cj,α,p,N e (R ) Wp p N + Cj,α,p,N e−ct t− 2 kv0 k (2j+|α|−N −1)+ n + kv0 k (|α|−1)+ Wp

(R )

Wp

(Rn )

;

k∂tj ∂xα E∞,v (t)(ρ0 , v0 )kLp (Rn ) −ct −N 2 5 Cj,α,n,N e t kρ0 k (2j+|α|−N −1)+ n + kρ0 k (|α|−1)+ n Wp (R ) Wp (R ) N + Cj,α,n,N e−ct t− 2 kv0 k (2j+|α|−N )+ n + kv0 k (|α|−2)+ n . Wp

(R )

Wp

(R )

3. Proof of Theorem 2.1 (1). To prove Theorem 2.1 (1), we put " # λ− (ξ)t − λ (ξ)eλ+ (ξ)t λ (ξ)e + − (3.1) L11 (t, x) = F −1 ϕ0 (ξ) (x), λ+ (ξ) − λ− (ξ) " # λ+ (ξ)t − eλ− (ξ)t e L12 (t, x) = −iγF −1 t ξ ϕ0 (ξ) (x), λ+ (ξ) − λ− (ξ) L21 (t, x) = t L12 (t, x), L22 (t, x) = K1 (t, x) + K2 (t, x) − K3 (t, x), h i 2 K1 (t, x) = F −1 e−α|ξ| t ϕ0 (ξ) (x)I, I is unit matrix, " # λ+ (ξ)t − λ (ξ)eλ− (ξ)t ξ ξ λ (ξ)e j + − k K2 (t, x) = F −1 ϕ0 (ξ) (x), λ+ (ξ) − λ− (ξ) |ξ|2 −1 −α|ξ|2 t ξj ξk K3 (t, x) = F e ϕ0 (ξ) (x), |ξ|2 and then, from (2.5) and (2.9) it follows that L11 (t, ·) L12 (t, ·) ρ (3.2) E0 (t)(ρ0 , v0 ) = ∗ 0 , L21 (t, ·) L22 (t, ·) v0

206


where ∗ denotes the spatial convolution. In view of the Young inequality, in order to get Theorem 2.1 (1) it suffices to show that for t > 0 (3.3)

k∂tj ∂xα L11 (t, ·)kL∞ (Rn ) 5 Cj,α,n t−(

j+|α| 3n−1 + 2 ) 4

,

(3.4)

k∂tj ∂xα L12 (t, ·)kL∞ (Rn ) 5 Cj,α,n t−(

j+|α| 3n−1 + 2 ) 4

,

k∂tj ∂xα K1 (t, ·)kL∞ (Rn ) 5 Cj,α,n t

|α| −( n + 2 +j) 2

(3.6)

k∂tj ∂xα K2 (t, ·)kL∞ (Rn ) 5 Cj,α,n t

j+|α| −( 3n−1 + 2 ) 4

(3.7)

k∂tj ∂xα K3 (t, ·)kL∞ (Rn ) 5 Cj,α,n t−( 2 +

(3.5)

n

|α| +j) 2

, ,

.

It is obvious that Z j β −1 −α|ξ|2 t ξ ξ 2 i k ∂ ∂x F ϕ (ξ) (x) 5 C e−α|ξ| t |ξ|2j+|β| dξ e δ − 0 j,β,n ik t 2 |ξ| Rn n

5 Cj,β,n t−( 2 +

|β| +j) 2

,

which show (3.5) and (3.7). In view of (3.1), we put # " λ+ (ξ)t − eλ− (ξ)t e (3.8) ψ(ξ)ϕ0 (ξ) (x), Kψ,0 (t, x) = F −1 λ+ (ξ) − λ− (ξ) " # λ+ (ξ)t − λ (ξ)eλ− (ξ)t − −1 λ+ (ξ)e Kψ,1 (t, x) = F (3.9) ψ(ξ)ϕ0 (ξ) (x), λ+ (ξ) − λ− (ξ) " # λ+ (ξ)t − λ (ξ)eλ− (ξ)t + −1 λ− (ξ)e (3.10) Kψ,2 (t, x) = F ψ(ξ)ϕ0 (ξ) (x), λ+ (ξ) − λ− (ξ) where ψ = ψ(ω) ∈ C ∞ (S n−1 ), S n−1 = { ξ ∈ Rn | |ξ| = 1} and ψ(ξ) = ψ(ξ/|ξ|). By (2.3) and (2.4), we know that (3.11)

λ+ (ξ)λ− (ξ) = A2 B 2 |ξ|2 , λ+ (ξ) + λ− (ξ) = −2A|ξ|2 , λ± (ξ)2 + 2Aλ± (ξ)|ξ|2 + γ|ξ|2 = 0,

and then (3.12)

( Kψ,1 (t, x) = ∂t Kψ,0 (t, x), Kψ,2 (t, x) = −∂t Kψ,0 (t, x) + 2A∆Kψ,0 (t, x).

Therefore, in order to show (3.3), (3.4) and (3.6) it suffices to show the following theorem: Theorem 3.1. Let n = 2. For any t = 0, we have k∂tj ∂xα Kψ,0 (t, ·)kL∞ (Rn )

−

5 Cj,α,n (1 + t)

“

j+|α| 3n−3 + 2 4

”

.


207

To prove this theorem, first of all, we shall estimate Kψ,0 (t, x) near the light cone. Namely, we shall show that for t = max(1, (R/R0 )4 ) and |x| = R0 t “ ” j+|α| − 3n−3 + 2 j α 4 (3.13) , ∂t ∂x Kψ,0 (t, x) 5 Cj,α,n (1 + t) where R is the number appearing in Lemma 3.2, below and R0 is the fixed number such that R0 5 γ/4. To obtain (3.13), we shall use the following lemma concerning the stationary phase method (cf. Vainberg [9, pp. 29-35]): Lemma 3.2. Let g(ω) ∈ C ∞ (S n−1 ), S n−1 = {ξ ∈ Rn | |ξ| = 1}. Then, there exist a R > 1 and a Cg such that Z ir(ˆ x·ω) 5 Cg r− n−1 2 , x ˆ ∈ S n−1 , r = R. e g(ω) dS ω n−1 S

If we put |ξ| = r, we have p λ± (ξ) = −A r2 ± ir B 2 − r2 = λ± (r). Since we may assume that ϕ0 (ξ) = ϕ0 (|ξ|) = ϕ0 (r), by using the polar coordinate we have n Z ∞ λ+ (r)eλ+ (r)t − λ− (r)eλ− (r)t |α|+n−1 1 j α r ϕ0 (r)dr ∂t ∂x Kψ,0 (t, x) = 2π λ+ (r) − λ− (r) 0 Z · ei(ˆx·ω)r|x| (iω)α ψ(ω) dSω , S n−1

where x ˆ = x/|x|. Let > 0 be a number determined later on. Let us consider the case where |x| = R, below. Since r|x| = |x| = R when r = , by Lemma 3.2 Z α i(ˆ x·ω)r|x| 5 Cα (r|x|)− n−1 2 . e (iω) ψ(ω) dS , ω n−1 S √ Noting that ϕ0 (r) = 0 when r = B/ 2 (cf. (2.10)), we have j α ∂t ∂x Kψ,0 (t, x) Z Z ∞ n−1 2 5C rn+j+|α|−2 dr + e−Ar t rn−2+j+|α| (r|x|)− 2 dr . 0

√ If we make the change of variable; r t = s in the last integration and if we use the assumption: |x| = R0 t, then we have “ ” j+|α| + 2 − 3n−3 j α n+j+|α|−1 4 +t . ∂t ∂x Kψ,0 (t, x) 5 Cj,α,n Choose > 0 in such a way that n+j+|α|−1

=t

−

“

j+|α| 3n−3 + 2 4

”

.

208


When |x| = R0 t and t = max(1, (R/R0 )4 ), we see that |x| = R0 t · t

−

“

j+|α| 3n−3 + 2 4

” /(n+j+|α|−1)

= R0 t

n−1 + j+|α| 4 2 n+j+|α|−1

1

= R0 t 4 = R.

Therefore, we have (3.13). Now, we shall show that for t = 1 and |x| 5 R0 t “ ” j+|α| − 3n−3 + 2 j α 4 (3.14) . ∂t ∂x Kψ,0 (t, x) 5 Cj,α,n t If we put f (ξ) =

p

1−

|ξ|2 B −2

2

1 g(s) = − 2 2B

2

= 1 + |ξ| g(|ξ| ),

Z

1

0

√

1 dθ, 1 − θsB −2

then by Taylor’s formula we have N

eλ+ (ξ)t − e−λ− (ξ)t X 1 = λ+ (ξ) − λ− (ξ) `!

(3.15)

sin γ|ξ|t ∂t`

2

` e−A|ξ| t |ξ|2 g(|ξ|2 )t γf (|ξ|)

|ξ|

`=0

−A|ξ|2 t

+e

RN (t, |ξ|),

where RN (t, |ξ|) 1

N +1 3 2 (1 − θ) eiγ|ξ|t+iγ|ξ| g(|ξ| )tθ iγ|ξ|3 g(|ξ|2 )t 0 N +1 3 2 − e−iγ|ξ|t−iγ|ξ| g(|ξ| )tθ −iγ|ξ|3 g(|ξ|2 )t dθ.

1 = 2iγ|ξ|f (|ξ|)N !

Z

N

In fact, 2

2

eλ± (ξ)t = e−A|ξ| t e∓iγ|ξ|f (|ξ|)t = e−A|ξ| t e∓iγ|ξ|t+iγ|ξ|

3 g(|ξ|2 )t

.

Put h(θ) = e±iγ|ξ|

3 g(|ξ|2 )tθ

,

h(k) (θ) =

dk h (θ). dθk

Since h(1) = h(0) + h0 (0) + · · · +

1 (N ) 1 h (0) + N! N!

Z

1

(1 − θ)N h(N +1) (θ) dθ,

0

we have e±iγ|ξ|

3 g(|ξ|2 )t

N 1 = 1 + ±iγ|ξ|3 g(|ξ|2 )t + . . . ±iγ|ξ|3 g(|ξ|2 )t N! Z 1 N +1 1 3 2 + (1 − θ)N e±iγ|ξ| g(|ξ| )tθ ±iγ|ξ|3 g(|ξ|2 )t . N! 0


Since N N iγ|ξ|3 g(|ξ|2 )t eiγ|ξ|t − iγ|ξ|3 g(|ξ|2 )t e−iγ|ξ|t n o N = ∂tN eiγ|ξ|t − e−iγ|ξ|t |ξ|2 g(|ξ|2 )t N = 2i ∂tN sin γ|ξ|t |ξ|2 g(|ξ|2 )t , noting that λ+ (ξ) − λ− (ξ) = −2iγ|ξ|f (|ξ|), we have (3.15). We shall use the following lemma. Lemma 3.3 (cf. Mizohata [5], Evans [1]). Put −1 sin |ξ|t ˆ h(ξ) (x). w(t, x) = F |ξ| Then, for suitable constants aα we have Z X |α|+1 aα t w(t, x) = |z|=1

05|α|5 n−3 2

z α (∂xα h) (x + tz) dS

for odd n = 3; and X

w(t, x) =

aα t|α|+1

05|α|5 n−2 2

Z |z|51

z α (∂xα h) (x + tz) p dz 1 − |z|2

for even n = 2. Regarding (3.15), we put " # −A|ξ|2 t e ` G` (t, x) = F −1 |ξ|2 g(|ξ|2 )t ψ(ξ)ϕ0 (ξ) (x), f (|ξ|) 1 −1 ` sin γ|ξ|t c ω` (t, x) = F ∂t G` (t, ξ) (x). γ |ξ| Since F

−1

` sin γ|ξ|t ˆ ` −1 sin γ|ξ|t ˆ ∂t h(ξ) (x) = ∂t F h(ξ) (x), |ξ| |ξ|

by Lemma 3.3 we have (3.16)

∂tj ∂xβ Kψ,0 (t, x)

=

N X

∂tj ∂xβ ω` (t, x)

`=0

h i 2 + ∂tj ∂xβ F −1 e−A|ξ| t RN (t, |ξ|) (x),

209

210


where (3.17) ∂tj ∂xβ ω` (t, x)  min(`+k,|α|+1) j X X  1X j  `+k   ∂tm (γt)|α|+1 aα   k m γ   k=0 m=0 |α|5 n−3  Z2   X   α+δ α+β+δ j−k  z ∂ ∂ G · (t, x + γtz) dS  ` x t   |z|=1  |δ|=`+k−m     for odd n = 3; min(`+k,|α|+1) = j X X  1X j `+k   ∂tm (γt)|α|+1 aα   k m γ   m=0 k=0 |α|5 n−2  2    α+δ ∂ α+β+δ ∂ j−k G Z  z (t, x + γtz)  X x ` t    p · dz    1 − |z|2  |δ|=`+k−m |z|51   for even n = 2. The following proposition and lemma play an essential role to prove Theorem 3.1. Proposition 3.4 (Shibata-Shimizu [7]). Let α be a number > −n and put α = N + σ − n where N = 0 is an integer and 0 < σ 5 1. Let f (ξ) be a function in C ∞ (Rn − {0}) such that ∂ξγ f (ξ) ∈ L1 (Rn ), γ ∂ξ f (ξ) 5 Cγ |ξ|α−|γ| ,

|γ| 5 N ; ξ 6= 0,

∀

γ.

Then, we have −1 F [f (ξ)](x) 5 Cα,n max Cγ |x|−(n+|α|) ,

x 6= 0,

|γ|5N +2

where Cα,n is a constant depending essentially only on n and α. Lemma 3.5. Let α be a nonnegative number and ψ(t, ξ) be a function such that ψ(t, ·) ∈ C ∞ (Rn − {0}), γ ∂ξ ψ(t, ξ) 5 Cγ |ξ|α−|γ| , Put

∀

t = 0,

ξ 6= 0,

∀

γ,

∀

t = 0.

h i 2 g(t, x) = F −1 e−β|ξ| t ψ(t, ξ) (x),

where β > 0. Then, we have (3.18)

|g(t, x)| 5 Cα,β,n |x|−(α+n) , |g(t, x)| 5 Cα,β,n t

− α+n 2

,

x 6= 0,

t > 0.


211

Moreover, kg(t, ·)kL1 (Rn ) 5 Cα,β,n t−

|α| 2

,

α > 0, t > 0,

Z |g(t, x)|dx 5 Cβ,n,A (1 + log(1 + t)),

α = 0, t > 0.

|x|5At

Proof. By the formula of derivative of composed function (cf. Simader [8, p. 202]):   (3.19)

∂ξγ h(g(ξ))

=

|γ| X ν=1

X

 h(ν) (g(ξ))  

α1 +···+αν =γ |αi |=1

 ∂ξα1 g(ξ) . . . ∂ξαν g(ξ)  ,

we have 

 2 ∂ξγ e−β|ξ| t

=

|γ| X ν=1

2  (βt)ν e−β|ξ| t  

X

α1 +···+αν =γ |αi |=1

 ∂ξα1 |ξ|2 . . . ∂ξαν |ξ|2  .

Since (3.20)

αi M ∂ξ |ξ| 5 Cαi |ξ|M −|αi | , β

2

ξ 6= 0,

2

(t|ξ|2 )M e−β|ξ| t 5 CM,β e− 2 |ξ| t , we have 



|γ|

X 2 2  (βt)ν e−β|ξ| t  (3.21) ∂ξγ e−β|ξ| t 5 Cγ 

α1 +···+αν =γ |αi |=1

ν=1

5 Cγ

|γ| X

X

 |ξ|2ν−(|α1 |+···+|αν |)  

2

(β|ξ|2 t)ν e−β|ξ| t |ξ|−|γ|

ν=1 β

2

5 Cβ,γ |ξ|−|γ| e− 2 |ξ| t ,

ξ 6= 0,

and the Leibniz’s rule we have β 2 γ −β|ξ|2 t (3.22) ψ(t, ξ) 5 Cα,β,γ e− 2 |ξ| t |ξ|α−|γ| , ∂ξ e Therefore, by Proposition 3.4 we have (3.18). By the assumptions, we have Z Z −β|ξ|2 t α − α+n 2 |g(t, x)| 5 C e |ξ| dξ = Ct (3.23) Rn

5 Cα,β,n t−

Rn

α+n 2

.

ξ 6= 0, ∀ γ.

2

e−β|η| |η|α dη

212


By (3.18) and (3.23), we have kg(t, ·)kL1 (Rn ) 5 Cα,β,n t

− α+n 2

Z √ |x|5 t

α

5 Cα,β,n t− 2 , and also Z |g(t, x)| dx 5 Cβ,n t

Z

α > 0,

−n 2

Z dx 5 Cβ,n,A ,

|x|5At

Z

α = 0, t < 1,

|x|5At n

|g(t, x)| dx 5 Cβ,n t− 2

|x|5At

−(α+n) dx √ |x| |x|= t

dx + Cα,β,n

Z

Z √

dx + Cβ,n

|x|5 t

5 Cβ,n,A (1 + log t),

√

|x|−n dx t5|x|5At

α = 0, t = 1

which completes the Proof of Lemma 3.5. Concerning the estimate G` (t, x), we have (3.24) j−k α+β+δ G` (t, x + γtz) 5 Cj,k,α,β,δ,` |x + γtz|−(2(j−k)+|α|+|β|+|δ|+n) . ∂t ∂x In fact, ∂tj−k ∂xα+β+δ G` (t, x) " min(j−k,`) 2 X j − k e−A|ξ| t −1 =F (−A|ξ|2 )j−k−m (iξ)α+β+δ m f (|ξ|) m=0 # ` m 2 2 · ∂t |ξ| g(|ξ| )t ψ(ξ)ϕ0 (ξ) (x). By (3.20), (3.21) and (3.22) we have A 2 min(j−k,`) X j − k e− 2 |ξ| t µ (−A|ξ|2 )j−k−m (iξ)α+β+δ ∂ξ m f (|ξ|) m=0 ! · ∂tm (|ξ|2 g(|ξ|2 )t)` ψ(ξ)ϕ0 (ξ) 5 Cj,k,α,β,δ,`,µ |ξ|2(j−k)+|α|+|β|+|δ|−|µ| . Therefore, by Lemma 3.5, we have (3.24). First we consider the case when n is an odd = 3. When |z| = 1, |x| 5 R0 t, and R0 5 γ/4, we have γ |x + γtz| = γt − |x| = (γ − R0 )t = t. 2


213

Therefore, applying (3.24) to (3.17), we have for |x| 5 R0 and t = 1 (3.25) j β ∂t ∂x ω` (t, x) min(`+k,|α|+1) j X X 1X j ` + k (|α| + 1)! |aα | 5 (γt)|α|+1 t−m k m γ m! m=0 k=0 |α|5 n−3 2 Z X Cj,k,α,β,δ,` |x + γtz|−(2(j−k)+|α|+|β|+|δ|+n) dS · |z|=1

|δ|=`+k−m

min(`+k,|α|+1) j X X 1X j ` + k (|α| + 1)! 5 |aα | (γt)|α|+1 t−m k m γ m! m=0 k=0 |α|5 n−3 2 Z X −(2(j−k)+|α|+|β|+|δ|+n) dS · Cj,k,α,β,δ,` t |z|=1

|δ|=`+k−m

5 Cj,β,`,n t−(j+|β|+`+n−1) . Next, we consider the case when n is even = 2. By (3.17) we have j β ∂t ∂x ω` (t, x) 5

min(`+k,|α|+1) j X X 1X j ` + k (|α| + 1)! |α|+1 |α|+1−m |aα | γ t k m γ m! n−2 m=0 k=0 |α|5 2 j−k α+β+δ Z (t, x + γtz) G ∂ ∂ t X x ` p dz. · Cj,k,α,β,δ,` 2 1 − |z| |z|51 |δ|=`+k−m

Put Z |z|51

j−k α+β+δ G` (t, x + γtz) ∂t ∂x p dz = I + II, 1 − |z|2

where j−k α+β+δ G` (t, x + γtz) ∂t ∂x p I= dz, 1 1 − |z|2 5|z|51 2 j−k α+β+δ Z G` (t, x + γtz) ∂t ∂x p dz. II = 1 − |z|2 |z|5 12 Z

214


When 1/2 5 |z| 5 1, |x| 5 R0 t, t = 1 and R0 5 γ/4, we have γ γt γ |x + γtz| = − |x| = − R0 t = t. 2 2 4 Then, by (3.24) we have I 5 Cj,k,α,β,δ,` t−(2(j−k)+|α|+|β|+|δ|+n)

Z

dz 1 5|z|51 2

p

1 − |z|2

.

When |z| 5 1/2, |x| 5 R0 t, t = 1 and R0 5 γ/4, we have γ 3 |x + γtz| 5 t + R0 t 5 γt, 2 4

√ p

1 − |z|2 =

3 . 2

Therefore, putting p = x + γtz, by Lemma 3.5 we have Z 2 j−k α+β+δ (t, p) ∂ ∂ G II 5 √ dp t ` x 3 3 |p|5 4 γt 5 Cj,k,α,β,δ,`,n  1  t−n t− 2 (2(j−k)+|α|+|β|+|δ|)     −n  t (1 + log t) ·    t−n    

when 2(j − k) + |α| + |β| + |δ| = 1, when 2(j − k) + |α| + |β| + |δ| = 0 and ` = 0, when 2(j − k) + |α| + |β| + |δ| = 0 and ` = 1.

Combining these estimations, we have for |x| 5 R0 t and t = 1 (3.26)

j β ∂t ∂x w` (t, x) 5

j 1X j k γ k=0

min(`+k,|α|+1)

|aα |

X

m=0

15|α|5 n−2 2

`+k m

(|α| + 1)! (γt)|α|+1 t−m m!

n o 1 Cj,k,α,β,δ,`,n t−(2(j−k)+|α|+|β|+|δ|+n) + t−n− 2 (2(j−k)+|β|+|δ|)

X

·

X

|δ|=`+k−m

+

j−1 min(`+k,1) X X ` + k |aα | j k=0

·

k

X |δ|=`+k−m

m=0

m

m!

t1−m

n o 1 Cj,k,β,δ,`,n t−(2(j−k)+|β|+|δ|+n) + t−n− 2 (2(j−k)+|β|+|δ|)

LINEARIZED COMPRESSIBLE NAVIER–STOKES EQUATIONS min(j+`,1)

+

X

m=0

215

j + ` |aα | 1−m t m m!

o n 1 · Cj,β,`,n t−(`+j−m+|β|+n) + t−n− 2 (`+j−m+|β|) (1 + log(1 + t)) “ ” “ ” j+|β| j+|β| − 3n−3 + 2 − n−1 − 2` − 3n−3 + 2 − 14 4 4 4 5 Cj,β,`,n (1 + log(1 + t))t . +t Next, we shall estimate the remainder term. By (3.15) and (3.16), we have h i 2 ∂tj ∂xβ F −1 e−A|ξ| t RN (t, ξ) (x) =

j X j

k

k=o

h i 2 F −1 e−A|ξ| t (−A|ξ|2 )k (i|ξ|)β ∂tj−k RN (t, ξ) (x),

and j−k ∂t RN (t, ξ) Z 1 j−k X 1 3 2 j−k = (1 − θ)N ∂tj−k−`1 e±iγ|ξ| g(|ξ| )tθ ` 2iγ|ξ|f (|ξ|)N ! 1 0 `1 =0 `1 X `1 `1 −`2 ±iγ|ξ|t `2 3 2 N +1 · ∂t e ∂t (|ξ| g(|ξ| )t) dθ `2 `2 =0

j−k X `1 X

5 Cj,β,N

|ξ|3(N +j−k)+2(1−`1 )−`2 tN +1−`2 .

`1 =0 `2 =0

Combining these estimations, we have (3.27) h i j β −1 −A|ξ|2 t e RN (t, ξ) (x) ∂t ∂x F 5 Cj,β,N

j X j−k X `1 X

t

N +1−`2

5 Cj,β,N

k=0 `1 =0 `2 =0

5 Cj,β,N t−

N +j+|β| 2

.

t−

2

|ξ|3(N +j)+2(1−`1 )−`2 −k+|β| e−A|ξ| t dξ

Rn

k=0 `1 =0 `2 =0 j X j−k X `1 X

Z

N +j+|β|+(j−k−`1 )+(j−`1 ) 2

216


By (3.25), (3.26), (3.27) and (3.13), we have “ ” j+|β| − 3n−3 + 2 4 (3.28) ∂tj ∂xβ Kψ,0 (t, x) 5 Cj,β,n t ,

t = max(1, (R/R0 )4 ).

To complete the Proof of Theorem 3.1, we have to estimate the case when 0 5 t 5 max(1, (R/R0 )4 ). But, it is obvious that j β ∂t ∂x Kψ,0 (t, x) n Z j eλ+ (ξ)t − λ (ξ)j eλ− (ξ)t λ (ξ) 1 + − (iξ)β 5 ψ(ξ)ϕ0 (ξ) Rn 2π λ+ (ξ) − λ− (ξ) Z √B 2 rj+|β|+n−2 dr 5 Cj,β 0

5 Cj,β,n . Therefore, the Proof of Theorem 3.1 is completed. 4. Proof of Theorem 2.1 (2). In this section, we shall show Theorem 2.1 (2). In view of (3.1), (3.2) and Young inequality, it suffices to show that (4.1)

k∂tj ∂xα Lij (t, ·)kL1 (Rn ) 5 Cj,α,n (1 + t)q(n)−

j+|α| 2

,

where q(n) = (n − 1)/4 for odd n = 3 and = n/4 for even n = 2. Since the kernel of L11 (t, x), L12 (t, x) = t L21 (t, x) are the same as those of (2.12), (4.1) directly follows from the results of [6, Theorem 2.1] when (i, j) = (1, 1), (1, 2) and (2.1). Therefore, our task is to show (4.1) when (i, j) = (2, 2). In view of (3.1), we put L0 (t, x) = K2 (t, x) − K3 (t, x) " ! # ξj ξk λ+ (ξ)eλ+ (ξ)t − λ− (ξ)eλ− (ξ)t −1 −α|ξ|2 t (x). =F −e λ+ (ξ) − λ− (ξ) |ξ|2 Then, we have h i 2 K1 (t, x) = F −1 e−α|ξ| t ϕ0 (ξ) (x)I. √ Noting that ϕ0 (ξ) = 0 when |ξ| = B/ 2, we have L22 (t, x) = K1 (t, x) + L0 (t, x),

(4.2)

k∂tj ∂xβ K1 (t, ·)kL1 (Rn ) 5 Cj,β,n (1 + t)−j−

|β| 2

In fact, putting χ(x) = F −1 [ϕ0 (ξ)] (x) ∈ S(Rn ), we have

1 K1 (t, x) = n (4παt) 2

Z Rn

e−

|x−y|2 2αt

χ(y) dy.

.


217

By Young inequality, we see that kK1 (t, x)kL1 (Rn ) 5 Cn ,

t > 0.

When j + |β| = 1, we have h i h i 2 2 ∂tj ∂xβ F −1 e−α|ξ| t ϕ0 (ξ) = F −1 (iξ)β (−α|ξ|2 )j e−α|ξ| t ϕ0 (ξ) , and µ ∂ξ (iξ)β (−α|ξ|2 )j ϕ0 (ξ) 5 Cj,β,n |ξ|2j+|β|−|µ| ,

ξ 6= 0.

Therefore, by Lemma 3.5 we have k∂tj ∂xβ K1 (t, x)kL1 (Rn ) 5 Cj,β,n t−j−

|β| 2

,

t > 0.

When 0 < t 5 1, since Z h i |x−y|2 1 j β −1 − 2αt −α|ξ|2 t j β e e ϕ0 (ξ) (x) = (α∆) ∂x χ(y) dy ∂t ∂x F n (4παt) 2 Rn Z √ 1 2 e−|z| ((α∆)j ∂xβ χ)(x − 2αtz) dz, = n (2π) 2 Rn we have k∂tj ∂xβ K1 (t, ·)kL1 (Rn ) 5 Cj,β,n ,

0 < t 5 1.

Combining these estimations, we have (4.2). Now, we shall show that (4.3)

k∂tj ∂xβ L0 (t, ·)kL1 (Rn ) 5 Cj,β,n tq(n)−

j+|α| 2

,

t = 1.

By (3.15), we have λ+ (ξ)eλ+ (ξ)t − λ− (ξ)eλ− (ξ)t λ+ (ξ) − λ− (ξ) ! eλ+ (ξ)t − eλ− (ξ)t = ∂t λ+ (ξ) − λ− (ξ) ) (N X 1 sin γ|ξ|t e−A|ξ|2 t 2 ∂` (|ξ|2 g(|ξ|2 )t)` + e−A|ξ| t RN (t, |ξ|) = ∂t `! t |ξ| γf (|ξ|) `=0 −A|ξ|2 t N X1 e `+1 sin γ|ξ|t ∂t (|ξ|2 g(|ξ|2 )t)` = `! |ξ| γf (|ξ|) `=0 ! 2 N X e−A|ξ| t 1 ` sin γ|ξ|t 2 2 ` + ∂ ∂t (|ξ| g(|ξ| )t) `! t |ξ| γf (|ξ|) `=0 2 + ∂t e−A|ξ| t RN (t, |ξ|) .

218


Since f (|ξ|) = 1 + |ξ|2 g(|ξ|2 ), we have 2 sin γ|ξ|t e−A|ξ| t ∂t |ξ| γf (|ξ|) sin γ|ξ|t sin γ|ξ|t |ξ|2 g(|ξ|2 ) −A|ξ|2 t ∂t =e − ∂t . γ|ξ| |ξ| γf (|ξ|) Combining these two estimations, we have L0 (t, x) = L1 (t, x) + L2 (t, x) −

M01 (t, x)

N X 1 1 + M (t, x) `! ` `=1

+

N X `=0

1 2 M (t, x) `! `

N X `=1

1 M 3 (t, x) + RN (t, x), (` − 1)! `

where sin γ|ξ|t −A|ξ|2 t ξj ξk −1 e ϕ0 (ξ) (x), L1 (t, x) = F ∂t |ξ| |ξ|2 ξ ξ j k −1 −A|ξ|2 t −α|ξ|2 L2 (t, x) = F e −e t ϕ0 (ξ) (x), |ξ|2 " # 2 sin γ|ξ|t e−A|ξ| t g(|ξ|2 ) 1 −1 M0 (t, x) = F ∂t ξj ξk ϕ0 (ξ) (x), |ξ| γf (|ξ|2 ) " # −A|ξ|2 t ` ξj ξk e `+1 sin γ|ξ|t −1 1 2 2 ∂t M` (t, x) = F |ξ| g(|ξ| )t ϕ0 (ξ) (x), |ξ| γf (|ξ|2 ) |ξ|2 " # −A|ξ|2 t sin γ|ξ|t −Ae ` M`2 (t, x) = F −1 ∂t` |ξ|2 g(|ξ|2 )t ξj ξk ϕ0 (ξ) (x), |ξ| γf (|ξ|2 ) " −A|ξ|2 t e g(|ξ|2 ) −1 ` sin γ|ξ|t 3 ∂t M` (t, x) = F |ξ| γf (|ξ|2 ) # `−1 2 2 · |ξ| g(|ξ| )t ξj ξk ϕ0 (ξ) (x), −1

and ξj ξk −A|ξ|2 t RN (t, x) = F ∂t e RN (t, |ξ|) 2 ϕ0 (ξ) (x). |ξ| First, we shall show that −1

(4.4)

kL1 (t.·)kL1 (Rn ) 5 Cn tq(n) ,

t = 1.

Put g0 (t.x) = F

−1

−A|ξ|2 t ξj ξk e ϕ0 (ξ) (x). |ξ|2


219

Since L1 (t, x) + g0 (t, x) sin γ|ξ|t −1 sin γ|ξ|t d −1 gb0 (t, ξ) (x) − F = ∂t F ∂t g0 (t, ξ) (x), γ|ξ| γ|ξ| by Lemma 3.3 we have (4.5)

L1 (t, x) + g0 (t, x)     Z  1 X α α |α|+1 z (∂x g0 )(t, x + γtz) dS = ∂t aα (γt)   |z|=1   γ |α|5 n−3 2 sin γ|ξ|t d − F −1 ∂t g0 (t, ξ) (x) |ξ|2 Z X z α (∂xα g0 )(t, x + γtz) dS = aα (|α| + 1)(γt)|α|+1 |z|=1

|α|5 n−3 2

X

+

XZ

aα (γt)|α|+1

|δ|=1 |z|=1

|α|5 n−3 2

z α+δ (∂xα+δ g0 )(t, x + γtz) dS

when n is an odd = 3. In view of Lemma 3.3, we see that Z (4.6) a0 dS = 1. |z|=1

In fact, putting w(t, x) = F

−1

sin |ξ|t ˆ h(ξ) (x), |ξ|

by Lemma 3.3, we have h(x) = wt (0, x) Z X = aα (1 + |α|)t|α| |z|=1

|α|5 n−3 2

X

+

aα t

|α|+1

!

Z a0

dS |z|=1

XZ |δ|=1 |z|=1

|α|5 n−3 2

=

z

α

h(x),

(∂xα h)(x

z

α+δ

+ tz) dS

(∂xα+δ h)(x

t=0

+ tz) dS

t=0

220


which implies (4.6). Combining (4.5) and (4.6), we have (4.7)

L1 (t, x) Z = a0

{g0 (t, x + γtz) − g0 (t, x)} dS Z X |α| z α (∂xα g0 )(t, x + γtz) dS aα (1 + |α|)(γt)

|z|=1

+

|z|=1

15|α|5 n−3 2

+

X

aα (γt)|α|+1

XZ |δ|=1 |z|=1

|α|5 n−3 2

z α+δ (∂xα+δ g0 )(t, x + γtz) dS.

Similarly, we see that (4.8)

L1 (t, x) Z = a0

g0 (t, x + γtz) − g0 (t, x) p dz 1 − |z|2 |z|51 Z X z α (∂xα g0 )(t, x + γtz) p dz aα (1 + |α|)(γt)|α| 1 − |z|2 |z|51 n−2

+

15|α|5

X

+

2

aα (γt)|α|+1

XZ |δ|=1 |z|51

|α|5 n−2 2

z α+δ (∂xα+δ g0 )(t, x + γtz) p dz 1 − |z|2

when n is an even = 2. Concerning the estimate g0 (t, x), we have (4.9)

k∂tj ∂xα g0 (t, ·)kL1 (Rn ) 5 Cj,α,n t

“ ” |α| − j+ 2

,

j + |α| = 1.

In fact, we have ∂t` ∂xα g0 (t, x)

=F

−1

−A|ξ|2 t 2 ` α ξj ξk e (−A|ξ| ) (iξ) ϕ0 (ξ) (x) |ξ|2

and µ ∂ (−A|ξ|2 )` (iξ)α ξj ξk ϕ0 (ξ) 5 Cj,k,α,µ |ξ|2`+|α|−|µ| , ξ 2 |ξ|

ξ 6= 0.

Therefore, (4.9) follows from Lemma 3.5. Since Z 1 d g0 (t, x + γtz) − g0 (t, x) = {g0 (t, x + γtzθ)} dθ 0 dθ Z 1 = (∇x g0 )(t, x + γtzθ) dθ · γtz 0


by (4.9) we have

(4.10)

Z

{g0 (t, · + γtz) − g0 (t, ·)} dS

a0

|z|=1 L1 (Rn ) Z Z Z 1 |z| |(∇x g0 )(t, x + γtzθ)| dθdSdx 5 a0 γt Rn

5 Cn t

|z|=1

0

1 2

when n is an odd = 3; and

(4.11)

Z

g (t, · + γtz) − g (t, ·)

0 0 p dz

a0 2

1 − |z| |z|51 L1 (Rn ) Z Z Z 1 |z| |(∇x g0 )(t, x + γtzθ)| p 5 a0 γt dθdzdx n 1 − |z|2 R |z|51 0 1

5 Cn t 2 when n is an even = 2. Therefore, by (4.7), (4.9) and (4.10) we have

kL1 (t, ·)kL1 (Rn )

     1 X X |α|+1  |α| 2 2 2 5 Cn t + t + t     n−3 n−3 15|α|5 |α|5 2

5 Cn t

2

n−1 4

when n is an odd = 3; and by (4.7), (4.8) and (4.11) we have

kL1 (t, ·)kL1 (Rn )

     1 X X |α|+1  |α| 5 Cn t 2 + t2 + t 2     n−2 n−2 15|α|5 |α|5 2

5 Cn t

2

n 4

when n is an even = 2, which implies (4.4). Next, we shall show that (4.12)

k∂tj ∂xβ L1 (t, ·)kL1 (Rn ) 5 Cj,β,n tq(n)−

j+|β| 4

,

t = 1, j + |β| = 1.

221

222


By (4.7) we have

(4.13)

∂tj ∂xβ L1 (t, x) Z ∂tj ∂xβ {g0 (t, x + γtz) − g0 (t, x)} dS = a0 |z|=1

+

j X k=0

·

j k

min(k+1,|α|)

X

X

k=0

k

X

aα

Z

|δ|=k−m+1 |z|=1

X m=0

05|α|5 n−3 2

X

·

|α|! (γt)|α| t−m m!

z α+δ (∂xα+β+δ ∂tj−k g0 )(t, x + γtz) dS min(k,|α|+1)

j X j

X m=0

15|α|5 n−3 2

Z

|δ|=k−m |z|=1

+

aα (1 + |α|)

(|α| + 1)! (γt)|α| t1−m m!

z α+δ (∂xα+β+δ ∂tj−k g0 )(t, x + γtz) dS

when n is an odd = 3; and by (4.8) we have

(4.14)

∂tj ∂xβ L1 (t, x) Z ∂tj ∂xβ {g0 (t, x + γtz) − g0 (t, x)} p = a0 dz 1 − |z|2 |z|51 min(k+1,|α|) j X X X |α|! j + aα (1 + |α|) (γt)|α| t−m k m! n−2 k=0

·

15|α|5

X

Z

|δ|=k−m |z|51

+

j X j k=0

·

k X

m=0

2

z α+δ (∂xα+β+δ ∂tj−k g0 )(t, x + γtz) p dz 1 − |z|2 min(k,|α|+1)

X

aα

05|α|5 n−2 2

Z

|δ|=k−m+1 |z|51

X m=0

(|α| + 1)! (γt)|α| t1−m m!

z α+δ (∂xα+β+δ ∂tj−k g0 )(t, x + γtz) p dz 1 − |z|2


223

when n is an even n = 2. By (4.9) we have

Z

(4.15) a0 ∂tj ∂xβ {g0 (t, · + γtz) − g0 (t, ·)} dS

|z|=1 L1 (Rn ) Z n o 5 a0 k∂tj ∂xβ g0 (t, · + γtz)kL1 (Rn ) + k∂tj ∂xβ g0 (t, ·)kL1 (Rn ) dS 5

|z|=1 ” “ |β| − j+ 2 ; Cj,β,n t

and

Z

j β

∂ ∂ {g (t, · + γtz) − g (t, ·)}

0 0 t x p dz

a0 2

1 − |z| |z|51

(4.16)

L1 (Rn ) k∂tj ∂xβ g0 (t, ·)kL1 (Rn )

k∂tj ∂xβ g0 (t, · + γtz)kL1 (Rn ) + p 1 − |z|2

Z 5 a0 5

dz

|z|=1 “ ” |β| − j+ 2 Cj,β,n t .

Putting ( p(n) =

n−3 2 , n−2 2 ,

when n is an odd = 3, when n is an even = 2,

by (4.9), (4.13), (4.14), (4.15) and (4.16), we have k∂tj ∂xβ L1 (t, ·)kL1 (Rn )  j  “ |β| ” X − j+ 2 + 5 Cj,β,n t 

min(k+1,|α|)

X

X m=0

k=0 15|α|5p(n)

+



min(k,|α|+1)

j X

X

X

j+|β| 2

t

” “ |β|−|α|+m−k−1  − j+ 2



m=0

k=0 05|α|5p(n)

5 Cj,β,n tq(n)−

t

“ ” |β|−|α|+m−k − j+ 2

,

which implies (4.12). Now we shall estimate ∂t` ∂xβ L2 (t, x) = (α − A)j F −1

2

e−A|ξ| t − e−α|ξ|

2t

|ξ|2` (iξ)β

ξj ξk ϕ (ξ) (x). 0 |ξ|2

Since −A|ξ|2 t

e

−α|ξ|2 t

−e

2

Z

= (α − A)|ξ| t 0

1

e−θA|ξ|

2 t−(1−θ)α|ξ|2 t

dθ,

224


we have η −A|ξ|2 t −α|ξ|2 t 2` β ξj ξk ∂ e − e |ξ| (iξ) ϕ (ξ) 0 ξ |ξ|2 5 C`,β,η |ξ|2`+|β|+2−|η| t,

ξ 6= 0.

Therefore, by Lemma 3.5 we have (4.17)

k∂t` ∂ξβ L2 (t, ·)kL1 (Rn )

5 C`,β,n t

” “ |β| − j+ 2

, t > 0.

Next, we shall show that for t = 1  j β j+|β|+1  k∂t ∂x M01 (t, ·)kL1 (Rn ) 5 Cj,β,n tq(n)− 2 ,    j+|β|+`−1 k∂ j ∂ β M 1 (t, ·)k q(n)− 2 , ` = 1, L1 (Rn ) 5 Cj,β,`,n t t x ` (4.18) j+|β|+` j β q(n)− 2  2 , ` = 0, k∂t ∂x M` (t, ·)kL1 (Rn ) 5 Cj,β,`,n t    j+|β|+`  j β 3 q(n)− 2 k∂t ∂x M` (t, ·)kL1 (Rn ) 5 Cj,β,`,n t , ` = 1. To do this, we put g(|ξ|2 ) ξj ξk ϕ0 (ξ), γf (|ξ|) A 1 2 (|ξ|2 g(|ξ|2 )t)` ξj ξk ϕ0 (ξ), ψ`1 (t, x) = e− 2 |ξ| t γf (|ξ|) A −A 2 ψ`2 (t, x) = e− 2 |ξ| t (|ξ|2 g(|ξ|2 )t)` ξj ξk ϕ0 (ξ), γf (|ξ|) 2 A 2 g(|ξ| ) ψ`3 (t, x) = e− 2 |ξ| t (|ξ|2 g(|ξ|2 )t)`−1 ξj ξk ϕ0 (ξ). γf (|ξ|) Then, we have  A 2 i h sin γ|ξ|t − 2 |ξ| t 1 1 (t, x) = F −1 ∂  e ψ (t, ξ) (x), M  t 0  |ξ|  h A 02 i   M 1 (t, x) = −F −1 ∂ `−1 sin γ|ξ|t e− 2 |ξ| t ψ 1 (t, ξ) (x), ` ` |ξ| h t A 2 i (4.19) sin γ|ξ|t − 2 |ξ| t 2 −1 ` 2  M (t, x) = F ∂ e (t, ξ) (x), ψ  `   h t |ξ| A 2 ` i   M 3 (t, x) = F −1 ∂t` sin γ|ξ|t e− 2 |ξ| t ψ 3 (t, ξ) (x). ` ` |ξ| A

ψ01 (t, x) = e− 2 |ξ|

2t

Concerning the estimate ψ`k (t, ξ), we have µ k ∂ξ ψ` (t, ξ) 5 Cµ,` |ξ|2−|µ| ,

∀

µ.

Therefore, if we put h i g`k (t, x) = F −1 ψ`k (t, ξ) (x), then, by Lemma 3.5 we have (4.20)

k∂tj ∂xβ g`k (t, ·)kL1 (Rn )

5 Cj,β,k,`,n t

“ ” |β| − 1+j+ 2

.


225

In view of (4.19) and (4.20), we consider the function: −1 ` sin γ|ξ|t ˆ G(t, ξ) (x), N` (t, x) = F ∂t |ξ| where G(t, x) satisfies the following conditions: k∂tj ∂xβ G(t, ·)kL1 (Rn )

(4.21)

5 Cj,β,n t

“ ” |β| − 1+j+ 2

.

In order to prove (4.18), it suffices to show that k∂tj ∂xβ N` (t, ·)kL1 (Rn ) 5 Cj,β,`,n tq(n)−

(4.22)

j+|β|+` 2

.

By (3.17), we have ∂tj ∂xβ N` (t, x) min(`+k,|α|+1) j X X X j `+k aα = ∂tm (γt)|α|+1 k m n−3 k=0

|α|5

X

·

m=0

Z2

|δ|=`+k−m |z|=1

z α+δ ∂xα+β+δ ∂tj−k G (t, x + γtz) dS

when n is an odd = 3; and ∂tj ∂xβ N` (t, x) min(`+k,|α|+1) j X X X j `+k = aα ∂tm (γt)|α|+1 k m n−2 k=0

·

|α|5

X

m=0

2

Z

|δ|=`+k−m |z|51

z α+δ ∂xα+β+δ ∂tj−k G (t, x + γtz) p dz 1 − |z|2

when n is an even = 2. Therefore, by (4.21) we have k∂tj ∂xβ N` (t, ·)kL1 (Rn ) 5 Cj,β,`,n

j X X

min(`+k,|α|+1)

X

j+|β|+` 2

|α|−|β|−`−j − j+m−k 2 2

m=0

k=0 |α|5p(n)

5 Cj,β,`,n tq(n)−

t

,

which implies (4.22). In order to estimate the remainder term RN (t, x), we consider the function: h i ± ± RN,ψ (t, x) = F −1 rN,ψ (t, ξ) (x),

226


where ± rN,ψ (t, ξ) 2

e−A|ξ| t = 2iγ|ξ|f (|ξ|)

Z

1

(1 − θ)N e±iγ|ξ|(1+θ|ξ|

2 g(|ξ|2 ))t

dθ

0

· (±iγ|ξ|3 g(|ξ|2 ))N +1 ψ(ξ)ϕ0 (ξ), and ψ ∈ C ∞ (Rn − {0}) satisfies the condition: γ ∂ξ ψ(ξ) 5 Cγ |ξ|−|γ| , ∀ ξ 6= 0. If ψ(ξ) =

ξj ξk , |ξ|2

then we have

RN (t, x) = ∂t

(4.23)

o n − + (t, x) tN +1 . (t, x) − RN,ψ RN,ψ

First, we observe that n o δ ± (t, ξ) (4.24) ∂ξ (−A|ξ|2 ± iγ|ξ|(1 + θ|ξ|2 )g(|ξ|2 ))j (iξ)β rN,ψ A

2

5 Cj,β,N,δ |ξ|3N +2+j+|β|−2|δ| e− 4 |ξ| t ,

0 5 θ 5 1, ξ 6= 0.

In fact, by the formula of derivative of composed function (cf. (3.19)), we have (4.25) ∂ξδ e±iγ|ξ|(1+θ|ξ| =

|δ| X

2 g(|ξ|2 )t)

(±iγt)` e±iγ|ξ|(1+θ|ξ|

2 g(|ξ|2 )t)

`=1

·

X

∂ξα1 |ξ|(1 + θ|ξ|2 g(|ξ|2 )) . . . ∂ξα` |ξ|(1 + θ|ξ|2 g(|ξ|2 )) .

|α1 |+···+|α` |=|δ| |αi |=1

Since αν ∂ξ |ξ|(1 + θ|ξ|2 g(|ξ|2 )) 5 Cα,ν |ξ|1−|αν | ,

ξ ∈ supp ϕ0 ,

by (4.25) we have |δ| X δ ±iγ|ξ|(1+θ|ξ|2 g(|ξ|2 )t) ∂ e 5 C (t|ξ|)` |ξ|−|δ| , ξ δ `=1

ξ ∈ supp ϕ0 .


227

Therefore, we have for ξ ∈ supp ϕ0 and ξ 6= 0 n o δ ± (t, ξ) ∂ξ (−A|ξ|2 ± iγ|ξ|(1 + θ|ξ|2 )g(|ξ|2 ))j (iξ)β rN,ψ |δ−ν|

5 Cδ

X

|ξ|2 t 3N +2+j+|β|−|ν| − A 2

|ξ|

e

X

(t|ξ|)` |ξ|−|δ−ν|

`=1

05ν5δ

|δ−ν|

5 Cδ

X

3N +2+j+|β|−|δ| − A |ξ|2 t 2

|ξ|

e

X

(t|ξ|)` |ξ|−` .

`=1

05ν5δ

Since |ξ|−` 5 Cδ |ξ|−|δ| (0 5 ` 5 |δ|) when ξ ∈ supp ϕ0 and ξ 6= 0, by the above inequality we have (4.24). Since eix·ξ =

(4.26)

n X xj ∂ξ eix·ξ , i|x|2 j j=1

in view of (4.24) by n + 1-times integration by parts, we have ± ∂tj ∂xβ RN,ψ (t, x) X ix δ 1 n Z 1 Z eix·ξ ∂ξδ = |x|2 2π n 0 R |δ|=n+1 n o j ± · −A|ξ|2 ± iγ|ξ|(1 + θ|ξ|2 g(|ξ|2 )) (iξ)β rN,ψ (t, ξ) dξdθ

when N > n/3. Therefore, by (4.24) we have Z A 2 j β ± |ξ|3N +2+j+|β|−2(n+1) e− 4 |ξ| t dξ ∂t ∂x RN,ψ (t, x) 5 Cj,β,N,n |x|−(1+n) Rn

5 Cj,β,N,n |x|−(n+1) t−

3N +j+|β|−n 2

.

On the other hand, by (4.24) with δ = 0 we have Z A 2 j β ± |ξ|3N +2+j+|α| e− 4 |ξ| t dξ ∂t ∂x RN,ψ (t, x) 5 Cj,β,N Rn

5 Cj,β,N t

−

3N +2+j+|β|+n 2

.

Combining these two estimations, we have (4.27) ± k∂tj ∂xβ RN,ψ (t, ·)kL1 (Rn ) (Z Z 3N +2+j+|β|+n − 2 5 Cj,β,N,n t dx + √

√ t |x|= t

|x|5 t

5 Cj,β,N,n t−

3N +|α|+j+1−n 2

,

t = 1.

−

3N +j+|α|−n 2

) −n+1

|x|

dx

228


By (4.23) and (4.27), we have n . 3 Combining (4.4), (4.12), (4.17), (4.18) and (4.28), we have (4.3). To complete the Proof of Theorem 2.1 (2), we have to show that k∂tj ∂xβ RN (t, ·)kL1 (Rn ) 5 Cj,β,N t−

(4.28)

N −n+j+|β−1 2

k∂tj ∂xα L0 (t, ·)kL1 (Rn ) 5 Cj,α,n ,

(4.29)

,

t = 1, N >

0 5 t 5 1.

Regarding the relations: λ± (ξ) = −A|ξ|2 ∓ iγ|ξ|f (|ξ|), we put L0 (t, x) =

3 X

F −1 [ψj (t, ξ)] (x),

j=1

where 2 ξ ξ A|ξ|e−A|ξ| t −iγ|ξ|f (|ξ|)t j k e − eiγ|ξ|f (|ξ|)t ϕ0 (ξ), ψ1 (t, ξ) = 2iγf (|ξ|) |ξ|2 ! ξj ξk e−iγ|ξ|f (|ξ|)t − eiγ|ξ|f (|ξ|)t −A|ξ|2 t −1 ϕ0 (ξ), ψ2 (t, ξ) = e 2 |ξ|2 −A|ξ|2 t −α|ξ|2 t ξj ξk ψ3 (t, ξ) = e −e ϕ0 (ξ). |ξ|2

First, we shall estimate F −1 [ψ1 (t, ξ)] (x). By the formula of derivative of composed function (cf. (3.19)), we have (4.30) ∂ξδ e±γ|ξ|f (|ξ|)t =

|δ| X

(±iγt)` e±iγ|ξ|f (|ξ|)t

X

∂ξα1 {|ξ|f (|ξ|)} . . . ∂ξα` {|ξ|f (|ξ|)} .

|α1 |+···+|α` |=|δ| |αi |=1

`=1

Since αν ∂ξ |ξ|f (|ξ|) 5 Cαν |ξ|1−|αν | ,

B |ξ| 5 √ , ξ 6= 0, 2

by (4.30) we have (4.31)

|δ| X δ ±iγ|ξ|f (|ξ|)t |ξ|`−|δ| 5 Cδ |ξ|−|δ| , ∂ξ e 5 Cδ `=1

By (4.31) and Leibniz’ rule, we have n o δ j ∂ξ ∂t ψ1 (t, ξ)(iξ)α 5 Cj,δ,α |ξ|1−|δ| ,

B |ξ| 5 √ , ξ 6= 0. 2

B |ξ| 5 √ , ξ 6= 0. 2


229

√ Since supp ψ1 (t, ·) ⊂ {ξ ∈ Rn | |ξ| 5 B/ 2}, by Proposition 3.4 with (α, N, σ) = (1, n, 1) we have j α −1 ∂t ∂x F [ψ1 (t, ξ)] (x) 5 Cj,α,n |x|−n−1 , ∀ x 6= 0. On the other hand, we have Z j α −1 ∂ ∂ F [ψ (t, ξ)] (x) 5 C t x 1 j,α,n

B |ξ|5 √

|ξ| dξ 5 Cj,α,n .

2

Therefore, combining these two estimations, we have (4.32) k∂tj ∂xα F −1 [ψ1 (t, ξ)] (·)kL1 (Rn )

(Z 5 Cj,α,n

Z dx +

|x|51

) |x|−(n+1) dx

|x|=1

5 Cj,α,n . Next, we shall estimate F −1 [ψ2 (t, ξ)] (x). By Taylor’s formula, we have Z 1 ξj ξk −A|ξ|2 t ϕ0 (ξ). ψ2 (t, ξ) = −iγf (|ξ|)te sin(θγ|ξ|f (|ξ|)t) dθ |ξ| 0 Therefore, we have n o δ j ∂ξ (iξ)α ∂t ψ2 (t, ξ) 5 Cj,α,δ |ξ|1−|δ| ,

ξ 6= 0.

Employing the same argument as in F −1 [ψ1 (t, ξ)] (x), we have (4.33)

k∂tj ∂xα F −1 [ψ2 (t, ξ)] (·)kL1 (Rn ) 5 Cj,α,n .

Finally, we shall estimate F −1 [ψ3 (t, ξ)] (x). Since Z 1 2 ψ3 (t, x) = (A − α)t e−((1−θ)A+θα)|ξ| t dθξj ξk ϕ0 (ξ), 0

we have n o δ j ∂ξ (iξ)α ∂t ψ3 (t, ξ) 5 Cj,α,δ |ξ|2−|δ| . Therefore, employing the same argument as in F −1 [ψ1 (t, ξ)] (x), we have (4.34)

k∂tj ∂xα F −1 [ψ3 (t, ξ)] (·)kL1 (Rn ) 5 Cj,α,n .

Combining (4.32), (4.33) and (4.34), we have (4.29), which completes the proof.

230


5. Proof of Theorem 2.2. In this section, we shall prove Theorem 2.2. First, we consider the part where p √ √ 2 4 2 2 |ξ| = 2B. Since λ± (ξ) = −A |ξ| ± |ξ| − B |ξ| when |ξ| = 2B, we write: λ+ (ξ) = −A|ξ|2 + 1 + µ(ξ),

(5.1)

λ− (ξ) = −1 − µ(ξ),

where µ(ξ) =

AB 4 g 4|ξ|2

B2 |ξ|2

Z ,

g(s) =

1

3

(1 − θs)− 2 (1 − θ) dθ.

0

Note that g(B 2 /|ξ|2 ) ∈ C ∞ when |ξ| = 2. In view of (2.5) and (2.9), we put # λ± (ξ)t λ (ξ)e ∓ ϕ∞ (ξ)ˆ u(ξ) (x), L± (t)u(x) = F −1 λ+ (ξ) − λ− (ξ) " # ξ β eλ± (ξ)t −1 M±,β (t)u(x) = F ϕ∞ (ξ)ˆ u(ξ) (x), |β| = 1, λ+ (ξ) − λ− (ξ) " # λ± (ξ)eλ± (ξ)t ξj ξk −1 K±,∞ (t)v0 (x) = F ϕ∞ (ξ)vb0 (ξ) (x), λ+ (ξ) − λ− (ξ) |ξ|2 ξj ξk −1 −α|ξ|2 t ϕ∞ (ξ)vb0 (ξ) (x). K1,∞ (t)v0 (x) = F e δjk − |ξ|2 "

By [6, Theorem 4.2.1], we have for p = 1 or ∞ (5.2) k∂tj ∂xα L+ (t)ukLp (Rn ) 5 Cj,k,α t−(j−k) e−ct kuk

2k+(|α|−1)+

Wp

(Rn )

,

k∂tj ∂xα M+,β (t)ukLp (Rn ) 5 Cj,k,α t−(j−k) e−ct kukW 2k+|α| (Rn ) , |β| = 1, p j α −t −ct k∂t ∂x L− (t)u − e u kLp (Rn ) 5 Cj,α e kuk (|α|−1)+ n , Wp

k∂tj ∂xα M−,β (t)ukLp (Rn )

−ct

5 Cj,α e

kukW |α| (Rn ) , p

(R )

|β| = 1.

Now we shall show that for p = 1 or ∞ (5.3) 1 k∂tj ∂xα k+,∞ (t)v0 kLp (Rn ) 5 Cj,α,n 1 + t− 2 t−(j−k) e−ct kv0 kW 2k+|α| (Rn ) , p j α − 12 −(j−k) −ct k∂t ∂x k1,∞ (t)v0 kLp (Rn ) 5 Cj,α,n 1 + t t e kv0 kW 2k+|α| (Rn ) . p


231

Put " K+,j,k (t, x) = F −1 " K1,j,k (t, x) = F −1

# λ+ (ξ)j eλ+ (ξ)t (1 + |ξ|2 )j−k ξm ξ` ϕ∞ (ξ) (x), (1 + |ξ|2 )j |ξ|2 2

(−α|ξ|2 )j e−α|ξ| t (1 + |ξ|2 )j−k (1 + |ξ|2 )j # ξm ξ` ϕ∞ (ξ) (x), · δm` − |ξ|2

for j 5 k 5 0, and then (5.4)

∂tj ∂xα K+,∞ (t)v0 = K+,j,k (t, ·) ∗ ∂xα (1 − ∆)k v0 ;

∂tj ∂xα K1,∞ (t)v0 = K1,j,k (t, ·) ∗ ∂xα (1 − ∆)k v0 . √ By (5.1) we have for |ξ| = 2B ( ) j eλ+ (ξ)t (1 + |ξ|2 )j−k ξ ξ λ (ξ) ν + m ` (5.6) ϕ (ξ) ∂ξ ∞ (1 + |ξ|2 )j |ξ|2 (5.5)

2

5 Cj,k,ν t−(j−k) e−c1 t |ξ|−|ν| e−c2 |ξ| t ,

∀

ν,

and also we have ( ) 2 ν (−α|ξ|2 )j e−α|ξ| t (1 + |ξ|2 )j−k ξm ξ` (5.7) ϕ (ξ) ∂ξ ∞ 2 j 2 (1 + |ξ| ) |ξ| 2

5 Cj,k,ν t−(j−k) e−c1 t |ξ|−|ν| e−c2 |ξ| t ,

∀

ν.

Therefore, using (4.26) and the integration by parts n + 1 times, by (5.6) we have Z t−(j−k) e−c1 t 2 (5.8) |ξ|−n−1 e−c2 |ξ| t dξ |L+,j,k (t, x)| 5 Cj,k,n √ n+1 |x| |ξ|= 2B 5 Cj,k,n

t−(j−k) e−c1 t , |x|n+1

and by (5.7) we have (5.9)

|L1,j,k (t, x)| 5 Cj,k,n

t−(j−k) e−c1 t . |x|n+1

On the other hand, by (5.6) we have (5.10)

|L+,j,k (t, x)| 5 Cj,k,n t−(j−k) e−c1 t

Z

2

√

|ξ|= 2B

5 Cj,k,n t

−(j−k)− n −c1 t 2

e

,

e−c2 |ξ| t dξ

232


and by (5.7) we have (5.11)

n

|L1,j,k (t, x)| 5 Cj,k,n t−(j−k)− 2 e−c1 t .

Therefore, by (5.8) and (5.10) we have (5.12)

kL+,j,k (t, ·)kL1 (Rn ) ( 5 Cj,k,n t−(j−k) e−c1 t

t

−n 2

Z √ |x|5 t

Z dx +

√ |x|= t

1 dx |x|n+1

)

1 5 Cj,k,n 1 + t− 2 t−(j−k) e−c1 t , and by (5.9) and (5.11) we have (5.13)

1 kL1,j,k (t, ·)kL1 (Rn ) 5 Cj,k,n 1 + t− 2 t−(j−k) e−c1 t .

By (5.12), (5.13) and the Young inequality, we have (5.3). Next, we shall show that for p = 1 or ∞ (5.14)

k∂tj ∂xα K−,∞ (t)v0 kLp (Rn ) 5 Cj,k,α,n t−(j−k) e−c1 t kv0 kW 2k+|α| (Rn ) . p

Put " `− (t, x) = F

−1

# λ− (ξ)eλ− (ξ)t ξj ξk ϕ∞ (ξ) (x), λ+ (ξ) − λ− (ξ) |ξ|2

and then (5.15)

K−,∞ (t)v0 (x) = `− (t, ·) ∗ v0 .

Now, we shall prove that k∂tj `− (t, ·)kL1 (Rn ) 5 Cj e−ct . √ By (5.1) we have for |ξ| = 2B ( ) β λ− (ξ)j+1 eλ− (ξ)t ξm ξk (5.17) ∂ξ ϕ (ξ) 5 Cj,β (1 + t)|β| e−ct |ξ|−2−|β| . ∞ 2 λ+ (ξ) − λ− (ξ) |ξ| (5.16)

Therefore, using (4.26) and the integration by parts n − 1 times, by (5.17) we have ( −(n−1) , 0 < |x| 5 1; j −ct |x| ∂t `− (t, x) 5 Cj,n e −(n+1) |x| , |x| = 1, which implies (5.16). By (5.16) and the Young inequality, we have (5.14). In order to complete the Proof of Theorem 2.2, we have to estimate the part where B/2 5 |ξ| 5 2B (cf. (2.10)). In view of (2.6) and (2.9), below, if we put


233

(5.18) I (z + |ξ|2 )ezt 1 −1 N0,ψ (t, x) = F dzψ(ξ)ϕM (ξ) (x); 2 2 2 2 2πi Γ z + (α + β)|ξ| z + γ |ξ| I 1 −1 ezt N1,ψ (t, x) = iγξ F dzψ(ξ)ϕM (ξ) (x); 2 2 2 2 2πi Γ z + (α + β)|ξ| z + γ |ξ| h i 2 N2,ψ (t, x) = F −1 e−α|ξ| t ψ(ξ)ϕM (ξ) (x), where ψ ∈ C ∞ (S n−1 ) and ψ = ψ(ξ/|ξ|), then we have F −1 [ϕM (ξ)ˆ ρ(t, ξ)] (x) = N0,ψ (t, ·) ∗ ρ0 + N1,ψ (t, ·) ∗ v0 ; F −1 [ϕM (ξ)ˆ v (t, ξ)] (x) = N1,ψ (t, ·) ∗ ρ0 + N0,ψ (t, ·) ∗ v0 + N2,ψ (t, ·) ∗ v0 . If we use (4.26) and (2.7), then we see easily that j α ∂t ∂x N`,ψ (t, x) 5 Cj,α,N e−ct |x|−N , ∀ N = 0, integer. Therefore, applying the Young inequality to (5.18) we have (5.19) kF −1 [ϕM (ξ)(ˆ ρ, vˆ)(t, ξ)] kLp (Rn ) 5 Cj,α,p e−ct k(ρ0 , v0 )kLp (Rn ) ,

1 5 p 5 ∞.

Combining (5.2), (5.3), (5.14) and (5.19), we have Theorem 2.2.

References [1] L.C. Evans, Partial Differential Equations, Grund. Math. Wiss., Berkeley Mathematics Lecture Notes, 3a, 1998, MR 99e:35001, Zbl 0902.35002. [2] D. Hoff and K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., 44(2) (1995), 603-676, MR 96j:35189, Zbl 0842.35076. [3]

, Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves, Z. Angew. Math. Phys., 48 (1997), 597-614, MR 98i:35145, Zbl 0882.76074.

[4] T.-P. Liu and W. Wang, The pointwise estimates of diffusion wave for the NavierStokes system in odd multi-dimensions, Comm. Math. Phys., 196 (1998), 145-173, MR 99i:35127, Zbl 0912.35122. [5] S. Mizohata, The Theory of Partial Differential Equations, Cambridge Univ. Press, 1973, MR 58 #29033, Zbl 0263.35001. [6] Y. Shibata, On the rate of decay of solutions to linear viscoelastic equation, Math. Meth. Appl. Sci., 23 (2000), 203-226, MR 2001d:35188 Zbl 0947.35020. [7] Y. Shibata and S. Shimizu, A decay property of the Fourier transform and its application to Stokes problem, J. Math. Fluid Mech., 3 (2001), 213-230, MR 2002i:35005. [8] C.G. Simader, On Dirichlet Boundary Value Problem; An Lp -Theory Based on a Generalization of G˚ arding’s Inequality, Springer-Verlag, Lecture Notes in Math., 268, Berlin, Heidelberg, New York, 1972, MR 57 #13169, Zbl 0242.35027.

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[9] B.R. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach Science Publishers, New York, London, Paris, Montreux, Tokyo, 1989, MR 91h:35081. Received February 27, 2001. Department of Mathematics Kyushu Institute of Technology 1-1 Sensui-cho, Tobata-ku Kitakyusyu 804-8550 Japan E-mail address: [email protected] Department of Mathematics School of Science and Engineering Waseda University 3-4-1 Ohkubo Shinjuku-ku Tokyo 169-8555 Japan E-mail address: [email protected]


PURE-PERIODIC MODULES AND A STRUCTURE OF PURE-PROJECTIVE RESOLUTIONS Daniel Simson Dedicated to Stanislaw Balcerzyk on the occasion of his seventieth birthday

We investigate the structure of pure-syzygy modules in a pure-projective resolution of any right R-module over an associative ring R with an identity element. We show that a right R-module M is pure-projective if and only if there exists an integer n ≥ 0 and a pure-exact sequence 0 → M → Pn → · · · → P0 → M → 0 with pure-projective modules Pn , . . . , P0 . As a consequence we get the following version of a result in Benson and Goodearl, 2000: A flat module M is projective if M admits an exact sequence 0 → M → Fn → · · · → F0 → M → 0 with projective modules Fn , . . . , F0 .

1. Introduction. Throughout this paper R is an associative ring with an identity element. We denote by Mod(R) the category of all right R-modules. We recall (see [12]) that an exact sequence · · · → Xn−1 → Xn → Xn+1 → . . . in Mod(R) is said to be pure (in the sense of Cohn [6]) if the induced sequence · · · → Xn−1 ⊗R L → Xn ⊗R L → Xn+1 ⊗R L → . . . of abelian groups is exact for any left R-module L. An epimorphism f : Y −→ Z in Mod(R) is said to f be pure if the exact sequence 0 −→ Ker f −→ Y −→ Z −→ 0 is pure. A submodule X of a right R-module Y is said to be pure if the exact sequence 0 → X → Y → Y /X → 0 is pure. A module P in Mod(R) is said to be pure-projective if for any pure epimorphism f : Y −→ Z in Mod(R) the induced group homomorphism HomR (P, f ) : HomR (P, Y ) −→ HomR (P, Z) is surjective. The following facts are well-known (see [14], [15], [29], [31]): (i) A module P in Mod(R) is pure-projective if and only if P is a direct summand of a direct sum of finitely presented modules. (ii) Every module M in Mod(R) admits a pure-projective pure resolution P∗ in Mod(R), that is, there is a pure-exact sequence (1.1)

d

dn−1

d

d

d

n 2 1 0 . . . −→ Pn −→ Pn−1 −→ . . . −→ P1 −→ P0 −→ M →0

where the modules P0 , . . . , Pn , . . . are pure-projective. The main results of the paper are the following two theorems: 235

236

DANIEL SIMSON

Theorem 1.2. Let M be a right R-module and (1.1) a pure-exact sequence in Mod(R) such that the modules P0 , . . . , Pn , . . . are pure-projective. Then, for each n ≥ 0, the n-th pure-syzygy module Ker dn of M is an ℵn -directed union of ℵn -generated pure-projective R-modules, which are pure submodules of Pn and of Ker dn . Theorem 1.3. Let M be a right R-module. If there exists a pure-exact sequence 0 → M −→ Pn −→ Pn−1 −→ . . . −→ P1 −→ P0 −→ M → 0 in Mod(R) such that the modules P0 , . . . , Pn are pure-projective, then M is pure-projective. In other words, every pure-periodic R-module M is pure-projective. As a consequence of Theorem 1.3 we get the following version of a recent result by Benson and Goodearl in [5]. Corollary 1.4. Let M be a right flat R-module. If there exists an exact sequence 0 → M −→ Fn −→ Fn−1 −→ . . . −→ F1 −→ F0 −→ M → 0 in Mod(R) such that the modules F0 , . . . , Fn are projective, then M is projective. Our Theorem 1.2 extends the main projective resolution structure theorem [23, Theorem 1.5] (see also [17, Theorem 3.3] for n = 0) from flat modules to arbitrary modules. The Corollary 1.4 with n = 0 coincides with [5, Theorem 2.5]. The main results of the paper are proved in Sections 3 and 4. In Section 2 we collect preliminary facts and notation we need throughout the paper. In Section 5 we show that Theorems 1.2 and 1.3 remain valid in any locally finitely presented Grothendieck category A. Throughout this paper we use freely the module theory terminology and notation introduced in [1] and [12]. The reader is referred to [8], [12], [14], [16], [26], [29], [31] and to the expository papers [9] and [28] for a basic background and historical comments on purity and pure homological dimensions. 2. Preliminaries on the pure-projective dimension. We start this section by collecting basic definitions, notation and elementary facts we need throughout this paper. Given right R-modules M and N the n-th pure extension group PextnR (M, N ) is defined to be the n-th cohomology group of the complex HomR (P∗ , N ), where P∗ is a pure-projective resolution of M in Mod(R).

STRUCTURE OF PURE-PROJECTIVE RESOLUTIONS

237

The pure-projective dimension P.pd M of M is defined to be the minimal integer m ≥ 0 (or infinity) such that Pextm R (M, −) = 0. The right pure global dimension r.P.gl.dim R of R is defined to be the minimal integer n ≥ 0 (or infinity) such that PextnR = 0. Following [27] we call the ring R right pure semisimple if r.P.gl.dim R = 0. The left pure global dimension l.P.gl.dim R of a ring R was introduced in 1967 by R. Kielpi´ nski [14] and in 1970 by P. Griffith [7]. It was shown in [26, Theorem 2.12] that the right pure global dimension r.P.gl.dim R of the ring R is the supremum of P.pd M , where M runs through all right R-modules M such that P.pd M is finite. This means that the right finitistic pure global dimension of R and the right pure global dimension of R coincide. Throughout this paper we denote by ℵ an infinite cardinal number and by ℵ0 the cardinality of a countable set. A right R-module M is said to be ℵ-generated if it is generated by a set of cardinality ℵ, and M is ℵ-presented if M is ℵ-generated and for any epimorphism f : L → M with ℵ-generated module L the kernel Ker f is ℵ-generated, or equivalently, M is a limit of a direct system {Mj , hij } of cardinality ℵ consisting of finitely presented modules Mj (see [18], [20], [22], [26]). We say that M is an ℵ-directed union of submodules Mj , j ∈ J, if for each subset J0 of J of cardinality ℵ there exist jS 0 ∈ J such that Mt ⊆ Mj0 for all t ∈ J0 . A union Mξ of submodules Mξ of M is well-ordered and continξ