algebraic result of GM]: Given a subgroup G of GL(V ) which contains a prox- ... for the group G(R) of real points of a connected semi{simple algebraic group.
Semigroups containing proximal linear maps by H. Abels, G.A. Margulis and G.A. Soifer
I. Introduction A linear automorphism of a nite dimensional real vector space is called proximal if it has a unique eigenvalue { counting multiplicities { of maximal modulus. Proximal elements play an important role in the study of the dynamics of linear maps because of their simple structure: The induced map on projective space has a unique attracting xed point whose basin consists of the whole space except for a hyperplane. This contraction property played an essential role in Tits' proof of what is now called the Tits alternative [Ti], in Furstenberg's work on boundary theory [F], in the paper of the two last named authors on maximal subgroups [MS], in the work of Guivar'ch and coauthors on random walks on groups (cf. [GR]). The authors were motivated to do this work by their hope for yet another application of proximality, namely in their work in progress on the Auslander conjecture. For work on this conjecture see [A, Mi, D, DG, FG, GK, GrM, Ma1-3, Me, To, S]. An ecient tool to nd proximal elements is the following corollary of the main algebraic result of [GM]: Given a subgroup G of GL(V ) which contains a proximal element. Then so does every Zariski dense subsemigroup H of G , provided V { considered as a G {module { is strongly irreducible (for a de nition see the paragraph preceding 3.12). Here we show that under these hypotheses H in fact provides for a rich supply of proximal elements: There is a nite subset M of H such that for every g 2 GL(V ) at least one of the elements g , 2 M , is proximal. We obtain such a subset M with at most (dim V )2 elements. We show the main result (theorem 4.1) for a quantitative version of proximality, called (r; ") {proximality. For another richness result see 4.10.
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In section 5 of the paper we extend our result in two directions, namely for direct sums of strongly irreducible representations and for other types of Lyapunov ltrations. The latter means that instead of requiring that G contains an element with exactly one eigenvalue of maximal modulus we look at the minimum number n1 of eigenvalues of maximal modulus among elements of G . Then the same minimum is attained in the Zariski dense subsemigroup H . Similarly for the eigenvalues with maximum and next to maximum modulus, etc. In the nal section we give necessary and sucient conditions when the image of a reductive group under an irreducible representation contains a proximal linear map. The conditions are phrased in terms of the highest weight of . As an application (actually of 4.1 and results of [GR]) we obtain (see 6.8) that for the group G(R) of real points of a connected semi{simple algebraic group G over R every Zariski dense subsemigroup H contains R {regular elements, where an element g 2 G(R) is called R {regular if the number of eigenvalues { counting multiplicities { of modulus 1 of Ad(g) is minimum possible among elements g 2 G(R) . This was known before for subgroups H ([BL], [P]). We also obtain a corresponding niteness result. The proofs of the niteness results 4.1 and 5.14 make essential use of the notion of quasiprojective transformation due to Furstenberg [F] and the methods of the algebraic part of [GM]. The relevant notions and results are given in section 3. The proof of 4.1 is contained in section 4. There the simple structure of proximal linear maps is expounded, technically speaking in the form of \contractions" and \contractive sequences" on projective space. In section 5 we give the generalization to several irreducible summands and more general Lyapunov ltrations. The authors would like to thank I. Goldsheid for fruitful discussions. They also thank di erent institutions and foundations for their support during the preparation of this paper: the MSRI Berkeley, the SFB 343 Bielefeld, the MPI Bonn, Yale University and the NSF under grant DMS 9204 720. Without their help this paper whose authors live on three di erent continents would not have seen the light of day.
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II. Proximality A linear automorphism g of a nite dimensional real vector space V is called proximal if g has a unique eigenvalue = (g) of maximal absolute value | hence is real | and the corresponding weight space
V = fv 2 V ; (g ? E )n v = 0 for some ng is one{dimensional. Fix a norm k k on V and let be the corresponding metric on V . If g is proximal with eigenvalue = (g) of maximal absolute value set V + = V + (g) = V and let V < = V < (g) be the unique g {invariant hyperplane complementary to V + . Put + r(g) = k g j V < k = j(g) 1; " > 0 , if g is proximal and r(g) r and "(g) " . So g 2 GL(V ) is (r; ") {proximal i there is a decomposition V = V + V < into g {invariant subspaces such that 1) dim V + = 1 2) k g j V + k r k g j V < k and 3) (x+ ; V < ) " for x+ 2 V + of norm 1. Note that \ g proximal" does not imply r(g) 1 . But if g is proximal then p n r(gn ) converges to = max fjj; eigenvalue of g j V < g > 1 , in particular lim r (gn ) = 1 . n!1 We take the following criterion for proximality from [Ti]. For g 2 GL(V ) put
g1 (x) = kgg((xx))k
for x 6= 0 :
4
So g1 maps the \sphere" S = fx 2 V ; kxk = 1g to itself. For a metric space (X; ) de ne the norm of a map f : X ?! X with respect to by
(f (x); f (y)) (x; y) x6=y in X
kf k = sup (= 0 if card X 1) .
2.1 Lemma Let g 2 GL(V ) and let K be a compact subset of S such that g1 (K ) K and k g1 jK k < 1 . Here K denotes the interior of K with respect to the topology of S . Then g is proximal, g1 has a unique xed point in K , namely the point in V + (g) \ K and V < (g) does not intersect K .
The proof of [Ti 3.8 (ii)] for projective space works for the sphere, too.
III. Quasiprojective transformations In this section we shall recall the notion of quasiprojective transformation due to Furstenberg [F] and in particular the results on contractions from [GM]. It will be convenient to use di erent metrics on projective space. So we shall rst of all x a class of metrics. Two metrics and 0 on a set X are called equivalent if there is a positive constant c 1 such that
c (x; y) 0 (x; y) c?1 (x; y) for every pair x; y of points of X . Let X be a connected di erentiable manifold with a Riemannian tensor g . Then there is an induced metric = x;g , namely (x; y) is the in mum of the arc lengths with respect to g of piecewise di erentiable curves from x to y . The following facts are easily proved. If g and g0 are two Riemannian tensors on X then X;g0 j K and X;g j K are equivalent for any compact subset K of X .
3.1
5
3.2 If U is an open connected neighbourhood of a compact subset K of X then X;g j K and U;g j K are equivalent. It follows that
3.3 On a compact di erentiable manifold X there is a unique equivalence class
of metrics containing the metrics induced by Riemannian tensors. This class is characterized by the following property: If ' is a di eomorphism from an open subset U of X to a nite dimensional normed vector space (V; k k) then for any compact subset K of U the metric 0 (x; y) = k'(x) ? '(y)k on K is equivalent to j K
We consider the projective space P = PV associated with a nite dimensional real vector space V and endow it with some metric of the distinguished class of metrics equivalent to the Riemannian metrics. We call such a metric admissible. A particularly pleasant admissible metric is the angle metric on P with
([x]; [y]) = arccos kxjk(x; yk)jyk for x; y nonzero in V supposing we are given a positive inner product ( ; ) on V with associated norm kxk = (x; x)1=2 .
3.4 A map b : P ?! P is called quasiprojective if there is a sequence of projective
transformations of P converging pointwise to b . Let M1 (b) be the closure of the set of points, where b is discontinuous and let M0 (b) be the b {image of the set of continuity points of b . It is straightforward that for h 2 GL(V ) we have
M0 (h b) = h M0 (b); M0 (b h) = M0 (b) M1 (h b) = M1 (b); M1 (b h) = h?1 M1 (b)
(3.5) (3.6)
A quasiprojective transformation b is called a contraction if M0 (b) consists of one point only. For every linear concept we denote the corresponding projective concept by writing a P in front. E.g. for g 2 GL(V ) the corresponding projective map is Pg : PV = P ?! P . For a linear subspace W 6= 0 of V the corresponding subspace of
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P will be denoted PW . For a non{zero vector x of V let Px be the point P(Rx) 2 P . Let us call a sequence s = (Bn )n2N in GL(V ) contractive if (C 1) P Bn converges pointwise on P (C 2) there is a sequence ( n )n2N in R such that ( n Bn )n2N converges to a linear map b1 of rank 1 . Note that C 2 does not imply C 1.
3.7 Lemma
If s = (Bn )n2N is a contractive sequence in GL(V ) then b = lim P Bn is a contraction. Conversely, if for a sequence s = (Bn )n2N in GL(V ) the sequence (P Bn) n2N converges pointwise on P to a contraction then s contains a contractive subsequence.
Proof For a contractive sequence (Bn)n2N in GL(V ) the limit b = lim P Bn is represented by b1 for points not in ker b1 , i.e. b (P x) = P (b1 x) for b1 x = 6 0.
Hence b is continuous on the dense open subset P r P ker b1 , hence M0 (b) = P image b1 . Thus b is a contraction. Conversely, let (Bn )n2N be a sequence in GL(V ) such that (P Bn) n2N converges to the contraction b . Then, passing to a convergent subsequence of Bn kBn k?1 , we may assume that there is a sequence ( n )n2N in R such that ( n Bn )n2N converges to a non{zero linear map b1 . So b(P x) = P (b1 x) for x 62 ker b1 and b is continuous on the dense open subset P r P ker b1 , hence b1 takes its image in the line corresponding to M0 (b) for points x 62 ker b1 . It follows that rank b1 = 1 . For a contractive sequence s put
M0 (s) = P image b1 and
L1 (s) = P ker b1 ; with notations as above. We have seen in the proof of 3.7 that
M0 (s) = M0 (b)
7
for b = lim P s , so M0 (s) depends only on s . To see that L1 (s) is well{de ned note that for the sequence n in C 2 we have a constant c 1 such that
c?1 < j n j kBn k < c : Hence
L1 (s) = v 2 V ; kBn vk kBn k?1 ?! 0 is independent of the sequence ( n )n2N . It follows that
3.8
b1 in C 2 is uniquely determined by s up to a nonzero scalar, since ker b1 and the line image b1 are unique. We have seen above that M0 (s) depends only on b . This is not true for L1 (s) , e.g. for b sending all of P to one point every hyperplane occurs as L1 (s) for some contractive sequence with lim P s = b . Note that for a contractive sequence s and h 2 GL we have
L1 (h s) = L1 (s) and L1 (s h) = h?1 L(s) :
(3.9)
3.10 Lemma
For a contractive sequence s the sequence P s of projective transformations converges uniformly on compact subsets of P r PL1(s) to the constant map with image M0 (b) . For every admissible metric on P we have
kPBn j K k ?! 0 on every compact subset K of P r PL1(s) .
Proof We have pointwise convergence of Ps on P r PL1(s) to M0 (b) . It thus
suces to prove the second claim. To show this choose coordinates as follows: Put L1 := L1 (s) , let e 62 L1 . Then the projection : V r f0g ?! P; (x) = Px , induces a map je + L1 ?! P r PL1 which is an ane coordinate system. Put e0 = b1 (e) and let L0 be a hyperplane complementary to e0 . We thus have an ane coordinate system : e0 + L0 ?! P r PL0 in some neighbourhood of M0 (b) . Decompose n Bn according to R e0 L0 :
n Bn e = n e0 + yn ; yn 2 L0 n Bn x = n (x)e0 + Bn0 (x); x 2 L; Bn0 (x) 2 L0
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Then n ?! 1; yn ?! 0 , since n Bn e ?! b1 e = e0 . And kn j L1 k ?! 0; kBn0 k ?! 0 . Hence in the coordinate systems above we can represent P Bn by 0 (x) y + B n n Dn (x) = + (x) n n
which is de ned on the complement of some hyperplane in L1 and takes values in L0 . It is now easy to see that on any compact subset K of L1 for n suciently large Dn is de ned on L1 and kDn k ?! 0 , which implies our claim.
3.11 Remark Note the di erence between M1 (b) and L1 (s) . If b sends P to one point M0 (b) then M1 (b) = ; but L1 (s) may be any hyperplane { as
noted above. It thus is not true that Ps converges uniformly to the constant map with value M0 (b) on the complement of M1 (b) , because that would imply that PBn(P ) is contained in a given neighbourhood of M0 (b) for n suciently large which is of course impossible, since PBn is a bijection of P .
We have not discussed the question yet if contractions exist. At this point we need the main algebraic result of [GM]. Let H be a subsemigroup of GL(V ) . Then V | considered as an H {module S | is called strongly irreducible if there is no nite union Wj of linear subspaces that is invariant with respect to H other than 0 and V . Let G be the algebraic closure of H and let G0 be the connected component of the identity in G with respect to the Euclidean or the Zariski topology, the following statement is true for both. Then V is strongly irreducible i V considered as a G0 {module is irreducible in the usual sense of the word, as is easy to see, cf. [GM, lemma 6.2]. For our subsemigroup H of GL(V ) , let H be the set of quasiprojective transformations which are pointwise limits of sequences of projective transformations induced by elements of H . The main algebraic result of [GM] is the following
3.12 Theorem Let H be a strongly irreducible subsemigroup of GL(V ) . If G contains a contraction, so does H .
This theorem is obtained from [GM Theorem 6.3] in view of [GM lemma 6.5 and the remark following that lemma]. The two concepts of proximality and contraction are of course related, as follows.
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3.13 Lemma If h is a proximal element of GL(V ) then for the semigroup H = fhn ; n 2 Ng every quasiprojective transformation of H not in H is a
contraction. Conversely, if H is an irreducible subsemigroup of GL(V ) and H contains a contraction, then H contains a proximal element.
If h is proximal then the sequence hn (hn )?1 converges to the projection onto V (h) in the decomposition V = V (h) V < (h) . Hence, any subsequence s of (hn )n2N for which Ps converges pointwise on P is a contractive sequence, hence its limit is a contraction by 3.7. Conversely, if H contains a contraction then there is a contractive sequence in H , by lemma 3.7. We may assume that M0 (s) 62 L1 (s) , by passing from s to h s for some h 2 H by 3.5 and 3.9 and using the hypothesis that H is irreducible on V . It now follows from Tits' criterion 2.1 that every element Bn of the contractive sequence s is proximal for n suciently large.
Proof
IV. The niteness result In this section we prove our basic niteness result. Consequences will be discussed in the following sections. Fix a norm k k on V .
4.1 Theorem Let H be a strongly irreducible subsemigroup of GL(V ) . Sup-
pose the algebraic closure G of H contains a proximal element. Then there is an " > 0 and for every r 1 there is a nite subset M of H such that for every g 2 GL(V ) there is a 2 M such that g is (r; ") {proximal.
If we let Xr;" be the set of (r; ") {proximal elements of GL(V ) , we thus have
M Xr;" = GL(V ) We shall check proximality using Tits' criterion (lemma 2.1). We thus need for every g 2 GL(V ) a uniform bound for the norm of Pg away from some set of singularities PW1(g ) . This set will be determined using the Cartan decomposition, as follows:
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4.2 Proposition We can associate with every g 2 GL(V ) a hyperplane W1(g)
of V with the following property. For every " > 0 there is a constant c = c(") such that for every g 2 GL(V ) for the map Pg induced on P we have
k P g j P r U (PW1(g); ") k c : Here U (A; ") = fy 2 P ; (A; y) < "g is the " {neighbourhood of a subset A of P with respect to a given admissible metric on P .
Proof
Since the claim holds for every admissible metric if it holds for one we may assume that V = Rn+1 and is the angle metric (see formula before 3.4) with respect to the standard Euclidean inner product on Rn+1 . Let g = k0 p k be a Cartan decomposition of g 2 GL(V ) , so k; k0 2 On+1 (R) and p is a diagonal matrix with positive entries 1 2 n+1 > 0 . Put W1 (g) = k?1 he2 ; : : : ; en+1 iR . To prove the claim we may assume that 1 = 1 by passing from g to ?1 1 g . Since k and k0 induce isometries of P we may assume furthermore that g = p and W := W1 (g) = he2 ; : : : ; en+1 iR . Let A be the ane plane x1 = 1 in Rn+1 . The map g j A maps A to itself and has norm 1 with respect to the Euclidean norm of Rn+1 , since 2 ; : : : ; n 1 . Let : A ?! P be the natural projection (x) = P(Rx) . Then induces a P r PW , whose inverse we call ?1 . We have di eomorphism A ?! Pg =
gjA ?1;
hence for every compact ball K = fx ; kxk Rg in A the norm of Pg is bounded on (K ) by a constant depending only on K , not on g , by 3.3 and since k g j A k 1 . This implies our claim since for every " there is such a K with P r U (PW; ") (K ) . Let us rst give a rough sketch of the proof of 4.1. We start by constructing a suciently rich nite set S of contractive sequences, see lemma 4.4. Then for a given element g 2 G we nd an s0 2 S such that M0 (s0 ) is " {distant from W1 (g) . We then nd an s 2 S such that gM0 (s0 ) is " {distant from L1 (s) . Thus n in s takes gM0 (s0 ) close to M0 (s) . The elements of S are so chosen
11
that M0 (s) is " {distant from L1 (s0 ) for every pair (s; s0 ) of elements of S . Hence m0 n g takes M0 (s0 ) close to M0 (s0 ) . Taking norms into account we obtain the theorem.
4.3 Lemma Let A: = (An)n2N and B: = (Bn )n2N be sequences in GL(V )
for which PA: = (PAn)n2N and PB: converge pointwise on P to a and b , respectively. a) Then there is a subsequence C: of (AnBn )n2N for which PC: converges to ab . b) If furthermore the sequences (An kAn k?1 )n2N and (Bn kBn k?1 )n2N converge to linear maps a0 and b0 , and a0 b0 6= 0 { equivalently, M0 (b) 6 L1 (A:) { then (An Bn kAn Bn k?1 ) converges to a non{zero multiple of a0 b0 . Hence
0 6= M0 (ab) M0 (a) and
P 6= L1 (C:) = ker(a0 b0 ):
c) If A: is contractive and M0 (b) 6 L1 (A:) then there is a contractive subsequence C: of (An Bn )n2N . For every such C: we have
lim PC: = ab; M0 (C:) = M0 (ab) = M0 (a) and
L1 (C:) = ker a0 b0
d) Similarly with the roles of A: and B: interchanged: If B: is contractive and M0 (b) 6 L1 (A:) then there is a contractive subsequence C: of (An Bn )n2N . For every such C: we have
lim PC: = ab; M0 (C:) = M0 (b) 2 M0 (a) and
L1 (C:) = ker(a0 b0 ) = ker b0 :
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Proof Part a) is proved in the proof of [GM lemma 2.7]. b) Since the sequence s = (An Bn kAn Bn k?1 ) is contained in the compact metric space of operators of norm 1 , it suces to prove that every convergent subse-
quence of s has limit a0 b0 . We thus may assume that s converges. Evaluating the convergent sequences An Bn kAn k?1 kBn k?1 and s at a vector 62 ker(a0 b0 ) , we see that kAn Bn kkAn k?1 kBn k?1 converges to a number with 0 < < 1 , which proves that s converges to ?1 a0 b0 . The other claims of b) follow easily. c) is a consequence of a) and b).
4.4 Lemma Suppose a subsemigroup H of GL(V ) is Zariski{connected and
acts irreducibly on V . If H contains a contractive sequence then there is a nite set S of contractive sequences s1 ; : : : ; st in H with the following properties: 1) The points M0 (sj ) are distinct. 2) M0 (si ) 62 L1 (sj ) for 1 i; j t . T 3) L1 (sj ) = ; 4) The lines corresponding to M0 (sj ) , j = 1; : : : ; t , span V . In fact we nd such a set S of cardinality dim V .
Proof
We prove by induction on m dim V that there are m contractive sequences s1 ; : : : ; sm in H such that 1) and 2) hold for i; j m , m 3m ) codim j\=1 L1 (sj ) = m and m
4m ) dim j_=1M0 (sj ) = m ? 1 . The case m = 0 is trivial. So suppose we have s1 ; : : : ; sm satisfying these conditions, m < dim V . We know that there is a contractive sequence s in H . There is an element 2 H such that m
M0 ( s) = M0 (s) 62 j_=1M0 (sj )
(4.5)
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since V is an irreducible H {module. Similarly for every j m there is a 2 H such that M0 ( s) = M0 (s) 62 L1 (sj ) ( 4:6j ) Every one of the conditions (4:5) and (4:6)j de nes a Zariski{open subset of 0 s in H , hence there is a 2 H satisfying all of them. A similar argument using 3.9 shows that there is an element 2 H such that s1 ; : : : ; sm ; s satisfy 1m+1 through 4m+1 .
4.7 Proof of theorem 4.1 We work our way along the following sketch of proof. Let S = fs1 ; : : : ; st g be a sequence as in the preceding lemma. Fix g 2 GL(V ) . For " suciently small there is an si 2 S such that M0 (si ) is " {distant from W1 (g) . Then there is an sj 2 S such that gM0 (si ) is " {distant from L1 (sj ) .
Now apply an element j in the contractive sequence sj which takes gM0 (si ) close to M0 (sj ) . Since M0 (sj ) is " {distant from L1 (si ) a i suciently close to si takes j gM0 (si ) close to M0 (si ) . Finally g takes i j gM0 (si ) back close to gM0 (si ) . It then follows that the set of maps M = f i j g has the desired properties. To get notations straight, look at the following picture:
j
gM0 (si )
y
M0 (sj )
i
y
M0 (si )
g
y
gM0 (si )
Here a y b means that maps a close to b . So let S = fs1 ; : : : ; st g be as in the preceding lemma. Fix an admissible metric P on P . Then "1 = inf x2P j =1 (x; L1 (sj )) is positive by 3), hence for every point x 2 P we have (x; L1 (sj )) > "21t for some j . Similarly, for every hyperplane L in P we nd a number "2 such that (L; M0 (sj )) > "22t for some j , using 4). So there is a number " > 0 with the following properties. For every point x 2 P we have
(x; L1 (sj )) > 2" for some j:
(a)
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For every hyperplane L in P we have
(L; M0 (sj )) > 2" for some j: (M0 (si ); L1 (sj )) > 2" for 1 i; j t:
(b) (c)
So for a given g 2 GL(V ) there are numbers 1 i; j t , such that
(M0 (si ); W1 (g)) > 2"
(d)
(gM0 (si ); L1 (sj )) > 2":
(e)
kg j U (M0 (si); ")k c
(f)
and
Hence
where c = c(") 1 is the constant of proposition 4.2, hence
g(U (M0 (si ); "c?1 ) U (gM0 (si ); "):
(g)
Now choose for every sk 2 S an element k of the sequence sk such that
k (P n U (L1 (sk ); ")) U (M0 (sk ); 21 "c?1 )
(h)
k k j P n U (L1 (sk ); ")k
(i)
and
where > 0 will be xed later on. Then
j U (gM0 (si ); ") U (M0 (sj ); 21 "c?1 )
(j)
i j U (gM0 (si ); "c?1 ) U (M0 (si ); 21 "c?1 )
(k)
and
15
by (e), (h)j , (c) and (h)i , hence
g i j U (gM0 (si ); ") U (gM0 (si ); 2" )
(l)
by (f) and
kg i j j U (gM0 (si); ")k c 2
(m)
by (i) and (f). Hence if we choose suciently small depending on r the element g i j is (r; ") {proximal by Tits' criterion 2.1.
4.8 Addendum Note that { independent of r { a set M of (dim V )2 elements
suces.
Since in 4.4 we can take t dim V and in the proof above we found #M = t2 . One cannot hope that a set M with fewer than (dim V )2 elements will work since the condition that \ g is proximal" corresponds to \ gV + ( ) is not near V < ( ) " and \ gV + ( ) 2 V < ( ) " is a codimension one condition on g . The following sharpening of the theorem may be useful (see [PR] lemma 3.7 .)
4.9 Addendum In the theorem we can replace \ g is (r; ") -proximal" by
\ n g is (r; ") -proximal for every natural number n ".
Proof The map = i j constructed in the proof sends U := U (M0 (si); 21 "c?1) to itself by (e) and (h) and has norm k j U k 2 by (e) and (i) , hence we have for n 2 N the inclusion n V U , where V := U (gM0 (si ); "c?1 ) , as in (k) , and hence g n V U as in (l) and kg n j V k c: 2n as in (m) . Recall the following result of the same sort: If there is one proximal element, there are many.
4.10 Theorem [Ti; Ma 4, Appendix B]
Let F be a subgroup of GL(W ) which is connected in the Zariski topology. Suppose W is irreducible as a F { module. If F contains a proximal element the set X = fx 2 F ; x and x?1 are proximalg is Zariski dense in F .
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We here obtain an analogous result for semigroups.
4.11 Corollary Let H be a subsemigroup of GL(V ) which is connected with
respect to the Zariski topology. Suppose V is irreducible as an H {module. If H contains a proximal element { hence if the Zariski closure of H contains a proximal element { then the set of proximal elements of H is Zariski dense.
Proof Let P be the set of (r; ") {proximal elements of H , where (r; ") is as in theorem 4.1. For every h 2 H there is an element of the nite subset M of H of theorem 4.1 such that h 2 P . So H is contained in the nite union S
?1 P , 2 M . Let P be contained in an algebraic set A . Then H ?1 A
2M ? 1 hence H is contained in A for one 2 M since H is Zariski connected. But H A implies that for the Zariski closure G of H { a group { we have G = G A , q.e.d.
V. The Lyapunov ltration In this section we extend our niteness result for proximal maps to other types of Lyapunov ltrations and to direct sums of strongly irreducible representations. Let (V; k k) be a normed vector space over a local eld k of nite dimension and let g be a linear endomorphism of V . For every vector v 2 V de ne the Lyapunov exponent g (v) := lim sup n?1 log kgn (v)k: (5.1) n!1
In the terminology of [GM] this is the \Lyapunov index of the sequence gn ; n 2 N " in case k = R . The Lyapunov exponent is independent of the norm chosen. It has the following formal properties of an
17
5.2 \Ultrametric norm over the trivially valued eld k " For v; w 2 V and = 6 0 in k we have g (v) = g (v); g (v + w) max ( g (v); g (w)); g (0) = ?1:
(a) (b) (c)
5.3 For r 2 R = R [ f1g de ne Vr = fv 2 V ; g (v) rg and
V 0 such that for every r > 1 there is a nite subset M of H such that M:P ( ; r; ") = GL(V ) .
The proof is implied by the following results. The proof of the rst claim follows L from the following lemma applied to e2 ^n?e V in view of the remark following the de nition of (r; ") {proximality in section two.
5.15 Lemma
Suppose a representation : H ?! GL(V ) decomposes into strongly irreducible subrepresentations = 1 t . If i (H ) contains a proximal element for every i = 1; : : : ; t , then there is an element h 2 H which is proximal for every i = 1; : : : ; t .
21
Proof For every i we have the representation i : H ?! GL(Vi) , the subsemi-
group Hi = i (H ) of GL(Vi ) and the corresponding semigroup Hi of quasiprojective maps of PVi . The set of all pointwise limits of (H ) in H 1 H t forms a subsemigroup H of H 1 H t , the proof is as for the case of one factor [GM, Lemma 2.7]. For every i there is an element hi 2 H for which i (hi ) is proximal. Hence there is a contractive sequence i (si ) on Vi with M0i := M0 (i (si )) consisting of one point and Li1 \ M0i = ; where Li1 = L1 (i (si )) . Now let a = (a1 ; : : : ; at ) 2 H be such that dim M0 (ai ) is minimal. We claim that this sum is zero. This implies the lemma as in 3.13. If a linear automorphism g is proximal, so are all its powers gn ; n 1 . We may assume that H is a subsemigroup of GL(V ) and hence assume that H is Zariski{connected. Suppose that dim M0 (ai ) is positive, say dim M0 (a1 ) > 0 . We shall compose a with an element b = (b1 ; : : : ; bt ) 2 H such that
M0 (ai bi) M0 (ai ) for every i with strict inclusion for i = 1 . It suces to nd a b 2 H such that b1 is a contraction and M0 (bi) 6 L1 (ai ) for every i; by 4.3 b) and 4.3 d). There is a b 2 H such that b1 is a contraction, since 1 (H ) contains a proximal element, by 3.13. The subspace L1 (ai ) is a proper subspace of P(Vi) . So by irreducibility of i there is an element h 2 H such that M0 (hbi ) = hM0 (bi) 6 L1 (ai ) . This is a Zariski{open condition. So there is an h 2 H such that these conditions are satis ed for every i = 1; : : : ; t , by Zariski{connectedness of H . So hb has all the required properties and we thus get a contradiction, proving that dim M0 (bi) = 0 . Under the assumption of 5.14, let S be a nite set of sequences in H such that W i (s) is contractive for every i = 1; : : : ; t . Put Mi0 (S ) = M 0 (i (s)) and s2S T 1 0 f Li (S ) = L1 (i (s)) and let Mi (S ) be the linear subspace of Vi corresponding s2S to Mi0 (S ) . We say that S is in general position if the following condition holds for every i = 1; : : : ; t .
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(GP) For every pair (S 0 ; S 00 ) of subsets of S we have
fi0 (S 0) = inf(card S 0; dim Vi); dim M codim L1i (S 00 ) = inf(card S 00 ; dim Vi) and
fi0 (S 0) + L1i (S 00 )) = inf(dim Vi; dim Mfi0 (S 0) + dim L1i (S 00)): dim(M
5.16 Lemma Suppose the hypotheses of 5.15 hold and H is Zariski{connected.
Then for every nite set S as above in general position there is a sequence s such that S [ fsg is in general position.
Proof
Recall that two subspaces A and B of a vector space U are called transversal , denoted A t B , if A \ B = 0 or A + B = U . So the last condition fi0 (S 0) t L1i (S 00 ) . Now there is a sequence s of general position is equivalent to M in H such that i (s) is contractive for every i = 1; : : : ; t , by 5.15 and 3.13. For every pair S 0 , S 00 of subsets of S there is a 2 H such that
fi0 ( s) = Mi0 (s) t Mfi0 (S 0) + L1i (S 00); M
(a)
by irreducibility of i . So there is a 2 H such that (a) holds for every pair S 0 , S 00 of subsets of S . Similarly, there is a 2 H such that
fi0 (S 0) \ L1i (S 00); L1i (s ) = ?1L1i (s) t M
(b)
fi0 (S 0) \ L1i (S 00) = 0 , and, if M fi0 (S 0 [ f sg) \ L1i (S 00 ) L1i (s ) = ?1L1i (s) t M
(c)
It then follows that S [f s g is in general position, the case (S 0 [f s g; S 00 ) by a), the case (S 0 ; S 00 [ f s g) by b), and the case (S 0 [ f s g; S 00 [ f s g) by a), b) and c). We are now ready to prove our theorem.
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5.17 Theorem
Let H be a semigroup and let : H ! GL(V ) be a representation. Suppose decomposes into strongly irreducible representations = 1 t . Assume H contains for every i an element hi such that i (hi ) is proximal. Then for a given norm on V there is an " > 0 and for every Q r > 1 a nite subset M of H such that for every g = (g1 ; : : : ; gt ) 2 GL(Vi ) there is a 2 M such that i ( ) gi is proximal for every i .
Proof We just give a sketch of proof, since the details are very similar to those
of 4.7. We may assume that (H ) is Zariski{connected. Let S be a set in general Pt position of cardinality greater than (di ? 1) , where di = dim Vi . For a given set i=1 H1 ; H2 ; : : : ; Ht of hyperplanes Hi in Vi there are at most di ? 1 elements s 2 S such that Mi0 (s) 2 Hi , hence there is an element s 2 S such that Mi0 (s) 62 Hi for every i . Similarly, for a given set x1 ; : : : ; xt of points xi 2 PVi there is an element Q s 2 S such that PL1i (s) 63 xi for every i . Now x g = (g1 ; : : : ; gt ) 2 i GL(Vi ) . For " suciently small there is an s 2 S such that Mi0 (s) is " {distant from W1 (gi ) for every i . Then there is an s0 2 S such that gi Mi0 (s) is " {distant from L1i (s0 ) . Now apply an element 0 in the sequence s0 which takes giMi0 (s) close to Mi0 (s0 ) . In fact we want a compact subset Ki0 of the complement of L1i (s0 ) taken to a small neighbourhood of Mi0 (s0 ) and the norm of 0 on Ki0 small. Now Mi0 (s0 ) is " {distant from L1i (s) , by the third condition of general position. So a properly chosen element of the sequence s takes a compact subset Ki of the complement of L1i (s) { and in particular Mi0 (s0 ) { close to Mi0 (s) and has small norm on Ki . Finally gi takes
0 Mi0 (s) back close to giMi0 (s) and there is a universal bound for the norm of gi outside a neighbourhood of W1 (gi) for every i . We thus prove our claim using Tits' criterion.
5.18 Remark The proof shows that there is such a set M of cardinality Pt (1 + (di ? 1))2 . i=1
5.19 Remark There is a similar theorem for a direct sum of representations i
and types i of ags for every i . The proof follows from 5.17.
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VI. Proximal elements in reductive groups In this section we give necessary and sucient conditions for when the image of a reductive group under an irreducible representation contains a proximal linear map. The conditions are given in terms of the highest weight of . At the beginning of this section we state in which sense we can restrict attention to irreducible representations. At the end of the section we apply our main niteness result to obtain many R -regular elements.
6.1
Let H be a subsemigroup of GL(V ) . Suppose H contains a proximal element g with corresponding line V + and hyperplane V < . Let W 0 be the T maximal H -submodule of V contained in V < . So W 0 = h2H hV < . Also
W 0 = fv 2 V ; g (v) < spectral radius of g0 for every g0 of the form g0 = h g h?1 ; h 2 H g: Let W be the smallest H -submodule of V=W 0 containing the image of V + . Then WC is the unique minimal H -submodule of (V=W 0 )C . In particular, the corresponding representation : H ?! GL(W ) is absolutely irreducible and the Zariski closure of (H ) is a reductive subgroup of GL(W ) .
Proof We may assume that W 0 = 0 . Then for every non{zero vector v 2 VC 2n hv g < there is an element h 2 H such that hv 62 (V )C . Hence kg2n hvk converges for n ! 1 to a non{zero vector of (V + )C , for some norm k : k on VC . So (V + )C is contained in every non{zero H -submodule of VC .
If the Zariski closure of H is reductive we obtain 6.2 Proposition Let H be a subsemigroup of GL(V ) whose Zariski closure He is reductive. Suppose He contains a proximal element with corresponding line V + and hyperplane V < . Let W be the smallest H submodule of V containing V + and let W 0 be the maximal H -submodule contained in V < . Then V = W W 0 and W is an absolutely irreducible H -submodule of V .
Proof In the claims we may suppose H = He , in which case everything follows from 6.1.
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Let G be a reductive linear algebraic group de ned over R and let : G ?! GL(V ) be an absolutely irreducible representation of G de ned over R . We want to give criteria for when (GR ) contains a proximal element. Since the center of G acts by homotheties on VC it suces to consider the case when G is semisimple. Recall the theory of dominant weights of representations of reductive groups. Let G be a complex linear algebraic semi{simple group and let B be a Borel subgroup of G containing a maximal torus T of G . Then for every irreducible representation : G ! GL(V ) de ned over C the restriction of to T decomposes into weight spaces V , 2 X (T ) = Hom(T; GL1 ) . Let () be the set of weights of the representation , i.e. () = f 2 X (T ); V 6= 0g . Let be an order on X (T ) for which the roots of T on B are positive. There is a unique maximal weight with respect to and the corresponding weight space V is one{dimensional. The line V can also be characterized as the unique line in V stabilized by B . Let be the root system of G with respect to T and let + be the set of roots of T on B . Let be the unique basis of the root system contained in + . Then every weight 2 () is of the form
=?
X
a2
na a with na 0:
Let us now turn to the real case. Suppose our group G , the vector space V and the representation are de ned over R , that T contains a maximal R -split torus S of G and that B is contained in a minimal parabolic R -subgroup P of G . Let R be the basis of the root system R of S on G which is contained in the set R + of roots of S on P . Let j : X (T ) ?! X (S ) be the restriction homomorphism and let = fa 2 ; j (a) = 0g . We sometimes write () instead of () . 6.3 Theorem The following conditions are equivalent 1) (GR ) contains a proximal element. 2) The multiplicity of j () = jS is one. 3) There is no weight 2 () such that ? is a linear combination of elements of . 4) is orthogonal to .
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5) is xed by the subgroup of the Weyl group W = W (T; G) generated by the re ections ra corresponding to elements a 2 . 6) V is stabilized by P . In particular, (GR ) contains a proximal element if G is R -split, i.e. if S = T , by 2). The proof will occupy most of the remainder of this section. Recall that every element g of GL(V ) has a multiplicative Jordan decomposition g = s u = u s which is uniquely determined by the conditions that s is semisimple and u is unipotent. Every semisimple element s 2 GL(V ) has a polar decomposition g = p k = k p uniquely determined by the following conditions: p and k are semisimple, the eigenvalues of p are positive real and the eigenvalues of k have modulus one. The following lemma is obvious. 6.4 Lemma If g 2 GL(V ) is proximal with corresponding line V + and hyperplane V < then so are its semisimple part s and its polar part p , with the same line and hyperplane. Furthermore, the moduli of the eigenvalues on V + and V < remain unchanged when passing from g to s to p .
Proof of the theorem
1) =) 2) Suppose g is a proximal element of (GR ) . We may suppose that g is an element of the (Euclidean-) connected component ((GR )) of (GR ) which is the same as ((GR ) ) , see [B] 7.4. It follows that the polar part of g is also proximal which is of the form (p) for some polar element p 2 GR . Since any polar element is contained in a maximal R -split torus and maximal R -split tori are conjugate by elements of GR we may assume that p 2 (SR ) . Using the action of the relative Weyl group R W = N (S )=Z (S ) = N (S )R =Z (S )R of G over R we may assume that p is actually in the Weyl chamber C = fs 2 (SR ) ; b(s) 1 for b 2 R g . But R
= j ( r )
by [BT] 6.3 and 6.8. So a(p) is positive real for every a 2 X (T ) and (p) (p) for every 2 () . Since (p) is proximal we must have (p) > (p) for every 2 () other than . In particular, the complex line V is the weight space
27
for j () = jS in the decomposition of the complex vector space V into weight spaces with respect to S . 2) =) 3)
Write 2 () in the form
= ?
X
a2
na a:
Then na are non{negative integers. The dominant weight has multiplicity 1 . For the restriction j ( ) of to S we thus have
j ( ) = j () = Now j () R [ f0g , hence
j ( ) = j () ? where k(b) =
P n(a) .
X
a2
na j (a):
X b 2 R
k(b) b;
(6.5)
j (a)=b
3) =) 1) Again, let C (SR ) be the Weyl chamber fs 2 (SR ) ; b(s) 1 for b 2 R g . Then the computation above shows that for s in the interior C of C , i.e. if s 2 (SR ) and b(s) > 1 for every b 2 R , we have (s) < (s) for 6= in () , provided ? is not a linear combination of . Hence (s) is proximal for s 2 C . 3) () 4) There is a criterion due to Satake for R and C and for arbitrary elds of characteristic zero to Borel{Tits (see [BT] 12.16 and the reference there) as to when for a given irreducible representation with dominant weight a subset P of is of the form (q) = fa 2 ; na 6= 0g for some weight q = ? na a , a2 as follows: A subset of is of the form (q) i fg [ is connected. Here connectedness is de ned using a graph de ned like the Dynkin diagram. So there is a subset of which is of the form (q) for some q 2 () i the pair (; a) is an edge for at least one a 2 , i.e. i is not orthogonal to at least one a 2 . This shows the equivalence of 3) and 4) .
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4) () 5) () 6) Let W 0 be the subgroup of the Weyl group W (T; G) in 5), i.e. generated by ra , a 2 . The equivalence of 4) and 5) is clear. The group PC is the standard parabolic corresponding to the subset of , by [BT] 6.3. S We thus have the Bruhat decomposition PC = w2W 0 BwB . Since B stabilizes V this shows the equivalence of 5) and 6).
6.6 Remark
The proof shows that if the group (GR ) contains a proximal map then every inner point of a Weyl chamber C maps under to a proximal map. There are representations, though, for which also elements on walls of C { hence singular elements { give proximal maps. More precisely, the proof yields the following corollary. For s 2 C de ne type(s) = fb 2 R ; b(s) > 1g and cotype(s) = fb 2 R ; b(s) = 1g and similarly type (s) = fa 2 ; a(s) > 1g and cotype (s) = fa 2 ; a(s) = 1g: We have by the results of [BT] cited in the proof of 1) =) 2) type (s) = fa 2 ; j (a) 2 type (s)g and cotype (s) = fa 2 ; j (a) 62 type (s)g; where j : X (T ) ?! X (s) is the restriction homomorphism.
29
Suppose the hypothesis of the theorem. So we do not assume that (GR ) contains a proximal element. The theorem may be considered as the special case type (s) = 0 R , or equivalently cotype (s) = .
6.7 Corollary Suppose ful lls the hypotheses of the theorem. Then for s 2 C
the following conditions are equivalent. 1) (s) is proximal 2) is xed by the standard parabolic subgroup of G of type cotype (S ) . 3) is xed by the standard parabolic R -subgroup R Pcotype(s) . 4) is xed by the centralizer ZG (s) of s in G . 5) is orthogonal to cotype (S ) . 6) For every 6= in () we have k(b) > 0 for at least one b 2 type(s) .
Proof The equivalence of 1) and 6) follows from 6.5. The equivalences of 2), 3)
and 5) are proved as the corresponding equivalences of 3) through 6) in the theorem. Finally, the standard parabolic P of G whose type is cotype (s) is generated by B and ZG (s) , which shows 2) () 4). Recall from 6.4 that (g) is proximal i (pol(g)) is proximal, where p = pol(g) is the polar part of g . Since p is conjugate in GR to a unique element s of C , we can de ne type (g) := type (p) := type (s); and 6.7 gives a complete answer to the question when (g) is proximal. The following theorem ts into the framework of the present chapter on representations of algebraic groups, but does not use any of its results. The proof uses some basic facts about R -regular elements from [PR], in particular a connection with proximality. Let G be a connected semi{simple algebraic group de ned over R and let G(R) be the group of R {points of G . Let Ad be the adjoint representation of G(R) on its Lie algebra g . We assume that G(R) is noncompact. An element g 2 G(R) is called R {regular if the number of eigenvalues, counted with multiplicity, of modulus 1 of Ad g , is minimum possible. It is known that every R {regular element is
30
semisimple (see [PR]) and it is easy to see that an element of SLn (R) is R { regular i all its eigenvalues in the natural n {dimensional representation are real and distinct. The rst two parts of the following theorem are also proved in [BL] and [P] for the case that ? is a subgroup.
6.8 Theorem
Let ? be a Zariski dense subsemigroup of G(R) . Then ? contains an R {regular element. In fact, the set of R {regular elements of ? is Zariski{dense. There is a nite subset M of ? such that for every g 2 G(R) at least one of the elements g; 2 M , is R {regular.
Proof Let G(R) = KAN be an Iwasawa decomposition of G(R) , let n be the Lie algebra of N and let k be its dimension. In the representation ^k Ad of G(R) on ^k g let V be the smallest G(R) {submodule containing the one{dimensional subspace ^k n and let : G(R) ! GL(V ) be the corresponding representation.
The representation is irreducible since V is a highest weight module, see [PR 3.1]. An element g 2 G(R) is R {regular i (g) is proximal, by [PR Lemma 3.4]. So the theorem follows from theorem 4.1.
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Herbert Abels Fakultat fur Mathematik Universitat Bielefeld Postfach 100 131 33501 Bielefeld Germany
G.A. Margulis Department of Mathematics Yale University New Haven, Connecticut 06520 U.S.A.
G.A. Soifer Kemerovo State University Krasnaya 6 650043 Kemerovo Russia
