Invent. math. https://doi.org/10.1007/s00222-018-0807-z

On distinguished square-integrable representations for Galois pairs and a conjecture of Prasad Raphaël Beuzart-Plessis1

Received: 24 August 2017 / Accepted: 12 May 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract We prove an integral formula computing multiplicities of squareintegrable representations relative to Galois pairs over p-adic fields and we apply this formula to verify two consequences of a conjecture of Dipendra Prasad. One concerns the exact computation of the multiplicity of the Steinberg representation and the other the invariance of multiplicities by transfer among inner forms. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . . 3 Definition of a distribution for all symmetric pairs 4 The spectral side . . . . . . . . . . . . . . . . . . 5 The geometric side . . . . . . . . . . . . . . . . . 6 Applications to a conjecture of Prasad . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction Let F be a p-adic field (that is a finite extension of Q p for a certain prime number p) and H be a connected reductive group over F. Let E/F be a quadratic extension and set G := R E/F H E where R E/F denotes Weil’s restriction of

B Raphaël Beuzart-Plessis [email protected]

1

I2M-CNRS(UMR 7373), Université d’Aix-Marseille, Campus de Luminy, 13288 Marseille Cédex 9, France

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scalars [so that G(F) = H (E)]. To every complex smooth irreducible representation π of G(F) and every continuous character χ of H (F) we associate a multiplicity m(π, χ ) (which is always finite by [18, Theorem 4.5]) defined by m(π, χ ) := dim Hom H (π, χ ) where Hom H (π, χ ) stands for the space of (H (F), χ )-equivariant linear forms on (the space of) π . Recently, Prasad [43] has proposed very general conjectures describing this multiplicity in terms of the Langlands parameterization of π , at least for representations belonging to the so-called ‘generic’ L-packets. These predictions, which generalize earlier conjectures of Jacquet [27,28], are part of a larger stream that has come to be called the ‘local relative Langlands program’ and whose main aim is roughly to describe the ‘spectrum’ of general homogeneous spherical varieties X = H \G over local fields in terms of Langlands dual picture and correspondence. In the paper [44], and under the assumption that G is split, Sakellaridis and Venkatesh set up a very general framework to deal with these questions by introducing a certain complex reductive group Gˇ X associated to the variety X , which generalizes Langlands construction of a dual group, together with a morphism Gˇ X → Gˇ (actually, in the most general case, this should also include an extra SL 2 factor) which, according to them, should govern a great part of the spectral decomposition of L 2 (H \G) (see [44, Conjecture 16.2.2]). In a similar way, in the case where G = R E/F H E as above (note that such a group is never split) Prasad introduces a certain L-group L H op (further explanations on this notation are given below) and a morphism L H op → L G which should govern, on the dual side, the behavior of the multiplicities m(π, χ ) for a very particular quadratic character χ (denoted by ω H,E below) that has also been defined by Prasad. The main goal of this paper is to present some coarse results supporting Prasad’s very precise conjectures in the particular case of stable (essentially) squareintegrable representations. In the rest of this introduction we will recall the part of Prasad’s conjecture that we are interested in as well as the two consequences of it that we have been able to verify. We will also say some words on the proofs which are based on a certain simple local trace formula adapted to the situation and which takes its roots in Arthur’s local trace formula [4] as well as in Waldspurger’s work on the Gross–Prasad conjecture for orthogonal groups [46,47]. Prasad associates a number of invariants to the situation at hand. First, he constructs a certain quadratic character ω H,E : H (F) → {±1} as well as a certain quasi-split group H op over F which is an E/F form of the quasi-split inner form of H . We refer the reader to [43, §7–8] for precise constructions of those and content ourself to give three examples here:

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On distinguished square-integrable representations

– If H = GL n , then H op = U (n)qs (quasi-split form) and ω H,E = (η E/F ◦ det)n+1 where η E/F is the quadratic character associated to E/F; – If H = U (n) (a unitary group of rank n), then H op = GL n and ω H,E = 1; – If H = SO(2n + 1) (any odd special orthogonal group), then H op = SO(2n + 1)qs (the quasi-split inner form) and ω H,E = η E/F ◦ Nspin where Nspin : SO(2n + 1)(F) → F × /(F × )2 denotes the spin norm. To continue we need to restrict slightly the generality by only considering characters χ that are of ‘Galois type’ i.e. which are in the image of a map constructed by Langlands H 1 (W F , Z ( Hˇ )) → Homcont (H (F), C× ) This map is always injective (because F is p-adic) but not always surjective (although it is most of the time, e.g. if H is quasi-split). We refer the reader to [37] for further discussion on these matters. The character ω H,E is always of Galois type and, to every character χ of Galois type of H (F), Prasad associates op a certain ‘Langlands dual group’ Hχ which sits in a short exact sequence op 1 → Hˇ op → Hχ → W F → 1 op

together with a group embedding ι: Hχ → L G (where L G denotes the L-group of G) compatible with the projections to W F and algebraic when op restricted to Hˇ op . In the particular case where χ = ω H,E , we have Hχ = L H op and ι is the homomorphism of quadratic base-change. Remark 1 Although the short exact sequence above always splits, there does op not necessarily exist a splitting preserving a pinning of Hˇ op and hence Hχ is not always an L-group in the usual sense. Let WD F := W F × SL 2 (C) be the Weil–Deligne group of F. An ‘Lop parameter’ taking values in Hχ is defined as usual: a continuous Frobenius op semi-simple morphism WD F → Hχ which commutes with the projections to W F and is algebraic when restricted to SL 2 (C). We are now ready to state (a slight generalization of) the stable version of Prasad’s conjecture for squareintegrable representations: G Conjecture 1 Let φ: WD F → L G be a discrete  L-parameter, Π (φ) ⊆ Irr(G) the corresponding L-packet and Πφ = π ∈Π G (φ) d(π )π the stable representation associated to φ. Then, we have

m(Πφ , χ ) = |ker 1 (F; H, G)|−1

 |Z (φ)| |Z (ψ)| ψ

where

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R. Beuzart-Plessis op

– The sum is over the set of ‘L-parameters’ ψ: WD F → Hχ (taken up to ˇ Hˇ op -conj) making the following diagram commute up to G-conj, i.e. there −1 exists g ∈ Gˇ such that ι ◦ ψ = gφg , op

Hχ ψ

WD F

ι φ

LG

  – ker 1 (F; H, G) := Ker H 1 (F, H ) → H 1 (F, G) (corresponds to certain twists of the parameter ψ that become trivial in L G); ˇ W F and Z (ψ) := Cent ˇ op (ψ)/Z ( Hˇ op )W F – Z (φ) := Cent ˇ (φ)/Z (G) G

H

As we said, this is only part of Prasad’s general conjectures which aim to describe (almost) all the multiplicities m(π, χ ) explicitly. This version of the conjecture (and far more) is known in few particular cases: for H = G L(n) by Kable [29] and Anandavardhanan and Rajan [1], for H = U (n) by Feigon et al. [21] and for H = GSp(4) by Lu [39]. The following theorems are both formal consequences of Conjecture 1 and are the main results of this paper (see Theorems 6.3.1, 6.7.1): Theorem 1 Let H , H  be inner forms over F, G := R E/F H , G  := R E/F H  and χ , χ  characters of Galois type of H (F) and H  (F) corresponding to each other [i.e. coming from the same element in H 1 (W F , Z ( Hˇ )) = H 1 (W F , Z ( Hˇ  ))]. Let Π, Π  be (essentially) square-integrable representations of G(F) and G  (F) respectively which are stable (but not necessarily irreducible) and transfer of each other [i.e. ΘΠ (x) = ΘΠ  (y) for all stably conjugate regular elements x ∈ G reg (F) and y ∈ G reg (F) where ΘΠ , ΘΠ  denote the Harish-Chandra characters of Π and Π  respectively]. Then, we have m(Π, χ ) = m(Π  , χ  ). Theorem 2 For π = St(G) the generalized Steinberg representation of G(F) and χ a character of Galois type we have  1 if χ = ω H,E m(St(G), χ ) = 0 otherwise Theorem 2 also confirms an older conjecture of Prasad [42, Conjecture 3] which was already known for split groups and tamely ramified extensions by work of Broussous–Courtès and Courtès [12,15,16] and for inner forms of GL n by work of Matringe [40]. The proof of Broussous and Courtès is

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mainly based on a careful study of the geometry of the building whereas Matringe’s work uses some Mackey machinery. Our approach is completely orthogonal to theirs and is based on a certain integral formula computing the multiplicity m(π, χ ) in terms of the Harish-Chandra character of π . This formula is reminiscent and inspired by a similar result of Waldspurger in the context of the so-called Gross–Prasad conjecture [46,47]. It can also be seen as a ‘twisted’ version (‘twisted’ with respect to the non-split extension E/F) of the orthogonality relations between characters of discrete series due HarishChandra [14, Theorem 3]. It can be stated as follows (see Theorem 6.1.1): Theorem 3 Let π be a square-integrable representation of G(F) and χ be a continuous character of H (F). Assume that χ and the central character of π coincide on A H (F) [the maximal split central torus in H (F)]. Then, we have  m(π, χ ) = D H (x)Θπ (x)χ (x)−1 d x Γell (H )

where Θπ denotes the Harish-Chandra character of π [a locally constant function on G reg (F)], D H is the usual Weyl discriminant and Γell (H ) stands for the set of regular elliptic conjugacy classes in H (F) := H (F)/A H (F) equipped with a suitable measure d x. Theorem 1 is an easy consequence of this formula and Theorem 2 also follows from it with some extra work. Let us give an outline of the proof of Theorem 2 assuming Theorem 3. For notational simplicity we will assume that H is semi-simple. We have the following explicit formula for the character of the Steinberg representation (see Sect. 6.5 for a reminder) 

D G (x)1/2 ΘSt(G) (x) =

P0 ⊆P=MU

×

(−1)a P −a P0 

D M (y)1/2 δ P (y)1/2

{y∈M(F); y∼con j x}/M−con j

where P0 is a minimal parabolic subgroup of G and we refer the reader to the core of the paper for other unexplained notations which are however pretty standard. Plugging this explicit formula in Theorem 3 and rearranging somewhat the terms we get   a P −a P0 (−1) D M (x)χ (x)−1 d x m(St(G), χ ) = (M,P)/con j

Γell (M)

where the sum runs over the H (F)-conjugacy classes of pairs (M, P) with

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R. Beuzart-Plessis

– M an elliptic twisted Levi subgroup of H by which we mean an algebraic subgroup of H with trivial split center such that R E/F M E is a Levi subgroup of G; – P a parabolic subgroup of G with Levi component R E/F M E . Using a particular case of Harish-Chandra orthogonality relations between characters of discrete series [14, Theorem 3], we can show that (see Sect. 6.6)  D M (x)χ (x)−1 d x = (χ|M , 1) Γell (M)

where χ|M denotes the restriction of χ to M(F), 1 the trivial character of M(F) and (·, ·) denotes the natural scalar product on the space of virtual characters of M(F). Then, in the above expression for m(St(G), χ ), we can group together pairs (M, P) according to their stable conjugacy classes ending up with an equality 

m(St(G), χ ) =

(−1)a P −a P0 |ker 1 (F; M, H )|(χ|M , 1)

(M,P)/stab

  where ker 1 (F; M, H ) := Ker H 1 (F, M) → H 1 (F, H ) , a set which naturally parametrizes conjugacy classes inside the stable conjugacy class of (M, P). Set Hab for the quotient of H (F) by the common kernel of all the characters of Galois type [in case H is quasi-split it is just the abelianization of H (F)] and let Mab denote, for all elliptic twisted Levi M, the image of M(F) in Hab . Then, using Frobenius reciprocity, the last identity above can be rewritten as the equality between m(St(G), χ ) and ⎛



⎝χ ,

(M,P)/stab

⎞



Hab (−1)a P −a ker 1 (F; M, H ) I ndM (1)⎠ ab

and thus Theorem 2 is now equivalent to the following identity in the Grothendieck group of Hab :  (M,P)/stab



Hab (−1)a P −a ker 1 (F; M, H ) Ind M (1) = ω H,E ab

(1.0.1)

The proof of this identity in general is rather long and technical (see Proposition 6.4.1), so we content ourself (again) with giving two examples here: – If H = GL n , we have Hab = F × and ω H,E = ηn+1 E/F . If n is odd, M = H is the only elliptic twisted Levi and then 1.0.1 reduces to 1 = 1. On the

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On distinguished square-integrable representations

other hand, if n is even there are two (stable) conjugacy classes of pairs (M, P): 

∗ GL n/2 (E) M0 = P0 = H and M1 = GL n/2 (E) ⊆ P1 = GL n/2 (E) we have M0,ab = Hab = F × ⊃ M1,ab = N (E × ) and 1.0.1 reduces to the identity ×

F Ind N (E × ) 1 − 1 = η E/F

– For H = U (n) (a unitary group of rank n) we have Hab = Ker N E/F and ω H,E = 1. In this case, stable conjugacy classes of pairs (M, P) are parametrized by (ordered) partitions (n 1 , . . . , n k ) of n as follows: (n 1 , . . . , n k ) → M = U (n 1 ) × · · · × U (n k ) ⊆ P ⎛ ⎞ GL n 1 (E) ∗ ∗ ⎜ ⎟ .. =⎝ ⎠ . ∗ GL n k (E) Moreover, |ker 1 (F; M, H )| = 2k−1 and Mab = Hab for all M as above. Thus, in this case 1.0.1 reduces to the following combinatorial identity 

(−1)n−k 2k−1 = 1

(n 1 ,...,n k ) n 1 +···+n k =n

As we said, for its part, Theorem 3 is a consequence of a certain simple local trace formula adapted to the situation and to the proof of which most of the paper is devoted. Let us state briefly the content of this formula by assuming again, for simplicity, that the group H is semi-simple. Starting with a function f ∈ Cc∞ (G(F)), we consider the following expression in two variables χ K f (x,

 y) :=

H (F)

f (x −1 hy)χ (h)−1 dh,

x, y ∈ G(F)

This function is precisely the kernel of the operator on L 2 (H (F)\G(F), χ ) given by convolution by f . Formally, the trace of such an operator should be given by the integral of this kernel over the diagonal that is  χ χ K f (x, x)d x J ( f ) := H (F)\G(F)

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Unfortunately, in general the convolution operator given by f isn’t of traceclass and the above expression diverges. Nevertheless, we can still restrict our attention to some ‘good’ space of test functions for which the above integral is absolutely convergent. Recall, following Waldspurger [46], that the function f is said to be strongly cuspidal if for all proper parabolic subgroups P = MU  G we have  f (xu)du = 0 U (F)

for all x ∈ M(F). We also say, following Harish-Chandra [23], that f is a cusp form if the above kind of integrals vanish for all x ∈ G(F) (thus, and contrary to what we might guess, being a cusp form is stronger than being strongly cuspidal). Actually, it will be more convenient for us to work with functions that are not necessarily compactly supported: we will take f in the so-called Harish-Chandra–Schwartz space (see Sect. 2.3 for a reminder) denoted by C (G(F)). The notions of strong cuspidality and of cusp forms extend verbatim to this bigger space. The following theorem, whose proof is scattered all over this paper (see Theorems 3.1.1, 4.1.1, 5.1.1), is our main technical result: Theorem 4 Let f ∈ C (G(F)) be a strongly cuspidal function. Then, the expression defining J χ ( f ) is absolutely convergent (see Theorem 3.1.1) and we have: (i) (see Theorem 5.1.1) A geometric expansion  D H (x)Θ f (x)χ (x)−1 d x Jχ( f ) = Γell (H )

where the function Θ f is defined using weighted orbital integrals of Arthur (see Sect. 2.6); (ii) (see Theorem 4.1.1) If f is moreover a cusp form, a spectral expansion  m(π, χ ) Trace π ∨ ( f ) Jχ( f ) = π ∈Irr sqr (G)

where Irr sqr (G) denotes the set of (equivalence classes of) irreducible square-integrable representations of G(F) and for all π ∈ Irr sqr (G), π ∨ is the smooth contragredient of π . We prove this theorem by following closely the general method laid down by [11,46,47]. In particular, a crucial point to get the spectral expansion in the above theorem is to show that for π square-integrable the abstract multiplicity m(π, χ ) is also the multiplicity of π in the discrete spectrum of

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On distinguished square-integrable representations

L 2 (H (F)\G(F)). This fact is established in the course of the proof of Proposition 4.2.1 using the simple adaptation of an idea that goes back to Sakellaridis and Venkatesh [44, Theorem 6.4.1] and Waldspurger [47, Proposition 5.6]. Here is an outline of the contents of the different parts of the paper. In the first part, we set up the main notations and conventions as well as collect different results that will be needed in the subsequent sections. It includes in particular a discussion of a natural generalization of Arthur’s (G, M)-families to symmetric pairs that we call (G, M, θ )-families. The second part contains the proof of the absolute convergence of J χ ( f ) for strongly cuspidal functions f and in the third part we establish a spectral expansion of this distribution when f is a cusp form. These two parts are actually written in the more general setting of tempered symmetric pairs (G, H ) (which were called strongly discrete in [22]) to which the proofs extend verbatim. The fourth part deals with the geometric expansion of J χ ( f ). There we really have to restrict ourself to the setting of Galois pairs (that is when G = R E/F H E ) since a certain equality of Weyl discriminants (see 5.1.1), which is only true in this particular case, plays a crucial role in allowing to control the uniform convergence of certain integrals. Finally, in the last part of this paper we prove the formula for the multiplicity (Theorem 3) and give two applications of it towards Prasad’s conjecture (Theorems 1, 2). 2 Preliminaries 2.1 Groups, measures, notations Throughout this paper we will let F be a p-adic field (i.e. a finite extension of Q p for a certain prime number p) for which we will fix an algebraic closure F. We will denote by |·| the canonical absolute value on F as well as its unique extension to F. Unless specified otherwise, all groups and varieties that we consider in this paper will be tacitly assumed to be defined over F and we will identify them with their points over F. Moreover for every finite extension K of F and every algebraic variety X defined over K we will denote by R K /F X Weil’s restriction of scalars (so that in particular we have a canonical identification (R K /F X )(F) = X (K )). Let G be a connected reductive group over F and A G be its maximal central split torus. We set G := G/A G and AG := X ∗ (A G ) ⊗ R

where X ∗ (A G ) denotes the abelian group of cocharacters of A G . If V is a real vector space we will always denote by V ∗ its dual. The space A∗G can naturally be identified with X ∗ (A G )⊗R = X ∗ (G)⊗R where X ∗ (A G ) and X ∗ (G) stand

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for the abelian groups of algebraic characters of A G and G respectively. More generally, for every extension K /F we will denote by X ∗K (G) the group of characters of G defined over K . There is a natural morphism HG : G(F) → AG characterized by χ , HG (g) = log(|χ (g)|) for all χ ∈ X ∗ (G). We set AG,F := HG (A G (F)). It is a lattice in AG . The same notations will be used for the Levi subgroups of G (i.e. the Levi components of parabolic subgroups of G): if M is a Levi subgroup of G we define similarly A M , A M , HM and A M,F . We will also use Arthur’s notations: P (M), F (M) and L(M) will stand for the sets of parabolic subgroups with Levi component M, parabolic subgroups containing M and Levi subgroups containing M respectively. Let K be a maximal special compact subgroup of G(F). Then, for all parabolic subgroups P with Levi decomposition P = MU the Iwasawa decomposition G(F) = M(F)U (F)K allows to extend HM to a map H P : G(F) → A M defined by H P (muk) := HM (m) for all m ∈ M(F), u ∈ U (F) and k ∈ K . For all Levi subgroups M ⊂ L there is a natural decomposition A M = A LM ⊕ A L L := where A LM is generated by HM (Ker(HL|M(F) )) and we will set a M dim(A LM ). The Lie algebra of G will be denoted by g and more generally for any algebraic group we will denote its Lie algebra by the corresponding Gothic letter. We will write Ad for the adjoint action of G on g. We denote by exp the exponential map which is an F-analytic map from an open neighborhood of 0 in g(F) to G(F). For all subsets S ⊂ G, we write Cent G (S), Cent G(F) (S) and Norm G(F) (S) for the centralizer of S in G, resp. the centralizer of S in G(F), resp. the normalizer of S in G(F). If S = {x} we will denote by G x the neutral connected component of Cent G (x) := Cent G ({x}). We define G reg as the open subset of regular semisimple elements of G and for all subgroups H of G we will write Hreg := H ∩ G reg . Recall that a regular element x ∈ G reg (F) is said to be elliptic if A G x = A G . We will denote by G(F)ell the set of regular elliptic elements in G(F). The Weyl discriminant D G is defined by



D G (x) := det (1 − Ad(x)|g/gx )

For every subtorus T of G we will write W (G, T ) := Norm G(F) (T )/ Cent G(F) (T )

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for its Weyl group. If A ⊂ G is a split subtorus we will denote by R(A, G) the set of roots of A in G i.e. the set of nontrivial characters of A appearing in the action of A on g. More generally, if H is a subgroup of G and A ⊂ H is a split subtorus we will denote by R(A, H ) the set of roots of A in H . In all this paper we will assume that all the groups that we encounter have been equipped with Haar measures (left and right invariant as we will only consider measures on unimodular groups). In the particular case of tori we normalize these Haar measure by requiring that they give mass 1 to their maximal compact subgroups. For any Levi subgroup M of G we equip A M with the unique Haar measure such that vol(A M /A M,F ) = 1. If M ⊂ L are two Levi subgroups then we give A LM  A M /A L the quotient measure. Finally, we will adopt the following slightly imprecise but convenient notations. If f and g are positive functions on a set X , we will write f (x)  g(x) for all x ∈ X and we will say that f is essentially bounded by g, if there exists a c > 0 such that f (x)  cg(x), for all x ∈ X We will also say that f and g are equivalent and we will write f (x) ∼ g(x) for all x ∈ X if both f is essentially bounded by g and g is essentially bounded by f . 2.2 Log-norms All along this paper, we will assume that g(F) has been equipped with a (classical) norm |·|g , that is a map |·|g : g(F) → R+ satisfying |λX |g = |λ| · |X |g , |X + Y |g  |X |g + |Y |g and |X |g = 0 if and only if X = 0 for all λ ∈ F and X, Y ∈ g(F). For any R > 0, we will denote by B(0, R) the closed ball of radius R centered at the origin in g(F). In this paper we will freely use the notion of log-norms on varieties over F. The concept of norm on varieties over local fields has been introduced by Kottwitz [36, §18]. A log-norm is essentially just the logarithm of a Kottwitz’s norm and we refer to [11, §1.2] for the basic properties of these log-norms. For convenience, we collect here the definition and basic properties of these objects. First, an abstract log-norm on a set X is just a real-valued function x → σ (x) on X such that σ (x)  1, for all x ∈ X . For two abstract log-norms σ1 and σ2 on X , we will say that σ2 dominates σ1 and we will write σ1  σ2 if

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σ1 (x)  σ2 (x) for all x ∈ X . We will say that σ1 and σ2 are equivalent if each of them dominates the other. For an affine algebraic variety X over F, choosing a set of generators f 1 , . . . , f m of its F-algebra of regular functions O(X ), we can define an abstract log-norm σ X on X by setting σ X (x) = 1 + log (max{1, | f 1 (x)|, . . . , | f m (x)|}) for all x ∈ X . The equivalence class of σ X doesn’t depend on the particular choice of f 1 , . . . , f m and by a log-norm on X we will mean any abstract lognorm in this equivalence class. Note that if U is the principal Zariski open subset of X defined by the non-vanishing of Q ∈ O(X ), then we have   σU (x) ∼ σ X (x) + log 2 + |Q(x)|−1 for all x ∈ U . More generally, for X any algebraic variety over F, choosing a finite covering (Ui )i∈I of X by open affine subsets and fixing log-norms σUi on each Ui , we can define an abstract log-norm on X by setting   σ X (x) = inf σUi (x); i ∈ I such that x ∈ Ui Once again, the equivalence class of σ X doesn’t depend on the various choices and by a log-norm on X we will mean any abstract log-norm in this equivalence class. We will assume that all varieties that we consider in this paper are equipped with log-norms and we will set σ := σG and σ := σG . Let p: X → y be a regular map between algebraic varieties then we have σY ( p(x))  σ X (x) for all x ∈ X . If p is a closed immersion or more generally if p is a finite morphism [36, Proposition 18.1(1)] we have σY ( p(x)) ∼ σ X (x) for all x ∈ X . We say that p has the norm descent property (with respect to F) if, denoting by p F the induced map on F-points, we have σY (y) ∼

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x∈ p −1 F (y)

σ X (x)

On distinguished square-integrable representations

for all y ∈ p F (X (F)). By Proposition 18.3 of [36], if T is a subtorus of G then the projection G  T \G has the norm descent property i.e. we have σT \G (g) ∼ inf σ (tg) t∈T (F)

(2.2.1)

for all g ∈ G(F). In Sect. 2.9 we will prove that for H an F-spherical subgroup of G (i.e. a subgroup such that there exists a minimal parabolic subgroup P0 of G with H P0 open) the projection G  H \G also has the norm descent property. Let T ⊂ G again be a maximal subtorus. As the regular map T \G × Treg → G reg , (g, t) → g −1 tg is finite we get   σT \G (g)  σ (g −1 tg) log 2 + D G (t)−1

(2.2.2)

for all g ∈ G and all t ∈ Treg . For every variety X defined over F, equipped with a log-norm σ X , and all M > 0 we will denote by X [< M], resp. X [ M], the set of all x ∈ X (F) such that σ X (x) < M, resp. σ X (x)  M. With this notation, if T is a torus over F and k := dim(A T ) we have vol (T [< M])  M k

(2.2.3)

for all M > 0. 2.3 Function spaces Let ω be a continuous character of A G (F). We define Sω (G(F)) := Cc∞ (A G (F)\G(F), ω) as the space of functions f : G(F) → C which are smooth (i.e. locally constant), satisfy f (ag) = ω(a) f (g) for all (a, g) ∈ A G (F) × G(F) and are compactly supported modulo A G (F). Assume moreover that ω is unitary and let Ξ G be Harish-Chandra Xi function associated to a special maximal compact subgroup K of G(F) (see [45, §II.1]). Then, we define the Harish-Chandra–Schwartz space Cω (G(F)) as the space of functions f : G(F) → C which are biinvariant by an open subgroup of G(F), satisfy f (ag) = ω(a) f (g) for all (a, g) ∈ A G (F) × G(F) and such that for all d > 0 we have | f (g)|  Ξ G (g)σ (g)−d for all g ∈ G(F). 2.4 Representations In this paper all representations we will consider are smooth and we will always use the slight abuse of notation of identifying a representation π with the

123

R. Beuzart-Plessis

space on which it acts. We will denote by Irr(G) the set of equivalence classes of smooth irreducible representations of G(F) and by Irr cusp (G), Irr sqr (G) the subsets of supercuspidal and essentially square-integrable representations respectively. If ω is a continuous unitary character of A G (F) we will also write Irr ω (G) [resp. Irr ω,cusp (G), Irr ω,sqr (G)] for the sets of all π ∈ Irr(G) [resp. π ∈ Irr cusp (G), π ∈ Irr sqr (G)] whose central character restricted to A G (F) equals ω. For all π ∈ Irr(G) we will denote by π ∨ its contragredient and by ·, ·: π × π ∨ → C the canonical pairing. For all π ∈ Irr ω,sqr (G), d(π ) will stand for the formal degree of π . Recall that it depends on the Haar measure on G(F) and that it is uniquely characterized by the following identity (Schur orthogonality relations)  1 v1 , v2∨ v2 , v1∨  π(g)v1 , v1∨ v2 , π ∨ (g)v2∨ dg = d(π ) A G (F)\G(F) for all v1 , v2 ∈ π and all v1∨ , v2∨ ∈ π ∨ . From this, we easily infer that for every coefficient f of π we have Trace(π ∨ ( f )) =

1 f (1). d(π )

(2.4.1)

Let π ∈ Irr(G) and let ω be the inverse of the restriction of the central character of π to A G (F). Then, for all f ∈ Sω (G(F)) we can define an operator π( f ) on π by  ∨ π( f )v, v  := f (g)π(g)v, v ∨ dg A G (F)\G(F)

for all (v, v ∨ ) ∈ π × π ∨ . For all f ∈ Sω (G(F)) this operator is of finite rank and a very deep theorem of Harish-Chandra [25, Theorem 16.3] asserts that the distribution f ∈ Sω (G(F)) → Trace(π( f )) is representable by a locally integrable function which is locally constant on G reg (F). This function, the Harish-Chandra character of π , will be denoted Θπ . It is characterized by  Trace(π( f )) = Θπ (g) f (g)dg A G (F)\G(F)

for all f ∈ Sω (G(F)). If moreover the representation π is square-integrable (or more generally tempered) with unitary central character, then the integral

123

On distinguished square-integrable representations

defining π( f ) still makes sense for all f ∈ Cω (G(F)), the resulting operator is again of finite rank and the above equality continues to hold. 2.5 Weighted orbital integrals Let M be a Levi subgroup and fix a maximal special compact subgroup K of G(F). Using K we can define maps H P : G(F) → A M for all P ∈ P (M) (cf Sect. 2.1). Let g ∈ G(F). The family {−H P (g); P ∈ P (M)} is a positive (G, M)-orthogonal set in the sense of Arthur (see [2, §2]). In Q particular, following loc. cit. using this family we can define a weight v M (g) Q for all Q ∈ F (M). Concretely, v M (g) is the volume of the convex hull of the set {H P (g); P ∈ P (M), P ⊂ Q} (this convex hull belongs to a certain affine subspace of A M with direction A LM where Q = LU with M ⊂ L and we define the volume with respect to the fixed Haar measure on A LM ). If Q = G G (g) for simplicity. For every character ω of A (F), every we set v M (g) := v M G function f ∈ Cω (G(F)) and all x ∈ M(F) ∩ G reg (F), we define, again following Arthur, a weighted orbital integral by  Q Q Φ M (x, f ) := f (g −1 xg)v M (g)dg G x (F)\G(F)

The integral is absolutely convergent by the following lemma which is an immediate consequence of 2.2.2 and Lemma 2.9.2 (which will be proved later). Lemma 2.5.1 Let x ∈ M(F) ∩ G reg (F). Then, for all d > 0 there exists d  > 0 such that the integral   Ξ G (g −1 xg)σ (g −1 xg)−d σG x \G (x)d dg G x (F)\G(F)

converges. G (x, f ) for simplicity. If Once again if Q = G, we will set Φ M (x, f ) := Φ M M = G (so that necessarily Q = G), ΦG (x, f ) reduces to the usual orbital integral.

2.6 Strongly cuspidal functions Let ω be a continuous unitary character of A G (F). Following [46], we say that a function f ∈ Cω (G(F)) is strongly cuspidal if for every proper parabolic

123

R. Beuzart-Plessis

subgroup P = MU of G we have  f (mu)du = 0, U (F)

∀m ∈ M(F)

(the integral is absolutely convergent by [45, Proposition II.4.5]). By a standard change of variable, f is strongly cuspidal if and only if for every proper parabolic subgroup P = MU and for all m ∈ M(F) ∩ G reg (F) we have  U (F)

f (u −1 mu)du = 0

We will denote by Cω,scusp (G(F)) the subspace of strongly cuspidal functions in Cω (G(F)) and we will set Sω,scusp (G(F)) := Sω (G(F)) ∩ Cω,scusp (G(F)). Let K be a maximal special compact subgroup of G(F). For x ∈ G reg (F) set M(x) := Cent G (A G x ) (it is the smallest Levi subgroup containing x). Then, by [46, Lemme 5.2], for all f ∈ Cω,scusp (G(F)), all Levi subgroups M, all Q Q ∈ F (M) and all x ∈ M(F) ∩ G reg (F) we have Φ M (x, f ) = 0 unless Q = G and M = M(x). For all x ∈ G reg (F) we set G

Θ f (x) := (−1)a M(x) Φ M(x) (x, f ) Then the function Θ f is independent of the choice of K and invariant by conjugation [46, Lemme 5.2 and Lemme 5.3]. Also by [46, Corollaire 5.9], the function (D G )1/2 Θ f is locally bounded on G(F). We say that a function f ∈ Cω (G(F)) is a cusp form if it satisfies one of the following equivalent conditions (see [45, Théorème VIII.4.2 and Lemme VIII.2.1] for the equivalence between these two conditions): – For every proper parabolic subgroup P = MU and all x ∈ G(F) we have  f (xu)du = 0; U (F)

– f is a sum of matrix coefficients of representations in Irr ω,sqr (G). We will denote by 0 Cω (G(F)) the space of cusp forms. Let f ∈ 0 Cω (G(F)) and set f π (g) := Trace(π ∨ (g −1 )π ∨ ( f )) for all π ∈ Irr ω,sqr (G(F)) and all g ∈ G(F). Then, f π belongs to 0 Cω (G(F)) for all π ∈ Irr ω,sqr (G(F)) [24, Theorem 29] and we have an equality f =

 π ∈Irr ω,sqr (G)

123

d(π ) f π

(2.6.1)

On distinguished square-integrable representations

(This is a special case of Harish-Chandra–Plancherel formula, see [45, Theorem VIII.4.2]). Let 0 Sω (G(F)) := Sω (G(F)) ∩ 0 Cω (G(F)) be the space of compactly supported cusp forms. Similar to the characterization of 0 Cω (G(F)), a function f ∈ Sω (G(F)) belongs to 0 Sω (G(F)) if and only if it satisfies one the following equivalent conditions: – For every proper parabolic subgroup P = MU and all x ∈ G(F) we have  f (xu)du = 0; U (F)

– f is a sum of matrix coefficients of representations in Irr ω,cusp (G). Moreover, for f ∈ 0 Sω (G(F)), we have f π ∈ Irr ω,cusp (G) and a spectral decomposition 

f =

d(π ) f π .

0 S (G(F)) ω

for all π ∈

(2.6.2)

π ∈Irr ω,cusp (G)

Finally, we will need the following proposition. Proposition 2.6.1 Let π ∈ Irr sqr (G) and let f be a matrix coefficient of π . Then, we have Θ f (x) =

1 f (1)Θπ (x) d(π )

for all x ∈ G reg (F). Proof Unfortunately, the author has been unable to find a suitable reference for this probably well-known statement (however see [14, Proposition 5] for the case where x is elliptic and [2] for the case where π is supercuspidal). Let us say that it follows from a combination of Arthur’s noninvariant local trace formula [5, Proposition 4.1] applied to the case where one of the test functions is our f and of Schur orthogonality relations. Note that Arthur’s local trace formula was initially only proved for compactly supported test functions, but see [6, Corollary 5.3] for the extension to Harish-Chandra Schwartz functions.   2.7 Tempered pairs Let H be a unimodular algebraic subgroup of G (e.g. a reductive subgroup). We say that the pair (G, H ) is tempered if there exists d > 0 such that the integral

123

R. Beuzart-Plessis

 H (F)

Ξ G (h)σ (h)−d dh

is convergent. This notion already appeared in [22] under the name of strongly discrete pairs. Following the referee suggestion we have decided to call these pairs tempered instead so that it is more in accordance with the notion of strongly tempered pairs introduced by Sakellaridis and Venkatesh [44, §6] (since the latter implies the former but not conversely). This terminology is also justified by the fact that (G, H ) is tempered if and only if the Haar measure on H (F) defines a tempered distribution on G(F) i.e. it extends to a continuous linear form on C (G(F)). Moreover, by a result of Benoist and Kobayashi [8], when H is reductive and in the case where F = R (which is not properly speaking included in this paper) a pair (G, H ) is tempered if and only if L 2 (H (F)\G(F)) is tempered as a unitary representation of G(F). Although the author has not checked all the details, the proof of Benoist and Kobayashi seems to extend without difficulties to the p-adic case. However, we propose here a quick proof of one of the implications (but we won’t use it in this paper). Proposition 2.7.1 Assume that the pair (G, H ) is tempered. Then, the unitary representation of G(F) on L 2 (H (F)\G(F)) given by right translation is tempered i.e. the Plancherel measure of L 2 (H (F)\G(F)) is supported on tempered representations. Proof We will use the following criterion for temperedness due to Cowling et al. [17]: Let (Π, H) be a unitary representation of G(F). Then (Π, H) is tempered if and only if there exists d > 0 and a dense subspace V ⊂ H such that for all u, v ∈ V we have |(Π(g)u, v)|  Ξ G (g)σ (g)d

(2.7.1)

for all g ∈ G(F) where (·, ·) denotes the scalar product on H. We will check that this criterion is satisfied for V = Cc∞ (H (F)\G(F)) ⊂ H = L 2 (H (F)\G(F)). Let ϕ1 , ϕ2 ∈ Cc∞ (H (F)\G(F)) and choose f 1 , f 2 ∈ Cc∞ (G(F)) such that  ϕi (x) =

H (F)

f i (hx)dh

for i = 1, 2 and all x ∈ H (F)\G(F). Then, denoting by R(g) the operator of right translation by g and by (·, ·) the L 2 -inner product on L 2 (H (F)\G(F)), we have

123

On distinguished square-integrable representations

 (R(g)ϕ1 , ϕ2 ) =

ϕ1 (xg)ϕ2 (x)d x   f 1 (h 1 xg) f 2 (h 2 x)dh 2 dh 1 d x = H (F)\G(F) H (F)×H (F)   f 1 (h 1 xg) f 2 (h 2 h 1 x)dh 2 dh 1 d x = H (F)\G(F) H (F)×H (F)   f 1 (γ g) f 2 (hγ )dhdγ = H (F)\G(F)

H (F)

G(F)

for all g ∈ G(F). Let d > 0 that we will assume sufficiently large in what follows. As f 1 and f 2 are compactly supported, there obviously exist C1 > 0 and C2 > 0 such that | f 1 (γ )|  C1 Ξ G (γ )σ (γ )−2d and | f 2 (γ )|  C2 Ξ G (γ )σ (γ )−d for all γ ∈ G(F). It follows that for all g ∈ G(F) we have 



|(R(g)ϕ1 , ϕ2 )|  C1 C2 G(F)

H (F)

Ξ G (γ g)Ξ G (hγ )σ (hγ )−d dhσ (γ g)−2d dγ

Since σ (γ1 γ2 )−1  σ (γ1 )−1 σ (γ2 ) for all γ1 , γ2 ∈ G(F), this last expression is essentially bounded by 



σ (g)2d G(F)

H (F)

Ξ G (γ g)Ξ G (hγ )σ (h)−d dhσ (γ )−d dγ

for all g ∈ G(F). Let K be the special maximal compact subgroup used to define Ξ G . Then Ξ G is invariant both on the left and on the right by K and since σ (k1 γ k2 )−1  σ (γ )−1 for all γ ∈ G(F) and k1 , k2 ∈ K , we see that the last integral above is essentially bounded by 



σ (g)2d G(F)

 H (F)

K ×K

Ξ G (γ k1 g)Ξ G (hk2 γ )dk1 dk2 σ (h)−d dhσ (γ )−d dγ

for all g ∈ G(F). By the ‘doubling principle’ [45, Lemme II.1.3], it follows that  G 2d |(R(g)ϕ1 , ϕ2 )|  Ξ (g)σ (g) Ξ G (γ )2 σ (γ )−d dγ G(F)  × Ξ G (h)σ (h)−d dh H (F)

123

R. Beuzart-Plessis

for all g ∈ G(F). By [45, Lemme II.1.5] and the assumption that (G, H ) is tempered, for d sufficiently large the two integrals above are absolutely convergent. Thus, the criterion of Cowling–Haagerup–Howe is indeed satisfied for V = Cc∞ (H (F)\G(F)) and consequently L 2 (H (F)\G(F)) is tempered.   Finally, we include the following easy lemma which gives an alternative characterization of tempered pairs because it is how the tempered condition will be used in this paper. H = (A ∩ H )0 . The pair (G, H ) is tempered if and only Lemma 2.7.1 Set A G G if there exists d > 0 such that the integral  Ξ G (h)σ (h)−d dh H (F)\H (F) AG

converges. Proof As Ξ G is A G (F) invariant, it clearly suffices to show: For d > 0 sufficiently large, we have  σ (h)−3d  σ (ah)−3d da  σ (h)−d H (F) AG

(2.7.2)

for all h ∈ H (F). For this, we need first to observe that σ (h) ∼

inf

H (F) a∈A G

σ (ah)

(2.7.3)

H \H is a closed subgroup of A \G, this is for all h ∈ H (F). Indeed, as A G G H \H has the norm descent equivalent to the fact that the projection H  A G property and this can be easily deduce from the existence of an algebraic H × H  → H is subgroup H  of H such that the multiplication morphism A G  surjective and finite (so that in particular H (F) has a finite number of orbits H (F)\H (F)). in A G By the inequalities σ (h)  σ (ah) and σ (a)  σ (ah)σ (h) for all a ∈ H A G (F) and all h ∈ H (F), for any d > 0 we have



−3d

H (F) AG

σ (ah)

−2d

da  σ (h)

 H (F) AG

−2d

 σ (h) σ (h) d

123

σ (ah)−d da

 H (F) AG

σ (a)−d da

On distinguished square-integrable representations

for all h ∈ H (F). For d sufficiently large, the last integral above is absolutely convergent. Moreover, as the left hand side of the above inequality is clearly H (F), by 2.7.3, for d sufficiently large we invariant by h → ah for any a ∈ A G get 



−3d

H (F) AG

σ (ah)

da 

d inf

H (F) a∈A G

σ (h)−2d  σ (h)−d

σ (ah)

for all h ∈ H (F) and this shows one half of 2.7.2. On the other hand, by the H (F) and h ∈ H (F), for any inequality σ (ah)  σ (a)σ (h) for all a ∈ A G d > 0 we have   −3d −3d σ (a) da  σ (ah)−3d da σ (h) H (F) AG

H (F) AG

for all h ∈ H (F). Once again, for d sufficiently large the two integrals above are absolutely convergent and, as the right hand side of the inequality is invariH (F), by 2.7.3 for d sufficiently large we ant by h → ah for any a ∈ A G get  −3d

σ (h)



−3d inf

H (F) a∈A G

σ (ah)

 

H (F) AG

σ (ah)−3d da

for all h ∈ H (F) and this proves the second half of 2.7.2.

 

2.8 Symmetric varieties 2.8.1 Basic definition, θ -split subgroups Let H be an algebraic subgroup of G. Recall that H is said to be symmetric if there exists an involutive automorphism θ of G (defined over F) such that (G θ )0 ⊂ H ⊂ G θ where G θ denotes the subgroup of θ -fixed elements. If this is the case, we say that H and θ are associated. The involution θ is not, in general, determined by H but by [26, Proposition 1.2], its restriction to the derived subgroup of G is. From now on and until the end of Sect. 2.8.2 we fix a symmetric subgroup H of G and we will denote by θ an associated involutive automorphism. Let T ⊂ G be a subtorus. We say that T is θ -split if θ (t) = t −1 for all t ∈ T and we say that it is (θ, F)-split if it is θ -split as well as split as a torus over F.

123

R. Beuzart-Plessis

For every F-split subtorus A ⊂ G we will denote by Aθ the maximal (θ, F)split subtorus of A. A parabolic subgroup P ⊂ G is said to be θ -split if θ (P) is a parabolic subgroup opposite to P. If this is the case, H P is open, for the Zariski topology, in G (this is because h + p = g) and similarly H (F)P(F) is open, for the analytic topology, in G(F). If P is a θ -split parabolic subgroup, we will say that the Levi component M := P ∩ θ (P) of P is a θ -split Levi subgroup. Note that this terminology can be slightly confusing since a torus can be a θ -split Levi without being θ -split as a torus (e.g.

for G =

G L(2),  1 1 T the standard maximal torus and θ given by θ (g) = g ). 1 1 Nevertheless, the author believe that no confusion should arise in this paper as the context will clarify which notion is being used. Actually, a Levi subgroup M ⊂ G is θ -split (i.e. it is the θ -split Levi component of a θ -split parabolic) if and only if M is the centralizer of a (θ, F)-split subtorus if and only if M is the centralizer of A M,θ . We will adapt Arthur’s notation to θ -split Levi and parabolic subgroups as follows: if M is a θ -split Levi subgroup we will denote by P θ (M), resp. F θ (M), resp. Lθ (M) the set of all θ -split parabolic subgroups with Levi component M, resp. containing M, resp. the set of all θ -split Levi subgroups containing M. Let M ⊂ G be a θ -split Levi subgroup. We set A M,θ := X ∗ (A M,θ ) ⊗ R

and a M,θ := dim A M,θ . Note that we have A M = A M,θ ⊕ AθM

where as before a θ superscript indicates the subset of θ -fixed points. This decomposition is compatible with the decompositions A M = A LM ⊕ A L

for all L ∈ Lθ (M). Hence, we also have A M,θ = A LM,θ ⊕ A L ,θ

for all L ∈ Lθ (M), where we have set A LM,θ := A M,θ ∩ A LM . Also, L we let a M,θ := dim A LM,θ = a M,θ − a L ,θ . We define an homomorphism HM,θ : M(F) → A M,θ as the composition of the homomorphism HM with the projection A M  A M,θ . For all P ∈ P θ (M), the roots R(A M,θ , U P ) of A M,θ in the unipotent radical U P of P can be considered as elements of the dual space A∗M,θ of A M,θ . There is a unique subset Δ P,θ ⊂ R(A M,θ , U P ) such that every element of (A M,θ , U P ) is in an unique way a nonnegative integral

123

On distinguished square-integrable representations

linear combination of elements of Δ P,θ . The set Δ P,θ is the image of Δ P by ∗ the natural projection A∗M  A∗M,θ and it forms a basis of (AG M,θ ) . We call it the set of simple roots of A M,θ in P. Define   A+ P,θ := X ∈ A M ; α, X  > 0 ∀α ∈ Δ P,θ Then, we have the decomposition A M,θ =



A+ Q,θ

Q∈F θ (M)

More precisely the set R(A M,θ , G) of roots of A M,θ in G divides A M,θ into θ certain facets which are exactly the cones A+ Q,θ where Q ∈ F (M). In a similar way, the subspaces supporting the facets of this decomposition are precisely the subspaces of the form A L ,θ , L ∈ Lθ (M), whereas the chambers θ (i.e. the open facets) are precisely the cones A+ P,θ for P ∈ P (M). Fixing a maximal special compact subgroup K of G(F), for all P ∈ P θ (M), we define a map H P,θ : G(F) → A M,θ as the composition of H P with the projection A M  A M,θ i.e. we have H P,θ (muk) = HM,θ (m) for all m ∈ M(F), u ∈ U P (F) and k ∈ K . Let A0 be a maximal (θ, F)-split subtorus and let M0 be its centralizer in G. For simplicity we set A0 := A M0 ,θ . It is known that the set of roots R(A0 , G)  ∗ of A0 in G forms a root system in the dual space A0G to A0G (Proposition 5.9 of [26]). The Weyl group associated to this root system is naturally isomorphic to W (G, A0 ) := Norm G(F) (A0 )/M0 (F) and is called the little Weyl group (associated to A0 ) (again Proposition 5.9 of [26]). Two maximal (θ, F)-split subtori are not necessarily H (F)-conjugate (e.g. for G = GL n and H = O(n)) but they are always G(F)-conjugate.1 Let M be a θ -split Levi subgroup and let α ∈ R(A M,θ , G). Then we define a ‘coroot’ α ∨ ∈ AG M,θ as follows. First assume that α is a reduced root (i.e. α ∈ / R(A , G)). Let Mα be the unique θ -split Levi containing M such that M,θ 2 A Mα ,θ = Ker(α). Let Q α be the unique θ -split parabolic subgroup of Mα 1 Indeed if A and A are two maximal (θ, F)-split tori, M := Cent (A ),M  := 0 0 G 0 0 0 Cent G (A0 ), P0 ∈ P θ (M0 ) and P0 ∈ P θ (M0 ) then by [26, Proposition 4.9], P0 and P0 are conjugated by an element of g ∈ G(F) ∩ H P0 and it suffices to show that we can take g in G(F) ∩ H M0 but this follows from the fact that H ∩ P0 = H ∩ M0 .

123

R. Beuzart-Plessis

with θ -split Levi M such that Δ Q α = {α}. Let P0Mα be a minimal θ -split parabolic subgroup of Mα contained in Q α and set M0 := P0Mα ∩ θ (P0Mα ), A0 := A M0 ,θ . Let Δ0Mα be the set of simple roots of A0 in P0Mα . Then there is an unique simple root β ∈ Δ0Mα whose projection to A∗M,θ equals α. Let β ∨ ∈ A0 be the corresponding coroot. Then we define α ∨ as the image of β ∨ by the projection A0  A M,θ . We easily check that this construction does not depend on the choice of P0Mα since for another choice P0 ,Mα with M0 := P0 ,Mα ∩ θ (P0 ,Mα ) and A0 := A M0 ,θ there exists m ∈ M(F) with m A0 m −1 = A0 and m P0 ,Mα m −1 = P0Mα . If α is nonreduced, there exists α∨

α0 ∈ R( M,θ , G) such that α = 2α0 and we simply set α ∨ = 20 . α ∨ ∈ A M the corresponding Let  α ∈ R(A M , G) be a root extending α and  ∨ coroot. Then, in general the projection of  α to A M,θ does not coincide with ∨ α as defined above but, however, the two are always positively proportional. Finally, we remark that when M is a minimal θ -split Levi subgroup, so that R(A M,θ , G) is a root system, then for all α ∈ R(A M,θ , G), α ∨ coincides with the usual coroot defined using this root system. Let P be a θ -split parabolic subgroup. We set A P,θ := A M,θ and a P,θ = a M,θ where M := P ∩ θ (P) and we let Δ∨P,θ ⊆ AG M,θ be the set of simple G ∗ ∗  coroots corresponding to Δ P,θ ⊆ (A M,θ ) and Δ P,θ ⊆ (AG M,θ ) be the basis ∨ dual to Δ P,θ . More generally, let Q ⊃ P be another θ -split Levi subgroup. Q

Q

L where L := Q ∩ θ (Q) and we let We set A P,θ := A LM,θ and a P,θ := a M,θ

Δ P,θ ⊆ (A P,θ )∗ be the set of simple roots of A M,θ in P ∩ L, (Δ P,θ )∨ ⊆ A P,θ Q ⊆ (A Q )∗ be the basis be the corresponding set of simple coroots and Δ Q

Q

Q

P,θ

dual to (Δ P,θ )∨ . We have decompositions

Q

P,θ

Q

Q

A P,θ = A P,θ ⊕ A Q,θ ,

A∗P,θ = (A P,θ )∗ ⊕ A∗Q,θ Q

for which Δ P,θ ⊆ Δ P,θ , (Δ P,θ )∨ ⊆ Δ∨P,θ and moreover Δ Q,θ (resp. Δ∨Q,θ ) is Q

Q

the image of Δ P,θ − Δ P,θ [resp. Δ∨P,θ − (Δ P,θ )∨ ] by the projection A∗P,θ  A∗Q,θ (resp. A P,θ  A Q,θ ). We define the following functions: Q

Q

Q

– τ P,θ : characteristic function of the set of X ∈ A M,θ such that α, X  > 0 Q

for all α ∈ Δ P,θ ; Q

– τ P,θ : characteristic function of the set of X ∈ A M,θ such that α , X  > 0 Q ; for all α ∈ Δ P,θ

–

Q δ M,θ :

123

characteristic function of the subset A L ,θ of A M,θ .

On distinguished square-integrable representations Q

We also define a function Γ P,θ on A M,θ × A M,θ , whose utility will be revealed in the next section, by 

Q

Γ P,θ (H, X ) :=

R (−1)a R,θ −a Q,θ τ P,θ (H ) τ R,θ (H − X ) Q

R∈F θ (M);P⊆R⊆Q

Let M be a θ -split Levi subgroup. Then, for all P ∈ P θ (M) we set   ∗ ∨ ∨ ∨ A+,∗ P,θ := λ ∈ A P,θ ; α , λ > 0 ∀α ∈ Δ P,θ As for A M,θ , the set of coroots R(A M,θ , G)∨ divides A∗M,θ into facets which θ are exactly the cones A+,∗ Q,θ for Q ∈ F (M) and the chambers for this decom+,∗ position are the A P,θ , P ∈ P θ (M). As usual, we say that two parabolics P, P  ∈ P θ (M) are adjacent if the intersection of the closure of their corresponding chambers contains a facet of codimension one. If this is the case, the hyperplane generated by this intersection is called the wall separating the two chambers. 2.8.2 (G, M, θ )-families and orthogonal sets As we have recalled in the previous section, the combinatorics of θ -split Levi and parabolic subgroups is entirely governed, as is the case for classical Levi and parabolic subgroups, by a root system. As a consequence, for M a θ -split Levi subgroup of G the classical theory of (G, M)-families due to Arthur extends without difficulty to a theory of (G, M, θ )-families indexed by θ split parabolics that we now introduce. By definition, a (G, M, θ )-family is a family (ϕ P,θ ) P∈P θ (M) of C ∞ functions on i A∗M,θ such that for any two adjacent parabolic subgroups P, P  ∈ P θ (M), the functions ϕ P,θ and ϕ P  ,θ coincide +,∗ on the wall separating the chambers i A+,∗ P,θ and i A P  ,θ . To a (G, M, θ )-family (ϕ P,θ ) P∈P θ (M) we can associate a scalar ϕ M,θ as follows: the function ϕ M,θ (λ) :=



ϕ P,θ (λ)ε P,θ (λ),

λ ∈ i A∗M,θ

P∈P θ (M)

where we have set   ∨ /Z[Δ ] ε P,θ (λ) := meas AG P,θ M,θ



λ, α ∨ −1

α ∨ ∈Δ∨P,θ

is C ∞ and we define ϕ M,θ := ϕ M,θ (0). Note that we need a Haar measure θ on AG M,θ for the definition of the functions ε P,θ (P ∈ P (M)) to make sense.

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R. Beuzart-Plessis

We fix one as follows. Let A M,θ be the image of A M,θ in G := G/A G and G let AM,θ be the inverse image of A M,θ in G. Let AG M,θ,F ⊆ A M,θ denote the G . It is a lattice of AG and we choose our measure image of AM,θ (F) by HM,θ M,θ G / A is of measure one. so that the quotient AG M,θ M,θ,F We will actually only need (G, M, θ )-families of a very particular shape obtained as follows. We say that a family of points Y M,θ = (Y P,θ ) P∈P θ (M) in A M,θ is a (G, M, θ )-orthogonal set if for all adjacent P, P  ∈ P θ (M) we have Y P,θ − Y P  ,θ = r P,P  α ∨ where r P,P  ∈ R and {α ∨ } = Δ∨P,θ ∩ −Δ∨P  ,θ . We say that the family is a positive (G, M, θ )-orthogonal set if it is a (G, M, θ )-orthogonal set and moreover r P,P   0 for all adjacent P, P  ∈ P θ (M). To a (G, M, θ )orthogonal set Y M,θ = (Y P,θ ) P∈P θ (M) we associate the (G, M, θ )-family (ϕ P,θ (·, Y M,θ )) P∈P θ (M) defined by ϕ P,θ (λ, Y M,θ ) := eλ,Y P,θ  ,

λ ∈ i A∗M,θ

and we let v M,θ (Y M,θ ) := ϕ M,θ (0, Y M,θ ) be the scalar associated to this (G, M, θ )-family. If Y M,θ is a positive (G, M, θ )-orthogonal set then v M,θ (Y M,θ ) is just the volume of the convex hull of the elements in the family Y M,θ (with respect to the fixed Haar measure on AG M,θ ). For any (G, M, θ )orthogonal set Y M,θ = (Y P,θ ) P∈P θ (M) and all Q ∈ F θ (M) we define Y Q,θ to be the projection of Y P,θ to A Q,θ for any P ∈ P θ (M) with P ⊂ Q (the result is independent of the choice of P) and more generally for any Q, R ∈ F θ (M) R be the projection of Y R θ with Q ⊂ R we let Y Q,θ P,θ to A Q,θ for any P ∈ P (M) with P ⊂ Q. Then, for any L ∈ Lθ (M) the family Y L ,θ := (Y Q,θ ) Q∈P θ (L) forms a (G, L , θ )-orthogonal set. Fixing a maximal special compact subgroup K of G(F) to define maps H P,θ (P ∈ P θ (M)), for all g ∈ G(F) the family Y M,θ (g) := (−H P,θ (g)) P∈P θ (M) is a positive (G, M, θ )-orthogonal set. Indeed, the family (−H P (g)) P∈P (M) is a positive (G, M)-orthogonal set in the classical sense of Arthur (see [2, §2]) and thus for all P = MU P , P  = MU P  ∈ P θ (M) we have −H P (g) + H P  (g) ∈



R+ α ∨ .

α∈R(A M ,U P )∩−R(A M ,U P  )

As the projection of R(A M , U P ) [resp. R(A M , U P  )] to A∗M,θ is R(A M,θ , U P ) [resp. R(A M,θ , U P  )] and for all α ∈ R(A M , G) the projection of the coroot α ∨ to A M,θ ∗ is positively proportional to α ∨ , where α ∈ R(A M,θ , G) denotes

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On distinguished square-integrable representations

the projection of α to A∗M,θ , it follows that 

−H P,θ (g) + H P  ,θ (g) ∈

R+ α ∨

α∈R(A M,θ ,U P )∩−R(A M,θ ,U P  )

for all P, P  ∈ P θ (M) i.e. Y M,θ (g) is a positive (G, M, θ )-orthogonal set. We define v M,θ (g) := v M,θ (Y M,θ (g)) There is another easier way to obtain (G, M, θ )-orthogonal sets. It is as follows. Let M0 ⊂ M be a minimal θ -split Levi subgroup with little Weyl group W0 . Fix P0 ∈ P θ (M0 ). Then, for all X ∈ A0 := A M0 ,θ we define a (G, M0 , θ )orthogonal set Y [X ]0 := (Y [X ] P0 ,θ ) P  ∈P θ (M0 ) by setting Y [X ] P0 ,θ := w P0 X 0 for all P0 ∈ P θ (M0 ) where w P0 ∈ W0 is the unique element such that w P0 P0 = P0 . By the general construction explained above this also yields a (G, M, θ )orthogonal set Y [X ] M,θ = (Y [X ] P,θ ) P∈P θ (M) . Let Y M,θ = (Y P,θ ) P∈P θ (M) be a (G, M, θ )-orthogonal set. For Q ∈ F θ (M) Q we define a function Γ M,θ (·, Y M,θ ) on A M,θ by 

Q

Γ M,θ (H, Y M,θ ) :=

Q

R δ M,θ (H )Γ R,θ (H, Y R,θ )

R∈F θ (M);R⊂Q Q

R and Γ where the functions δ M,θ R,θ have been defined in the previous section. Let L = Q ∩ θ (Q). Fixing a norm |·| on A LM,θ , we have the following basic property concerning the support of this function (see [38, Corollaire 1.8.5]): There exists c > 0 independent of Y M,θ such that for all H ∈ A M,θ with Q Γ M,θ (H, Y M,θ )  = 0 we have

|H Q |  c

sup P∈P θ (M);P⊂Q

Q

|Y P,θ |

(2.8.1)

where H Q denotes the projection of H to A LM,θ . Q

Moreover, if the (G, M, θ )-orthogonal set Y M,θ is positive then Γ M,θ (·, Y M,θ ) is just the characteristic function of the set of H ∈ A M,θ such that H Q belongs Q to the convex hull of (Y P,θ ) P∈P θ (M);P⊂Q ([38, Proposition 1.8.7]). Without assuming the positivity of our (G, M, θ )-orthogonal set, we have the identity [38, Lemme 1.8.4(3)]

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R. Beuzart-Plessis



Q

G Γ M,θ (H, Y M,θ )τ Q,θ (H − Y Q,θ ) = 1

(2.8.2)

Q∈F θ (M)

for all H ∈ A M,θ . Let R be a free Z-module of finite type. Recall that a exponential-polynomial on R is a function on R of the following form  χ (Y ) pχ (Y ) f (Y ) =  χ ∈R

 denotes the group of complex (not necessarily unitary) characters of where R , pχ is a ‘complex polynomial’ function on R, i.e. an R and for all χ ∈ R . If f element of Sym((C ⊗ R)∗ ),which is zero for all but finitely many χ ∈ R is an exponential-polynomial on R then a decomposition as above is unique,  such that pχ  = 0 is called the set of exponents of the set of characters χ ∈ R f and p1 (corresponding to χ = 1 the trivial character) is called the purely polynomial part of f . Finally, we define the degree of f as the maximum, , of the degree of the polynomials pχ . We record the following over all χ ∈ R lemma whose proof is elementary: Lemma 2.8.1 Let R be a free Z-module of finite type, let f be a exponentialpolynomial on R and let C ⊂ R ⊗ R be an open cone (with a vertex possibly different from the origin). Then,if the limit lim

Y ∈R∩C |Y |→∞

f (Y )

exists it equals the constant term of the purely polynomial part of f . Let M0 ⊂ M be a minimal θ -split Levi subgroup, A0 := A M0 ,θ and fix P0 ∈ P θ (M0 ). For all X ∈ A0 we dispose of the (G, M, θ )-orthogonal set Y [X ] M,θ = (Y [X ] P,θ ) P∈P θ (M) defined above. We let Y M,θ + Y [X ] M,θ := (Y P,θ + Y [X ] M,θ ) P∈P θ (M) be the sum of the two (G, M, θ )-orthogonal sets Y M,θ and Y [X ] M,θ . Obviously, it is also a (G, M, θ )-orthogonal set. We let  v M,θ (Y M,θ + Y [X ] M,θ )  G Γ M,θ (HM,θ (a), Y M,θ + Y [X ] M,θ )da := A G (F)\AM,θ (F)

where we recall that AM,θ is the subtorus generated by A G and A M,θ . Let A0,F denote the image of A M0 ,θ (F) by HM0 ,θ . It is a lattice in A0 and we have the following lemma (combine [41, Lemme 1.7(ii)] with equalities 1.5(2) and 1.3(7) of loc.cit.):

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On distinguished square-integrable representations

Lemma 2.8.2 For every lattice R ⊂ A0,θ,F ⊗ Q the function X ∈ R →  v M,θ (Y M,θ + Y [X ] M,θ ) is an exponential-polynomial whose degree and exponents belong to finite sets which are independent of Y M,θ . Moreover, if we denote by  v M,θ,0 (Y M,θ , R) the constant coefficient of the purely polynomial part of this exponential-polynomial there exists c > 0 depending only on R such that for all k  1 we have  a G



 M,θ



1 −1



sup |Y P,θ | .

v M,θ,0 Y M,θ , k R − v M,θ (Y M,θ )  ck P∈P θ (M) Let now Y M = (Y P ) P∈P (M) be a usual (G, M)-orthogonal set. This induces a (G, M, θ )-orthogonal set Y M,θ := (Y P,θ ) P∈P θ (M) where, for all P ∈ P θ (M), we denote by Y P,θ the projection of Y P to A M,θ . The subspace A M,θ + AG of A M being special in the sense of [3, §7],2 we have a descent formula (Proposition 7.1 of loc.cit.) v M,θ (Y M,θ ) =



Q

G d M,θ (L)v M (Y M )

(2.8.3)

L∈L(M)

where for all L ∈ L(M), Q is a parabolic with Levi component L which G (L) is a coefficient depends on the choice of a generic point ξ ∈ A M and d M,θ G,θ G,θ L which is nonzero only if AG M = A M ⊕ A M . Moreover if A M = 0 then we G (G) = 1. Let K be a special maximal compact subgroup of G(F) have d M,θ that we use to define the maps H P for P ∈ P (M) and H P,θ for P ∈ P θ (M). Then, the formula 2.8.3 applied to the particular case where Y P = H P (g) for all P ∈ P (M) and for some g ∈ G(F) yields

v M,θ (g) =



Q

G d M,θ (L)v M (g).

(2.8.4)

L∈L(M)

2.9 Estimates In this section we collect some estimates that we will need in the core of the paper. We start with four lemmas concerning maximal tori and integrals over regular orbits in G.

2 Indeed with the notations of loc.cit. we need to check that for every root β ∈ R(A M,θ , G)  the sum α∈Σ(β) m α α is trivial on AθM but this is trivial since for all α ∈ Σ(β) we have ι(α) := −θ (α) ∈ Σ(β) and m ι(α) = m α .

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R. Beuzart-Plessis

Lemma 2.9.1 Let T ⊂ G be a maximal torus . Then, we have σ (t)  σ (g −1 tg) for all t ∈ T and all g ∈ G. Proof Let W := W (G F , TF ) be the absolute Weyl group of T and set B := G//G − Ad (i.e. the GIT quotient of G acting on itself by the adjoint action). Let p: G → B be the natural projection. By Chevalley theorem, the inclusion T → G induces an isomorphism T //W  B and thus the restriction of p to T is a finite morphism. Hence, we have σ (t) ∼ σB ( p(t)) for all t ∈ T and it follows that σ (t) ∼ σB ( p(t)) = σB ( p(g −1 tg))  σ (g −1 tg) for all t ∈ T and all g ∈ G.

 

Lemma 2.9.2 (Harish-Chandra, Clozel) Let T ⊂ G be a maximal torus. Then, for all d > 0 there exists d  > 0 such that   D G (t)1/2 Ξ G (g −1 tg)σ (g −1 tg)−d dg  σ (t)−d T (F)\G(F)

for all t ∈ Treg (F). Proof By Corollary 2 of [14] there exists d0 > 0 such that  G 1/2 sup D (t) Ξ G (g −1 tg)σ (g −1 tg)−d0 dg < ∞ T (F)\G(F)

t∈Treg (F)

Thus by Lemma 2.9.1, for all d > 0 we have  D G (t)1/2 Ξ G (g −1 tg)σ (g −1 tg)−d0 −d dg  σ (t)−d T (F)\G(F)

for all t ∈ Treg (F).

 

Lemma 2.9.3 Let T ⊂ G be a subtorus such that Treg := T ∩G reg is nonempty (i.e. T contains nonsingular elements). Then, for all k > 0, there exists d > 0 such that the integral  log(2 + D G (t)−1 )k σ (t)−d dt T (F)

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On distinguished square-integrable representations

converges. Proof We denote by X ∗ (T ) the group of regular characters of T defined over F F. There exists a multiset Σ of nontrivial elements in X ∗ (T ) such that F

D G (t) =



|α(t) − 1|

α∈Σ

for all t ∈ Treg (F) where we have denoted by |·| the unique extension of the absolute value over F to F. We have    log(2 + D G (t)−1 )  log 2 + |α(t) − 1|−1 α∈Σ

Thus, by Cauchy–Schwartz, it suffices to prove the following claim: For all α ∈ X ∗ (T ) − {1} and all k > 0 there exists d > 0 such that the F integral   k log 2 + |α(t) − 1|−1 σ (t)−d dt (2.9.1) T (F)

converges. Let α ∈ X ∗ (T ) − {1} and let Γα ⊂ Γ F be the stabilizer of α for the natural F Galois action. Write Γα = Gal(F/Fα ) where Fα /F is a finite extension. By the universal property of restriction of scalars, α induces a morphism  α: T → α ) the kernel of  α , for all k > 0 and all d > 0 R Fα /F Gm . Denoting by Ker( we have  k  log 2 + |α(t) − 1|−1 σ (t)−k dt T (F)     −1 k × log 2 + |α(t) − 1| σ (tt  )−d dt  dt = T (F)/ Ker( α )(F)

Ker( α )(F)

As there exists a subtorus T  ⊂ T such that √ the multiplication map α ) → T is an isogeny (so that σ (t)σ (t  )  σ (tt  ) and T  × Ker(  α ) × T  ), we see that for all d σ (t) ∼ σT / Ker( α ) (t) for all (t, t ) ∈ Ker( sufficiently large (i.e. so that the integral below converges) we have  −d/2 σ (tt  )−d dt   σT / Ker( α ) (t) Ker( α )(F)

for all t ∈ T (F)/ Ker( α )(F). Since T (F)/ Ker( α )(F) is an open subset of α )) (F) we are thus reduced to the case where  α is an embedding. (T / Ker(

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R. Beuzart-Plessis

Define N (α) ∈ X ∗ (T ) by N (α) :=



σ (α)

σ ∈Γ F /Γα

We distinguish two cases. First, if N (α)  = 1 then we have an inequality     log 2 + |α(t) − 1|−1  log 2 + |N (α)(t) − 1|−1 for all t ∈ T (F) with N (α)(t)  = 1. Hence, up to replacing α by N (α) we may assume that α ∈ X ∗ (T ) in which case by the previous reduction we are left to prove 2.9.1 in the particular case where T = Gm and α = I d in which case it is easy to check. Assume now that N (α) = 1. Since  α is an embedding this implies that T is anisotropic and we just need to prove that for all k > 0 the function k  t ∈ T (F) → log 2 + |α(t) − 1|−1 is locally integrable. Using the exponential map we are reduced to proving a similar statement for vector spaces where we replace α(t) − 1 by a linear form which is easy to check directly.   Combining 2.2.2 with Lemmas 2.9.2 and 2.9.3 we get the following: Lemma 2.9.4 Let T ⊂ G be a subtorus such that Treg := T ∩G reg is nonempty and let T G be the centralizer of T in G (a maximal torus). Then, for all k > 0 there exists d > 0 such that the integral 

 D (t) G

T (F)

1/2 T G (F)\G(F)

Ξ G (g −1 tg)σ (g −1 tg)−d σT G \G (g)k dgdt

converges. The following lemma will be needed in the proof of the next proposition. As it might be of independent interest we present it separately. Lemma 2.9.5 Let G be an anisotropic group over F and Y an affine G-variety. Set Y  = Y /G for the GIT quotient (it is an affine algebraic variety over F) and denote by p: Y → Y  the natural projection. Then we have σY (y) ∼ σY  ( p(y)) for all y ∈ Y (F).

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On distinguished square-integrable representations

Proof First, we have σY  ( p(y))  σY (y) for all y ∈ Y since p is a morphism of algebraic varieties. Let f ∈ F[Y ], we need to show that log(2 + | f (y)|)  σY  ( p(y)) for all y ∈ Y (F). Let W be the G-submodule of F[Y ] generated by f and V be its dual. There is a natural morphism ϕ: Y → V and we have a commutative diagram Y  = Y /G

Y ϕ

V

ϕ

V  := V /G

By definition there is a function f V ∈ F[V ] such that f = f V ◦ ϕ and moreover σV  (y  )  σY  (y  ) for all y  ∈ Y  . Hence, we are reduced to the case where Y = V and we may assume that f is homogeneous. By Kempf’s extension of the stability criterion of Mumford over any perfect field [30, Corollary 5.1] and since G is anisotropic, for every v ∈ V (F) the G-orbit G.v ⊂ V is closed. It follows that there exist homogeneous polynomials P1 , . . . , PN ∈ F[V  ] = F[V ]G whose only common zero in V (F) is 0. We only need to show that for some R, C > 0 we have max(1, | f (v)|)  C max(1, |P1 (v)|, . . . , |PN (v)|) R

(2.9.2)

for all v ∈ V (F). Up to replacing f, P1 , . . . , PN by some powers, we may assume that they are all of the same degree. Then, for every 1  i  N , f /Pi is a rational function on the projective space P(V ) and the map [v] ∈ P(V )(F) → min (|( f /P1 )([v])|, . . . , |( f /PN )([v])|) ∈ R+ is continuous for the analytic topology hence bounded (as P(V )(F) is compact) and this proves that inequality 2.9.2 is true for R = 1 and some constant C.   Following [31, Definition 4.9], we say that a subgroup H ⊂ G is Fspherical if there exists a minimal parabolic subgroup P0 of G such that H P0 is open, in the Zariski topology, in G. For example symmetric subgroups (see Sect. 2.8.1) are F-spherical. Recall that in Sect. 2.2 we have defined a ‘norm descent property’ for regular maps between F-varieties. Proposition 2.9.1 Let H ⊂ G be an F-spherical subgroup. Then, the natural projection p: G → H \G has the norm descent property.

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R. Beuzart-Plessis

Proof Set X := H \G. By [36, Proposition 18.2 (1)], it suffices to show that X can be covered by Zariski open subsets over which the projection p has the norm descent property. Since G acts transitively on X it even suffices to construct only one such open subset (because its G-translates will have the same property). By the local structure theorem [31, Corollary 4.12], there exists a parabolic subgroup Q = LU of G such that – U = HQ is open in G; – H ∩ Q = H ∩ L and this subgroup contains the non-anisotropic factors of the derived subgroup of L. Obviously, to show that the restriction of p to U has the norm descent property it is sufficient to establish that L → H ∩ L\L has the norm descent property. We are thus reduced to the case where H contains all the non-anisotropic factors of the derived subgroup of G. Let G der denote the derived subgroup of G, G der,nc the product of the non-anisotropic factors of G der , G der,c the product of the anisotropic factors of G der and set G  = G der,c Z (G)0 , H  = H ∩ G  . Then, we have H = G der,nc H  and the multiplication map G der,nc × G  → G is an isogeny. It follows that there exists a finite set {γi ; i ∈ I } of elements of G(F) such that G(F) =



G der,nc (F)G  (F)γi

i∈I

and H (F)\G(F) =



H  (F)\G  (F)γi

i∈I

From these decompositions, we infer that we only need to prove the norm descent property for G  → H  \G  i.e. we may assume that G der,nc = 1. Let G c be the product of G der,c with the maximal anisotropic subtorus of Z (G)0 and consider the projection p  : X := H \G → X  := H G c \G We claim that σ X (x) ∼ σ X  ( p  (x))

(2.9.3)

for all x ∈ X (F). As H is reductive (since G(F) contains no unipotent element), X is affine and the claim follows from the Lemma 2.9.5. Now because of 2.9.3, we may replace X by X  i.e. we may assume that H contains G c . As the multiplication map G c × A G → G is an isogeny, by a

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On distinguished square-integrable representations

similar argument as before we are reduced to the case where G is a split torus for which the proposition is easy to establish directly.   3 Definition of a distribution for all symmetric pairs 3.1 The statement Let G be a connected reductive group over F, H be a symmetric subgroup of G and θ be the involution of G associated to H (see Sect. 2.8.1). Set H = (A ∩ H )0 , G := G/A , H := H/A , X := A (F)H (F)\G(F), AG G G H G X := H A G \G, σ X := σX and σ := σG . Note that X is an open subset of X(F). Let χ and ω be continuous unitary characters of H (F) and A G (F) respectively such that χ|A H (F) = ω|A H (F) . Then, for all f ∈ Sω (G(F)) we χ

G

G

define a function K f on X by χ

K f (x) :=

 H (F)\H (F) AG

f (x −1 hx)χ (h)−1 dh χ

If the pair (G, H ) is tempered then the expression defining K f makes sense for all f ∈ Cω (G(F)) (by Lemma 2.7.1). The goal of this chapter is to show that if f is strongly cuspidal then the expression J χ ( f ) :=



χ

X

K f (x)d x

is convergent. More precisely we will prove the following Theorem 3.1.1 For all f ∈ Sω,scusp (G(F)), the expression defining J χ ( f ) is absolutely convergent. Moreover, if the pair (G, H ) is tempered then the expression defining J χ ( f ) is also absolutely convergent for all f ∈ Cω,scusp (G(F)). 3.2 Some estimates Let A ⊂ G be a (θ, F)-split subtorus, set M := Cent G (A) and let Q = LU Q ∈ F θ (M) where L := Q ∩ θ (Q). Let Q = θ (Q) = LU Q be the opposite parabolic subgroup and set   A+ := a ∈ A; |α(a)|  1 ∀α ∈ R(A, U Q ) Q

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R. Beuzart-Plessis

and

  A+ (δ) := a ∈ A; |α(a)|  eδσ (a) ∀α ∈ R(A, U Q ) Q

for all δ > 0. Recall that if Y is an algebraic variety over F and M > 0 we denote by Y [< M] the subset of y ∈ Y (F) with σY (y) < M. We also recall that we have fixed a (classical) norm |·|g and that for any R > 0, B(0, R) denotes the closed ball of radius R centered at the origin for this norm (see Sect. 2.2). Lemma 3.2.1 (i) Let  > 0 and δ > 0. Then, we have   σ (a)  sup σ (g), σ (a −1 ga)   for all a ∈ A+ (δ) and all g ∈ G(F)\ Q(F)aU Q [< σ (a)]a −1 ; Q (ii) Let 0 < δ  < δ and c0 > 0. Then, if  > 0 is sufficiently small we have    aU Q [< σ (a)]a −1 ⊆ exp B(0, c0 e−δ σ (a) ) ∩ u Q (F) for all a ∈ A+ (δ). Q (iii) We have σ (h)  σ (a −1 ha) and σ (h) + σ (a)  σ (ha) for all a ∈ A and all h ∈ H . (iv) Set L := Q ∩ θ (Q), HL := H ∩ L and H Q := HL  U Q where U Q denotes the unipotent radical of Q. Then, we have σ (h Q )  σ (a −1 h Q a) for all a ∈ A+ and all h Q ∈ H Q . Q

Proof (i) and (ii) are essentially [11, Lemma 1.3.1] (i) and (ii) applied to the group G := A G \G. To prove (iii), we first observe that θ (a −1 ha) = aha −1

and

θ (ha)−1 ha = a 2

for all a ∈ A, h ∈ H . Hence, we have sup(σ (aha −1 ), σ (a −1 ha))  σ (a −1 ha) and σ (a 2 )  σ (ha)

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On distinguished square-integrable representations

for all a ∈ A, h ∈ H . Since σ (a) ∼ σ (a 2 ) for all a ∈ A and σ (h)  σ (ha) + σ (a), this already suffices to establish the second inequality. To prove the first one, it only remains to show the following We have   σ (g)  sup σ (aga −1 ), σ (a −1 ga)

(3.2.1)

for all a ∈ A and all g ∈ G. Fix an embedding ι: G → SL n for some n  1 which sends the torus A into the standard maximal torus An of SL n . Then, we are reduced to proving 3.2.1 in the particular case where G = SL n and A = An . For every matrix g ∈ SL n , denote by gi, j (1  i, j  n) the (i, j)th-entry of g and for a ∈ An , set ai = ai,i . Then, we have σ (g) ∼ sup log(2 + |gi, j |) i, j

Hence, it suffices to show that for all 1  i, j  n we have   log(2 + |gi, j |)  sup log(2 + |(aga −1 )i, j |), log(2 + |(a −1 ga)i, j |) −1 for all g ∈ SL n and a ∈ An . However, (aga −1 )i, j = ai a −1 j gi, j , (a ga)i, j =

−1 a j ai−1 gi, j and at least one of the quotients ai a −1 j , a j ai is of absolute value greater than 1. The result follows. We now prove (iv). Every h Q ∈ H Q (F) can be written h Q = h L u Q where h L ∈ HL ⊂ L and u Q ∈ U Q . Moreover, we have σ (lu Q ) ∼ σ (l) + σ (u Q ) and σ (u Q )  σ (a −1 u Q a) for all l ∈ L, u Q ∈ U Q and a ∈ A+ . Besides, as Q

HL ⊂ H , by (iii) we have σ (h L )  σ (a −1 h L a) for all h L ∈ HL and a ∈ A. It follows that σ (h Q ) ∼ σ (h L ) + σ (u Q )  σ (a −1 h L a) + σ (a −1 u Q a) ∼ σ (a −1 h Q a) for all h Q = h L u Q ∈ H Q = HL  U Q and all a ∈ A+ . This proves (iv). Q

 

3.3 Weak Cartan decompositions and Harish-Chandra–Schwartz space of X Let A0, j , j ∈ J , be representatives of the H (F)-conjugacy classes of maximal (θ, F)-split tori of G. There are a finite number of them and by a result of Benoist and Oh [9] and Delorme and Sécherre [19], there exists a compact

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R. Beuzart-Plessis

subset KG ⊂ G(F) such that G(F) =



H (F)A0, j (F)KG

(3.3.1)

j∈J

This decomposition is called a weak Cartan decomposition. Let C ⊂ G(F) be a compact subset with nonempty interior and set ΞCX (x) = vol X (xC)−1/2 for all x ∈ X and where vol X refers to a G(F)-invariant measure on X (which exists as H is reductive hence unimodular). If C  ⊂ G(F) is another compact subset with nonempty interior, the functions ΞCX and ΞCX are equivalent and we will denote by Ξ X any such function (for some choice of C). Proposition 3.3.1 (i) For every compact subset K ⊆ G(F), we have the following equivalences Ξ X (xk) ∼ Ξ X (x) σ X (xk) ∼ σ X (x)

(3.3.2) (3.3.3)

for all x ∈ X and all k ∈ K. (ii) Let A0 be a (θ, F)-split subtorus of G. Then, there exists d > 0 such that Ξ G (a)σ (a)−d  Ξ X (a)  Ξ G (a) σ X (a) ∼ σ (a) for all a ∈ A0 (F). (iii) There exists d > 0 such that the integral 

Ξ X (x)2 σ X (x)−d d x X

is absolutely convergent. Assume moreover that the pair (G, H ) is tempered, then we have (iv) For all d > 0 there exists d  > 0 such that 



H (F)\H (F) AG

for all x ∈ X .

123

Ξ G (hx)σ (hx)−d dh  Ξ X (x)σ X (x)−d

(3.3.4) (3.3.5)

On distinguished square-integrable representations

(v) There exist d > 0 and d  > 0 such that   Ξ G (x −1 hx)σ (x −1 hx)−d dh  Ξ X (x)2 σ X (x)d H (F)\H (F) AG

for all x ∈ X . (vi) More generally, let Q be a θ -split parabolic subgroup of G and set L := Q ∩ θ (Q), HL := H ∩ L and H Q := HL  U Q where U Q denotes the unipotent radical of Q. Let A0 ⊂ L be a maximal (θ, F)-split subtorus and set   A+ := a ∈ A0 (F); |α(a)|  1 ∀α ∈ R(A0 , U Q ) Q

where U Q denotes the unipotent radical of Q. Then, H Q is a unimodular algebraic group, (G, H Q ) is a tempered pair and, fixing a Haar measure dh Q on H Q (F), there exists d > 0 and d  > 0 such that   Ξ G (a −1 h Q a)σ (a −1 h Q a)−d dh Q  Ξ X (a)2 σ X (a)d H (F)\H Q (F) AG

for all a ∈ A+ . Q

Proof (i) Is easy and left to the reader. (ii) Let M0 be the centralizer of A0 in G. For all P0 ∈ P θ (M0 ) set A+ P0 := {a ∈ A0 (F); |α(a)|  1 ∀α ∈ R(A0 , P0 )} Then 

A0 (F) = P0

∈P θ (M

A+ P0

(3.3.6)

0)

Thus, we may fix P0 ∈ P θ (M), set A+ := A+ P0 and prove 3.3.4 and 3.3.5 + for those a belonging to A . Since P0 is θ -split, H P0 is open in G. The proof of (ii) is now the same as Proposition 6.7.1(ii) of [11] after replacing Proposition 6.4.1(iii) of loc. cit. by Lemma 3.2.1 (iii). (iii) The proof is exactly the same as for Proposition 6.7.1 (iii) of [11]: we use the weak Cartan decomposition 3.3.1 to show that X has polynomial growth in the sense of [10] and then we conclude as in loc. cit. (iv) Let A0 be a maximal (θ, F)-split subtorus of G, M0 := Cent G (A0 ) and P0 ∈ P θ (M0 ). By the decompositions 3.3.1 and 3.3.6, points (i) and (ii) and Lemma 3.2.1(iii) it suffices to show the existence of d > 0 such that

123

R. Beuzart-Plessis

 H (F)\H (F) AG

Ξ G (ha)σ (h)−d dh  Ξ G (a)

for all a ∈ A+ P0 . Since P0 is θ -split, H (F)P0 (F) is open in G(F). It follows that if K is a maximal compact subgroup of G(F) by which Ξ G is right invariant there exists an open-compact subgroups J ⊂ G(F) and J H ⊂ H (F) such that J ⊂ J H a K a −1 for all a ∈ A+ P0 . Hence, for all + d > 0, all k ∈ J and all a ∈ A P0 , writing k = k H ak G a −1 with k H ∈ J H and k G ∈ K , we have  H (F)\H (F) AG

Ξ G (hka)σ (h)−d dh =

 H (F)\H (F) AG

−d Ξ G (ha)σ (hk −1 H ) dh

It follows that for all d > 0 we have  Ξ G (ha)σ (h)−d dh H (F)\H (F) AG







H (F)\H (F) AG

Ξ G (hka)dkσ (h)−d dh K

for all a ∈ A+ P0 and we conclude by the ‘doubling principle’ (see [11, Proposition 1.5.1]) and the fact that the pair (G, H ) is tempered. (v) Once again, the proof is very similar to the proof of Proposition 6.7.1 (v) of [11] so that we shall only sketch the argument. Let A0 be a maximal (θ, F)-split torus of G and P0 ∈ P θ (M0 ) where M0 := Cent G (A0 ). By the weak Cartan decomposition, (i), (ii), the inequality σ (h)  σ (a −1 ha)σ (a) and the decomposition 3.3.6, we are reduced to proving the existence of d, d  > 0 such that  H (F)\H (F) AG

Ξ G (a −1 ha)σ (h)−d dh  Ξ G (a)2 σ (a)d



for all a ∈ A+ P0 . Using the fact that H (F)P0 (F) is open in G(F) we show as in the proof of (iv) that if K is a maximal compact subgroup of G(F) we have  Ξ G (a −1 ha)σ (h)−d dh H (F)\H (F) AG





123

H (F)\H (F) AG

 K ×K

Ξ G (a −1 k1 hk2 a)σ (h)−d dk1 dk2 dh

On distinguished square-integrable representations

for all a ∈ A+ P0 and then we conclude again by the ‘doubling principle’ (see [11, Proposition 1.5.1 (vi)]) and the fact that the pair (G, H ) is tempered. (vi) The proof that H Q is unimodular is similar to the proof of Proposition 6.8.1 (ii) of [11] noticing that Q is a good parabolic subgroup with respect to H (that is Q H is open in G) and that HL = H ∩ Q. Moreover, the fact that the pair (G, H Q ) is tempered and the estimate can be proved in much the same way as Proposition 6.8.1 (iv)–(vi) of [11]. Indeed, if we denote by M0 the centralizer of A0 in G, we have A+ = Q

 P0 ∈P θ (M0 ) P0 ⊂Q

A+ P

0

and thus, fixing P0 ∈ P θ (M0 ) with P0 ⊂ Q, it suffices to show the existence of d, d  > 0 such that  The integral

H (F)\H Q (F) AG

Ξ G (h Q )σ (h Q )−d dh Q converges; (3.3.7)

and  H (F)\H Q (F) AG

Ξ G (a −1 h Q a)σ (a −1 h Q a)−d dh Q 

 Ξ X (a)2 σ X (a)d for all a ∈ A+ . P0

(3.3.8)

Then, 3.3.7 can be proved exactly as Proposition 6.8.1 (iv) of [11] where the only inputs used are the facts that Q H is open in G and that the pair (G, H ) is tempered. Also, 3.3.8 can be proved exactly as Proposition 6.8.1 (vi) of loc. cit. where this time the only inputs are the estimates of (ii) and of Lemma 3.2.1 (iv), the convergence of 3.3.7 and the fact that P 0 H is open in G.   3.4 Proof of Theorem 3.1.1 By Proposition 3.3.1 (iii), it suffices to establish the two following claims. χ

For all f ∈ Sω,scusp (G(F)) the function x → K f (x) is compactly supported. (3.4.1)

123

R. Beuzart-Plessis

Assume that the pair (G, H ) is tempered. Then, for all d > 0 and all f ∈ Cω,scusp (G(F)), we have χ

|K f (x)|  Ξ X (x)2 σ X (x)−d

(3.4.2)

for all x ∈ X . We will only show 3.4.2, the proof of 3.4.1 being similar and actually slightly easier. Moreover, the proof of 3.4.2 is also very similar to the proof of Theorem 8.1.1 (ii) of [11]. We will thus content ourself with outlining the main steps. Let A0 be a maximal (θ, F)-split subtorus of G, M0 := Cent G (A0 ) and P0 = M0 U0 ∈ P θ (M0 ). Let P 0 = θ (P0 ) be the opposite parabolic subgroup and set   A+ := a ∈ A0 (F); |α(a)|  1 ∀α ∈ R(A0 , P 0 ) P0

By the weak Cartan decomposition 3.3.1 as well as Proposition 3.3.1(i), it suffices to show 3.4.2 only for x = a ∈ A+ . For all Q ∈ F θ (M0 ) and δ > 0 P0 set   A+ (δ) := a ∈ A0 (F); |α(a)|  eδσ (a) ∀α ∈ R(A0 , U Q ) Q

where Q := θ (Q) and U Q denotes the unipotent radical of Q. Then, if δ is sufficiently small we have A+ = P0



A+ (δ) ∩ A+

Q∈F θ (M0 )−{G},P0 ⊂Q

Q

P0

Thus, fixing Q ∈ F θ (M0 ) − {G} with P0 ⊂ Q and δ > 0, it suffices to prove the estimate 3.4.2 only for x = a ∈ A+ (δ) ∩ A+ . We fix such a Q and such Q

P0

a δ henceforth. Let U Q be the unipotent radical of Q and set L := Q ∩ Q, HL := H ∩ L and H Q := HL  U Q . We define a unitary character χ Q of H Q (F) by setting χ Q (h L u Q ) = χ (h L ) for all h L ∈ HL (F) and u Q ∈ U Q (F). Then by Proposition 3.3.1(vi) , H Q is a unimodular algebraic group and the pair (G, H Q ) is tempered. Thus, fixing a Haar measure dh Q on H Q (F), we can define  χ ,Q f (x −1 h Q x)χ Q (h Q )−1 dh Q K f (x) := H (F)\H Q (F) AG

for all f ∈ Cω (G(F)) and all x ∈ G(F). As U Q ⊂ H Q ⊂ Q, for any χ ,Q strongly cuspidal function f ∈ Cω,scusp (G(F)), the function K f vanishes

123

On distinguished square-integrable representations

identically. Therefore, it is sufficient to show the existence of c > 0 such that for every f ∈ Cω (G(F)) and d > 0 we have



χ

χ ,Q

K f (a) − cK f (a)  Ξ X (a)2 σ X (a)−d

(3.4.3)

for all a ∈ A+ (δ) ∩ A+ . We prove this following closely the proof of PropoQ P0 sition 8.1.4 of [11]. We henceforth fix f ∈ Cω (G(F)) and d > 0. The F-analytic map H (F) ∩ P 0 (F)U0 (F) → U0 (F) h = pu → u is submersive at the origin (this follows from the fact that P 0 (F)H (F) is open in G(F)). Therefore, we may find a compact-open neighborhood U0 of 1 in U0 (F) together with an F-analytic map h: U0 → H (F) such that h(u) ∈ P 0 (F)u for all u ∈ U0 and h(1) = 1. Set U Q := U0 ∩U Q (F), H := HL (F)h(U Q ) and fix Haar measures dh L and du Q on HL (F) and U Q (F) whose product is the fixed Haar measure on H Q (F). The following fact is easy and can be proved exactly the same way as point (8.1.7) of [11] (note that here H ∩ Q = H ∩ L): The map HL (F) × U Q → H (F), (h L , u Q ) → h L h(u Q ), is an Fanalytic open embedding with image H and there exists a smooth function j ∈ C ∞ (U Q ) such that 

 H

ϕ(h)dh =

 HL (F) U Q

ϕ(h L h(u Q )) j (u Q )du Q dh L

(3.4.4)

for all ϕ ∈ L 1 (H). Fix  > 0 that we will assume sufficiently small in what follows. By Lemma 3.2.1(ii), for  small enough we have aU Q [< σ (a)] a −1 ⊆ U Q for all a ∈ A+ (δ). This allows us to define Q

  H <,a := HL (F)h aU Q [< σ (a)] a −1 , H Q,<,a := HL (F)aU Q [< σ (a)] a −1

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R. Beuzart-Plessis

and the following expressions  χ ,< K f (a) := χ ,Q,< (a) Kf

 :=

H (F)\H <,a AG

f (a −1 ha)χ (h)−1 dh

H (F)\H Q,<,a AG

f (a −1 h Q a)χ Q (h Q )−1 dh Q

for all a ∈ A+ (δ). Set c = j (1) (where the function j (·) is the one appearing Q in 3.4.4). Obviously 3.4.3 will follows if we can show that for  sufficiently small we have:



χ

χ ,< (3.4.5)

K f (a) − K f (a)  Ξ X (a)2 σ X (a)−d



Q

χ ,Q,< (a)  Ξ X (a)2 σ X (a)−d (3.4.6)

K f (a) − K f χ ,<

Kf

χ ,Q,<

(a) = cK f

(a)

(3.4.7)

for all a ∈ A+ (δ) ∩ A+ . Q P0 First we prove 3.4.5 and 3.4.6. By Lemma 3.2.1 (i)–(iii)–(iv) and noticing that H <,a = H (F) ∩ Q(F)aU Q [< σ (a)] a −1 and H Q,<,a = H Q (F) ∩ Q(F)aU Q [< σ (a)] a −1 , we see that σ (a)  σ (a −1 ha) and σ (a)  σ (a −1 h Q a) for all a ∈ A+ (δ), h ∈ Q

H (F)\H <,a and h Q ∈ H Q (F)\H Q,<,a . Thus, by definition of Cω (G(F)), for every d1 > 0 and d2 > 0, the left hand sides of 3.4.5 and 3.4.6 are essentially bounded by  −d1 σ (a) Ξ G (a −1 ha)σ (a −1 ha)−d2 dh H (F)\H (F) AG

and −d1

σ (a)

 H (F)\H Q (F) AG

Ξ G (a −1 h Q a)σ (a −1 h Q a)−d2 dh Q

for all a ∈ A+ (δ) respectively. By Proposition 3.3.1 (v)–(vi), there exists Q d3 > 0 such that for d2 sufficiently large the last two expressions above are . Finally, by essentially bounded by Ξ X (a)2 σ X (a)d3 σ (a)−d1 for all a ∈ A+ Q Proposition 3.3.1 (ii), if d1 is sufficiently large we have σ X (a)d3 σ (a)−d1  σ X (a)−d for all a ∈ A0 (F). This proves 3.4.5 and 3.4.6.

123

On distinguished square-integrable representations

It only remains to prove 3.4.7. By 3.4.4 and the choice of Haar measures, we have χ ,<

Kf



 (a) =

H (F)\H (F) aU [<σ (a)]a −1 AG L Q −1 f (a h L h(u Q )a)χ (h L h(u Q ))−1 j (u Q )du Q dh L

(3.4.8)

and χ ,Q,< (a) Kf



 =

H (F)\H (F) AG L

aU Q

[<σ (a)]a −1

f (a −1 h L u Q a)χ (h L )−1 du Q dh L (3.4.9)

for all a ∈ A+ (δ). Since the function u Q → χ (h(u Q ))−1 j (u Q ) is smooth, by Q Lemma 3.2.1(ii) for  sufficiently small we have χ (h(u Q ))−1 j (u Q ) = j (1) = c

(3.4.10)

for all a ∈ A+ (δ) and all u Q ∈ aU Q [< σ (a)]a −1 . Let J ⊂ G(F) be a Q compact-open subgroup by which f is right invariant. By definition, the map u Q → h(u Q )u −1 Q is F-analytic, sends 1 to 1 and takes values in P 0 (F). By Lemma 3.2.1(ii) again, it follows that for all 0 < δ  < δ and all c0 > 0 if  is sufficiently small we have   −δ  σ (a) a ∈ exp B(0, c e ) ∩ p (F) a −1 h(u Q )u −1 0 0 Q for all a ∈ A+ (δ) ∩ A+ and all u Q ∈ aU Q [< σ (a)]a −1 . Moreover, there Q

P0

exists α > 0 such that |Ad(g −1 )X |g  eασ (g) |X |g for all g ∈ G(F) and all X ∈ g(F). Hence, if  is sufficiently small we have −1 −1 −1 −1 −1 a −1 u −1 Q h(u Q )a = (a u Q a) (a h(u Q )u Q a)(a u Q a) ∈ J

(3.4.11)

for all a ∈ A+ (δ) ∩ A+ and all u Q ∈ aU Q [< σ (a)]a −1 . It is clear that 3.4.7 Q P0 follows from 3.4.8, 3.4.9, 3.4.10 and 3.4.11 and this ends the proof of Theorem 3.1.1.  

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R. Beuzart-Plessis

4 The spectral side 4.1 The statement In this chapter G is a connected reductive group over F, G := G/A G and H H as the is a symmetric subgroup of G. As in the previous chapter, we let A G connected component of H ∩ A G . Let ω and χ be continuous unitary characters of A G (F) and H (F) respectively such that ω|A H (F) = χ|A H (F) . Set G

G

ν(H ) := [H (F) ∩ A G (F): A H (F)] . In Sect. 3.1, we have defined a linear form f ∈ Sω,scusp (G(F)) → J χ ( f ) which, if (G, H ) is a tempered pair, extends to a continuous linear form f ∈ Cω,scusp (G(F)) → J χ ( f ). For all π ∈ Irr(G) we define a multiplicity m(π, χ ) by m(π, χ ) := dim Hom H (π, χ ) where Hom H (π, χ ) denotes the space of linear forms : π → C such that  ◦ π(h) = χ (h) for all h ∈ H (F). By Theorem 4.5 of [18], we know that this space is always finite dimensional so that the multiplicity m(π, χ ) is well-defined. Recall that Irr ω,cusp (G), resp. Irr ω,sqr (G), denote the sets of equivalence classes of irreducible supercuspidal, resp. square-integrable, representations of G(F) with central character ω. Define the following linear forms χ (f) f ∈ Sω,scusp (G(F)) → Jspec,cusp  m(π, χ ) Trace(π ∨ ( f )) := ν(H ) π ∈Irr ω,cusp (G)

and χ

f ∈ Cω,scusp (G(F)) → Jspec,disc ( f )  m(π, χ ) Trace(π ∨ ( f )) := ν(H ) π ∈Irr ω,sqr (G)

Notice that the sums defining these linear forms are always finite by the result of Harish-Chandra that for every compact-open subgroup J ⊂ G(F) the set of

123

On distinguished square-integrable representations

π ∈ Irr ω,sqr (G) with π J  = 0 is finite [45, Théorème VIII.1.2]. Recall that in Sect. 2.6 we have introduced certain spaces 0 Sω (G(F)), 0 Cω (G(F)) of cusp forms. The goal of this chapter is to prove the following Theorem 4.1.1 For all f ∈ 0 Sω (G(F)) we have χ (f) J χ ( f ) = Jspec,cusp

Moreover, if (G, H ) is a tempered pair, for all f ∈ 0 Cω (G(F)) we have χ

J χ ( f ) = Jspec,disc ( f ). 4.2 Explicit description of the intertwinings For all π ∈ Irr ω,cusp (G(F)) we define a bilinear form Bπ : π × π ∨ → C

by ∨

Bπ (v, v ) :=

 H (F)\H (F) AG

π(h)v, v ∨ χ (h)−1 dh

for all (v, v ∨ ) ∈ π × π ∨ . If (G, H ) is a tempered pair and π ∈ Irr ω,sqr (G(F)) then the above integral is also absolutely convergent and thus also defines a bilinear form Bπ : π × π ∨ → C. In all cases, we have Bπ (π(h 1 )v, π ∨ (h 2 )v ∨ ) = χ (h 1 )χ −1 (h 2 )Bπ (v, v ∨ )

for all (v, v ∨ ) ∈ π × π ∨ and all h 1 , h 2 ∈ H (F). Thus Bπ factorizes through a bilinear form Bπ : πχ × πχ∨−1 → C

where πχ and πχ∨−1 denote the spaces of (H (F), χ )- and (H (F), χ −1 )coinvariants in π and π ∨ respectively i.e. the quotients of π and π ∨ by the subspaces generated by vectors of the form π(h)v −χ (h)v (h ∈ H (F), v ∈ π ) and π ∨ (h)v ∨ − χ (h)−1 v ∨ (h ∈ H (F), v ∨ ∈ π ∨ ) respectively. The following proposition has been proved in more generality in [44, Theorem 6.4.1] when the subgroup H is strongly tempered (in the sense of loc. cit.). The same kind of idea already appears in [47, Proposition 5.6]. Proposition 4.2.1 Bπ induces a nondegenerate pairing between πχ and πχ −1 .

123

R. Beuzart-Plessis

Proof We will prove the proposition when (G, H ) is a tempered pair and π ∈ Irr ω,sqr (G), the case where π ∈ Irr ω,cusp (G) being similar and easier. Fix a G(F)-invariant scalar product (·, ·) on π . We can define the following sesquilinear version of Bπ Lπ : π × π → C   (v, v ) →

H (F)\H (F) AG

(π(h)v, v  )χ (h)−1 dh

which factorizes through a sesquilinear pairing Lπ : πχ × πχ → C. Obviously, it suffices to show that this pairing is non degenerate. Since πχ is finite dimensional, this is equivalent to saying that the map v ∈ π → Lπ (·, v) ∈ Hom H (π, χ ) is surjective. To continue we need the following lemma, a consequence of the weak Cartan decomposition (3.3.1).   Lemma 4.2.1 For all  ∈ Hom H (π, χ ) and all v ∈ π we have  |(π(x)v)|2 d x < ∞ X

and moreover for all f ∈ Cω−1 (G(F)) the integral  A G (F)\G(F)

f (g)(π(g)v)dg

is absolutely convergent and equals (π( f )v). Proof For every compact-open subgroup J ⊂ G(F) we will denote by e J ∗   1 (π(k)v)dk. the smooth linear form (i.e. an element of π ∨ ) v ∈ π → vol(J ) J Let A0 be a maximal (θ, F)-split subtorus of G, M0 := Cent G (A0 ) (a minimal θ -split Levi subgroup) and P0 ∈ P θ (M0 ). Set A+ P0 := {a ∈ A0 (F); |α(a)|  1 ∀α ∈ Δ P0 } Then, by the weak Cartan decomposition 3.3.1 and Proposition 3.3.1(i)–(iv), in order to prove that the two integrals of the proposition are convergent it suffices to show that for all d > 0 we have |(π(a)v)|  Ξ G (a)σ (a)−d

123

(4.2.1)

On distinguished square-integrable representations J for all a ∈ A+ P0 . Let J ⊂ G(F) be a compact-open subgroup such that v ∈ π . Since P0 is θ -split, H (F)P0 (F) is open in G(F) and consequently there exists a compact-open subgroup J  ⊂ G(F) such that

J  ⊂ H (F)a (J ∩ P0 (F)) a −1 + for all a ∈ A+ P0 . Thus, for all a ∈ A P0 we have (π(a)v) = e J  ∗ , π(a)v and the inequality 4.2.1 now follows from the known asymptotics of smooth coefficients of square-integrable representations. To prove the last part of the proposition, choose J ⊂ G(F) a compact-open subgroup by which f is invariant on the left. Then, we have    1 f (g)(π(g)v)dg = f (g) (π(kg)v)dkdg vol(J ) AG (F)\G(F) A G (F)\G(F) J  f (g)e J ∗ , π(g)vdg = A G (F)\G(F)

= e J ∗ , π( f )v = (π( f )v)   By the lemma we can define a scalar product, also denoted (·, ·), on Hom H (π, χ ) characterized by  (π(x)v) (π(x)v  )d x = (,  )(v, v  ) X

for all ,  ∈ Hom H (π, χ ) and all v, v  ∈ π . Let  ∈ Hom H (π, χ ) which is orthogonal for this scalar product to all the forms Lπ (·, v) for v ∈ π . To conclude it suffices to show that this implies  = 0. Since for all v, v  ∈ π we have (v, π(·)v  ) ∈ Cω−1 (G(F)), by the lemma we have  0= (π(x)v  )Lπ (π(x)v  , v)d x X  ν(H ) (v)(v  , v  ) (π(g)v  )(v, π(g)v  )dg = = ν(H ) d(π ) A G (F)\G(F) for all v, v  ∈ π and where d(π ) stands for the formal degree of π . Hence,  = 0.   4.3 Proof of Theorem 4.1.1 Once again we will prove the theorem in the case where (G, H ) is a tempered pair and f ∈ 0 Cω (G(F)). The case where f ∈ 0 Sω (G(F)) is completely

123

R. Beuzart-Plessis

similar since compactly supported cusp forms are linear combinations of matrix coefficients of supercuspidal representations whereas a general cusp form f ∈ 0 Cω (G(F)) is a linear combination of matrix coefficients of squareintegrable representations. More precisely, let f ∈ 0 Cω (G(F)) and for all π ∈ Irr ω,sqr (G(F)), set f π (g) := Trace(π ∨ (g −1 )π ∨ ( f )) for all g ∈ G(F). Then, we have f π ∈ 0 C (G(F)) for all π ∈ Irr ω ω,sqr (G(F)) and by the Harish-Chandra–Plancherel formula for cusp forms 2.6.1 we have 

Jχ( f ) =

d(π )J χ ( f π )

π ∈Irr ω,sqr (G(F))

Thus, it suffices to show that J χ ( f π ) = ν(H )d(π )−1 m(π, χ ) Trace(π ∨ ( f )) for all π ∈ Irr ω,sqr (G(F)). Fix π ∈ Irr ω,sqr (G(F)). As f π is a sum of coefficients of π , the equality above is equivalent to J χ ( f v,v ∨ ) = ν(H )d(π )−1 m(π, χ ) f v,v ∨ (1) for all (v, v ∨ ) ∈ π × π ∨ where f v,v ∨ (g) := π(g)v, v ∨  for all g ∈ G(F). Fix (v, v ∨ ) ∈ π × π ∨ . Then, we have χ

Kf

v,v ∨

(x) = Bπ (π(x)v, π ∨ (x)v ∨ )

for all x ∈ X . Choose a basis (v 1 , . . . , v N ) of πχ (where N = m(π, χ )) and ∨ ∨ let (v ∨ 1 , . . . , v N ) be the dual basis of πχ −1 with respect to Bπ (such a dual basis exists thanks to Proposition 4.2.1). Let (v1 , . . . , v N ) and (v1∨ , . . . , v ∨ N) be any lifting of these basis to π and π ∨ respectively. Then we have Bπ (π(x)v, π ∨ (x)v ∨ ) =

N 

Bπ (π(x)v, vi∨ )Bπ (vi , π ∨ (x)v ∨ )

i=1

for all x ∈ X . Now by Lemma 4.2.1, we have χ

J (f

 v,v ∨

)= =

X N  i=1

123

Bπ (π(x)v, π ∨ (x)v ∨ )d x

X

Bπ (π(x)v, vi∨ )Bπ (vi , π ∨ (x)v ∨ )d x

On distinguished square-integrable representations

= ν(H ) = ν(H )

N   i=1 N  i=1

= ν(H )N

A G (F)\G(F)

π(g)v, vi∨ Bπ (vi , π ∨ (g)v ∨ )dg

v, v ∨  Bπ (vi , vi∨ ) d(π )

v, v ∨  = ν(H )d(π )−1 m(π, χ ) f v,v ∨ (1) d(π )  

5 The geometric side 5.1 The statement In this chapter E/F is a quadratic extension, H is a connected reductive group over F and G := R E/F H E . We have a natural inclusion H → G and we shall denote by θ involution of G induced by the nontrivial element of Gal(E/F). Hence H = G θ . Note that in this case, with the notations of Sect. 3.1 we have H = A . Set AG H ν(H ) := [H (F) ∩ A G (F): A H (F)] As in Sect. 3.1, we let G := G/A G , H := H/A H , X := A G (F)H (F)\G(F), X := H A G \G, σ X := σX and σ := σG . Note that X is an open subset of X(F). We have the following identity between Weyl discriminants D H (h) = D G (h)1/2 ,

h ∈ Hreg (F)

(5.1.1)

which will be crucial in what follows. Let ω and χ be continuous unitary characters of A G (F) and H (F) respectively such that ω|A H (F) = χ|A H (F) . In Sect. 3.1, we have defined a continuous linear form f ∈ Cω,scusp (G(F)) → J χ ( f ). We define a second continuous linear χ form f ∈ Cω,scusp (G(F)) → Jgeom ( f ) by setting χ Jgeom (f)

:= ν(H )

 T ∈Tell (H )

−1

|W (H, T )|

 T (F)

D H (t)Θ f (t)χ (t)−1 dt

for all f ∈ Cω,scusp (G(F)), where Tell (H ) denotes a set of representatives of the H (F)-conjugacy classes of maximal elliptic tori in H , we have set T := T /A H for all T ∈ Tell (H ) and we recall that T (F) is equipped with the Haar measure of total mass 1. Since for all f ∈ Cω,scusp (G(F)) the function

123

R. Beuzart-Plessis

(D G )1/2 Θ f is locally bounded, by 5.1.1 we see that the expression defining χ Jgeom ( f ) is absolutely convergent. The goal of this chapter is to show the following. Theorem 5.1.1 For all f ∈ Cω,scusp (G(F)), we have χ J χ ( f ) = Jgeom (f)

We fix a function f ∈ Cω,scusp (G(F)) until the end of this chapter. 5.2 Truncation and first decomposition We fix a sequence (κ N ) N 1 of functions κ N : X (F) → {0, 1} satisfying the two following conditions: There exist C1 , C2 > 0 such that for all x ∈ X (F) and all N  1, we have : σ X (x)  C1 N ⇒ κ N (x) = 1 κ N (x)  = 0 ⇒ σ X (x)  C2 N

(5.2.1)

There exists an open-compact subgroup K  ⊂ G(F) such that the function κ N is right-invariant by K  for all N  1.

(5.2.2)

Such a sequence of truncation functions is easy to construct (see [11, §10.9]). Set  χ JN ( f ) = K χ ( f, x)κ N (x)d x X

for all N  1. Then we have χ

J χ ( f ) = lim J N ( f )

(5.2.3)

N →∞

Let T (H ) be a set of representatives of the H (F)-conjugacy classes of maximal tori in H . By the Weyl integration formula for H , we have K χ ( f, x) =



|W (H, T )|−1

T ∈T (H )



×

123

T (F)\H (F)

 T (F)

D H (t)

f (x −1 h −1 thx)dhχ (t)−1 dt

On distinguished square-integrable representations

where we have set T := T /A H for all T ∈ T (H ). At least formally, it follows that for all N  1   χ −1 |W (H, T )| D H (t) JN ( f ) = T (F)

T ∈T (H )



×

T G (F)\G(F)

f (g −1 tg)κ N ,T (g)dgχ (t)−1 dt

(5.2.4)

where for all T ∈ T (H ) we have denoted by T G the centralizer of T in G (a maximal torus in G) and we have set  κ N ,T (g) := κ N (ag)da A G (F)T (F)\T G (F)

for all g ∈ G(F). Define χ J N ,T ( f )

 :=

 D (t) H

T (F)

T G (F)\G(F)

f (g −1 tg)κ N ,T (g)dgχ (t)−1 dt

for all N  1 and all T ∈ T (H ). The equality 5.2.4 can thus be restated as χ

JN ( f ) =



χ

|W (H, T )|−1 J N ,T ( f )

(5.2.5)

T ∈T (H )

The previous formal manipulations are justified a posteriori by the following lemma: Lemma 5.2.1 (i) There exists k  1 such that κ N ,T (g)  N k σT G \G (g)k for all N  1 and all g ∈ G(F). χ (ii) For all T ∈ T (H ), the expression defining J N ,T ( f ) is absolutely convergent and the identity 5.2.5 is valid. Proof (i) Since the natural inclusion A G T \T G ⊂ X is a closed immersion (essentially because T G is θ -stable), we have σ X (a) ∼ σ AG T \T G (a) for all a ∈ T G (F). As σ X (x)  σ X (xg)σ (g) for all (x, g) ∈ X × G(F), it follows from 5.2.1 that there exists c1 > 0 such that for all N  1, all a ∈ T G (F) and all g ∈ G(F) we have κ N (ag)  = 0 ⇒ σ AG T \T G (a) < c1 N σ (g)

123

R. Beuzart-Plessis

Hence, since the function κ N is nonnegative and bounded by 1, by 2.2.3 there exists k > 0 such that   κ N ,T (g)  meas (A G T \T G )[< c1 N σ (g)]  N k σ (g)k for all N  1 and all g ∈ G(F). The function g → κ N ,T (g) being left invariant by T G (F) we may replace σ (g) in the inequality above by inf a∈T G (F) σ (ag) which by 2.2.1 is equivalent to σT G \G (g). This proves (i). (ii) Since f belongs to the Harish-Chandra–Schwartz space Cω (G(F)), this follows from a combination of (i), 5.1.1 and Lemma 2.9.4.   From now on and until the end of Sect. 5.5, we fix a torus T ∈ T (H ). 5.3 Change of truncation G

Set T := T G /A G (a maximal torus of G) and let A be the maximal (θ, F)G split subtorus of T . Let A ⊂ T G be the inverse image of A and set  κ N ,A (g) :=

A G (F)\A(F)

κ N (ag)da

for all g ∈ G(F) and all N  1. We define the following quantity ν(T ) := [H (F) ∩ A G (F): A H (F)] × [A(F) ∩ A T (F): A H (F)]−1 Then, we have χ J N ,T ( f )

 = ν(T )

 D (t) H

T (F)

A T (F)A(F)\G(F)

f (g −1 tg)κ N ,A (g)dgχ (t)−1 dt (5.3.1)

Indeed, by our choices of Haar measures on tori (see Sect. 2.1) and noting that T (F) ∩ A G (F) = H (F) ∩ A G (F), we have  κ N ,T (g) =

A G (F)T (F)\T G (F)

κ N (ag)da = [(T ∩ A G )(F): (A T ∩ A G )(F)]



×

123

A G (F)A T (F)\T G (F)

κ N (ag)da

On distinguished square-integrable representations

 = [(H ∩ A G )(F): (A T ∩ A G )(F)] A T (F)A(F)\T G (F)  × κ N (at G g)dadt G A G (F)(A∩A T )(F)\A(F)

= [(H ∩ A G )(F): (A G ∩ A T )(F)] [(A ∩ A T )(F): (A G ∩ A T )(F)]−1  × κ N ,A (t G g)dt G A T (F)A(F)\T G (F)  = ν(T ) κ N ,A (t G g)dt G A T (F)A(F)\T G (F)

for all g ∈ G(F) and all N  1, hence the result. Since A(F)A T (F)\T G (F) is compact by 2.2.1 we have σT G \G (g) ∼

inf

a∈A(F)A T (F)

σ (ag)

(5.3.2)

for all g ∈ G(F) and hence the same proof as that of Lemma 5.2.1(i) shows that there exists k > 0 such that κ N ,A (g)  N k σT G \G (g)k

(5.3.3)

for all N  1 and all g ∈ G(F). Let M be the centralizer in G of A. It is a θ -split Levi subgroup with A M,θ = Aθ . Indeed, the inclusion Aθ ⊂ A M,θ is obvious and T G is a maximal torus of M hence A M,θ is included in A T G ,θ = Aθ . Let A0 be a maximal (θ, F)-split subtorus of G containing A and denote by A0 its inverse image in G. Let M0 be the centralizer in G of A0 . It is a minimal θ -split Levi subgroup, we again have A0,θ = A M0 ,θ and we set A0,θ := A M0 ,θ . Let K be a special maximal compact subgroup of G(F). We use K to define the functions H Q,θ for all Q ∈ F θ (M0 ) (see Sect. 2.8.1). Fix P0 ∈ P θ (M0 ) and let Δ0 be the set of simple roots of A0 in P0 . To every Y ∈ A+ P0 ,θ we associate a positive (G, M0 , θ )orthogonal set (Y P0 ) P  ∈P θ (M0 ) by setting Y P0 = wY where w is the unique 0 element in the little Weyl group W (G, A0 ) such that w P0 = P0 . By the general constructions of Sect. 2.8.2, this also induces a positive (G, M, θ )-orthogonal set (Y P ) P∈P θ (M) . For all g ∈ G(F), we define another (G, M, θ )-orthogonal set Y (g) = (Y (g) P ) P∈P θ (M) by setting Y (g) P := Y P − H P,θ (g)

for all P ∈ P θ (M) where P := θ (P). Recall that this (G, M, θ )-orthogonal G (·, Y (g)) on A set induces a function Γ M,θ M,θ (see Sect. 2.8.2). If Y (g) is a

123

R. Beuzart-Plessis

positive (G, M, θ )-orthogonal set then this is just the characteristic function of the convex hull of {Y (g) P ; P ∈ P θ (M)}. Define   v M,θ (Y, g) :=

A G (F)\A(F)

G Γ M,θ (HM,θ (a), Y (g))da

for all Y ∈ A+ P0 ,θ and all g ∈ G(F). Fixing a norm |·| on A0,θ , by 2.2.3 there exists k > 0 such that we have an inequality | v M,θ (Y, g)|  (1 + |Y |)k σT G \G (g)k

(5.3.4)

for all Y ∈ A+ P0 ,θ and all g ∈ G(F). Define the following expression χ

JY,T ( f ) := ν(T )



 T (F)

D H (t)

A T (F)A(F)\G(F)

f (g −1 tg) v M,θ (Y, g)dgχ (t)−1 dt

for all Y ∈ A+ P0 ,θ . Using 5.3.4 and reasoning as in the proof of Lemma 5.2.1(ii), we can show that this expression is absolutely convergent. Proposition 5.3.1 Let 0 < 1 < 2 < 1. Then, for all k > 0 we have



χ

χ

J N ,T ( f ) − JY,T ( f )  N −k for all N  1 and all Y ∈ A+ P0 ,θ satisfying the two inequalities N 1  inf α(Y )

(5.3.5)

sup α(Y )  N 2

(5.3.6)

α∈Δ0

α∈Δ0

Proof Let 0 < 1 < 2 < 1. For M > 0 we will denote by 1 0, we can write χ

χ

χ

χ

χ

χ

J N ,T ( f ) = J N ,T,
123

On distinguished square-integrable representations

for all N  1 and all Y ∈ A+ P0 ,θ , where χ J N ,T,
 := ν(T )

 D (t) H

T (F)

A T (F)A(F)\G(F)

−1

× 1
A T (F)A(F)\G(F)

× 1M (g) f (g χ

−1

tg)κ N ,A (g)dgχ (t)−1 dt

χ

and JY,T, 0 and all k > 0 we have χ

|J N ,T,N  ( f )|  N −k and

χ |JY,T,N  ( f )|

for all N  1 and all Y ∈

A+ P0 ,θ

(5.3.7) 

N −k

satisfying inequality 5.3.6.

By 5.3.3 and 5.3.4 and the fact that f belongs to the Harish-Chandra–Schwartz space Cω (G(F)), we only need to show that for all k, k  > 0 and  > 0 there exists d > 0 such that 

 D (t) H

T (F)

A(F)A T (F)\G(F) −k 

1N  (g)Ξ G (g −1 tg)σ (g −1 tg)−d

σT G \G (g)k dgdt  N

for all N  1. By 5.3.2, for all r > 0 this integral is essentially bounded by N

−r



 D (t) H

T (F)

T G (F)\G(F)

Ξ G (g −1 tg)σ (g −1 tg)−d σT G \G (g)k+r dgdt

for all N  1. By 5.1.1 and Lemma 2.9.4 for all k, r > 0 there exists d > 0 making the last integral above convergent. The claim follows. Choose  > 0 such that  < 1 . By 5.3.7, it suffices to show that for all k > 0 we have





χ χ

J N ,T,
(5.3.8)

123

R. Beuzart-Plessis

for all N  1 and all Y ∈ A+ P0 ,θ satisfying inequalities 5.3.5 and 5.3.6. For + θ Q ∈ F (M), Y ∈ A P0 ,θ and N  1, we set 

Y,Q

κ N ,A (g) :=

Q

A G (F)\A(F)

G Γ M,θ (HM,θ (a), Y (g))τ Q,θ (HM,θ (a)

−Y (g) Q )κ N (ag)da Q

G have been defined for all g ∈ G(F) where the functions Γ M,θ (·, Y (g)) and τ Q,θ in Sect. 2.8.2. Note that Q

Q

Γ M,θ (·, Y (ag)) = Γ M,θ (· + HM,θ (a), Y (g)) Y,Q

for all a ∈ A(F). Hence the functions κ N ,A are A(F)-invariant on the left and this allows us to define the following expressions   χ ,Y,Q D H (t) J N ,T,
A(F)A T (F)\G(F) Y,Q × 1
θ for all N  1, all Y ∈ A+ P0 ,θ and all Q ∈ F (M). By 2.8.2, we have χ

J N ,T,


χ ,Y,Q

Q∈F θ (M)

J N ,T,
for all N  1 and all Y ∈ A+ P0 ,θ . Thus, to show 5.3.8 it suffices to establish the two following facts There exists N0  1 such that χ ,Y,G

χ

J N ,T,
(5.3.9)

for all N  N0 and all Y ∈ A+ P0 ,θ satisfying inequality 5.3.6. θ For all Q ∈ F (M), Q  = G, and all k > 0 we have



χ ,Y,Q

J N ,T,
(5.3.10)

for all N  1 and all Y ∈ A+ P0 ,θ satisfying inequality 5.3.5. χ ,Y,G

χ

First we prove 5.3.9. By definition of J N ,T,
123

On distinguished square-integrable representations

for all N  N0 , all Y ∈ A+ P0 ,θ satisfying inequality 5.3.6 and all g ∈ G(F)

with σ (g) < N  . Unraveling the definitions of κ N ,A (g) and  v M,θ (Y, g), we see that it would follow if we can show the implication Y,Q

  G HM,θ (a), Y (g)  = 0 ⇒ κ N (ag) = 1 Γ M,θ

(5.3.11)

for all N  1, all Y ∈ A+ P0 ,θ satisfying inequality 5.3.6, all g ∈ G(F) with σ (g) < N  and all a ∈ A(F). By 2.8.1, there exists C > 0 such that for all Y ∈ A+ P0 ,θ , all g ∈ G(F) and all a ∈ A(F) we have 

G Γ M,θ

 HM,θ (a), Y (g)  = 0 ⇒ σ AG \A (a)  C



 sup α(Y ) + σ (g)

α∈Δ0

As σ AG \A (a) ∼ σ X (a) (this is a consequence of the facts that Aθ A G \A is closed in X and A G \Aθ A G is finite) and σ X (ag)  σ X (a) + σ (g) for all a ∈ A(F) and all g ∈ G(F), it follows that there exists C  > 0 such that for all N  1, all Y ∈ A+ P0 ,θ satisfying inequality 5.3.6, all g ∈ G(F) with σ (g) < N  and all a ∈ A(F) we have     G HM,θ (a), Y (g)  = 0 ⇒ σ X (ag)  C  N 1 + N  Γ M,θ Since , 1 < 1, by property 5.2.1 of our sequence of truncation functions the last inequality above implies κ N (ag) = 1 whenever N  1. This shows 5.3.11 and ends the proof of 5.3.9. It only remains to prove claim 5.3.10. Fix Q ∈ F θ (M), Q  = G, with Levi decomposition Q = LU Q where L := Q ∩ θ (Q). Let Q = θ (Q) = LU Q be the opposite parabolic subgroup. We have the Iwasawa decomposition G(F) = L(F)U Q (F)K and accordingly we can decompose the integral 

1
A(F)A T (F)\G(F)



=

1
A(F)A T (F)\L(F)×U Q (F)×K

for all N  1, all Y ∈ A+ P0 ,θ and all t ∈ T reg (F). To continue we need the following fact which we will establish after we finish the proof of 5.3.10: There exists N0  1 such that for all N  N0 , all Y ∈ A+ P0 ,θ satisfying inequality 5.3.5 and all l ∈ L(F), u ∈ U Q (F), k ∈ K with σ (luk) < N  we have Y,Q

Y,Q

κ N ,A (luk) = κ N ,A (lk)

(5.3.12)

123

R. Beuzart-Plessis

Taking 5.3.12 for granted we get 

1
A(F)A T (F)\G(F)



=

1
A(F)A T (F)\L(F)×U Q (F)×K

for all N  1, all t ∈ T reg (F) and all Y ∈ A+ P0 ,θ satisfying inequality 5.3.5. As f is strongly cuspidal if we forget the term 1
1
A(F)A T (F)\G(F)



=−

1N  (luk) f (k −1 u −1l −1 tluk)κ N ,A (lk)dldudk Y,Q

A(F)A T (F)\L(F)×U Q (F)×K

for all N  1, all t ∈ T reg (F) and all Y ∈ A+ P0 ,θ satisfying inequality 5.3.5. Hence, to get 5.3.10 it only remains to show that for all k > 0 we have 

 D (t) H

T (F)

A(F)A (F)\L(F)×U (F)×K

T Q





Y,Q

−1 −1 −1



1N  (luk) f (k u l tluk) κ N ,A (lk) dldudkdt  N −k

(5.3.13)

for all N  1 and all Y ∈ A+ θ,P0 . Since for all (G, M, θ )-orthogonal set Z = Q

(Z P ) P∈P θ (M) the function X ∈ Aθ → Γ M,θ (X, Z ) is uniformly bounded G independently of Z (this follows from the definition of this function) and τ Q,θ is a characteristic function, we have



Y,Q

κ N ,A (g)  κ N ,A (g) for all N  1, all Y ∈ A+ P0 ,θ and all g ∈ G(F). Hence, by 5.3.3 and the fact that f belongs to the Harish-Chandra–Schwartz space Cω (G(F)) there exists k > 0 such that for all d > 0 the left hand side of 5.3.13 is essentially bounded by the product of N k with 

 D (t) H

T (F)

A(F)A T (F)\L(F)×U Q (F)×K G −1 −1 −1 −1 −1 −1

1N  (luk)Ξ (k

u

σT G \G (lk)k dldudkdt

123

l

tluk)σ (k

u

l

tluk)−d

On distinguished square-integrable representations G ⊂ L we have σ for all N  1 and all Y ∈ A+ T G \G (lk)  P0 ,θ . Since T σT G \G (luk) for all l ∈ L(F), u ∈ U Q (F) and all k ∈ K . Thus, the last expression above is essentially bounded by



 D (t) H

T (F)

A(F)A T (F)\G(F)

1N  (g)Ξ G (g −1 tg)σ (g −1 tg)−d σT G \G (g)k dg

for all N  1. We already saw that for all k  > 0 we can find d > 0 such  that this last integral is essentially bounded by N −k for all N  1. This shows 5.3.13 and ends the proof of 5.3.10 granting 5.3.12. Q G (· − We now prove 5.3.12. For g ∈ G(F), the function Γ M,θ (·, Y (g))τ Q,θ Y (g) Q ) depends only on the points Y (g) P for all P ∈ P θ (M) with P ⊂ Q and those points remain invariant by left translation of g by U Q (F). Hence, it suffices to show the following: There exists N0  1 such that for all N  N0 , all Y ∈ A+ P0 ,θ satisfying inequality 5.3.5, all l ∈ L(F), u ∈ U Q (F), k ∈ K with σ (luk) < N  and all a ∈ A(F) we have Q

G (HM,θ (a) − Y (l) Q )  = 0 ⇒ κ N (aluk) = κ N (alk) Γ M,θ (HM,θ (a), Y (l))τ Q,θ (5.3.14)

Let N  1 and Y , l, u, k be as above (in particular Y satisfies condition 5.3.5 and σ (luk) < N  ). We will show that the conclusion of 5.3.14 holds provided N is sufficiently large. Let a ∈ A(F) be such that  G Q  (Hθ,M (a) − Y (l) Q )  = 0 Γ M,θ HM,θ (a), Y (l) τ Q,θ We need to show that κ N (aluk) = κ N (alk). There exists C > 0 such that for all g ∈ G(F) and all Y ∈ A+ P0 ,θ if σ (g)  C inf α(Y ) α∈Δ0

θ then Y (g) P ∈ A+ P,θ for all P ∈ P (M) and thus Y (g) is a positive (G, M, θ )orthogonal set. As  < 1 it follows that for N sufficiently large the (G, M, θ )orthogonal set Y (l) is positive. In particular, again for N sufficiently large, the function Q

G (X − Y (l) Q ) X ∈ A M,θ → Γ M,θ (X, Y (l))τ Q,θ

123

R. Beuzart-Plessis

is the characteristic function of the sum of A+ Q,θ with the convex hull of the family (Y (l) P ) P⊂Q . As  < 1 and σ (l)  N  , it follows that   log|β(a)|  inf β Y P − H P,θ (l)  inf α(Y ) − σ (l)  N 1 (5.3.15) α∈Δ0

P⊂Q

for all β ∈ R(A, U Q ). Fix a norm |·| on g(F) and let us denote by B(0, r ) the open ball of radius r centered at the origin for all r > 0. Since σ (lul −1 )  N  and  < 1 , we deduce from 5.3.15 that there exists a constant c1 > 0 such that for N big enough we have   1 alul −1 a −1 ∈ exp B(0, e−c1 N ) Let Pa ∈ P θ (M) be such that HM,θ (a) ∈ A+ Pa ,θ (the closure of the positive chamber associated to Pa ). Since Pa is θ -split, the multiplication map H (F)× Pa (F) → G(F) is submersive at the origin and hence this map admits an F-analytic section defined on a neighborhood of 1 in G(F). It follows that there exists c2 > 0 (independent of a since there is only a finite number of possibilities for Pa ) so that for N large enough   1 alul −1 a −1 ∈ H (F) exp B(0, e−c2 N ) ∩ pa (F) 

Choose X ∈ B(0, e−c2 N 1 ) ∩ pa (F) with alul −1 a −1 ∈ H (F) exp(X ). Since κ N is left invariant by H (F) we have κ N (aluk) = κ N (exp(X )alk)  As a ∈ A+ Pa ,θ ,  < 1 and σ (l)  N , there exists a constant c3 > 0 such that 1

k −1l −1 a −1 Xalk ∈ B(0, e−c3 N ) By property 5.2.2 of our sequence of truncation functions, we deduce that for N sufficiently large κ N is right invariant by exp(k −1l −1 a −1 Xalk). Hence, κ N (exp(X )alk) = κ N (alk) This proves claim 5.3.14 and ends the proof of the proposition.

123

 

On distinguished square-integrable representations

5.4 First computation of the limit Recall the function g ∈ G(F) → v M,θ (g) introduced in Sect. 2.8.2. Its value at g ∈ G(F) is given by the volume of the convex hull of the set {H P,θ (g); P ∈ P θ (M)}. Proposition 5.4.1 We have χ lim J ( f ) N →∞ N ,T

G a M,θ

= (−1)

× f (g

−1

 ν(T )

 D (t) H

T (F)

tg)v M,θ (g)dgχ (t)

A T (F)A(F)\G(F) −1

dt

Proof Let 0 < 1 < 2 < 1, 0 < δ < 1 and set  + A+ P0 ,θ (δ) := Y ∈ A P0 ,θ ; inf α(Y )  δ sup α(Y ) α∈Δ0

α∈Δ0

Then A+ P0 ,θ (δ) is a cone in A0,θ with nonempty interior. By Proposition 5.3.1 for all k > 0 we have





χ χ ( f ) − J ( f ) J

 N −k

N ,T Y,T 1 −1 2 for all N  1 and all Y ∈ A+ P0 ,θ (δ) with N  inf α(Y )  δ N . As for α∈Δ0

N sufficiently large the two sets !  + 1 −1 2 Y ∈ A P0 ,θ (δ); N  inf α(Y )  δ N α∈Δ0

and  Y ∈

A+ P0 ,θ (δ);

1

(N + 1)

 inf α(Y )  δ α∈Δ0

−1

2

!

(N + 1)

intersect, it follows that the two limits χ χ lim J ( f ), lim JY,T ( f ) + N →∞ N ,T Y ∈A P ,θ (δ)→∞ 0

χ

exist and are equal. We will denote by J∞,T ( f ) this common limit. Let A0,θ,F denote the image of A0,θ (F) by HM0 ,θ . Then by Lemma 2.8.2, we know that for every lattice R ⊂ A0,θ,F ⊗Q and all g ∈ G(F) the function Y ∈ R ∩ A+ v M,θ (Y, g) coincides with the restriction of an exponentialP0 ,θ (δ) →  polynomial of bounded degree and with exponents in a fixed finite set (both

123

R. Beuzart-Plessis

independent of g). Let us denote by  v M,θ,0 (R, g) the constant term of the purely polynomial part of this exponential-polynomial. Then by Lemma 2.8.1, we have χ J∞,T ( f )

 = ν(T )

 D (t) H

T (F)

A T (F)A(F)\G(F)

f (g −1 tg) v M,θ,0 (R, g)dgχ(t)−1 dt

(5.4.1) for every lattice R ⊂ A0,θ,F ⊗ Q. Fix such a lattice R. By Lemma 2.8.2, there exists r > 0 such that







G 1 a M,θ

 v M,θ (g)

 σ (g)r k −1

v M,θ,0 k R, g − (−1) for all k  1 and all g ∈ G(F). Since the left hand side is invariant by left translation of g by T G (F), we also have







G 1 a M,θ

 v M,θ (g)

 σT G \G (g)r k −1

v M,θ,0 k R, g − (−1) for all k  1 and all g ∈ G(F). By Lemma 2.9.4, this implies that 

 D (t)  H

lim

k→∞ T (F)

G

= (−1)a M,θ

T (F)

A T (F)A(F)\G(F)

f (g

−1

tg) v M,θ,0



D H (t)

A T (F)A(F)\G(F)

 1 R, g dgχ (t)−1 dt k

f (g −1 tg)v M,θ (g)dgχ (t)−1 dt

From this and 5.4.1 (which is of course also true if we replace R by k1 R, k  1) we deduce the proposition.   5.5 End of the proof By the descent formula 2.8.4 and Proposition 5.4.1 we have " # G χ lim J N ,T ( f ) = (−1)a M,θ ν(T ) A T G (F): A T (F)A(F) N →∞   Q G × d M,θ (L) Φ M (t, f )χ (t)−1 dt L∈L(M)

123

T (F)

On distinguished square-integrable representations

Since f is strongly cuspidal only the term corresponding to L = G can contribute to the sum above so that " # G G χ (G) lim J N ,T ( f ) = (−1)a M,θ ν(T ) A T G (F): A T (F)A(F) d M,θ N →∞  Φ M (t, f )χ (t)−1 dt (5.5.1) × T (F)

Assume first that T is not elliptic in H . We distinguish two cases: – If M  = Cent G (A T G ) we have Φ M (t, f ) = 0 for all t ∈ T reg (F) as M  = M(t) so that the limit 5.5.1 vanishes. H – If M = Cent G (A T G ), we have AG,θ M = AT  = 0 (as T is not elliptic in G (G) = 0 and the limit 5.5.1 also vanishes in this case. H ), thus d M,θ Hence, in both cases the limit 5.5.1 equals zero for T nonelliptic in H . Now, if T is elliptic in H we have A T = A H , A T G = A and AG,θ M = 0 so that " # G d M,θ (G) = 1, A T G (F) : A T (F)A(F) = 1 and ν(T ) = ν(H ) and we get χ lim J ( f ) N →∞ N ,T

G aM

= (−1)

 ν(H )

 = ν(H )

T (F)

Φ M (t, f )χ (t)−1 dt

D (t)Θ f (t)χ (t)−1 dt H

T (F)

Theorem 5.1.1 now follows from the above equality, 5.2.3 and 5.2.5.

 

6 Applications to a conjecture of Prasad In this chapter E/F is a quadratic extension, H is a connected reductive group over F and G := R E/F H E . We will denote by θ the involution of G induced by the nontrivial element of Gal(E/F). Hence H = G θ . As before we set G := G/A G and H := H/A H . If Q is an algebraic subgroup of H then R E/F Q E is an algebraic subgroup of G. Also, note that if P (resp. M) is a parabolic (resp. Levi) subgroup of G which is θ -stable [i.e. θ (P) = P, resp. θ (M) = M] then there exists a parabolic (resp. Levi) subgroup P (resp. M) of H such that P = R E/F P E (resp. M = R E/F M E ). We will denote by R(G) the space of virtual representations of G(F) that is the complex vector spaces with basis Irr(G). Similarly, if A is an abelian group we will denote by R(A) the space of virtual characters of A. We will write H i (F, ·) for the functors of Galois cohomology and if H, G are algebraic groups over F with H a subgroup of G we will set   ker 1 (F; H, G ) := Ker H 1 (F, H) → H 1 (F, G )

123

R. Beuzart-Plessis

By [35, Theorem 1.2], for every connected reductive group over F there exists a natural structure of abelian group on H 1 (F, G ) which is uniquely characterized by the fact that for every elliptic maximal torus T ⊂ G the natural map H 1 (F, T ) → H 1 (F, G ) is a group morphism. Moreover, for every connected reductive groups H, G and every morphism H → G the induced map H 1 (F, H) → H 1 (F, G ) is a group morphism. Indeed, if T is an elliptic maximal torus in H (whose existence is guaranteed by [33, p. 271]) and T  is a maximal torus of G containing the image of T , then we have a commuting square H 1 (F, T )

H 1 (F, T  )

H 1 (F, H)

H 1 (F, G )

where the upper, left and right arrows are morphisms of groups and moreover the map H 1 (F, T ) → H 1 (F, H) is surjective [35, Lemma 10.2]. From these, it easily follows that H 1 (F, H) → H 1 (F, G ) is a morphism of abelian groups. Finally, for every π ∈ Irr(G) and every continuous character χ of H (F) we recall that in Sect. 4.1 we have defined a multiplicity m(π, χ ) := dim Hom H (π, χ ) which is always finite by [18, Theorem 4.5]. The function π ∈ Irr(G) → m(π, χ ) extends by linearity to R(G). 6.1 A formula for the multiplicity We will denote by Γell (H ) the set of regular elliptic conjugacy classes in H (F) and we equip this set with a topology and a measure characterized by the fact that for all x ∈ H reg (F) the map t ∈ G x (F) → t x ∈ Γell (H ), which is welldefined in a neighborhood of the identity, is a local isomorphism preserving measures near 1 (recall that in Sect. 2.1 we have fixed Haar measures on the F-points of any torus and in particular on G x (F)). More concretely, if we fix a set Tell (H ) of representatives of the H (F)-conjugacy classes of elliptic maximal tori in H , then for every integrable function ϕ on Γell (H ) we have the following integration formula  Γell (H )

123

ϕ(x)d x =

 T ∈Tell (H )

−1

|W (H, T )|

 T (F)

ϕ(t)dt

On distinguished square-integrable representations

where we have set T := T /A H for all T ∈ Tell (H ) and we recall since T is anisotropic, the Haar measure on T (F) is of total mass 1. For all π ∈ Irr(G) and every continuous character χ of H (F) with ωπ |A H (F) = χ|A H (F) , set  m geom (π, χ ) :=

Γell (H )

D H (x)Θπ (x)χ (x)−1 d x

This expression makes sense since semisimple regular elements of H are also semisimple and regular in G and the function x ∈ Hreg (F) → D H (x)1/2 Θπ (x) is locally bounded on H (F) by [25, Theorem 16.3] and the identity D H (x) = D G (x)1/2 . Recall that we are denoting by Irr sqr (G) the set of (equivalence classes of) irreducible essentially square-integrable representations of G(F). The main theorem of this section is the following. Theorem 6.1.1 For all π ∈ Irr sqr (G) and every continuous character χ of H (F) with ωπ |A H (F) = χ|A H (F) we have m(π, χ ) = m geom (π, χ ) Proof Up to twisting π and χ by real unramified characters, we may assume that ωπ and χ are unitary. Set ω := ωπ |AG (F) . By Theorems 4.1.1 and 5.1.1, for all f ∈ 0 Cω (G) we have   m(σ, χ ) Trace(σ ∨ ( f )) = D H (x)Θ f (x)χ (x)−1 d x σ ∈Irr ω,sqr (G)

Γell (H )

By 2.4.1 and Proposition 2.6.1, when we apply this equality to a coefficient of π we get the identity of the theorem.   6.2 Galoisian characters and Prasad’s character ω H,E Let Hˇ denote the complex dual group of H , Z ( Hˇ ) be its center and W F be the Weil group of F. Denoting by Homcont (H (F), C× ) the group of continuous characters of H (F), Langlands has defined an homomorphism α H : H 1 (W F , Z ( Hˇ )) → Homcont (H (F), C× ) which is injective since F is p-adic but is not always surjective although it is most of the time (e.g. if H is quasi-split). We refer the reader to [37] for discussion of these matters. We will call the image of α H the set of Galoisian characters (of H (F)). Assume that H is semi-simple. Let Hsc

123

R. Beuzart-Plessis

be the simply connected cover of H and π1 (H ) be the kernel of the projection Hsc → H . By Tate–Nakayama duality, we have an isomorphism H 1 (W F , Z ( Hˇ ))  H 1 (F, π1 (H )) D , where (·) D denotes duality for finite abelian groups, and the morphism α H is the composition of this isomorphism with the (dual of the) connecting map H (F) → H 1 (F, π1 (H )). In particular, in this case, a character of H (F) is Galoisian if and only if it factorizes through H 1 (F, π1 (H )). In [43], Prasad has defined a quadratic character ω H,E : H (F) → {±1} which depends not only on H but also on the quadratic extension E/F. It is a Galoisian character whose simplest definition is as the image by α H of the cocycle c defined by c(w) = 1 if w ∈ W E (the Weil group of E) and c(w) = z ifw ∈ W F \W E , where z denotes the image of the central element

−1 ∈ SL 2 (C) by any principal SL 2 -morphism SL 2 (C) → Hˇ . In −1 what follows we shall need another description of Prasad’s character (see [43, §8] for the equivalence between the two definitions). First of all, ω H,E is the pullback by H (F) → Had (F) of ω Had ,E , where Had denotes the adjoint group of H . Thus, to describe ω H,E we may assume that H is semisimple. We introduce notations as before: Hsc is the simply connected cover of H and π1 (H ) stands for the kernel of the projection Hsc → H . Let B ⊂ Hsc,F and T ⊂ B be a Borel subgroup and a maximal torus thereof (both a priori only defined over F). Let ρ ∈ X ∗ (T ) be the half sum of the positive roots of T with F respect to B (this belongs to the character lattice of T since Hsc is simplyconnected). Then, it can be easily shown that the restriction of ρ to π1 (H ) induces a morphism π1 (H ) → μ2 defined over F. Pushing this through the inclusion μ2 → Ker N E/F we get a morphism π1 (H ) → Ker N E/F and ω H,E is simply the composition of the connecting map Had (F) → H 1 (F, π1 (H )) with the corresponding homomorphism between Galoisian H 1 ’s: H 1 (F, π1 (H )) → H 1 (F, Ker N E/F )  {±1}. 6.3 First application: comparison between inner forms Let H  be another connected reductive group over F and let ψ H : H F  H  F be an inner twisting. Set G  := R E/F H  . Then ψ H induces an inner twisting ψ G : G F  G  (actually, there are natural isomorphisms G F  H F × H F , F G   H  × H  and using these as identifications we just have ψ G = ψ H × F F F ψ H ). Recall that two regular elements x ∈ G reg (F) and x  ∈ G reg (F) are stably conjugate if there exists g ∈ G(F) such that x  = ψ G (gxg −1 ) and the isomorphism ψ G ◦ Ad(g): G x,F  G   is defined over F. Similarly, two regular x ,F

123

On distinguished square-integrable representations

elements of G(F) are stably conjugate if they are conjugate by an element of G(F) which induces an isomorphism defined over F between their connected centralizers. We say that a virtual representation Π ∈ R(G) (or Π  ∈ R(G  )) is stable if its character ΘΠ (or ΘΠ  ) is constant on regular stable conjugacy classes in G(F) [resp. in G  (F)]. Two stable virtual representations Π ∈ R(G) and Π  ∈ R(G  ) are said to be transfer of each other if for all pairs (x, x  ) ∈ G reg (F) × G reg (F) of stably conjugate regular elements we have ΘΠ (x) = ΘΠ  (x  ). By the main results of [7], every stable virtual representation Π ∈ R(G) is the transfer of a stable virtual representation Π  ∈ R(G  ) and conversely. We define similarly the notion of stable conjugacy for regular elements in H (F) and H  (F) and of transfer between (virtual) representations of H (F) and H  (F). The inner twist ψ H allows to identify the L-groups of H and H  and thus to get an identification H 1 (W F , Z ( Hˇ )) = H 1 (W F , Z ( Hˇ  )). We say that two Galoisian characters χ , χ  of H (F) and H  (F) correspond to each other if they originate from the same element of H 1 (W F , Z ( Hˇ )). Galoisian characters are always stable and if χ , χ  are Galoisian characters of H (F), H  (F) respectively that correspond to each other then they are also transfer of each other. Theorem 6.3.1 Let Π and Π  be stable virtual essentially square-integrable representations of G(F) and G  (F) respectively. Let χ and χ  be Galoisian characters of H (F) and H  (F) respectively. Then, if Π, Π  are transfer of each other and χ , χ  correspond to each other, we have m(Π, χ ) = m(Π  , χ  ) Proof Let Γell (H )/stab be the set of stable conjugacy classes in Γell (H ). It is easy to see that we can equip Γell (H )/stab with a unique topology and a unique measure such that the natural projection p: Γell (H )  Γell (H )/stab  is a local isomorphism preserving measures locally. We define Γell (H )/stab  and equip it with a topology and a measure in a similar way. Let p  : Γell (H )   Γell (H )/stab be the natural projection. Since Π and Π  are stable and essentially square-integrable, by Theorem 6.1.1 we have  m(Π, χ ) =

Γell (H )/stab

| p −1 (x)|D H (x)ΘΠ (x)χ (x)−1 d x

and 



m(Π , χ ) =

 

Γell (H )/stab

| p

−1



(y)|D H (y)ΘΠ  (y)χ  (y)−1 dy

123

R. Beuzart-Plessis

Since we can always transfer elliptic regular elements to all inner forms (see [35, §10]), there is a bijection 

Γell (H )/stab  Γell (H )/stab characterized by: x → y if and only if x and y are stably conjugate. It is not hard to see that this bijection preserves measures locally and hence glob (F) be two stably conjugate elements. ally. Let x ∈ Hreg (F) and y ∈ Hreg As ΘΠ , ΘΠ  on the one hand and χ , χ  on the other hand are transfer of each other, we have ΘΠ (x) = ΘΠ  (y) and χ (x) = χ  (y). Moreover, we  also have D H (x) = D H (y). Therefore, to get the theorem it only remains cohomological to show that | p −1 (x)| = | p  −1 (y)|. By

standard

1 arguments

−1 1 −1 



we have | p (x)| = ker (F; T, G) and | p (y)| = ker (F; T  , G  )

where T := G x and T  := G y . Since F is p-adic, by [35, Theorem 1.2] there exist structures of abelian groups on H 1 (F, G) and H 1 (F, G  ) such that the natural maps H 1 (F, T ) → H 1 (F, G) and H 1 (F, T  ) → Lemma 10.2] these H 1 (F, G  ) are morphisms of groups.

by [35,

1 Moreover, 1



are surjective. It follows that ker (F; T, G) = |H (F, T )||H 1 (F, G)|−1

1

  and ker (F; T , G ) = |H 1 (F, T  )||H 1 (F, G  )|−1 . As T and T  are Fisomorphic we have H 1 (F, T )  H 1 (F, T  ) and by [35, Theorem 1.2] again  G and G  have isomorphic L-groups). we have H 1 (F, G)  H 1 (F, G

) (since 1 This suffices to conclude that ker (F; T, G) = ker 1 (F; T  , G  ) and therefore that | p −1 (x)| = | p  −1 (y)|.   6.4 Elliptic twisted Levi subgroups In this section we assume for simplicity that H is semi-simple and quasi-split (All the results presented in this section are still true, with obvious modifications, in general. However, the assumption that H is semi-simple and quasi-split simplifies a lot the proofs and, in any case, we will only need to apply them for such groups.) We say that an algebraic subgroup M of H is a twisted Levi subgroup if R E/F M E is a Levi subgroup of G. If M is a twisted Levi subgroup of H , we say that it is elliptic if AM = {1}. Lemma 6.4.1 Let M be a Levi subgroup of G and set M := M ∩ H . Then, the following assertions are equivalent: (i) (ii) (iii) (iv)

M is an elliptic twisted Levi subgroup of H ; M contains an elliptic maximal torus of H ; A M is θ -split; M is θ -split and P (M) = P θ (M).

123

On distinguished square-integrable representations

Proof (i) ⇒ (ii) By [33, p. 271], M contains a maximal torus T such that A T = AM = {1}. Thus, T is elliptic and since M is of the same (absolute) rank as H , it is also maximal in H . This proves the first implication. (ii) ⇒ (iii) Assume that M contains an elliptic maximal torus T of H and set T := R E/F T E . Then we have A M ⊂ A T and A T is θ -split [as (AθT )0 is a split torus contained in T and so is trivial] from which it follows that A M is also θ -split. (iii) ⇒ (iv) Assume that A M is θ -split. Then, M is θ -split since it is the centralizer of A M . Moreover θ acts on A M as −I d thus sending any positive chamber A+ P corresponding to P ∈ P (M) to its opposite. This shows that θ P (M) = P (M). (iv) ⇒ (i) Assume that M is θ -split and P (M) = P θ (M). In particular M is θ -stable and since M comes by restriction of scalars from a subgroup of H E (as is any evi subgroup of G), it follows that M = R E/F M E i.e. M is a twisted Levi. It only remains to show that M is elliptic. Assume, by way of contradiction, that it is not the case i.e. AM  = {1}. Then there exists a parabolic P ∈ P (M) such that HM (AM (F)) contains a nonzero element of the closure of the positive chamber associated to P. Since this element is θ fixed and the intersection of the closures of the positive chambers associated to P and P (the parabolic subgroup opposite to P) is reduced to {0} we cannot   have θ (P) = P thus contradicting the fact that P (M) = P θ (M). Let P0 be a minimal θ -split parabolic subgroup of G. We claim that P0 is also a minimal parabolic subgroup of G (hence a Borel subgroup since G is quasi-split). Proof Let B be a Borel subgroup of G. Since two minimal θ -split parabolic subgroups are always G(F)-conjugate [26, Proposition 4.9], it suffices to show the existence of g ∈ G(F) such that g Bg −1 is θ -split. Over the algebraic closure we have G F  H F × H F with θ exchanging the two copies. Since B is in the same class as its opposite Borel this shows the existence of g1 ∈ G(F) such that g1 B F g1−1 is θ -split. Then, the set U := H F g1 B F is a Zariski open subset of G F with the property that for all g ∈ U the Borel g B F g −1 is θ -split. Since G(F) is dense in G for the Zariski topology [20, Exp XIV, 6.5, 6.7] we   can find g ∈ G(F) ∩ U and it has the desired property. Set T0 := P0 ∩ θ (P0 ), Amin := A T0 , A0 := Amin,θ and denote by Δmin and Δ0 the sets of simple roots of Amin and A0 in P0 . All the parabolic subgroups that we will consider in this section will be standard with respect to P0 (i.e. contain P0 ) and when we write P = MU for such a parabolic subgroup we always mean that U is the unipotent radical and M the unique Levi component containing T0 . We have a natural projection Δmin  Δ0 and θ naturally acts on R(Amin , G) sending Δmin to −Δmin . Let Δ− be the set of simple roots

123

R. Beuzart-Plessis

α ∈ Δmin such that θ (α) = −α. It can be identified with a subset of Δ0 through he projection Δmin  Δ0 (i.e. the restriction of this projection to Δ− is injective). Let I ⊂ Δ− . We will denote by PI = M I U I the unique parabolic subgroup containing P0 such that the set of simple roots of Amin in M I ∩ P0 is precisely Δmin − I . Then PI is θ -split and we have M I = PI ∩ θ (PI ). Moreover, as A∗M I is generated by the restrictions of the roots in I , θ acts as −I d on this space showing that A M I is θ -split and thus by point (ii) of the previous lemma that M I := M I ∩ H is an elliptic twisted Levi subgroup of H . Let Hab be the quotient of H (F) by the common kernel of all the Galoisian characters χ of H (F). It is an abelian group and for all I ⊂ Δ− we will denote by M I,ab the image of M I (F) in Hab . Define C as the set of pairs (M, P) with M an elliptic twisted Levi subgroup of H and P ∈ P (M) where M := R E/F M E and D as the set of triples (T, M, P) with (M, P) ∈ C and T ⊂ M an elliptic maximal torus. We let C and D denote the H (F)-conjugacy classes in C and D respectively. We say that two pairs (M, P), (M , P  ) ∈ C are stably conjugate, and we will write (M, P) ∼stab (M , P  ), if there exists h ∈ H (F) such that h M F h −1 = M F and h PF h −1 = P  . Write C /stab for the set of stable conjugacy classes in C . F We let Tell (H ) be a set of representatives of the H (F)-conjugacy classes of elliptic maximal tori in H . Finally for all P = MU ⊃ P0 and all T ∈ Tell (H ), we define   Γ M (T ) := γ ∈ G(F); γ −1 T γ ⊂ M /M(F) Proposition 6.4.1 (i) For all T ∈ Tell (H ), all P = MU ⊇ P0 and all γ ∈ Γ M (T ) we have (T, γ Mγ −1 ∩ H, γ Pγ −1 ) ∈ D; (ii) The map 



Γ M (T ) → D

T ∈Tell (H ) P0 ⊆P=MU

γ ∈ Γ M (T ) → (T, γ Mγ −1 ∩ H, γ Pγ −1 ) [which is well-defined by (i)] is surjective and the fiber over (T, M, P) ∈ D is of cardinality |W (H, T )||W (M, T )|−1 . (iii) For all (M, P) ∈ C the fiber of the map C → C /stab containing (M, P) is of cardinality |ker 1 (F; M, H )|. (iv) For all (M, P), (M , P  ) ∈ C we have (M, P) ∼stab (M , P  ) if and only if P and P  are in the same class and the map I ⊆ Δ− → (M I , PI ) ∈ C /stab is a bijection. (v) Let (M, P), (M , P  ) ∈ C be such that (M, P) ∼stab (M , P  ). Then, for every Galoisian character χ of H (F) we have χ|M = 1 if and only if

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On distinguished square-integrable representations

χ|M = 1, where we have denoted by χ|M and χ|M the restrictions of χ to M(F) and M (F) respectively. (vi) We have the following identity in R(Hab ):  Hab (−1)|Δ− −I | |ker 1 (F; M I , H )|I ndM (1) = ω H,E I,ab I ⊂Δ−

where ω H,E : Hab → {±1} is Prasad’s character (see Sect. 6.2). Proof (i) For all T ∈ Tell (H ), all P = MU ⊃ P0 and all γ ∈ Γ M (T ), the subgroup γ Mγ −1 ∩ H contains an elliptic maximal torus of H (namely T ) and thus by Lemma 6.4.1 is an elliptic twisted Levi subgroup. This shows that (T, γ Mγ −1 ∩ H, γ Pγ −1 ) ∈ D. (ii) Let (T, M, P) ∈ D. Up to conjugation we may assume that T ∈ Tell (H ). As P0 is a minimal parabolic subgroup of G, there exist γ ∈ G(F) such that γ −1 Pγ ⊃ P0 and γ −1 R E/F M E γ ⊃ T0 . This shows the surjectivity. The claim about the cardinality of the fibers is a consequence of the two following facts: Let T, T  ∈ Tell (H ), P = MU ⊃ P0 , P  = M U  ⊃ P0 , γ ∈ Γ M (T ) and γ  ∈ Γ M  (T  ). Then the two triples (T, γ Mγ −1 ∩ H, γ Pγ −1 ) and (T  , γ M  γ −1 ∩ H, γ P  γ −1 ) are H (F)-conjugate if and only if T = T  , M = M  , P = P  and γ  ∈ Norm H (F) (T )γ M(F). (6.4.1) Let T ∈ Tell (H ), P = MU ⊃ P0 and γ ∈ Γ M (T ). Then, the image of the map Norm H (F) (T ) → Γ M (T ) h → hγ is of cardinality |W (H, T )||W (M, T )|−1 where M := γ Mγ −1 ∩ H. (6.4.2) Proof of 6.4.1 Assume that (T, γ Mγ −1 ∩ H, γ Pγ −1 ) and (T  , γ M  γ −1 ∩ H, γ P  γ −1 ) are H (F)-conjugate. By definition of Tell (H ) we have T = T  and since P and P  are both standard with respect to P0 we also have P = P  and M = M  . Thus there exists h ∈ H (F) such that     −1 h T, M, γ Pγ −1 h −1 = T, M , γ Pγ  where M := γ Mγ −1 ∩ H and M := γ  Mγ  −1 ∩ H . This equality immediately implies that h ∈ Norm H (F) (T ). Moreover, since γ Mγ −1 =

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R. Beuzart-Plessis

R E/F M E and γ  Mγ  −1 = R E/F ME we also have hγ Mγ −1 h −1 = γ  Mγ  −1 and hγ Pγ −1 h −1 = γ  Pγ  −1 proving that γ  −1 hγ normalizes both M and P i.e. γ  ∈ hγ M(F) and hence γ  ∈ Norm H (F) (T )γ M(F). This proves one direction of the claim the other being obvious. Proof of 6.4.2 As |W (H, T )||W (M, T )|−1 is the cardinality of the quotient Norm H (F) (T )/ NormM(F) (T ) it suffices to show that for all h, h  ∈ Norm H (F) (T ) we have hγ = h  γ in Γ M (T ) if and only if h  −1 h ∈ M(F). But as M = γ Mγ −1 ∩ H this immediately follows from the definition of Γ M (T ). (iii) This follows from a standard cohomological argument by noticing that M is the normalizer of the pair (M, P) in H . (iv) If (M, P) ∼stab (M , P  ) then in particular P and P  are conjugate in G(F) and thus are in the same class (i.e. are G(F)-conjugate). Conversely, assume that P and P  are in the same class and set M := R E/F M E , M  := R E/F ME . Then, there exists g ∈ G(F) such that g Pg −1 = P   and g Mg −1 = M  . Let P and P be the parabolic subgroups opposite to  P and P  with respect to M and M  respectively. Then g P g −1 = P and  since θ (P) = P, θ (P  ) = P , θ (M) = M and θ (M  ) = M  we also have θ (g)Pθ (g)−1 = θ (g Pg −1 ) = P



and θ (g)Mθ (g)−1 = θ (g Mg −1 ) = M  Therefore g −1 θ (g) normalizes both M and P and thus g −1 θ (g) ∈ M(F). Since the map   M(F) → m ∈ M(F); θ (m) = m −1 m → m −1 θ (m) is surjective, there exist m ∈ M(F) and h ∈ H (F) such that g = hm. We have h PF h −1 = P  and h M F h −1 = M showing that (M, P) and F F (M , P  ) are stably conjugate. This shows the first part of (iv). Notice that for all P ∈ P θ (M0 ) with P ⊃ P0 the torus A M , where M := P ∩ θ (P), is θ -split if and only if P = PI for some I ⊂ Δ− . Hence, the second part of (iv) follows from the first and Lemma 6.4.1 since every class of

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On distinguished square-integrable representations

parabolic subgroups contains a unique element which is standard with respect to P0 .  . Let H be the simply connected (v) We need to show that Mab = Mab sc cover of H and π1 (H ) be the kernel of the projection Hsc → H . Then we have Hab = H 1 (F, π1 (H )). Let Msc and Msc denote the inverse images of M and M in Hsc . From the short exact sequences 1 → π1 (H ) → Msc → M → 1 and 1 → π1 (H ) → Msc → M → 1, we get exact sequences 1 → Mab → Hab → H 1 (F, Msc )  → H H 1 (F, Msc ). Thus, we need and 1 → Mab ab →   to show that 1 1  Ker Hab → H (F, Msc ) = Ker Hab → H (F, Msc ) . By hypothesis, there exists h ∈ H (F) such that h M F h −1 = MF and h PF h −1 = P  . Set M := R E/F M E and M  := R E/F ME . Then, we also have F h M F h −1 = M F and it follows that for all σ ∈ Γ F we have h σ h −1 ∈ H (F) ∩ M  (F) = M (F). Choose h sc ∈ Hsc (F) which lifts h. Then, there is a bijection ι: H 1 (F, Msc )  H 1 (F, Msc ) given at the level of cocycles by   c → σ ∈ Γ F → h sc c(σ )σ h −1 sc  Let c0 be the 1-cocycle σ ∈ Γ F → h sc σ h −1 sc ∈ Msc (F) and denote by 1  1 [c0 ] its class in H (F, Msc ). Then ι−[c0 ]: H (F, Msc )  H 1 (F, Msc ) is a bijection of pointed sets (and even an isomorphism of abelian groups) making the following square commute

Hab

H 1 (F, Msc ) 

Hab

H 1 (F, Msc )

This immediately implies that the kernels of the upper and bottom arrows are identical. (vi) Let I ⊂ Δ− and denote by M I,sc the inverse image of M I in Hsc . Then, from the short exact sequence 1 → π1 (H ) → M I,sc → M I → 1 we get an exact sequence 1 → M I,ab → Hab → H 1 (F, M I,sc ) → H 1 (F, M I ) → H 2 (F, π1 (H )) By [32] the natural connecting map H 1 (F, H ) → H 2 (F, π1 (H )) is an isomorphism and it follows that the previous exact sequence can be rewritten as 1 → M I,ab → Hab → H 1 (F, M I,sc ) → ker 1 (F; M I , H ) → 1

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R. Beuzart-Plessis

From this exact sequence, we deduce the following equality in R(Hab ): Hab |ker 1 (F; M I , H )|I ndM (1) = Res IHab I nd1I (1) I,ab

(6.4.3)

where Res IHab denotes the restriction functor with respect to the morphism Hab → H 1 (F, M I,sc ) and I nd1I denotes the induction functor with respect to the morphism 1 → H 1 (F, M I,sc ). Moreover the morphism H 1 (F, MΔ− ,sc ) → H 1 (F, M I,sc ) induced by the inclusion MΔ− ⊂ M I , makes the following square commute Hab

H 1 (F, MΔ− ,sc )

Hab

H 1 (F, M I,sc ) Δ

Δ

Hab Hence, we have a factorization Res IHab = ResΔ ◦ Res I − where Res I − − denotes the restriction functor with respect to the morphism H 1 (F, MΔ− ,sc ) → H 1 (F, M I,sc ). Combining this with 6.4.3, we get the identity

 I ⊂Δ−

Hab (−1)|Δ− −I | |ker 1 (F; M I , H )|I ndM (1) I,ab

⎛

Hab ⎝ = ResΔ −



⎞ Δ (−1)|Δ− −I | Res I − I nd1I (1)⎠

(6.4.4)

I ⊂Δ−

To continue, we need to compute the groups H 1 (F, M I,sc ) and the morphisms H 1 (F, MΔ− ,sc ) → H 1 (F, M I,sc ) explicitly for all I ⊂ Δ− . Let T0,sc and P0,sc denote the inverse image of T0 and P0 in G sc := R E/F Hsc,E . Since T0,sc is θ -stable, there exists a maximal torus T0,sc of H such that T0,sc = R E/F T0,sc,E . Moreover, there exists a Borel subgroup P0,sc of Hsc,E such that P0,sc = R E/F P0,sc . In what follows, we fix an algebraic closure F of F containing E and we set Γ E := Gal(F/E). Let Δmin,F be the set of simple roots of T0,sc,F in P0,sc,F . It is a subset of X ∗ (T0,sc ) which is Γ E -stable (as P0,sc is defined over E) and we have a F natural surjection Δmin,F  Δmin (obtained by restriction to the maximal split subtorus of T0,sc,E ) whose fibers are precisely the Γ E -orbits in Δmin,F . For all β ∈ Δmin,F , we will denote by β ∈ X ∗ (T0,sc ) the corresponding F weight and for all α ∈ Δmin we define

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On distinguished square-integrable representations



α :=

β

β∈Δmin,F ;β →α

where the sum is over the set of simple roots β ∈ Δmin,F mapping to α through the projection Δmin,F  Δmin . We always have α ∈ X ∗E (T0,sc ) but we warn the reader that in general α is NOT the weight associated to the simple root α in the usual sense (although it is proportional to it). Let I ⊂ Δ− . For all α ∈ I the character α extends to M I,sc,F and is defined over E and thus gives rise to a character R E/F M I,sc,E → R E/F Gm,E . Since θ (α ) = −α , this last character induces a morphism M I,sc → Ker N E/F that we will also  I denote by α . Consider the torus TI := Ker N E/F and the morphism κ I := (α )α∈I : M I,sc → TI Then we claim that The induced map H 1 (κ I ): H 1 (F, M I,sc ) → H 1 (F, TI )

(6.4.5)

is an isomorphism. Let M I,sc,der be the derived subgroup of M I,sc and set TI := M I,sc /M I,sc,der Then, we have an exact sequence H 1 (F, M I,sc,der ) → H 1 (F, M I,sc ) → H 1 (F, TI ) and H 1 (F, M I,sc,der ) is trivial by [32] since M I,sc,der is simply connected. Moreover the morphism H 1 (F, M I,sc ) → H 1 (F, TI ) is surjective. Indeed, if T ⊂ M I,sc is a maximal elliptic torus (which exists by [33, p. 271]) then the kernel of the projection T → TI is a maximal anisotropic torus of M I,sc,der and thus by Tate–Nakayama duality its H 2 vanishes and the morphism H 1 (F, T ) → H 1 (F, TI ), which obviously factorizes through H 1 (F, M I,sc ), is surjective. Therefore, the morphism H 1 (F, M I,sc ) → H 1 (F, TI ) is an isomorphism and it only remains to show that the natural map H 1 (F, TI ) → H 1 (F, TI ) is also an isomorphism. Let TI be the kernel of the projection TI → TI . Then TI is connected since it is a quotient of the common kernel of all the α ’s, α ∈ I , which is a connected group [this follows from the fact that {α ; α ∈ I } generates X ∗E (M I,sc )]. Therefore, TI is an anisotropic torus (since TI is) and by Tate–Nakayama duality again we just need to prove the injectivity of H 1 (F, TI ) → H 1 (F, TI ) or, equivalently, that the map H 1 (F, TI ) → H 1 (F, TI ) has trivial image. We have norm maps

123

R. Beuzart-Plessis   → T  giving rise to a commuting N : R E/F TI,E → TI and N : R E/F TI,E I square

H 1 (F, TI ) N

H 1 (F, TI ) N

 ) H 1 (F, R E/F TI,E

 ) H 1 (F, R E/F TI,E

I Since TI,E  Gm,E is the maximal split torus quotient of M I,sc,E , the  torus R E/F TI,E is anisotropic and therefore so is the kernel of the norm  → T  (which is automatically connected). Hence, by Tate– map R E/F TI,E I  ) → H 1 (F, T  ) is surjective and Nakayama again, the map H 1 (F, R E/F TI,E I it follows, by the above commuting square, that to conclude we only need  ) is trivial. By an argument similar to what to show that H 1 (F, R E/F TI,E  ), where we have done before, the map H 1 (F, M I,sc ) → H 1 (F, R E/F TI,E M I,sc := R E/F M I,sc,E , is surjective. Since M I,sc is a Levi subgroup of G sc , the map H 1 (F, M I,sc ) → H 1 (F, G sc ) has trivial kernel and by [32] it follows  ) = 1 also and this ends the that H 1 (F, M I,sc ) = 1. Hence, H 1 (F, R E/F TI,E proof of 6.4.5. By 6.4.5, we have isomorphisms

H 1 (F, M I,sc )  H 1 (F, TI )  (Z/2Z) I

(6.4.6)

for all I ⊂ Δ− such that the maps H 1 (F, MΔ− ,sc ) → H 1 (F, M I,sc ) correspond to the natural projections (Z/2Z)Δ− → (Z/2Z) I (eα )α∈Δ− → (eα )α∈I We can now compute the right hand side of 6.4.4. For all I ⊂ Δ− , let A I denote the kernel of the projection H 1 (F, MΔ− ,sc ) → H 1 (F, M I,sc ). In particular we have A∅ = H 1 (F, MΔ− ,sc )  (Z/2Z)Δ− and A I is the subgroup (Z/2Z)Δ− \I . For all I ⊂ Δ− we have  Δ Res I − I nd1I (1) = χ χ ∈Irr(A∅ );χ|A I =1

where Irr(A∅ ) denotes the set of characters of A∅ . Thus the sum  Δ (−1)|Δ− −I | Res I − I nd1I (1) I ⊂Δ−

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On distinguished square-integrable representations

is equal to 



(−1)|Δ− −I |

I ⊂Δ−

χ=

χ ∈Irr(A∅ );χ|A I =1

⎛





⎝

χ ∈Irr(A∅ )

⎞ (−1)|Δ− −I | ⎠ χ

I ⊂Δ− ;χ|A I =1

Let χ ∈ Irr(A∅ ). By the above description of the subgroups A I ⊂ A∅ , we see that A I + A J = A I ∩J for all I, J ⊂ Δ− . Hence there exists a smallest subset Iχ ⊂ Δ− such that χ|A Iχ = 1 and we have 

(−1)|Δ− −I | =



(−1)|Δ− −I |

Iχ ⊂I ⊂Δ−

I ⊂Δ− ;χ|A I =1

This sum is zero unless Iχ = Δ− in which case it is equal to 1. Using again the explicit determination of the subgroups A I ⊂ A∅ , it is easy to see that there is only one character χ ∈ Irr(A∅ ) with Iχ = Δ− , namely the character ω defined (via the isomorphism 6.4.6) by (Z/2Z)Δ− → {±1}



(eα )α∈Δ− → (−1)

α∈Δ− eα

Therefore by 6.4.4, we get  Hab Hab (−1)|Δ− −I | |ker 1 (F; M I , H )|I ndM (1) = ResΔ (ω) − I,ab I ⊂Δ−

Hab and it only remains to show that ResΔ (ω) = ω H,E . Set −

ρ :=



α =

α∈Δmin

 β∈Δmin,F

β , ρ1 :=





α and ρ2 :=

α∈Δ−

α

α∈Δmin \Δ−

Then, by restriction ρ, ρ1 and ρ2 define three morphisms π1 (H ) → Ker N E/F . Hab (ω) are the morphisms By definition, ω H,E and ResΔ − Hab = H 1 (F, π1 (H )) → H 1 (F, Ker N E/F )  {±1} induced by ρ and ρ1 respectively. Thus it suffices to show that the morphism H 1 (ρ2 ) : H 1 (F, π1 (H )) → H 1 (F, Ker N E/F ) induced by ρ2 is trivial. By definition of Δ− we can find a subset S ⊂ Δmin \Δ− such that Δmin \Δ− = S  −θ (S) (disjoint union). For all α ∈ S, the character

123

R. Beuzart-Plessis

α induces a morphism π1 (H ) → R E/F Gm,E  R E/F (Ker N E/F ) E . By the decomposition Δmin \Δ− = S −θ (S), we see that H 1 (ρ2 ) is the composition of 

H 1 (α ): H 1 (F, π1 (H )) → H 1 (F, R E/F (Ker N E/F ) E )

α∈S

with the norm map H 1 (F, R E/F (Ker N E/F ) E ) → H 1 (F, Ker N E/F ). By Hilbert 90, we have H 1 (F, R E/F (Ker N E/F ) E ) = 1 and thus H 1 (ρ2 ) = 0. This ends the proof of the proposition.

 

6.5 Reminder on the Steinberg representation Fix a minimal parabolic subgroup P0 of G with Levi decomposition P0 = M0 U0 . Then, the Steinberg representation of G(F) is by definition the following virtual representation 

St(G) :=

(−1)a M −a M0 i PG (δ P ) 1/2

P0 ⊂P=MU

where i PG denotes the functor of normalized parabolic induction. It follows from [13] that St(G) is in fact a true representation of G(F) which is moreover irreducible and square-integrable. Obviously, the Steinberg representation has trivial central character. Moreover, if G ad denotes the adjoint group of G then St(G) is the pullback of St(G ad ) by the projection G(F) → G ad (F). For all x ∈ G reg (F) and all P = MU ⊃ P0 , let us set   Γ M (x) := γ ∈ G(F); γ −1 xγ ∈ M(F) /M(F) Then, the character ΘSt(G) of St(G) is given by the following formula [24, Theorem 30] 

D G (x)1/2 ΘSt(G) (x) =

(−1)a M −a M0

P0 ⊂P=MU

×



γ ∈Γ M (x)

123

D M (γ −1 xγ )1/2 δ P (γ −1 xγ )1/2

(6.5.1)

On distinguished square-integrable representations

for all x ∈ G reg (F). In particular, we have ΘSt(G) (x) = (−1)aG −a M0

(6.5.2)

for all x ∈ G(F)ell The representation St(G) is stable: this follows from the fact that parabolic induction sends stable distributions to stable distributions. Let H  be another connected reductive group over F and ψ H : H F  H  be an inner twisting. F Following notations of Sect. 6.3 we define G  := R E/F H E and the inner twisting ψ G : G F  G  . Let P0 = M0 N0 be a minimal parabolic subgroup F of G  . Since transfer is compatible with parabolic induction we have that a  (−1)a M0 St(G) and (−1) M0 St(G  ) are transfer of each other. Moreover, in a M −a  our situation we have (−1) 0 M0 = 1. Indeed, by the main result of [34] a M −a  we have (−1) 0 M0 = e(G)e(G  ) where e(G) and e(G  ) are the so-called Kottwitz signs of G and G  respectively and it follows from points (4) and (5) of the Corollary of loc.cit. that e(G) = e(H )2 = 1 and e(G  ) = e(H  )2 = 1. Thus,we have that St(G) and St(G  ) are transfer of each other.

(6.5.3)

6.6 Harish-Chandra’s orthogonality relations for discrete series Let π and σ be essentially square-integrable representations of H (F) with central characters coinciding on A H (F). Then, we have the following orthogonality relation between the characters of π and σ ∨ (the smooth contragredient of σ ) which is due to Harish-Chandra (see [14, Theorem 3]): 

 Γell (H )

D H (x)Θπ (x)Θσ ∨ (x)d x =

1 0

if π  σ otherwise

where the measure on Γell (H ) is the one introduced in Sect. 6.1. These relations can be seen as an analog of Theorem 6.1.1 in the case where E = F × F. Let χ be a continuous character of H (F). In the particular case where π = St(H ) and σ = St(H ) ⊗ χ , by Sect. 6.5.2 we get the relation 

 D (x)χ (x)d x = H

Γell (H )

1 0

if χ = 1 otherwise

(6.6.1)

Indeed if χ is nontrivial then St(H ) ⊗ χ   St(H ) since St(H ) and St(H ) ⊗ χ have different cuspidal supports.

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R. Beuzart-Plessis

6.7 Second application: multiplicity of the Steinberg representation Theorem 6.7.1 For every Galoisian character χ of H (F) we have  m(St(G), χ ) =

1 0

if χ = ω H,E otherwise.

Proof Let χ be a Galoisian character of H (F). If the restriction of χ to the center of H (F) is nontrivial then obviously m(St(G), χ ) = 0 since St(G) has trivial central character. We assume now that χ restricted to the center of H (F) is trivial. Let Had be the adjoint group of H and set G ad := R E/F Had,E . Let H (F)ad denote the image of H (F) by the projection H (F) → Had (F) (i.e. the quotient H (F)/Z H (F)). Then, since St(G) is the pullback of St(G ad ) to G(F), by Frobenius reciprocity we have H (F)

m(St(G), χ ) = dim Hom H (F)ad (St(G), χ ) = dim Hom Had (F) (St(G ad ), I nd H ad (F)ad χ ) H (F)

The representation I nd H ad (F)ad χ is a multiplicity-free sum of Galoisian characters containing ω Had ,E if and only if χ = ω H,E . This shows that the statement of the theorem for Had implies the statement of the theorem for H . Thus, we may assume that H is adjoint. Moreover, by Sect. 6.5.3 and Theorem 6.3.1, up to replacing H by its quasi-split inner form, we may also assume that H is quasi-split. We will now use freely the notations introduced in Sect. 6.4. By Theorem 6.1.1 and Sect. 6.5.1, we have 

m(St(G), χ ) =

|W (H, T )|−1

T ∈Tell (H )

×





(−1)a M −a M0

P0 ⊆P=MU



γ ∈Γ M (T ) T (F)

D M (γ −1 tγ )1/2 χ (t)−1 dt

(Note that we have δ P (γ −1 tγ ) = 1 for all γ and t as in the expression above since γ −1 tγ is a compact element). By Proposition 6.4.1(ii), it follows that   aM −a M0 (−1) D M (x)χ (x)−1 d x m(St(G), χ ) = Γell (M)

(M,P)∈C

where for all (M, P) ∈ C we have set aM := a M with M := R E/F M E . Thus, by 6.6.1 we also have m(St(G), χ ) =

 (M,P)∈C

123

(−1)aM −a M0 (χ|M , 1)

On distinguished square-integrable representations

where χ|M denotes the restriction of χ to M(F) and (·, ·) denotes the natural scalar product on the space of virtual characters of Mab . By Proposition 6.4.1(iii)–(v), this can be rewritten as m(St(G), χ ) =



  (−1)aM I −a M0 |ker 1 (F; M I , H )| χ|M I , 1

I ⊂Δ−

By Frobenius reciprocity, it follows that m(St(G), χ ) =

 I ⊂Δ−

Hab (−1)aM I −a M0 |ker 1 (F; M I , H )|(χ , I ndM 1) I,ab

It is easy to see that (−1)aM I −a M0 = (−1)|Δ− −I | for all I ⊂ Δ− and therefore by Proposition 6.4.1(vi) the last expression above is equal to 1 if χ = ω H,E and 0 otherwise.   Acknowledgements I am grateful to Jean-Loup Waldspurger for a very careful proofreading of a first version of this paper. I also thank the referee for correcting many inaccuracies and for the numerous comments to make the text more readable. The author has benefited from a Grant of Agence Nationale de la Recherche with reference ANR-13-BS01-0012 FERPLAY.

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