on the unramiﬁed spectrum of spherical varieties over p...

On the unramified spectrum of spherical varieties over

p-adic fields

Yiannis Sakellaridis

Abstract

The description of irreducible representations of a group G can be seen as a questionin harmonic analysis; namely, decomposing a suitable space of functions on G into irre-ducibles for the action of G×G by left and right multiplication.

For a split p-adic reductive group G over a local non-archimedean field, unramifiedirreducible smooth representations are in bijection with semisimple conjugacy classes inthe “Langlands dual” group.

We generalize this description to an arbitrary spherical variety X of G as follows: Ir-reducible quotients of the “unramified” Bernstein component of C∞c (X) are in naturalalmost bijection with (a number of copies of) the quotient of a complex torus by the“little Weyl group” of X. This leads to a weak analog of results of D. Gaitsgory andD. Nadler on the Hecke module of unramified vectors, and an understanding of the phe-nomenon that representations “distinguished” by certain subgroups are functorial lifts. Inthe course of the proof, rationality properties of spherical varieties are examined and anew interpretation is given for the action, defined by F. Knop, of the Weyl group on theset of Borel orbits.

2000 Mathematics Subject Classification 22E50 (Primary); 11F85, 14M17 (Secondary)


Contents

1 Introduction 31.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Parametrization of irreducible quotients . . . . . . . . . . . . . . . . . . . 31.3 The Hecke module of unramified vectors . . . . . . . . . . . . . . . . . . . 41.4 Interpretation of Knop’s action . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Rationality results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 Endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.7 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Spherical varieties over algebraically closed fields 72.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Knop’s action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Parabolically induced spherical varieties . . . . . . . . . . . . . . . . . . . 82.4 Non-homogeneous spherical varieties . . . . . . . . . . . . . . . . . . . . . 92.5 Non-degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Rationality properties 103.1 Homogeneous spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Rationality of the open orbit . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Splitting in B-orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 Spherical varieties for SL2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.5 Rationality of orbits of maximal rank . . . . . . . . . . . . . . . . . . . . . 123.6 The Zariski and Hausdorff topologies . . . . . . . . . . . . . . . . . . . . . 133.7 Invariant differential forms and measures . . . . . . . . . . . . . . . . . . . 14

4 Mackey theory and intertwining operators 154.1 Unramified principal series . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 The Bernstein decomposition and centre . . . . . . . . . . . . . . . . . . . 164.3 Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.4 Distributions on a single orbit . . . . . . . . . . . . . . . . . . . . . . . . . 184.5 Convergence and rationality . . . . . . . . . . . . . . . . . . . . . . . . . . 204.6 Jacquet modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.7 Discussion of the poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.8 Non-trivial line bundles and standard intertwining operators . . . . . . . . 25

5 Interpretation of Knop’s action 255.1 Avoidance of “bad” divisors . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2 The theorem for SL2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.3 Statement and proof of the main theorem . . . . . . . . . . . . . . . . . . 275.4 Corollaries and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.5 Parabolic induction with an additive character . . . . . . . . . . . . . . . 32

6 Unramified vectors and endomorphisms 346.1 Spectral support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.2 The Hecke module of unramified vectors . . . . . . . . . . . . . . . . . . . 366.3 A commutative ring of endomorphisms . . . . . . . . . . . . . . . . . . . . 37

2

On the unramified spectrum of spherical varieties

1. Introduction

1.1 MotivationLet G be a split reductive linear algebraic group over a local non-archimedean field k in characteristiczero. A k-variety X with a k-action1 of G is called spherical if the Borel subgroup B of G has anopen orbit X on X.2 This includes, but is not limited to, symmetric (fixed points of involutionsof G) and horospherical varieties (varieties where the stabilizer of each point contains a maximalunipotent subgroup). The group G itself can be considered as a spherical variety under the actionof G×G on the left and right, and in fact many well-known theorems for algebraic groups can beseen as special cases of more general theorems for spherical varieties under this perspective (e.g.[Kn94b]).3

The importance of the open orbit condition becomes apparent in the following:

Theorem Vinberg and Kimel’feld [VK78]. Let X be a quasi-affine G-variety over an algebraicallyclosed field k. The space k[X] of regular functions on X, considered as a representation of G byright translations, is multiplicity-free if and only if X is spherical.

If X = H\G is quasi-affine, (which, we show in §2.1, can be assumed without serious loss ingenerality) the above theorem states that X is spherical if and only if (G,H) is a Gelfand pair inthe category of algebraic representations.

One goal of the present work is to examine to what extent a similar result is true in the cate-gory of smooth representations of p-adic groups. Spherical varieties are ubiquitous in the theory ofautomorphic forms, albeit their importance has not been fully appreciated. Providing candidatesfor Gelfand pairs (but even, sometimes, when the Gelfand condition fails), spherical varieties playan essential role in the theory of integral representations of L-functions [PS75, GPR87], in the rel-ative trace formula [Ja97, La] and other areas such as explicit computations of arithmetic interest[Ca80, HiY99, Sa]. It turns out that the unramified part of the spectrum of X, where k is localnon-archimedean, is always of finite multiplicity, and I have been able to express the “generic”multiplicity in terms of invariant-theoretic data associated to the spherical variety.

Hence, another goal is to establish a connection between the representation theory over p-adicfields and the rich algebro-geometric structure of spherical varieties which has been discovered inthe works of Brion, Knop, Luna, Vinberg, Vust and others. This may allow to replace explicit,hands-on methods such as double coset decompositions with more elegant ones. More importantly,it leads to a fascinating general picture, which parallels results that lie at the heart of the Langlandsconjectures and which I shortly describe now.

1.2 Parametrization of irreducible quotientsThe main phenomenon that the current work reveals is the local analog of a global statement ofthe following form, very often arising in the theory of the relative trace formula and elsewhere: “Anautomorphic representation π of G is a functorial lift from (a certain other group) G′ if and onlyif it is distinguished by (a certain subgroup) H.” Instead of explaining the global notion of being“distinguished”, we describe its local analog which is the object of study here: π is distinguished byH = H(k) if it appears as a quotient of C∞c (X), the representation of G = G(k) on the space ofsmooth, compactly supported functions on X = (H\G)(k).

1Our convention will be that the action of the group is on the right.2It is enough to assume that there exists an open Borel orbit over the algebraic closure; we show later that it willthen have a point over k.3Notice, though, that one usually makes use of a theorem for G in order to prove its generalization to an arbitraryspherical variety; this is the case in our present work, too.

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Recall that irreducible unramified representations of G are in “almost bijection” with semisimpleconjugacy classes (' A∗/W ) in the Langlands dual group G of G – and they can be realized as(subquotients of) unramified principal series I(χ) = IndG

B(χδ12 ). (Here A∗ is the maximal torus of the

dual group, or equivalently the complex torus of unramified characters of the Borel subgroup, andW is the Weyl group; δ is the modular character of the Borel subgroup.) To each spherical varietyX, Brion [Br90] associates a finite reflection group (the “little Weyl group”) WX . An alternativeconstruction by Knop [Kn94a] proves, among others, that WX ⊂ W canonically; and WX acts onthe vector space a∗X := X (X) ⊗Z k, where X (X) denotes the weights of B-semiinvariants (regulareigenfunctions) on the open orbit. The complex analog of a∗X is the Lie algebra a∗X := X (X) ⊗Z Cof a subtorus A∗X ⊂ A∗. Let us also denote by P = P(X) the standard parabolic g ∈ G|Xg = Xand by [W/WP ] the canonical set of representatives of minimal length of W/WP -cosets (where WP

is the Weyl group of the Levi of P). With this notation, a simplified version of our main theoremis:

1.2.1 Theorem. i) There exists a non-zero morphism M : C∞c (X)→ I(χ) only if χ belongs to

a [W/WP ]-translate of δ−12A∗X ; the image of such an M , for generic χ, is irreducible.

ii) Assume that the k-points of the open B-orbit all belong to the same B-orbit. There is a natural

non-zero rational family of morphisms: Sχ : C∞c (X) → I(χ) for χ ∈ δ− 12A∗X such that: The

quotients Sχ and Swχ, for generic χ, are isomorphic if and only if w ∈WX ; and for w in a set

of representatives of WX\NW (δ−12A∗X) and generic χ the quotients Swχw form a basis for

the space of morphisms into the corresponding irreducible representation. Hence, for genericχ ∈ δ− 1

2A∗X , we have dimHom(C∞c (X), I(χ)) = (NW(δ−12 A∗X) : WX).

(We assume throughout that X is quasi-affine and that X – the set of k-points of its open orbit –admits a B-invariant measure. Neither of these assumptions, as we show, causes harm in generality.)

In other words, there is a natural almost one-to-one correspondence between a “basis of irre-ducible quotients” of the “unramified” Bernstein component (cf. §4.2) (C∞c (X))ur and the complexspace δ−

12A∗X/WX . In the phenomenon of “distinguished” lifts that I alluded to before, A∗X and WX

are the maximal torus and the Weyl group of the “Langlands dual” G′ of G′.

1.3 The Hecke module of unramified vectorsIt should be mentioned that in the present work only the maximal torus and the Weyl group of thisdual group appear. The whole group has been identified in recent work of Gaitsgory and Nadler[GN1]–[GN4] in the context of the geometric Langlands program. In that work, it is proven that acertain category of G(o)-equivariant perverse sheaves on X (where the spherical variety X is nowdefined over a global complex curve and o = C[[t]]) is equivalent to the category of finite-dimensionalrepresentations of G′. I prove the following weak analog of their results in the p-adic setting:

1.3.1 Theorem. Assume that the k-points of the open B-orbit all belong to the same B-orbit.Let K denote a hyperspecial maximal compact subgroup of G. Let HX denote the quotient of theHecke algebra H(G,K) ' C[A∗]W corresponding to the image of δ−

12A∗X under A∗ → A∗/W and

let KX denote the quotient field of HX .

The space C∞c (X)K is a finitely-generated, torsion-free module for HX .

Moreover, we have: C∞c (X)K ⊗HX KX '(C(δ−

12A∗X)WX

).

Notice that the invariants A∗X ,WX appearing in the theorems above only depend on the openG-orbit H\G ⊂ X, although the representations considered depend on X itself.

As is usually the case with spherical varieties, one recovers classical results by considering G asa spherical G×G variety under left and right multiplication; in this case, we recover the (generic)

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description of irreducible unramified representations of G by semisimple conjugacy classes in itsLanglands dual group, and the Satake isomorphism.

1.4 Interpretation of Knop’s action

Theorem 1.2.1 comes as a corollary to an analysis that we perform and an interpretation in thecontext of the representation theory of p-adic groups that we give to an action4, defined by F. Knop[Kn95a], of the Weyl group W on the set of Borel orbits on X. We recall the definition of it inSection 2; for now, let wY denote the image of a B-orbit Y under the action of w ∈ W . Usingstandard “Mackey theory” [BZ76, Cas] I define for every B-orbit of maximal rank Y a rationalfamily of morphisms: SYχ : C∞c (X)→ I(χ), given by rational continuation of a suitable integral on

Y ; what was denoted by Sχ in the formulation of Theorem 1.2.1 is now SXχ . (Notice that, whilethis form of “Mackey theory” has been used extensively in the past, it has probably never beenapplied in this generality, and the technical results that we collect or prove for that purpose maybe of independent interest.)

Recall now that we also have the standard intertwining operators Tw : I(χ)→ I(wχ). The heartof the current work is the proof of the following theorem on the effect of composing the operatorsTw with SYχ :

1.4.1 Theorem. We have Tw SXχ 6= 0 if and only if w ∈ [W/WP ]. In that case, Tw SXχ ∼ SwXwχ ,

where ∼ denotes equality up to a non-zero rational function of χ.

1.5 Rationality results

The simplifying assumption made above, that the k-points of the open B-orbit all belong to thesame B-orbit, fails to be true in many interesting cases. In order to consider the general case, wherethe k-points of a B-orbit may split into several B-orbits, we need some rationality results. In section3 I prove:

1.5.1 Theorem. Every B-orbit of maximal rank has a point over k. In addition, all B-orbits ofmaximal rank split into the same (finite) number of B-orbits over k, and the splitting of each ofthem is naturally parametrized by a finite abelian group Γ (same for all).

The rationality properties of the structure of Borel orbits on X that we examine may be ofindependent interest. They generalize a portion of work of Helminck and Wang on symmetricvarieties [HW93].

If the k-points of a B-orbit split into several B-orbits, we still define morphisms: C∞c (X)→ I(χ)using Mackey theory, but the dimension of the space SYχ of morphisms associated to an orbit of

maximal rank Y is equal to the order of Γ. Theorem 1.4.1 still holds in the form Tw SXχ ∼ SwXwχ ,

but it is desirable to exhibit bases of the spaces SYχ which map explicitly to each other whencomposed with Tw. I have been able to describe such bases (whose elements are denoted by SY,θχ ) –the definition is rather complicated to be stated here and we therefore refer the reader to Theorem5.3.1 for details. The general statements of Theorems 1.2.1 and 1.3.1 can be found in Corollaries5.4.1, 5.4.2 and Theorem 6.2.1, respectively.

4The action defined by Knop is now a right action, due to our convention that the group acts on the right. However,

we modify it to a left action by defining wY := Yw−1, where Yw−1

denotes the action of w−1 on Y as defined byKnop.

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1.6 EndomorphismsFinally, I discuss a (partly conjectural) ring of endomorphisms of the Hecke module (C∞c (X))K

which bears a remarkable similarity to the algebra of invariant differential operators on a sphericalvariety. For simplicity, assume here that Γ = 1. We recall Knop’s generalization of the Harish-Chandra homomorphism [Kn94b]:

Theorem Knop. The algebra of invariant differential operators on a spherical variety X (overan algebraically closed field k of characteristic 0) is commutative and isomorphic to k[ρ + a∗X ]WX .This generalizes the Harish-Chandra homomorphism for the center z(G) of the universal envelopingalgebra of g (if we regard the group G as a spherical G×G variety) and the following diagram iscommutative:

z(G) −−−−→ D(X)G∥∥∥∥∥∥

k[a∗]W −−−−→ k[ρ+ a∗X ]WX

Our description of unramified vectors in C∞c (X) leads easily to a conjectural description of acommutative subalgebra of their endomorphism algebra as an H(G,K)-module, which should benaturally isomorphic the C[δ−

12A∗X ]WX , the ring of regular functions on δ−

12A∗X/WX . Assume here

that Γ = 1. The precise statement of the conjecture is:

Conjecture. Call “geometric” an endomorphism of (C∞c (X))K that preserves up to a rational

multiple the family of morphisms SXχ . There is a canonical isomorphism(EndH(G,K)C∞c (X)K

)geom 'C[δ−

12A∗X ]WX such that the following diagram commutes:

H(G,K) −−−−→ (EndH(G,K)C∞c (X)K

)geom

∥∥∥∥∥∥

C[A∗]W −−−−→ C[δ−12A∗X ]WX

In fact, it is easy to prove this conjecture in many cases:

1.6.1 Theorem. The above conjecture is true if:

i) the unramified spectrum of X is generically multiplicity-free, in which case the geometricendomorphisms are all the endomorphisms of (C∞c (X))K , or

ii) the spherical variety X is “parabolically induced” from a spherical variety whose unramifiedspectrum is generically multiplicity-free.

1.7 NotationWe will always be working over a field of characteristic zero. By a local non-archimedean field wewill mean a locally compact one (hence, with finite residue field.) A reductive group is an affinealgebraic group with trivial unipotent radical; in addition, for the whole paper “reductive” will alsomean (geometrically) connected.

Given a scheme Y over k, we denote by Y the base change Y×spec k spec k. The set of k-pointswill be denoted by Y or by Y(k). We generally fix a Borel subgroup B ⊂ G (with unipotent radicalU) and, when necessary, a maximal torus A ⊂ B. We denote by Gm, Ga the multiplicative andadditive group, respectively, over k, by N (•) the normalizer of •, by L(•) the Lie algebra of •, by UP

the unipotent radical of a group P, and by Uα the one-parameter unipotent subgroup associatedto the root α of A. For any root α of A, α will denote the correponding co-root: Gm → A. Weuse additive, exponential notation for roots and co-roots – for example, we write aα for the value

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of α on a ∈ A. If Y is a B-variety with an open B-orbit, then Y denotes the open B-orbit. Givena B-variety Y, we denote by k(Y)(B) (resp. k[Y](B)) the set of B-semiinvariants (eigenfunctions)on the rational (resp. regular) functions on Y, and by X (Y) the corresponding group of weights(eigencharacters). Finally, the space of an one-dimensional complex character χ of a group H isdenoted by Cχ.

Acknowledgements

I would like to thank professors Joseph Bernstein, Daniel Bump, Dennis Gaitsgory, Herve Jacquet,Friedrich Knop, Peter Sarnak and Akshay Venka-tesh for useful discussions, correspondence orreferences on this and related problems. Also, David Nadler for making available to me his preprintswith Dennis Gaitsgory.

2. Spherical varieties over algebraically closed fields

2.1 Basic notionsLet G be an algebraic group over an arbitrary field k in characteristic zero. By a G-variety (overk) we will mean a geometrically integral and separated k-scheme of finite type with a morphicaction of G over k. A G-variety X is called homogeneous if G(k) acts transitively on X(k) – thenX is automatically non-singular. If X has a point over k, its stabilizer H is a subgroup over kand X ' H\G, the geometric quotient of G by H. Conversely, for any subgroup H the geometricquotient H\G is a homogeneous variety under the action of G.

Now, let G be a reductive group over a field k. A G-variety X over k (not necessarily homoge-neous) is called spherical if B (where B is a Borel subgroup of G) has a Zariski open orbit on X.This is equivalent ([Br86],[Vi86]) to there being finitely many B-orbits. As a matter of convention,when we say “a B-orbit on X” we will mean “a B-orbit on X” – then one naturally has to examinequestions such as whether a “B-orbit” is defined over k, which will be the object of the next section.

For the whole paper, we will assume that X is quasi-affine. This is not a really serious restric-tion: By [Bo91, Theorem 5.1], given a subgroup H of G there exists a finite-dimensional algebraicrepresentation of G over k in which H is the stabilizer of a line. If H has trivial k-character group(i.e. group of homomorphisms H → Gm over k), then this implies that H\G is embedded in thespace of this representation and hence is quasi-affine. (Recall [Bo91, Proposition 1.8] that an orbitof an algebraic group is always locally closed.) Hence, for an arbitrary H, we may replace H by thekernel H0 of all its k-characters and consider the quasi-affine variety H0\G, which is spherical forthe (H/H0)×G action.

From this point until the end of the present section we assume that k is algebraically closed.Given a B-orbit Y, the group of weights X (Y) of B acting on k(Y) is the character group ofA/AY , where A = B/U and AY is the image modulo U of the stabilizer of any point y ∈ Y. Therank of X (Y) is called the rank of the orbit Y. If Y is the open orbit, we will denote AY by AX ;the corresponding rank is the rank of the spherical variety. The rank of the open orbit is maximalamong all B-orbits, as we explain below.

We recall the classification and properties of spherical subgroups H for PGL2.

2.1.1 Theorem (A classic). The spherical subgroups H of G = PGL2 over an algebraically closedfield k in characteristic zero are:

Type G: PGL2; in this case there is a single B-orbit on H\G.

Type T: a maximal torus T; there are three B-orbits: the open one, and two closed orbits of smallerrank.

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Type N: N (T); there are two B-orbits: the open one, and a closed one of smaller rank.

Type U: S ·U, where U is a maximal unipotent subgroup and S ⊂ N (U); there are two B-orbits,an open and a closed one, both of the same rank.

2.2 Knop’s actionIn [Kn95a], F. Knop defines an action of the Weyl group on the set of Borel orbits on a homogeneousspherical variety. This action is defined explicitly for simple reflections, and then it is shown thatthis description induces an action of the Weyl group (i.e. satisfies the braid relations). For the simplereflection wα corresponding to a simple root α it is defined as follows:

Let Pα denote the minimal5 parabolic associated to α (for a fixed choice of maximal torusA → B) and let Y be a B-orbit; we will define an action of wα on the set of B-orbits containedin the Pα-orbit Y · Pα. Consider the quotient Pα → PGL2 = Aut(P1) where P1 = B\Pα. Theimage of the stabilizer (Pα)y of a point y ∈ Y is a spherical subgroup of PGL2, and according tothe classification above we say that “(Y, α) is of type G, T, N or U”. (As a matter of language, wealso say that “α raises Y” if Y is not open in YPα.) We define the action according to the type ofthat spherical subgroup:

If it is of type G, wα will stabilize the unique B-orbit in the given Pα-orbit. In the case of typeT, wα stabilizes the open orbit and interchanges the other two. In the case of type N, wα stabilizesboth orbits. Finally, in the case of type U, wα interchanges the two orbits. Since this defines aright action in our case that the group acts on the right, we modify it to a left action by definingwY := Yw−1

, where Yw−1denotes the action of w−1 on Y as defined by Knop; of course, in the

case of simple reflections the description does not change. Notice that in every case the action ofwα preserves the rank of the orbit; more precisely, Knop proves:

2.2.1 Lemma. Let Y denote the open orbit of YPα, and let Z∗ denote the closed orbits (if any).There exist the following relations between their character groups:

Type G: X (Y) ⊂ X (A)wα .

Type U: wαX (Z) = X (Y).

Type T: wαX (Z1) = X (Z2) ⊂ X (Y).

Type N: wαX (Z) ⊂ X (Y).

(An exponent on the right denotes “invariants”. An exponent on the left denotes the action ofthe Weyl group. Due to our modification of the definition, the lemma is true as stated, with theleft action of W on the characters.) In particular, X (wY) = wX (Y) for every w and the set B00 oforbits of maximal rank is stable under the action of the Weyl group.

2.3 Parabolically induced spherical varietiesThere is an “inductive” process of constructing spherical subgroups: Given a Levi subgroup L ofa parabolic P ⊂ G and a spherical subgroup M of L, we can form the subgroup H = M nUP,which is a spherical subgroup of G. The structure of the B-orbits of X := H\G, relevant to theBorel orbits of M\L, has been investigated by Brion [Br01]. Each orbit Y of X can be writtenuniquely as Y′wB for w ∈ [WP \W ], where [WP \W ] denotes the set of representatives of minimallength for right cosets of WP (the Weyl group of L) and Y′ a Borel orbit of X′ := M\L. We haveX (X) = X (X′) and WX = WX′ .

5Although, strictly speaking, the Borel is a minimal parabolic, we will be using this term to refer to those parabolicsubgroups whose Levi has semisimple rank one, as is usual in the literature.

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2.4 Non-homogeneous spherical varietiesNow we examine spherical varieties X which are not necessarily homogeneous, i.e. may have morethan one G-orbit. It is known then [Kn91] that X contains a finite number of G-orbits, and thateach of them is also spherical. Let Y be a G-orbit. To Y one associates [Kn91, §2] the coneCY(X) ⊂ Q := HomZ(X (X),Q) spanned by the valuations induced by B-stable prime divisorswhich contain Y. This cone is non-trivial (more precisely [Kn91, Theorem 3.1], there exists abijection between isomorphism classes of simple embeddings of H\G and “colored cones”) and wehave:

2.4.1 Theorem. Let X be a quasi-affine spherical variety, Y a G-orbit and f ∈ k[Y](B). Thereexists f ′ ∈ k[X](B) with f ′|Y = f . Hence, the group of weights of B on Y is a subgroup of theweights of B on X. More precisely, X (Y) = CY(X)⊥ = χ ∈ X (X)|v(χ) = 0 for every v ∈ CY(X).In particular, every non-open G-orbit on X has strictly smaller rank than X itself.

Proof. cf. [Kn91, Theorem 6.3].

From this we deduce:

2.4.2 Lemma. Let Z ⊂ X be the closure of a non-open G-orbit. There exists an irreducible B-stableand closed V ⊂ X such that: (i) the map k[V](B) → k[Z](B) is surjective (every B-eigenfunctionon Z extends to Y) and (ii) Z is a divisor in V and it is the only irreducible divisor inducing thevaluation vZ on k[V](B).

Proof. It is known (Theorem 2.4.1) that every B-eigenfunction on Z extends to X. The complementof the open B-orbit in any B-stable closed subset of X is a union of B-stable irreducible divisors,therefore by induction on the dimension we can find an irreducible B-stable and closed V suchthat Z is a divisor of V. By [Kn91, Corollary 1.7] (which is stated for V = X but the proof worksmore generally given the extension property for B-eigenfunctions), there exists a B-eigenfunction fwhich vanishes on all B-stable divisors of V but Z. This proves that the valuation vZ characterizesZ uniquely.

2.5 Non-degeneracyWe recall the notion of a non-degenerate spherical variety [Kn95a, §6],[Kn94a]: We denote thestandard parabolic g|X · g = X (the elements of G which preserve the open B-orbit) by P(X).The spherical variety X is called non-degenerate if for every root α appearing in the unipotentradical of P(X) there exists χ ∈ X (X) such that χα 6= 1. This implies that P(X) is the largestparabolic subgroup P such that every character in X (X) extends to a character of P(X). It is provenin [Kn94a, Lemma 3.1] that every quasi-affine variety is non-degenerate. We will need a variant ofthis statement which includes the character groups of smaller B-orbits:

2.5.1 Lemma. Let X be a quasi-affine spherical variety, and let Y be a B-orbit. Let α be a simplepositive root that either does not raise Y (i.e. YPα = Y) or raises Y of type U. Then either (Y, α)is of type G (i.e. YPα = Y) or there exists χ ∈ X (Y) with 〈χ, α〉 6= 0.

Proof. Assume 〈χ, α〉 = 0 for every χ ∈ X (Y). Recall that X (Y) = χ | χ|AY= 1; hence α(Gm) ⊂

AY . By examining the distinct cases for SL2, this already excludes the possibility of Y being theopen orbit in YPα, unless (Y, α) is of type G. Given a point y ∈ Y with α(Gm) ⊂ By (such apoint must exist since all maximal tori of B are conjugate inside of B), the Lie algebra of By splitsinto a sum of eigenspaces of α(Gm); if α raises Y of type U this implies that Uα ⊂ By. Hencethe stabilizer of y in [Pα,Pα] ' SL2 is a Borel subgroup BSL2 and we get an embedding of thecomplete variety BSL2\SL2 = P1 into the quasi-affine variety X, a contradiction. Therefore (Y, α)has to be of type G.

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3. Rationality properties

3.1 Homogeneous spaces

The main questions that we examine in this section have to do with whether B-orbits are definedover a non-algebraically closed field k and to what extent Knop’s action makes sense on the set of k-rational B-orbits. The results will be used in the next sections to examine the unramified spectrumover p-adic fields. We start by recalling certain classical results: We use the terminology of [Bo91],according to which a solvable k-group is “k-split” (or simply “split”) if it admits a normal seriesover k whose successive quotients are k-isomorphic to Gm or Ga.

3.1.1 Theorem. Let G be an algebraic group and H a solvable algebraic subgroup. Assume thatthe maximal reductive quotient of H is (connected and) k-split.

i) If X is a homogeneous H-variety then X is affine and X(k) 6= ∅.ii) G(k) acts transitively on (H\G)(k).

Proof. These are [Bo91, Theorem 15.11 and Corollary 15.7]. Notice that in characteristic zero, everyunipotent group is connected and k-split, therefore we only needed to assume that the quotient ofH by its unipotent radical was k-split in order to deduce that H is k-split.

3.2 Rationality of the open orbit

From now on, assume that G is a split reductive group over a field k. This means that it has aBorel subgroup which is defined over k and k-split. Let X be a spherical G-variety (not necessarilyhomogeneous) over k. We assume that X is quasi-affine (cf. §2.1).

As a generalization of Theorem 3.1.1, (1), we prove:

3.2.1 Proposition. i) Every B-eigenfunction on k(X) is defined (up to a k-multiple) over k.

ii) The open B-orbit has a point (in particular, is defined) over k.

The proposition is true in general for any quasi-affine variety X over k with a k-action of a splitsolvable group B over k such that B has an open orbit on X.

Proof. The first claim follows from the fact that B is split, hence all weights are defined over k,hence the (one-dimensional) eigenspaces for B on k(X) are Galois invariant and, therefore, definedover k.

For the second, notice that there is a non-zero regular B-eigenfunction which vanishes on thecomplement of the open B-orbit. Indeed, the space of regular functions which vanish on the comple-ment is non-zero (because X is quasi-affine) and B-stable. As a representation of B it decomposesinto the direct sum of finite-dimensional ones. Let V be such a finite-dimensional component. Thespace of U-invariants V U (where U is the unipotent radical of B) is then non-zero, and since everyfinite-dimensional representation of A = B/U is completely reducible, it follows that there exists anonzero B-eigenfunction which vanishes on the complement of the open orbit.

Now, it follows from the first claim that this eigenfunction can be assumed to be in k[X]. Hencethe open orbit is k-open (and, therefore, defined over k); by Theorem 3.1.1, (1), the open B-orbithas a point over k.

Because of this proposition, the open G-orbit on X is isomorphic to H\G over k, where H is aclosed subgroup over k (cf. §3.1).

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3.3 Splitting in B-orbitsNow we examine the splitting of the k-points of a k-rational B-orbit Y in B = B(k)-orbits. Themain result is that they are naturally parametrized by an abelian group ΓY (more precisely: theyform ΓY -torsor), which is finite for k a locally compact field.

Let Y be a B-orbit which is defined (hence, has a point y) over k. It is k-isomorphic to By\B,where By is the stabilizer of y in B. Recall that AY denotes the image of By under the quotientmorphism B→ A = B/U.

3.3.1 Lemma. i) For every B-orbit Y, the quotient of Y by U exists and is naturally isomorphicto A/AY . A U-orbit has a k-point if and only if the point of A/AY corresponding to it is ak-point.

ii) There is a one-to-one correspondence between B-orbits on (By\B)(k) and A-orbits on(AY \A)(k).

Proof. The fact that By\B/U exists and is equal to A/AY is well-known. We need to prove thatthe quotient map is surjective on k-points. The fiber over every k-point is a homogeneous k-varietyfor U. By Theorem 3.1.1, (1), the fiber has a point, hence the map on k-points is surjective.

Now, according to Theorem 3.1.1, (2), U acts transitively on the k-points of each k-rationalU-orbit. Therefore, the second claim follows immediately by tracing back the isomorphism of theset of U-orbits with A/AY .

3.3.2 Corollary. The set of B-orbits on Y is naturally a torsor for the abelian group ΓY =(A/AY )(k)/Im(A), where Im(A) denotes the image of A = A(k) under the quotient morphismA→ A/AY .

Remark. Despite the fact that the identification of the set of B-orbits with ΓY depends on thechoice of an initial point y ∈ Y , we will implicitly fix such an identification in most cases in orderto simplify the language of the arguments.

We are left with examining the map on k-points for a surjective map of tori A1 → A2. Rememberthat the categories of diagonalizable k-groups and of finitely generated Z-modules with a continuousGalois action are equivalent, via the contravariant functor which assigns to each group its charactergroup ([Bo91, Proposition 8.12]).

3.3.3 Lemma. Let A1 → A2 be a surjective morphism of split tori, defined over a local non-archimedean field k. Then:

i) Im(A1) is open and closed in A2.

ii) The map of k-points A1 → A2 is surjective if and only if the quotient X (A1)/X (A2) has notorsion.

iii) The quotient A2/Im(A1) is finite.

Proof. For the first claim, the corresponding map on Lie algebras L(A)→ L(A/AY ) is surjective.Hence the image of A is open. It is also closed, since its complement is a union of cosets, which areopen. This proves the first claim.

For the second and third, by assumption the character group X (A2) = X (A2)k injects intoX (A1) (and both are lattices, i.e. free Z-modules of finite rank). By the elementary divisors theorem,we can find a basis v1, . . . , vm for the latter such that c1v1, . . . , cnvn is a basis for the former(ci ∈ N, n 6 m). Equivalently, we can write A1 = Gmm, A2 = Gnm such that the map A1 → A2 israising the i’th factor to the ci’th power for i 6 n and discarding the i’th factors for i > n. Thisreduces the problem to A1 = A2 = Gm and the map being the c-th power map, where we see thatA2/Im(A1) = k×/(k×)c, a finite group, trivial if and only if c = 1.

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3.4 Spherical varieties for SL2

Spherical varieties for SL2 are of dimension at most 2. Therefore, homogeneous spherical varietiesfor SL2 (over an arbitrary field k) belong to the homogeneous varieties classified by F. Knop in[Kn95b, Theorem 5.2]; it is easily seen that all the varieties in loc. cit. are spherical. We recallthis classification according to the classification of the corresponding homogeneous spaces over thealgebraic closure (Theorem 2.1.1) and examine some basic rationality properties.

3.4.1 Case G The subgroup H = G = SL2. There is a single B-orbit.

3.4.2 Case T The subgroup H = T, a (maximal) torus.

By the equivalence of categories between diagonalizable k-groups and lattices with a Galoisaction, isomorphism classes of 1-dimensional tori over k are classified by Hom(Gal(k/k),Z/2) '(k×/(k×)2).

One way to describe the homogeneous space T\G is as Q(1, β) := A ∈ gl2|tr(A) = 1,det(A) =β where β ∈ k and 4β − 1 6= 0, under the adjoint action of SL2.

The space X has, in general, several G-orbits. However, notice that we can naturally extend theaction of SL2 to an action of PGL2, and PGL2 acts transitively on X.

We examine the splitting of the open Borel orbit in B-orbits: We have AX = ±1, and hencethe orbits of B on the open B-orbit are parametrized by k×/(k×)2. (Don’t confuse this with theparametrization of isomorphism classes of tori, mentioned above.) However, if we extend the actionto PGL2 then its Borel subgroup acts transitively on X – and this will be important later.

Notice that if the torus is non-split, it does not embed over k into a Borel subgroup and thereforethe smaller B-orbits do not have a point over k.

3.4.3 Case N The subgroup H is the normalizer of a maximal torus. It turns out that for alltori we get the same homogeneous variety: Indeed, the space N (T)\SL2 can be identified with theopen subset of P(pgl2) defined by 4 det(A) − (tr(A))2 6= 0. It can then be seen that for every T,N (T) appears as a stabilizer of a k-point. Again, the action extends to PGL2. Notice also that thek-points of N (T) coincide with the k-points of T for T non-split. This implies that the PGL2-orbitof a k-point with stabilizer N (T), for T non-split, is isomorphic as a PGL2-space with T\PGL2.The splitting of X in B-orbits is parametrized by k×/(k×)4, while if we consider the action of PGL2

and let B denote its Borel subgroup the orbits under B are parametrized by k×/(k×)2 (and theB-orbits are related to B-orbits through the natural map k×/(k×)4 → k×/(k×)2).

3.4.4 Case U The subgroup H is equal to S ·U, where U is a maximal unipotent subgroupand S ⊂ N (U).

As a G-space, X splits into a disjoint union of spaces isomorphic to SU\G. The k-points ofthe open B-orbit may split into several B-orbits. However, because of the Bruhat decompositionover k, every one of them belongs to a different G-orbit. Notice also that (because of the Bruhatdecomposition) both B-orbits have k-points.

3.5 Rationality of orbits of maximal rank

3.5.1 Proposition. If a B-orbit Y is defined (equivalently: has a point) over k, then wY is, too,for every w ∈W . In particular, all the B-orbits of maximal rank are defined over k. Moreover, theassociated finite groups ΓY and ΓwY describing the splitting of each orbit into B-orbits are naturallyisomorphic.

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Proof. Consider the Pα-orbit of Y. Dividing by UPα we get a spherical variety for Lα, the Leviof Pα. Further dividing by the connected component of the center of Lα we get a spherical varietyXα for SL2 (or PGL2). In both steps, the quotient maps are surjective on k-points since we aredividing by a unipotent group and a split torus, respectively. Therefore, a B-orbit on X has a pointover k if its image in Xα is defined over k. By examining now the SL2-spherical varieties whichwere classified above, the first claim follows.

The open orbit is, by Proposition 3.2.1, defined over k. Moreover, all orbits of maximal rankbelong to the open G-orbit and are in the W -orbit of the open orbit. This proves the second claim.

The isomorphism between ΓY and ΓwY comes from the k-rational map w : A→ A and the factthat AwY = wAY (Lemma 2.2.1).

Because of the last isomorphism, all of the groups ΓY for Y of maximal rank will be identifiedwith ΓX := ΓX .

3.6 The Zariski and Hausdorff topologiesFor any topological field k, the k-points of a variety X over k naturally inherit a topology fromthat of k. Indeed, since X = Homk−alg(k[X], k) (we assume for simplicity of notation here that Xis affine, the general case can be recovered by covering X by affine neighborhoods), every pointcan in particular be viewed as a map from k[X] to k, and therefore the set of points inherits thecompact-open topology from the space of such maps. (With k having its given topology and k[X]considered discrete.) If the topology on k is locally compact, totally disconnected and Hausdorff, sowill be the induced topology on X. We will conveniently refer to that topology as the “Hausdorff”topology.

For a spherical variety, we wish to examine the relation between closures of B-orbits in boththe Zariski and the Hausdorff topology. By definition, essentially, the “Hausdorff” topology is finerthan the Zariski topology, therefore a Zariski-open set is also Hausdorff-open. Does the Hausdorffclosure of a B-orbit coincide with the Zariski closure? The example below shows that this is notthe case, at least not in non-homogeneous varieties:

3.6.1 Example. Let X be the subvariety of A2×Ar0 defined by the equation: x2−ay2 = 0 (a 6= 0).Consider the following action of G = B = G2

m: (x, y, a) · (r, k) = (rk3x, rk2y, k2a). Then the B(k)orbits are:

– (x, y, x2

y2)|x, y 6= 0,

– (0, 0, a)|a ∈ (k×)2,– (0, 0, a)|a /∈ (k×)2.

The second and the third form together the k-points of the same B-orbit, but only the second is inthe Hausdorff closure of the first.

Contrary to the previous example, for a homogeneous spherical variety we have:

3.6.2 Lemma. Any neighborhood of a point x ∈ Y (in the Hausdorff topology), where Y is a Borelorbit of dimension j < dimX contains k-points belonging to orbits of dimension j + 1.

Proof. Let α be a simple root raising Y, and consider the spherical variety YPα/UPα for the LeviLα. (We have seen that the map YPα → YPα/UPα is surjective on k-points.) From examining thedistinct cases for spherical varieties of groups of semisimple rank one, we deduce the statement.

Our main object of study will be spaces of locally constant, compactly supported functions onspherical varieties. We need to decide whether, in the non-homogeneous case, we will allow the

13


support of our functions to extend beyond the Hausdorff closure of X (the k-points of the openB-orbit); or whether we will redefine X as the Hausdorff closure of X. For the discussion of thenext section and the results of section 5 it does not make a difference, since the smaller G-orbitshave smaller rank and the results that we prove are “generic” and are not influenced by the smallerorbits. However, for the study of unramified vectors in section 6 we require that the support of allfunctions is contained in the closure of X. Of course, the Hausdorff closure of X is G-stable. Weexamine the phenomenon of the example more closely:

3.6.3 Lemma. For every k-rational G-orbit Y ⊂ X and every B-orbit closure V containing Y wehave a natural homomorphism of groups: ΓV → ΓY coming from a map between B-orbits on V andY . Each B-orbit on V is mapped under this map to the unique B-orbit on Y which is contained inits closure (and hence the image of the map corresponds to the B-orbits of Y which belong to theHausdorff closure of V ).

Proof. Recall (Theorem 2.4.1) that every B-eigenfunction on Y extends to X (in particular, to V).This implies that X (Y) → X (V) or, equivalently, AV ⊂ AY . This induces:

A/AV → A/AY

andΓV → ΓY

as claimed.Now, if we fix a set of generators for the B-semiinvariants on k[V](B), we simultaneously fix

an isomorphism of the quotient V/U with A/AV and of Y/U with A/AY . This shows that thehomomorphism on Γ-groups comes from a natural map between B-orbits.

The implication: γ ∈ ΓY r Im(ΓV) ⇒ “the orbit represented by γ does not lie in the closureof V ” follows immediately from the definition of the Hausdorff topology: Indeed, neighborhoods inthis topology are determined by the values attained by regular functions, and if γ does not belongto the image of ΓV then there exist B-eigenfunctions whose values on the orbit of Y representedby γ are not approximated by their values on V . To show the converse, let y ∈ Y , let γy be theelement of ΓY corresponding to the B-orbit to which it belongs and let γv ∈ ΓV be a preimage ofγy under ΓV → ΓY . Assume that a neighborhood N of y does not meet a B-orbit V1 ⊂ V whichcorresponds to γv. Then the same is true for every U -translate of N , therefore we may assume thatN is U -invariant. But a fundamental system of U -invariant neighborhoods of y is determined bythe values of all f ∈ k[V](B); therefore if, for every such f , f(y) can be approximated by f(x) forx ∈ V1 then N must meet V1, a contradiction.

3.7 Invariant differential forms and measuresGiven a linear algebraic group G, its unipotent radical UG carries a (left and right) invariant topform ω. It is unique up to scalar, and the adjoint action: Adg : u 7→ g−1ug of G transforms it by acharacter d : G→ Gm (the “modular character”); in other words, Ad∗g(ω) = d(g)ω. This characteris the sum of all roots of G on the Lie algebra of UG, and it is also equal to the ratio between aright- and a left-invariant top form on G (which agree at the identity).

The group of isomorphism classes of G-line bundles on a homogeneous variety X = H\G (overthe algebraic closure) is naturally: PicG(X) := X (H) [KKV89]. Let Lψ denote the correspondingline bundle for the character ψ of H; its sections can be identified with global sections f of thetrivial bundle on G such that f(hg) = ψ(h)f(g) for all h ∈ H, g ∈ G. If ψ is a k-character, thenLψ is defined over k. There is a non-zero G-invariant global section of L∗ψ ⊗ Ω (the sheaf of top-degree differential forms valued in the dual of Lψ) if and only if ψ = dH

dG|H ; in particular, there is aG-invariant top form on X if and only if the modular characters of G and of H agree on H.

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Given a smooth variety X over a local field k, any k-rational top differential form ω on Xgives rise to a positive Borel measure on the topological space of its k-rational points [We82]. Thismeasure will be denoted by |ω|. The complex character δG := |dG| of G = G(k) is equal to the ratiobetween right and left Haar measure on G and is also called “the modular character”.

We will show that without loss of generality we may assume the k-points of the open orbitX in our spherical variety X to possess a B-invariant measure. For this, we may take X to behomogeneous.

As discussed in §2.1, we can assume that G = G1 × T over k, where T is a torus, G1 actstransitively on X: X = H0\G1, where H0 has no k-characters. Then X possesses a G1-invariantk-rational top form ω. The idea is to replace T, if necessary, by a subtorus.

Let B1 = G1 ∩ B, a Borel subgroup of G1; then B = B1 × T. For the open orbit we have:X = (H∩(B1×T))\(B1×T) (assuming that HB is open). Then the quotient X/B1 ' (HB1∩T)\T.Let HT = HB1 ∩T. Then T is a finite quotient over k of T′ ×HT , where T′ is some subtorus andX is still a spherical G1×T′-variety. Let χ be the k-character under which ω transforms under theaction of T. Since HT∩T′ is finite, it follows that |ω| (which is a positive measure on X) is invariantunder (HT ∩T′)(k). Hence |ω| varies by a positive (unramified) character of (TH ∩T′)(k)\T′(k),and by twisting it by the inverse of that character (which is constant on the orbits of B1) we obtaina B1 × T ′-invariant measure on X.

4. Mackey theory and intertwining operators

In this section we summarize the method of intertwining operators (cf. [BZ76, Cas]). It is usuallyreferred to as “Mackey theory” by analogy to Mackey’s theorem for representations of finite groups.The existence of an open orbit is very important here. We use work of Denef, Deshommes andGarrett to establish general properties of the intertwining operators and we discuss the preciserelationship between intertwining operators constructed analytically and the Jacquet modules.

4.1 Unramified principal series

From now on, k will always denote a locally compact non-archimedean local field. We work inthe abelian category S of smooth representations of G, which means that every vector has openstabilizer.

An unramified character of an algebraic group A over a p-adic field k is a complex character ofA which is of the form |f1|s1 · · · |fr|sr , where f1, . . . , fr are algebraic characters of A (i.e. homomor-phisms into Gm), defined over k, the sign | · | denotes the p-adic absolute value and s1, . . . , sr ∈ C.

The group of unramified characters of A a natural structure of a complex algebraic torus: Iff1, . . . , fm form a basis for the group of algebraic characters modulo torsion and if χ = |f1|s1 · · · |fm|smthen the association χ 7→ (qs1 , . . . , qsm) ∈ (C×)m, where q is the order of the residue field of k, definesthe structure of a complex torus on the group of unramified characters.

If X = H\G is a homogeneous G-variety over k, an unramified character ψ of H gives rise to acomplex G-line bundle over X, to be denoted by Lψ. If X = H\G then sections of this line bundlecan be described as complex functions f on G such that f(hg) = ψ(h)f(g) for every h ∈ H, g ∈ G. Ingeneral, choose a normal subgroup H1 ⊂ H such that H1 is in the kernel of all unramified charactersof H and H/H1 is a k-split torus (this is always possible: choose a quotient H/H1 of H/H0 – whereH0 is as in §2.1 – such that X (H/H1)k is isomorphic to the quotient of X (H/H0)k by its torsion)then sections of Lψ can be described as functions f on (H1\G)(k) such that f(hx) = ψ(h)f(x) forh ∈ (H/H1)(k). (Recall that the quotient map H1\G → H\G is surjective on k-points.) There isan L∗ψ-valued G-invariant measure on X if and only if ψ = δH

δG.

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Let B be the Borel subgroup of G; we will denote the complex torus of its unramified charactersby A∗. This is the maximal torus in the “Langlands dual” group of G. Let d (resp. δ) be the algebraic(resp. complex) modular character of the Borel; it is equal to the sum 2ρ of all the positive roots.Given an unramified character χ of B, we define the unramified principal series I(χ) := IndG

B(χδ12 ).

(Since we are working in the smooth category, IndGB(χδ

12 ) is the space of smooth sections of L

χδ12

over

B\G.) We recall its properties: For a hyperspecial maximal compact subgroup K of G, it contains aunique (up to scalar multiple) vector invariant under K (called “unramified”). For generic χ ∈ A∗,it is irreducible. Generic, when talking about points on complex varieties, will mean “everywhere,except possibly on a finite number of divisors”. For generic χ, again, and w ∈ W we have anisomorphism Tw : I(χ) ' I(wχ). We will recall the construction of the intertwining operator Twbelow. Also, the spaces I(χ) can be identified to each other as vector spaces by considering therestriction of f ∈ I(χ) to K. If we call this common underlying vector space V , and we have afamily of maps πχ from a set S to I(χ) for χ varying on a subvariety D of A∗, then we say thatthe family is regular if for every s ∈ S we have πχ(s) ∈ V ⊗C[D]. Similarly we define the notion of“rational” family of maps. We write πχ,1 ∼ πχ,2 to denote that πχ,1 = c(χ)πχ,2 for some non-zerorational function c of χ.

We will need to recall more information on the divisors on which the above statements (irre-ducibility of I(χ) and isomorphism with I(wχ)) may fail to be true: First, there are the “irregular”characters, i.e. those given by an equation χ = wχ, w ∈ W . Those are precisely the charactersbelonging to one of the divisors Rα := χ|χα = 1, where α is a root and α the corresponding co-root. More precisely, the representation I(χ) may be reducible for χ irregular, and the intertwiningoperator Tw “blows up” on the divisor

⋃α>0,wα<0Rα. Then, there are the divisors Qα (α a root)

described by the equation χα = q. It is known that, for such χ, I(χ) is reducible and Tw ceases tobe an isomorphism on the divisor:

⋃

α>0,wα<0

Qα ∪⋃

α<0,wα>0

Qα.

Returning to our spherical variety, the complex torus of unramified characters of B supportedby an orbit Y (i.e. generated by complex powers of the modulus of k-rational B-semiinvariants onY ) will be denoted by A∗Y . If Y = the open B-orbit on X we will be denoting A∗Y by A∗X . (Its Liealgebra is a∗X = X (X)⊗Z C.)

4.2 The Bernstein decomposition and centreBy the theory of the Bernstein centre [Be84], the category S is the direct sum of categories SP,σ,indexed by equivalence classes of pairs of data (“parabolic subgroup”, “quasi-cuspidal representationof its Levi”). The “simplest” of these categories is indexed by the data (“Borel subgroup”, “trivialrepresentation”). It will, by abuse of language, be called the “unramified Bernstein component”,although not all representations belonging to it are unramified (i.e. possess a vector invariant undera maximal compact subgroup). Given a smooth representation V , its “unramified” direct summandVur admits the following equivalent characterizations:

i) Every irreducible subquotient of Vur and no irreducible subquotient of its complement is iso-morphic to a subquotient of some unramified principal series.

ii) Vur is the space generated by the vectors of V which are invariant under the Iwahori subgroup.

Moreover, the centre z(S) of S is described in [Be84]: This is, by definition, the endomorphismring of the identity functor; in other words, every element of this ring is a collection of endomor-phisms, one for each object in the category, such that when applied simultaneously they commutewith all morphisms in the category. The centre can also be identified with the convolution ring of all

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conjugation-invariant distributions on G whose support becomes compact when they are convolvedwith the characteristic measure of any open-compact subgroup.

By the above decomposition, one evidently has z(S) =∏P,σ z(SP,σ). Each of the factors in this

product is naturally isomorphic to the space of regular functions on a complex variety (and thedisjoint union of these varieties is called the “Bernstein variety”). The centre of Sur is naturally iso-morphic to C[A∗]W by mapping each element to the scalar by which it acts on I(χ), for all χ ∈ A∗.Convolving the corresponding distributions with the characteristic measure of a hyperspecial maxi-mal compact subgroup K, we get an isomorphism of rings with the spherical Hecke algebra H(G,K)of K-biinvariant measures on G. The fact that H(G,K) ' C[A∗]W is the Satake isomorphism.

4.3 FiltrationsLet X be a locally compact, totally disconnected space with a continuous (right) action of G. Thenthe space C∞c (X) of locally constant, compactly supported complex functions on X furnishes asmooth representation of G. The discussion below applies more generally to the space C∞c (X,Lψ)of smooth, compactly supported sections of Lψ, but for simplicity we will work with the trivialbundle here and make a few comments on the general case in §4.8.

By Frobenius reciprocity,6

HomG (C∞c (X), I(χ)) = HomB

(C∞c (X),C

χδ12

).

If Y ⊂ X is open and B-stable, and Z = X r Y , then we have an exact sequence:

0→ C∞c (Y )→ C∞c (X)→ C∞c (Z)→ 0

which gives rise to an exact sequence of distributions (by definition, the linear dual of C∞c ):

0→ D(Z)→ D(X)→ D(Y )→ 0.

By applying the functor of (B,χ−1δ−12 )-equivariance we get a sequence on the spaces D(•)(B,χ−1δ−

12 )

= HomB

(C∞c (•),C

χδ12

)(recall that, by definition, the action of g on a distribution D is given by

π∗(g)D(f) = D(π(g−1)f)), but we might lose right exactness:

0→ D(Z)(B,χ−1δ−

12 ) → D(X)(B,χ

−1δ−12 ) → D(Y )(B,χ

−1δ−12 ). (1)

We apply the above in the setting ofX= the k-points of our spherical variety X. As we remarked,the Zariski topology is coarser than the induced Hausdorff topology, hence the set of k-points oforbits of dimension > d is open in the set of k-points of orbits of dimension > d. More precisely, wehave the following filtration:

0 → C∞c (X) → C∞c (∪dim(Y)>dim(X)−1Y ) → . . . → C∞c (X)

with successive quotients isomorphic to C∞c (∪dim(Y)=dY ) for the appropriate d.

It follows that the dimension of the space of (B,χ−1δ−12 )-equivariant distributions on X is less

than or equal to the sum of the dimensions of (B,χ−1δ−12 )-equivariant distributions on Y , for all

k-rational orbits Y. Part of what we prove below is that, for generic χ, we actually have equality.In any case, the problem now has been divided in two parts: examine the question of (B,χ−1δ−

12 )-

equivariant distributions on any single B-orbit, and then determine whether these distributionsextend to the whole space.

6The isomorphism asserted by Frobenius reciprocity is given as follows: Given a morphism into I(χ), compose with“evaluation at 1” to get a functional into C

χδ12. For the whole paper, we will avoid distinguishing between the

morphism and the functional whenever possible, and we will be using the same letter to denote both.

17


4.4 Distributions on a single orbit4.4.1 The case where B acts transitively Now we examine the question of

(B,χ−1δ−12 )-equivariant distributions on C∞c (Y ), where Y = y · B is some orbit of B (not of

B). We will use the natural projection: C∞c (B) ³ C∞c (yB) = C∞c (By\B), given by integration onthe left over By with respect to right Haar measure:

p(f)(x) =∫

By

f(bx)drb , f ∈ C∞c (B)

in order to pull-back such a distribution to B. If we pull back a (B,χ−1δ−12 )-equivariant distribution

to B, we get a (B,χ−1δ−12 )-equivariant distribution on B, which has to be χ−1δ−

12 times right Haar

measure, hence equal to χ−1δ12 · dlb, where dlb denotes left Haar measure. In other words the

distribution is given by

SYχ : φ 7→∫

Bf(b)χ−1δ

12 (b)dlb (2)

where f ∈ C∞c (B) such that p(f) = φ. This distribution is well defined (i.e. will factor through thesurjection p : C∞c (B) ³ C∞c (By\B)) if and only if:

χ−1δ12

∣∣∣By

= δBy (3)

where δBy is the modular character of By.A character satisfying the above condition will be called Y -admissible. Notice that two Y -

admissible characters χ differ by an element of A∗Y . Hence, the characters such that C∞c (Y ) admitsa non-zero morphism into I(χ) form a complex subvariety AdmY ⊂ A∗ of dimension equal tothe rank of the orbit Y . Moreover, it is immediate that the family of functionals SYχ is regular inχ ∈ AdmY (cf. §4.1).

In case that the orbit Y admits a B-invariant measure, or equivalently that δ|By = δBy , thecondition of admissibility takes the nicer form:

χδ12 ∈ A∗Y

and the distribution can be expressed as an integral on Y :

SYχ : φ 7→∫

Yφ(y)|f1(y)|s1 · · · |fm(y)|smdy (4)

where f1, . . . , fm are a basis for the k-rational semiinvariants of B on Y (modulo torsion) ands1, . . . , sm ∈ C such that |f1|s1 · · · |fm|sm is of weight χ−1δ−

12 . More generally, even if there is no

B-invariant measure there will always be a measure dy that varies by unramified character ψ ofB,7 and then the same expression will give SYχ except that the weight of |f1|s1 · · · |fm|sm should be

ψ · χ−1δ−12 .

Finally, notice that our distribution SYχ is only well defined up to a constant; this constantdepends on the choice of y (and the measure on B) in the presentation of (2), and on the choice ofsemiinvariants and the measure dy in the presentation of (4).

4.4.2 Non-transitive action of B and weighted distributions This paragraph is slightly technicalin nature; it introduces certain weighted sums of the morphisms Sχ which correspond to differentB-orbits on the same B-orbit; these weighted sums will later appear in the formulations of ourtheorems.

7The reason is that all line bundles Lψ on Y are isomorphic if we forget the B-action, as follows from the fact thatevery unramified character of By can be extended to a character of B.

18


In the case that Y contains multiple B-orbits Yγγ∈ΓY we can, of course, apply the above toeach B-orbit separately (recall that each B-orbit is open-closed inside of Y ) and get morphisms SYγχfor each γ. On the other hand, since ΓY has the structure of an abelian group, we can consider adifferent basis of the space of morphisms: C∞c (Y )→ I(χ) consisting of sums of the SYγχ ’s weightedby characters η of ΓY :

SY,ηχ :=∑

γ∈ΓY

η(γ)SYγχ .

(We will explain the meaning of this sum below, since up to now the SYγχ ’s are only defined up toa constant.)

It turns out, however, that for our purposes the “correct” distributions will be given by differentweighted sums. Namely, we group together B-orbits which belong to a certain subgroup Γ(0) ofΓ = ΓY and weigh together only the distributions supported on them, using a character of Γ(0). Theprecise definition of Γ(0) will be discussed later, here we just introduce notation given an arbitrarysubgroup Γ(0) ⊂ Γ. Let η be a character of Γ(0), let ζ denote a coset of Γ(0) in Γ and denote thedata (ζ, η) by θ. Choosing a representative γ0 ∈ Γ for ζ, we get a function supported on the coset ζ:ηζ(γ) := η(γ0γ). The dependence on the choice of γ0 will be suppressed from the notation, becauseour definitions will only depend on it up to a scalar. We define:

SY,θχ :=∑

γ∈ζηζ(γ)S

Yγχ . (5)

However, we need to make more precise what we mean by this sum, since, as mentioned, thedistributions SYγχ are only defined up to a scalar factor.

While there is no canonical way to normalize the SYγχ ’s, we can canonically normalize themrelative to each other by using expression (4) with a measure dy coming from a suitable differentialform on Y. Namely, if ωY is a top form on Y which varies by an algebraic character Ψ of B (with|Ψ| = ψ in the above notation), then SY,θχ can be expressed as:

SY,θχ : φ 7→∫

Yφ(y)|f1(y)|s1 · · · |fm(y)|sm · ηζ(f1(y), . . . , fm(y))|ωY | (6)

where the fi’s and the si’s are as in (4). (Recall that the fi’s define a k-isomorphism of A/AY withGmm and hence a map from Y to Γ; through this the expression ηζ(f1(y), . . . , fm(y)) makes sense.)

We have defined above rational families of morphisms SY,γχ := SYγχ , SY,ηχ and SY,θχ : C∞c (Y )→ I(χ)

for all Y -admissible unramified characters χ. As a matter of notation, when we do not need to specifywhich of the above families we are refering to we will be writing SY,∗χ .

4.4.3 Comparison between admissible characters for different orbits We may also describe therelations between the varieties of admissible characters on the Borel orbits of a Pα-orbit using 2.2.1.

Case G : We have χα = q for every χ ∈ AdmY . (Recall that α denotes the positive root in the Leviof Pα.)

Case U : Notice that the stabilizer Bz of the closed orbit has unipotent radical of dimension one

larger than the stabilizer By of the open orbit. In fact, δ12 |Bz = eα · wαδ

12By

and δBz = wαδBy(as characters on AZ = wαAY ), and this implies that AdmwαY = wαAdmY .

Case T: If Y denotes the open orbit and Z∗ the closed ones (in the case of a split torus, for we haveseen that if T is non split then the closed orbits are not defined over k) then AdmY ⊃ AdmZ∗ .Notice, however, that for the small orbits it does not hold that AdmZ1 = wαAdmZ2 – the correctrelation is δ−

12 AdmZ1 = wα

(δ−

12 AdmZ2

).

19


Case N: If Y denotes the open orbit and Z the closed one then AdmY ⊃ AdmZ and δ−12 AdmZ is

wα-invariant.

4.4.4 Corollary. For every B-orbit Y, we have AdmY ⊂ wAdmX for some w ∈ [W/WP ].

Recall that P = P(X) denotes the standard parabolic g ∈ G|Xg = X and WP the Weylgroup of its Levi.

Proof. The orbit of X under the Weyl group action is precisely the set of orbits of maximal rank,and elements of WP stabilize the open orbit. Therefore, admissible characters on orbits of maximalrank are of the form wAdmX for some w ∈ [W/WP ]. Every B-orbit within the open G-orbit is“raised” to an orbit of maximal rank by a sequence of simple roots – hence, for those the claimfollows from the description of admissible characters above. Having proven it for B-orbits withinthe open G-orbit, it is enough now to prove the claim for Y=the open B-orbit inside each smallerG-orbit. Recall (Lemma 2.4.2) that Y is a divisor inside of a B-stable closed V ⊂ X, and thatevery semiinvariant on Y extends to V, therefore, AY ⊃ AV of codimension one. Moreover, sincethe unipotent radical of Bv acts trivially on V, it must act trivially on its closure, hence mustbe contained in By. For dimension reasons we now deduce that By and Bv must have the sameunipotent radical; it follows that AdmY ⊂ AdmV , and by induction on the dimension, AdmV iscontained in a [W/WP ]-translate of AdmX .

4.5 Convergence and rationality

We now discuss whether a sequence of the form (1) is surjective on the right. The integral expression(2) defines an equivariant extension of the distribution SYχ to the whole space C∞c (X) if it convergesfor all φ ∈ C∞c (X). In general, one proves convergence for χ in a certain region and then showsthat the resulting functional is rational in χ ∈ AdmY (cf. §4.1), hence providing an extension foralmost every χ ∈ AdmY . In this paragraph we review a theorem of Denef [Den85, Theorem 3.1] andDeshommes [Des96, Theoreme 2.5.1] which establishes rationality of those integrals in general. Wealso use the method of Denef and Deshommes (which originates in Igusa) to deduce some additionalestimates which will be needed later. As a corollary of rationality, for all χ which do not belong tothe “poles” of the intertwining operators, we deduce that the sequence (1) is surjective on the right.

Recall that, by Hironaka’s embedded resolution of singularities [HiH64], given a k-rational B-orbit closure Y there exists a (canonical) regular k-scheme Y and proper k-morphism π : Y → Y,which is an isomorphism on Y and such that the inverse image E of Y rY is an effective divisor(the “exceptional divisor”) whose irreducible components have normal crossings.

We can now pull back φ, the semiinvariants fi and the differential form ωY (where dy = |ωY |)of (4) to Y in order to express the integral as an integral on Y . Let y0 ∈ Y ∩ supp(π∗φ) and letD1, . . . , Dk be the prime components of E which contain y0. Then there exist local coordinatesx1, . . . , xn identifying a neighborhood of y0 in the Hausdorff topology with a neighborhood of 0 inkn such that Di = xi = 0 for 1 6 i 6 k and |π∗ωY | =

∏ki=1 |xi|midx1 · · · dxn (up to a constant)

for some integers mi. By shrinking that neighborhood, if necessary, we may assume that if thepull-back of any B-semiinvariant f is non-zero on all Di’s then |f | is constant in a neighborhoodof y0. Therefore, if φ is supported on such a neighborhood of y0, the integral (4) is equal (up to aconstant) to

∫π∗φ(x)

k∏

i=1

|xi|rik∏

i=1

|xi|midx1 · · · dxn (7)

for suitable complex numbers r1, . . . , rk. More precisely, the ri’s are related to the si’s of (4) as

20


follows:

ri =∑

j

sjvDi(fj) (8)

where vDi denotes the valuation induced by Di. Recall also that π∗φ is locally constant.From this we deduce that:

i) The integral is rational in the parameters qri (or, equivalently, in qsi). Its poles are:k∏

i=1

11− qri+mi+1

.

ii) Let Sr1,...,rk(φ) denote the rational continuation of the above integral. For j = (j1, . . . , jl)(l 6 k) a set of integers, let φj denote the restriction of φ to the set where |xi| = q−ji (for all1 6 i 6 l). Then

Sr1,...,rk(φj)¿rl+1,...,rk

l∏

i=1

q(ri+mi+1)ji

where ¿rl+1,...,rk means that the implied constant depends on rl+1, . . . , rk.

iii) More generally, if hi =∏km=1 x

eimm (1 6 i 6 l), j = (j1, . . . , jl) and φj denotes the restriction

of φ to the set where |hi| = q−ji (for all 1 6 i 6 l) then there exists κ (depending on the ri’s)such that

Sr1,...,rk(φj)¿ qκ·max ji .

Since π∗φ is locally constant and compactly supported on Y , we can write it as a sum of functionssupported in neighborhoods as above. Then from the above observations we deduce immediately:

4.5.1 Proposition. i) The integral (4), representing SYχ (φ), converges for all φ ∈ C∞c (Y ), for χin an open subregion of AdmY . It is rational in χ, and its poles are products of factors of theform:

11− q−νDχvD

whereD denotes an irreducible component of the exceptional divisorE which meets the supportof π∗φ, vD is the valuation corresponding to D and νD is an integer that depends only on D.

ii) Let f ∈ k[Y](B) of weight ψ and let φi denote the restriction of φ to the set where |f | = q−j .For κÀ 0,

SYχψκ =∑

j

SYχψκ(φj)

(with the above sum converging absolutely).

iii) Consider the lattice A/A0U . The image of φ ∈ C∞c (Y ) in C∞(A/A0Ay) under the map t :φ 7→ ∫

(A0U)y\A0Uφ(ya)dy is supported on a translate of a cone 〈a, ψ〉 < 0 for some ψ ∈

Hom(A/A0U,Z) and satisfies

t(φ)(a)¿ eκ〈a,ψ〉

for some κ.

The expression χvD requires some clarification. First, consider the valuation vD as an elementof Hom(X (Y),Z) by its restriction to k(Y)(B) as follows: 〈ψ, vD〉 := vD(fψ), where fψ ∈ k(Y)(B)

of weight ψ. (Such an fψ is unique up to a constant.) Next, recall that X (Y)⊗C is the Lie algebraof A∗Y ; this allows us to consider χvD as a complex character of A∗Y : namely, if χ = |ψ| where ψ isan algebraic character, we set χvD = q〈ψ,vD〉. Finally, by translating AdmY by a suitable character,

21


we may identify it with A∗Y and then χvD makes sense for χ ∈ AdmY . This interpretation dependson the character chosen to translate AdmY , but this dependence is absorbed in the (non-canonical)factor q−νD .

Proof. The fact that one can make sense of SYχ as a rational function of χ follows from observation(i) above. However, if we split π∗φ into pieces so that the integral looks like (7), we have no a prioriguarantee that all these integrals will converge absolutely for the same χ. Recall, however, thatthere exists a non-zero f ∈ k[Y](B) which vanishes on YrY. Therefore, convergence for χ in someopen region will follow if we prove the second assertion. The stated form of the poles follows fromobservation (i) and relation (8).

The second assertion follows from observation (iii) above, by setting h = f .Finally, for the third assertion, notice that the integral under consideration is the integral of φ

restricted to the set where |fi| have a fixed value, for all fi ranging in a set of generators for k[Y](B).Therefore, the estimate follows again from observation (iii) above and the existence of an f as above(of weight ψ).

Remark. By the approach of Denef and Deshommes to p-adic integrals, there is not even the needfor our integral to be convergent for χ in some region in order to make sense out of it. However, ourproof of the fact that there exists a region of convergence will be useful in §5.3.6 and may also beof interest for explicit calculations as in [Sa].

Combining all the results above, we have proven the following:

4.5.2 Theorem. A necessary condition for the existence of a non-zero morphism: C∞c (X)→ I(χ)is that χ ∈ w

(δ−

12A∗X

)for some w ∈ [W/WP ]. For every B-orbit Y on X there exists a natural

such family SYχ : C∞c (X)→ I(χ), rational in χ ∈ AdmY . For almost all χ satisfying the condition,

the space of morphisms C∞c (X)→ I(χ) admits a basis consisting of all SYχ with χ ∈ AdmY .

4.6 Jacquet modulesAny morphism V → I(χ) (for (π, V ) a smooth representation of G) factors through the Jacquetmodule VU : This is, by definition the maximal quotient of V where U acts trivially; equivalently, itis equal to the quotient of V by the span of v − π(u)v|u ∈ U, v ∈ V . It is well-known that theA-equivariant functor V 7→ VU is exact, due to the fact that U is filtered by compact subgroups.In fact, since we are only considering unramified principal series, we may as well compose with thefunctor VU 7→ VA0U ((co-)invariants for the maximal compact subgroup A0 of A), which is alsoexact; we will call VA0U the unramified Jacquet module.

In what follows we examine the Jacquet modules for some basic GL1-and GL2-spherical varieties.We do this in order to demonstrate how the method of intertwining operators gives us informationon the Jacquet module; to show that the Jacquet module does not, in general, have a very simplegeometric description; and to discuss what happens at characters χ on the poles of the intertwiningoperators, where the above method fails to prove surjectivity of (1) on the right.

4.6.1 Example. Let X = A1, as a GL1-spherical variety. From the two orbits X = k× and Z = 0we have the sequence:

0→ C∞c (k×)→ C∞c (k)→ C→ 0.

The corresponding sequence of unramified Jacquet modules is:

0→ Cc(Z)→ C∞c (Z ∪ −∞)→ C→ 0 (9)

22


where Cc(Z) denotes compactly supported sequences on Z and C∞c (Z ∪ −∞) denotes sequencessupported away from −∞ which stabilize in a neighborhood of −∞.

Tate’s thesis shows that the intertwining operator SXχ has a pole at χ = 1, which is exactlywhere the intertwining operator SZχ appears. (This is a general phenomenon which will be discussedbelow.) The element 1 − χ of the Bernstein center (respectively: the Hecke algebra element whichmaps the sequence (an)n to the sequence (an−an−1)n) is clearly injective from C∞c (k) onto C∞c (k×)(resp. on the corresponding unramified parts), and this implies that the Jacquet module of C∞c (k) isequal to that of C∞c (k×). Its unramified part is, as mentioned, Cc(Z) ' C[T, T−1] and if we rewrite(9):

0→ C[T, T−1]→ C[T, T−1]→ C→ 0then the map on the right is just evaluation at 1.

We deduce, in particular, that the sequence (1) is not surjective on the right in this case.

4.6.2 Example. Let X = T\PGL2, where T = A is a k-split torus. Let Y1,Yw denote the twoclosed orbits represented by the elements 1 and w (a representative for the non-trivial Weyl groupelement), respectively. By the Bruhat decomposition, PGL2 = B t BwB, the orbit Yw has anopen, B-stable neighborhood A\(BwB) = A\(UwU) ' UwU. Hence as a B-variety the open setA\(BwB) is isomorphic to A1 ×U (with the usual action of A ' Gm on the first factor), and itsquotient by U exists and is equal to A1. Since w acts on the left as a G-automorphism, it carriesthe open Bruhat cell to an open neighborhood of Y1, and induces a U-isomorphism of the latterwith A1×U. The open B-orbit A\U×wB (where U× = Ur 1) is included in both of those opensets, and under the isomorphisms that we described it corresponds to A1 r 0 ×U. Therefore wehave the following maps of quotients:

A\UwB ³ A1 ← A1 r 0yA\wUwB ³ A1 ←A1 r 0

(10)

where (it can easily be checked) the vertical map is multiplication by −1.Hence the quotient of X by U is non-separated. It is easy to see now that the Jacquet module

of C∞c (A\UwB) is equal to C∞c (k) as a k×-module (where k× = A = B/U), and similarly that ofC∞c (A\wUwB). Since:

C∞c (X) = C∞c (A\UwB)+C∞c (A\wUwB) = (C∞c (A\UwB)⊕ C∞c (A\wUwB)) /C∞c (A\U×wB)diag

it follows that its Jacquet module is (C∞c (k)⊕ C∞c (k))/C∞c (k×)diag.Hence, using the previous example, the sequence (1) is not surjective on the right; at χ = 1 the

dimension of intertwining operators is equal to two, coming from the two closed orbits.

4.6.3 Example. Let now X = U\PGL2. The Bruhat decomposition gives us a filtration:

0→ C∞c (U\BwB)→ C∞c (X)→ C∞c (U\B)→ 0

with corresponding Jacquet modules (cf. [Cas, Proposition 6.2.1]):

0→ C∞c (k×)→ C∞c (X)U → C∞c (k×)→ 0

and unramified Jacquet modules:

0→ C[T, T−1]→ C∞c (X)A0U → C[T, T−1]→ 0.

The latter is a sequence of H(G,K) = C[T, T−1]-modules (where convolution with elements of theHecke algebra corresponds to multiplication of polynomials), and since this ring is a principal ideal

23


domain and the modules are free, the sequence splits, so we have (non-canonically):

C∞c (X)A0U ' C[T, T−1]⊕ C[T, T−1].

Therefore we see that although the intertwining operator SXχ has a pole at χ = 1, the correspondingsequence (1) is surjective on the right in this case, and the dimension of intertwining operators isconstantly equal to two.

In any case, our intertwining operators are enough to characterize the image of a φ in the Jacquetmodule:

4.6.4 Lemma. A vector φ ∈ C∞c (X) lies in the kernel of the Jacquet morphism C∞c (X)→ C∞c (X)Uif and only if the integral of φ over every U -orbit is zero. Similarly, it lies in the kernel of theunramified Jacquet morphism if and only if its integral over every A0U -orbit is zero. The integralof φ over all A0U -orbits in a given B-orbit Y is zero if and only if SYχ (φ) = 0 for generic χ ∈ AdmY .

Proof. For φ|Y , where Y is a B-orbit, compactly supported, all claims are obvious. (The first andsecond claim concerning the image of φ in the Jacquet module – resp. unramified Jacquet module– of C∞c (Y ).)

Since the kernel of the Jacquet morphism is generated by elements of the form f − R(u)f , thedirection⇒ of the first claim is obvious. Conversely, suppose that the integral of φ over any U -orbitis zero. We will prove the claim using the standard filtrations of Jacquet modules and by inductionon the orbit dimension. Let m be the minimal dimension of an orbit which intersects the supportof φ. Then the image of φ under

C∞c (X)U ³ C∞c (∪dimY6mY )U

is zero. The kernel of the above map is equal to the Jacquet module ofC∞c (∪dimY >mY ), hence φ differs from a φ′ ∈ C∞c (∪dimY >mY ) by a function of the form f −R(u)f .Since the latter has integral zero over any U -orbit, we reduce the problem to φ′, which allows us tocomplete the proof by induction. The claim about the unramified Jacquet module follows similarly.

For the last claim, the direction⇒ is, again, obvious. For the inverse, use part (iii) of Proposition4.5.1: Multiplying t(φ) by a suitable character of B, it lands in L1(A/A0Ay). By standard Fourieranalysis on the discrete abelian group A/A0Ay, if its Fourier transform is zero then the functionitself is zero.

The importance of the above lemma is that we do not have to worry about intertwining operatorswhich may not be expressible in terms of our SYχ ’s.

4.7 Discussion of the polesLet SY,∗χ : C∞c (X)→ I(χ) be as above. By the “poles” of SY,∗χ we mean the smallest divisor D whichcontains the polar divisors of SY,∗χ (φ) for all φ ∈ C∞c (X). We have proven above that there existssuch a divisor, i.e. that there is only a finite number of distinct irreducible polar divisors appearingfor all φ.

In Proposition 4.5.1 we gave a description of the poles of SY,∗χ in terms of geometric data of ourspherical variety. There is also a representation-theoretic understanding of the poles, discussed in[Ga99], which leads to necessary conditions for the poles to appear.

Let D be a closed prime divisor of A∗Y . The local ring oA∗Y ,D is principal. Hence, if D is containedin the subvariety where SY,∗χ has poles, there is f ∈ mA∗Y ,D (the maximal ideal) such that SY,∗χ :=f(χ)SY,∗χ is regular and nonzero on a dense subset of D.8 However, the functional SY,∗χ was regular

8In fact, as we saw in Proposition 4.5.1, the polar divisors in our case are always principal.

24


when restricted to C∞c (Y ), the functional SY,∗χ will vanish on C∞c (Y ) for χ ∈ D and will be supportedon Y r Y . We deduce that, for D to be a polar divisor of SY,∗χ , it has to be contained in the varietyof admissible characters of a smaller orbit in the closure of Y .

Since we already know (by Proposition 4.5.1) that some of the possible poles will appear, andusing some other results from the theory of spherical varieties, we will be able later to slightlyextend Garrett’s results in some cases (Lemma 6.1.4) and identify certain “anomalous” intertwiningoperators linked to smaller G-orbits.

4.8 Non-trivial line bundles and standard intertwining operatorsAs mentioned above, exactly the same arguments apply to intertwining operators: C∞c (X,Lψ) →I(χ), where ψ is some complex character of H. The condition of admissibility with respect to aB-orbit Y is now: χ−1δ

12 |By = δBy

yψ−1|By , where yψ denotes the character by which the stabilizerof y (a conjugate of H) acts on the fiber of the map G→ H\G over y.

As a special case of this, the filtration of B\G defined by the Bruhat decomposition gives riseto the standard intertwining operators for unramified principal series:

Tw : I(χ)→ I(wχ)

which are rational in χ ∈ A∗ and are given by the rational continuation of the integral:∫Qα>0,w−1α<0 Uα

φ(w−1u)du.

The above integral expression depends on the choice of a representative for w in N (A), but onlyup to a character of A∗, therefore we will ignore this dependence whenever we can. Their poles area union of “irregular” divisors as described in §4.1, and one can verify that those are the characterswhere a “smaller” Shubert cell can support an intertwining operator into I(χ).

5. Interpretation of Knop’s action

5.1 Avoidance of “bad” divisorsThe object of this section is to investigate what happens when one composes the morphisms SY,∗χwith the intertwining operators for principal series Tw. Before we do that, we need to examine issuesthat might arise from the set of our characters χ being contained in some of the hypersurfaces whereI(χ) is reducible (and where some of the Tw annihilate a subrepresentation). It turns out that thiscan only happen for trivial reasons. These trivial reasons are best exhibited in the example of theSL2-spherical variety of type G, namely X=a point. Then C∞c (X) is the trivial representation of

G; it is contained as a proper subrepresentation in I(χ) where χ :(a ∗

a−1

)7→ |a|−1 and as

a quotient (but not a subrepresentation) in I(wχ). Hence, Sχ maps into that subrepresentationof I(χ) and Tw : I(χ) → I(wχ) annihilates its image. This is essentially the only way things cango wrong. We recall from §4.1 the definition of the “bad” divisors Rα = χ ∈ A∗|χα = 1 andQα = χ ∈ A∗|χα = q, where α is a root and q is the order of the residue field.

5.1.1 Lemma. The variety δ−12A∗X is never contained in one of the “irregular” divisors Rα. It is

contained in Qα if and only if α is a simple, positive root of the Levi of P(X).

Proof. Since δ−12 ∈ δ− 1

2A∗X and δ−12 is regular, δ−

12AX∗ is not contained in any of the Rα.

We have: δ−12A∗X ⊂ Qα ⇒ e−ρ ∈ Qα ⇐⇒ 〈ρ, α〉 = 1 ⇐⇒ α ∈ ∆. In that case, we see that wα

has to centralize A∗X which, by the assumption of non-degeneracy (§2.5), implies that α ∈ ∆P(X).The converse is easily checked.

25


5.1.2 Corollary. For generic χ ∈ δ− 12A∗X the image of SX,∗χ in I(χ) is irreducible.

Proof. Indeed, since a generic χ is contained only in those Qα with α simple, positive and appearingin the Levi quotient L of P(X), and since the stabilizer inside P(X) of a generic point contains,modulo the unipotent radical of P(A), the commutator subgroup of L, it follows that for such χ theimage of SX,ηχ in I(χ) belongs to the irreducible subspace induced from the trivial representationof the commutator of L.

5.2 The theorem for SL2

Our basic theorem, Theorem 5.3.1 below, will be proved by reducing it to the case of SL2. Since itsmore general version has a complicated formulation which aims to control the splitting of B-orbitsinto B-orbits, we present the case of SL2 first in order to motivate that formulation. Although thegeneral theorem will be stated only for sections of the trivial complex line bundle over X, in theproof we will need the SL2-formulation for an arbitrary bundle.

Recall from §2.2 Knop’s action of the Weyl group on orbits of the Borel subgroup, in particular,on orbits of maximal rank.

5.2.1 Theorem. Let X = H\G be a quasi-affine homogeneous spherical variety for G = SL2, letY be its open B-orbit, Γ the group describing its splitting into B-orbits, ψ an unramified characterfor H and SY

χ the corresponding space of intertwining operators: C∞c (X,Lψ) → I(χ) (the span of

all SY,∗χ ’s). Then Tw SYχ is identically zero if and only if X is of type G, and in all other cases forgeneric χ ∈ A∗X = AdmX we have:

Tw SYχ = SwYwχ .

More precisely, the action of Tw can be described explicitly as follows:

Case U: We have Tw SYχ ∼ SwYwχ for every element SYχ ∈ SYχ .

Case T: Let Γ(0) = Γ, η a character of Γ and SY,ηχ as in §4.4.2. Then Tw SY,ηχ ∼ SY,ηwχ .

Case N: Let Γ(0) = (k×)2/(k×)4 ⊂ k×/(k×)2 = Γ and θ = (ζ, η) a set of data consisting of a coset

and a character of Γ0. Then Tw SY,θχ ∼ SY,θwχ .

Proof. We consider case by case:

5.2.2 Case G As already remarked, C∞c (X) is the trivial representation of G, and that appearsin the subspace of the reducible representation I(δ−

12 ) which is annihilated by Tw.

5.2.3 Case U Notice that the stabilizer in G = G(k) of a k-point is conjugate to S · U withS ⊂ N (U) a finite group, by the discussion of non-degeneracy (§2.5). In particular, χ is w-regularand Tw is an isomorphism for generic χ ∈ A∗X ; also, there is a B-invariant measure.

We denote by Z the closed B-orbit and by Y the open one. Choose a representative w ∈ N (A)for the non-trivial element of the Weyl group (also denoted by w), which will be used to define theintertwining operator Tw. The choice of w defines a bijection between orbits of U (= points) on Zand orbits of U on Y. If z ∈ Z, the corresponding orbit on U is z · wU. Of course, the bijectioncarries k-points to k-rational orbits. If f ∈ k[Z](B) and we use this bijection to pull it back to aU-invariant function F on Y, we immediately see that F ∈ k[Y](B).

Notice that SZγχ converge for all χ since Z is closed. Now use F to write:

SYγχ (φ) =

∫

Yγ

φ(y)|F (y)|sdy =

26

On the unramified spectrum of spherical varieties∫

U0\U(k)

∫

Zγ

φ(z · wu)|f(z)|sdzdu = Tw SZγwχ(φ).

This is an exact equality and hence can be extended to any linear combination of the SYγχ ’s.Since, for generic χ, Twα Twα ∼ Id , it also follows that TwαS

Y,∗χ ∼ SZ,∗wαχ.

5.2.4 Case T It may seem surprising at first that cases T and N are the most complicated ones,since there is only one B-orbit of maximal rank, and hence the morphism into I(χ) defined with thisorbit should (generically) map to itself when composed with Tw. However, the k-points of the openorbit may split into several B-orbits, giving rise to several morphisms. In order to describe theircomposition with Tw one needs to pick a suitable basis indexed by characters of Γ; the proofs willnot be as straightforward as in Case U, since the integrals involved will not converge simultaneously.Therefore, we use representation-theoretic arguments.

Notice that the subgroup H contains the center of SL2. Therefore, we may let SL2/±1 =PGL2 act on X. Set G = PGL2 and B= the Borel of G. The important feature that will allowus to control the splitting in B-orbits is that they all belong to the same B-orbit. This is becausewe do not have an equality B/Z = B: in fact, one readily sees that B/(B/Z) = Γ. The characterχ := χη−1 can therefore be seen as a (ramified, if η is non-trivial) character of B. One can, of course,define morphisms SY

χ , Tw for such a ramified character in exactly the same way, and it follows that,since for a generic χ there is a unique such morphism SY

χ into the representation I(χ) of G (theone defined with Y= the open orbit), we must have Tw SY

χ ∼ SYwχ.

Remark. Compare our treatment of “Case T” above with explicit calculations performed for aparticular spherical variety in [HiY05]; the calculations there correspond to the case H=a splittorus of SL2.

5.2.5 Case N In this case, as we saw, the G-space X splits into a disjoint union of spaces ofthe form (T\SL2)(k) or finite quotients of those, with T ranging through all isomorphism classesof tori in SL2. We group B-orbits according to this decomposition: two B-orbits belong together ifand only if they correspond to the same coset of Γ(0). Another way to describe the grouping is asfollows: We can again extend the action to G = PGL2, and two orbits belong together if and onlyif they belong to the same G-orbit. Hence, the morphisms SY,θχ are supported on a single G-orbit,and there they are weighted according to the character η of Γ(0). Hence, we can treat each G-orbitseparately: If the torus is non-split then N (T ) = T , hence as a G-space the corresponding orbit willbe isomorphic to T\G and the claim follows directly from Case T. If the torus is split, the claimfollows again easily, by noticing that a section f ∈ C∞c (N (T )\G,ψ) back to C∞c (T\G, ψ) and usingCase T.

Remark. The theorem does not hold for the orbits of lesser rank in X. Indeed, as seen in §4.4.3, forZ an orbit of lesser rank in “Case T” or “Case N” we do not have AdmwZ = wAdmZ and thereforewe cannot have Tw SZ,∗χ ∼ SwZ,∗wχ .

5.3 Statement and proof of the main theorem

Similarly to the SL2-case, the main theorem will say, roughly, that the space SYχ of morphismsdefined by the orbit Y , when acted upon by Tw, is mapped to the space S

wYχ of morphisms defined

by the orbit wY . The proof will consist of two steps: The first will focus on the orbit of a minimalparabolic and will be reduced to the SL2-case. The second will be to ensure that orbits outside ofthis parabolic do not contribute. In order to be able to identify the action of Tw more explicitly ona distinguished basis, we need to make sure that the basis we choose, every time we reduce to the

27


SL2 case, reduces to one of the distinguished bases described above.

This is accomplished by using the weighted morphisms of §4.4.2 and the following definition ofthe subgroup Γ(0):

Let Z0 ⊂ A be the subgroup generated by the images of −1 under all simple coroots: α : Gm →A. Let Y be a Borel orbit with corresponding group of splittings ΓY . Recall that ΓY is the quotientof A/AY (k) by the image of A. Consider instead the quotient of A/(AY Z0)(k) by the image of thegroup (A/Z0)(k), and denote that by Γ′Y . The map A/AY → A/(AY Z0) induces a homomorphism:Γ→ Γ′. We let Γ(0) be the kernel of this homomorphism. Now for every data θ = (ζ, η) as in §4.4.2we have morphisms SY,θχ . The general statement of the theorem is:

5.3.1 Theorem. Let Y be of maximal rank in Y · Pα. Then Twα SY,∗χ is nonzero if and only if(Y, α) is not of type G, and in that case we have:

Twα SY,θχ ∼ SwαY ,θwαχ . (11)

Proof. In the course of the proof, we will need the following lemma:

5.3.2 Lemma. The homomorphism Γ→ Γ′ is surjective.

Notice that this holds although the map on k-points: A/AY (k) → A/(AY Z0)(k) may not besurjective.

Proof. It suffices to show that, given a split torus A and two subgroups A1,A2 over k the mapA/A1 ×A/A2 → A/(A1A2) induces a surjection on k-points. Recall (Lemma 3.3.3, (2)) that thisis equivalent to showing that the corresponding quotient of character groups has no torsion. If Xdenotes X (A) and Xi denotes X (A/Ai) (i = 1, 2), then Xi is a subgroup of X and the image ofX (A/(A1A2)) in X1 × X2 is the diagonal copy of X1 ∩ X2. It is clear, then, that the quotient hasno torsion, for if χ ∈ X1 × X2 ⊂ X × X maps to a torsion element in the quotient, then χ belongsto the diagonal copy of X , hence has to belong to the diagonal copy of X1 ∩ X2.

5.3.3 Proof of the theorem for the orbit of a minimal parabolic It is obvious that both SY,∗χ andTwαS

Y,∗χ are zero on functions supported away from the closure of (Y · Pα)(k). Therefore, these

functionals factor through C∞c ((Y ·Pα)(k)). The first step is to prove the theorem restricted tofunctions which are compactly supported in (Y ·Pα)(k); then, in §5.3.6 we extend it to the wholeC∞c ((Y ·Pα)(k)):

5.3.4 Proposition. Theorem 5.3.1 is true for w = wα and the Sχ’s restricted to φ with supp(φ)∩(YPα)(k) ⊂ (YPα)(k).

Proof. Notice that the quotient of Y · Pα by the unipotent radical of Pα is well defined and allof the functionals SYχ , Twα SYχ and S

wYwχ , when restricted to C∞c ((Y ·Pα)(k)), factor through the

map:

ια : C∞c ((Y ·Pα)(k))→ C∞c ((Hα\Lα)(k)) : f 7→∫

UPα,ξ\UPαf(ξu·)du. (12)

Here Lα is the Levi quotient of Pα and Hα is the stabilizer of a k-point on (Y ·Pα)/UPα . (Recallalso that the map YPα → YPα/UPα is surjective on k-points (Theorem 3.1.1)). Therefore the firststep in the proof will be completed if we prove Theorem 5.3.1 for homogeneous spherical varietiesof groups of semisimple rank one.

28


5.3.5 The next reduction is in order to be able to divide by central tori. There are serioustechnical issues to overcome here – namely, we need to ensure that every time that we reduce tothe SL2-case the weighted averages of our functionals will reduce as they should to the functionalsof Theorem 5.2.1.

Assume that Z1 is a split torus in the center of L = Lα, and let Xα = Hα\L. The map:

Hα\L→ (HαZ1)\Lis surjective on k-points, since the fiber over each k-point is acted upon transitively by a split torus.The fiber over each k-point is isomorphic to (Hα ∩ Z1)\Z1 over k; let us denote the quotient of((Hα ∩Z1)\Z1)(k) by the image of Z1 – in other words the set of orbits of Z1 on the fibers – by Γ1

(it has a natural group structure).For our B-orbit Y, denote Γ = ΓY . Let Yα be the image of Y in (HαZ1)\L, and let Γα be the

corresponding group of B-orbits. We have a natural exact sequence:

1→ Z1/(Z1 ∩AY )→ A/AY → A/(Z1AY )→ 1. (13)

This sequence splits (non-canonically): Indeed, consider the dual sequence of character lattices forthe former. Since Z1/(Z1 ∩AY ) is a torus, its character group is torsion-free and the sequence ofcharacter groups splits. Fix such a splitting:

A/AY ' Z1/(Z1 ∩AY )×A/(Z1AY ). (14)

This allows us to identify the bundle Hα\L over the open orbit Yα with the trivial bundle: Yα ×(Z1/(Z1 ∩AY )).

The sequence above gives rise to:

1→ Γ1 → Γ→ Γα → 1 (15)

with a corresponding splitting:

Γ ' Γ1 × Γα. (16)

Now consider the subgroup Γ(0) ⊂ A defined previously. We have a commutative diagram withexact rows:

1 −−−−→ Z1/(Z1 ∩AY ) −−−−→ A/AY −−−−→ A/(Z1AY ) −−−−→ 1yy

y1 −−−−→ Z1/Z1 ∩AY Z0 −−−−→ A/(AY Z0) −−−−→ A/(Z1AY Z0) −−−−→ 1

(17)

which gives rise to a commutative diagram with exact rows and columns:1 1 1y

yy

1 −−−−→ Γ(0)1 −−−−→ Γ(0) −−−−→ Γ(0)

α −−−−→ 1yy

y1 −−−−→ Γ1 −−−−→ Γ −−−−→ Γα −−−−→ 1y

yy

1 −−−−→ Γ′1 −−−−→ Γ′ −−−−→ Γ′α −−−−→ 1yy

y1 1 1

(18)

29


The elements of the first row are defined by the exactness of the vertical sequences (given Lemma5.3.2). Exactness of the second and third row follow from the previous diagram. Then the only thingwhich remains to be shown is exactness of the first row on the right, which follows from the exactnessof the rest of the diagram.

The splitting (16) induces:

Γ(0) ' Γ(0)1 × Γ(0)

α . (19)

In our setting, L will be split of semisimple rank one, Z1 will be the connected component of itscenter. Then Z0 also belongs to the center, and L/(Z1Z0) = PGL2. The right column of (17) willbe that of a spherical variety for a split semisimple group of rank one, hence of SL2 or PGL2. Thegroup Γ(0)

α will be non-trivial exactly if the right column corresponds to a non-trivial SL2-sphericalvariety such that AYα contains its center. We have also seen that the orbits corresponding to Γ(0)

α

admit the following description: We may let SL2/±1 = PGL2 act on this spherical variety; someorbits of BSL2(k) will belong to the same orbit of BPGL2(k), and those which belong to the sameorbit as that represented by “1” are the ones that Γ(0)

α corresponds to.Given data θ = (ζ, η) as in §4.4.2, the splittings (16) and (19) give rise to factorizations: η =

η1×ηα and ζ = ζ1×ζα, where η1 is a character of Γ(0)1 , ηα a character of Γ(0)

α , ζ1 a coset of Γ(0)1 in Γ1

and ζα a coset of Γ(0)α in Γα. We may now write the morphism SY,θχ : C∞c ((Hα\L)(k))→ I(χ) as a

composition of two morphisms S2 S1: First, we integrate over the fibers of the map (Hα\L)(k)→((HαZ1)\L)(k) against the function η1,ζ1 (cf. §4.4.2) on Γ1 to get a map into a certain subspace ofC∞((Hα\L)(k)):

S1(φ)(x) =∫

(Z1/(Z1∩AY ))(k)φ(zx)χ−1δ

12 (z)η1,ζ1(z)dz.

Consider the restriction of S1(φ) to ((Hα ∩ L′)\L′)(k), where L′ = [L,L]. Then S1(φ) is a sectionof the induced line bundle from the character χδ

12 of (Z1Hα ∩ L′)(k). Having, by (14), fixed a

trivialization of this bundle and a set of data θα = (ζα, ηα) of Γ(0)α , we may now identify S2 with

the morphism:

SY,θαχ : C∞c (((Hα ∩ L′)\L′)(k), χδ 12 |(Z1Hα∩L′)(k))→ I(χ).

We first reduced the proof of the first step (Proposition 5.3.4) to proving it for the Levi quotientof the minimal parabolic. By our assumption that G is split and simply connected, this is isomorphicto the product of SL2 or GL2 with a split torus. Dividing by the connected component of the center,we are reduced to the case of a homogeneous quasi-affine variety of SL2, which has already beendone.

5.3.6 Orbits in the closure do not contribute To conclude the proof of Theorem 5.3.1, we needto show that what we just proved for φ compactly supported on (Y ·Pα)(k) continues to hold forφ supported in its closure. The idea is to split the intersection of the support of φ with Pα intoinfinitely many compact pieces, let φi denote the restriction of φ to the i-th piece by φi (henceφ =

∑i φi when restricted to (Y · Pα)(k)) and use the fact that SY,θχ (φ) =

∑i S

Y,θχ (φi) when χ is

such that the integral expression for SY,θχ converges. The problem is that Twα and SYχ will not, ingeneral, converge simultaneously so we cannot use their integral expressions to prove directly thatTwα

∑i S

Y,θχ (φi) =

∑i TwαS

Y,θχ (φi). To solve this problem, we could make use of the asymptotic

estimates of Proposition 4.5.1, part (ii), with a suitable f (as we will do later in Lemma 6.1.4).However, asymptotic estimates are unnecessary here:

5.3.7 Lemma. Let K1 be an open compact subgroup of Pα. Let g1, . . . , gm be representatives forthe orbits of K1 on B\Pα. Then there are rational functions r1, . . . rm of χ such that for φ ∈ I(χ)K1

30


we have:

Twαφ =m∑

j=1

rj(χ)φ(gj).

The lemma is a direct consequence of the rationality of Tw and the fact that IndPαB (χδ

12 )K1 is

finite-dimensional. It is important that we only fix a compact open subgroup of Pα, not of the wholegroup G.

Now, given φ ∈ C∞c ((Y ·Pα)(k)) fix a compact-open K1 ⊂ Pα such that φ is K1-invariant andrepresentatives g1, . . . , gm as above and enumerate the K1-orbits on (Y ·Pα)(k): A1, A2, A3, . . . . Letφi = φ · 1Ai ∈ C∞c ((Y ·Pα)(k)).

Notice that for g ∈ Pα the sets Aig define a partition of (Y ·Pα)(k) in g−1K1g-orbits.For χ in the region of convergence of the integral expression for SYχ we have: SYχ (R(gj)φ) =∑i S

Yχ (R(gj)φi) for every j. Using the previous lemma:

Twα∑

i

SYχ (φi) =m∑

j=1

∑

i

R(gj)SYχ (φi) =∑

i

m∑

j=1

R(gj)SYχ (φi) =∑

i

TwαSYχ (φi).

Using Proposition 5.3.4, we have TwαSY,θχ = S

wY,θwχ (φi). Hence: TwαSYχ = S

wYwχ . This completes

the proof of Theorem 5.3.1.

5.4 Corollaries and examplesWe discuss the implications of Theorem 5.3.1 for elements of the Weyl group of length greater thanone:

5.4.1 Corollary. Tw SX,∗χ 6= 0 if and only if w ∈ [W/WP ]. For w ∈ [W/WP ], Tw SX,θχ ∼ SwX,θwχ .

Proof. Every w = w1w2 with w1 ∈ [W/WP ] and w2 ∈WP (uniquely). It follows from Theorem 5.3.1that Tw2 SX,ηχ = 0. The elements of [W/WP ] are characterized by the fact that wα > 0 for every(simple) positive root α in the Levi of P. From Lemma 5.1.1 and the properties of intertwiningoperators (§4.1) it follows that Tw1 is an isomorphism for almost every χ on δ−

12A∗X . The second

statement follows immediately from Theorem 5.3.1.

5.4.2 Corollary. Denote by Sθχ the operator SX,θχ . For generic χ ∈ δ− 12A∗X the quotients Sθ1χ1

and

Sθ2χ2are isomorphic if and only if χ1 = wχ2 for some w ∈WX and θ1 = θ2. Consequently, for generic

χ ∈ δ− 12A∗X we have dimHom(C∞c (X), I(χ)) = (NW(−ρ+ a∗X) : WX)× |Γ|.

Proof. By [Kn95a], the stabilizer of the open orbit is WX nWP , where P = P(X). The points ofA∗X are left stable by WP , hence the first assertion.

Recall that the space Hom(C∞c (X), I(χ)) has as a basis, for almost all χ, the morphisms SY,θχ

supported on orbits Y of maximal rank. Recall also that all orbits of maximal rank are linked tothe open B-orbit under the action of the Weyl group by a series of simple reflections of type U, andthat AdmwY = wAdmY . The latter implies that the orbits of maximal rank supporting morphismsinto I(χ) are those in the orbit of δ−

12A∗X under NW (−ρ + a∗X). The second assertion now follows

from the first.

Let us now compare our results with a few well-known examples:

5.4.3 Example. The spherical variety X = U\G. It is known that the little Weyl group of ahorospherical variety is trivial (and vice versa: if the little Weyl group of a spherical variety is

31


trivial then the variety is horospherical) and it is easy to check that AdmX = A∗X = A∗. Therefore,our results tell us that all irreducible representations in the unramified spectrum appear, at leastgenerically, but the generic multiplicity is equal to the order of the Weyl group. (This is, of course,expressed in the fact that I(χ) ' I(wχ) for generic χ.)

5.4.4 Example. The subgroup H = GLdiagn of G = GLn × GLn. In this case AdmX = A∗X is

equal to A∗GLnembedded in A∗ as a 7→ diag(a, a−1 ) and WX = W diag

GLn. Therefore, generically in

the unramified spectrum, X = H\G distinguishes irreducible representations of the form τ ⊗ τ ,where τ is an irreducible representation of GLn and τ denotes its contragredient. This is, of course,well-known to hold not only generically and not only for the unramified spectrum.

5.4.5 Example. The space X = Matn under the G = GLn ×GLn action by multiplication on theleft and right. The open G-orbit is equal to the spherical variety of the previous example, thereforethe generic description of the unramified spectrum is identical to the previous case.

As follows immediately from Corollary 5.4.2, the generic multiplicity may be greater than 1 (i.e.the Gelfand condition may fail to hold) for two reasons: The k-points of the open B-orbit split intoseveral B-orbits; or the little Weyl group of X does not coincide with the normalizer of −ρ+ a∗X . Inaddition to the simple SL2-examples that we have seen, we mention another instance of the former:

5.4.6 Example. In [HiY05], Y. Hironaka examines Sp4 as a spherical homogeneous Sp4 × (Sp2)2-space over a local non-archimedean field. It is discovered that the generic multiplicity is equal tothe order of k×/(k×)2; this is due to the splitting of the X in B-orbits.

The non-coincidence of WX with N (−ρ+a∗X) is very common in parabolically induced examples,since, as we already mentioned, the little Weyl group of the parabolically induced spherical varietyis equal to the little Weyl group of the original spherical variety for the Levi. The example of U\G,mentioned above, is an instance of this. However, parabolically induced spherical varieties do notexhaust the list of such examples:

5.4.7 Example. The group SL2×SL2 embeds naturally in G = Sp4. Let H be theGm×SL2 subgroupthereof (where Gm is a maximal split torus in SL2). It is easy to see that AdmX = A∗X = A∗, howeverit is known that WX is not the whole Weyl group, but a subgroup of W of index 2.

5.5 Parabolic induction with an additive characterIn applications one often comes across representations induced from “parabolically induced” spheri-cal subgroups, but not from the trivial (or the modulus of an algebraic) character of those subgroupsbut from a complex character of its unipotent radical.

5.5.1 Example. The Whittaker model is the line bundle LΨ over U\G, where Ψ : U → C× is ageneric character of U ; this means that, if we identify the abelianization of U with the directproduct of the one-parameter subgroups Uα, for α ranging over all simple positive roots, thenΨ = ψ Λ, where Λ : U → Ga is a functional which does not vanish on any of the Uα and ψ isa nontrivial complex character of Ga(k) = k. Hence, the Whittaker model is parabolically inducedfrom the trivial subgroup of A; if Ψ were the trivial character, then its spectrum would only containrepresentations whose Jacquet module with respect to U is non-trivial, and with generic multiplicityequal to the order of the Weyl group. On the contrary, for Ψ a generic character, the spectrum isknown to be much richer (e.g. it contains all supercuspidals), and multiplicity free for every (notonly generic) irreducible representation.

5.5.2 Example. The Shalika model for GL2n is the line bundle LΨ overH\G, where H is parabolicallyinduced from the maximal parabolic P with Levi L = GLn × GLn and the spherical subgroup

32


M = GLdiagn thereof; and Ψ is the character ψ(tr(X)) of UP , where ψ is as above a complex

character of k and X ∈ Matn(k) under the isomorphism UP 'Matn. It is known that the Shalikamodel, too, is multiplicity-free, and it distinguishes lifts from SO2n+1.

We will see that even those cases can be linked to Knop’s theory – more precisely, to an extensionof the Weyl group action to non-spherical varieties.

Let H = M n UP be a parabolically induced spherical subgroup of G, with notation as in§2.3. Let Ψ : UP → C× be a character. Any such character of UP factors through a morphism:Λ : UP → Ga, composed with a complex character ψ of Ga(k) = k. Now, assume that Λ isnormalized by M, and by abuse of notation use the same letter to denote the induced morphism:H→ Ga. Let H0 = kerΛ; the variety H0\G is the total space of a Ga-torsor over H\G (no longerspherical), and the map

π : H0\G→ H\Gis surjective on k-points for the usual reasons. One is interested in the space C∞c (H\G,LΨ), i.e. thespace of smooth complex functions on H0\G which satisfy f(h · x) = Ψ(h)f(x) for h ∈ H/H0(k)and such that π(supp(f)) is compact.

By repeating exactly the same Mackey-theoretic arguments that we used before, one sees directlythat the k-rational B-orbits on H\G which give rise to a morphism into I(χ), for some unramifiedcharacter χ, are those represented by elements ξ such that

H ∩ ξB ⊂ H0. (20)

One sees also that if an orbit Y satisfies this condition, then one can define a rational family SYχ ofmorphisms into I(χ) for exactly the same χ’s as before. Also, by the description of B-orbits in §2.3,one sees that the open B-orbit satisfies (20). Denote by BΛ

00 the set of orbits of maximal rank whichsatisfy (20). Our goal is to describe the unramified quotients of C∞c (X,Ψ) in a similar manner aswe did with C∞c (X); more precisely, we will link it with Knop’s Weyl group action on H0\G.

Since the latter space is not spherical, we need to revisit Knop’s theory and recall the necessaryfacts regarding its extension to non-spherical varieties. The complexity of a variety Y with a B-action is defined as c(Y) = maxy∈Ycodim(yB). Let X be a G-variety, not necessarily spherical.We have c(X) = 0 if and only if X is spherical. We let B0(X) denote the set of closed, irreducible,B-stable subsets with complexity equal to the complexity of X. Then Knop defines an action ofthe Weyl group W on B0 – it leaves stable the subset B00 of those B-stable subsets whose generalB-orbit is of maximal rank. In the case of spherical varieties, this action coincides with the one thatwe discussed, and B00 is in bijection with the set of B-orbits of maximal rank, hence the use ofthe same symbol to denote those. To see how the action is defined in the general case, one repeatsthe same steps, by letting Pα act on the B-stable set Y ∈ B0, examining the image of a generalstabilizer Py in Aut(B\Pα) ' PGL2 and considering cases. Additionaly to the cases that we sawin the spherical case, one now has the case F\PGL2, where F is a finite subgroup. But in thatcase, there is only one closed, irreducible, B-stable subset of complexity equal to the complexityof F\PGL2 (namely, the whole space F\PGL2 itself), and the corresponding element of the Weylgroup will be fixing it.

Now let us return to our parabolically induced spherical variety. Let us consider inverse images ofB-orbits under π. The set π−1Y|Y ∈ B00(H\G) is precisely the set of closed, irreducible, B-stablesubsets of H0\G whose generic B-orbit has maximal rank. Which of those belong to B00(H0\G)?One sees easily that, for Y a B-orbit on H\G, π−1Y has complexity 1 (the complexity of H0\G) ifand only if π−1Y is not a single B-orbit, which is the case if and only if (20) is satisfied. Thereforewe have a natural isomorphism of sets: BΛ

00(H\G) ' B00(H0\G). The Weyl group action on thelatter induces a Weyl group action on the former, which differs from the action of W on B00(X).

33


Our result will be:

5.5.3 Theorem. In the above setting, let SY,∗χ denote the family of morphisms into I(χ) defined bythe orbit Y ∈ BΛ

00(X) and let wY denote the image of Y under the Weyl group action on BΛ00(X).

We have Tw SX,∗χ 6= 0 if and only if w ∈ [W/WP ], and for w ∈ [W/WP ], Tw SX,θχ ∼ SwX,θwχ .

The proof is similar to the case of Ψ = trivial and is omitted.

6. Unramified vectors and endomorphisms

6.1 Spectral support

Since the results of this paper are all stated for “generic” quotients of the “unramified” Bernsteincomponent, it is natural to ask to what extent those “generic” quotients are enough to characterizea vector in our representation. Given a smooth representation V , let us call spectral support (orsimply support) of V its support as a module for the Bernstein center z(S). In other words, it is thesubvariety of the Bernstein variety corresponding to the ideal of z(S) which annihilates V . Given avector v ∈ V , we will call (spectral) support of v the support of the smallest subrepresentation of Vcontaining v. Our question can be reformulated as follows: To what extent is the spectral supportof a vector v ∈ C∞c (X)ur (or one of its quotients) equal to the image of δ−

12A∗X in A∗/W (the

unramified component of the Bernstein variety)? We will say that v is of “generic support” if itssupport is the image of δ−

12A∗X in A∗/W .

It is easy to see that not all vectors in C∞c (X)ur have generic support in general. For instance,let X = T\PGL2 as in Example 4.6. Recall our description of its Jacquet module: We can haveφ ∈ C∞c (X) whose image in the Jacquet module is non-zero, but is zero when restricted to k×. Infact, we can generate φ as follows: Choose a suitable φ1 supported in a neighborhood of the divisorZ1, and apply the automorphism “w” to it (action of the non-trivial Weyl group element on theleft). Let φ = φ1 − wφ1. Since “w” is G-equivariant, we will have R(g)φ = R(g)φ1 − wR(g)φ1, andtherefore our description of the image of φ in the Jacquet module will hold for all its G-translates,as well. As a result, the support of φ is not generic.

Let K be a hyperspecial maximal compact subgroup of G. The following theorem gives anassertive answer to our question for K-invariant vectors. For what follows we will be denoting bySχ the B-equivariant functional into C

χδ12

defined by the open B-orbit X.

The following theorem is clearly false in the case of anomalous non-homogeneous varieties suchas that of Example 3.6.1. Therefore, for the rest of the paper we re-define X to mean the Hausdorffclosure of X (cf. Lemma 3.6.3):

6.1.1 Theorem. The support of every φ ∈ C∞c (X)K is generic. In fact, if Sθχ(φ) = 0 for every χand θ then f = 0.

Proof. The second statement, although it appears stronger, is actually equivalent. First, by ourmain theorem if Sχ(φ) = 0 as a morphism into I(χ) (not as a functional) then SY,∗χ (φ) = 0 for everyorbit Y of maximal rank. Moreover I(χ) contains a unique (up to scaling) non-zero K-invariantvector whose value at 1 is non-zero, so to say that the value of the functional Sχ applied to φ iszero, for a K-invariant φ, is the same as saying that Sχ(φ) = 0 as an element of I(χ).

Assume Sθχ(φ) = 0 for every χ and θ. By Lemma 4.6.4, it suffices to show that the functionalsSYχ , for Y of smaller rank, vanish on φ and all its translates. We will consider two cases separately:Y a Borel-orbit in the open G-orbit, and Y a Borel-orbit in a different G-orbit.

34


6.1.2 Proof within the open G-orbit By Theorem 5.3.1, we may assume that Y is not raisedof type U for any simple root α, for otherwise wαY would be of larger dimension with the sameproperty, and we can apply induction on the dimension of Y. (It is for this application that weneeded to state and prove Theorem 5.3.1 in a more general context than just orbits of maximalrank.) Since Y cannot be the open orbit, there exists a simple root α raising it of type T or N. LetZ denote the open orbit of Y ·Pα; by assumption, SZχ f = 0 for every χ or, equivalently, the integralof f along every orbit of A0U on Z(k) is equal to zero.

We may first integrate over the unipotent radical as in (12) to reduce the problem to ι(f) ∈C∞c ((YPα/UPα)(k)), i.e. to a spherical variety of type T or N for Lα. We need only examine thecase T(k)\SL(k) = (T\SL)(k), with T split – take T = the torus of diagonal elements. Indeed,recall that The Jacquet module of this variety has been described in Example 4.6; it can be identifiedwith the space of smooth functions on k×∪01, 02, whose value as x→ 0 stabilizes to the differenceof the values at 01 and 02. By assumption, the image φU of φ in the Jacquet module is zero on k×,therefore it suffices to show that φU (01) = φU (02). By the Iwasawa decomposition, AUK = G, φ isdetermined by its values on the closed orbit Z1, and its integral over Z1 (which is also a U -orbit)is equal to its integral over G. This will not change if we apply the “w”-automorphism, thereforeφU (01) = φU (02).

This finishes the case where Y is contained in the open G-orbit.

6.1.3 Proof on smaller G-orbits Let Y be a k-rational Borel orbit, belonging to a non-openG-orbit Z. Since Z itself is a spherical variety, it suffices by the proof of the previous case to assumethat Y = Z (the open B-orbit in Z). Also, we may inductively assume that the theorem has beenproven for all larger G-orbits containing Z in their closure. The point now is that the morphismassociated to the smaller G-orbit is “picked up” by the morphism associated to a larger orbit afterregularization of a pole:

Let Z be the closure of a k-rational G-orbit and let V be as in Lemma 2.4.2 the closure of a(k-rational) B-orbit, containing Z as a divisor. Recall from Lemma 3.6.3 that there exists a naturalhomomorphism ΓV → ΓZ coming from a map between B-orbits. Let γ 7→ γ′ under this map; thismeans that the orbit Zγ′ ⊂ Z corresponding to γ′ is the unique B-orbit in the Hausdorff closure ofthe orbit Vγ ⊂ V corresponding to γ. Let P (χ) = 1− q−νZχvZ be the pole of SVχ associated to Z.

6.1.4 Lemma. The zero set Z(P ) of P coincides with AdmZ and for every χ ∈ AdmZ we have:

P (χ)SV,γχ ∼ SZ,γ′χ .

Proof. We know already that P (χ)SVχ will be a nonzero distribution which is supported on V r V ,so we just need to identify it. By Lemma 2.4.2, there exists a B-eigenfunction f which vanishes onevery B-stable divisor in V but Z, hence its zero locus will be precisely the complement of V ∪ Z.

Clearly, for φ ∈ C∞c (V ∪ Z) we have P (χ)SV,γχ (φ) ∼ SZ,γ′χ (φ) for χ ∈ Z(P ). This proves alreadythat AdmZ is the zero set of Z(P ), since their dimensions satisfy:

dimAdmZ = rk(X (Z)) = rk(X (V))− 1 = dimZ(P)

(due to the fact that every B-eigenfunction on Z extends to V).Applying Proposition 4.5.1, (ii), with f as above and ψ denoting the weight of f , we have that

for large m and χ ∈ AdmZ :

P (χψm)SV,γχψm(φ) =∑

j

P (χψm)SV,γχψm(φj) ∼∑

j

SZ,γ′

χψm(φj) = SZ,γ′

χψm(φ).

By the rationality of these distributions, we deduce that the claim is true for every χ ∈ AdmZ .

35


Remark. The statement is not true for Z any B-stable set; indeed recall the case of T\PGL2, whereboth closed orbits define the same valuation on k[X](B) and renormalizing the intertwining operatorof X at its pole “picks up” a sum of the intertwining operators of the closed orbits.

To finish the proof of the theorem, we have by assumption (and by induction on the dimension)that SV,γχ (φ) = 0 for V as in the lemma above. According to the lemma, P (χ)SV,γχ (φ) coincides withSZ,γ

′χ (φ) for generic χ in the set of admissible characters for Z, therefore SZ,γ

′χ (φ) is also identically

equal to zero. (Notice that Z, γ′ will indeed belong to the Hausdorff closure of some V, γ – that wasthe point of redefining X as the Hausdorff closure of X.)

6.2 The Hecke module of unramified vectorsThe result of the previous section allows us to present a weak analog of the main result of [GN1]–[GN4], namely a description of the Hecke module of K-invariant vectors. (These vectors are com-monly called “spherical”, but to avoid a double use of this word we will only call them “unramified”.)Notice that C∞c (X)K ⊂ C∞c (X)ur.

Let H(G,K) denote the convolution algebra of K-biinvariant measures on G. Recall the Satakeisomorphism:H(G,K) ' C[A∗]W . Since all vectors in C∞c (X)ur have spectral support over the imageof δ−

12A∗X inA∗/W ,H(G,K) acts on C∞c (X)K through the corresponding quotient, which will be de-

noted by HX . Let KX denote the fraction field of HX , hence naturally: KX ' C(δ−12A∗X)NW (−ρ+a∗X).

6.2.1 Theorem. The space C∞c (X)K is a finitely-generated, torsion-free module for HX .

Moreover, we have: C∞c (X)K ⊗HX KX '(C(δ−

12A∗X)WX

)|Γ|.

Remark. The isomorphism above is not a canonical one, since it depends, as we shall see, on thechoice of one K-invariant vector.

Proof. The fact that C∞c (X)K is torsion-free over HX follows from Theorem 6.1.1.

Fix a φ0 ∈ C∞c (X)K and define a map C∞c (X)K →(C(δ−

12A∗X)

)|Γ|by

φ 7→(Sθχ(φ)Sθχ(φ0)

)

θ

.

We claim, first, that the image of this map lies in the WX -invariants. Indeed, by Corollary 5.4.2,for w ∈WX the quotients Sχ and Swχ are isomorphic (through Tw). Since there is a unique line ofK-invariant vectors in I(χ), it follows that Sχ(φ) = c · Sχ(φ0) for some constant c (depending onχ), and since Sχ and Swχ are isomorphic, it follows that that this constant is the same for χ and

for wχ. This proves that we have a map: C∞c (X)→(C(δ−

12A∗X)WX

)|Γ|.

We have shown in Theorem 6.1.1 that for φ ∈ C∞c (X)K it is not possible that Sθχ(φ) = 0 forevery χ, θ – this establishes that the map is injective.

We prove surjectivity of the map when tensored with KX . Notice that the space of morphisms:C∞c (X) → I(χ) has generically dimension equal to r := (N (ρ + a∗X) : WX) · |Γ|. Suppose thatC∞c (X)K ⊗HX KX had smaller dimension over KX . Then a basis S1

χ, . . . , Srχ of the space of

morphisms would be linearly dependent when restricted to the subspace generated by C∞c (X)K ,say

∑i ci(χ)Siχ = 0 on that subspace. Given φ ∈ C∞c (X)ur, I claim that

∑i ci(χ)Siχ(φ) = 0 for

generic (and hence every) χ. Indeed, for generic χ the image of the Siχ’s in I(χ) is irreducible andunramified, hence if

∑i ci(χ)Siχ(φ) is non-zero then one of its G-translates, when convolved with

the characteristic function of K, should be non-zero. Since the Siχ’s are G-equivariant, we have:

36


R(Kg)∑

i ci(χ)Siχ(φ) =∑

i ci(χ)Siχ(R(Kg)φ) = 0 by assumption, since R(Kg)φ ∈ C∞c (X)K . Itfollows that

∑i ci(χ)Siχ = 0 on the whole space, contradicting what we know about the dimension

of the space of morphisms into I(χ). This proves the stated isomorphism.Finally, recall from §4.7 that we may regularize Sχ so that it is regular for every χ. Then

φ 7→ Sχ(φ) defines an injection C∞c (X)K → C[δ−12A∗X ]. Since the latter is a finitely generated

HX -module, it follows that C∞c (X)K is also finitely generated.

6.3 A commutative ring of endomorphismsIn this section we assume, for simplicity, that Γ = 1, i.e. each B-orbit of maximal rank containsonly one rational B-orbit.

6.3.1 Definition of the map Let D ∈ EndH(G,K)(C∞c (X)K). It induces an endomorphism of

C∞c (X)K⊗HX KX '(C(δ−

12A∗X)WX

)which is KX -linear. If it is also C(δ−

12A∗X)WX -linear (in other

words, if Sχ D ∼ Sχ on C∞c (X)K), then we will call D “geometric”. Of course, if (C∞c (X))ur

is generically multiplicity-free (i.e. WX = NW (−ρ + a∗X)), then every endomorphism is geometric,but this will not be the case in general. The map D 7→ c(χ), where c(χ) is given by the relationSχ D = c(χ)Sχ, defines a ring homomorphism EndH(G,K)(C∞c (X)K)geom → C(δ−

12 A∗X). In fact, by

Theorem 5.3.1, the image lies in invariants of the little Weyl group WX . Moreover, since by Theorem6.2.1 C∞c (X)K is a finitely generated HX -module, every HX -algebra of endomorphisms is a finitelygenerated module over HX ; and since the integral closure of HX in C(δ−

12A∗X)WX is C[δ−

12A∗X ]WX

(the variety A∗X/WX is normal), it follows that the image of the above homomorphism must lie inC[δ−

12A∗X ]WX . We conjecture that the image is the whole ring:

Conjecture. There is a canonical isomorphism:(EndH(G,K)C

∞c (X)K

)geom ' C[δ−12A∗X ]WX .

The reason that we believe that these endomorphisms exist in general is the following analogywith invariant differential operators on X:

As was proven by F. Knop in [Kn94b], the algebra of invariant differential operators on a sphericalvariety (over an algebraically closed field K) is commutative, and isomorphic to K[ρ+a∗X ]WX . Here,a∗X is isomorphic to the Lie algebra of what we denote by A∗X . This generalizes the Harish-Chandrahomomorphism (if we regard the group G as a spherical G×G variety), and in fact the followingdiagram is commutative:

z(G) −−−−→ D(X)G∥∥∥∥∥∥

K[a∗]W −−−−→ K[ρ+ a∗X ]WX

(21)

What we propose is a similar diagram for the p-adic group, which on the left side will have theSatake isomorphism for H(G,K) (or equivalently, the unramified factor of the Bernstein centre) andon the right side the “geometric endomorphisms” that we defined above, which should be viewedas an analog for the invariant differential operators:

H(G,K) −−−−→ (EndH(G,K)C∞c (X)K

)geom

∥∥∥∥∥∥

C[A∗]W −−−−→ C[δ−12A∗X ]WX

(22)

37


Remark. The reader should not be confused by the fact that in Knop’s result the Lie algebra a∗Xis offset by ρ while in ours the torus A∗X is offset by δ−

12 , which is equal to −ρ: The discrepancy is

a matter of definitions, and to fix it we could have denoted by I(χ−1) what we denoted by I(χ) –but of course this would contradict the conventions in the literature.

Our interest in this conjecture comes from the fact that the analog of invariant differentialoperators suggests the possibility of a “geometric” construction of these endomorphisms, whilespectral methods do not seem to suffice in general.

However, it is easy to prove the conjecture in the cases that X is generically multiplicity-free orparabolically induced from a multiplicity-free one. Then these endomorphisms will be provided bythe Hecke algebra of G or, respectively, of a Levi subgroup acting “on the left”:

6.3.2 Theorem. Conjecture 6.3.1 is true if:

i) the unramified spectrum of X is generically multiplicity-free, in which case the geometricendomorphisms are all the endomorphisms of C∞c (X)K , or

ii) the spherical variety X is “parabolically induced” from a spherical variety whose unramifiedspectrum is generically multiplicity-free.

Proof. In the first case, C[A∗]W surjects onto C[δ−12A∗X ]WX . The claim that these are all the endo-

morphisms follows from Theorem 6.1.1.In the second case, as we saw in §2.3, the subtorus A∗X and the Weyl group WX coincide with

those associated to the corresponding spherical variety of the Levi. Hence C[δ−12A∗X ]WX is surjected

upon by the Bernstein center of L acting “on the left”.

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Yiannis Sakellaridis [email protected] of Mathematics , Stanford University , Stanford, CA 94305-2125 , USA

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on the unramiﬁed spectrum of spherical varieties over p...

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