erikp.vandenban, job j. kuit* and henrik schlichtkrull k ...ban00101/manus/bks_kinvcusp2018.pdf ·...

Forum Math. ��; aop

Research Article

Erik P. van den Ban, Job J. Kuit* and Henrik Schlichtkrull

K-invariant cusp forms for reductivesymmetric spaces of split rank onehttps://doi.org/��.��/forum-��-��Received June ��, ��; revised August ��, ��

Abstract: Let G/H be a reductive symmetric space of split rank one and let K be amaximal compact subgroupof G. In a previous article the first two authors introduced a notion of cusp forms for G/H. We show that thespace of cusp forms coincides with the closure of the space of K-finite generalized matrix coe�cients of dis-crete series representations if and only if there exist no K-spherical discrete series representations. Moreover,we prove that every K-spherical discrete series representation occurs with multiplicity one in the Planchereldecomposition of G/H.Keywords: symmetric space, cusp form, discrete series representation

MSC ��: ��E��, ��E��Communicated by: Karl-Hermann Neeb

� Introduction

By refining a suggestion of M. Flensted-Jensen, the first two authors introduced a notion of cusp forms forreductive symmetric spaces of split rank one in [�]. For reductive groups of split rank one this definition ofcusp forms coincides with Harish-Chandra’s definition. It further generalizes the definition of cusp formsfor hyperbolic spaces given in [�, �]. The definition of cusp forms does not straightforwardly generalize toreductive symmetric spaces of higher split rank as the cuspidal integrals are not always convergent; see [��,Section �].

Let G/H be a reductive symmetric space of split rank one. We write C(G/H) for the space of Harish-Chandra–Schwartz functions on G/H. In [�] a class Ph of minimal parabolic subgroups is identified suchthat the cuspidal integrals

RQ�(g) := �NQ/NQ∩H �(gn) dn (g ∈ G)

are absolutely convergent for every Q ∈ Ph and � ∈ C(G/H). Here NQ is the unipotent radical of Q. A function� ∈ C(G/H) is said to be a cusp form ifRQ� = � for all Q ∈ Ph. Let Ccusp(G/H) denote the space of cusp formsand let Cds(G/H) be the closure in C(G/H) of the span of K-finite generalized matrix coe�cients of discreteseries representations for G/H. It is shown in [�, Theorem �.��] that

Ccusp(G/H) ⊆ Cds(G/H).Erik P. van den Ban,Mathematical Institute, Utrecht University, PO Box �� , �� TA Utrecht, Netherlands,e-mail: [email protected]*Corresponding author: Job J. Kuit, Institut für Mathematik, Universität Paderborn, Warburger Straße ��, �� Paderborn,Germany, e-mail: [email protected] Schlichtkrull, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken �,�� København Ø, Denmark, e-mail: [email protected]

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� � E. P. van den Ban, J. J. Kuit and H. Schlichtkrull, K-invariant cusp forms

Let K be amaximal compact subgroup of G so that K ∩ H is amaximal compact subgroup ofH. For a finiteset � of irreducible unitary representations of K we write C(G/H)� for the subspace of C(G/H) of K finitefunctions with K-isotypes contained in �. In [�, Theorem �.�] it is established that

Cds(G/H)� := Cds(G/H) ∩ C(G/H)�admits an L�-orthogonal decomposition

Cds(G/H)� = Ccusp(G/H)� ⊕ Cres(G/H)� ,where Cres(G/H)� is spanned by certain residues of Eisenstein integrals defined in terms of parabolic sub-groups in Ph.

It is a fundamental result of Harish-Chandra that for reductive Lie groups no residual discrete seriesrepresentations occur, i.e., if G is a reductive Lie group, then

Cds(G) = Ccusp(G); (�.�)

see [��], [��, Thm. ��], [��, Sections �� and ��] and [��, Thm. ��.�.��]. In [�, Theorem �.��] the followingcriterion was given for the analogue of (�.�) for reductive symmetric spaces of split rank one:

Cres(G/H)K = {�} implies Ccusp(G/H) = Cds(G/H). (�.�)

The main result of this article is that this is actually an equivalence.

Theorem �.�. There exist no non-zero K-invariant cusp forms, i.e.,

Ccusp(G/H)K = {�}. (�.�)

Moreover, the following assertions are equivalent:(i) Cds(G/H) = Ccusp(G/H).(ii) Cds(G/H)K = {�}.Note that for the group case equality (�.�) is valid without any assumptions on the split rank of the group,which is a result of Harish-Chandra [��, Lemma ��].

The analysis needed for the proof of Theorem �.� is further used to prove the following theorem, whichconfirms some special cases of the multiplicity one result of [��, p.�, Theorem �].

Theorem �.�. Let G/H have split rank one. Every K-spherical discrete series representation occurs with multi-plicity one in the Plancherel decomposition of G/H.The article is organized as follows: We start by introducing the necessary notation in Section �. In Sections �and � we set up the machinery needed for the proof of Theorem �.�. The proof is given in Section �. Finally,Theorem �.� is proved in Section �.

� Notation and preliminaries

Throughout the paper, Gwill be a reductive Lie group of the Harish-Chandra class, σ an involution of G andHan open subgroup of the fixed point subgroup for σ. We assume that H is essentially connected as definedin [�, p.��]. The involution of the Lie algebra g of G obtained by deriving σ is denoted by the same symbol.Accordingly, wewrite g = h ⊕ q for the decomposition of g into the+�- and−�-eigenspaces for σ. Thus, h is theLie algebra of H. Here and in the rest of the paper, we adopt the convention to denote Lie groups by Romancapitals, and their Lie algebras by the corresponding Fraktur lower cases.

We fix a Cartan involution θ that commutes with σ and write g = k ⊕ p for the corresponding decomposi-tion of g into the +�- and −�-eigenspaces for θ. Let K be the fixed point subgroup of θ. Then K is a σ-stablemaximal compact subgroupwith Lie algebra k. In addition, we fix amaximal abelian subspace aq of p ∩ q anda maximal abelian subspace a of p containing aq. Then a is σ-stable and

a = aq ⊕ ah,Brought to you by | Utrecht University Library

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E. P. van den Ban, J. J. Kuit and H. Schlichtkrull, K-invariant cusp forms � �

where ah = a ∩ h. This decomposition allows us to identify a∗q and a∗h with the subspaces (a/h)∗ and (a/q)∗,respectively, of a∗.

Let A be the connected Lie group with Lie algebra a. We define M to be the centralizer of A in K. The setof minimal parabolic subgroups containing A is denoted by P(A).

If Q is a parabolic subgroup, then its nilpotent radical will be denoted by NQ. Furthermore, we agree towrite �Q = θQ and �NQ = θNQ. Note that if Q ∈ P(A), then MA is a Levi subgroup of Q and Q = MANQ is theLanglands decomposition of Q.

The root system of a in g is denoted by Σ = Σ(g, a). For Q ∈ P(A) we putΣ(Q) := �α ∈ Σ : gα ⊆ nQ�.

Let Zg(aq) denote the centralizer of aq in g. We define the elements ρQ and ρQ,h of a∗ byρQ( ⋅ ) = �� tr(ad( ⋅ )|nQ ) and ρQ,h( ⋅ ) = �� tr(ad( ⋅ )|nQ∩Zg(aq)).

We say that Q is h-compatible if �α, ρQ,h� ≥ � for all α ∈ Σ(Q).We write Ph(A) for the subset of P(A) consisting of all h-compatible parabolic subgroups.

� τ-spherical cusp formsLet (τ, Vτ) be a finite-dimensional representation of K. We write C∞(G/H : τ) for the space of smooth func-tions � : G/H → Vτ satisfying the transformation rule

�(kx) = τ(k)�(x) (k ∈ K, x ∈ G/H)and we write C(G/H : τ) for the space of � ∈ C∞(G/H : τ) that are Schwartz (see [�, Section �.�]).

Let W(aq) be the Weyl group of the root system of g in aq. Then W(aq) can be realized as the quotientW(aq) = NK(aq)/ZK(aq). LetWK∩H(aq) be the subgroup ofW(aq) of elements that can be realized inNK∩H(aq).We choose a setW of representatives ofW(aq)/WK∩H(aq) inNK(aq) ∩NK(ah) such that e ∈W. This is possiblebecause of the identity

NK(aq) = (NK(aq) ∩NK(ah))ZK(aq);see [��, top of p.��].

Leta� := �

α∈Σ∩a∗h

ker(α)and define

m� := Zg(aq) ∩ a⊥� .Let m�n be the direct sum of all non-compact ideals in m� and let M�n be the connected subgroup of G withLie algebram�n. We define τM to be the restriction of τ to M and write τ�M for the subrepresentation of τM on(Vτ)M�n∩K . We further define

AM,�(τ) :=�v∈W C∞(M/M ∩ vHv−� : τ�M).

We equipAM,�(τ) with the natural Hilbert space structure and note that it is finite-dimensional.Given v ∈W and Q ∈ P(A), we define the parabolic subgroup Qv ∈ P(A) by

Qv := v−�Qv.Let Q ∈ Ph(A). For � ∈ C(G/H : τ) defineHQ,τ� : Aq → AM,�(τ) to be the function given by(HQ,τ�(a))v(m) = aρQ−ρQ,h �

NQv /H∩NQv

�(mavn) dn (v ∈W, m ∈ M, a ∈ Aq).Brought to you by | Utrecht University Library



By [�, Theorem �.��], the integral is absolutely convergent for every � ∈ C(G/H). Furthermore, the mapHQ,τ : C(G/H : τ)→ C∞(Aq) ⊗AM,�(τ) thus obtained is continuous. We call � ∈ C(G/H : τ) a τ-sphericalcusp form if for every Q ∈ Ph(A),

HQ,τ� = �.We will now describe the relation between the τ-spherical cusp forms and the cusp forms defined in the

previous section. Let � be a finite subset of �K. For a representation of K on a vector space V, we denote thesubspace of K-finite vectors with isotypes in � by V�. Consider C(K) equipped with the left-regular represen-tation of K. Define Vτ := C(K)�, i.e., let Vτ be the space of K-finite functions on K whose isotopy types forthe left regular representation are contained in �. We define τ to be the unitary representation of K on Vτobtained from the right action. Then there is a canonical isomorphism

� : C(G/H)� → C(G/H : τ)given by

��(x)(k) = �(kx) (� ∈ C(G/H)� , k ∈ K, x ∈ G/H).By [�, Remark �.�] we now have

�(Ccusp(G/H)�) = Ccusp(G/H : τ).� A formula forHQ,τ

In [�] Eisenstein integrals were constructed which were then used in [�] to derive a formula for HQ,τ. Thisformula is very useful to analyze the relation between cusp forms and discrete series representations. Wewillnow recall this formula and all relevant objects. For details we refer to the two mentioned articles.

We fix Q ∈ Ph(A). We further choose a minimal σθ-stable parabolic subgroup P� containing A, with theproperty that Σ(Q) ∩ σθΣ(Q) ⊆ Σ(P�). (It is easy to see that such a minimal σθ-stable parabolic subgroupalways exists.)

Given ψ ∈ AM,�(τ), λ ∈ a∗q� and v ∈W, we define the function ψv,Q,λ : G → Vτ by

ψv,Q,λ(kman) = aλ−ρQ−ρQ,hτ(k)ψv(m).Letωv beanon-zerodensity onh/h ∩ Lie(v−�Qv). If−�Re λ, α� is su�ciently large for every α ∈ Σ(Q) ∩ σθΣ(Q),then for each x ∈ G and v ∈W the function

h �→ ψQ,λ(xhv−�) dlh(e)−�∗ωdefines an integrable Vτ-valued density on H/H ∩ v−�Qv (see [�, Proposition �.�]). For these λ we define theEisenstein integral Eτ(Q : ψ : λ) ∈ C∞(G/H : τ) by

Eτ(Q : ψ : λ)(x) := �v∈W �

H/H∩v−�Qv ψv,Q,λ(xhv−�) dlh(e)−�∗ωv (x ∈ G).The function λ �→ Eτ(Q : ψ : λ) extends to a meromorphic C∞(G/H : τ)-valued function on a∗q�. This defini-tion of Eisenstein integrals coincides with the definition in [�, Section �]. We write Eτ(Q : ⋅ ) for the map

AM,�(τ) ∋ ψ �→ Eτ(Q : ψ : ⋅ ).We define

a∗+q = a∗+q (P�) := �λ ∈ a∗q : �λ, α� > � for all α ∈ Σ(P�)�.Let SQ,τ be the set of λ ∈ a∗+q + ia∗q such that Eτ(Q : − ⋅ ) is singular at λ. By [�, Lemma �.�], this set is finiteand contained in a∗+q . It follows from [�, Theorem �.�� (b)] that all poles of Eτ(Q : − ⋅ ) are simple.

Let ξ be the unique vector in a∗+q of unit length with respect to the Killing form. For a meromorphicfunction φ : a∗q� → � and a point � ∈ a∗q� we define the residue

Resλ=� φ(λ) := Resz=� φ(� + zξ).




Here, z is a variable in the complex plane, and the residue on the right-hand side is the usual residue fromcomplex analysis, i.e., the coe�cient of z−� in the Laurent expansion of z �→ φ(� + zξ) around z = �. For� ∈ SQ,τ we define Resτ(Q : �) = Resτ(Q : � : ⋅ ) to be the function G/H → Hom(AM,�(τ), Vτ) given by

Resτ(Q : � : x)(ψ) = − Resλ=−� E(Q : ψ : λ)(x).

By [�, Theorem �.�� (a)],

Resτ(Q : �)(ψ) ∈ Cds(G/H : τ) (� ∈ SQ,τ , ψ ∈ AM,�(τ)). (�.�)

Following [�, Section�.�], we define for� ∈ C∞c (G/H : τ) the smooth function IQ,τ� : Aq → AM,�(τ) thatis determined by the equation�IQ,τ�(a), ψ� = lim

�↓� ��ν+ia∗

q

�G/H ��(x), Eτ(Q : ψ : − �λ)(x)�aλ dx dλ

for every ψ ∈ AM,�(τ) and a ∈ Aq. Here ν is any choice of element of a∗+q ; the definition is independent of thischoice. The map IQ,τ : C∞c (G/H : τ)→ C∞(Aq) ⊗AM,�(τ) extends to a continuous map

IQ,τ : C(G/H : τ)→ C∞(Aq) ⊗AM,�(τ);see [�, Proposition �.�]. The image of IQ,τ is contained in the tempered AM,�(τ)-valued functions on Aq andis called the tempered term of the Harish-Chandra transform. This map has the following properties.

Proposition �.� ([�, Corollaries �.� and �.��]). (i) Assume � ∈ C(G/H : τ). Then for every ψ ∈ AM,�(τ) anda ∈ Aq one has �HQ,τ�(a) − IQ,τ�(a), ψ� = �

�∈SQ,τ a� �G/H ��(x), Resτ(Q : � : x)(ψ)� dx. (�.�)

(ii) Cds(G/H : τ) = ker(IQ,τ).� Proof of Theorem �.�

From (�.�) it follows that (ii) implies (i) in Theorem�.�. Moreover, if (�.�) holds, then (i) implies (ii). It remainsto prove (�.�).

Let Q ∈ Ph(A). Let further �K be the trivial representation of K and let � ∈ Cds(G/H : �K) = Cds(G/H)K .Then IQ,�K� = �. HenceHQ,�K� = � if and only if the right-hand side of (�.�) vanishes for all a ∈ Aq and allψ ∈ AM,�(�K). The latter is true if and only if�

G/H ��(x), Res�K (Q : � : x)(ψ)� dx = � (� ∈ SQ,�K , ψ ∈ AM,�(�K)),i.e.,HQ,�K� = � if and only if � is perpendicular to

VQ := span�Res�K (Q : �)(ψ) : � ∈ SQ,�K , ψ ∈ AM,�(�K)�.To show (�.�) it thus su�ces to prove the following proposition.

Proposition �.�. VQ = Cds(G/H)K .To prove the proposition we will study the orthogonal projection (with respect to the inner product onL�(G/H : �K))

Tds : C∞c (G/H : �K)→ Cds(G/H : �K).To this end we first recall a formula for Tds.




Let the minimal σθ-stable parabolic P� be as before (see Section �). For λ ∈ a∗q� and ψ ∈ AM,�(�K) wedefine the Eisenstein integral E�K ( �P� : ψ : λ) = E(P� : ψ : λ) like Eτ(Q : ψ : λ) in the previous section, butwith τ and Q replaced by �K and �P� = θP�, respectively. Note that in order to replace Q by P� in this construc-tion, we need to replace the spaceAM,�(τ) by

AM� ,�(τ) :=�v∈W C∞(M�/M� ∩ vHv−� : τM� ),

where τM� is the restriction of τ to M� ∩ K. However, in view of [�, Lemma �.�] applied with vHv−� in placeof H, for v ∈W we have

AM,�(τ) � AM� ,�(τ).Wenormalize these Eisenstein integrals as in [�, Section�], and thusweobtain the normalized Eisenstein

integralE�( �P� : ψ : λ) ∈ C∞(G/H : �K)

for ψ ∈ AM,�(�K) and generic λ ∈ a∗q�.We define

A−q := �a ∈ A : aα < � for all α ∈ Σ(P�)�.For w ∈W let δw ∈ AM,�(�K) be the element satisfying�ψ, δw� = ψw(e) (ψ ∈ AM,�(�K)).Observe thatAM,�(�K) is spanned by {δw : w ∈W}. For w ∈W and generic λ ∈ a∗q� we write

Φw(λ : ⋅ ) = Φ �P� ,w(λ : ⋅ ) : A−q → End(�) = �for the function introduced in [�, Section ��]. From [�, (��) and Remark �.�] it follows thatΦw(λ, a) dependsholomorphically on λ for λ ∈ a∗+q + ia∗q. Moreover, it can be seen from [�, (��) and Proposition �.�] thatΦw(λ : a) is real for λ ∈ a∗+q .

Let � = {−α} be the set of simple roots in Σ( �P�). From [�, Theorem ��.� (c)] (see also [�, Definition ��.�])it follows that Tds coincides with the operator T� defined in [�, (�.�)]. In our setting it is straightforwardto rewrite this equation and thus obtain the following formula for Tds: For � ∈ C∞c (G/H : �K), w ∈W anda ∈ A−q,

Tds�(w−�aw) = �G/H �(x)�

�∈S Resλ=� (Φw(λ : a)E�( �P� : δw : −λ)(x)) dx. (�.�)

Note that Tds� is completely determined by this formula as KA−qWH is a dense open subset of G.We now compare the residues occurring in (�.�) to the residues Res�K (Q : �). This is done in the following

lemma.

Lemma �.�. The set S := SQ,�K is equal to the set of λ ∈ a∗+q + ia∗q such thatλ �→ Φw(λ : a)E�( �P� : δw : −λ)

is singular at λ for some w ∈W and a ∈ A−q. The poles which occur are simple. Moreover, for every � ∈ S thereexists a constant c� > � so that for every w ∈W and a ∈ A−q,

Resλ=� (Φw(λ : a)E�( �P� : δw : −λ)) = c�Φw(� : a)Res�K (Q : �)(δw). (�.�)

Proof. Let P ∈ P(A) be the unique minimal parabolic subgroup contained in P� with Σ(P) ∩ a∗h = Σ(Q) ∩ a∗h.For generic λ ∈ a∗� the standard intertwining operator A(σ(P) : Q : �M : λ) maps the space C∞(Q : �M : λ)Kto the space C∞(σ(P) : �M : λ)K . Both of these spaces are �-dimensional. Let �Q,λ ∈ C∞(Q : �M : λ)K and�σ(P),λ ∈ C∞(σ(P) : �M : λ)K be determined by

�Q,λ(e) = �σ(P),λ(e) = �.Brought to you by | Utrecht University Library



Then the action of A(σ(P) : Q : �M : λ) on C∞(Q : �M : λ)K is determined by the identity

A(σ(P) : Q : �M : λ)�Q,λ = c(σ(P), Q : λ)�Q,λ .Here c := c(σ(P), Q : ⋅) is the partial c-function which for λ in the set�λ ∈ a∗� : Re�λ, α� > � for all α ∈ Σ(P�) ∩ Σ(Q)�is given by the integral

c(λ) = �θNP�∩θNQ

�Q,λ(n) dn, (�.�)

and for other λ ∈ a∗� by meromorphic continuation. It follows from [�, Proposition �.�] that for genericλ ∈ a∗q�,

E�K (Q : ψ : −λ) = c(λ + ρQ,h)E�( �P� : ψ : −λ) (ψ ∈ AM,�(�K)).By assumption, Q ∈ Ph(A), and hence �ρQ,h, α� ≥ � for all α ∈ Σ(Q). Therefore, �λ + ρQ,h, α� > � for all

α ∈ Σ(P�) ∩ Σ(Q) if λ ∈ a∗+q + ia∗q, and thus λ �→ c(λ + ρQ,h) is holomorphic on a∗+q + ia∗q and given by the inte-gral representation (�.�). Note that for λ ∈ a∗+q (P) the integrand is strictly positive, and hence c(λ + ρQ,h) > �.

Let � ∈ S. Since the pole of E�K (Q : ψ : −λ) at λ = −� is simple and the function

λ �→ Φw(λ : a)c(λ + ρQ,h)

is holomorphic on a∗+q + ia∗q, it follows thatResλ=� (Φw(λ : a)E�( �P� : δw : −λ)) = Φw(� : a)

c(� + ρQ,h) Res�K (Q : �)(δw),and hence (�.�) follows with c� = �

c(�+ρQ,h) .Proof of Proposition �.�. Since Tds is the restriction to C∞c (G/H)K of the orthogonal projection

C(G/H)K → Cds(G/H)K(with respect to the L�-inner product), it follows from the formula (�.�) for Tds and Lemma �.� that

Cds(G/H)K = span � ��∈S Resλ=� (Φw(λ : a)E�( �P� : δw : −λ)) : a ∈ A−q , w ∈W�= span � ��∈S c�Φw(� : a)Res�K (Q : �)(δw) : a ∈ A−q , w ∈W�⊆ VQ .

The other inclusion VQ ⊆ Cds(G/H)K is a consequence of (�.�).� Multiplicity of K-spherical discrete series representationsIn this final sectionwe use the analysis that has been used for the proof of Theorem �.� to prove Theorem �.�.

We begin with a lemma. If π is a discrete series representation for G/H, then we write Cπ(G/H) for theclosure of the span of the K-finite generalized matrix coe�cients of π in C(G/H). Note that the closure ofCπ(G/H) in L�(G/H) decomposes into a direct sum of representations equivalent to π.

Lemma �.�. For every K-spherical discrete series representation π of G/H there exists a unique � ∈ SQ,�K sothat

Cπ(G/H)K ⊆ span�Res�K (Q : �)(ψ) : ψ ∈ AM,�(�K)�. (�.�)




Moreover, if �, ν ∈ S and � �= ν, then for every ψ, χ ∈ AM,�(�K),�G/H Res�K (Q : � : x)(ψ)Res�K (Q : ν : x)(χ) dx = �. (�.�)

Proof. Let π beaK-spherical discrete series representation forG/H. ThenCπ(G/H)K is non-zero andCπ(G/H)Kis canonically identified with a subspace Cπ(G/H : �K) of C(G/H : �K). Let � ∈ Cπ(G/H : �K). Let �G/Hand �Aq be the Laplacian on G/H and Aq, respectively. Since � is a joint-eigenfunction of �(G/H), thereexists a c ∈ � such that

�G/H� = c�.The constant c depends only on π, not on the particular choice of �. By [�, Lemma �.�], the functionHQ,�K�satisfies

�AqHQ,�K� = (c + �ρP� , ρP��)HQ,�K�. (�.�)

It follows from Proposition �.� that HQ,�K� is a finite sum of exponential functions, all with non-zeroreal exponents � in the set SQ,�K . Together with (�.�) this implies that there exists a unique � ∈ SQ,�K (onlydepending on π, not on the function �) with ��, �� = c + �ρP� , ρP��, and a ψ� ∈ AM,�(�K) such that

HQ,�K�(a) = a�ψ�.

In view of (�.�) it follows that � is orthogonal to Res�K (Q : ν)(ψ) for every ν ∈ Swith ν �= � and ψ ∈ AM,�(�K).We conclude that for every K-spherical discrete series representation π there exists a unique � ∈ S such that

Cπ(G/H)K ⊆ � �ν∈S\{�} span�Res�K (Q : ν)(ψ) : ψ ∈ AM,�(�K)��⊥.

For � ∈ S let D� be the set of discrete series representations π such that �G/H acts on Cπ(G/H) by thescalar ��, �� − �ρP� , ρP��. It follows from Proposition �.� that�

π∈D�

Cπ(G/H)K = � �ν∈S\{�} span�Res�K (Q : ν)(ψ) : ψ ∈ AM,�(�K)��⊥,

and hence for every � ∈ S,�π∈D�

Cπ(G/H)K = � �ν∈S\{�}�π∈Dν

Cπ(G/H)�⊥= �ν∈S\{�} �χ∈S\{ν} span�Res�K (Q : χ)(ψ) : ψ ∈ AM,�(�K)�= span�Res�K (Q : �)(ψ) : ψ ∈ AM,�(�K)�.

This proves the assertions in the proposition.

Proof of Theorem �.�. Let π be a K-spherical discrete series representation. If |W| = �, then the right-handside of (�.�) is �-dimensional, and hence dimCπ(G/H)K = � and the multiplicity with which π occurs in thePlancherel decomposition is equal to �.

Now assume that |W| = �. In view of Lemma �.� we may rewrite (�.�) as

Tds�(w−�aw) = ��∈S c�Φw(� : a) �

G/H �(x)Res�K (Q : � : x)(δw) dx,with a ∈ A−q and w ∈W. We used in the derivation of this formula that Φw(� : ⋅ ) is real-valued for � ∈ a∗+q .Since Res�K (Q : �)(δw) ∈ Cds(G/H)K, it follows in view of (�.�) that for v, w ∈W and a ∈ A−q,

Res�K (Q : � : w−�aw)(δv) = c�Φw(� : a) �G/H Res�K (Q : � : x)(δv)Res�K (Q : � : x)(δw) dx.




In particular, it follows that there exist constants cv,w ∈ � so thatRes�K (Q : � : kawh)(δv) = cv,wΦw(� : a) (k ∈ K, a ∈ A−q , h ∈ H, v, w ∈W).

Let v� be the non-trivial element inW. Note that for every w ∈W the restricted functions

Res�K (Q : �)(δe)|KA−qwH and Res�K (Q : �)(δv� )|KA−

qwH

are linearly dependent. Since the Res�K (Q : �)(δv) are K-fixed (hence K-finite) vectors in an irreducible sub-representation of L�(G/H), they are analytic vectors and hence real analytic functions on G/H. It follows thatcv,w is independent of w ∈W, and thus that Res�K (Q : �)(δe) and Res�K (Q : �)(δv� ) are linearly dependent.Therefore, the right-hand side of (�.�) is�-dimensional. This implies that dimCπ(G/H)K = � and that π occursin the Plancherel decomposition of G/H with multiplicity one.

Funding: The second author was supported by Deutsche Forschungsgemeinschaft (DFG, German ResearchFoundation) – ��.

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