A twisted invariant Paley-Wiener theorem for real reductive groups

A twisted invariant Paley-Wiener theorem for real reductive groups Patrick Delorme and Paul Mezo November 13, 2007 Abstract Let G+ be the group of real points of a possibly disconnected linear reductive algebraic group defined over R, which is generated by the real points of a connected component G0 . Let K be a maximal compact subgroup of the group of real points of the identity component of this algebraic group. We characterize the space of maps π 7→ tr(π(f )), where π is an irreducible tempered representation of G+ , and f varies over the space of smooth, compactly supported functions on G0 , which are left- and right-K-finite. This work is motivated by applications to the twisted Arthur-Selberg trace formula.

1

Introduction

Let G+ (resp. G) be the group of real points of a possibly disconnected linear reductive algebraic group G+ defined over R (resp. of its identity component). We assume that G+ is generated by one of its connected components, whose set of real points G0 is assumed to be non-empty. Then G0 generates G+ . Let K + be a maximal compact subgroup of G+ and let K be the maximal compact subgroup K + ∩ G of G. The intersection G0 ∩ K + in non-empty. We may therefore fix σ ∈ G0 ∩ K + so that G0 = Gσ. In fact, we may choose σ so that it fixes a minimal parabolic subgroup P m = M m Am N m . The present article has four principal goals: A To classify the irreducible tempered representations π of G which are σ-stable, i.e. equivalent to their σ-conjugates π σ . This is equivalent to classifying the irreducible representations of G+ whose restrictions to G are irreducible and tempered. The set of equivalence classes of irreducible tempered σ-stable representations will be ˆ σtemp . denoted by G B To classify (using A) the irreducible admissible representations of G which are σstable. C To construct a canonical operator Sπ which intertwines π σ with π for every irreducible tempered σ-stable representation π. We will denote by Θσπ the distribution on G defined by Θσπ (f ) = tr(π(f )Sπ ), f ∈ Cc∞ (G). 1

ˆ σtemp → C which are σ-twisted invariant Fourier D To characterize the functions F : G transforms of some left- and right-K-finite element f ∈ Cc∞ (G), i.e. functions of the form ˆ σtemp . π 7→ Θσπ (f ), π ∈ G This characterization is required for the general proof of the invariant trace formula as given in [Art88] (see p. 505). Let us briefly recall previous works in the case that G+ is connected, i.e. when G = G+ . A was achieved by Knapp and Zuckerman ([KZ82]) using limits of discrete series (see also [She82] when G is disconnected as a Lie group), and by Vogan ([Jr.79]) using minimal K-types. B is known as the Langlands classification ([Lan89]). When G = G+ , C is trivial. D is the work of Clozel with the first author ([CD84], [CD90]). These two articles used the work of Knapp and Zuckerman and the work of Vogan. The result of the first author in [Del84] on Harish-Chandra homomorphisms and transitional spaces was also a crucial ingredient, as was Arthur’s Paley-Wiener theorem ([Art83]). Turning to the case of G 6= G+ , A-D were solved in [Del91] for complex groups G and σ equal to complex conjugation relative to a quasi-split inner form. The solution relied mainly on a very nice classification for A, and a restriction theorem from [CD90], which is generalized in the present work. Later, the second author treated the case of G = GL(n, R) and some automorphisms in the case of G = SL(n, R), exhibiting some new phenomena ([Mez04]). More recently, he treated the case of general G when σ is an involution ([Mez07]). The proof assumed that all R groups were trivial, but included an approach to the general case. This work is the starting point of the present article. We would like to point out that Bouaziz ([Bou94]) has proved an invariant PaleyWiener theorem and that Renard ([Ren97]), has proved a twisted invariant PaleyWiener theorem for functions which are not K-finite. Neither of the results may be deduced from the other. To apply the Paley-Wiener theorem to the Arthur-Selberg trace formula ([Art88], [Art89]) K-finite functions are needed. This is our main motivation in studying K-finite functions. Our solutions to A-D unfold as follows. We obtain the classification of A (Theorem 1) by combining the classifications of Vogan and Knapp-Zuckerman with the characterization of σ-stable representations given in [Mez07]. The latter characterization uses automorphisms of G which are attached to generalized principal series representations. The solution of B follows from A and the Langlands classification (cf. Proposition 3.1 [Mez07]). Any generalized principal series representation of A depends on a continuous parameter. The canonical intertwining operators of C (Lemma 6) are conjugates of operators which are independent of these continuous parameters (Lemma 5). The operators of conjugation are normalized intertwining operators as in [KS80]. This independence from the continuous parameters plays a key role in the growth estimates of the twisted invariant Fourier transforms of compactly supported functions on G (Lemma 8). A slight change in perspective, carries us from twisted invariant Fourier transforms to twisted characters ((5.2)). The relations between the twisted characters arising from A and C are studied systematically in Proposition 2. These relations are sufficient 2

for us to formulate our twisted invariant Paley-Wiener theorem (Theorem 3). The proof of our theorem generalizes the proof of Theorem 1 [CD90] to the twisted context. This generalization presents several obstacles. One must correlate the properties of the R-groups with those of the automorphisms attached to the generalized principal series of A. One must also generalize a restriction theorem for polynomials invariant under an automorphism of a Dynkin diagram (Theorem 5). The heart of Theorem 3 is Proposition 4, where the relations of Proposition 2 again play an important part. Theorem 3 has a corollary (Theorem 4) motivated by the twisted Arthur-Selberg trace formula (see p. 505 [Art88] and section 11 [Art89]). We do not need to list the relations of Proposition 2 to state it, but we do need to introduce some notations. Let P = M AN a parabolic subgroup of G, and P 0 be the intersection of its normalizer P + in G+ with G0 . We assume in the following that P 0 is non-empty. Similarly, let L+ be the intersection of the normalizer of L = M A in G+ with P + . Let M + be the subgroup of L+ generated by τ and M . Then L+ = M + A, P + = L+ N and N and A are normal subgroups of P + and L+ respectively. Note that A is not necessarily in the center of L+ . Let a be the Lie algebra of A. We will repeatedly use the convention that if a group J acts on a vector space E, and X ⊂ J, then E X denotes the space of elements of E fixed by all of the elements of X. Now, given a tempered unitary representation ε+ of M + , whose restriction to + M is irreducible, and λ ∈ ia∗L , then ε+ ⊗ eλ ⊗ 1N is a unitary representation of P + . + We denote by πεP+ ,λ the corresponding unitarily induced representation from P + to G+ . Theorem: + ˆ+ Let φ be a function defined on the tempered dual G temp of G , which is non-zero only ˆ0 temp of (equivalence classes) of representations of G+ whose restrictions on the subset G to G are irreducible and tempered. We identify φ with its Z-linear extension to the the set of tempered representations of G+ of finite length. Then there exists a left- and right-K-finite f ∈ Cc∞ (G) such that ˆ+ tr(π + (f )π + (σ)) = φ(π + ), π + ∈ G temp if and only if φ satisfies the following conditions: ˆ such that φ(π + ) = 0 if the (i) There exists a finite subset Γ of the unitary dual K + restriction of π to G does not contain any K-type in Γ. ˆ0 temp have the same restriction to G, then π + (σ) = c π 0+ (σ), (ii) If π + and π 0+ ∈ G where c is a root of unity. In this case, φ(π + ) = c φ(π 0+ ). (iii) Let Q = MQ AQ NQ be a parabolic subgroup of G with Q0 non-empty. Assume that ε+ is a representation of MQ+ whose restriction to MQ is tempered and irreducible, and +

M+

+

λ ∈ a∗MQ . Then λ 7→ φ(πεQ+ ,λ ) is the Fourier transform of a function in Cc∞ (aQ Q ). Theorem 4 is a more precise version of this theorem, taking into account the size of the support of the function f . Acknowledgment: The first author would like to thank J. Carmona for many useful conversations during the elaboration of this article.

3

2 2.1

Preliminaries Generalized principal series

Given a group J, elements g, x ∈ J, and X ⊂ J, we set g.x := gxg −1 and g.X := gXg −1 . If π is a map defined on a a subgroup H of J, and H is normalized by some x ∈ G, then π x denotes the map defined on H by π x (h) = π(x−1 .h) for h ∈ H. Note that π may be a representation of H. Recall from the introduction that E J is the subspace of J-fixed elements in a vector space E. If J is a Lie group, j will denote its Lie algebra and Ad will denote the adjoint representation of J on j. Let us reconsider the objects defined from G+ as in the introduction. The set G0 generates G+ , indeed for any σ ∈ G0 one has G+ = ∪ni=1 σ i G, where n is the least positive integer such that σ n ∈ G. We choose a maximal compact subgroup K + of G+ , which is the fixed point group of some Cartan involution θ, in the sense of Proposition 1.10 [BHC62]. Set K = K + ∩ G. According to Proposition 1.10 [BHC62], K is a maximal compact subgroup of G and K 0 = K + ∩ G0 is non-empty. Clearly, σ.K = K for any σ ∈ K 0 . We fix such a σ from now on. Let 0 G be the intersection of the kernels of the continuous characters of G with values in R∗+ . Let AG be the analytic subgroup of G whose Lie algebra is the subspace of the anti-invariant elements under θ in the center of the Lie algebra g. Actually, with our definition of a Cartan involution, AG is just the identity component of the group of real points of a maximally split torus in the center of G. One calls AG the split component of the center of G. We fix a symmetric bilinear form B on g, which is invariant under the adjoint group Ad G, and under θ. In addition, we may assume that the quadratic form kXk2 = −B(X, θX) is positive definite. Suppose P is a parabolic subgroup of G. Then L = P ∩ θ(P ) is its θ-stable Levi subgroup. The decomposition P = M AN , where M = 0 L, A = AL , and N is the unipotent radical of P , is called the Langlands decomposition of P . We fix a minimal parabolic subgroup Pm = Mm Am Nm and recall that a parabolic subgroup is said to be standard if it contains Pm . Let W (A) be the quotient of the normalizer of A in K by its centralizer. The group W (A) acts naturally on A and on the (equivalence classes of) representations of M . If δ is a unitary representation of M and λ ∈ ia∗ , we denote by Wδ (resp. Wλ ) the stabilizer in W (A) of δ (resp. λ). We also define Wδ,λ := W (A)δ ∩ W (A)λ . Let ρP be the half-sum of the roots of a∗ determined by the root spaces in n. We P denote by Iδ,λ the space of measurable functions ϕ from G to the space of δ, such that ϕ(gman) = a−λ−ρP δ(m−1 )ϕ(g), g ∈ G, m ∈ M, a ∈ A, n ∈ N, and the integral kϕk2 := K |ϕ(k)|2 dk is finite. This space is endowed with the scalar product defined from k.k. The group G acts unitarily on this space by left-translations, P and the corresponding representation is denoted by π ˜δ,λ . P Let I(δ) be the space formed by the restriction of the elements of Iδ,λ to K. Observe that I(δ) is independent of λ. The restriction is bijective and the representation R

4

P P obtained from π ˜δ,λ by “transport de structure” will be denoted by πδ,λ . This version of P P π ˜δ,λ is called the compact realization of π ˜δ,λ . For any ϕ ∈ I(δ), we will take ϕλ to be P P the unique element of Iδ,λ whose restriction to K is ϕ. The equivalence class of πδ,λ , ∗ λ ∈ ia does not depend on P with Levi subgroup M A ([KS80]). As a result, we will MA P sometimes write πδ,λ instead of πδ,λ .

2.2

A review of the classification of irreducible tempered representations: a dictionary between two points of view

Let P = M AN be a parabolic subgroup of G, δ be a discrete series representation of M , and λ ∈ ia∗ . Define A(δ) to be the set of minimal K-types of I(δ) ([Jr.79], and the references after equation (1.8) in [Del05] for G not connected as a Lie group). For µ ∈ A(δ), let I µ (δ) be the corresponding isotypic component of µ in I(δ). It is irreducible as a representation of K. We fix an element µ0 of A(δ). We shall make extensive use of the intertwining operators of Knapp and Stein ([KS80]), and of the particular normalization introduced by the first author ([Del84], [Del05] for G not connected as a Lie group). For a parabolic subgroup Q = M AV with Levi subgroup M A, A(Q, P, δ, λ) is an analytic family of unitary operators in λ ∈ ia∗ , Q P intertwining πδ,λ with πδ,λ . In addition, for any µ ∈ A(δ) there exists cµδ (Q, P ) ∈ C, µ0 with cδ (Q, P ) = 1, such that For all λ ∈ ia∗ , A(Q, P, δ, λ)|I µ (δ) is multiplication by cµδ (Q, P ).

(2.1)

If R is an another parabolic subgroup with Levi subgroup M A, then A(R, Q, δ, λ)A(Q, P, δ, λ) = A(R, P, δ, λ), λ ∈ ia∗ .

(2.2)

Notice that we are using the letter A for the normalized intertwining operators instead of A as in [KS80] or [Del84]. Let M 0 A0 be a Levi subgroup of G with M A ⊂ M 0 A0 . We assume that θ(V ) ∩ N ⊂ 0 M . Let a0⊥ be the orthogonal complement of a0 in a. We may decompose λ ∈ ia∗ as ⊥

λ = λ0 + λ00 , λ0 ∈ a0∗ , λ00 ∈ (a0 )∗ . We will require the following fact: A(Q, P, δ, λ) depends only on λ00 .

(2.3)

Indeed, by Theorem 7.6 and Theorem 8.4 [KS80], it suffices to prove (2.3) in the case where P and Q are adjacent. In that case (cf. l.c. Proposition 7.5 and definition of the γ factors in section 8 [KS80]), the unnormalized operators, as well as the normalizing factors depend only on λ00 , so (2.3) follows. For w ∈ Wδ , A(P, w, δ, λ) is an analytic family of unitary operators in λ ∈ ia∗ , P P intertwining πδ,λ with πδ,wλ . In addition, for any µ ∈ A(δ) there exists a character χµ of Wδ , such that For all λ ∈ ia∗ and w ∈ Wδ , A(P, w, δ, λ)|I µ (δ) is multiplication by χµ (w) and χµ0 is trivial. 5

(2.4)

A(P, w, δ, w0 λ)A(P, w0 , δ, λ) = A(P, ww0 , δ, λ), λ ∈ ia∗ , w, w0 ∈ Wδ .

(2.5)

Let w˜ be a representative in K of w ∈ Wδ and let uw˜ be a unitary operator intertwining δ w˜ with δ. Then there exists c ∈ C of modulus one such that A(P, w, δ, λ) = cuw˜ Rw˜ A(w−1 P w, P, δ, λ), λ ∈ ia∗ .

(2.6)

P Indeed, both sides of this equation are unitary self-intertwining operators of πδ,λ and P are analytic in λ. When πδ,λ is irreducible, these operators have to be proportional. This is true for λ in an open and dense set of ia∗ . Furthermore, the two families are constant on minimal K-types. Hence the proportionality factor is constant on this set and the assertion follows by analytic continuation. Recall that Wδ0 is the normal subgroup of Wδ consisting of those elements w for which A(P, w, δ, 0) is trivial. Because of this, each character χµ as mentioned above is a character of Wδ which is trivial on Wδ0 . Recall also that Wδ0 is the Weyl group of a root system ∆δ in a and P determines a + set of positive roots ∆+ δ of ∆δ . The positive roots ∆δ determine a unique chamber Cδ in a, i.e. the set of elements in a on which the elements of ∆+ δ are greater than zero. If ¯δ . ∆+ is empty, it is equal to a. We denote its closure by C δ Let Rδc be the subgroup of Wδ leaving ∆+ δ invariant. The group Wδ is the semidirect c 0 ˆ δ described above may product of Rδ and Wδ . Consequently, every character χµ ∈ W c be identified with a character of Rδ . Moreover, the map µ 7→ χµ is a bijection between ˆ δc . A(δ) and R 0 For λ ∈ ia∗ , let Wδ,λ be the subgroup of elements w of Wδ,λ for which A(P, w, δ, λ) is trivial (it is already trivial on the K-type µ0 ). It is the Weyl group of the root system ∆δ,λ = {α ∈ ∆δ |(α, λ) = 0}. We let ∆+ δ,λ be the subset of positive roots determined by c P . Let Rδ,λ be the subgroup of Wδ,λ preserving ∆+ δ,λ . We now introduce representations attached to the objects we have just defined (cf. [Jr.79] and [Del84] for G connected as a Lie group, and equations (1.6)-(1.8), (1.15), (1.16) [Del05] for G not connected as a Lie group). c P ˆ and πδ,H,χ,λ Let λ ∈ iC δ , H = Rδ,λ . Let χ ∈ H be the subrepresentation P of πδ,λ generated by the set of minimal K-types, µ ∈ A(δ), such that P P χµ |H = χ. Then πδ,H,χ,λ is irreducible and πδ,H,χ,λ contains the minimal K-type µ if and only if χµ |H = χ. Moreover,

(2.7)

P P πδ,λ = ⊕χ∈Hˆ πδ,H,χ,λ . P Every irreducible tempered representation arises as such a πδ,H,χ,λ , for a standard parabolic subgroup P . The data (M, δ, λ) are determined modulo conjugacy by K. Once δ is given, H and χ are unique, and λ is unique up to the action of Rδc . The notions of (2.7) actually hold in a more general setting: P Every subrepresentation of πδ,λ , λ ∈ ia∗ is characterized by the minimal K-types it contains.

6

(2.8)

c Let Diag(δ) be the set of subgroups {Rδ,λ |λ ∈ iC δ } of Rδc . For H ∈ ∗H P P Diag(δ) and λ ∈ ia , let πδ,H,χ,λ be the subrepresentation of πδ,λ generated by the minimal K-types µ such that χµ |H = χ. The subrepresentation P πδ,H,χ,λ contains the minimal K-type µ if and only if χµ |H = χ. P P πδ,λ = ⊕χ∈Hˆ πδ,H,χ,λ , λ ∈ ia∗H .

(2.9)

(2.10)

For H, H 0 ∈ Diag(δ) with H ⊂ H 0 , one has P πδ,H,χ,λ = ⊕χ0 ∈Hˆ 0 ,χ0

|H

P =χ πH 0 ,χ0 ,λ ,

0

λ ∈ ia∗H .

(2.11)

Let us justify (2.9)-(2.11). From (2.7) one sees that (2.9) and (2.10) are true for λ ∈ P iC δ and H = Rδc ,λ . After decomposing πδ,H,χ,λ into irreducible representations, using c (2.7) it is apparent that (2.9)-(2.10) also hold for λ ∈ iC δ and H ⊂ Rδ,λ . Decomposing both sides of (2.11) in a similar manner reveals that (2.11) holds for λ ∈ iC δ . This establishes (2.9)-(2.11) when λ ∈ iC δ . Now suppose λ ∈ ia∗H for some H ∈ Diag(δ). As Wδ0 is the Weyl group of ∆δ , there exists w ∈ Wδ0 such that ν := wλ ∈ iC δ . Let us c c c first show that Rδ,λ ⊂ Rδ,ν . If r ∈ Rδ,λ then rν = rwλ = rwr−1 λ = rwr−1 w−1 ν. As Wδ0 is normal in Wδ the element rwr−1 lies in Wδ0 . Therefore rν and ν are conjugate by an element of Wδ0 and both belong to iC δ . This implies rν = ν, and so the inclusion c c P P Rδ,λ ⊂ Rδ,ν is proven. Now, taking into account that πδ,λ is equivalent to πδ,ν , one sees P P P that πδ,H,χ,λ is equivalent to πδ,H,χ,ν . Assertions (2.9)-(2.11), for πδ,H,χ,λ are therefore P consequences of the parallel statements for πδ,H,χ,ν , which we proved for ν ∈ iC δ . We carry on by listing some facts relating discrete series representations to nondegenerate limits of discrete series (see section 2 [CD90] for references, essentially [KZ82] for G connected and [She82] for G not connected as a Lie group). Let λ ∈ iC δ and c Aλ be the fixed-point set in A of Rδ,λ . The centralizer of Aλ in G admits Aλ as split λ λ component and is written as M A . It is the Levi subgroup of a parabolic subgroup of G, which does not necessarily contain P or Pm . The element λ may be regarded as an element of i(aλ )∗ and one has the finite decomposition λ λ P ∼ πδ,λ = ⊕j πδMjλ ,λA ,

(2.12)

where each induced representation on the right is irreducible and λ

P ∩M πδ,0 = ⊕j δjλ ,

(2.13)

is a decomposition into nondegenerate limits of discrete series δjλ of M λ . The set {δjλ |λ ∈ iC δ } will be called the set of nondegenerate limits of discrete series which are strongly affiliated to δ. The set of Levi subgroups M λ Aλ , as λ varies over iC δ , will be called the set of Levi subgroups strongly affiliated to δ. Every irreducible tempered representation π of G occurs in a decomposition as λ λ above, i.e. is of the form πδMλ ,λA , where λ ∈ iC δ , and δ λ is a nondegenerate limit of discrete series of M λ strongly affiliated to δ. The data (M λ Aλ , δ λ , λ) are determined up to conjugacy under K. 7

Let M1 A1 be a Levi subgroup strongly affiliated to δ and δ1 be a nondegenerate limit of discrete series of M1 strongly affiliated to δ. By definition, we may choose an element ν ∈ iC δ ∩ ia∗ such that Aν = A1 and πδM1 ,ν1 A1 is irreducible. To these data ˆ in the following one may associate a subgroup H of Rδc , a character of H, and χ ∈ H c ˆ manner. Recall that the dual group Rδ acts simply transitively on A(δ), and by fixing 1 A1 µ0 we may identify A(δ) with Rδc . The set A(δ1 ) of minimal K-types πδM1 ,λ , λ ∈ ia∗1 c ˆ δc , where H = Rδ,ν corresponds to an orbit in A(δ) of the complement H ⊥ of H in R . The elements of this orbit are characterized by their restriction χ to H. This may be P seen by applying the results of (2.7) to the irreducible subrepresentation πδM1 ,ν1 A1 of πδ,ν . ∗H ν ∗ ∗ For all λ in ia = i(a ) = ia1 , one has P 1 A1 ∼ . πδM1 ,λ = πδ,H,χ,λ

(2.14)

P In fact, both sides are equivalent to subrepresentations of πδ,λ containing the same minimal K-types and so the assertion follows from (2.8). We henceforth fix a set of representatives of conjugacy classes under K of pairs (M, δ) (often abbreviated simply as δ), where L = M A is the Levi subgroup of a standard parabolic subgroup of G. This set will be called set of discrete data and denoted by DD. For (M, δ) ∈ DD, there exists by definition a unique standard parabolic subgroup with Levi subgroup M A. It will be called the standard parabolic subgroup of (M, δ). This fixes a choice of ∆+ δ and C δ .

3 3.1

The classification of irreducible tempered σ-stable representations A choice of σ

Lemma 1 There exists σ in K 0 = K + ∩ G0 such that (i) Conjugation by σ commutes with θ and Adσ commutes with the differential of θ. (ii) σ normalizes Mm , Am and Pm . (iii) Adσ is of finite order on am . (iv) We may choose our bilinear form B to be Adσ-invariant. Proof. The set K + ∩G0 is non-empty (Proposition 1.10 [BHC62]). Let σ 0 be any element of K + ∩ G0 . Then σ 0 .Pm is a minimal parabolic subgroup of G. Hence it is of the form k.Pm for some k ∈ K. Set σ = k −1 σ 0 . Clearly, σ is fixed by θ. This implies assertion (i). By definition, σ.Pm = Pm , and by (i) it is clear that Mm Am = Pm ∩ θ(Pm ) is preserved by σ. Thus σ preserves Mm and Am in view of their definitions. This proves (ii). As σ preserves the roots of am , we need only prove that σ has finite order on aG to obtain (iii). As G+ has finitely many connected components, the element σ n lies in G for some positive integer n. Hence, Adσ n is trivial on the center of G and (iii) is proven. Finally, recall that B restricted to [g, g] is a multiple of the Killing form. Hence it is invariant under Adσ. Also, the center z of g is orthogonal to [g, g] with respect to B. 8

After possibly averaging B over the finite group of automorphisms generated by Adσ restricted to z, one may assume that B is Adσ-invariant.  3.2

The definition of DDT

The next lemma is a slight improvement of Proposition 4.1 [Mez07]. Lemma 2 Let (M, δ) ∈ DD and P = M AN be the standard parabolic subgroup of G with Levi P subgroup M A. If there exists a σ-stable irreducible subrepresentation of πδ,λ for some λ ∈ iC δ then there exists kδ in K such that (i) conjugation by τδ := kδ−1 σ leaves Am , A, M , ∆+ δ , as well as the equivalence class of δ invariant. (ii) The automorphism Adτδ is of finite order on a, commutes with θ, and preserves B. P σ Proof. (i) For λ ∈ iC δ , the representation (πδ,λ ) is equivalent to πδσ.P σ ,σλ (cf. the proof of Proposition 3.1 [Mez07], the intertwining map being given by ϕ 7→ ϕσ with the notation of section 2.1 in the compact realizations). The Langlands Disjointness Theorem (pp. 149-150 [Lan89]) provides k ∈ K with kσ.M = M , kσ.A = A, and δ σ equivalent to δ k . After possibly multiplying by a representative in K of a suitable element of Wδ0 , and then by an element of K ∩ M , one finds the desired kδ . (ii) The automorphism Adτδ preserves B, since the actions of Adkδ and Adσ do. The action of Adτδ on the center is equal to that of σ, as Adkδ acts trivially. Moreover Adτδ preserves the Killing form. Hence it commutes with θ and preserves B ((iv) Lemma 1). The assertion on the finiteness order of Adτδ on a follows from the fact that it permutes the roots of a and agrees with Adσ on the center of g (cf. (iii) Lemma 1). 

We denote by DDT the set of elements (M, δ) ∈ DD satisfying the hypotheses of Lemma 2. The Langlands Disjointness Theorem tells us that an element (M, δ) ∈ DD P σ P is in DDT if and only if (πδ,0 ) is equivalent to πδ,0 . For each (M, δ) ∈ DDT we fix a kδ and τδ as in Lemma 2. Remark 1 There are several choices for DD and DDT . In the case of base change ([Del91]), the set DDT was chosen so that kδ was always trivial and hence τδ was equal to σ. Such a choice may not possible in general. 3.3

The action of τδ on Rδc and A(δ)

Fix δ ∈ DDT and let P = M AN be its standard parabolic subgroup. We choose a unitary operator Uδ on the space of δ such that δ τδ (m) = Uδ−1 δ(m)Uδ , m ∈ M.

(3.1)

Recall that we have fixed µ0 ∈ A(δ) in section 2.2. ¿From now on, we denote the differential of Adτδ by τδ whenever the action on a is required. With the same abuse, we denote Adkδ by kδ , and Adσ by σ. 9

c Lemma 3 Suppose r ∈ Rδc . Then the element τδ (r) := τδ rτδ −1 |a belongs to Rδ as a group of automorphisms of a.

Proof. As conjugation by τδ preserves K and A (Lemma 2), it is clear that τδ (r) is in W (A). Moreover, as τδ preserves δ and ∆+ δ (Lemma 2), it follows that τδ (r) preserves them as well.  Lemma 4 For every µ ∈ A(δ), one has µτδ ∈ A(δ). Proof. Suppose kC is the complexification of the Lie algebra of K, and b is a Borel subalgebra of kC containing the Lie algebra t of a Cartan subgroup of K. Let γ be a highest weight of µ with respect to b, i.e. a highest weight of some irreducible constituent of µ restricted to the identity component of K. Let ρc be the half-sum of the roots of t in b. As Cartan subgroups of K, and Borel subalgebras of kC are conjugate by elements of K, there exists k in K such that Ad(kτδ )b = b and Ad(kτδ )t = t. It is clear that kτδ (γ) is a highest weight of µτδ . As k and τδ preserve B, kγ + 2ρc k = kkτδ γ + 2kτδ ρc k = kkτδ γ + 2ρc k ¿From the definition of minimal K-types (cf. [Jr.79], Definition 5.1 for G connected as a Lie group, and [CD84] p.433 in general) this implies that µτδ is an element of A(δ).  ˆ δc on A(δ), we define Using the previous lemma and the simply transitive action of R c ˆ δ such that χ0 ∈ R µτ0δ = χ0 .µ0 , or equivalently, χµ0 τδ = χ0 . 3.4

(3.2)

τδ -stable and σ-stable representations

We are now prepared to determine exactly which of the representations defined in (2.7) are τδ -stable or σ-stable. Lemma 5 (i) The following defines a unitary operator Tδ on I(δ), (Tδ (ϕ))(k) := Uδ ϕτδ (k) = Uδ ϕ(τδ−1 .k), ϕ ∈ I(δ), k ∈ K (see (3.1)). τδ .P P τδ (ii) For all λ ∈ ia∗ , Tδ intertwines (πδ,λ ) with πδ,τ . δλ τ δ µ µ (iii) For all µ ∈ A(δ), Tδ sends I (δ) to I (δ). (iv) For w ∈ Wδ , one has Tδ A(P, w, δ, λ)Tδ−1 = χ0 (w)A(τδ .P, τδ (w), δ, τδ λ), λ ∈ ia∗ . (v) For µ ∈ A(δ), χµτδ = χ0 χτµδ . τδ .P P (vi) If H ∈ Diag(δ) and λ ∈ ia∗H then Tδ intertwines (πδ,H,χ,λ )τδ with πδ,H 0 ,χ0 ,τ λ , where δ H 0 = τδ (H), and χ0 = χ0 |τδ (H) χτδ . 10

Proof. (i) If ψ = Tδ (ϕ), k ∈ K, and m ∈ K ∩ M then, using (3.1), one has ψ(km) = Uδ δ(τδ−1 .m−1 )(ϕ(τδ−1 .k)) = δ(m−1 )Uδ ϕ(τδ−1 .k) = δ(m−1 )ψ(k). This proves that Tδ is an operator on I(δ). It is evident that it is unitary. (ii) The operator Tλ corresponding to Tδ in the non-compact realization is given by P (Tλ (ϕ))(g) := Uδ (ϕ(τδ−1 .g)), ϕ ∈ Iδ,λ , g ∈ G. τδ .P P τδ Its image lies in Iδ,τ by computations similar to those in (i), and intertwines (˜ πδ,λ ) δλ τδ .P with π ˜δ,τδ λ . Assertion (ii) now follows easily. ˆ and (iii) Let πδ be the representation of K on I(δ). Let (µ, Vµ ) be a model for µ ∈ K, µ Tµ : Vµ → I (δ) be a unitary intertwining operator. Then Tδ ◦ Tµ intertwines (µτδ , Vµ ) with the image of I µ (δ) under Tδ as may be seen from

πδ (k)Tδ Tµ v = Tδ πδ (τδ−1 (k))Tµ v = Tδ Tµ µτδ (k)v, k ∈ K, v ∈ Vµ . τδ .P τδ .P (iv) Both sides of the equation to prove intertwine πδ,τ , with πδ,τ . To see this one δλ δ wλ may use the intertwining properties of the various operators, and the fact that if an operator intertwines π and π 0 then it also intertwines π τδ and π 0τδ . On the open dense set of ia∗ where these representations are irreducible, both operators are proportional. To see that they are equal, it is therefore enough to verify that they are equal on the isotypic component of µ0 τδ . This is obvious from (iii), (2.4), and (3.2). (v) The desired equation follows by restricting the equation of (iv) to the isotypic component of µτδ ∈ A(δ), recalling the definitions of χµ and χµτδ , and applying (iii). P (vi) From (iii), the minimal K-types of (πδ,H,χ,λ )τδ are of the form µτδ , where µ runs P through the minimal K-types of πδ,H,χ,λ . Using (v), they are seen to be the elements 0 µ ∈ A(δ), such that χµ0 |τδ (H) = χ0 |τδ (H) χτδ . The claim follows from (2.7). 

The following theorem is a classification of the σ-stable irreducible tempered representations. Theorem 1 P (i) Every irreducible tempered σ-stable representation of G is equivalent to πδ,H,χ,λ for c ˆ some δ ∈ DDT , λ ∈ iC δ , H = Rδ,λ , χ ∈ H such that (1) H is τδ -stable. (2) The character χ satisfies χτδ = χ0 |H χ. The set of such characters of H will be ˆ δ ). denoted by H(τ (3) There exists r ∈ Rδc such that rτδ λ = λ. The set of such subgroups of Rδc , corresponding to r, will be denoted Diag(τδ , r). 0 P (ii) Two representations πδ,H,χ,λ and πδP0 ,H 0 ,χ0 ,λ0 as in (i) are equivalent if and only if δ = δ 0 , H = H 0 , χ = χ0 and there exists r ∈ Rδc such that λ0 = rλ.

11

Proof. (i) By Lemma 2 and section 2.2, every irreducible tempered σ-stable represenP tation of G is is equivalent to πδ,H,χ,λ for some δ ∈ DDT , λ ∈ iC δ . Since τδ = kδ σ, P σ P P the representation (πδ,H,χ,λ ) is equivalent to (πδ,H,χ,λ )τδ . By Lemma 5 (vi), (πδ,H,χ,λ )τδ τδ .P 0 0 τδ is equivalent to πδ,H 0 ,χ0 ,τ λ , where H = τδ (H), and χ = χ0 |τ (H) χ . As A is τδ -stable δ δ τδ .P (Lemma 2), the parabolic subgroup τδ .P has M A as a Levi subgroup and πδ,H 0 ,χ0 ,τ λ δ P P P is equivalent to πδ,H Thus, π is τ -stable if and only if π 0 ,χ0 ,τ λ ([KS80]). δ δ,H,χ,λ δ,H,χ,λ δ P and πδ,H 0 ,χ0 ,τ λ are equivalent. Using the remark following (2.7) and the fact that τδ δ preserves C δ (Lemma 2), this is true if and only if H = H 0 , χ = χ0 and τδ λ = rλ for some r ∈ Rδc . (ii) The assertion follows from the classification of tempered irreducible representations in section 2.2.  The σ-stable representations of Theorem 1 are induced from parabolic subgroups which are not necessarily stable under the actions of any of σ, τδ or rτδ . We close this section by relating our σ-stable representations to some representations which are induced from parabolic subgroups which are stable under the action of rτδ . Proposition 1 Let (M, δ) ∈ DDT , λ ∈ iC δ and let π be an irreducible subrepresenP 1 A1 tation of πδ,λ . Suppose that π is equivalent to πδM1 ,λ , where M1 A1 is equal to M λ Aλ , and (M1 , δ1 ) is a nondegenerate limit of discrete series strongly affiliated to δ as follows from section 2.2. Recall that a∗1 may be viewed as a subspace of a∗ , and λ ∈ ia∗1 (see (2.12)). Let Q be the parabolic subgroup of G whose Lie algebra is the sum of the a root spaces for roots α with Im(α, λ) ≥ 0. Its Levi subgroup MQ AQ is the centralizer in G of λ and contains M1 A1 . Then the representation π is σ-stable if and only if there exists P ∩M r ∈ Rδc , such that rτδ λ = λ, r˜τδ normalizes MQ AQ and Q, and (πδ1 ,0 Q )r˜τδ is equivalent P ∩M Q to ε := πδ1 ,0 Q , where r˜ is a representative in K of r. Moreover, π is equivalent to πε,λ and r˜τδ .Q = Q. Proof. We must determine when π is τδ -stable. First, Q contains a parabolic subgroup P1 with Levi subgroup M1 A1 . Indeed, let P10 = M1 A1 N10 be a parabolic subgroup of G with Levi subgroup M1 A1 . Then P1 = (P10 ∩ MQ )NQ has the required property. 1 A1 Thus, πδM1 ,λ is equivalent to πδP11,λ . Applying induction in stages, π is equivalent to π Q P ∩MQ . Let us show that the conditions of the theorem are sufficient. Suppose they (πδ

1 ,0

),λ

are satisfied. As rτδ λ = λ, the parabolic subgroup Q satisfies r˜τδ .Q = Q. The resulting representation is induced from from an r˜τδ -stable parabolic subgroup and an r˜τδ -stable representation. This implies that the induced representation is r˜τδ -stable. It is then also τδ - and σ-stable, as r˜, kδ ∈ G. This proves that the conditions are sufficient. Let us prove that they are necessary. According to (i) Theorem 1, there exists c r ∈ Rδc such that rτδ λ = λ. It follows from Lemma 2 that τδ normalizes Rδ,λ . Hence, λ r˜τδ normalizes A1 = A (section 2.2) and M1 . It therefore suffices to determine when 1 A1 1 A1 r πδM1 ,λ is r˜τδ -stable. The representation (πδM1 ,λ ) ˜τδ is easily seen to be equivalent to 1 A1 1 A1 1 A1 πδM0 ,λ , where δ10 is equal to δ1r˜τδ . Thus, π is τδ -stable if and only if πδM0 ,λ and πδM1 ,λ 1 1 are equivalent. By section 2.2, this true if and only if there exists a w ∈ NK (A1 ) fixing λ, and conjugating δ10 with δ1 . Finally, if w fixes λ, it is in MQ by definition and this P ∩M P ∩M implies that (πδ1 ,0 Q )r˜τδ is equivalent to πδ1 ,0 Q .  12

4

The classification of irreducible admissible σ-stable representations

We now turn to the resolution of B, our second principal goal. The results of this section appear essentially in section 3 [Mez07]. We include a brief review for the sake of completeness and convenience. Suppose P = M AN is a parabolic subgroup and ρ is an irreducible tempered representation of M . Suppose further that λ lies in the complexification of a∗ and that its real part lies in the positive chamber of a∗ determined by P . Langlands has shown (section P P 3 [Lan89]) that the induced representation πρ,λ has a unique irreducible quotient Jρ,λ and that every irreducible admissible representation of G is (infinitesimally) equivalent P to some such Langlands quotient. He also proved that two Langlands quotients, Jρ,λ 0 and JρP0 ,λ0 , are equivalent if and only if there exists h ∈ G such that h.P = P 0 , ρh is equivalent to ρ0 , and hλ = λ0 ; i.e. Langlands quotients are unique up to conjugation. This allows us to restrict our classification to standard parabolic subgroups P , as all parabolic subgroups are conjugate to a (unique) standard parabolic subgroup. P Theorem 2 Suppose P is a standard parabolic subgroup of G and Jρ,λ is a Langlands P σ quotient as above. Then Jρ,λ is σ-stable if and only if σ.P = P , ρ is equivalent to ρ and σλ = λ.

Proof. We provide a sketch, leaving the details to the proof of Proposition 3.1 [Mez07]. P σ Composition by σ provides an intertwining operator between (πρ,λ ) and πρσ.P σ ,σλ . This P σ equivalence induces an equivalence between the Langlands quotients (Jρ,λ ) and Jρσ.P σ ,σλ . P The “if” part of the Theorem is now immediate. For the converse, we see that Jρ,λ is equivalent to Jρσ.P σ ,σλ , and since σ.Pm = Pm , the parabolic subgroup σ.P remains standard. The Theorem then follows from the uniqueness of Langlands quotients up to conjugation.  Taken together, Theorem 2 and Theorem 1, constitute a classification of irreducible admissible σ-stable representations of G.

5 5.1

Intertwining operators and twisted characters Representations of G+ and twisted characters

The group G+ /G is cyclic and generated by the coset of any element τ ∈ G0 . Let us look at an irreducible unitary representation π + of G+ whose restriction to G is an irreducible representation π of G. Let τ ∈ G0 and let n be the order of the image of τ in G+ /G, so that τ n = g ∈ G. Then T = π + (τ ) is an intertwining operator between π τ and π, such that T n = π + (τ n ) = π(g). Conversely, if T is such an intertwining operator then it defines a representation of G+ . If T 0 is another operator which intertwines π τ with π then T 0n is proportional to π(g), by Schur’s lemma. Thus, there are exactly n distinct choices for a constant c ∈ C such that T = cT 0 and T n = π(g).

13

The τ -twisted character Θτπ of π + is the distribution on G defined by Θτπ (f ) = tr(π(f )π + (τ )), f ∈ Cc∞ (G). Now suppose τ, σ belong to G0 and σ = kτ for k ∈ K. Then one has Θσπ = Rk Θτπ

(5.1)

where R denotes the right-regular representation of G. We will speak of a τ -twisted character of π more generally by replacing π + (τ ) above with any intertwining operator between π τ and π. 5.2

The definition of the operators T (δ, H, χ, λ)

Lemma 6 (i) The operator T (δ, r, λ), defined by T (δ, r, λ) := A(P, δ, r, τδ λ)A(P, τδ .P, δ, τδ λ)Tδ , r ∈ Rδc , λ ∈ ia∗rτδ . P τδ P intertwines (πδ,λ ) with πδ,rτ . δλ τ (ii) T (δ, r, λ) sends the isotypic component I µ (δ) to I µ δ (δ) for every µ ∈ A(δ). It is independent of λ, and is equal to a scalar multiple of Tδ on each I µ (δ). (iii) Let r˜ be a representative in K of r ∈ Rδc and ur˜ be a unitary intertwining operator between r˜δ and δ. Then there exists c ∈ C such that

T (δ, r, λ) = cur˜Rr˜A(r.P, τδ .P, δ, τδ λ)Tδ , λ ∈ ia∗rτδ . P (iv) Let T (δ, H, χ, λ) be the restriction of T (δ, r, λ) to the space of πδ,H,χ,λ for each ∗rτ P ˆ δ ) and λ ∈ ia δ . T (δ, H, χ, λ) intertwines (πδ,H,χ,λ )τδ with H ∈ Diag(τδ , r), χ ∈ H(τ P πδ,H,χ,λ .

Proof. τδ .P P τδ (Lemma 5 (i) and (ii)), assertion (i) follows from As Tδ intertwines (πδ,λ ) with πδ,τ δλ the properties of intertwining operators of [KS80]. The first part of assertion (ii) follows from Lemma 5 (iii). The second part follows from the intertwining properties of our normalized intertwining operators ((2.1), (2.4)). Using equation (2.6), and taking into account r = r−1 , one has T (δ, r, λ) = c ur˜ Rr˜ A(r.P, P, δ, τδ λ) A(P, τδ .P, δ, τδ λ) Tδ . Assertion (iii) is therefore results from the properties of our normalized intertwining P operators ((2.2)). As πδ,H,χ,λ is τδ -stable, its set of minimal K-types is also τδ -stable P (Lemma 4). Assertion (iv) therefore follows from (i), (ii) and the definitions of πδ,H,χ,λ ˆ δ ) (cf. Theorem 1 (i)). and H(τ  Remark 2 In the preceding lemma, one may replace P by any other parabolic subgroup with the same Levi subgroup. This is also the case for Lemma 5. Lemma 6 allows us to define the twisted character τδ P Θδ,r,H,χ,λ (f ) = tr(πδ,H,χ,λ (f ) T (δ, H, χ, λ)), f ∈ Cc∞ (G).

14

(5.2)

5.3

Some auxiliary operators This section is inspired by section 5 [Mez07]. We begin with a simple remark: Let L ⊂ L0 be two Levi subgroups of G which contain Am . Let P 0 = L0 N 0 be a parabolic subgroup of G with Levi subgroup L0 . Then there exists a parabolic subgroup P of G with Levi subgroup L such that P ⊂ P 0 .

(5.3)

Indeed, if PL0 is any parabolic subgroup of L0 with Levi subgroup L then P = PL0 N 0 has the required property. Let (M, δ) ∈ DDT , P = M AN be a standard parabolic subgroup of G, r ∈ Rδc and H ∈ Diag(τδ , r). By the definition of Diag(τδ , r), there exists λ ∈ ia∗ such that c Rδ,λ = H, rτδ λ = λ, and λ ∈ iC¯δ . Furthermore, the centralizer of aH is a Levi subgroup L1 = M1 A1 , with a1 = aH (see section 2.2). Thus the set O of λ ∈ ia∗hrτδ ,Hi with c Rδ,λ = H, is non-empty. Hence, the fixed-point set of r0 ∈ Rδc \ H in ia∗hrτδ ,Hi is a proper subspace. The complement of O in ia∗hrτδ ,Hi is equal to the possibly trivial finite union of these proper subspaces. According to Baire’s theorem, the set O is open and dense in ia∗hrτδ ,Hi . The finite group hrτδ , Hi preserves Cδ . Therefore we may average an element of iCδ under the action of hrτδ , Hi to conclude that iCδ ∩ iahrτδ ,Hi 6= ∅. The density of O now implies that iCδ ∩ O 6= ∅. Let us show that there exists λ0 ∈ iCδ ∩ O such that for any a-root α, α vanishes on ia∗hrτδ ,Hi if and only if it vanishes on λ0 . Suppose α is a root of a in g which does not vanish on ahrτδ ,Hi and let Oα be the complement of the kernel of α in ahrτδ ,Hi . It is the complement of a hyperplane and as such is open and dense. The intersection of all the sets Oα is dense, by Baire’s theorem, and therefore intersects the non-empty open set iCδ ∩ O. We may therefore take λ0 in this intersection. The element λ0 satisfies the desired property. Let Q = MQ AQ NQ be the parabolic subgroup of G whose Lie algebra is the sum of of the root spaces of the roots α of a such that Im(α, λ0 ) ≥ 0. This implies that the Levi subgroup MQ AQ is the centralizer in G of λ0 . Since λ0 is rτδ -invariant, one has r˜τδ .Q = Q,

(5.4)

where r˜ is a representative in K of r. Notice that the set of roots of a whose root spaces lie in nQ contains ∆+ δ , as λ0 ∈ Cδ . Notice also that aQ is the intersection of the kernels of those roots which vanish on λ0 . Recalling the main property of λ0 above, this implies that aQ contains ahrτδ ,Hi . Let M1 A1 = M λ0 Aλ0 be a strongly affiliated Levi subgroup as defined in section 2.2. The group M1 A1 centralizes ahrτδ ,Hi , which contains λ0 . As MQ AQ is the centralizer of λ0 , one has M A ⊂ M 1 A1 ⊂ M Q AQ . By (5.3), there exist parabolic subgroups P 0 and P1 of G, with Levi subgroups M A and M1 A1 respectively, such that P 0 ⊂ P1 ⊂ Q. As P 0 ⊂ Q and Q is r˜τδ -stable, one has θ(P 0 ) ∩ r˜τδ .P 0 ⊂ θ(Q) ∩ r˜τδ .Q = θ(Q) ∩ Q = MQ AQ . 15

(5.5)

Lemma 7 (i) The operator T 0 := A(P 0 , δ, r, τδ λ)A(P 0 , τδ .P 0 , δ, λ)Tδ , λ ∈ ia∗hrτδ ,Hi 0

0

P τδ P is independent of λ ∈ ia∗hrτδ ,Hi and intertwines (πδ,λ ) with πδ,λ . 0 P ∗hrτδ ,Hi (ii) The space of πδ,H,χ,λ is independent of λ ∈ ia and 0

Q P πδ,H,χ,λ = πε,λ , λ ∈ ia∗hrτδ ,Hi

where Q is r˜τδ -stable and

P 0 ∩(MQ AQ )

ε = πδ1 ,0

is a r˜τδ -stable representation of MQ . P0 (iii) Let T 0 (δ, r, H, χ) be the restriction of T 0 to the space of πδ,H,χ,λ . It intertwines P0 τδ P0 (πδ,H,χ,λ ) with πδ,H,χ,λ . (iv) There exists c ∈ C such that for all λ ∈ ia∗hrτδ ,Hi the operator T (δ, r, H, χ, λ) of Lemma 6 is equal to the restriction of the operator cA(P 0 , P, δ, λ)−1 T 0 A(P 0 , P, δ, λ) to P the space of πδ,H,χ,λ . ˆ δ ), and λ ∈ ia∗hrτδ ,Hi , one has (v) For H ∈ Diag(τδ , r), χ ∈ H(τ 0

P δ Θτδ,r,H,χ,λ (f ) = c tr(πδ,λ (f )T 0 (δ, r, H, χ)), f ∈ Cc∞ (G).

Proof. (i) By Remark 2 and Lemma 6 (iii) applied to P 0 , one sees that T 0 has the desired intertwining property and that A(P 0 , δ, r, τδ λ)A(P 0 , τδ .P 0 , δ, λ) = c ur˜ Rr˜ A(r.P 0 , τδ .P 0 , δ, τδ λ), λ ∈ ia∗hrτδ ,Hi . As r(ahrτδ ,Hi ) ⊂ raQ , the element τδ λ = rλ lies in i(raQ )∗ for any λ ∈ ia∗hrτδ ,Hi . It follows from (2.3) and (5.5) that A(r.P 0 , τδ .P 0 , δ, τδ λ) is an intertwining operator which does not depend on λ ∈ ia∗hrτδ ,Hi . This shows that T 0 is independent of λ ∈ ia∗hrτδ ,Hi . (ii) For λ ∈ O ∩ iCδ , one has M λ Aλ = M1 A1 (see section (2.2)). As P 0 ⊂ P1 ⊂ Q, we may apply induction in stages as in equation (2.14). In the context at hand, this equation takes the form 0

Q P = πε,λ πδ,H,χ,λ , λ ∈ (O ∩ iCδ ) ⊂ ia∗hrτδ ,Hi ,

where

P 0 ∩(MQ AQ )

ε = πδ1 ,0

.

When λ ∈ iCδ ∩ O, the first assertion of (ii) follows from an analysis of the induction in stages in the compact realization. It follows for all λ ∈ ia∗hrτδ ,Hi by analytic continuation. The r˜τδ -stability of Q has been shown in (5.4). Finally, for λ = λ0 the parabolic subgroup Q and the representation ε are as in Proposition 1, and so the r˜τδ -stability of ε follows. (iii) This assertion follows from Remark 2 and Lemma 5 applied to P 0 . (iv) We will make use of the following fact. Let π, π 0 be two unitary representations of 16

G of finite length, A (resp. S) be an invertible intertwining operator between π and π 0 (resp. π σ and π). Then S 0 = ASA−1 is an intertwining operator between π 0σ and π 0 , and tr(π 0 (f )S 0 ) = tr(π(f )S), f ∈ Cc∞ (G).

(5.6)

The operators T (δ, r, H, χ, λ) and A(P, P 0 , δ, λ)−1 T 0 A(P, P 0 , δ, λ) are intertwining operP P ators between (πδ,H,χ,λ )τδ and πδ,H,χ,λ for λ ∈ ia∗hrτδ ,Hi . For λ in the non-empty open P set iCδ ∩ O (see above) of ia∗hrτδ ,Hi , the representation πδ,H,χ,λ is irreducible (see section 2.2). These two operators are proportional by Schur’s lemma. The two operators are independent of λ on each minimal K-type, and are proportional to Tδ (equation (2.1) and Lemma 6 (ii)). Therefore the proportionality factor is independent of λ in this open set. The assertion follows by analytic continuation. (v) This assertion follows from (iii), (iv) and equation (5.6). 

6

Some properties of twisted characters

The properties we prove here will reappear in Theorem 3 as those properties which characterize functions derived from twisted characters. The first property we prove hearkens back to the classical theorem of Paley and Wiener. Let Cc∞ (G, K)t be the space of smooth functions on G, which are left- and right-K-finite, and whose support is contained in K exp(Bt )K, where Bt is the closed ball of radius t > 0 about the origin in am . Define Cc∞ (ahrτδ ,Hi )t to be the space of smooth functions on ahrτδ ,Hi with support in the closed ball of radius t, and define PW(ahrτδ ,Hi )t to be the image of Cc∞ (ahrτδ ,Hi )t under the Fourier transform. Lemma 8 For each f ∈ Cc∞ (G, K)t , the function on ia∗hrτδ ,Hi defined by, δ λ 7→ Fδ,r,H,χ (λ) := Θτδ,r,H,χ,λ (f )

is an element of the space PW(ahrτδ ,Hi )t . Proof. Let f ∈ Cc∞ (G, K)t . The K-finiteness of f implies that there exists an orthogonal projection p onto a finite sum of isotypic components of K in I(δ) such that P0 P0 πδ,H,χ,λ (f ) = p πδ,H,χ,λ (f ) p. Substituting the right-hand side into the equality of Lemma 7 (ii), Fδ,r,H,χ is seen to be the restriction to ia∗hrτδ ,Hi of a finite sum of coefficients of the matrix Fourier transform of f . The lemma follows from the properties of the Fourier transform of elements of Cc∞ (G, K)t as given in [Art83] (see also 2.1 [CD84]).  Proposition 2 (i) Let r ∈ Rδc and H ∈ Diag(τδ , r). If w ∈ Wδ0 satisfies w(ia∗hrτδ ,Hi ) = ia∗hrτδ ,Hi then δ δ ˆ δ ), λ ∈ ia∗hrτδ ,Hi . Θτδ,r,H,χ,wλ = Θτδ,r,H,χ,λ , χ ∈ H(τ

(ii) Let r ∈ Rδc . If H, H 0 are subgroups in Diag(τδ , r), with H ⊂ H 0 , then δ Θτδ,r,H,χ,λ =

X

hrτδ ,H 0 i δ ˆ Θτδ,r,H . 0 ,χ0 ,λ , χ ∈ H(τδ ), λ ∈ a

ˆ 0 (τδ ),χ0 =χ χ0 ∈H |H

17

(iii) If r, s ∈ Rδc and H ∈ Diag(τδ , s), with rs ∈ H, then ia∗hrτδ ,Hi = ia∗hsτδ ,Hi and δ δ ˆ δ ), λ ∈ ia∗hrτδ ,Hi . Θτδ,r,H,χ,λ = χ(rs)Θτδ,s,H,χ,λ , χ ∈ H(τ

(iv) Suppose r, s ∈ Rδc and H ∈ Diag(τδ , r). Recall that τδ (s) is the action of τδ on s. Then H is an element of Diag(τδ , rsτδ (s)), and δ δ ˆ δ ), λ ∈ ia∗hrτδ ,Hi . , χ ∈ H(τ = χ0 (τδ (s)) Θτδ,r,H,χ,λ Θτδ,rsτ δ (s),H,χ,sλ

(The left-hand side is defined, as our hypothesis on λ implies that sλ ∈ ia∗hrsτδ (s),Hi .) Proof. (i) Let λ ∈ ia∗hrτδ ,Hi . The restriction A(λ) of A(P, w, δ, λ) to the space of πδ,H,χ,λ interP P twines πδ,H,χ,λ with πδ,H,χ,wλ . By Lemma 6 (i), the operator T (δ, r, H, χ, λ) intertwines P τδ P (πδ,H,χ,λ ) with πδ,H,χ,λ . Taking into account equation (5.6), is suffices to prove that T 0 (λ) := A(λ)−1 T (δ, r, H, χ, wλ)A(λ) P P is equal to T (λ) := T (δ, r, H, χ, λ). Both operators intertwine (πδ,H,χ,λ )τδ with πδ,H,χ,λ . P P τδ As the representation πδ,H,χ,λ and its equivalent representation (πδ,H,χ,λ ) are generated by their minimal K-types, it is enough to prove this equality on the minimal K-types P of πδ,H,χ,λ . On these minimal K-types T (δ, H, χ, λ) does not depend on λ (Lemma 6 (ii)). On the other hand, A(λ) is trivial on all of the minimal K-types as w ∈ Wδ0 (see (2.4)). The equality T 0 (λ) = T (λ) follows. (ii) From decomposition (2.11), one has δ Θτδ,r,H,χ,λ (f ) =

X

P tr(πδ,H ) 0 ,χ0 ,λ (f )T (δ, r, H, χ, λ)|I δ,H 0 ,χ0 ,λ

ˆ 0 ,χ0 =χ χ0 ∈H |H P where Iδ,H 0 ,χ0 ,λ denotes the space of πδ,H For λ in the non-empty open subset 0 ,χ0 ,λ . ∗hrτδ ,H 0 i c 0 ∗hrτδ ,H 0 i P {ν ∈ ia ∩iC δ |Rδ,ν = H } of ia , the representation πδ,H 0 ,χ0 ,λ is irreducible. If 0 0 ˆ χ ∈ / H (τδ ), this representation is neither σ-stable (Theorem 1 (i)) nor τδ -stable. In this case the operator T (δ, r, H, χ) sends the space of this subrepresentation to an orthogonal P subspace, as the multiplicities of irreducible representations in πδ,λ is one. Hence, the 0 contribution of this χ to the sum above is zero. As the operators T (δ, r, H, χ, λ) and ˆ 0 (τδ ) are restrictions of T (δ, r, λ), assertion (ii) follows. T (δ, r, H 0 , χ0 , λ), χ0 ∈ H (iii) Since rs ∈ H, one has a∗hrτδ ,Hi = a∗hsτδ ,Hi . As in (i), it suffices to prove

T (δ, r, H, χ, λ) = χ(rs)T (δ, s, H, χ, λ), λ ∈ ia∗hrτδ ,Hi = ia∗hsτδ ,Hi .

(6.7)

P P Since both sides are intertwining operators between (πδ,H,χ,λ )τδ and πδ,H,χ,λ , and the sum P of the isotypic components of the minimal K-types of πδ,H,χ,λ generates this representation of G, one has only to check the equality of these two operators on each such component I µ (δ). These µ ∈ A(δ) satisfy χµ |H = χ. Using equations (2.1), (2.4) and Lemma 6(i)-(ii), one sees that the restriction of T (δ, r, H, χ, λ) to such an I µ (δ) is equal to τ χµτδ (r) cµ δ (P, τδ .P ) Tδ |I µ (δ) .

18

Similarly, the restriction of T (δ, s, H, χ, λ) to such an I µ (δ) is equal to τ

χµτδ (s) cµ δ (P, τδ .P ) Tδ |I µ (δ) . P P As πδ,H,χ,λ is τδ -stable, µτδ is a minimal K-type of πδ,H,χ,λ (Lemma 4). Hence, χµτδ |H = χ and χµτδ (r) = χ(rs)χµτδ (s). Identity (6.7) follows. (iv) First, if λ ∈ ia∗hrτδ ,Hi , one has sλ ∈ ia∗hrsτδ (s),Hi by

rsτδ (s)τδ sλ = rsτδ sτδ−1 τδ sλ = rsτδ λ = s(rτδ λ) = sλ. c = H. Then sλ satisfies As H ∈ Diag(τδ , r) there exists λ ∈ iC δ ∩ ia∗hrτδ ,Hi with Rδ,λ ∗hrsτδ (s),Hi c sλ ∈ iC δ ∩ ia and Rδ,sλ = H. Hence, H ∈ Diag(τδ , rsτδ (s)). P P The restriction A(λ) of A(P, δ, s, λ) to the space of πδ,H,χ,λ intertwines πδ,H,χ,λ with P P πδ,H,χ,sλ . Its inverse is the restriction of A(P, δ, s, sλ) to the space of πδ,H,χ,sλ (see (2.5)). Using (5.6), assertion (iv) follows once we prove

A(λ)−1 T (δ, rsτδ (s), H, χ, λ) A(λ) = χ0 (τδ (s)) T (δ, r, H, χ, λ).

(6.8)

It is enough to prove this equation on the isotypic component of each minimal K-type P . Using (2.4), (2.1) and Lemma 6 (i)-(ii), one sees that the restriction to µ of πδ,H,χ,λ µ I (δ) of the operator on the left-hand side of (6.8) is equal to τ

χµτδ (s) χµτδ (rsτδ (s)) cµ δ (P, τδ (P )) Tδ |I µ (δ) χµ (s). This operator is equal to τ

χµτδ (r) χµτδ (τδ (s)) χµ (s) cµ δ (P, τδ .P ) Tδ |I µ (δ) .

(6.9)

On the other hand, by Lemma 5 (v), one has in turn that χµτδ = χ0 χτµδ and χµτδ (τδ (s)) = χ0 (τδ (s))χµ (s). Therefore (6.9) is equal to τ

χµτδ (r) χ0 (τδ (s)) cµ δ (P, τδ .P ) Tδ |I µ (δ) . The restriction to I µ (δ) of the operator on the right-hand side of (6.8) to is seen, by parallel computations, to be equal to the previous expression.  6.1

σ-twisted characters

P Proposition 3 Define S(δ, r, H, χ, λ) = πδ,H,χ,λ (kδ )T (δ, r, H, χ, λ), where kδ is the elˆ δ ), and λ ∈ ia∗hrτδ ,Hi the following ement of Lemma 2. For H ∈ Diag(τδ , r), χ ∈ H(τ hold. P P (i) The operator S(δ, r, H, χ, λ) intertwines (πδ,H,χ,λ )σ with πδ,H,χ,λ . (ii) The corresponding twisted character P Θσδ,r,H,χ,λ (f ) = tr(πδ,H,χ,λ (f ) S(δ, r, H, χ, λ)), f ∈ Cc∞ (G) τδ satisfies the same relations as Θδ,r,H,χ,λ does in Proposition 2, as well as the analogue of Lemma 8.

19

Proof. (i) This assertion is clear, as σ = kδ τδ . (ii) This assertion follows easily from (5.1).

7



The main theorem

Theorem 3 Suppose that we are given functions Fδ,r,H,χ : ia∗hrτδ ,Hi → C, for every δ ∈ DDT , ˆ and let t > 0. Then there exists f ∈ Cc∞ (G, K)t such r ∈ Rδc , H ∈ Diag(τδ , r), χ ∈ H that Θσδ,r,H,χ,λ (f ) = Fδ,r,H,χ (λ), λ ∈ ia∗hrτδ ,Hi for all of the above data if and only if: (i) The functions are identically zero except for a finite number of δ ∈ DDT . (ii) Each function Fδ,r,H,χ belongs to PW(ahrτδ ,Hi )t . (iii) Let r ∈ Rδc and H ∈ Diag(τδ , r). If w ∈ Wδ0 and r ∈ Rδc satisfy w(ia∗hrτδ ,Hi ) = ia∗hrτδ ,Hi then ˆ δ ), λ ∈ ia∗hrτδ ,Hi . Fδ,r,H,χ (wλ) = Fδ,r,H,χ (λ), χ ∈ H(τ (iv) Let r ∈ Rδc . If H, H 0 are elements of Diag(τδ , r) for r ∈ Rδc , with H ⊂ H 0 , then Fδ,r,H,χ (λ) =

ˆ δ ), λ ∈ ahrτδ ,H 0 i . Fδ,r,H 0 ,χ0 (λ), χ ∈ H(τ

X ˆ 0 (τδ ),χ0 =χ χ0 ∈H |H

(v) If r, s ∈ Rδc , H ∈ Diag(τδ , s), with rs ∈ H, one has: ˆ δ ), λ ∈ ia∗hrτδ ,Hi = ia∗hsτδ ,Hi Fδ,r,H,χ (λ) = χ(rs)Fδ,s,H,χ (λ), χ ∈ H(τ (vi) Let r, s ∈ Rδc , H ∈ Diag(τδ , r). Then H is an element of Diag(τδ , rsτδ (s)), and ˆ δ ), λ ∈ ia∗hrτδ ,Hi . Fδ,rsτδ (s),H,χ (sλ) = χ0 (τδ (s))Fδ,r,H,χ (λ), χ ∈ H(τ Proof. The “only if” part of the Theorem follows from Propositions 2 and 3, together with Lemma 8. We therefore turn to the proof of the “if” part. In so doing, we follow the inductive reasoning of section 2.3 [CD84] and the following analogue of Proposition 1 [CD84]. 0 Let us introduce the transitional spaces Cc∞ (G)µµ , where µ, µ0 are equivalence classes of irreducible unitary representations of K. Let θµ , θµ0 be the complex conjugates of the normalized characters of µ and µ0 . We define 0

Cc∞ (G)µµ = θµ ∗ Cc∞ (G) ∗ θµ0 .

(7.10)

An element f of this space will be said to be of type (µ, µ0 ). For each representation π of G, the operator π(f ) sends the isotypic component of type µ0 to the isotypic component of type µ. It annihilates the other isotypic components. 20

Proposition 4 Let t > 0. Suppose (Fδ,r,H,χ ) is a family of functions satisfying conditions (i)-(vi) of Theorem 3 for a fixed δ ∈ DDT . Then there exist functions hµ ∈ Cc∞ (G, K)t , µ ∈ A(δ), of type (µ, µτδ ), such that P Fδ,r,H,χ (ν) = tr(πH,χ,ν (

X

ˆ δ ). hµ ) T (r, H, χ, ν)), r ∈ Rδc , ν ∈ ia∗hrτδ ,Hi , χ ∈ H(τ

µ∈A(δ)

We postpone the proof of Proposition 4 until the next section and continue our proof by induction. We shall perform this proof by induction using a partial ordering on DDT . Define δ < δ 0 to mean that kµk < kµ0 k for all µ ∈ A(δ) and µ0 ∈ A(δ 0 ) (see Definition 5.1 [Jr.79] and p. 433 [CD84] for the definition of kµk). Define δ ≤ δ 0 to mean that either δ < δ 0 or δ = δ 0 . Now, suppose we are given functions satisfying the conditions (i)-(vi) of Theorem 3. Define the support ΓF of these functions to be the collection of δ ∈ DDT such that Fδ,r,H,χ does not vanish for some r, H, and χ. ¯ F be the Condition (ii) implies that ΓF is a finite collection of representations. Let Γ collection of representations in DDT which are less than or equal to some representation ¯ F |. If Γ ¯ F is empty then f = 0 in ΓF . We shall prove Theorem 3 by induction on |Γ ¯ F is not empty and δ 0 ∈ DDT is a maximal solves the problem. Now suppose that Γ ¯ F . Proposition 4 tells us that element of ΓF . Clearly, δ 0 is also a maximal element of Γ there exists h ∈ Cc∞ (G, K)t , which is a sum of functions of type (µ0 , µ0δ ), µ0 ∈ A(δ 0 ), such that 0

0

0

P 0 0 0 0 ∗hr τδ0 ,H i Fδ0 ,r0 ,H 0 ,χ0 (λ) = tr(πH 0 ,χ0 ,λ (h) T (r , H , χ )), λ ∈ i(a )

ˆ 0 (τδ0 ). Define a new family functions by for any r0 ∈ Rδc0 , H 0 ∈ Diag(τδ0 , r0 ) and χ0 ∈ H 0 P Fδ,r,H,χ (λ) = Fδ,r,H,χ (λ) − tr(πH,χ,λ (h) T (r, H, χ)), λ ∈ ia∗hrτδ ,Hi ,

(7.11)

ˆ δ ). for δ ∈ DDT , r ∈ Rδc , H ∈ Diag(τδ , r) and χ ∈ H(τ This new family apparently also satisfies conditions (i)-(vi) of Theorem 3. In addition, ¯F 0 ( Γ ¯ F . Suppose δ ∈ ΓF 0 . Fδ00 ,r0 ,H 0 ,χ0 vanishes by construction. We wish to show that Γ 0 Then there exist r, H, and χ such that Fδ,r,H,χ 6= 0. From (7.11), we evidently have P that Fδ,r,H,χ 6= 0 or tr(πH,χ,λ (h) T (r, H, χ)) 6= 0. The former inequality implies that δ ∈ ΓF − {δ 0 }. The latter inequality implies that some µ0 ∈ A(δ 0 ) is a K-type of δ. By definition, any µ ∈ A(δ) satisfies kµk ≤ kµ0 k. If |µk = kµ0 k then µ0 ∈ A(δ) and 0 Proposition D.1 [CD90] implies that δ is equal to δ 0 , which contradicts Fδ,r,H,χ 6= 0. In 0 0 consequence, kµk < kµ k for all µ ∈ A(δ), that is, δ < δ . This proves in turn that ¯ F −{δ 0 }, ΓF 0 ( Γ ¯ F and Γ ¯F 0 ( Γ ¯ F . We may now appeal to the induction hypothesis δ∈Γ 0 ∞ to obtain a function f ∈ Cc (G, K)t such that P 0 Fδ,r,H,χ (λ) = tr(πH,χ,λ (f 0 ) T (r, H, χ)), λ ∈ ia∗hrτδ ,Hi .

Substituting this equation into (7.11), it is clear that f = f 0 + h satisfies the desired properties. 

21

8

The proof of the Proposition 4

In this section we fix δ ∈ DDT and for simplicity often drop the lower index δ from much of the previous notation. For example, we write ∆ instead of ∆δ , and τ for τδ . Moreover, Rδc will be denoted by R. 8.1

An extension result

Lemma 9 Suppose Fδ,r,H,χ satisfies conditions (ii)-(vi) of Theorem 3 for every r ∈ R, ˆ ). Then Fδ,r,H,χ extends to a W 0 -invariant function H ∈ Diag(τδ , r) and χ ∈ H(τ F˜δ,r,H,χ ∈ PW(a)t . Proof. Suppose {r1 , . . . , rm } is a minimal set of generators for H ⊂ R. Let E1 be the subspace of fixed points under r1 , ∆1 = {α|E1 |α ∈ ∆, α|E1 6= 0} and ∆+ 1 be the set + ∗ of elements of ∆1 which are restrictions of elements of ∆ to E1 . By Proposition A.2 [CD90], ∆1 is a root system and the Weyl group of ∆1 is W1 = {w|E1 |w ∈ W 0 , w(E1 ) = E1 }. By Proposition A.1 and Lemme C.1 [CD84], the restriction map 0

1 PW(a)W → PW(E1 )W t t

is surjective. Arguing inductively, we obtain surjections Wδ0

PW(a)t

m m 1 → PW(E1 )W → · · · → PW(Em )W = PW(aH )W , t t t

(8.1)

where Wm is the Weyl group of the root system of ∆m = {α|E H |α ∈ ∆δ , α|E H 6= 0}. + Thus, ∆+ m = {α|E H |α ∈ ∆δ , α|E H 6= 0} is a set of positive roots of ∆m . Now, consider the automorphism rτ of a. For any s ∈ H, τ (s) = τ −1 sτ lies in H, as H is τ -stable, and srτ λ = rτ (τ −1 sτ )λ = rτ (λ), λ ∈ a∗H . This shows that a∗H is rτ -stable. Since ∆+ is also τ - and r-stable, ∆+ m is rτ -stable. We may apply Corollary 1 of the Appendix, to conclude that the restriction map m PW(aH )W → PW(ahrτ,Hi )W t t

0

(8.2)

is surjective, where W 0 is a certain subgroup of W 00 := {w|a∗hrτ,Hi |w ∈ Wδ0 , w(a∗hrτ,Hi ) = a∗hrτ,Hi }. By condition (iii) of Theorem 3, the function Fδ,r,H,χ is invariant under W 00 . 0 As a result, the function Fδ,r,H,χ belongs to PW(ahrτ,Hi )W t . Combining the surjections of (8.1) and (8.2), we obtain a function F˜δ,rH,χ as desired. 

22

8.2

Some τ -stable subgroups of R

Let R0 be the subgroup of the automorphism group of a∗ generated by τ and R. It is a finite group, as τ is of finite order on a and normalizes R. Given λ ∈ a∗ , define Rλ0 = {γ ∈ R0 : γλ = λ}. Lemma 10 Suppose s ∈ R. Then there exists a unique subgroup R0 [sτ ] of R0 which satisfies the following. 0

(i) a∗R [sτ ] = a∗sτ . (ii) There exists λ ∈ C¯ ∩ a∗sτ such that Rλ0 = R0 [sτ ] (here C = Cδ ). (iii) The element sτ belongs to R0 [sτ ]. (iv) Suppose H 0 is a subgroup of R0 such that sτ ∈ H 0 and Rλ0 = H 0 for some λ ∈ a∗ . Then R0 [sτ ] ⊂ H 0 . Proof. The proof is essentially that of Lemme C.2 [CD90] with R replaced by R0 . We include it for the sake of completeness. Recall that C is a Weyl chamber of ∆ = ∆δ . For each subgroup H of R0 set AH = {λ ∈ C¯ ∩ a∗sτ |Rλ0 = H}. It is immediate that AH ⊂ a∗H ⊂ a∗sτ and C¯ ∩ a∗sτ = ∪H⊂R0 AH . One may average an element of C over the finite group generated by sτ to obtain an element in C ∩ a∗sτ (cf. section 5.3). As a result, the set C¯ ∩ a∗sτ has non-empty interior, as a subset of a∗sτ . According to Baire’s Theorem, one of the subgroups in the above union has an invariant subspace which is open in, and therefore equal to, a∗sτ . Denote such a subgroup by R0 [sτ ]. It is apparent from its definition that sτ ∈ R0 [sτ ]. This proves assertions (i), (ii) and (iii). Now suppose λ ∈ a∗ 0 is as in the hypothesis of (iv). Then λ ∈ a∗sτ = a∗R [sτ ] , and so R0 [sτ ] ⊂ Rλ0 = H 0 . This proves (iv), from which the uniqueness assertion also follows.  Lemma 11 Suppose s ∈ R and set R[sτ ] = R0 [sτ ] ∩ R. Then R[sτ ] is stable under conjugation by τ . Proof. Suppose r ∈ R[sτ ]. By Lemma 3, τ (r) = τ rτ −1 belongs to R. Since R is abelian, the element τ rτ −1 is equal to (sτ )r(sτ )−1 . The latter element belongs to R0 [sτ ] ( Lemma (iii) 10).  Lemma 12 Suppose s ∈ R. Then R[sτ ] belongs to Diag(τ, s).

23

Proof. In view of Lemma 11, it remains only to show that R[sτ ] = Rλ for some ¯ According to Lemma 10, there exists λ ∈ a∗sτ ∩ C¯ such that Rλ0 = R0 [sτ ]. λ ∈ a∗sτ ∩ C. It is easily verified that Rλ = Rλ0 ∩ R = R0 [sτ ] ∩ R = R[sτ ].  Lemma 13 Suppose s ∈ R, H ∈ Diag(τ, s) and h ∈ H. Then H contains R[shτ ]. Proof. By the definition of Diag(τδ , s), there exists λ ∈ a∗sτ ∩ C¯ such that Rλ = H. Since λ is fixed by shτ = hsτ , the element shτ belongs to Rλ0 . According to Lemma 10 (iv), the group R0 [shτ ] is contained in Rλ0 . Finally, H = Rλ = Rλ0 ∩ R ⊃ R0 [shτ ] ∩ R = R[shτ ].  8.3

The statement and proof of a key lemma

Lemma 14 ˆ ), is a family of functions satisfying Suppose Fδ,r,H,χ , r ∈ R, H ∈ Diag(τ, r), χ ∈ H(τ ˆ there exists a function Φα ∈ conditions (ii)-(vi) of Theorem 3. Then for each α ∈ R, W0 PW(a)t such that Fδ,r,H,α (λ) =

χ0 (r) ατ (r) Φα (λ), λ ∈ ia∗hrτ,Hi ,

X ˆ |H =χ α∈R,α

ˆ ). for any r ∈ R, H ∈ Diag(τ, r), χ ∈ H(τ Proof. In what follows, we generalize the proof of the similar non-twisted statement, Proposition C.1 [CD84]. Let r ∈ R. The subgroup R[rτ ], defined in Lemma 10, belongs to Diag(δ, r) (Lemma 12). Consequently, the function Fδ,r,R[rτ ],η ∈ PW(ahrτ,R[rτ ]i )t is [](τ ) . By Lemma 9, each of these functions extends defined for each character η ∈ R[rτ 0 ∗ to a function F˜δ,r,R[rτ ],η ∈ PW(a)W t . Define ϕr : ia → C by ϕr =

X

F˜δ,r,R[rτ ],η .

\](τ ) η∈R[rτ

ˆ define For each α ∈ R, Φα =

1 X χ0 (s) ατ (s) ϕs . |R| s∈R

Clearly, the function Φα lies in PW(a)W t . In order to prove that these objects satisfy ˆ ). We proceed by rearranging the lemma, we choose r ∈ R, H ∈ Diag(τ, r) and χ ∈ H(τ 0

24

the right-hand side of the identity in the lemma as χ0 (r) ατ (r) Φα (λ) =

X

χ0 (r)ατ (r)

X ˆ |H =χ α∈R,α

ˆ |H =χ α∈R,α

1 X ϕs (λ) |R| s∈R

=

1 X χ0 (s)ατ (s)ϕs (λ) |R| s∈R (χ0 ατ )(rs), λ ∈ ia∗hrτδ ,Hi .

X

ˆ |H =χ α∈R,α

Let us consider the inner sum in more detail. Fix representatives 1, r2 , . . . , r` of the cosets in R/H. Then the map (ri H, h) 7→ ri h, h ∈ H is a group isomorphism from R/H × H to R. This induces the dual isomorphism [ × H, ˆ∼ ˆ as all of the groups are abelian. Suppose that rs = ri h for some h ∈ H. R = R/H Then the summand (χ0 ατ )(rs) may be decomposed according to the dual isomorphism [ Furthermore, since H ∈ Diag(τ, r) and as α0 (ri H)(χ0 ατ )(h) for some α0 ∈ R/H. ˆ ) (see Theorem 1 (2)) one has χ ∈ H(τ (χ0 ατ )(h) = χ0 (h)α(τ −1 (h)) = χ0 (h)χ(τ −1 (h)) = χ0 (h)χτ (h) = χ0 (h)χ0 (h)χ(h) = χ(h), so that (χ0 ατ )(ri h) = α0 (ri H)χ(h). If ri 6= 1 then the inner sum reduces to X

χ(h)

α0 (ri H) = χ(h) × 0 = 0,

[ α0 ∈R/H

thanks to the orthogonality relations of characters. On the other hand, if rs = h then [ the sum is equal to |R/H|χ(h). Taking these identities into account, we may continue our earlier computation by writing X

χ0 (r) ατ (r) Φα (λ) =

ˆ |H =χ α∈R,α

1 X [ |R/H|χ(h)ϕrh (λ) |R| h∈H

1 X χ(h)ϕrh (λ) |H| h∈H X 1 X = χ(h) |H| h∈H \

(8.3)

=

F˜δ,rh,R[rhτ ],η (λ), λ ∈ ia∗hrτδ ,Hi .

η∈R[rhτ ](τ )

Lemma 13 tells us that H ⊃ R[rhτ ] for every h ∈ H. As Fδ,rh,R[rhτ ],η satisfies condition (iv) of Theorem 3, we have F˜δ,rh,R[rhτ ],η (λ) = Fδ,rh,R[rhτ ],η (λ) =

X ˆ ),χ0 χ0 ∈H(τ =η |R[rhτ ]

25

Fδ,rh,H,χ0 (λ), λ ∈ ia∗hrτδ ,Hi .

Substituting the expression on the right into (8.3), summing over the characters η, we obtain (χ0 ατ )(r)Φα (λ) =

X

ˆ |H =χ α∈R,α

X 1 X χ(h) Fδ,rh,H,χ0 (λ), λ ∈ ia∗hrτδ ,Hi . |H| h∈H 0 ˆ χ ∈H(τ )

Now, condition (v) of Theorem 3 tells us that Fδ,rh,H,χ0 (λ) = χ0 (h)Fδ,r,H,χ0 (λ). Combining this with the orthogonality relations, we conclude that (χ0 ατ )(r)Φα (λ) =

X

ˆ |H =χ α∈R,α

X 1 X χ(h) χ0 (h)Fδ,r,H,χ0 (λ) |H| h∈H 0 ˆ χ ∈H(τ )

=

X

Fδ,r,H,χ0 (λ)(

ˆ ) χ0 ∈H(τ

1 X χ(h)χ0 (h)) |H| h∈H

= Fδ,r,H,χ (λ), as desired. 8.4



The behaviour of Φα under R

ˆ to define ¿From now on, we shall use the bijection µ 7→ χµ between A(δ) and R Φµ = Φχµ for any function Φχµ as defined in Lemma 14. By Lemma 5 (v), the equation of Lemma 14 may be rewritten as Fδ,r,H,α (λ) =

X

χ0 (r) χτµ (r) Φµ (λ) =

µ∈A(δ),χµ |H=χ

X

χµτ (r) Φµ (λ),

(8.4)

µ∈A(δ),χµ |H=χ

for any λ ∈ ia∗hrτ,Hi . Now suppose s ∈ R, t > 0 and Φ is any collection (Φµ )µ∈A(δ) of functions in 0 PW(a)W t . Then (s · Φ)µ (λ) = χµ (s)χµτ (s) Φµ (sλ), λ ∈ ia∗ defines an action on the set of such collections, thanks to the commutativity of R. Lemma 15 Suppose a collection Φ = (Φµ )µ∈A(δ) of functions in PW(a)W satisfies t c equation (8.4) of Lemma 14, and s ∈ Rδ . Then s · Φ is also a collection of functions in 0 PW(a)W which satisfies equation (8.4). t 0

Proof. By Lemma 5 (v), we have χµτ (τ (s)) = χ0 (τ (s)) χµ (τ −1 (τ (s))) = χ0 (τ (s)) χµ (s). Consequently, χµτ (r) χµ (s) χµτ (s) = χµ (s) χµτ (rs) = χµ (s) χµτ (τ (s)) χµτ (τ (s)) χµτ (rs) = χ0 (τ (s)) χµτ (rsτ (s)). 26

Therefore, for λ ∈ ia∗hrτ,Hi , X

χµτ (r)(s · Φ)µ (λ) = χ0 (τ (s))

X

χµτ (rsτ (s))Φµ (sλ).

µ∈A(δ),χµ|H =χ

µ∈A(δ),χµ|H =χ

By hypothesis the right-hand side is equal to χ0 (τ (s))Fδ,rsτ (s),H,χ (sλ). By condition (vi) of Theorem 3, this expression is equal to Fδ,r,H,χ (λ).  Lemma 15 further shows that if Φ = (Φµ )µ∈A(δ) satisfies the conclusion of Lemma P 14 then this is also the case for |R|−1 s∈R s · Φ. This being the case, we may assume without loss of generality that our Φ satisfies Φµ (sλ) = χµ (s)χµτ (s) Φµ (λ), λ ∈ ia∗

(8.5)

for any s ∈ Rδc . 8.5

The conclusion of the proof of Proposition 4

Suppose the hypotheses of Proposition 4 are satisfied. In Lemma 14, we have shown 0 the existence of functions Φµ ∈ PW(a)W satisfying (8.5) for every minimal K-type t µ ∈ A(δ), and Fδ,r,H,χ (λ) =

χµτ (r) Φµ (λ), λ ∈ ia∗hrτ,Hi .

X µ∈A(δ),χµ|H =χ

We wish to express the right-hand side as P tr(πδ,H,χ,λ (

X

hµ ) T (δ, r, H, χ, λ)),

µ∈A(δ) τ

for some functions hµ ∈ Cc∞ (G, K)µµ . To obtain this expression, it is sufficient to have P tr(πδ,H,χ,λ (hµ ) Tr,µ ) = χµτ (r)Φµ (λ), λ ∈ ia∗hrτδ ,Hi ,

(8.6)

where Tr,µ is the restriction of T (δ, r, H, χ, λ) to I µ (δ). In fact, it is sufficient for this equation to hold for r = 1 and all λ ∈ ia∗H . Indeed, by Lemma 6 (i)-(ii) and (2.4), the τ operator Tr,µ is bijective from I µ (δ) to I µ δ (δ), independent of λ ∈ ia∗hrτδ ,Hi and Tr,µ = A(P, δ, r, 0) T1,µ = χµτ (r) T1,µ . Having reduced the proof of Proposition 4 to finding these specific functions hµ satisfying (8.6), we may take advantage of the action of R on Φµ as in (8.5). For any 0 η W0 ˆ define (PW(a)W t > 0 and η ∈ R t ) to be the subspace of functions Φ in PW(a)t which satisfy Φ(sλ) = η(s) Φ(λ), λ ∈ ia∗ , s ∈ Rδc . 0

It is obvious from (8.5) that Φµ belongs to (PW(a)W )ηµ , where ηµ = χµ χµτ . The existence of the hµ as stated in Proposition 4 is a consequence of the next result with µ0 = µτ . This result appears in equation (1.38) [Del05] in the required generality. The 27

particular case of µ = µ0 , was proven for G connected as a Lie group in Proposition 1 [CD84]. Note that the proof given in [Del05] uses [DFJ91] instead of [Art83]. ˆ Suppose further that Suppose t > 0, µ, µ0 ∈ A(δ) and η = χµ0 χµ ∈ R. 0 0 PW(a)tµµ is the space of functions from ia∗ to Hom(I µ (δ), I µ (δ)) defined by P λ 7→ πδ,λ (h)|I µ0 (δ) , λ ∈ ia∗ ,

(8.7)

for h ∈ Cc∞ (G, K)t of type (µ, µ0 ). Then 0

0

0

η µ µ PW(a)µµ = (PW(a)W t t ) ⊗ Hom(I (δ), I (δ)).

9

A corollary of the main theorem

Let P = M AN be a parabolic subgroup of G, and P 0 be the intersection of G0 with its normalizer P + in G+ . We assume in the following that it is non-empty and contains τ . Thus, τ.P = P , which implies that σ.P is conjugate under G to P . This implies that there exists k ∈ K with kσ ∈ P 0 so that P 0 ∩ K + is non-empty. We may choose τ ∈ P 0 ∩ K + . Hence, the map P + /P → G+ /G is surjective. It is bijective, as the normalizer of P in G is P . The normalizer P + is generated by P 0 . Similarly, let L+ be the intersection of the normalizer of L = M A in G+ with P + . It is an algebraic group and has non-empty intersection L0 with G0 , which generates L+ . In fact, τ ∈ K + ∩ P 0 is in L+ , as τ normalizes P + and θ(P + ). Let M + be the subgroup of L+ generated by τ and M . Then L+ = M + A, and A is a normal subgroup of L+ but is not necessarily in the center of L+ . If X is equal to either K, P , L, or M then B 0 := B ∩ G0 is equal to Bτ , B 0 generates B + and the canonical map B + /B → G+ /G is surjective. Moreover, P + = L+ N , and + A and N are normal subgroups of L+ and P + respectively. The fixed-point spaces aL and aτ are equal. If ε+ is a tempered unitary representation of M + , whose restriction ε to M is + irreducible and λ ∈ ia∗L = ia∗τ , then ε+ ⊗ eλ ⊗ 1N is a unitary representation of P + . + We denote by πεP+ ,λ the corresponding unitarily induced representation from P + to G+ . +

The unitarily induced representation πεP+ ,λ from P + to G+ restricts to G P as a representation canonically equivalent to πε,λ .

(9.8)

Lemma 16 Let (Θλ ) and (Θ0λ ), be two families of σ-twisted characters (cf. section 5.1) of representations πλ and πλ0 of G+ respectively for λ in an open connected subset Ω of a finite-dimensional subspace of ia∗ , such that: (i) The restrictions of πλ and πλ0 to G are equivalent. (ii) The families of twisted characters are analytic. P (iii) Θλ0 = Θ0λ0 6= 0 for some λ0 ∈ Ω such that πδ,λ is irreducible in a neighborhood of λ0 . Then the two families of twisted characters are identical. Proof. Recall from section 5.1 that n is the least positive integer such that σ n ∈ K. It is also the order of the coset of σ in G+ /G or K + /K. As mentioned in section 5.1, there 28

exist n equivalence classes of representations of G+ with a given irreducible restriction to G. They differ by an nth root of unity on σ ∈ G0 . Thus, there exist nth roots of unity c(λ) such that in a connected neighborhood of λ0 Θλ = c(λ)Θ0λ . Since Θ and Θ0 are analytic, c(λ) is constant. The constant c(λ) equals 1, as c(λ0 ) = 1. The lemma follows by analytic continuation.  + ˆ+ Theorem 4 Let φ be a function defined on the tempered dual G temp of G , which ˆ0 temp of equivalence classes of representations of G+ is non-zero only on the subset G whose restrictions to G are irreducible and tempered. We also denote by φ the Z-linear extension of φ to the the set of tempered representations of G+ of finite length. Then there exists f ∈ Cc∞ (G, K)t with

ˆ+ tr(π + (f )π + (σ)) = φ(π + ), π + ∈ G temp if and only if φ satisfies the following conditions: ˆ such that φ(π + ) = 0 if the (i) There exists a finite subset Γ of the unitary dual K restriction π of π + to G does not contain any K-type in Γ. ˆ0 temp have the same restriction π to G and π + (σ) = cπ 0+ (σ), for (ii) If π + and π 0+ ∈ G some nth root of unity c, then φ(π + ) = cφ(π 0+ ). (iii) Let Q = MQ AQ NQ be a parabolic subgroup of G with Q0 non-empty. Assume that + + ε+ is a tempered representation of MQ+ , and λ ∈ a∗MQ . Then λ 7→ φ(πεQ+ ,λ ) is the M+

Fourier transform of a function on aQ Q of support contained in the closed ball of radius t. In such a case we will say that φ is the twisted invariant Fourier transform of f . Proof. Let us show first that the conditions are necessary. If φ is the twisted invariant Fourier transform of f ∈ Cc∞ (G, K)t , it evidently satisfies (i) and (ii). Since σ ∈ K + + it follows that in the compact realization the operator πεQ+ ,λ (σ) does not depend on λ. We may therefore prove condition (iii) by imitating the proof Lemma 8. Let us show that the conditions are sufficient. First, we wish to define Fδ,r,H,χ,λ P as in Theorem 2. The operator S(δ, r, H, χ, λ) (section 6.1) intertwines (πδ,H,χ,λ )σ P P with πδ,H,χ,λ . Therefore, when πδ,H,χ,λ is irreducible, the operators S(δ, r, H, χ, λ)n P n and πδ,H,χ,λ (τδ ) are proportional (cf. section 5.1). As τδn ∈ K, the second operator is independent of λ. By Lemma 6 (ii), this is true also for the first operator restricted to the minimal K-types. Thus, the proportionality factor is independent P of λ when πδ,H,χ,λ is irreducible. Since these operators are analytic in λ, the proportionality factor is always independent of λ. Consequently, there exists c ∈ C that P S 0 (δ, r, H, χ, λ) = cS(δ, r, H, χ, λ) verifies (S 0 (δ, H, χ, λ))n = πδ,H,χ,λ (τδn ). As a conseP quence, S 0 (δ, H, χ, λ) determines a representation (πδ,H,χ,λ )+ of G+ . It is clearly analytic in λ. We define P FH,δ,r,χ (λ) = c−1 φ((πδ,H,χ,λ )+ ). The relations between the Θσδ,r,H,χ,λ (see Propositions 2 and 3) carry over to the twisted P characters of the (πδ,H,χ,λ )+ , by Lemma 16. The fact that φ is Z-linear on the set of 29

tempered representations of G+ , therefore implies that the family of functions Fδ,r,H,χ satisfies conditions (iii)-(vi) of Theorem 3. Condition (i) from Theorem 3 follows from condition (i) for φ. It remains to verify condition (ii) of Theorem 3. To achieve this we + P show that the family of representations (πδ,H,χ,λ )+ is equivalent to a family πεQ+ ,λ for a suitable parabolic subgroup Q of G. In fact, Proposition 1 tells us that for λ ∈ a∗rτδ , Q P the representation πδ,H,χ,λ is equivalent to πε,λ (with Q and ε as in Proposition 1). In particular, Q, ε and λ are r˜τδ -stable. As a result, r˜τδ ∈ Q0 , ε extends to an irreducible + + P representation of MQ+ and λ ∈ a∗MQ . The representation πεQ+ ,λ is, like (πδ,H,χ,λ )+ , a P representation of G+ whose restriction to G is equivalent to πδ,H,χ,λ . One may multiply + ε by a root of unity so that they are equivalent representations of G+ for some λ0 P for which πδ,H,χ,λ is irreducible. By Lemma 16, one sees that the twisted characters of 0 + Q P (πδ,H,χ,λ )+ and πε+ ,λ are identical for all λ. This means that the two representations are equivalent and + Fδ,r,H,χ (λ) = c−1 φ(πεQ+ ,λ ), λ ∈ ia∗hrτδ ,Hi . Condition (ii) of Theorem 2 for Fδ,r,H,χ follows from condition (iii) for φ. Thus Theorem 3 applies and furnishes a function f ∈ Cc∞ (G, K)t , such that P P φ((πδ,H,χ,λ )+ ) = tr(πδ,H,χ,λ (f ) π + (σ)),

for all data (r, H, χ, λ). According to Theorem 1, every irreducible tempered σ-stable P representation of G appears as such a πδ,H,χ,λ . This implies that for every σ-stable irreducible representation π of G, there exists an extension π + to G+ with φ(π + ) = tr(π(f ) π + (σ)).

(9.9)

ˆ 0temp . Condition (ii) implies that this is also true for all π + ∈ G + + Finally, if π is an irreducible representation of G , whose restriction to G is reducible, then both sides of (9.9) are zero. This shows that f has the required properties. 

10

Appendix

Theorem 5 Suppose ∆ is a root system of a subspace of a finite-dimensional real vector space E, W is its Weyl group, and β is a non-trivial automorphism of E, of finite order, preserving ∆ and a subset of positive roots. Suppose further that Wβ = {w|E β |w ∈ W, w(E β ) = E β }. Let S(E) denote the algebra of polynomial functions on E. Then the restriction map from S(E) to S(E β ) induces a surjection from S(E)W to S(E β )Wβ . Proof. The theorem reduces easily to the case where ∆ generates E. We first treat the case when the Dynkin diagram of ∆ is connected. In this case, the automorphism β is either an involution or is an automorphism of order three of the Dynkin diagram of type D4 . As the theorem is known to hold when β is an involution (Appendix A [CD90]), we assume that the latter holds. Following section 4.8 chapter VI [Bou02], E = R4 and {ε1 , ε2 , ε3 , ε4 } is the canonical basis of R4 . The roots system ∆ is equal to {±εi ± εj |1 ≤ i < j ≤ 4}, 30

and the base for ∆ is α1 = ε1 − ε2 , α2 = ε2 − ε3 , α3 = ε3 − ε4 , α4 = ε3 + ε4 . The elements of the Weyl group W are of the form w = sgn ◦ s, where sgn denotes a sign change on the εi of product 1, and s is a permutation of ε1 , ε2 , ε3 , ε4 . Without loss of generality, β(α1 ) = α3 , β(α3 ) = α4 , β(α4 ) = α1 , β(α2 ) = α2 . The subspace E β is generated by α2 and α1 + α3 + α4 . It is accordingly also generated by α1 + 2α2 + α3 + α4 = ε1 + ε2 and α1 + α2 + α3 + α4 = ε1 + ε3 . At this point, it is convenient to conjugate β by the element x ∈ W , which fixes ε1 , ε4 0 and negates both ε2 and ε3 . The the fixed point set E β of the resulting automorphism β 0 = xβx−1 is generated by α1 = ε1 − ε2 = x(ε1 + ε2 ) and α2 = ε2 − ε3 = x(ε1 + ε3 ) − (ε1 − ε2 ). β0

If E is stable under w = sgn ◦ s, then s must fix ε4 , and sgn = ±Id. Thus, Wβ 0 is isomorphic to the direct product S3 × {±1}, where S3 is the permutation group of 0 ε1 , ε2 , ε3 . Using the canonical coordinates, S(E β )S3 is isomorphic to the polynomial 0 algebra generated by the restriction of X12 + X22 + X32 and X1 X2 X3 to E β . Let u1 and u2 denote the respective restrictions. As u1 is invariant under −1, the subalgebra 0 S(E β )Wβ0 is generated by u1 and u22 . On the other hand, according to [Bou02], S(E)W is generated by the symmetric polynomials t1 (X) = (X1 )2 + . . . + (X4 )2 t2 (X) = (X1 )2 (X2 )2 + . . . + (X3 )2 (X4 )2 t3 (X) = (X1 )2 (X2 )2 (X3 )2 + . . . + (X2 )2 (X3 )2 (X4 )2 t4 (X) = X1 X2 X3 X4 . The desired surjectivity now follows from the fact that (t1 )|E β0 = u1 , and (t3 )|E β0 = u22 . We now prove the theorem in the case of disconnected Dynkin diagrams. In this circumstance, the automorphism β may permute the connected components. The decomposition of a permutation into a product of disjoint cycles allows us to reduce the problem to the case where β permutes n isomorphic copies of a connected Dynkin diagram cyclically among themselves. The cyclic permutation given by β on these connected components allows us to identify each of them with a given one D0 , generating a space E 0 and a root system R0 with Weyl group W 0 . The nth power of β induces an automorphism β 0 of the Dynkin diagram D0 . One has E β = {(x, . . . , x)|x ∈ E 0 , β 0 (x) = x} As a result, the theorem reduces to the connected case proven above. Corollary 1 The natural restriction map from PW(E) to PW(E β ) induces a surjection from PW(E)W to PW(E β )Wβ Proof. The corollary follows from Theorem 5 by the argument given in Lemma C.1 [CD90].  31

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Patrick Delorme E-mail: [email protected] Institut de Math´emathiques de Luminy UMR6206 CNRS Universit´e de la M´editerran´ee 163 Avenue de Luminy Case 907 13288 Marseille Cedex 09 France

Paul Mezo E-mail: [email protected] School of Mathematics and Statistics Carleton University 1125 Colonel By Drive Ottawa, ON K1S 5B6 Canada

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