The Descent Map from Automorphic Representations of Gl (n) to Classical Groups

THE DESCENT MAP FROM AUTOMORPHIC REPRESENTATIONS OF GL(n) TO CLASSICAL GROUPS

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THE DESCENT MAP FROM AUTOMORPHIC REPRESENTATIONS OF GL(n) TO CLASSICAL GROUPS David Ginzburg Tel Aviv University, Israel

Stephen Rallis The Ohio State University, USA

David Soudry Tel Aviv University, Israel

World Scientific NEW JERSEY

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TA I P E I

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CHENNAI

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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THE DESCENT MAP FROM AUTOMORPHIC REPRESENTATIONS OF GL(n) TO CLASSICAL GROUPS Copyright © 2011 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN-13 978-981-4304-98-6 ISBN-10 981-4304-98-0

Printed in Singapore.

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Preface

In this work we give a detailed account of the descent method, which constructs in an explicit way, an inverse to the (standard) Langlands functorial lift from irreducible, automorphic, cuspidal, generic representations of a classical group to the appropriate general linear group. We include metaplectic groups as well. The main idea is to apply Fourier coefficients of Gelfand-Graev type, or of Fourier-Jacobi type to certain residual Eisenstein series, built from the given automorphic representation on the general linear group. The heart of our descent method is the study of these Fourier coefficients and their analogous local Jacquet modules. The descent method is closely related to Rankin-Selberg integrals and Shimura type integrals, which represent the standard L-functions for pairs of irreducible, automorphic, cuspidal, generic representations, one on the classical (or metaplectic) group, and the other on the general linear group. We published the details of this theory of descent in the case of metaplectic groups. The details of this theory in all other cases appear here for the first time. We consider, in addition to metaplectic groups, symplectic groups, special orthogonal groups (split, or quasi-split over a given number field) and quasi-split unitary groups. In some parts of this book (mainly in Chapters 5–7) we work in the more general framework of classical groups, which are not necessarily split, or quasi-split. This book is intended for graduate students and researchers in automorphic representations and representation theory of p-adic groups. We hope that it will serve as a useful text.

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Contents

Preface

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1. Introduction

1

1.1 1.2 1.3

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formulas for the Weil representation . . . . . . . . . . . . . . . . . The case, where H is unitary and the place v splits in E . . . . . .

2. On Certain Residual Representations 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

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3. Coefficients of Gelfand-Graev Type, of Fourier-Jacobi Type, and Descent

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3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

The groups . . . . . . . . . . . . . . . . . . . . . . The Eisenstein series to be considered . . . . . . . L-groups and representations related to Pϕ . . . . The residue representation . . . . . . . . . . . . . . The case of a maximal parabolic subgroup (r = 1) A preliminary lemma on Eisenstein series on GLn . Constant terms of E(h, fτ,¯s ) . . . . . . . . . . . . . Description of W (Mϕ , Dk ) . . . . . . . . . . . . . . Continuation of the proof of Theorem 2.1 . . . . .

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Gelfand-Graev coefficients . . . . . . . . . . . . . . . . . . . . . . . Fourier-Jacobi coefficients . . . . . . . . . . . . . . . . . . . . . . . Nilpotent orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Global integrals representing L-functions I . . . . . . . . . . . . . . Global integrals representing L-functions II . . . . . . . . . . . . . Definition of the descent . . . . . . . . . . . . . . . . . . . . . . . . Definition of Jacquet modules corresponding to Gelfand-Graev characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of Jacquet modules corresponding to Fourier-Jacobi characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

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4. Some double coset decompositions 4.1 4.2 4.3 4.4

The space Qj \h(V )k /Qℓ . . . . . . . . . . A set of representatives for Qj \h(V )k /Qℓ Stabilizers . . . . . . . . . . . . . . . . . . The set Q\h(Wm,ℓ )k /Lℓ,w0 . . . . . . . .

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5. Jacquet modules of parabolic inductions: Gelfand-Graev characters 5.1 5.2

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The case where K is a field . . . . . . . . . . . . . . . . . . . . . . 81 The case K = k ⊕ k . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6. Jacquet modules of parabolic inductions: Fourier-Jacobi characters 6.1 6.2

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The case where K is a field . . . . . . . . . . . . . . . . . . . . . . 121 The case K = k ⊕ k . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7. The tower property 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8

A general lemma on “exchanging roots” . . . . . . . . . . . . . . A formula for constant terms of Gelfand-Graev coefficients . . . . Global Gelfand-Graev models for cuspidal representations . . . . The general case: H is neither split nor quasi-split . . . . . . . . Global Gelfand-Graev models for the residual representations Eτ¯ A formula for constant terms of Fourier-Jacobi coefficients . . . . Global Fourier-Jacobi models for cuspidal representations . . . . Global Fourier-Jacobi models for the residual representations Eτ¯

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8. Non-vanishing of the descent I 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10

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The Fourier coefficient corresponding to the partition (m, m, m′ − 2m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conjugation of Sm by the element αm . . . . . . . . . . . . . . . . Exchanging the roots y1,2 and x1,1 (dimE V = 2m , m > 2) . . . . First induction step: exchanging the roots yi,j and xj−1,i , for 1 ≤ i < j ≤ [ m+1 2 ]; dimE V = 2m . . . . . . . . . . . . . . . . . . . . . . First induction step: odd orthogonal groups . . . . . . . . . . . . . Second induction step: exchanging the roots yi,j and xj−1,i , for i + j ≤ m + 1, j > [ m+1 2 ] (dimE V = 2m) . . . . . . . . . . . . . . . Completion of the proof of Theorems 8.1, 8.2; dimE V = 2m . . . . Completion of the proof of Theorem 8.3 . . . . . . . . . . . . . . . Second induction step: odd orthogonal groups . . . . . . . . . . . . Completion of the proof of Theorems 8.1, 8.2; h(V ) odd orthogonal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

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9. Non-vanishing of the descent II f 4n+2 (A) . . . . . . . . . . . . . . . . . . . . . . . 9.1 The case HA = Sp 9.2 The case H = SO4n+1 . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Whittaker coefficients of the descent corresponding to GelfandGraev coefficients: the unipotent group and its character; h(V ) 6= SO4n+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Conjugation by the element ηˆm . . . . . . . . . . . . . . . . . . . . 9.5 Exchanging roots: h(V ) = SO4n , U4n . . . . . . . . . . . . . . . . 9.6 Nonvanishing of the Whittaker coefficient of the descent corresponding to Gelfand-Graev coefficients: h(V ) = SO4n , U4n . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Nonvanishing of the Whittaker coefficient of the descent corresponding to Gelfand-Graev coefficients: h(V ) = U4n+2 , SO4n+3 . . . . . . . . . . . . . . . . . . . . . . . . 9.8 The Whittaker coefficient of the descent corresponding to Fourierf 4n+2 (A) . . . . . . . . . . . . . . . . . Jacobi coefficients: HA 6= Sp 9.9 The nonvanishing of the Whittaker coefficient of the descent corresponding to Fourier-Jacobi coefficients: f 4n (A), U4n (A) . . . . . . . . . . . . . . . . . . . HA = Sp4n (A), Sp 9.10 Nonvanishing of the Whittaker coefficient of the descent corresponding to Fourier-Jacobi coefficients: h(V ) = U4n+2 . . . . 10. Global genericity of the descent and global integrals 10.1 10.2 10.3 10.4 10.5 10.6

Statement of the theorems . . . . . Proof of Theorem 10.3 . . . . . . . Proof of Theorem 10.4 . . . . . . . A family of dual global integrals I . A family of dual global integrals II L-functions . . . . . . . . . . . . .

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The cuspidal part of the weak The image of the weak lift . . On generalized endoscopy . . Base change . . . . . . . . . . Automorphic induction . . .

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11. Langlands (weak) functorial lift and descent 11.1 11.2 11.3 11.4 11.5

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Bibliography

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Index

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Chapter 1

Introduction

1.1

Overview

The essence of the descent method for automorphic forms is the application of certain Fourier coefficients of Gelfand-Graev type or of Fourier-Jacobi type to some appropriate residues of Eisenstein series. In this book, we will study this method, in detail, when these Eisenstein series are induced from “Siegel type” maximal parabolic subgroups. As a result, we will construct an explicit inverse map to the weak Langlands functorial lift from irreducible, automorphic, cuspidal, generic representations of GA to irreducible, automorphic representations of GLm (AE ), where G is a classical group, defined over a number field F , which is split or quasi-split, over F , such that the standard representation of L G0 has degree m, and E is either F , or a quadratic extension of F (in case of quasi-split unitary groups); A is the Adele ring of F . This descent is applicable to irreducible, automorphic representations of GLm (AE ), which are in the so called “tempered” part of the image of the functorial lift from GA . We prove that these are precisely the representations of GLm (AE ) obtained as functorial lifts from generic (cuspidal) representations of GA . The existence of the weak functorial lift above, when restricted to generic representations of GA , was proved in [Cogdell, Kim, Piatetski-Shapiro and Shahidi (2004)], [Kim and Krishnamurthy (2005)], [Cogdell, Piatetski-Shapiro and Shahidi (2011)]. The residual Eisenstein series that we consider are on a classical group, H, defined over F , which admits a “Siegel type” maximal parabolic subgroup P , defined over F , namely, its Levi subgroup M is isomorphic to ResE/F GLm . The Eisenstein series is induced from an irreducible automorphic representation of MA , having the form τ1 × · · · × τr , 1

(1.1)

tensored by | det ·|s− 2 , for a complex parameter s. The residue is at s = 1. For 1 ≤ i ≤ r, τi is an irreducible, automorphic representation of GLmi (AE ), cuspidal whenever mi > 1, and m1 +· · ·+mr = m. We assume that the representations τi are pairwise inequivalent and that there is a certain type of an L-function, depending on H, such that the partial L-function of this type of each representation τi has a 1

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pole at s = 1. The type of L function to consider is the same as the one determined by the finite dimensional representation of L M in the unipotent radical of L P . In detail, when H is the split even orthogonal group SO2m , then we consider exterior square partial L-functions LS (τi , ∧2 , s). In this case, for LS (τi , ∧2 , s) to have a pole at s = 1, mi must be even. When H is the split odd orthogonal group SO2m+1 , the L-functions above are the symmetric square L-functions LS (τi , sym2 , s). Each one is assumed to have a pole at s = 1. When H is the symplectic group Sp2m , we assume that LS (τi , s − 12 )LS (τi , ∧2 , 2s − 1) has a pole at s = 1, for all i, and when H is the quasi-split (even) unitary group U2m , corresponding to the quadratic extension E/F , the Asai L-functions LS (τi , Asai, s) are all assumed to have a pole f 2m (A) is the metaplectic at s = 1. We will also consider the case where HA = Sp S 2 cover of Sp2m (A), and then assume that L (τi , sym , s) has a pole at s = 1, for all i. The Eisenstein series on HA induced from 1

1

τ1 | det ·|s1 − 2 × · · · × τr | det ·|sr − 2

(1.2)

has a pole at (s1 , ..., sr ) = (1, ..., 1) (Chapter 2). Consider the resulting residual representation of HA , and denote it by Eτ¯ . This is the same as the residue at s = 1 of the Eisenstein series of HA induced from (1.1). We apply to the residual representation Eτ¯ Fourier coefficients of Gelfand-Graev type, or of Fourier-Jacobi type. These coefficients correspond to nilpotent orbits which correspond to partitions of the form (j, 1i ) (Sec. 3.3). Let ℓ = [ 2j ]. When we apply such a Fourier coefficient to an automorphic function on HA , we obtain an automorphic function on a group which stabilizes the corresponding character. Denote this group, for this overview, by Gℓ . It turns out that when H is orthogonal, Gℓ is also orthogonal, but in a number of variables of an opposite parity; when H is (even) unitary, Gℓ is also even unitary, unless we take a coefficient of Gelfand-Graev type, in which case Gℓ is odd unitary. Finally, when HA is symplectic (resp. metaplectic), Gℓ (A) is metaplectic (resp. symplectic). Let us denote the space obtained by application of the Fourier coefficients above, to the residual representation Eτ¯ , by σψ,ℓ (¯ τ ), and consider it as a module over Gℓ (A). Here, ψ is a nontrivial character of F \A, which enters in the definition of the Fourier coefficients above. We fix it throughout this book. The notation σψ,ℓ (¯ τ ), just used, is a little simplified, since it depends on more parameters, for example, when H is odd orthogonal, it depends on a square class α in F ∗ . The spaces σψ,ℓ (¯ τ ) are the main object of study in this book. We prove that there τ ) is nontrivial and cuspidal, and beyond ℓm , these is an index ℓm , such that σψ,ℓm (¯ spaces are identically zero. This is the tower property (Chapter 7). The complete determination of ℓm follows from the work in Chapters 5–9. Denote G = Gℓm and τ ) (this notation is a little simplified for the sake of exposition). In σψ (¯ τ ) = σψ,ℓm (¯ Chapters 5–6, we find the local unramified parameters of σψ (¯ τ ), and indeed they lift to those of the representation (1.1). The representation σψ (¯ τ ) is the descent of the representation (1.1). We also say that it is the descent of Eτ¯ . The descent construction is intimately related to the Rankin-Selberg integrals which represent the standard L-function for pairs of generic representations of GA

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and GLm (AE ). Assume, for simplicity, that G is split, and τ is cuspidal (r = 1). The definition of the descent σψ (τ ) is tailored to give at once the property that LS (σ × τ, s) has a pole at s = 1, for any irreducible summand σ of the descent of τ . Even more, the residue at s = 1, of the Rankin Selberg integral, corresponding to LS (σ × τ, s), becomes the L2 -pairing between σ ¯ and the descent of τ . The descent σψ (τ ) is canonic, in the sense that for any irreducible, automorphic, cuspidal, generic representation σ of GA , which lifts to τ , σ ¯ has a nontrivial L2 -pairing with the descent of τ . We review the theory of the global integrals above in Chapter 10. Since we know the precise image of the weak functorial lift from cuspidal generic representations to automorphic representations of GLm (AE ), we see immediately that each one of the factors τi in (1.1) is lifted from a generic, cuspidal representation σi on the Adele points of an appropriate classical group G(i). In this way we establish generalized endoscopy, starting with σ1 ⊗ · · · ⊗ σr on GA (1) × · · · × GA (r), then lift each σi (generic, cuspidal) to τi on GLmi (AE ), and next form the parabolic induction τ1 × · · ·× τr (isobaric sum). Assume, for simplicity, that τi are all cuspidal and pairwise inequivalent. Then the descent exhibits an irreducible automorphic, cuspidal, generic representation σ on an appropriate group GA , which is a lift of σ1 ⊗ · · · ⊗ σr . For example, let σ1 ,...,σr be pairwise inequivalent, irreducible, automorphic, cuspidal representations of SO2,1 (A) ∼ = PGL2 (A). Then there is an irreducible, automorphic, cuspidal, generic representation σ of SO2r+1 (A), which is a lift of σ1 ⊗ · · · ⊗ σr , corresponding to the standard homomorphism of L-groups (direct sum embedding) SL2 (C) × · · · × SL2 (C) 7→ Sp2r (C).

Similarly, we obtain base change and automorphic induction, for G (and generic, cuspidal representations). Given a residual representation Eτ¯ of HA , as above, we examine all possible Fourier coefficients of Gelfand-Graev type, or Fourier-Jacobi type, when applied to Eτ¯ , and obtain the representations (descents) σψ (¯ τ ) of GA . Sometimes the groups (G, ResE/F GLm ) do not correspond to a case of functoriality. For example, assume that τ is an irreducible, self-dual, automorphic, cuspidal representation of GL2n+1 (A), whose central character is trivial. Then LS (τ, sym2 , s) has a pole at f 4n+2 (A) s = 1. When we form the corresponding residual representation on Sp and apply the descent, given by Fourier-Jacobi coefficients, we obtain a cuspidal, generic representation σ of Sp2n (A), which lifts to τ . However, we may consider the corresponding residual Eisenstein series on SO4n+3 (A), and apply descent via Gelfand-Graev coefficients. In this case, we get a cuspidal, generic representation π of SO2n+2 (A). Standard functoriality takes representations of SO2n+2 (A) to representations of GL2n+2 (A) (and not GL2n+1 (A)). Still, π and τ are nicely related. The lift of π to GL2n+2 (A) is the parabolic induction τ × 1. The representations π and σ are related by the theta correspondence (at least up to isomorphism). In Chapters 5, 6 we consider a local analogue of the descent construction. We replace the notion of a Fourier coefficient, with respect to a character of a unipo-

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The Descent Map from Automorphic Representations of GL(n) to Classical Groups

tent group, being of Gelfand-Graev type or Fourier-Jacobi type, by the notion of a Jacquet module with respect to the analogous local unipotent group and its character. We obtain general decompositions of these Jacquet modules, when applied to parabolic inductions. These decompositions have the form of a Leibniz rule, where these Jacquet modules play the role of derivatives on the classical group side, and, on the GL-group side, we find the Bernstein-Zelevinsky derivatives. These calculations are used in a crucial way, first, to assert that the spaces σψ,ℓ (¯ τ ) are zero, if ℓ is large (larger than ℓm , defined in (3.36), (3.38)), and second, to compute the unramified parameters of σψ (¯ τ ). The calculations done in these two chapters cover more cases than needed in this book, but they will definitely become useful in further studies. Much of the work carried out in this book can be done at the local level. Namely, let H be considered over a p-adic field k, and consider the parabolic induction (1.2), where, now, the representations τi are supercuspidal, pairwise inequivalent, and we replace the pole conditions at s = 1, with a similar pole condition at s = 0, for the analogous local L-functions. Instead of the residual representation above, we take the Langlands quotient πτ¯ of the parabolic induction (1.2) at s1 = · · · = sr = 1. Now, apply the Jacquet modules above to πτ¯ , and denote the resulting spaces by (the same notation) σψ,ℓ (¯ τ ), viewed as modules over Gℓ (k). In [Ginzburg, Rallis and Soudry I (1999)] and [Ginzburg, Rallis and Soudry (2002)], we proved that τ ) is irreducible, supercuspidal and generic, in case Hk = Sp4n (k) σψ (¯ τ ) = σψ,ℓm (¯ f (k)). In [Jiang and Soudry (2003)], and [Jiang and Soudry (and then G = Sp 2n (2004)], this was translated to Hk = SO4n (k) and G = SO2n+1 (k). Similar results can be proved for quasi-split unitary groups ([Tanai (2011)]). The study of local descent in the remaining cases is a little more delicate. See [Jiang and Soudry (2011)], [Liu (2011)]. As written in the opening sentence of this introduction, Fourier coefficients of Gelfand-Graev type, or of Fourier-Jacobi type may be applied to residual Eisenstein series, and, in this work, we do this for residual Eisenstein series coming off “Siegel type” parabolic subgroups and nice “tempered” representations of GLm (AE ), of the form (1.1). One may, of course, consider other residual representations. For example, the residual Eisenstein series comes off other maximal parabolic subgroups, or Siegel type parabolic subgroups and the representation of the form (1.1) is replaced a “Speh representation”. At present, we cannot formulate general theorems, for this theory, as we do in this work. The research here is still in progress. The descent construction in these more general cases, is tied to the more general Rankin-Selberg integrals for G × ResE/F GLm . See [Ginzburg, Piatetski-Shapiro and Rallis (1997)], [Ginzburg, Jiang, Rallis and Soudry (2009)], [Soudry (2006)]. These involve arbitrary cuspidal representations of GA . The notion of (global) Whittaker model (on the G-side) is replaced by the notion of (global) Gelfand-Graev models, or FourierJacobi models. In Chapter 7, we prove that every irreducible, automorphic, cuspidal representation of GA has such models. Their uniqueness is now known by [Aizenbud,

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Gurevitch, Rallis and Schiffmann (2010)], [Sun (2009)]. The descent construction, in these cases, should yield irreducible, automorphic, cuspidal representations of GA , with a prescribed global Gelfand-Graev (or Fourier-Jacobi) model, which lifts to a given representation of GLm (AE ). If this representation has the form (1.1), then this would construct, in theory, the full near equivalence class on GA , which lifts to (1.1), and, in particular, also such non-generic representations. More generally, the descent method should construct for us, CAP representations of GA , with prescribed models, which lift to given “non-tempered” representations of GLm (AE ). Here are some references where such results had been obtained: [Ginzburg (2003)], [Ginzburg, Rallis and Soudry (2005)], [Ginzburg (2008)], [Ginzburg, Jiang, and Soudry (2010)], [Ginzburg, Jiang, and Soudry (2011)]. The basic ideas of the descent method appeared first in [Ginzburg, Rallis and Soudry (1997)] and in [Ginzburg, Rallis and Soudry I (1997)] (where we called the descent “backward lifting”...). Our first detailed study of the descent appeared in [Ginzburg, Rallis and Soudry (1999)], where we focused on the descent from cuspidal representations τ of GL2n (A), such that LS (τ, ∧2 , s) has a pole at s = f 2n (A). The 1 and LS (τ, 12 ) 6= 0, to (ψ-) generic cuspidal representations on Sp descent, in this case, was completed in the papers [Ginzburg, Rallis and Soudry I (1999)], [Ginzburg, Rallis and Soudry II (1999)], [Ginzburg, Rallis and Soudry (2002)]. The case where G = SO2n+1 with the lift to GL2n appeared in [Ginzburg, Rallis and Soudry (2001)]. Here, the descent is known to be irreducible. This was proved in [Jiang and Soudry (2004)]. Another different proof was obtained in [Ginzburg, Jiang, and Soudry (2011)]. A survey of the descent to cuspidal, generic representations of GA , from representations of the form (1.1), when G is a split, or quasi-split classical group (over F ) appeared in [Soudry (2005)]. See also [Soudry (2006)]. The detailed study of the descent in the cases where G is symplectic, even (split, or quasi-split) orthogonal, or quasi-split unitary, appears in detail, for the first time, here, in this book. Recently, the same descent construction was carried out, with similar proofs, for GSpin groups, in [Hundley and Sayag (2009)]. This book is organized as follows. In Chapter 2, we prove the existence of the residual representations Eτˆ , as above, in Theorem 2.1. (In case HA is metaplectic, this residual representation depends also on the additive character ψ.) In Chapter 3, we define Fourier coefficients of Gelfand-Graev type and of Fourier Jacobi type for classical groups. These enter in the global integrals of Rankin-Selberg type, or of Shimura type, which represent standard L-functions for G×ResE/F GLm . We survey the theory of these integrals for the case of irreducible, automorphic, cuspidal, generic representations. Then we define the spaces, denoted above by σψ,ℓ (¯ τ ) (in the text, we use a slightly different notation). The descent is defined by assigning a special value ℓm to ℓ. See (3.36), (3.38). In Theorem 3.1, we state the main theorem concerning the descent σψ (¯ τ ), and then we outline the strategy of proof. Chapter 4 contains information about double coset decompositions, such as double cosets of H modulo two maximal parabolic subgroups, representatives, stabilizers,

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and more. The material here is known. We use it for many computations needed in Chapters 5, 6, 7, 10. In the first section of Chapter 5 we carry out a detailed analysis, over a p-adic field k, of the semi-simplification, as Gℓ (k)-modules, of the Jacquet modules mentioned above, corresponding to a Gelfand-Graev character, when applied to parabolic induction, from a maximal parabolic subgroup of Hk , and a representation τ ⊗ σ of its Levi part; τ being a representation of the general linear group factor (of the Levi part), and σ being a representation of the factor which is a classical group (or a metaplectic group) of the same type as that of Hk . We do this in a quite general set-up, where the Witt index of the form preserved by Hk is arbitrary. The main result here is Theorem 5.1, which has the form of a Leibniz rule; the various summands of the semisimplification (except at most one) are parabolic inductions to Gℓ (k) from tensor products of Bernstein Zelevinsky derivatives of order t of τ and Gℓ−t (k)- Jacquet modules of the same type of σ; t varies in a certain range, depending on ℓ, the parabolic subgroup of Hk , and the Witt index. We apply the theorem to various cases, among which we take the unramified local factors of our global residual representation Eτ¯ . The main theorem here is Theorem 5.6, which tells us that the corresponding Jacquet modules vanish for ℓ > ℓm , and that, for ℓ = ℓm , if they are nontrivial, then, in most cases, they sit inside a representation of Gk , induced from the Borel subgroup, from the “right” unramified character. A more general form of this theorem appears in Theorem 5.5, which is useful for the more general theory of descent. In the second section of Chapter 5, we work out the split version of the content of Sec. 5.1. This is needed when we consider unitary groups over F , which split at a place v of F , where Fv = k, and then the groups G become, at the place v, general linear groups. The notion of a Jacquet module, with respect to a Gelfand-Graev character becomes here an object which looks like a Bernstein-Zelevinsky “mixed derivative”. In Chapter 6 we do the same things for Fourier-Jacobi characters. This completes the local ground work needed for the descent. In Chapter 7, we return to the global set-up. Here, we obtain a general formula for constant terms along unipotent radicals of maximal parabolic subgroups of Gℓ of Fourier coefficients of Gelfand-Graev type, or of Fourier-Jacobi type of automorphic forms on HA . These are Theorem 7.2 and Theorem 7.8. As a result, we prove that irreducible, automorphic, cuspidal representations of HA have global Gelfand-Graev models, or global Fourier-Jacobi models. See Theorem 7.5 and Theorem 7.10. Finally, we apply the general formula above, and the results of Chapters 5, 6, to prove the tower property of the modules σψ,ℓ (¯ τ ). More precisely, these vanish, for ℓ > ℓm , and all constant terms along unipotent radicals of parabolic subgroups of G = Gℓm τ ) are zero (Theorems 7.6, 7.11), so that if we know that applied to σψ (¯ τ ) = σψ,ℓm (¯ σψ (¯ τ ) is nontrivial, then it is cuspidal. (Once again, in text our notation is a little different.) The main work in Chapters 8, 9 is to show that σψ (¯ τ ) is nontrivial. In several cases we obtain explicit formulae relating a certain Whittaker coefficient of the descent σψ (¯ τ ), with a Whittaker coefficient of the representation (1.1). Once

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we know that σψ (¯ τ ) is nontrivial, then the results of Chapters 5, 6 imply that any irreducible summand of σψ (¯ τ ) lifts, almost everywhere to the representation (1.1). In Chapter 10, we prove that all irreducible summands of σψ (¯ τ ) are (globally) generic with respect to a certain Whittaker character. As a result we conclude that the descent σψ (¯ τ ) is multiplicity free (Theorem 10.2). As we mentioned before, the descent construction σψ (¯ τ ) is naturally tied to the global integrals which represent the standard L-function for pairs of irreducible, automorphic, representations σ and τ of GA × GLm (AE ), where σ is cuspidal and generic, and τ has the form (1.2). We explain with detailed proofs the unfolding process of these global integrals to Eulerian integrals, and we do this in general, for Gℓ × ResE/F GLm , for any ℓ and m. In particular, we show, that if we take σ to be non-generic, then these global integrals vanish identically. In the last chapter, we determine the image of the functorial lift from irreducible, automorphic, cuspidal generic representations of GA to GLm (AE ). Since the descent construction allows us to go back from the general linear group to the classical group, we get as applications the existence of generalized endoscopy, for cuspidal, generic representations of classical groups, as well as base change and automorphic induction (with respect to cyclic extensions of F of odd prime degree). In the following two sections we write down formulae for the Weil representation, which will be needed in the sequel. 1.2

Formulas for the Weil representation

We record here some formulas for the Weil representation, which will be used in this book. See, for example, [Rao (1993)]. Let W be a symplectic space of dimension 2n over the number field F . Denote by h, i a non-degenerate, anti-symmetric bilinear form on W . Let W = W+ + W−

(1.3)

be a decomposition of W into a direct sum of two maximal isotropic subspaces W + , W − , which are in duality under h, i. We realize the Heisenberg group HW , corresponding to (W, 2h, i), as HW = W ⊕ F , with multiplication rule (x, t) · (y, z) = ((x + y), t + z + hx, yi). We let Sp(W ) ∼ = Sp2n (F ) act on W from the right. It also acts on HW by (x, t) · g = (x · g, t). When we form the semi-direct product HW ⋊ Sp(W ), the conjugation g −1 (x, t)g becomes (x · g, t), for x ∈ W, t ∈ F, g ∈ Sp(W ). We will also view HW and Sp(W ) as algebraic groups over F . Let ψ be a nontrivial character of F \A. We denote f by ωψ the Weil representation of HW (A) · Sp(W )A , corresponding to ψ. It can be

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realized in the Schrodinger model acting in the space S(W + (A)) of Schwartz-Bruhat functions on W + (A). Here are the formulas which we will need. ωψ ((w+ , 0; 0))φ(ξ) = φ(ξ + w+ ) ωψ ((0, w− ; 0))φ(ξ) = ψ(2hξ, w− i)φ(ξ)

ωψ ((0, 0; t))φ(ξ) = ψ(t)φ(ξ)   g 0 , ǫ)φ(ξ) = ǫγψ (det g)| det g|1/2 φ(ξ · g) ωψ ( 0 g∗   Iy ωψ ( , ǫ)φ(ξ) = ǫψ(hξ, ξ · yi)φ(ξ) . 0I

(1.4)

+ In (1.4), we wrote an element of HW (A) as (w+ , w− ; t), where w+  ∈ W (A), w− ∈ ab W − (A), t ∈ A. Also, ǫ = ±1, and we write an element of Sp(W ) as , following cd (1.3), so that a ∈ HomF (W + , W + ), b ∈ HomF (W + , W − ) etc. We realize the local metaplectic groups Sp(W )v by using the normalized Rao cocycle, corresponding to the Siegel parabolic subgroup preserving W − . See [Rao (1993)]; the global double f cover Sp(W )A of Sp(W )A is compatible with the local double covers at each place. See, for example, [Jiang and Soudry (2007)], Sec. 2.2. Finally γψ is the Weil factor associated to ψ. f The corresponding theta series is (for h ∈ HW (A), g ∈ Sp(W )A ) X φ θψ (h · g) = ωψ (h · g)φ(ξ) . ξ∈X + (F )

Remark 1: The semi-direct product HW ⋊ Sp(W ) will usually appear as an “almost” parabolic subgroup of Sp2n+2 (F ) (a one-dimensional torus is missing)      1  1x t  x ∈ W, g ∈ Sp(W )  I x′  ·  g   t∈F  1 1

Here x′ is the following linear functional on W , x′ (y) = hy, xi , The product rule

y∈W .



    1x t 1y z 1 x + y t + z + hx, yi  I x′   I y ′  =  I x′ + y ′  1 1 1

explains why we consider HW with respect to 2h, i.

Let E be a quadratic extension of F , and let Y be a 2n dimensional space over E, equipped with a non-degenerate, anti-Hermitian form (, ). Assume that Y has a (maximal) isotropic subspace of dimension n over E. Let U(Y ) be the corresponding unitary group. Let h, i = 21 trE/F (, ). This is a symplectic form on Y , viewed now as

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a 4n dimensional space over F . Denote this 4n dimensional F -space by Y ′ . Then, since U(Y ) preserves h, i, we have an F - embedding U(Y ) ֒→ Sp(Y ′ ). It is known that the metaplectic cover of Sp(Y ′ )A splits over U(Y )A . See [Moeglin Vigneras and Waldspurger (1987)], p. 56, [Gelbart and Rogawski (1991); Kudla (1994)]. In order to write such a splitting, we have to choose a character λ of E ∗ \A∗E , whose restriction to A∗ is ωE/F – the nontrivial quadratic character of F ∗ \A∗ , associated to E/F . The splitting also depends on ψ. We will use the splitting given in [Gelbart and Rogawski (1991)], Sec. 3. Denote this splitting by sψ,λ . We should remark c ′ )A , a covering of Sp(Y ′ )A , where the cocyle that sψ,λ embeds U(Y )A inside Sp(Y takes values in the unit circle (in C∗ ), and not in {±1}. However, this cocyle is cohomologically of order two. When we compose the Weil representation, using this cocyle, and sψ,λ on the unitary group, we get the Weil representation ωψ,λ of U(Y )A . We extend it, as usual, to HY ′ (A). Let us write Y = Y++Y− as a direct sum of two maximal isotropic subspaces of Y , which are in duality under (, ). Of course, Y + and Y − , viewed as 2n dimensional subspaces of Y ′ , are also isotropic for h, i, and in duality under h, i. Let us write the elements of HY ′ as (y + , y − ; t), where y + ∈ Y + , y − ∈ Y − , t ∈ F . We have, for φ ∈ S(Y + (AE )), ωψ,λ ((y + , 0; 0))φ(ξ) = φ(ξ + y + ) ωψ,λ ((0, y − ; 0))φ(ξ) = ψ(trE/F ((ξ, y− ))φ(ξ) ωψ,λ ((0, 0; t))φ(ξ) = ψ(t)φ(ξ) (1.5)   g 0 1/2 ωψ,λ φ(ξ) = γ (N (det g))λ(det g))| det g| φ(ξ · g) ψ E/F 0 g∗   1 Iu ωψ,λ φ(ξ) = ψ( trE/F ((ξ, ξ · u)))φ(ξ). I 2 + + − − Here, y ∈ Y (A ), y ∈ Y (AE ), t ∈ A, and we write the elements of U(Y ) E   ab as , where a ∈ HomE (Y + , Y + ), b ∈ HomE (Y + , Y − ) etc. Note that cd γψ (NE/F (x))λ(x) is a character of E ∗ \A∗E , whose restriction to A∗F is ωE/F . Let us denote γ(x) = γψ (x)λ(x), and re-denote ωψ,λ as ωψ,γ , so that   g 0 ωψ,γ φ(ξ) = γ(det g)| det g|1/2 φ(ξ · g). (1.6) 0 g∗ Remark 2: As in the previous case, the semi-direct product HY ′ ⋊U(Y ) will usually appear as an “almost” parabolic subgroup of U(Y ⊕ Y1,1 ), where Y1,1 = Ey1 + Ey−1 is orthogonal to Y , (y1 , y−1 ) = 1, and y±1 are isotropic. This subgroup has the form     1  1 x t   IY x′   g  x ∈ Y, t¯ = t − (x, x), g ∈ U(Y ) ,   1 1

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where we write matrices with respect to Ey1 + Y + Ey−1 , and x′ is the linear   1 x t functional on Y , which takes y to hy, xi. The subgroup of elements  IY x′ , as 1 above, is isomorphic to the Heisenberg group HY ′ , by taking the last element to (x; t − 12 (x, x)). Note that, for t ∈ E, such that t¯ = t − (x, x), t − 12 (x, x) lies in F . Denote the last map by j. Then we have,     1 x t 1 u z 1 1 j  IY x′  ·  IY u′  = (x + u, (t − (x, x)) + (z − (u, u) + hx, ui) . 2 2 1 1 Thus, j is an isomorphism. 1.3

The case, where H is unitary and the place v splits in E

√ Let E = F [ ρ] be a quadratic extension of F . Fix a place v of F . Denote k = Fv and K = Ev = Fv ⊗F E. Assume that ρ is a square in k, say ρ = d2 , d ∈ k. Then K∼ = k ⊕ k by √ (1.7) i : a ⊗ 1 + b ⊗ ρ 7→ (a + bd, a − bd), a, b ∈ k. Here, we will identify k with i(k) = {(x, x)|x ∈ k}. Note that the Galois conjugation of E/F becomes, at the place v, (x, y) = (y, x), and the trace map becomes trK/k ((x, y)) = x + y. Consider a δ-Hermitian space (V, b), dimE V = m′ ; δ = ±1 (b is the corresponding δ′ Hermitian form). Choose a basis of V over E, {ǫj }m j=1 , and denote its corresponding Gram matrix by J; t J¯ = δJ. Write √ J = J1 + ρJ2 , where Ji are matrices over F . Then t J1 = δJ1 and t J2 = −δJ2 . Consider the space Vk = k ⊗F V . We use (1.7) to view Vk as X ⊕ X, where X is a vector space over ′ k, of dimension m′ . Take X to be the space spanned over k by the basis {ǫj }m j=1 . Then the isomorphism (we keep denoting it by i) Vk ∼ = X ⊕ X is given by   m′ m′ m′ X X X √ (aj ⊗ǫj +bj ⊗ ρǫj ) 7→  (aj + dbj )ǫj , (aj − dbj )ǫj  = (i1 (u), i2 (u)), i:u= j=1

j=1

j=1

(1.8)

where, aj , bj ∈ k.

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Let us write the K - valued form β = bv , on Vk , in terms of the isomorphisms (1.7), (1.8). We have, on Vk , ′

′

m m X X √ √ β( (ai ⊗ ǫi + bi ⊗ ρǫi ), (αi ⊗ ǫi + βi ⊗ ρǫi )) = i=1

t

i=1

t

( aJ1 α + ρ bJ2 α − ρ aJ2 β − ρt bJ1 β) ⊗ 1

(1.9)

t

+ (−t aJ1 β − ρt bJ2 β +t aJ2 α +t bJ1 α) ⊗

√ ρ,

where a is the column vector of coordinates a1 , ..., am′ , and similarly for b, α, β. The map i takes (1.9) to (t (a + db)(J1 + dJ2 )(α − dβ),t (a − db)(J1 − dJ2 )(α + dβ)). ′

(1.10) ′

Let us identify X with the column space k m , through the basis {ǫj }m j=1 . Denote −1 ˜ β = i ◦ β ◦ i , i.e. we view β as a bilinear form on X ⊕ X, with values in k ⊕ k. Denote, as in (1.8), i1 (J) = J1 + dJ2 ,

i2 (J) = J1 − dJ2 .

Then (1.9) reads as ˜ β((x, y), (u, w)) = (t xi1 (J)w,t yi2 (J)u)),

(1.11)

for x, y, u, w ∈ X. Let H = U(V ) be the unitary group corresponding to (V, b), regarded as an algebraic group over F . Consider the group Hk = HFv . It consists of K-linear transformations g of Vk , which can be described as follows. The maps g are given by pairs of k-linear maps (g1 , g2 ), on X, such that, for a ∈ k, and 1 ≤ i ≤ m′ , √ g(a ⊗ ǫi ) = a ⊗ g1 ǫi + a ⊗ ρ(g2 ǫi ) √ √ g(a ⊗ ρǫi ) = ρa ⊗ g2 ǫi + a ⊗ ρ(g1 ǫi ), and, for u, w ∈ Vk , β(gu, gw) = β(u, w),

(1.12) √ where β is given by (1.9), (1.10). We may denote g = g1 ⊗ 1 + g2 ⊗ ρ, and identify ′ g1 , g2 with their matrices, with respect to the basis {ǫi }m i=1 . Now, it is easy to see that, for x, y ∈ X, i ◦ g ◦ i−1 (x, y) = ((g1 + dg2 )x, (g1 − dg2 )y),

(1.13)

and (1.12) becomes ˜ ˜ β(((g 1 + dg2 )x, (g1 − dg2 )y), ((g1 + dg2 )w, (g1 − dg2 )z)) = β((x, y), (w, z)), for all (x, y), (w, z) ∈ X ⊕ X. This, by (1.11), means that t t

x (g1 + dg2 )(J1 + dJ2 )(g1 − dg2 )z =t x(J1 + dJ2 )z,

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and t t

y (g1 − dg2 )(J1 − dJ2 )(g1 + dg2 )w =t y(J1 − dJ2 )w,

for all x, y, z, w ∈ X. These two conditions are equivalent to

g1 − dg2 = (g1 + dg2 )∗ = (J1 + dJ2 )−1 t (g1 + dg2 )−1 (J1 + dJ2 ).

(1.14)

Thus, the map r′ (g1 ⊗ 1 + g2 ⊗

√ ρ) = (g1 + dg2 , g1 − dg2 ) = (g1 + dg2 , (g1 + dg2 )∗ )

(1.15)

realizes an isomorphism of Hk with {(h, h∗ )|h ∈ GL(X)}, which is isomorphic to GL(X). Note, again, that by (1.13), r′ (g) = i ◦ g ◦ i−1 .

Finally, denote by r(g) the first coordinate of r′ (g), in (1.15), i.e. r(g) = g1 + dg2 . This establishes an isomorphism of Hk with GL(X) ∼ = GLm′ (k). Let m ˜ denote the Witt index of the form b. Assume that it is positive. Let V ± ⊂ V be two maximal isotropic subspaces, in duality under b, and let V0 = ± (V + + V − )⊥ . Choose bases over E, {e±1 , ..., e±m ˜ }, for V , respectively, such that b(ei , e−j ) = δi,j , for all 1 ≤ i, j ≤ m. ˜ Let B0 be a basis of V0 , over E. Order the resulting basis of V , so that the first m ˜ elements are e1 , ..., em ˜ elements ˜ , the last m are e−m , ..., e , and the elements of B stand in the middle. Use this basis as ˜ −1 0 m′ {ǫi }i=1 above. The corresponding Gram matrix has the form   wm ˜ , J = J0 (1.16) δwm ˜

where J 0 is the Gram matrix of B0 , and wm ˜ ×m ˜ matrix obtained by ˜ is the m writing the rows of the identity matrix Im ˜ in reverse order. Consider, for 1 ≤ j ≤ m, ˜ the isotropic subspaces Vj+ = SpanE {e1 , ..., ej }.

+ Let Vj,k = Fv ⊗F Vj+ = k ⊗F Vj+ . Then

where Xj+

+ i(Vj,k ) = Xj+ ⊕ Xj+ ,

(1.17)

+ + gVj,k = Vj,k ⇐⇒ (g1 ± dg2 )(Xj+ ) = Xj+ .

(1.18)

√ = Spank {e1 , ..., ej }. For g = g1 ⊗ 1 + g2 ⊗ ρ ∈ Hk , we have

Consider a flag of the form ϕ : Vm+1 ⊂ Vm+1 +m2 ⊂ ... ⊂ Vm+1 +m2 +···+mr , for m1 , ..., mr ≥ 1, m1 + · · · + mr ≤ m. ˜ Let Pϕ ⊂ H be the parabolic subgroup Pϕ ⊂ H, which stabilizes ϕ. Then, by (1.18), r′ (Pϕ (k)) is the subgroup of all pairs (h, h∗ ) ∈ GL(X)2 , satisfying h(Xi+ ) = h∗ (Xi+ ) = Xi+ ,

(1.19)

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for i = m1 + ... + ms ; s ≤ r. Here h∗ is taken with respect to J in (1.16). Thus, with respect to the standard basis, r(Pϕ (k)) is the standard parabolic subgroup of GLm′ (k), which corresponds to the partition (m1 , ..., mr , m′′ , mr , ..., m1 ) of m′ , where m′′ = m′ − 2(m1 + ... + mr ). Let us go back to the quasi-split even unitary group U(Y ) of Sec. 1.2 (right after Remark 1). We will write the formulas of the Weil representation ωψv ,γv of HY ′ ,v ⋊ U(Y )v , where we use the notation from Remark 2, at the end of Sec. 1.2 (so, we assume now that U(Y ) acts from the right on Y .) Here m′ = 2n and m ˜ = n. Consider the basis {e1 , ..., en , e−n , ..., e−1 } of Y over E, as above. Its Gram matrix J2n is given by (1.16), with m ˜ = n, δ = −1, and omitting J 0 . Take ± Y = SpanE {e±1 , ..., e±n }. As before, we have an isomorphism Yk± = Yv± = k ⊗F Y± ∼ = Z ± ⊕Z ± , where Z ± = Spank {e±1 , ..., e±n }. We let ωψv ,γv act in S(Z + ⊕Z + ). We realize the corresponding Heisenberg group, at v, as (Z + ⊕ Z + )⊕ (Z − ⊕ Z − )⊕ k, with the product rule (y1 , y2 ; z1 , z2 ; u) · (y1′ , y2′ ; z1′ , z2′ ; u′ ) = 1 (y1 + y1′ , y2 + y2′ ; z1 + z1′ , z2 + z2′ ; u + u′ + (y1 · z2′ − y2′ · z1 + y2 · z1′ − y1′ · z2 )), 2 n n P P ci e−i , xi ei , and z = where, for y = i=1

i=1

y·z = For φ ∈ S(Z + ⊕ Z + ), we have

n X

xi ci .

i=1

ωψv ,γv ((y1 , y2 ; 0, 0; 0))φ(ξ1 , ξ2 ) = φ(ξ1 + y1 , ξ2 + y2 ) ωψv ,γv ((0, 0; z1 , z2 ; 0))φ(ξ1 , ξ2 ) = ψv (ξ1 · z2 + ξ2 · z1 ))φ(ξ1 , ξ2 ) ωψv ,γv ((0, 0; 0, 0; u))φ(ξ1 , ξ2 ) = ψv (u)φ(ξ1 , ξ2 )   det a 1/2 a0 ωψv ,γv | φ(ξ1 · a, ξ2 · b∗ ) φ(ξ1 , ξ2 ) = γ1 (det(ab))| 0b det b   In s ωψv ,γv φ(ξ1 , ξ2 ) = ψv (ξ2 · (ξ1 s)))φ(ξ1 , ξ2 ). In

(1.20)

Here, γv = (γ1 , γ1−1 ), where γ1 is a character of k ∗ . The matrices a, b are in GLn (k), acting from the right on Z + ; b∗ = wnt b−1 wn . Similarly, s is in Mn (k), and is viewed as a linear map, acting from the right, from Z + to Z − . These formulas are the direct translations of (1.5) to the split version. It will be convenient to use another realization of ωψv ,γv . For this, note that the subspaces Z = (Z + ⊕ 0) ⊕ (Z − ⊕ 0) and Z ′ = (0 ⊕ Z + ) ⊕ (0 ⊕ Z − ) are also maximal isotropic subspaces, in duality, of (Z + ⊕ Z + ) ⊕ (Z − ⊕ Z − ). Let us realize ωψv ,γv in S(Z) =

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S(Z + ⊕ Z − ). To get this realization, consider the isomorphism j described in Remark 2, in the end of Sec. 1.2. Then its version at v (i.e. i ◦ jv ◦ r−1 ) is   1 u z 1 (1.21) j ′  I2n v  = (u,t vJ2n ; z − uv). 2 1 The product rule in the Heisenberg group is now 1 (u,t vJ2n ; z)(u′ ,t vJ2n ; z ′ ) = (u + u′ ,t (v + v ′ )J2n ; z + z ′ + (uv ′ − u′ v)). (1.22) 2 Let us write the Heisenberg group as H4n = {(u1 , u2 ; z)|u1 , u2 ∈ k 2n , z ∈ k}. The symplectic form on k 4n = k 2n ⊕ k 2n ∼ = Z ⊕ Z ′ is 1 < (u1 , u2 ), (u′1 , u′2 ) >= (u1 ·t u′2 − u′1 ·t u2 ), 2 and the map, which we keep denoting by j,   1 u z 1 j  I2n v  = (u,t v; z − uv), 2 1

is an isomorphism onto H4n . Note again, that {(u1 , 0) ∈ k 2n ⊕ k 2n |u1 ∈ k 2n } and {(0, u2 ) ∈ k 2n ⊕k 2n |u2 ∈ k 2n } are maximal isotropic subspaces, which are in duality.   1 Conjugation by  g , g ∈ GL2n (k), acts on H4n , so that 1       1 1 u z 1 1 ug z j  g −1   I2n v   g  = j  I2n g −1 v  = (ug,t (g −1 v); z), 1 1 1 1 and the symplectic form <, > is preserved. Recall now, that (g, g ∗ ) also preserves <, > (see (1.14)). If we write the elements of Sp4n (k) in matrix form, following the decomposition (Z + ⊕ Z + ) ⊕ (Z − ⊕ Z − ), then the form of the image i(g) of (g, g ∗ ) inside Sp4n (k) is   a 0 b 0 0 d′ 0 b′   i(g) =  c 0 d 0  , 0 c′ 0 a′    ′ ′ ab d b −1 t −1 + − (following Z = Z ⊕ Z ) and J2n · g J2n = ′ where g = . Consider cd c a   In  In  . Then conjugation by w =    −In In

i(g)w =

 g

wn ·t g −1 wn



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15

lies in the Levi part of the Siegel parabolic subgroup. We have ωψv ,γv (g) = ωψv (i(g), sψv ,γv (i(g)) = ωψv (w, 1)ωψv [(i(g), sψv ,γv (i(g))(w,1) ]ωψν (w, 1)−1 . Thus, ωψv ,γv is equivalent to GL2n (k)

ωψv [(i(g), sψv ,γv (i(g))(w,1) ] = ωψv (i(g)w , αw (g)sψv ,γv (i(g))), 1

which acts on φ ∈ S(k 2n ) by αw (g)sψv ,γv (i(g))γψv (det(g))| det(g)| 2 φ(xg). It follows that αw (g)sψv ,γv (i(g))γψv (det(g)) is a character of GL2n (k), and hence, it is of the form µ(det(g)), where µ  is a character of k ∗ . From the formulae (1.20), it is easy  a to conclude that for g = , a ∈ GLn (k), we have In αw (g)sψv ,γv (i(g))γψv (det(g)) = γ1 (det(a)), and hence µ = γ1 . Let us re-denote γ1 by γ. Thus ωψv ,γv , as a representation of GL2n (k)H4n can be realized in S(k 2n ) = S(Z), with the following formulae   1 u z ωψv ,γv  I2n 0 φ(x) = ψv (z)φ(x + u) 1   1 0 0 (1.23) ωψv ,γv  I2n e φ(x) = ψv (xe)φ(x) 1   1 1 ωψv ,γv  g  φ(x) = γ(det(g))| det(g)| 2 φ(xg). 1 These formulae make sense also for GLN (k)H2N , acting on S(k N ), for any N , even, or odd.

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Chapter 2

On Certain Residual Representations

In this chapter, we introduce certain Eisenstein series on classical (or metaplectic) groups H, and establish the existence of certain multi-residues. The corresponding residual representations will be the main objects of study of this book.

2.1

The groups

Let F be a number field and let E ⊇ F be an extension of degree e ≤ 2. Let V be a vector space, of dimension m′ over E, equipped with a (non-degenerate) δ-Hermitian form b, where δ = ±1. Thus, b(u, v) = δb(v, u).

Here, for x ∈ E, x ¯ = x, in case e = 1 (i.e. E = F ), and in case e = 2, x ¯ denotes the Galois conjugate of x over F . In case (V, b) is quadratic, we assume that dimF V ≥ 3. Denote by h(V ) the connected component of the isometry group of (V, b), viewed as an algebraic group over F . If b is symplectic, we denote, for a place v of F , and ε = 0, 1, ( f v ), ε = 1 Sp(V h(Vv )ε = (2.1) Sp(Vv ), ε = 0 , f v ) is the metaplectic cover of Sp(Vv ). Let H = H(V ) be any one of the where Sp(V groups h(V ), if V is not symplectic, or h(V )ε , ε = 0, 1, if V is symplectic. We apply to H the standard language of algebraic groups over F , although the metaplectic f ), we group is not algebraic. Still, when we mention a parabolic subgroup Q of Sp(V mean that we take a parabolic subgroup P of Sp(V ), and then the Fv -points of Q f v ). Sometimes we denote Q = P˜ . We define are the inverse image of P (Fv ) in Sp(V f the dual group of Sp(V ) to be Sp2m (C), where dimF V = 2m. In this chapter, we make the following assumptions on (V, b). Assumption 2.1.

(1) When E is a quadratic extension of F , the δ-Hermitian space (V, b) is even dimensional, i.e. dimE V = m′ is even. 17

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(2) The Witt index of (V, b) is

1

2 dimE V



= m.

These assumptions will hold through large parts of this book (in other parts, mainly in Chapter 4 and Chapter 5, we will not make these assumptions). Let V + be a maximal isotropic subspace of V (dimE V + = m), and let V − be a maximal isotropic subspace of V , which is dual to V + , with respect to b. Thus, either V = V + ⊕ V − , or (V + ⊕ V − )⊥ is one dimensional. Let V0 = (V + ⊕ V − )⊥ . Then V = V + ⊕ V0 ⊕ V − , (2.2) where, by assumption, V0 = 0, in case b is not symmetric, and in case b is symmetric, dimF V0 ≤ 1. Fix a maximal flag in V + , 0 ⊂ V1+ ⊂ V2+ ⊂ . . . ⊂ Vm+ = V + , (2.3) and choose a basis {e1 , ..., em } of V + over E, such that Vi+ = SpanE {e1 , ..., ei }. Let {e−1 , ..., e−m } be the basis of V − , which is dual to {e1 , ..., em }, i.e. b(ei , e−j ) = δi,j , for all1 ≤ i, j ≤ m. The parabolic subgroups of H, which stabilize flags of the form (2.4) ϕ : Vm+1 ⊂ Vm+1 +m2 ⊂ ... ⊂ Vm+1 +m2 +···+mr , for m1 , ..., mr ≥ 1, m1 + · · · + mr ≤ m, will be our standard parabolic subgroups, and similarly for their Levi subgroups. In case of an even orthogonal group, we have to allow the following cases as well. Denote + + Vm = Vm−1 ⊕ F e−m , and consider parabolic subgroups which stabilize flags of the form (2.5) ϕ : Vm+1 ⊂ Vm+1 +m2 ⊂ . . . ⊂ Vm+1 +···+mr−1 ⊂ +Vm . These are also counted among the standard parabolic subgroups (when V is even dimensional over E = F , and b is symmetric). Here, we will denote mr = m − (m1 + · · · + mr−1 ). For the flag ϕ in (2.4), denote, for 1 ≤ i ≤ r, o n Vϕ+ (i) = SpanE ePi−1 mj +1 , ePi−1 mj +2 , . . . , ePij=0 mj j=0 j=0 o n − P (2.6) Vϕ (i) = SpanE e− i−1 mj −1 , e− Pi−1 mj −2 , . . . , e− Pij=0 mj . j=0

j=0

Here, m0 = 0. We will consider flags ϕ of the form (2.4), such that m1 + · · · + mr = m . (2.7) In case of an even orthogonal group and a flag ϕ of the form (2.5), we denote by Vϕ+ (i), Vϕ− (i), for 1 ≤ i ≤ r − 1 the subspaces in (2.6), and for i = r, we denote o n Vϕ+ (r) = SpanF ePr−1 mj +1 , ePr−1 mj +2 , . . . , em−1 , e−m j=0 j=0 o n − (2.8) Vϕ (r) = SpanF e− Pr−1 mj −1 , e− Pr−1 mj −2 , . . . , e−(m−1) , em . j=0

j=0

Denote by Pϕ the standard parabolic subgroup of H, which stabilizes ϕ. Let Pϕ = Mϕ · Nϕ be its Levi decomposition, where Mϕ is the Levi subgroup. Except in the metaplectic case, Mϕ is isomorphic to GL(Vϕ+ (1)) × · · · × GL(Vϕ+ (r)) (viewed as an algebraic group over F ), i.e. Mϕ is isomorphic to ResE/F GLm1 ×· · ·×ResE/F GLmr . f v ), at every In the metaplectic case, we have to take the full inverse image in Sp(V place v.

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On Certain Residual Representations

2.2

19

The Eisenstein series to be considered

Let ϕ be a flag as in (2.4), satisfying (2.7); in case H is even orthogonal, we do not consider ϕ as in (2.5). Let τ1 , . . . , τr be irreducible, unitary, automorphic representations of GL(Vϕ+ (1))A , . . . , GL(Vϕ+ (r))A , respectively. If mi ≥ 2, we assume that τi is cuspidal (i = 1, ..., r). In case H is not metaplectic, consider, for τ¯ = (τ1 , . . . , τr ), s¯ = (s1 , . . . , sr ) ∈ Cr , s −1/2

1 A Rτ¯,¯s = IndH Pϕ (A) τ1 | det ·|E

s −1/2

⊗ · · · ⊗ τr | det ·|Er

(2.9)

s −1/2

where we view ⊗ri=1 τi | det ·|Ei as a representation of Pϕ (A) extended from Mϕ (A), so that it is trivial on Nϕ (A). In case H is metaplectic, we fix a nontrivial character ψ of F \A, and consider f ) Sp(V

Rτ¯,¯s,ψ = IndPϕ (A)A (τ1 | det ·|s1 −1/2 ⊗ · · · ⊗ τr | det ·|sr −1/2 ) · µψ .

(2.10)

The inducing representation of Mϕ (A) in (2.10) acts as follows. (diag(a1 , . . . , ar ), ε) 7→ εµψ (a1 · . . . · ar ) · ⊗ri=1 τi (ai )| det(ai )|si −1/2 , r Y Y γψ (det ai ). µψ (a1 , . . . , ar ) = (det(ai ), det aj )

(2.11)

µψα (a1 , . . . , ar ) = µψ (a1 , . . . , ar )χα (det(a1 ) · ... · det(ar )),

(2.12)

i=1

1≤i<j≤r

Here, ai ∈ GL(Vϕ+ (i))A , ε = ±1; ( , ) denotes the Hilbert symbol, and γψ is the Weil factor attached to ψ. We realize the local metaplectic groups Sp(V )v by using the normalized Rao cocycle, corresponding to the standard Siegel parabolic subgroup. f )A of Sp(V )A is compatible with the See, [Rao (1993)]; the global double cover Sp(V local double covers at each place. See, for example, [Jiang and Soudry (2007)], Sec. 2.2. Note that, for α ∈ F ∗ , where χα (x) = (x, α) is the quadratic character corresponding to α. Thus, Rτ¯,¯s,ψα = Rτ¯⊗χα ,¯s,ψ ,

(2.13)

where τ¯ ⊗ χα = (τ1 ⊗ χα , ..., τr ⊗ χα ). We denote by ρτ¯,¯s either one of the representations (2.9), (2.10). If needed, and we want to stress the presence of ψ in case H is metaplectic, then we will go back to the notation in (2.10). For a smooth holomorphic section fτ¯,¯s , we denote by E(h, fτ¯,¯s ) the corresponding Eisenstein series on HA . If needed, then, in case H is metaplectic, we will denote fτ¯,¯s,ψ and E(h, fτ¯,¯s,ψ ). 2.3

L-groups and representations related to Pϕ

Assume, first, that dimE V is even. In this case, let Vϕ,i = Vϕ+ (i) ⊕ Vϕ− (i) ,

i = 1, . . . , r .

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The only case where dimE V is odd, which we consider in this chapter, is when E = F and b is symmetric. In this case, let Vϕ,i = Vϕ+ (i) ⊕ V0 ⊕ Vϕ− (i) ,

i = 1, . . . , r .

Define in all cases, Hϕ,i = H(Vϕ,i ) ,

i = 1, . . . , r .

Denote by LHϕ,i the complex dual group of Hϕ,i , in case E = F . (In fact, this 0 is really the connected component of the L-group, LHϕ,i , but since, in this case the group Hϕ,i is split over F , it suffices to take this as “the L-group”.) In case [E : F ] = 2, we denote by LHϕ,i the semidirect product of the complex dual group of Hϕ,i with Z2 ∼ = Gal(E/F ). Consider the parabolic subgroup Pϕ,i of Hϕ,i , which corresponds to Vϕ+ (i), 1 ≤ i ≤ r. It has a Levi decomposition Pϕ,i = Mϕ,i · Nϕ,i , where Mϕ,i is the Levi subgroup. Let LPϕ,i = LMϕ,i · LNϕ,i be the corresponding parabolic subgroup of LHϕ,i . Denote by αϕ,i the representation of LMϕ,i in the Lie algebra of L Nϕ,i . We list the various cases in detail. ∼ Sp f 2m , LHϕ,i = Sp2m (C) (recall that (1) If H is metaplectic, then Hϕ,i = i i dimF Vϕ,i = 2mi , in this case), LMϕ,i = GLmi (C), and αϕ,i is the symmetric square representation of GLmi (C). (2) If (V, b) is odd orthogonal, then Hϕ,i is isomorphic to the split orthogonal group SO2mi +1 , LHϕ,i = Sp2mi (C), and LMϕ,i , αϕ,i are as in the previous case. (3) If (V, b) is even orthogonal, then Hϕ,i is isomorphic to the split orthogonal group SO2mi , LHϕ,i = SO2mi (C), LMϕ,i = GLmi (C), and αϕ,i is the exterior square representation of GLmi (C). (4) If H is symplectic, then Hϕ,i ∼ = Sp , LHϕ,i = SO2m +1 (C), LMϕ,i = GLm (C), and αϕ,i is the direct sum of

(1) αϕ,i

2mi

i

i

– the standard representation of GLmi (C) in

(2)

Cmi , and αϕ,i – the exterior square representation of GLmi (C). (5) If [E : F ] = 2 and (V, b) is δ-Hermitian and even dimensional, then Hϕ,i is isomorphic to the quasi-split unitary group U2mi , corresponding to E, LHϕ,i ∼ = GL2mi (C) ⋉ Z2 , and LMϕ,i ∼ = (GLmi (C) × GLmi (C)) ⋉ Z2 . Denote the nontrivial element of Z2 by η. Then, in LHϕ,i , the action of η on g in GL2mi (C) is by g η = w2mi · tg −1 w2mi .

(2.14)

Here, wk is the k × k matrix, which has 1 along the main anti- diagonal, and zeroes elsewhere. In LMϕ,i , the action of η on (g1 , g2 ) in GLmi (C) × GLmi (C) is by (g1 , g2 )η = (g2 , g1 ) .

(2.15)

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The representation αϕ,i of (GLmi (C) × GLmi (C)) ⋉ Z2 acts in the space of mi × mi complex matrices by αϕ,i (g1 , g2 )(x) = g1 x · tg2 , αϕ,i (η)(x) = tx .

(2.16)

This is the Asai representation. 2.4

The residue representation

We keep the notation of the previous sections. We denote by S any finite set of places of F , including those at infinity, such that τ1 , . . . , τr , E, and ψ are unramified outside S. The next theorem is the main theorem of this chapter. Here, if H is not the (2) (1) symplectic group, we denote LS (τi , αϕ,i , s) = 1, and LS (τi , αϕ,i , s) = LS (τi , αϕ,i , s), (i.e. this applies to H metaplectic, orthogonal, or even unitary), LS denotes a partial L-function (outside S). Theorem 2.1. Let τ1 , . . . , τr be irreducible, unitary, automorphic representations of GL(Vϕ+ (1))A , . . . , GL(Vϕ+ (r))A , respectively. We assume that τi is cuspidal, if dimE Vϕ+ (i) > 1. Assume that (1)

(2)

(1) For each 1 ≤ i ≤ r, LS (τi , αϕ,i , s) has a pole at s = 1, and LS (τi , αϕ,i , 12 ) 6= 0. (2) The representations τ1 , . . . , τr are pairwise different. Then, for any holomorphic smooth section fτ¯,¯s in ρτ¯,¯s , and h ∈ HA , the function s¯ 7→ (s1 − 1) · . . . · (sr − 1)E(h, fτ¯,¯s ) is holomorphic at (1, . . . , 1), and nontrivial, as the section varies. If fτ¯,¯s is K finite, then lims¯→(1,...,1) (s1 − 1) · . . . · (sr − 1)E(h, fτ¯,¯s ) is square integrable. Finally, consider the representation Eτ¯ of HA generated by these residues. Then Eτ¯ is irreducible. Remarks (1)

(1) Note that the condition LS (τi , αϕ,i , 12 ) 6= 0 is automatic if H is not symplec(1)

tic, since then, by definition, LS (τi , αϕ,i , s) ≡ 1. If H is symplectic, then (1)

LS (τi , αϕ,i , s) = LS (τi , s), and then the condition means that LS (τi , 12 ) 6= 0. (2) In case [E : F ] = 2, so that (V, b) is δ-Hermitian, (it is even dimensional over (2) E), LS (τi , αϕ,i , s) = LS (τi , αϕ,i , s) is the Asai (partial) L-function of τi . (2)

(3) Note that the condition that LS (τi , αϕ,i , s) has a pole at s = 1 implies that τi∗ = τi , where τi∗ = τˆi , if E = F , and if [E : F ] = 2, then τi∗ = τˆ′ i , where τi′ is the composition of τi with the Galois conjugation of E/F . Indeed, in the first case, note that LS (τi × τi , s) = LS (τi , ∧2 , s)LS (τi , sym2 , s).

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Here, αϕ,i is either the exterior square, or the symmetric square representation. Each of these L-functions is nontrivial at s = 1, as proved by Shahidi in general [Shahidi, F. (1981)], and so, we conclude that LS (τi × τi , s) has a pole at s = 1, which implies that τi = τˆi . This last assertion is a theorem of Jacquet and Shalika [Jacquet and Shalika I (1981)]. If E/F is quadratic, and b is δHermitian, then we use the equality LS (τi × τi′ , s) = LS (τi , Asai, s)LS (τi ⊗ γ, Asai, s) .

(4)

(5)

(6) (7)

Here, γ is an extension of ωE/F , the quadratic character associated to the quadratic extension E/F , to E ∗ \A∗E . Again, by the result above of Shahidi, LS (τi ⊗ γ, Asai, s) does not vanish at s = 1, and so we conclude, as before, that LS (τi × τi′ , s) has a pole at s = 1, which implies (by [Jacquet and Shalika I (1981)], as above), that τi∗ = τi . Note that in case H is symplectic, or even orthogonal, the assumption on τi , 1 ≤ i ≤ r, is that LS (τi , ∧2 , s) has a pole at s = 1. This implies, by [Jacquet and Shalika II (1981)] that mi is even. In the statement of the theorem, we may replace all partial L-functions with respect to S, by partial L-functions corresponding to any finite set of places, containing S. Indeed, by [Jacquet and Shalika I (1981)], the real parts of the exponents of the unramified parameters of the representations τi , at a given unramified place, lie in the open interval (−1/2, 1/2), and so, any local Lfunction of the type appearing in the theorem does not have a pole at Re(s) ≥ 1/2, and, in particular, at s = 1, or at s = 1/2. Similarly, using the formulas in [Jacquet Piatetski-Shapiro and Shalika (1983); Jacquet (2009)], of local Lfunctions of irreducible, unitary, generic representations, we can replace the partial L-functions with complete L-functions. The irreducibility of Eτ¯ was observed by Erez Lapid, and we thank him for pointing it to us. If needed, then, in case H is metaplectic, we will denote the residue representation by Eτ¯,ψ . The proof of Theorem 2.1 will occupy the rest of this chapter.

2.5

The case of a maximal parabolic subgroup (r = 1)

The proof of Theorem 2.1 is by induction on r. We start with the case r = 1. Here, Pϕ is the parabolic subgroup of H, which corresponds to V + = Vm+ . Denote, in this case, Pϕ = P , Mϕ = M , and τ1 = τ . Note that τ is a representation of GLm (AE ), and Hϕ,1 = H. Denote, as in [Moeglin and Waldspurger (1995)], I.1.7, by W (M ) the set of Weyl elements w in H, such that w(α) > 0, for all positive roots inside M , and such that wM w−1 is a standard Levi subgroup of H. Due to the maximality of P , W (M ) has just one nontrivial element, which we denote

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23

by w. Denote by M (w) the corresponding intertwining operator on ρτ,s . Then E(h, fτ,s ) has a pole at s = 1 (as the section varies), if and only if M (w)(fτ,s ) has a pole at s = 1 (as the section varies). See [Moeglin and Waldspurger (1995)], II.1.7, IV.3,4. We also know that if this pole exists, then it is a simple pole. See [Moeglin and Waldspurger (1995)], IV.1.11. Write τ ∼ = ⊗τv . Assume that fτ,s is a decomposable section, corresponding to ⊗v fτv,s , where fτv,s is a section for ρτv,s , defined analogously to (2.9), (2.10) over Ev , and such that for v ∈ / S, fτv,s = fτ0v,s w – a nontrivial fixed unramified vector for ρτv,s . Let ρτv,s be the representation s−1/2 w

of Hv induced from Pvw and (τv | det ·|Ev

) . Here, P w is the standard parabolic s−1/2 w

subgroup of H, whose Levi part is wM w−1 , and (τv | det ·|Ev

) is the composition

s−1/2 with conjugation by w−1 . (In the metaplectic case, we have to take of τv | det ·|Ev s−1/2 τv | det ·| µψv (det ·) instead of τv | det ·|s−1/2 .) Fix an unramified (nontrivial) 0 0 0 vector f(τv,s )w for ρw τv,s . Then fτv,s and f(τv ,s)w can be chosen so that (up to the

isomorphism τ ∼ = ⊗τv )

M (w)(fτ,s )(I) = LS (τ,α(1) ,s− 12 )LS (τ,α(2) ,2s−1) LS (τ,α(1) ,s+ 12 )LS (τ,α(2) ,2s) (1)

(2)

·

N

(2.17)

S v∈S Mv (w)(fτv,s )(I) ⊗ f(τ,s)w (I).

Here, α(1) = αϕ,1 , α(2) = αϕ,1 , where ϕ is the flag which consists of Vm+ only, S 0 f(τ,s) By w = ⊗v6∈S f(τ w , and Mv (w) is the local intertwining operator at v. v,s )

assumption, LS (τ, α(1) , s − 21 ) is (holomorphic) nonzero at s = 1. Since τ is unitary, LS (τ, α(1) , s + 12 ) and LS (τ, α(2) , 2s) are clearly holomorphic and nonzero at s = 1. Thus, the quotient in front of the r.h.s. of (2.17) has a pole at s = 1. Since Mv (w)(fτv ,s )(I) is not identically zero, as the section varies, we conclude that there is a choice of fτv,s for v ∈ S, such that M (w)(fτ,s )(I) has a pole at s = 1. This proves the first two assertions of Theorem 2.1, in case r = 1. Assume now that fτ,s is K - finite. We apply the square integrability criterion of [Moeglin and Waldspurger (1995)], Lemma I.4.11. For this, we have to show that the cuspidal support of Ress=1 E(h, fτ,s ), along any standard parabolic subgroup has negative exponents. In this case, it is easy to see that our residue has only one exponent. Here, all constant terms of the residue, along radicals of standard parabolic subgroups vanish, except the radical N of P itself. (This follows as a special easy case from Lemma 2.2 below.) We have E N (h, fτ,s ) = fτ,s (h) + M (w)(fτ,s (h). Hence, [Ress=1 E(·, fτ,s )]N = Ress=1 M (w)(fτ,s ). Thus, the cuspidal support of the 1 residue consists only of τ | det ·|− 2 on GLm (AE ), and, of course, it has a negative exponent. Finally, in order to get the irreducibility of the square-integrable residual 1 A 2 representation Eτ , we note that Eτ is a unitary quotient of IndH PA τ | det ·| . Since at 1 v 2 each place v, the Langlands quotient πv of IndH Pv τv | det ·| is its unique semisimple ∼ quotient, we conclude that Eτ = ⊗v πv , and, in particular, Eτ is irreducible. When

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H is metaplectic, we have to replace τ by τ · µψ in the last two arguments. This completes the proof of Theorem 2.1 in case r = 1. Remark In the proof, P w = P in all cases. This is clear if H is not even orthogonal. If H is even orthogonal, then P w = P , if and only if m is even. This is indeed the case, by our assumptions, as explained in Remark 4, Sec. 2.4. 2.6

A preliminary lemma on Eisenstein series on GLn

The following lemma will be needed repeatedly, and is derived easily from [Moeglin and Waldspurger (1995)]. We bring it for completeness sake. The proof is replicated from [Ginzburg, Rallis and Soudry (2002)], Lemma 1.14. Lemma 2.1. Let τ1 , . . . , τr be irreducible, automorphic, unitary, representations of GLm1 (AE ), . . . , GLmr (AE ), respectively. Assume, for each 1 ≤ i ≤ r, that τi is cuspidal if mi > 1, and if j 6= i, then there is no x ∈ C, such that τj = τi | det ·|xE . Then the Eisenstein series on GLm (AE ), m = m1 + · · · + mr , corresponding to the representation induced from the standard parabolic subgroup Pm1 ,...,mr , which corresponds to the partition (m1 , ..., mr ) of m, and the representation τ1 | det ·|zE1 ⊗ · · · ⊗ τr | det ·|zEr , is holomorphic at (z1 , ..., zr ), satisfying Re(z1 ) ≥ · · · ≥ Re(zr ). Proof. The Levi subgroup L of Pm1 ,...,mr is isomorphic to GLm1 × · · · × GLmr , (all groups are now considered over E). By the general theory of Eisenstein series, it is enough to show that at s¯ = (s1 , ..., sr ) = (z1 , ..., zr ), as above, all intertwining operators applied to smooth holomorphic sections ϕτ¯,¯s , A(w)(ϕτ¯ ,¯s ), on GL (AE ) τ1 | det ·|sE1 ⊗ · · · ⊗ τr | det ·|sEr , are holomorphic, for all w ∈ WGLm (L), IndPmm,...,m r (AE ) 1 where WGLm (L) is the set of Weyl elements w of WGLm , the Weyl group of GLm , of minimal length in their left coset, modulo the Weyl group of L, such that wLw−1 is a standard Levi subgroup of GLm . Here, we use the notation of [Moeglin and Waldspurger (1995)], II.1.7. We have   (i) w(α) > 0, for all positive roots inside L WGLm (L) = w ∈ WGLm (ii) wLw−1 is a standard Levi subgroup of GLm (2.18) WGLm (L) is in bijection with the permutation group Sr . Given w ∈ WGLm (L), the corresponding permutation ε ∈ Sr is defined by     gε−1 (1) g1     .. w  . . .  w−1 =   , gi ∈ GLmi . . gr

gε−1 (r)

Let us realize the elements of WGLm as m × m permutation matrices. From (2.18), it is easy to see that if we write w ∈ WGLm (L) in the form w = (w1 , w2 . . . , wr ),

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where wi has mi columns, then wi has the form   0  ..   .    Im   i  .   ..  0

i.e. wi has the identity block Imi appearing somewhere (and has zeroes elsewhere). i−1 P mε−1 (j) rows, The permutation ε ∈ Sr is such that wε−1 (i) has zeroes in the first j=0

and then appears the identity block Imε−1 (i) , and again the following rows are zero rows. Here ε−1 (0) = 0 = m0 . Denote (as in [Moeglin and Waldspurger (1989)], p.607), for w corresponding to ε, inv(w) = {(i, j) | 1 ≤ i < j ≤ r, and ε(i) > ε(j)} .

Assume that the section ϕτ¯,¯s is decomposable as ⊗v ϕτ¯v ,¯s – a product of local holomorphic sections, where τ¯v = (τ1,v , . . . , τr,v ). The product is taken over all places v of E. Let S be a finite set of places of E, including those at infinity, outside of which τi,v is unramified, for 1 ≤ i ≤ r, and ϕτ¯v ,¯s is the unramified section, such (0) (0) that ϕτ¯v ,¯s (I) is the tensor product of fixed spherical vectors v1,v ⊗ · · · ⊗ vr,v . For example, if mi > 1, realize τi,v as a representation, induced from an unramified character of the corresponding Borel subgroup, and take the normalized unramified vector in this induced representation, i.e. which is 1 at Imi , and if mi = 1, take (0) vi,v = 1. We get that A(w)(ϕτ¯,¯s )(Im ; Im1 , . . . , Imr ) is equal to Y Y LS (τi × τˆj , si − sj ) Av (w)(ϕτ¯v ,¯s )(Im ; Im1 , . . . , Imr ) S L (τi × τˆj , si − sj + 1) v∈S

=

Y

v∈s

Here,

(i,j)∈inv(w)

A∗v (w)(ϕτ¯v ,¯s )(Im ; Im1 , . . . , Imr ) ·

A∗v (w)(ϕτ¯,v,¯s ) =

Y

(i,j)∈inv(w)

Y

(i,j)∈inv(w)

L(τi × τˆj , si − sj ) . L(τi × τˆj , si − sj + 1)

L(τi,v × τˆj,v , si − sj + 1) Av (w)(ϕτ¯,v,¯s ) . L(τi,v × τˆj,v , si − sj )

Our assumptions imply that, at the point in question, and, for i < j, (si = zi , i = L(τi ׈ τj ,si −sj ) 1, ..., r), L(τi ׈ τj ,si −sj +1) is holomorphic, since Re(zi − zj ) ≥ 0. (See [Moeglin and Waldspurger (1989)], Appendix). By [Moeglin and Waldspurger (1989)], Prop. I.10, A∗v (w)(ϕτ¯v ,¯s )(Im ; Im1 , . . . , Imr ) is holomorphic at z¯, for all v ∈ S. This shows that A(w)(ϕτ¯,¯s ) is holomorphic at z¯.  Remark f m (A). Here, Lemma 2.1 applies to similar Eisenstein series on the double cover GL at a place v, f m (Fv ) = {(g, ε)|g ∈ GLm (Fv ), ε = ±1} , GL

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and the multiplication is given by (g1 , ε1 )(g2 , ε2 ) = (g1 g2 , ε1 ε2 (det g1 , det g2 )), where (x, y) is the Hilbert symbol in Fv . Let τ1 , . . . , τr be as in Lemma 2.1. Let φτ¯,¯s f m (A) GL

be a smooth holomorphic section for IndP˜

m1 ,...,mr (A)

(τ1 | det ·|s1 ⊗· · ·⊗τr | det ·|sr )·µψ ,

where (τ1 | det ·|s1 ⊗ · · · ⊗ τr | det ·|sr ) · µψ takes (diag(a1 , ..., ar ), 1) to

γψ (det a1 · . . . · det ar ) ⊗ri=1 | det ai |si τi (ai ), ai ∈ GLmi (A). Then ϕτ¯,¯s (g) = γψ (det g)−1 φτ¯,¯s (g, 1) is a smooth holomorphic section for GL (A) IndPmm,...,m (A) (τ1 | det ·|s1 ⊗· · ·⊗τr | det ·|sr ). We have, first in the domain of absolute r 1 convergence, X X γψ (det γg)−1 φτ,¯s (γg, 1) ϕτ¯,¯s (γg) = γ∈Pm1 ,...,mr \GLm (F )

γ∈Pm1 ,...,mr \GLm (F )

=

P

(det γ, det g)γψ (det g)−1 φτ,¯s (γg, 1)

γ∈Pm1 ,...,mr \GLm (F )

= γψ (det g)−1

P

φτ,¯s ((γ, 1)(g, 1)).

γ∈Pm1 ,...,mr \GLm (F )

Here, we used the fact that γψ is trivial on F ∗ . We conclude that the Eisenstein f m (A)), corresponding to φτ,¯s , evaluated at (g, 1), is equal to γψ (det g) series (on GL times the Eisenstein series (on GLm (A)), corresponding to ϕτ,¯s , evaluated at g. In particular, Lemma 2.1 applies again. 2.7

Constant terms of E(h, fτ,¯s )

We get back to our Eisenstein series E(h, fτ,¯s ), and now consider a general flag ϕ as in (2.4), r ≥ 1, such that (2.7) is satisfied. Let 1 ≤ k ≤ m. Consider the (standard maximal) parabolic subgroup Qk of H, which corresponds to Vk+ . In case H is even + orthogonal, we let Q+ m and Qm denote the parabolic subgroups which preserve + + Vm and Vm , respectively. Denote, also, in this case, by Qm , either one of these two parabolic subgroups. Write the Levi decomposition Qk = Dk Uk . The Levi subgroup Dk is isomorphic (over F ) to GL(Vk+ ) · H(Wm,k ) (direct product, except in the metaplectic case), where + − Wm,k = Xm−k ⊕ V0 ⊕ Xm−k ,

(2.19)

and + − Xm−k = SpanE {ek+1 , . . . , em }, Xm−k = SpanE {e−(k+1) , . . . , e−m }.

(2.20)

In case H is even orthogonal, and Qm = +Qm , we have to replace GL(Vm+ ) by GL(+Vm ) (which are isomorphic), and we will also denote, in this case, the Levi

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+ part by Dm , or by +Dm , accordingly. Note that here H(Wm,k ) = {I}. We need to study, for 1 ≤ k ≤ m, the constant terms Z E(uh, fτ¯,¯s )du. E Uk (h, fτ¯,¯s ) = Uk (F )\Uk (A)

We have (see [Moeglin and Waldspurger (1995)], II.1.7), for h ∈ HA , g ∈ Dk (A), X EDk (g, M (w)(ρτ¯,¯s (h)(fτ¯,¯s )). (2.21) E Uk (gh, fτ¯,¯s ) = w∈W (Mϕ ,Dk )

Here, denoting by WH the Weyl group of H,  (i) w(α) > 0, for all positive roots inside Mϕ   −1 W (Mϕ , Dk ) = w ∈ WH (ii) w (α) > 0 for all positive roots inside Dk     (iii) wMϕ w−1 is a standard Levi subgroup inside Dk   

The interpretation in the metaplectic case is self-evident. For w in W (Mϕ , Dk ), M (w) is the corresponding intertwining operator. Recall that M (w)(ρτ¯,¯s (h)fτ¯,¯s ) lies in Dk (A)

D (A)

s −1/2

k IndQϕ,w (τ1 | det ·|E1

s −1/2 w

⊗ · · · ⊗ τr | det ·|Er

)

if H is not metaplectic, and in case it is, we have to insert µψ as in (2.10). Here, Qϕ,w is the standard parabolic subgroup of Dk , whose Levi part is wMϕ w−1 . Finally, EDk (g, M (w)(ρ τ¯,¯s (h)fτ¯,¯s ) is the Eisenstein series on Dk (A) corresponding to M (w)(ρτ¯,¯s (h)fτ¯,¯s ) . Dk (A)

In Chapter 4, we will explain in detail, using only linear algebra, the structure of the sets Qℓ \h(V )/Qj , compute Weyl representatives and stabilizers. 2.8

Description of W (Mϕ , Dk )

In this section, in the metaplectic case, we take the original standard Levi subgroups f ). We of h(V ), corresponding to ϕ, or Vk+ , rather than their inverse image in Sp(V will keep denoting them by Mϕ , Dk . Of course, W (Mϕ , Dk ) remains unchanged. Let us, first, show that, for a given w ∈ W (Mϕ , Dk ), w permutes the blocks of Mϕ , in such a way that the permuted blocks fit into a standard Levi subgroup of Dk . In order to formulate this precisely, let us denote, for a permutation ξ of {1, 2, ..., r}, ξ(ϕ) : Vm+ξ(1) ⊂ Vm+ξ(1) +mξ(2) ⊂ . . . ⊂ Vm+ξ(1) +···+mξ(r) = Vm+ .

(2.22)

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Lemma 2.2. (1) Assume that when h(V ) is even orthogonal, Dk 6= +Dm . Let w ∈ W (Mϕ , Dk ). Then there is a permutation ξ of {1, 2, ..., r}, such that, for all 1 ≤ i ≤ r, we have + − {w(Vϕ+ (ξ(i))), w(Vϕ− (ξ(i))} = {Vξ(ϕ) (i), Vξ(ϕ) (i)} .

(2.23)

± Moreover, with respect to the standard bases of Vϕ± (ξ(i)), Vξ(ϕ) (i), given in + (2.6), w can be chosen so that its matrix of action from Vϕ (ξ(i)) to the image − (i). In this last space is Imξ(i) , except in case δ = −1 and w(Vϕ+ (ξ(i)) = Vξ(ϕ) case, the matrix is (chosen to be) −Imξ(i) . Finally, there is 1 ≤ jw ≤ r, such that

mξ(1) + · · · + mξ(jw ) = k .

(2.24)

+

(2) If h(V ) is even orthogonal, then W (Mϕ , Dm ) = ∅. Proof. Denote the set of simple roots of Mϕ by ∆Mϕ , and denote the set of positive roots by φ+ Mϕ . (Here, we take the set of roots determined by the (standard) Borel subgroup which preserves the maximal isotropic flag obtained from the basis e1 , ..., e−1 .) Similarly, consider ∆Dk and φ+ Dk . For α ∈ ∆Mϕ , (i) and (iii) in the definition of W (Mϕ , Dk ) imply that, for w ∈ W (Mϕ , Dk ), X aβ · β, w(α) = β∈∆Dk

for some non-negative integers aβ . Thus, X α= aβ w−1 (β) .

(2.25)

β∈∆Dk

By (ii) in the definition of W (Mϕ , Dk ), all w−1 (β), which appear in (2.25), are positive roots. Since α is simple, we conclude that aβ in (2.25) are all zero, except for one β ′ ∈ ∆Dk , where aβ ′ = 1. Thus, w(α) = β ′ ∈ ∆Dk . This shows that w(∆Mϕ ) ⊂ ∆Dk .

(2.26)

Consider the spaces Vϕ,i , 1 ≤ i ≤ r, introduced in Sec. 2.3. Put hϕ,i = h(Vϕ,i ), and regard it as a subgroup of h(V ), by letting its elements act as the identity on L + − j6=i (Vϕ (j) ⊕ Vϕ (j)). Consider the standard Levi subgroup Mϕ,i of hϕ,i , which + preserves Vϕ (i). We regard ∆Mϕ,i – the set of simple roots of Mϕ,i , as a subset of ∆Mϕ . Since ∆Mϕ,i is connected in the Dynkin diagram of h(V ), so is w(∆Mϕ,i ). By (2.26), we conclude that w(∆Mϕ,i ) is a connected subset of ∆Dk (1 ≤ i ≤ r). Since the roots in ∆Mϕ have the same length, it is clear that w(∆Mϕ ) lies in the set of simple roots of the Levi subgroup of Qm . (If h(V ) is even orthogonal, we have to consider the two possibilities for Qm .) Let ∆ = {α1 , α2 , . . . , αm } be the set of simple roots of h(V ), such that ∆Mϕ,i = {αm1 +···+mi−1 +1 , . . . , αm1 +···+mi −1 } ,

i = 1, ..., r.

(2.27)

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Here, if mi = 1, then ∆Mϕ,i = ∅. It follows that there is a permutation ξ ∈ Sr , such that wMϕ,ξ(i) w−1 = Mξ∗ϕ,i

(2.28)

and moreover, w(∆Mϕ,ξ(i) ) = ∆Mξ∗ϕ,i ,

i = 1, ..., r ,

(2.29)

where ξ ∗ ϕ is the flag

Vm+ξ(1) ⊂ Vm+ξ(1) +mξ(2) ⊂ . . . ⊂ Vm+ξ(1) +···+mξ(r−1) ⊂ Vm× .

(2.30)

Here, Vm× = Vm+ , except, possibly, when h(V ) is even orthogonal, and then we should allow Vm× = +Vm , as well. From (2.29), it follows that if ∆Mϕ,ξ(i) 6= ∅, then either w(αm1 +···+mξ(i)−1 +t ) = αmξ(1) +···+mξ(i−1) +t ,

1 ≤ t ≤ mξ(i) − 1,

(2.31)

or w(αm1 +···+mξ(i)−1 +t ) = αmξ(1) +···+mξ(i) −t ,

1 ≤ t ≤ mξ(i) − 1.

(2.32)

In case h(V ) is even orthogonal, and Vm× = +Vm , we have to allow (2.31), (2.32) with one difference; for i = r, replace in (2.31), for t = mξ(r) − 1, w(αm1 +···+mξ(r) −1 ) = αm

(2.33)

and replace in (2.32), for t = 1, w(αm1 +···+mξ(r)−1 +1 ) = αm .

(2.34)

By (2.28), we have w(Vϕ± (ξ(i))) = Mξ∗ϕ · w(Vϕ± (ξ(i))), and we conclude, that for each 1 ≤ i ≤ r,

+ − w(Vϕ+ (ξ(i))) = Vξ∗ϕ (i), and w(Vϕ− (ξ(i))) = Vξ∗ϕ (i),

(2.35)

− + w(Vϕ+ (ξ(i)))) = Vξ∗ϕ (i), and w(Vϕ− (ξ(i))) = Vξ∗ϕ (i).

(2.36)

or

This is clear if mξ(i) = 1, since w is a Weyl element. If mξ(i) > 1, i.e. ∆Mϕ,ξ(i) 6= ∅, then (2.35), (2.36) follow from (2.31)–(2.34). Moreover, we can choose w (modulo diagonal elements), such that, in case (2.35), if we consider w as a lin+ (i), and choose the standard bases in ear transformation from Vϕ+ (ξ(i)) to Vξ∗ϕ each space (i.e. {em1 +···+mξ(i)−1 +1 , em1 +···+mξ(i)−1 +2,... } and {emξ(1) +···mξ(i−1) +1 , emξ(1) +···+mξ(i−1) +2,... }), then the matrix of w, with respect to these bases, is Imξ(i) . − In case (2.36), if we take the standard basis of Vξ∗ϕ (i), then the matrix of w is Imξ(i) , except in case δ = −1, where the matrix is −Imξ(i) . This follows from (2.31)–(2.34). Since w(Mϕ ) is a standard Levi subgroup of Dk , (2.24) is clear. Note that unless h(V ) is even orthogonal, and Vm× = +Vm in (2.30), we have ξ ∗ ϕ = ξ(ϕ), and

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the proof of the lemma is complete. Finally, assume that h(V ) is even orthogonal. Let us show that Vm× = +Vm in (2.30) is impossible. The reason is that, in this case, the determinant of w as in (2.35), (2.36) must be −1, and this will be a contradiction. Indeed, by Remark 4 in Sec. 2.4, mi is even for all 1 ≤ i ≤ r. Thus, the determinant of w in (2.35), (2.36) is equal to the determinant of the passage from the basis {e1 , . . . , em−1 , e−m , em , e−(m−1) , . . . , e−1 } to the standard basis {e1 , . . . , em , e−m , . . . , e−1 }, which is −1. This completes the proof of Lemma 2.2. From now on, we will re-denote, in case h(V ) is even orthogonal, Qm = Q+ m and + Dm = Dm . Denote, for 1 ≤ j ≤ 2r, Vϕ (j) =

 V + (j), ϕ

V − (2r + 1 − j), ϕ

1≤j≤r r + 1 ≤ j ≤ 2r.

The following lemma is clear.

Lemma 2.3. Let u ∈ h(V ) be an element of the unipotent radical of the standard Borel subgroup of h(V ). If there are 1 ≤ j1 6= j2 ≤ 2r, such that there is 0 6= v ∈ Vϕ (j1 ), for which u(v) − v ∈ Vϕ (j2 ), then j2 < j1 . Let w ∈ W (Mϕ , Dk ), and let ξ be the permutation from Lemma 2.2. Define, for 1 ≤ i ≤ r,  ± ξ(i), (i) if w(Vϕ± (ξ(i))) = Vξ(ϕ) (2.37) ξ ∗ (i) = 2r + 1 − ξ(i), if w(V ± (ξ(i))) = V ∓ (i). ϕ ξ(ϕ)

Note that, by definition, for 1 ≤ i ≤ r,

+ w(Vϕ (ξ ∗ (i))) = Vξ(ϕ) (i).

(2.38)

Lemma 2.4. Let w and ξ be as above. Consider jw from (2.24). Then ∗

ξ ∗ (1) < ξ ∗ (2) < · · · < ξ ∗ (jw )

ξ (i) = ξ(i),

for

(2.39)

i = jw + 1, . . . , r

(2.40)

ξ(jw + 1) < ξ(jw + 2) < · · · < ξ(r).

(2.41)

+ + Proof. Let i ≤ jw − 1. Let x ∈ HomE (Vξ(ϕ) (i + 1), Vξ(ϕ) (i)), and let u(x) be the unipotent element of h(V ) defined by u(x) = idVξ(ϕ),ℓ , ℓ 6= i, i + 1 Vξ(ϕ),ℓ

u(x)

+ − Vξ(ϕ) (i)⊕Vξ(ϕ) (i+1)

= idV +

ξ(ϕ)

− (i)⊕Vξ(ϕ) (i+1) ;

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+ for v ∈ Vξ(ϕ) (i + 1),

u(x)(v) = v + x(v) ,

(2.42)

− and for y ∈ Vξ(ϕ) (i),

u(x)(y) = y + x′ (v), − − where x′ ∈ HomE (Vξ(ϕ) (i), Vξ(ϕ) (i + 1)) is the dual of x (with respect to the form b). Since i ≤ jw − 1, it is clear that the subgroup of elements u(x), as above, is generated by positive root subgroups inside Dk , and hence w−1 u(x)w is upper unipotent. We have, for v ∈ Vϕ (ξ ∗ (i + 1)),

w−1 u(x)w(v) = v + w−1 [x(w(v))].

+ Indeed, by (2.38), w(v) ∈ Vξ(ϕ) (i + 1), and by (2.42)

u(x)(w(v)) = w(v) + x(w(v), and now apply w−1 . Note that w−1 [x(w(v))] ∈ Vϕ (ξ ∗ (i)). By Lemma 2.3, we must have ξ ∗ (i) < ξ ∗ (i + 1). This proves (2.39). If k = m, then we are done. Assume that k < m. The same argument proves that ξ ∗ (jw + 1) < ξ ∗ (jw + 2) < · · · < ξ ∗ (r).

(2.43)

We just need to apply it for jw + 1 ≤ i ≤ r − 1. It remains to show that ξ ∗ (r) ≤ r, and then (2.43) becomes (2.41). Assume then that ξ ∗ (r) > r. Then by definition (see (2.37)) − w(Vϕ+ (ξ(r))) = Vξ(ϕ) (r).

(2.44)

Assume, first, that either h(V ) is not odd orthogonal, or h(V ) is odd orthogonal, + − (r)) be such that (r), Vξ(ϕ) but mξ(r) > 1. In this case, let x ∈ HomE (Vξ(ϕ) b(x(v1 ), v2 ) + b(v1 , x(v2 )) = 0 ,

− ∀v1 , v2 ∈ Vξ(ϕ) (r) .

Let, as before, u(x) be the following unipotent element of h(V ): u(x) = idVξ(ϕ),ℓ ⊕V + (r) , ℓ < r, + Vξ(ϕ),ℓ ⊕Vξ(ϕ) (r)

u(x)(v) = v + x(v)

ξ(ϕ)

− v ∈ Vξ(ϕ) (r).

(2.45)

Note that the subgroup of these elements u(x) is generated by positive root subgroups inside Dk , and hence w−1 u(x)w is upper unipotent. Let v ∈ Vϕ+ (ξ(r)). By − (2.44), w(v) ∈ Vξ(ϕ) (r), and by (2.45), w−1 u(x)w(v) = v + w−1 [x(w(v))] . Note that w−1 [x(w(v))] ∈ Vϕ− (ξ(r)), and so w−1 u(x)w(v) − v ∈ Vϕ− (ξ(r)), for v ∈ Vϕ+ (ξ(r)). This contradicts Lemma 2.3. Finally, assume that h(V ) is odd orthog+ onal, and mξ(r) = 1. Again, the argument is similar. Let x ∈ HomF (V0 , Vξ(ϕ) (r)).

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+ Recall that V0 is one dimensional, say V0 = F · e0 . Since mξ(r) = 1, Vξ(ϕ) (r) = F em . Write x(e0 ) = tem . Now define u(x) ∈ h(V ) by u(x) = idVξ(ϕ),ℓ ⊕V + (r) , ℓ
ξ(ϕ)

u(x)(e0 ) = e0 + tem

u(x)(e−m ) = e−m − te0 − 12 t2 em .

Then u(x) is a one parameter subgroup corresponding to the simple root αm . Of course, u(x) ∈ Dk and so w−1 u(x)w is upper unipotent. We have, as before, for − v ∈ Vϕ+ (ξ(r)), such that w(v) = e−m ∈ Vξ(ϕ) (r) w−1 u(x)w(v) = w−1 u(x)(e−m ) = w−1 (e−m − te0 − 12 t2 em ) = v ± te0 − 12 t2 w−1 (em ) .

Note again that w−1 (em ) ∈ Vϕ− (ξ(r)), and so w−1 u(x)w(v)−v ∈ V0 ⊕Vϕ− (ξ(r)). Exactly, as in Lemma 2.3, this is impossible, for an upper unipotent element w−1 u(x)w. The proof of Lemma 2.4 is now complete.  Let 0 ≤ iw ≤ jw be the last integer, such that ξ ∗ (iw ) ≤ r. Put mξ(1) + · · · + mξ(iw ) = bw .

(2.46)

Recall that Mϕ ⊂ Dm , and Dm is naturally isomorphic to GL(V + ). Using the standard basis, Mϕ is isomorphic to the standard Levi subgroup of GL(V + ) of type (m1 , ..., mr ). The permutation ξ defines an element wξ′ ∈ WDm (Mϕ ), where WDm is the Weyl group of Dm , and we use the notation of Sec. 2.6. The element wξ′ is defined by + wξ′ (Vξ(ϕ) (i)) = Vϕ+ (ξ(i)),

i ≤ r,

(2.47)

+ and the matrix of wξ′ , as a linear map from Vξ(ϕ) (i) to Vϕ+ (ξ(i)), with respect to the standard bases, is Imξ(i) . From (2.38) and (2.47), we have + + (i)) = Vξ(ϕ) (i) , if 1 ≤ i ≤ iw , or jw + 1 ≤ i ≤ r wwξ′ (Vξ(ϕ)

(2.48)

+ − wwξ′ (Vξ(ϕ) (i)) = Vξ(ϕ) (i) , if iw + 1 ≤ i ≤ jw .

(2.49)

and

+ + (i) to Vξ(ϕ) (i) in Recall that w is chosen so that the matrix of w · wξ′ from Vξ(ϕ) − (2.48), or to Vξ(ϕ) (i) in (2.49), with respect to the standard bases, is Imξ(i) . Thus, representing wwξ′ as a matrix, we get that   Ibw   0 0 Ik−bw   ′  . wwξ =  (2.50) 0 I 0    δIk−bw 0 0 Ibw

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The middle identity block in (2.50) is of size dimE V − 2k. Put ℓ = [ 12 (dimE V − 2k)]. Let us decompose the matrix (2.50) as w1 w2 w1−1 , where   Ibw     ω   Ik−bw   , ω= w1 =  IV0  Iℓ   ω∗ 

w2 = 

Thus,

Ibw +ℓ

µ



Ibw



 , µ= IV0 Ibw +ℓ δIk−bw

Ik−bw



.

w = w1 w2 w1−1 wξ′−1 . 2.9

(2.51)

Continuation of the proof of Theorem 2.1

We now return to (2.21), where we may take h = 1. We also return to the notation used prior to the last section (where we apply the standard notation to metaplectic groups as well). In this section, we complete the proof of Theorem 2.1. By [Moeglin and Waldspurger (1995)], IV.4.1, we have, for w ∈ W (Mϕ , Dk ), written as in (2.51) M (w) = M (w1 ) ◦ M (w2 ) ◦ M (w1−1 wξ′−1 ).

(2.52)

Note that w1−1 wξ′−1 ∈ WDm (Mϕ ) ⊂ W (Mϕ ), where W (Mϕ ) consists of all Weyl elements w of H, such that w(α) > 0, for all positive roots inside Mϕ , and such that wMϕ w−1 is a standard Levi subgroup. By Sec. 2.6, M (w1−1 wξ′−1 )(fτ¯,¯s ) is s −1/2

holomorphic at (1, . . . , 1). M (w1−1 wξ′−1 ) permutes the data τ1 | det ·|E1

s −1/2 τr | det ·|Er s

τξ(1) | det ·|Eξ(1) s

for Mϕ (A) to

−1/2

τξ(r) | det ·|Eξ(r)

s

⊗ · · · ⊗ τξ(iw ) | det ·|Eξ(iw )

−1/2

−1/2

s

⊗ τξ(jw +1) | det ·|Eξ(jw +1) s

−1/2

⊗ τξ(iw +1) | det ·|sξ(iw +1) −1/2 ⊗ · · · ⊗ τξ(jw ) | det ·|Eξ(jw )

⊗ ··· ⊗

⊗ ···⊗

−1/2

, (2.53)

for Mξ′ (ϕ) (A), where ξ ′ (ϕ) is the flag corresponding to the permutation ξ ′ which sends 1, 2, ..., r to ξ(1), ..., ξ(iw ), ξ(jw + 1), ..., ξ(r), ξ(iw + 1), ..., ξ(jw ). In the metaplectic case, we have to “correct” the data above by µψ appropriately. Note that Lemma 2.1 applies here. Indeed, if τj = τi | det ·|xE , then, using the fact that τi and τj are self conjugate, we conclude that τi | det ·|x = τi | det ·|−x . This implies that |a|2mi x = 1, for all a ∈ A∗E , and hence x = 0. Thus, τi = τj , and so i = j.

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Next, we apply M (w2 ) to M (w1−1 wξ′−1 )(fτ¯,¯s ), which is a holomorphic section at (1, ..., 1), induced from data corresponding to (2.53). Denote, for an integer 1 ≤ t ≤ m,   Im−t   0 0 It  , ωt =    0 IV 0 0

δIt 0 0 Im−t

and for two integers 1 ≤ t1 , t2 , such that t1 + t2 ≤ m 

ωt1 ,t2 = 

where

Im−(t1 +t2 )

wt′ 1 ,t2

ωt′ 1 ,t2 Im−(t1 +t2 )



 ,



 0 It2 It1 0    . = I V0    0 It1  It2 0

We will repeatedly use the identity

ωt1 +t2 = ωt1 ωt1 ,t2 ωt2 .

(2.54)

Clearly, w2 =

jY w −1

ωmξ(i) ωmξ(i) ,mξ(i+1) +···+mξ(jw )

i=iw +1

and we have M (w2 ) =

jY w −1

i=iw +1

!

ωmξ(jw ) ,

!

M (ωmξ(i) ) ◦ M (ωmξ(i) ,mξ(i+1) +···+mξ(jw ) )

(2.55)

◦ M (ωmξ(jw ) ). (2.56)

Thus, apply first (2.54) to t1 = mξ(jw −1) , t2 = mξ(jw ) . From Sec. 2.5, we know that (sξ(jw ) − 1)M (ωmξ(jw ) ) (acting on the inducing data (2.53)) is holomorphic at (1, ..., 1). This operator changes, in (2.53), τξ(jw ) | det ·|sξ(jw ) −1/2 to ∗ τξ(j | det ·|−sξ(jw ) +1/2 . Note that by Remark 3 in Sec. 2.4, τi∗ = τi , for all w) i ≤ r. Next, M (ωmξ(jw −1) ,mξ(jw ) ) acts on a section induced from data, which is s

−1/2

−s

+1/2

obtained from (2.53), by replacing τξ(jw ) | det ·|Eξ(jw ) by τξ(jw ) | det ·|E ξ(jw ) , so that, at (1, ..., 1), the vector of exponents becomes ( 21 , ..., 12 , − 21 ), which by Sec. 2.6, is a point of holomorphy, and now the data (2.53) is changed, so that all factors remain the same, except the last two factors, which are now −s +1/2 s −1/2 τξ(jw ) | det ·|E ξ(jw ) ⊗ τξ(jw −1) | det ·|Eξ(jw −1) . Now, apply M (ωmξ(jw −1) ).

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Again, by Sec. 2.5, (sξ(jw −1) − 1)M (ωmξ(jw −1) ) is holomorphic at (1, ..., 1). Thus, (sξ(jw −1) − 1)(sξ(jw ) − 1)M (ωmξ(jw −1) ) ◦ M (ωmξ(jw −1) ,mξ(jw ) ) ◦ M (ωξ(jw −1) )

=(sξ(jw −1) − 1)(sξ(jw ) − 1)M (ωmξ(jw −1) +mξ(jw ) ) ,

applied to M (w1−1 wξ′−1 )(fτ¯,¯s ) is holomorphic at (1, ..., 1), and it permutes the original inducing data for Mϕ (A) to the data obtained from (2.53) by changing sξ(i) − 12 to −sξ(i) + 12 in the last two factors, and then reversing their order, i.e. the data (2.53) becomes  !  r iw O O sξ(i) −1/2 sξ(i) −1/2 ⊗ ⊗ τξ(i) | det ·|E τξ(i) | det ·|E i=1

jO w −2

i=iw +1

s

τξ(i) | det ·|Eξ(i)

−1/2

!

i=jw +1

−sξ(jw ) +1/2

⊗ τξ(jw ) | det ·|E

−sξ(jw −1) + 21

⊗ τξ(jw −1) | det ·|E

,

(2.57)

for Mξ1′ (ϕ) (A), where ξ1′ = ξ ′ ◦ (r, r − 1). From (2.54)–(2.56), (sξ(jw −1) − 1)(sξ(jw ) − 1)M (w2 ) =

jY w −2

i=iw +1

!

M (ωmξ(i) ) ◦ M (ωmξ(i) ,mξ(i+1) +···+mξ(jw ) )

◦ (sξ(jw −1) − 1)(sξ(jw ) − 1)M (ωmξ(jw −1) +mξ(jw ) ) .

(2.58)

Now repeat the last argument to deduce that (sξ(jw −2) − 1)(sξ(jw −1) − 1)(sξ(jw ) − 1)M (ωmξ(jw −2) ) ◦ M (ωmξ(jw −2) ,mξ(jw −1) +mξ(jw ) ) ◦ M (ωmξ(jw −1) +mξ(jw ) )

= (sξ(jw −2) − 1)(sξ(jw −1) − 1)(sξ(jw ) − 1)M (ωmξ(jw −2) +mξ(jw −1) +mξ(jw ) ) applied to M (w1−1 wξ′−1 )(fτ¯,¯s ) is holomorphic at (1, ..., 1), and now (2.57) is changed as follows: the first two products remain the same, and the rest of the product is ! jO w −3 −s +1/2 sξ(i) −1/2 −s (j )+1/2 ⊗ τξ(jw ) | det ·|E ξ w ⊗ τξ(jw −1) | det ·|E ξ(jw −1) τξ(i) | det ·|E i=iw +1

−sξ(jw −2) +1/2

⊗ τξ(jw −2) | det ·|E

,

for Mξ2′ (ϕ) (A), where ξ2′ = ξ1′ ◦ (r − 2, r)(r − 2, r − 1). Substitute this in (2.58) and jw Q (sξ(i) − 1)M (w2 ) ◦ M (w1−1 wξ′−1 )(fτ¯,¯s ) is holomorphic so on. We conclude that i=iw +1

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at (1, ..., 1). This operator takes the original inducing data for Mϕ (A) to  !  r iw O O −s +1/2 s −1/2 ⊗ τξ(i) | det ·|sξ(i)−1/2  ⊗ τξ(jw ) | det ·|E ξ(jw ) τξ(i) | det ·|Eξ(i) i=jw +1

i=1

⊗

−s +1/2 τξ(jw −1) | det ·|E ξ(jw −1)

on Mξj′

w −iw

ξj′ w −iw −1

−sξ(iw +1) +1/2

⊗ · · · ⊗ τξ(iw +1) | det ·|E

,

(2.59)

(A), where −1(ϕ)

= ξ ′ ◦ (r, r − jw + iw + 1) ◦ (r − 1, r − jw + iw + 2) ◦ (r − 2, r − jw + iw + 3) ◦ · · ·

Finally (see (2.52)) we apply M (w1 ) to

jw Q

(sξ(i) − 1)M (w2 ) ◦ M (w1−1 wξ′−1 )(fτ¯,¯s ).

i=iw +1

Note that, at (1, ..., 1), the exponents in (2.59) are ( 21 , ..., 12 , − 12 , ..., − 21 ), − 12 occurring jw − iw times, so that, as before, Lemma 2.1 is applicable. We conclude that M ′ (w) =

jw Y

(sξ(i) − 1)M (w)

i=iw +1

is holomorphic at (1, ..., 1). It takes the original inducing data for Mϕ (A) to ! iw O sξ(i) −1/2 −s +1/2 −s +1/2 τξ(i) | det ·|E ⊗ τξ(jw ) | det ·|E ξ(jw ) ⊗ · · · ⊗ τξ(iw +1) | det ·|E ξ(iw +1) i=1

r O

i=jw +1

s

τξ(i) | det ·|Eξ(i)

−1/2

,

on wMϕ (A)w−1 . Recall that wMϕ w−1 is a standard Levi subgroup of Dk . Denote by Pϕ,w the standard parabolic subgroup of Dk , whose Levi part is wMϕ w−1 . Denote its unipotent radical by Nϕ,w . Clearly, (1) (2) (1) (2) (1) (2) Pϕ,w = Pϕ,w · Pϕ,w , wMϕ w−1 = Mϕ,w · Mϕ,w , Nϕ,w = Nϕ,w · Nϕ,w , (i)

(i)

the products being direct, except in case H is metaplectic. Here, Pϕ,w = Mϕ,w · (i) f + ) in the Nϕ,w , i = 1, 2, is a standard parabolic subgroup of GL(Vk+ ) (or GL(V k metaplectic case), in case i = 1, and of H(Wm,k ), in case i = 2. (See beginning of Sec. 2.7 for notation.) We may assume that M ′ (w)(fτ¯,¯s ), restricted to Dk (A), is (1) (2) (1) a sum of smooth meromorphic sections ϕτ¯,ξ,¯siw ,jw · ϕτ¯,ξ,¯sjw , where ϕτ¯,ξ,¯siw ,jw is a f + (AE ) in the metaplectic case), induced from section on GL(Vk+ (AE ) (or GL(V k ! ! jw −i i w −1 w O O −sξ(jw −i) +1/2 sξ(i) −1/2 1/2 τξ(jw −i) | det ·|E . τξ(i) | det ·|E ⊗ · δQk (1) Mϕ,w (A)

i=1

i=0

(2.60) (In case H is metaplectic, we have to multiply (2.60) by γψ (det ·).) It is holomorphic at sξ(i) = 1, for 1 ≤ i ≤ jw . Here, s¯iw ,jw = (sξ(1) − 12 + n0 , . . . , sξ(iw ) − 21 +

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n0 , −sξ(jw ) +

1 2

+ n0 , . . . , −sξ(iw +1) +

1 2

37

1/2

+ n0 ), where δQk (a) = | det a|nE0 , for a ∈ (2)

GL(Vk+ )A . Note that n0 is positive. The section ϕτ¯,ξ,¯sjw is on H(Wm,k )A , with inducing data r O

i=jw +1

s

τξ(i) | det ·|Eξ(i)

−1/2

.

(2.61)

It is holomorphic at sξ(i) = 1, for jw + 1 ≤ i ≤ r. Here, s¯jw = (sξ(jw +1) , . . . , sξ(r) ). Getting back to (2.21), where it is enough to take h = g = 1, we see that EDk (1, M ′ (w)(fτ¯,¯s )) is a sum of products of the corresponding Eisenstein se(1) (2) ries EGL(V + )ǫ (1, ϕτ¯,ξ,¯siw ,jw )EH(Wm,k ) (1, ϕτ¯,ξ,¯sjw ). Here, GL(Vk+ )ǫ is GL(Vk+ ), for k (1) f + ), for ǫ = 1. At s¯ = (1, ..., 1), E + ǫ (1, ϕ ǫ = 0, and GL(V ) is k

GL(Vk )

τ¯,ξ,¯ siw ,jw

the Eisenstein series on GL(Vk+ )ǫA , induced from (2.60), at the point s¯iw ,jw = ( 21 + n0 , . . . , 12 + n0 , − 21 + n0 , . . . , − 21 + n0 ), where 21 + n0 occurs iw times. By (1) Lemma 2.1, this is a point of holomorphy for EGL(V + )ǫ (1, ϕτ¯,ξ,¯siw ,jw ). Next, k

(2)

EH(Wm,k ) (1, ϕτ¯,ξ,¯sjw ) is an Eisenstein series on H(Wm,k )A , with (2.61) as inducing data. By induction, (2)

(sξ(jw +1) − 1) · · · · · (sξ(r) − 1)EH(Wm,k ) (1, ϕτ¯,ξ,¯sjw ) is holomorphic at sξ(jw +1) = · · · = sξ(r) = 1. This proves that r Y

(sξ(i) − 1)EDk (1, M (w)(fτ¯,¯s ))

i=iw +1

is holomorphic at s¯ = (1, ..., 1). In particular,

r Q

(si − 1)EDk (1, M (w)(fτ¯,¯s ) is

i=1

holomorphic at s¯ = (1, ..., 1), for all w ∈ W (Mϕ , Dk ), and from (2.21), we get r r Q Q (si − (si − 1)E Uk (1, fτ¯,¯s ) is holomorphic at (1, . . . , 1). This proves that that i=1

i=1

1)E(1, fτ¯,¯s ) is holomorphic at s¯ = (1, . . . , 1). Let us also record, at this point, the following corollary.

Corollary 2.1. Let w ∈ W (Mϕ , Dk ). Assume that bw > 0, i.e. iw ≥ 1 (see (2.46)). Then r Y (si − 1)EDk (1, M (w)(fτ¯,¯s )) = 0. (2.62) lim s¯→(1,...,1)

i=1

Only w ∈ W (Mϕ , Dk ) with bw = 0 (i.e. iw = 0) contribute to the limit (2.62), i.e. lim

s¯→(1,...,1)

r Y

i=1

(si−1)E Uk (1, fτ¯,¯s ) =

X

w∈W (Mϕ ,Dk ) bw =0

lim

s¯→(1,...,1)

r Y

i=1

(si−1)EDk (1, M (w)(fτ¯,¯s )). (2.63)

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Finally, it remains to show that

r Q

i=1

(si − 1)E(1, fτ¯,¯s ) is nontrivial. Let us consider

the constant term along Um . Then it is clear that, in (2.63), there is only one w ∈ W (Mϕ , Dm ), such that bw = 0, namely   Im  w= ±IV0 δIm

(If H is odd orthogonal, δ = 1, ±IV0 = ±1, and choose the sign so that det w = 1. If H is even orthogonal, then V0 = 0, δ = 1, and we have seen that our assumptions imply that m is even.) Note that iw = 0, jw = r, and ξ is the permutation, which takes (1, 2, ..., r) to (r, ..., 2, 1). Here, r Y (si − 1)M (w) . M ′ (w) = i=1

′

We have seen that M (s)(fτ¯,¯s ) is holomorphic at (1, ..., 1). Let us show that it is nontrivial, at this point. For this, it is enough to consider a decomposable section fτ¯,¯s , which corresponds to ⊗v fτ¯v ,¯s . Using similar notation as in Sec. 2.5, (2.17), we get (again, up to the isomorphisms τi ∼ = ⊗τi,v ) M ′ (w)(fτ¯,¯s )(I) = r Y LS (τi × τj , si + sj − 1) Y LS (τi , α(1) , si − 21 )LS (τi , α(2) , 2si − 1) (si − 1) LS (τi × τj , si + sj ) i=1 LS (τi , α(1) , si + 21 )LS (τi , α(2) , 2si ) i6=j O S Mv (w)(fτ¯v ,¯s )(I) ⊗ f(¯ τ ,¯ s)w (I). v∈S

(2.64) By our assumptions, the expression above, which consists of the product of ratios r Q (si − 1), is holomorphic and nonzero at s¯ = of partial L-functions, multiplied by i=1 N (1, . . . , 1). Also, it is clear that the restrictions to Dm (AE ) of v∈S Mv (w)(fτ¯v ,¯s ) ⊗ S f(¯ τ ,¯ s)w , in the r.h.s. of (2.64), contain (as S varies) all decomposable smooth holomorphic sections on Dm (AE ) ∼ = GL(Vm+ (AE )), corresponding to the inducing data 1/2 −s +1/2 −s +1/2 δQm (1) τr | det ·|E r . ⊗ τr−1 | det ·|−sr−1 +1/2 ⊗ · · · ⊗ τ1 | det ·|E 1 Mϕ,w

(1) Mϕ,w

is the standard Levi subgroup of GL(Vm+ )ǫ of type (mr , mr−1 , . . . , m1 ). Here Thus, with substitutions as above, we get that EDm (1, M ′ (w)(fτ¯,¯s )), evaluated at −1/2 1/2 times an Eisenstein series on GL(Vm+ (AE ))ǫ , (1, ..., 1) is | det ·|E · δQm + GL(Vm )

corresponding to an arbitrary decomposable smooth holomorphic section induced from τr | det ·|zEr ⊗ τr−1 | det ·|zr−1 ⊗ · · · ⊗ τ1 | det ·|zE1

evaluated at the point (0, ..., 0)). In particular, it is nontrivial. This proves the first two assertions of Theorem 2.1.

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Let us prove the assertion about the square integrability. As in the case r = 1, we prove that the multi-residue of our Eisenstein series has cuspidal support with negative exponents along standard parabolic subgroups. Let U be a unipotent radical of a standard parabolic subgroup of H. Denote E1 (h, fτ¯,(1,...,1) ) =

lim

s¯→(1,...,1)

r Y (si − 1)E(h, fτ¯,¯s ).

i=1

Since we have to consider the constant term E1U (h, fτ¯,(1,...,1) ), we may assume, by Lemma 2.2, that U ⊃ Uk , for some 1 ≤ k ≤ m (i.e. in case H is an even orthogonal group, we don’t need to consider the radical of +Qm ). Corollary 2.1 and (2.63) show us how to compute the constant term along Uk . Let us write U = (U (1) × U (2) ) · Uk , where U (1) is a unipotent radical of a standard parabolic subgroup of GL(Vk+ ), and U (2) is a unipotent radical of a standard parabolic subgroup of H(Wm,k ). Then by (2.60), (2.61), and the arguments above, we get that the constant term of E1 , along −1/2 U , times δQk , can be written as a sum of expressions of the following form: X

(1)

U (1) EGL(V ) + ǫ (a, ϕ )

w∈W (Mϕ ,Dk )

k

lim

s¯jw →(1,...,1)

bw =0

r Y

(2)

(2)

U (sξ(i) − 1)EH(W (b, ϕτ¯,ξ,¯sjw ) m,k )

i=jw +1

(2.65) Here, ϕ is the value at (1, ..., 1) of a (K-finite holomorphic) section on Njw −1 GL(Vk+ (AE )) (or the double cover), induced from i=0 τξ(jw −i) | det · (1)

−sξ(jw −i) +1/2

|E

(2)

, and ϕτ¯,ξ,¯sjw is (K - finite) as in (2.61). Thus, we have to show (1)

U (1) that, for each w in the summation of (2.65), the exponents of EGL(V ), for + ǫ (a, ϕ ) k

the relevant U (1) , and the exponents of lim

s¯jw →(1,...,1)

r Y

(sξ(i) − 1)EH(Wm,k ) U

(2)

(2)

(b, ϕτ¯,ξ,¯sjw ),

i=jw +1

(1)

U (1) for the relevant U (2) , are all negative. The exponents of EGL(V ) (for the + ǫ (a, ϕ ) k

relevant U (1) ) are clearly always of the form (− 21 , − 21 , ...), and, in particular, are negative. The exponents of r Q (2) U (2) (sξ(i) − 1)EH(W (b, ϕτ¯,ξ,¯sjw ) (for the relevant U (2) ) are allims¯jw →(1,...,1) m,k ) i=jw +1

ways negative, by induction on r. (The case r = 1 was checked in Sec. 2.5.) Finally, the irreducibility of the residual representation Eτ¯ follows exactly as in case r = 1 (Sec. 2.5). The proof of Theorem 2.1 is now complete.

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Chapter 3

Coefficients of Gelfand-Graev Type, of Fourier-Jacobi Type, and Descent

We introduce here the notions of Gelfand-Graev coefficients and of Fourier-Jacobi coefficients for automorphic forms on our classical groups. These will be applied later to the residual representations Eτ¯ of Chapter 2 (Theorem 2.1) when we introduce the descent construction. This descent is naturally related to a family of Rankin-Selberg integrals, which represent standard L-functions of pairs. In the end of this chapter we define the corresponding Jacquet modules in the analogous local setting. We will keep the notation used in Chapter 2. We will not always impose Assumption 2.1 from Chapter 2. However, we assume that h(V ) is split or quasi-split, over F . Thus, when h(V ) is unitary, we assume that it is quasi-split; when h(V ) is odd orthogonal, we assume that it is split, and when it is even orthogonal, we assume that is either split or quasi-split. We will denote the Witt   index (dimension of maximal isotropic subspace) of V by m. ˜ We keep denoting 12 dimE V = m. We fix a non-trivial character ψ of F \A. We denote by ψE,δ the character ψ, in case E = F , and in case [E : F ] = 2, this is the character of E\AE , defined as follows. If δ = −1, this is ψ( 21 trE/F (x)), and if δ = 1, this is ψ( 12 trE/F ( √xρ )), where √ E = F [ ρ].

3.1

Gelfand-Graev coefficients

In this section, we let the form b on V be symmetric, when E = F , or Hermitian (δ = 1) when [E : F ] = 2. As in Chapter 2, we fix maximal isotropic subspaces V ± , in duality, with respect to b, and consider the decomposition (2.2). Now, V0 = (V + ⊕ V − )⊥ may be two dimensional, in case b is symmetric, m′ = 2m and m ˜ = m − 1. Otherwise, V0 maybe zero, or one dimensional. Fix a maximal flag in V +, + + 0 ⊂ V1+ ⊂ V2+ ⊂ . . . ⊂ Vm (3.1) ˜ = V + + and choose a basis {e1 , ..., em over E, such that Vi = SpanE {e1 , ..., ei }, and ˜ } of V − a dual basis {e−1 , ..., e−m ˜ } of V , as in Chapter 2. When V0 is one dimensional, we 41

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choose a basis vector e0 for V0 . We assume, for simplicity, that b(e0 , e0 ) = 1. When V0 is two dimensional (i.e. b is symmetric, m′ = 2m and m ˜ = m − 1), we choose (1) (2) an orthogonal basis {e0 , e0 } of V0 . We assume, for simplicity, that we may take these vectors, with (1)

(1)

b(e0 , e0 ) = 1, ∗

(2)

(2)

b(e0 , e0 ) = −c,

(3.2)

where c ∈ F is not a square. These assumptions on V0 do not restrict our orthogonal groups. For 1 ≤ ℓ ≤ m, ˜ let ϕℓ be the flag ϕℓ : 0 ⊂ V1+ ⊂ V2+ ⊂ · · · ⊂ Vℓ+

(3.3)

Consider the (standard) parabolic subgroup of h(V ), which stabilizes ϕℓ . Denote it by Pℓ . Let Pℓ = Mℓ ·Nℓ be its Levi decomposition, Mℓ being the Levi subgroup. It is isomorphic to GL(Vϕ+ℓ (1)) × · · · × GL(Vϕ+ℓ (ℓ)) × h(Wm,ℓ ) ∼ = [ResE/F GL1 ]ℓ × h(Wm,ℓ ), where, as in Sec. 2.7,
⊥ Wm,ℓ = Vℓ+ + Vℓ− . (3.4) Note that Vϕ+ℓ (i) = SpanE {ei }, 1 ≤ i ≤ ℓ. We assume that Wm,ℓ 6= 0, i.e. ℓ < m, in case dimE V = 2m, and m ˜ = m. Let us choose an anisotropic vector w0 ∈ Wm,ℓ . Define, for u in Nℓ χℓ,w0 (u) =

ℓ X i=2

b(u · ei , e−(i−1) ) + b(u · w0 , e−ℓ ).

(3.5)

Clearly, χℓ,w0 is an F - rational homomorphism from Nℓ to the additive algebraic group ResE/F (Ga ). Consider the character ψℓ,w0 = ψE,1 ◦ χℓ,w0

(3.6)

of Nℓ (A); it is trivial on Nℓ (F ). Here, we consider χℓ,w0 also as a homomorphism from Nℓ (A) to AE . Embed h(Wm,ℓ ), inside h(V ), as the subgroup of elements of h(V ) which act as the identity on Vℓ+ + Vℓ− . Note that this is the subgroup corresponding to 1 × · · · × 1 × h(Wm,ℓ ) of Mℓ . Consider the action, by conjugation, of h(Wm,ℓ ) on Nℓ , and on its rational homomorphisms to ResE/F (Ga ). Then the stabilizer, Lℓ,w0 , of χℓ,w0 , in h(Wm,ℓ ), is the fixator of w0 , in h(Wm,ℓ ), i.e. h(w0⊥ ∩ Wm,ℓ ). Denote Rℓ,w0 = Lℓ,w0 · Nℓ .

(3.7)

Let π be an automorphic representation of HA = h(V )A , acting in a space of automorphic forms Vπ . Define, for ξπ ∈ Vπ , h ∈ HA , Z ψ −1 (v)dv. (3.8) ξπ (vh)ψℓ,w ξπ ℓ,w0 (h) = 0 Nℓ (F )\Nℓ (A)

We call (3.8) the Gelfand-Graev coefficient of ξπ , with respect to ψℓ,w0 . Note that ψ

ψ

ξπ ℓ,w0 (γh) = ξπ ℓ,w0 (h) , ∀γ ∈ Rℓ,w0 (F ).

(3.9)

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ψ In particular, the function ξπ ℓ,w0

Lℓ,w0 (A)

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43

is automorphic. Denote by σψℓ,w0 (π) the

representation of Lℓ,w0 (A), acting by right translations, in the space of automorphic ψ functions, spanned by all functions of the form ξπ ℓ,w0 , as ξπ varies in the Lℓ,w0 (A)

space of π. Note that (3.8) is a standard Whittaker coefficient, in case h(V ) is even orthogonal, ℓ = m − 1, and in case h(V ) is odd orthogonal, or odd unitary, and ℓ = m. These notions and definitions make sense also when ℓ = 0. In this case, N0 is the trivial group, R0,w0 = L0,w0 = h(w0⊥ ) is the fixator of w0 in h(V ), and σ0,w0 (π) acts on the space of restrictions to L0,w0 ((A) of all functions in the space of π. In the sequel, we will make the following choices of w0 . Assume that 0 ≤ ℓ < m. ˜ m′ +1 α e . Note that b(y , y ) = Let α ∈ F ∗ . We will choose w0 = yα = em − m ˜ α α ˜ + (−1) 2 ′ (−1)m +1 α. The vector yα lies in Wm,m−1 and hence lies in Wm,ℓ , for all ℓ < m. ˜ In ˜ this case, we will denote ψℓ,α = ψℓ,yα Lℓ,α = Lℓ,yα Rℓ,α = Rℓ,yα

(3.10)

σψℓ,α (π) = σℓ;ψ,yα (π). Let ℓ = m, ˜ and so, by our assumption, Wm,m ˜ = V0 6= 0. Then w0 is a nonzero vector in V0 . In case h(V ) is (split) odd orthogonal, or (quasi-split) odd unitary, we will choose w0 = e0 , and in case h(V ) is quasi-split even orthogonal, we will choose (2) w0 = e0 . Finally, since the group Lℓ,w0 is determined uniquely, up to conjugation by an element in h(Wm,ℓ ), by b(w0 , w0 ), it will sometimes be convenient to make the following choice. Assume that V is odd dimensional, ℓ < m and α is a square α′2 , in case E = F , or a norm NE/F α′ , in case [E : F ] = 2. Then, sometimes, we may replace yα by y(α′ ) = α′ e0 . Remark: The notion of Fourier coefficients of Gelfand-Graev type with respect to ˜ as long (Nℓ , ψℓ,w0 ) is, of course, valid without any restriction on the Witt index m, as ℓ ≤ m. ˜ In this book the groups h(V ) that we work with are split, or quasi-split over F . However, in several places, we will treat these notions in general, mainly in the analogous local setting (see Chapters 4, 5). Note that our notations and definitions are general and make sense for any Witt index m ˜ ≤ m, that is even when h(V ) is not split or quasi-split. 3.2

Fourier-Jacobi coefficients

Assume that V is even dimensional (m′ = 2m), and the form b is either symplectic, or anti-Hermitian (i.e. δ = −1). Thus, h(V ) is either symplectic, or quasi-split even unitary. We use the same notation as in Chapter 2, and as in Sec. 3.1. Let ℓ < m,

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and consider the subspaces Vℓ± , Wm,ℓ , as in (3.3), (3.4). Thus, + − V = Vℓ+ + Wm,ℓ + Vℓ− = Vℓ+1 + Wm,ℓ+1 + Vℓ+1 .

Here, we take a non-zero vector w0 ∈ Wm,ℓ , which is isotropic (this is automatic when b is symplectic). Let us simply choose w0 = eℓ+1 . Consider the rational homomorphism χℓ,w0 of Nℓ , given as in (3.5), and, again, denote ψℓ,w0 = ψE,−1 ◦ χℓ,w0 (in this section δ = −1). Denote the stabilizer of χℓ,w0 in h(Wm,ℓ ) by Lℓ,w0 . It is the contained in the parabolic subgroup of h(Wm,ℓ ), which preserves the isotropic line E · eℓ+1 , and it is the stabilizer of eℓ+1 , in this parabolic subgroup. In case ℓ < m − 1, Lℓ,w0 has Levi decomposition Hℓ Nℓ′ . Here, Hℓ , the Levi subgroup, is isomorphic to h(Wm,ℓ+1 ), and the unipotent radical, Nℓ′ , is naturally isomorphic to the Heisenberg group of Wm,ℓ+1 , HWm,ℓ+1 , over F ; HWm,ℓ+1 = Wm,ℓ+1 ⊕ F . We will denote the center of Nℓ′ by Cℓ . We identify it with the center of HWm,ℓ+1 . Clearly, Nℓ+1 is generated by Nℓ and Nℓ′ . We have: Nℓ \Nℓ+1 ∼ = HWm,ℓ+1 . Fix such an isomorphism jℓ . Denote, as in the previous case, Rℓ,w0 = Lℓ,w0 Nℓ .

(3.11)

Since w0 is already chosen to be eℓ+1 , we will usually abbreviate Lℓ,w0 to Lℓ , ψℓ,w0 to ψℓ , etc. If ℓ = m − 1, then Lℓ,em is just the one dimensional root subgroup, which corresponds to the simple root in the Siegel radical, so that, in this case, Rℓ,em = Nm is the standard maximal unipotent subgroup. Assume that ℓ < m − 1. Then the subspace Wm,ℓ+1 has a natural symplectic structure over F . See Sec. 1.2 for the case when h(V ) is unitary. Let ωψ be the f m,ℓ+1 )A associated to ψ; its restriction to Weil representation of HWm,ℓ+1 (A) ⋊ Sp(W HWm,ℓ+1 (A) is irreducible, and its central character is ψ. Recall that the metaplectic cover of Sp(Wm,ℓ+1 )A splits over h(Wm,ℓ+1 )A , in case h(V ) is unitary. See Sec. 1.2. For this, we have to choose a character γ of E ∗ \A∗E , such that γ|A∗ = ωE/F – the nontrivial quadratic character of F ∗ \A∗ , associated to E. This gives us the Weil representation ωψ,γ of HWm,ℓ+1 (A) ⋊ h(Wm,ℓ+1 )A , as in Sec. 1.2. See (1.5), (1.6). (m−ℓ−1) We will sometimes write ωψ,γ = ωψ,γ . If h(V ) is symplectic, we let γ = 1 and f ωψ,1 = ωψ (on HW (A) ⋊ Sp(Wm,ℓ+1 )A ). Denote, in case h(V ) is unitary, m,ℓ+1

ˆ ℓ = h(Wm,ℓ+1 ), H

and in case h(V ) is symplectic, denote ( Sp(Wm,ℓ+1 ), ˆ Hℓ = f m,ℓ+1 ), Sp(W

f ) if H = Sp(V

if H = Sp(V ).

(3.12)

(3.13)

Finally, in case ℓ = m − 1, Wm,ℓ+1 = {0}, we replace HWm,ℓ+1 by the simple root subgroup in the Siegel radical, and we replace ωψ,γ by ψE,−1 . We define, in this

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ˆ m−1 = {1}. Note, that the definitions above are meaningful for ℓ = 0. case, H Let π be an automorphic representation of HA , acting in a space Vπ of automorf )A . Assume that phic functions. We assume that π is genuine, in case HA = Sp(V ℓ < m − 1. Let φ be a Schwartz function in a Schrodinger model of ωψ−1 ,γ −1 , and (m−ℓ−1),φ ˆ ℓ (A) let θψ−1 ,γ −1 be the corresponding theta series. Define, for ξπ ∈ Vπ , h ∈ H Z (m−ℓ−1),φ (3.14) ξπ (vh′ )ψℓ−1 (v)θψ−1 ,γ −1 (jℓ (v)h′′ )dv. F Jψφℓ ,γ (ξπ )(h) = Nℓ+1 (F )\Nℓ+1 (A)

Here, the character ψℓ , which is a character of Nℓ (A), is extended trivially to Nℓ+1 (A). (Nℓ+1 is the semidirect product of Nℓ and the unipotent radical Nℓ′ , of Lℓ , which we identify with HWm,ℓ+1 .) In (3.14), h′ = h′′ = h, if h(V ) is f ), then unitary; if h(V ) is symplectic, then we consider two cases. If H = Sp(V ′ ′′ ˆ f Hℓ (A) = Sp(Wm,ℓ+1 )A , and h = h ∈ Sp(Wm,ℓ+1 )A project to h. If H = Sp(V ), ˆ ℓ (A) = Sp(W f m,ℓ+1 )A , and h = h′′ project to h′ . then H φ F Jψℓ ,γ (ξπ ) is called the Fourier-Jacobi coefficient of ξπ , with respect to ωψ,γ

(and φ). We define, also F Jψφℓ ,γ (ξπ ), for the case ℓ = −1 In this case, we define Nℓ+1 = N0 to be the identity group, and Wm,0 = V , so that (3.12), (3.13) make sense, and (3.14) reads as (m),φ

F Jψφ−1 ,γ (ξπ )(h) = ξπ (h′ )θψ−1 ,γ −1 (h′′ ).

(3.15)

ˆ ℓ (F ), F Jψφℓ ,γ (ξπ )(rh) = F Jψφℓ ,γ (ξπ )(h), ∀r ∈ H

(3.16)

Note that

ˆ ℓ (A). Denote by σψ ,γ (π) the so that F Jψφℓ ,γ (ξπ ) is an automorphic function on H ℓ ˆ ℓ (A), acting by right translations, in the space of automorphic representation of H functions, spanned by all functions of the form F Jψφℓ ,γ (ξπ ), as ξπ varies in the space of π. If ℓ = m − 1, then by our convention (3.14) is a Whittaker coefficient of ξπ . (0),φ −1 applied to the simple root subgroup in the Siegel (θψ−1 ,γ −1 is replaced by ψE,−1 ′ ′′ radical; h = h = h = 1.) Fourier-Jacobi coefficients can be defined for any unitary group (odd or even, quasi-split, or otherwise) as well, but they will not be needed here. 3.3

Nilpotent orbits

The Fourier coefficients of Gelfand-Graev type, or of Fourier-Jacobi type correspond to nilpotent orbits in the Lie algebra of H. Recall that such orbits are parametrized by certain partitions of dimE V . See [Humphreys (1995)]; [Collingwood and McGovern (1993)], Chapter 5; [Springer and Steinberg (1970)], Chapter IV. The partition, which corresponds to the Gelfand-Graev coefficients, i.e. to χℓ,w0 in (3.5), when ′ w0 is anisotropic, is (2ℓ + 1, 1m −(2ℓ+1) ), i.e. 1 is repeated m′ − (2ℓ + 1) times (m′ = dimE V ). The partition, which corresponds to the Fourier-Jacobi coefficients,

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i.e. to χℓ,w0 , when w0 is isotropic, is (2ℓ + 2, 1m −(2ℓ+2) ). Let us explain this in more detail, and relate it to the general set-up of degenerate Whittaker models as in [Moeglin and Waldspurger (1987)]. In order to define a degenerate Whittaker model, we have to choose a nilpotent element f in Lie(H), and a one parameter subgroup d of H, whose image is a one dimensional split torus (over F ), such that Ad(d(t)) · f = t−2 f ,

t ∈ F ∗.

(3.17)

Let us realize the Lie algebra of H as Lie(H) = {A ∈ EndE (V ) | b(Av1 , v2 ) + b(v1 , Av2 ) = 0 ,

∀v1 , v2 ∈ V } .

Write again V = Vℓ+ + Wm,ℓ + Vℓ− . We write the elements of Lie(H) in matrix form, following this decomposition. Let us choose   zℓ 0 0 f = fℓ,w0 = f1 (w0 ) n 0  ∈ Lie(H) . ′ 0 f1 (w0 ) z˜ℓ

Here, we think of f1 (w0 ) as an element of HomE (Vℓ+ , Wm,ℓ ), which we identity with Wm,ℓ × · · · × Wm,ℓ (ℓ times). We take

1 f1 (w0 ) = (0, ..., 0, w0 ). 2 The element zℓ is in HomE (Vℓ+ , Vℓ+ ), which we identify with Mℓ (E), through the basis {e1 , ..., eℓ }. We take 2zℓ to be the lower nilpotent Jordan block of order ℓ. The elements f1 (w0 )′ , z˜ℓ are uniquely determined by the condition that fℓ,w0 ∈ Lie(H). Finally, n = 0Wm,ℓ , if w0 is anisotropic, and if w0 is isotropic, then, as a matrix,   0 ... 0  ..  . n =  ... . 1 ... 0

This is a basis element of the the opposite of the Lie algebra of Cℓ (the center of HWm,ℓ+1 ). Clearly, f is nilpotent. Let   aℓ (t)  ∈ h(V ), d′ℓ (t) =  IWm,ℓ ∗ aℓ (t) where t¯ = t, and

aℓ (t) = diag(t2ℓ , t2ℓ−2 , t2ℓ−4 , ..., t2 ). Define d = dℓ = d′ℓ , if w0 is anisotropic. If w0 is isotropic (recall that we then take w0 = eℓ+1 and h(V ) is either symplectic, or even unitary), we define d(t) = dℓ (t) = diag(taℓ (t), t, IWm,ℓ+1 , t−1 , t−1 aℓ (t)∗ ).

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It is easy to verify (3.17). Now we decompose the Lie algebra Lie(H) according to the weights of Ad(d(t)), M Lie(H) = Lie(H)i , where

Lie(H)i = {S ∈ Lie(H) | Ad(d(t))S = ti S , Note that

∀t ∈ F ∗ }.

    x y z  Lie(Nℓ ) =  0Wm,ℓ y ′  ∈ Lie(H) | x is upper nilpotent ,   x ˜

and, clearly, if w0 is anisotropic, then M M Lie(Nℓ ) = Lie(H)i = Lie(H)i i≥2

and if w0 is isotropic, then Lie(Nℓ Cℓ ) =

i≥1

M

Lie(H)i

i≥2

Lie(Nℓ+1 ) =

M

Lie(H)i .

i≥1

Next, note that the character χℓ,w0 , is also obtained by χℓ,w0 (exp S) = tr(fℓ,w0 · S) ,   x y z Here, for S =  0Wm,ℓ y ′  ∈ Lie(Nℓ ), x˜

S ∈ Lie(Nℓ) .

tr(fℓ,w0 · S) = 2(tr(zℓ · x) + tr(f1 (w0 ) · y)) =

ℓ−1 X

xi,i+1 + yℓ (w0 ) .

i=1

We think of y as an element of HomE (Wm,ℓ , Vℓ+ ), which we identify as the space ∗ of column vectors of ℓ coordinates in Wm,ℓ . Thus, y is a column of elements ∗ y1 , ..., yℓ ∈ Wm,ℓ . Note that in the Fourier-Jacobi coefficient (3.14), when we factor the integration through Nℓ Cℓ , the inner integration is a Fourier coefficient of the automorphic form ξπ , along Nℓ Cℓ , with respect to the extension to Cℓ (A) of ψℓ , given by the character ψ itself (which is the central character of ωψ,γ on HWm,ℓ+1 (A)). Note, also, that this extension, still denoted by ψℓ (w0 is now isotropic) is given by ψℓ (exp S) = ψE,−1 (tr(fℓ,w0 S), for S ∈ Lie(NℓCℓ )(A). Finally, the elementary divisors of the nilpotent elements fℓ,w0 are x2ℓ+1 , x, x, ..., x (dimE Wm,ℓ+1 + 1 times), in case w0 is anisotropic, and in case w0 is isotropic, these are x2ℓ+2 , x, x, ..., x (dimE Wm,ℓ+1 times).

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3.4

Global integrals representing L-functions I

In this section (and the next one), we review briefly the theory of global integrals, which represent standard L-functions for pairs of cuspidal, generic representations on our classical (or metaplectic) group and on ResE/F GLn . The purpose of this short review is to motivate our descent construction. In Chapter 10, we will explain in detail the unwinding process of the global integrals to “Eulerian” integrals, which depend on certain suitable Whittaker coefficients of the data involved. In this section, we take (V, b) is as in Sec. 2.1, i.e. Assumption 2.1 holds. Let G be the group Lℓ,w0 , in case w0 is anisotropic (Sec. 3.1), and, in case w0 is isotropic, ˆ ℓ ((3.12), (3.13)). Note that, in the first case, G may be split let G be the group H odd orthogonal, split, or quasi-split even orthogonal, or (quasi-split) odd unitary. In the second case, (w0 isotropic), G may be symplectic, metaplectic, or (quasi-split) even unitary. This due to Assumption 2.1. Let σ be an irreducible, automorphic, cuspidal representation of GA . We assume that σ is genuine, in case G is metaplectic. Consider the representations Rτ¯,¯s of HA , or Rτ¯,¯s,ψ , given in (2.9), (2.10). Recall that this is the representation of HA parabolically induced from s − 21

τ1 | det ·|E1

s − 21

⊗ · · · ⊗ τr | det ·|Er

,

where τ1 , ..., τr are irreducible, automorphic, unitary representations of GL(Vϕ+ (1))A , ..., GL(Vϕ+ (r))A , respectively, such that τi is cuspidal, if mi = dimVϕ+ (i) ≥ 2. In case H is metaplectic, we also have to insert µψ in the induction data, as in (2.9). Let E(h, fτ¯,¯s ) be the Eisenstein series, corresponding to a smooth holomorphic section fτ¯,¯s . Let us apply Fourier coefficients of Gelfand-Graev type, or of Fourier-Jacobi type to E(h, fτ¯,¯s ). Thus, if w0 is anisotropic, consider, as in (3.8), Z −1 ψℓ,w0 (v)dv. (3.18) E(vh, fτ¯,¯s )ψℓ,w E (h, fτ¯,¯s ) = 0 Nℓ (F )\Nℓ (A)

This Fourier coefficient is of Gelfand-Graev type. Recall from (3.9) that, when restricted to GA , it becomes an automorphic function. Let ϕσ be a cusp form in the space of σ. Define Z ϕσ (g)E ψℓ,w0 (g, fτ¯,¯s )dg. (3.19) L(ϕσ , fτ¯,¯s ) = GF \GA

This the L2 -pairing between the complex conjugate of σ and σψℓ,w0 (Rτ¯,¯s ). The integrals (3.19) are the global integrals of Rankin-Selberg type, which represent the partial L - function r Y

i=1

LS (σ × τi , si ),

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provided σ is globally generic, with respect to an appropriate Whittaker character, which depends on the coefficient of the Eisenstein series (3.18). As usual, S denotes a finite set of places of F , containing the Archimedean ones, such that outside S, the representations σ, τ1 ,...,τℓ , the extension E/F are all unramified, and ψ is normalized (outside S). Similarly, when h(V ) is either symplectic, or even unitary, where we now take w0 ˆ ℓ (A), to be isotropic, consider the Fourier-Jacobi coefficient of E(h, fτ¯,¯s ), for h ∈ H −1 ≤ ℓ ≤ m − 1, Z (m−ℓ−1),φ E(vh′ , fτ¯,¯s )ψℓ−1 (v)θψ−1 ,γ −1 (jℓ (v)h′′ )dv F Jψφℓ ,γ (E(·, fτ¯,¯s ))(h) = Nℓ+1 (F )\Nℓ+1 (A)

(3.20) as in (3.14). It will be convenient to re-denote F Jψφℓ ,γ (E(·, fτ¯,¯s ))(h) = F Jψφℓ ,γ (E(h, fτ¯,¯s )).

(3.21)

ˆ ℓ (A). At this By (3.16), F Jψφℓ ,γ (E(h, fτ¯,¯s ) is an automorphic function on GA = H point, we should remember that, in case H is metaplectic, ψ enters in the data of fτ¯,¯s , and hence in the data of the Eisenstein series; denote it now by E(h, fτ¯,¯s,ψ ). If we change ψ to ψ α , for α ∈ F ∗ , then by (2.13), F Jψφℓ ,γ (E(h, fτ¯,¯s,ψα )) = F Jψφℓ ,γ (E(h, fτ¯⊗χα ,¯s,ψ )).

Let ϕσ be a cusp form in the space of σ. Define Z ϕσ (g ′ )F Jψφℓ ,γ (E(g ′ , fτ¯,¯s ))dg. L(ϕσ , fτ¯,¯s , φ) =

(3.22)

(3.23)

G′F \G′A

Here, G′ = G, if h(V ) is unitary, and G′ = Sp(Wm,ℓ+1 ), if h(V ) is symplectic. In f m,ℓ+1 ), the integrand in (3.23), as a function on Sp(W f m,ℓ+1 )A this case, if G = Sp(W is trivial on the center, and hence factors through Sp(Wm,ℓ+1 )A . This is due to our assumption that σ is genuine. In (3.23), when GA is metaplectic, g ′ ∈ GA projects to g ∈ G′A , and when GA is symplectic, g ′ = g ∈ GA . The integral (3.23) is the L2 -pairing between the complex conjugate of σ and σψℓ ,γ (Rτ¯,¯s,ψ ). If needed, when we we want to stress the presence of ψ in the inducing data, when H is metaplectic, then we will denote the l.h.s. of (3.23) by L(ϕσ , fτ¯,¯s,ψ , φ). By (3.22), we have (in case H is metaplectic), for α ∈ F ∗ , L(ϕσ , fτ¯,¯s,ψα , φ) = L(ϕσ , fτ¯⊗χα ,¯s,ψ , φ).

(3.24)

The integrals (3.23) are the global integrals of Shimura type, which represent the partial L - function r Y

i=1

LS (σ × (τi ⊗ γ −1 ), si ),

in case G is symplectic, or even unitary; in case G is metaplectic, we have to take the L-functions LSψ (σ × τi , si ), with respect to ψ, instead. As before, σ must

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be globally generic, with respect to an appropriate Whittaker character. Here are some references. When G is orthogonal, see [Ginzburg, Piatetski-Shapiro and Rallis (1997)], where Gelfand-Graev coefficients of general Eisenstein series, induced from cuspidal data on maximal parabolic subgroups, are considered. The more special case of Siegel Eisenstein series can be found in [Soudry (1993)], for G odd orthogonal. The case where G is even orthogonal is worked out in [Kaplan (2011)], [Kaplan I (2011)]. The case where G is symplectic or metaplectic can be found in [Ginzburg, Rallis and Soudry (1998)]. The case where G is even unitary can be found in [BenArtzi and Soudry (2011)] (the case where G is odd unitary is in preparation by the same authors). See also the survey [Soudry (2005)]. In all of the above references, only the case r = 1 is treated, so that τ¯ = τ1 . However, all other cases (i.e. r > 1) work the same, with exactly the same proof, when τ1 | det ·|s−1/2 is replaced by an Eisenstein series on GLm (A), induced from cuspidal data. We will repeat, in Chapter 10, the proof that the global integrals (3.19), (3.23) are “Eulerian”, and depend on certain Whittaker coefficient of ϕσ . This will be needed in order to show that our descent map, produces globally generic automorphic representations. The integrals (3.19) and (3.23) are not identically zero, if and only if σ is globally generic with respect to an appropriate Whittaker character (this character will be specified in Chapter 10). In this case, for decomposable data, these integrals are “Eulerian” in the sense that, for Re(s1 ) ≫ . . . ≫ Re(sr ) ≫ 0, they decompose as Y L(¯ s) = Lv (¯ s), (3.25) v

where v varies over the places of F . We omitted here the dependence on ϕσ , fτ¯,¯s , etc., in order to simplify notation. The local factors can be represented by absolutely convergent integrals (in an open domain of s¯, as above) and they depend on the appropriate local Whittaker function of ϕσ , on fτ¯v ,¯s , φv etc. They have a meromorphic continuation, and are not identically zero at each point s¯. Finally, for all finite places v, where all local data are unramified (and normalized), one has

Lv (¯ s) =

r Q

i=1

L(σv × (τi,v ⊗ γv−1 ), si )

where L(¯ τv , ϕ, s¯) =

Y

1≤i<j≤r

L(τi,v ×τj,v , si +sj )

L(¯ τv , ϕ, s¯) r  Y

i=1

,

(1) L(τi,v , αϕ,i , si

(3.26)  1 (2) + )L(τi,v , αϕ,i , 2si ) . 2

(3.27) f m,ℓ+1 ), we have to replace L(σv × τi,v , si ) in (3.26) by In case G = Sp(W Lψv (σv × τi,v , si ). Recall that, in this case, there is no canonical way to assign unramified parameters to the unramified representation σv . In order to assign such parameters, we have to first make a choice of a nontrivial character of Fv . We choose ψv . Recall, that by definition, γv = 1, except in case G is even unitary.

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Global integrals representing L-functions II

The integrals (3.19), (3.23) have “dual” analogs, where the cuspidal representation on G and the Eisenstein series on H switch roles. Thus, in this section, (V, b) is either as in Sec. 3.1, with the extra requirement, that if b is Hermitian, then dimE V is odd, or (V, b) is as in Sec. 3.2. Let σ be an irreducible, automorphic, cuspidal representation of HA , and assume it is genuine, in case H is metaplectic. Consider the group G, as in the previous section. We will assume that, in case h(V ) is odd orthogonal, or odd unitary, then the anisotropic vector w0 is such that b(w0 , w0 ) is a square, or, respectively, a norm of an element in E ∗ (e.g. b(w0 , w0 ) = 1). Thus, when w0 is anisotropic, the group G = Lℓ,w0 is split orthogonal, when h(V ) is orthogonal, and, respectively, it is quasisplit, even unitary, when h(V ) is odd unitary. The remaining cases are, as before, those of Sec. 3.2, that is H is symplectic, metaplectic, or (quasi-split) even unitary; ˆ ℓ . Note, that now, the spaces Wm,ℓ ∩ w0⊥ , w0 is isotropic (w0 = eℓ+1 ), and G = H when w0 is anisotropic, or Wm,ℓ+1 , when w0 is isotropic, satisfy Assumption 2.1. Consider an Eisenstein series of the form E(g, fτ¯,¯s ), as in Sec. 2.2, but on the group GA . Now, consider the integrals, which are “dual” to (3.19), (3.23). Let w0 be anisotropic. Define Z ψ ϕσ ℓ,w0 (g)E(g, fτ¯,¯s )dg. (3.28) L(ϕσ , fτ¯,¯s ) = GF \GA

For w0 isotropic, define L(ϕσ , fτ¯,¯s , φ) =

Z

F Jψφℓ ,γ (ϕσ )(g ′ )E(g ′ , fτ¯,¯s )dg.

(3.29)

G′F \G′A

See the explanation right after (3.23). These integrals represent the L2 -pairing between the complex conjugate of Rτ¯,¯s (respectively Rτ¯,¯s,ψ ) with σψℓ,w0 (π) (respectively σψℓ ,γ (π)). As before, these integrals are not identically zero, if and only if σ is globally generic, with respect to an appropriate Whittaker character, and then, for decomposable data, we have, for Re(s1 ) ≫ . . . ≫ Re(sr ) ≫ 0, a decomposition as in (3.25), as well as the unramified computation (3.26), (3.27). As in (3.24), when G is metaplectic, if we want to stress the dependence on ψ of the section, and hence of the integral, we will use the notation L(ϕσ , fτ¯,¯s,ψ , φ), and then, for α ∈ F ∗ , we have L(ϕσ , fτ¯,¯s,ψα , φ) = L(ϕσ , fτ¯⊗χα ,¯s,ψ , φ).

(3.30)

For references, see [Ginzburg (1990)], for orthogonal groups, [Ginzburg, Rallis and Soudry (1998)], for symplectic groups, [Ben-Artzi and Soudry (2009)], for unitary groups. See also [Gelbart and Piatetski-Shapiro (1987); Tamir (1991); Watanabe (2000)], for the “close rank” cases. Again, in these references, only the case r = 1 is

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treated, so that τ¯ = τ1 . However, the proofs work exactly same when τ1 | det ·|s−1/2 is replaced by an Eisenstein series on GLm (A), induced from cuspidal data. We will repeat, in Chapter 10, the proof that the global integrals (3.28), (3.29) are “Eulerian”, and depend on certain Whittaker coefficient of ϕσ . 3.6

Definition of the descent

In order to motivate the descent construction, let H and G be as in Sec. 3.4. Consider an irreducible, automorphic, cuspidal representation σ of GA , such that it has a weak functorial lift to GLm (AE ), to an irreducible, automorphic representation τ . Assume, for simplicity, that τ is cuspidal, and that G is split (so that E = F and γ = 1). The fact that τ is a weak functorial lift of σ means that, at almost all finite places v, where σv and τv (and ψv in case G is metaplectic) are unramified, the semi-simple conjugacy class in L G0 , corresponding to σv is mapped to the semi-simple conjugacy class in GLm (C), corresponding to τv , under the natural embedding L G0 ֒→ GLm (C). Here, L G0 is the connected component of the L-group of G. In case G is metaplectic, in 2n variables, recall that we define its L-group to be that of the split group SO2n+1 , that is Sp2n (C). Of course, L GL0m = GLm (C). We conclude that τ is self-dual, at almost all places, and hence, by the strong multiplicity one theorem for cuspidal representations of GLm (A) (see [Jacquet and Shalika II (1981)]), we get that τˆ = τ . Let S be a finite set of places of F , which includes those which are Archimedean, and outside which, σ and τ are unramified. Then LS (σ × τ, s) = LS (τ × τ, s),

and hence LS (σ × τ, s) has a pole at s = 1 (in case G is metaplectic, we assume that ψ is normalized outside S, and we have to replace LS (σ × τ, s) by LSψ (σ × τ, s).) This follows from the fact that LS (τ × τˆ, s) always has a pole at s = 1 (see [Jacquet and Shalika I (1981)]) and τ is self-dual. Let us use one of the integrals (3.19), (3.23) to represent LS (σ × τ, s) (or LSψ (σ × τ, s).) This determines the group H (as in Sec. 3.4) and ℓ uniquely. For example, if G = SO2n+1 , then m = 2n, H = SO4n , f and ℓ = n − 1, or if G = Sp2n , then m = 2n + 1, H = Sp 4n+2 , and ℓ = n. Note that the denominator (3.27) is 1 LS (τ, α(1) , s + )LS (τ, α(2) , 2s), 2 where the first factor is 1, unless G is metaplectic, in which case the first factor is LS (τ, s + 21 ); the second factor is the partial exterior square, or the symmetric square L - function of τ , at 2s. Since τ is unitary, this denominator is nonzero at s = 1 ([Jacquet and Shalika I (1981); Jacquet and Shalika II (1981)]). Thus, one of the corresponding integrals (3.19), (3.23) has a pole at s = 1, for some choice of data, and we conclude that the Eisenstein series, on HA , E(·, fτ,s ), corresponding to the representation Rτ,s , (resp. Rτ,s,ψ ) of HA , induced from the parabolic subgroup,

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whose Levi part is isomorphic to GLm , and the representation τ | det ·|s− 2 , has a pole (simple) at s = 1. It follows (and we will explain this in detail in Chapter 10 that LS (τ, α(2) , s) has a (simple) pole at s = 1, and, in case G is metaplectic, we also have LS (τ, 12 ) 6= 0. This puts us in the case r = 1 of Theorem 2.1. Moreover, the corresponding integral (3.19), or (3.23) is non-trivial, when we replace the Eisenstein series E(h, fτ,s ) by its residue at s = 1. This means that the L2 - pairing, along GF \GA (resp. G′F \G′A ), between the cusp forms of σ and the complex conjugates of the Gelfand-Graev coefficient, with respect to ψℓ,w0 , (resp. the Fourier-Jacobi coefficient, with respect to ψℓ and ωψ ) of Ress=1 E(h, fτ,s ) is nontrivial. Once again, note that given G and m (i.e. GLm ), such that L G0 ֒→ GLm (C), this determines the group H, with a maximal parabolic subgroup, whose Levi part is isomorphic to GLm , and the coefficient, applied to the residue (and, in particular, this determines ℓ). Now, we face the question, which represents the main idea of this work. Question 3.1. Does the space generated by the (complex conjugates of the) GelfandGraev coefficients (resp. Fourier-Jacobi coefficients) of the residual representation Eτ , spanned by the residues of the Eisenstein series above, on HA , considered as a GA -module, reconstruct σ back (at least, up to an isomorphism at almost all places)? The answer is yes, and the proof will occupy the next six chapters. Now, we are ready to define the descent. We keep notations and assumptions as in Chapter 2 and Sec. 3.4. Let τ1 , ..., τr be pairwise different irreducible, unitary automorphic representations of GL(Vϕ+ (1))A , ..., GL(Vϕ+ (r))A respectively. Assume that τi is cuspidal, (2)

if dimE Vϕ+ (i) > 1, and assume that LS (τi , αϕ,i , s) has a pole at s = 1, and (1) LS (τi , αϕ,i , 21 )

6= 0, for 1 ≤ i ≤ r. These are the assumptions of Theorem 2.1. Consider the residue representation Eτ¯ of HA , whose existence is asserted by Theorem 2.1. Let us apply Gelfand-Graev coefficients, or Fourier-Jacobi coefficients to Eτ¯ . Denote, for 0 ≤ ℓ ≤ m, ( Lℓ,w0 , if w0 is anisotropic ∗ Hℓ,w0 = (3.31) ˆ ℓ, H if w0 is isotropic. See Sec. 3.1, (3.12) and (3.13). Recall that we consider isotropic w0 only when the group is symplectic, metaplectic, or even unitary, corresponding to an antiHermitian form, and in these cases, we chose w0 = eℓ+1 . Assume that w0 is anisotropic. Denote (see Sec. 3.5) τ ) = σψℓ,w0 (Eτ¯ ). σψℓ,w0 (¯

(3.32)

∗ This is a representation of Hℓ,w (A) = Lℓ,w0 (A) (by right translations) in a space 0 ∗ of automorphic functions on Hℓ,w (A). Assume, next, that w0 is isotropic. Denote 0 (see Sec. 3.7)

σψℓ ,γ (¯ τ ) = σψℓ ,γ (Eτ¯ ).

(3.33)

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ˆ ℓ (A) (by right translations) in a space of automorphic This is a representation of H ˆ functions on Hℓ (A). Recall that, in all cases, where we consider Fourier-Jacobi coefficients, we assume that dimE V is even. When H is metaplectic, our residual representation depends on ψ, and we define σψℓ ,γ (¯ τ ) = σψℓ ,γ (Eτ¯,ψ ).

(3.34)

When we change ψ to ψ α , for α ∈ F ∗ , we get, by (2.12), that Eτ¯,ψα = Eτ¯⊗χα ,ψ . Note that, in this case, the conditions that all L-functions LS (τi , sym2 , s) have a pole at s = 1, do not change when we replace each τi by τi ⊗ χα , since χ2α = 1, and so Eτ¯⊗χα ,ψ exists as well. In this case, γ = 1, and τ ⊗ χα ). σψℓ ,γ (Eτ¯,ψα ) = σψℓ ,γ (Eτ¯⊗χα ,ψ ) = σψℓ ,γ (¯

(3.35)

τ ), σψℓ ,γ (¯ τ ), where ℓ is roughly The main objects of our study are the spaces σψℓ ,w0 (¯ half the Witt index of (V, b). Recall that we assume now that the Witt index of V  1 is 2 dimE V = m (Assumption 2.1). Assume that w0 is anisotropic. Denote (see Sec. 2.1)   E V0 (3.36) ℓm = m−1+dim 2 τ ). τ ) = σψℓm ,w0 (¯ σψ,w0 (¯

Assume that w0 is isotropic. Denote ( m−1 if m is odd and H is metaplectic 2 , ℓm =  m−2  , otherwise 2 σψ,γ (¯ τ ) = σψℓm ,γ (¯ τ ).

(3.37)

(3.38) (3.39)

Note that, if H is metaplectic, then, by (3.35), the representation in (3.39) becomes σψ,γ (¯ τ ⊗ χα ), when we replace, for α ∈ F ∗ , Eτ¯,ψ by Eτ¯,ψα . τ ) the Eisenstein series on GLm (AE ) corresponding to the Denote by EGLm (¯ parabolic induction from τ1 ⊗ . . . ⊗ τr (that is, we take τ1 | det ·|zE1 ⊗ . . . ⊗ τr | det ·|zEr , and evaluate at (z1 , ..., zr ) = (0, ..., 0)). If r = 1, we mean that EGLm (¯ τ ) = τ1 . We τ ), in the first case, or σψ,γ (¯ τ ), in the second case, the call the representation σψ,w0 (¯ descent of EGLm (¯ τ ) from GLm to Hℓ∗m ,w0 . ′ For the choice w0 = yα = em + (−1)m +1 α2 e−m , made at the end of Sec. 3.1, denote (see (3.10)) σψℓ,α (¯ τ ) = σψℓ,α (Eτ¯ ) ∗ ∗ Hℓ,α = Hℓ,y α ∗ Hα,m

=

Hℓ∗m ,yα ,

(3.40) τ ). σψ,α (¯ τ ) = σψ,yα (¯

When V is symplectic, or when V is (even) anti-Hermitian, denote ∗ ˆ ℓm . Hm = Hℓ∗m ,eℓm +1 = H

(3.41)

Our notation of the descent is a little ambiguous, since it does not mention the target ∗ ∗ group Hα,m , or Hm . Adding this to the notation will make it too cumbersome. In

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the context, it will always be clear, where do we descend to, since the group H will always be known, and then the choice of the Gelfand-Graev coefficient, or Fourier∗ ∗ Jacobi coefficient will determine the group Hα,m , or Hm . ∗ Let us write in detail, in the following two tables, all cases of groups (H, Hm ) for which we construct a descent. The first table contains the cases, where the descent is defined by Gelfand-Graev coefficients (w0 is anisotropic).

1) 2) 3) 4) 5) 6) 7)

(2)

H

ResE/F GLm

∗ Hα,m

αϕ,i = αϕ,i

SO4n SO4n+1 SO4n+1 SO4n+3 SO4n+3 U4n U4n+2

GL2n GL2n GL2n GL2n+1 GL2n+1 ResE/F GL2n ResE/F GL2n+1

SO2n+1 SO2n SO2n,α SO2n+2 SO2n+2,α U2n+1 U2n+1

∧2 sym2 sym2 sym2 sym2 Asai Asai

(3.42)

(2)

The fourth column indicates our requirement that LS (τi , αϕ,i , s) has a pole at s = 1, for 1 ≤ i ≤ r. This is why the case H = SO4n+2 is not included. Here, GLm = GL2n+1 , and hence there is an index 1 ≤ i ≤ r, for which mi = dimF Vϕ+ (i) is odd, so that LS (τi , ∧2 , s) is entire. See [Jacquet and Shalika (1990)]. Thus, there is no multi-residue at (1 ..., 1) coming from τ¯. In Case (2), SO2n denotes the split orthogonal group in 2n variables, while, in Case (3), we assume that α is not a square in F ∗ , and SO2n,α denotes the quasi-split orthogonal group, corresponding to the quadratic form in 2n variables, with Witt index n − 1 and discriminant (−1)n α. Similarly, in Cases (6), (7). Note that Cases (1), (2), (3), (7) are cases of functoriality, in the sense that we ∗ have a standard L-homomorphism from the L-group of Hα,m to that of ResE/F GLm . The second table contains the cases, where the descent is defined by FourierJacobi coefficients.

8) 9) 10) 11) 12)

(1)

(2)

H

ResE/F GLm

∗ Hm

αϕ,i = αϕ,i ⊕ αϕ,i

Sp4n f Sp 4n f Sp4n+2

GL2n GL2n GL2n+1 ResE/F GL2n ResE/F GL2n+1

f 2n Sp Sp2n Sp2n U2n U2n+2

st ⊕ ∧2 sym2 sym2 Asai Asai

U4n U4n+2

(3.43)

Here, in case (8), st ⊕ ∧2 indicates the requirement that LS (τi , ∧2 , s) has a pole at s = 1, and LS (τi , 12 ) 6= 0. We did not include the case H = Sp4n+2 , since then GLm = GL2n+1 , and αϕ,i = st ⊕ ∧2 . But, since mi is odd, for some 1 ≤ i ≤ r, LS (τi , ∧2 , s) is entire.

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Note that Cases (8), (10), (11), are cases of functoriality. We now state our main theorem. Theorem 3.1. Notations and assumptions are as in Chapter 2 and Sec. 3.4. Consider (V, b) and the group H = H(V ) in the cases of functoriality, in the last two tables (Cases (1), (2), (3), (7), (8), (10), (11)). Let τ1 , ..., τr be pairwise different, irreducible, unitary automorphic representations of GL(Vϕ+ (1))A ∼ = GLmr (AE ), respectively. = GLm1 (AE ),...,GL(Vϕ+ (r))A ∼ Assume that (1) each representation τi is cuspidal, when dimE Vϕ+ (i) > 1; (1)

(2)

(2) for each i, LS (τi , αϕ,i , s) has a pole at s = 1, and LS (τi , αϕ,i , 12 ) 6= 0; r Q ωτi is trivial, when restricted to A∗F , in all (3) the product of central characters i=1

cases above, except when H = SO4n+1 (and then LS (τi , sym2 , s) has a pole at s = 1, for all i; in this case, we know that ωτ2i = 1, for each i.)

Consider the descent σψ,α (¯ τ ), in Cases (1), (2), (3), (7), and σψ,γ (¯ τ ), in Cases r Q ωτi . In Cases (8), (10), (11). In Cases (2), (3), we choose α, such that χα = i=1

∗ ∗ (1), (7), we choose any α. Let us denote by G the group Hα,m , or the group Hm , in all the cases of this theorem. Then

(1) the descent is a nontrivial automorphic representation of GA ; (2) the descent is cuspidal; (3) each irreducible summand of the descent has a weak lift (with respect to ψ, in Case (8), and with respect to γ, in Case (11); see Theorem 9.8) to the representation of GLm (AE ), parabolically induced from (τ1 ⊗ ... ⊗ τr ) ⊗ γ −1 , i.e. to the isobaric sum τ = (τ1 ⊗ γ −1 ) ⊞ ... ⊞ (τr ⊗ γ −1 ). (4) Each irreducible summand of the descent is globally generic, with respect to a certain Whittaker character, which will be specified in the proof; denote it, for a moment, by ψG . (5) The descent is a multiplicity free representation of GA . (6) Let σ be an irreducible, automorphic, cuspidal representation of GA , which is −1 globally generic with respect to the Whittaker character ψG . Assume that σ 2 has a weak lift to τ on GLm (AE ). Then σ ¯ has a nontrivial L -pairing with the descent of τ . (See Theorem 11.1 for a precise formulation.) The descent construction and its main properties appear in [Ginzburg, Rallis and Soudry (1999)], where we outlined the main ideas, formulated the main conjectures and initiated the proof of the main theorem for the case of symplectic groups. The proof was completed in [Ginzburg, Rallis and Soudry I (1999)], [Ginzburg, Rallis and Soudry II (1999)], [Ginzburg, Rallis and Soudry (2002)]. The remaining cases were

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surveyed, with proofs in low rank examples in [Soudry (2005)]. See also [Ginzburg, Rallis and Soudry (2001)], [Ginzburg, Rallis and Soudry (1997)], [Ginzburg, Rallis and Soudry I (1997)]. What about the remaining cases of the two tables above? They represent certain endoscopic (weak) lifts, which come from lifts, which already appear in the main theorem. They will be studied together with the cases of functoriality. For example, in Case (9), the descent of τ to Sp2n (A) is the theta lift from SO2n,α (A) of the descent σψ,α (¯ τ ) give by Cases (2) or (3). Our plan is to study, first, local analogues of the representations σψℓ,w0 (π), σψℓ ,γ (π) (see Sec. 3.1 and Sec. 3.2). In the local non-Archimedean setting, say over a finite place v of F , these local analogues are defined through corresponding Jacquet modules with respect to the characters (ψv )ℓ,w0 , applied to smooth representations π of HFv . The main result here is a sort of Leibniz rule that these Jacquet module satisfy, when we take π to be induced from a maximal parabolic subgroup of H, from a tensor product of representations (one of the GL part and one of the H(W ) part). The terms which appear in this Leibniz decomposition involve BernsteinZelevinsky derivatives of the GL-representation and “deeper” Jacquet modules of the same type (Gelfand-Graev type, or Fourier-Jacobi type). We will do this in general, without restricting H, as in Sec. 3.4, for future reference. Then we apply these results to representations π = πτ,v which are unramified, and appear as the factor at the place v of our residual representation Eτ¯ . We will show that these Jacquet modules vanish on πτ,v , for all ℓ > ℓm . This will imply that, according to the case at hand from tables (3.42), (3.43), for ℓ > ℓm , σψℓ,α (¯ τ ) = 0,

σψℓ ,γ (¯ τ ) = 0.

(3.44)

Thus, the “first possible occurrence” in this case (namely π = πτ,v ) is ℓ = ℓm . Moreover, our study of these Jacquet modules will also tell us that when ℓ = ℓm , they occur, up to semi-simplification, in a certain unramified representation of ∗ ∗ Hα,m (Fv ), or Hm (Fv ), according to the case at hand. In the cases of functoriality, this unramified representation turns out to be exactly the one, which lifts to the parabolic induction τv from τ1,v ⊗ ... ⊗ τr,v to GLm (Ev ). This is the local part of this work. It will be carried out in the next three chapters. The global part consists of a study of Gelfand-Graev, or Fourier-Jacobi coefficients of automorphic representations of HA , in general, and of the residual representations Eτ¯ , in particular. We will derive general formulas for constant terms, along ∗ ˆ ℓ ), of Gelfandunipotent radicals of maximal parabolic subgroups of Hℓ,α (resp. H Graev coefficients (resp. Fourier-Jacobi coefficients) of automorphic forms on HA , with respect to the character ψℓ,α (resp. with respect to (ψℓ , γ)). These formulas involve coefficients of the same type, which are “deeper” than the one we start with, and they also involve similar coefficients applied to certain constant terms of our automorphic form. When we apply these formulas to the residual representation Eτ¯ , we get a tower property satisfied by the representations σψℓ,α (¯ τ ) (respectively,

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σψℓ ,γ (¯ τ )) as ℓ varies from m downwards. This tower property says that, at the first index ℓ0 , counting from m down, where the representation σψℓ0 ,α (¯ τ ) (respectively, σψℓ0 ,γ (¯ τ )) is nontrivial, this representation must be cuspidal. Thus, from (3.44), we conclude that if σψℓm ,α (¯ τ ) (respectively, σψℓm ,γ (¯ τ )) is nontrivial, then it is already cuspidal. Also, in such a case, our local study will show that the unramified factors, say at a place v, of any given irreducible summand do, indeed, lift to τv . All this will be done in Chapter 7. In Chapters 8 and 9, we will prove that the spaces of σψℓm ,α (¯ τ ) (respectively, σψℓm ,γ (¯ τ )) are nontrivial. Moreover, we will prove that a certain Whittaker coefficient is not identically zero on these spaces, and in several cases we will get very nice and interesting identities relating these Whittaker coefficients to GLm Whittaker coefficients on constant terms along the radical Um applied to Eτ¯ . This will achieve the proof of the main theorem, except for the assertion that each irreducible summand of the descent is globally generic. This we do in Chapter 10, where we derive the Eulerian nature of the global integrals, introduced in Sec. 3.4, Sec. 3.5. In the last chapter we give some applications. We describe the image of the weak functorial lift (in the cases of functoriality from tables (3.42), (3.43)) from ∗ ∗ generic, cuspidal representations of Hα,m (respectively Hm ) to ResE/F GLm . We also obtain generalized endoscopic weak lifts for generic cuspidal representations of our classical groups, as well as base change and automorphic induction. We end this chapter by the analogous definitions, in the local setting, of Jacquet modules with respect to characters of Gelfand-Graev type, or Fourier-Jacobi type. 3.7

Definition of Jacquet modules corresponding to Gelfand-Graev characters

We fix a finite place v of F . Denote, as in Sec. 1.3, k = Fv and K = Ev = k ⊗F E. √ Recall that, in case [E : F ] = 2, E = F [ ρ]. Thus, if ρ is not a square in k, then √ K = k[ ρ] is a quadratic extension of k. If ρ is a square in k, say ρ = d2 , d ∈ k, then K ∼ = k ⊕ k by the isomorphism i in (1.7). Let (V, b) be a quadratic space over F (E = F ), or a Hermitian space over E ([E : F ] = 2). Denote by m ˜ the Witt index of b. Unlike Sec. 3.1, we now do not ′ restrict m, ˜ and let it be arbitrary. We still denote dimE V = m′ , and m = [ m2 ]. Consider the space Vk = k ⊗F V . If K is a field, then we view this space as a vector space over K; dimK V = m′ . If K ∼ = k ⊕ k, then we view Vk as X ⊕ X, where X is a vector space over k, of dimension m′ . See (1.8). We will use the notation of Sec. 3.1 (see the remark right after (3.10)). Thus, V ± are maximal isotropic subspaces of V , in duality with respect to b; we have the isotropic flag (3.1) of subspaces Vj+ , ± j ≤ m, ˜ the dual bases {e±1 , ..., e±m ˜ } etc. Similarly, we consider the spaces Vk , + V0 (k), Vℓ (k), the flag ϕℓ,k , as in (3.3) etc. Let Hk be the group of k-rational points

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∼ GL(X) ∼ ∼ k ⊕ k, then Hk = of H. When H is unitary and K = = GLm′ (k). The isomorphism g 7→ r(g) is written explicitly in Sec. 1.3. If K is a field, then Hk is orthogonal, or unitary. Consider the anisotropic vector w0 ∈ Wm,ℓ = Wm,ℓ (F ), the characters ψℓ,w0 , ψℓ,α (α ∈ F ∗ ), and the groups Rℓ,w0 = Lℓ,w0 Nℓ , Rℓ,α = Lℓ,α Nℓ . Let π be a smooth representation of Hk , acting in a space Vπ . Let J(ψv )ℓ,w0 (π) be the Jacquet module of π, with respect to Nℓ (k) and its character (ψv )ℓ,w0 . Thus, the space of J(ψv )ℓ,w0 (π) is Vπ /Span{π(v)ξ − (ψv )ℓ,w0 (v)ξ|v ∈ Nℓ (k),

ξ ∈ Vπ }.

˜ This is a module over Lℓ,w0 (k). Denote, when ℓ < m, J(ψv )ℓ,α (π) = J(ψv )ℓ,yα (π). Let us describe, in detail, the pair (Nℓ (k), (ψv )ℓ,w0 ). Assume that K is a field. Let B0 be a basis of V0 over E. Then {e1 , ..., em ˜ } ∪ B0 ∪ {e−m ˜ , ..., e−1 }

(3.45)

is a basis of Vk over K. Let us represent the elements of Hk as matrices according to our chosen basis. Then     z y x   Nℓ (k) = u =  Im′ −2ℓ y ′  ∈ H|z ∈ Zℓ (K) . (3.46)   ∗ z

Zℓ denotes the standard maximal unipotent subgroup of GLℓ . Let us denote by ψK,1 the character ψv , in case K = k, and the character ψv ( 12 trK/k ( √xρ )), in case √ K = k[ ρ] is a quadratic extension of k. For u ∈ Nℓ (k), written in the form (3.46), ℓ−1 X zi,i+1 )ψK,1 (yℓ · w0′ ), (ψv )ℓ,w0 (u) = ψK,1 (

(3.47)

i=1

′

where we identify Wm,ℓ with the column space K m −2ℓ , through the sub-basis above, ′ which is the complement of {e±1 , ..., e±ℓ }, and w0′ is the vector in K m −2ℓ , corresponding to w0 . For the choice w0 = yα , we get ℓ−1 X ′ α (ψv )ℓ,α (u) = ψK,1 ( zi,i+1 )ψK,1 (yℓ,m−ℓ + (−1)m +1 yℓ,m′ −m−ℓ+1 ). ˜ ˜ 2 i=1

The stabilizer of (ψv )ℓ,w0 is, in matrix form, the subgroup     Iℓ  Lℓ,w0 ,v =  γ  ∈ h(V )(k)|γw0′ = w0′ ∼ = h(V ∩ w0⊥ )k .   Iℓ

(3.48)

(3.49)

Assume that b is Hermitian and K = k ⊕ k. Then we view Vk as X ⊕ X, as in Sec. 1.3; X is a vector space over k, of dimension m′ . We take the basis (3.45), as in Sec. 1.3, for the space X. See (1.8). Similarly, we view Wm,ℓ (k) as Ym,ℓ ⊕ Ym,ℓ , and we take the basis {eℓ+1 , ..., e−(ℓ+1) }, for the subspace Ym,ℓ ⊂ X. Recall, also,

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the subspaces Xi+ = Spank {e1 , ..., ei }, for 1 ≤ i ≤ m. ˜ By (1.19), for u ∈ Nℓ (k), r(u)(Xi+ ) = r(u)∗ (Xi+ ) = Xi+ , for 1 ≤ i ≤ ℓ, and hence r(u) has the form   z1 y1 x (3.50) r(u) =  Im′ −2ℓ y2  ∈ GLm′ (k), z2

where z1 , z2 ∈ Zℓ (k). Write the vector w0 as (w0′ , w0′ ), where w0′ ∈ Ym,ℓ . In order to write the the character (ψv )ℓ,w0 , applied to r(u), recall the Gram matrix J of the basis (3.45). It has the form (1.16). We have     wℓ wm−ℓ ˜ . J =  J˜  , J˜ =  J0 wm−ℓ wℓ ˜ √ Write J˜ = J˜1 + ρJ˜2 , where J˜i are matrices over F . Recall the form β˜ from (1.11). Then β˜ is given by a pair of bilinear forms β1 , β2 on X, such that ˜ β((x, y), (u, w)) = (β1 (x, w), β2 (y, u)); the bilinear forms βi are written explicitly in (1.11). Our assumption that w0 is anisotropic (over F ) means here that β1 (w0′ , w0′ ) = x0 6= 0.

(3.51)

Now, it is straightforward to see that, for r(u) of the form (3.50), (ψv )ℓ,w0 (r(u)) =  ℓ−1 P 1 ((z1 )i,i+1 + (z2 )i,i+1 ) + ψv 2d i=1

Note that

1 ′ 2d (β1 (r(u)w0 , e−ℓ )

∗

− β2 (r(u)



w0′ , e−ℓ )

.

(3.52)

 ∗  z2 −wℓt (y2 z2−1 )(J˜1 + dJ˜2 ) ∗ r(u)∗ =  Im′ −2ℓ −(J˜1 − dJ˜2 )t (z1−1 y1 )wℓ  , z1∗

where zi∗ = wℓt zi−1 wℓ . Let us identify Ym,ℓ , through its given basis, with the column ′ ′ space k m −2ℓ , and think of w0′ as an element of k m −2ℓ . Then β1 (r(u)w′ , e−ℓ ) − β2 (r(u)∗ w′ , e−ℓ ) = (y1 )ℓ · w′ + (t y2 )1 (J˜1 + dJ˜2 ) · w′ , (3.53) 0

0

0

0

t

where (y1 )ℓ denotes the last row of y1 and ( y2 )1 denotes the first row of t y2 . Let us write the stabilizer of the last character, i.e. the corresponding subgroup Lℓ,w0 ,v . We will identify this stabilizer with its image under the isomorphism r. Denote w0′′ = (J˜1 + dJ˜2 ) · w0′ . Then     Iℓ  Lℓ,w0 ,v =  γ  ∈ GLm′ (k)|γw0′ = w0′ , t w0′′ γ = t w0′′ . (3.54)   Iℓ Note that

Lℓ,w0 ,v ∼ = GLm′ −2ℓ−1 (k).

(3.55)

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The isomorphism is obtained as follows. Write ′ Ym,ℓ = Ym,ℓ ⊕ kw0′ , ′ where Ym,ℓ = {y ∈ Ym,ℓ |t w0′′ · y = 0}. This is the kernel of the linear functional, on Ym,ℓ , y 7→t w0′′ · y. Note that, by (3.51), t w0′′ · w0′ = x0 6= 0. Now, γ ∈ GLm′ −2ℓ (k) = GL(Ym,ℓ ), satisfies γw0′ = w0′ , and t w0′′ γ =t w0′′ , if and only if γw0′ = w0′ and ′ ′ ′ γ(Ym,ℓ ) = Ym,ℓ . This identifies Lℓ,α,v with GL(Ym,ℓ )∼ = GLm′ −2ℓ−1 (k). Let f0 be a matrix in GLm′ −2ℓ (k), whose first column is w0′ and the remaining ′ columns are a basis of Ym,ℓ . Then (see (3.51)) we clearly have

  1 0   f0−1 w0′ =  .   ..  0 t ′′ w0 f0 = x0 (1, 0, ..., 0).

(3.56)

Let  Iℓ f =  f0

Let

Iℓ



.

(ψ˜v )ℓ,w0 (r(u)) = (ψv )ℓ,w0 (f r(u)f −1 ). Then (ψ˜v )ℓ,w0 (r(u)) = ψv

! ℓ−1 1 X ( ((z1 )i,i+1 + (z2 )i,i+1 ) + (y1 )ℓ,1 + x0 (y2 )1,1 ) . (3.57) 2d i=1

A further conjugation by the diagonal matrix whose diagonal is ′

′

′

′

(2d)m , (2d)m −1 , ..., (2d)m −ℓ , x0 (2d)m −ℓ−1 , ..., x0 (2d), will take the character (3.57) to the character ℓ−1 X (ψ˜v )ℓ (r(u)) = ψv ( ((z1 )i,i+1 + (z2 )i,i+1 ) + (y1 )ℓ,1 + (y2 )1,1 ).

(3.58)

i=1



The stabilizer of the character (3.58) inside    Iℓ ˜ℓ =  γ L 

Iℓ



 ∈ GLm′ (k)



Iℓ GLm′ −2ℓ (k)

Iℓ



 is

    1 γ= , g ∈ GLm′ −2ℓ−1 (k) .  g

(3.59)

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3.8

Definition of Jacquet modules corresponding to Fourier-Jacobi characters

We keep the notation of the last section, only that now, we assume that the form (V, b) is as in Sec. 3.2. Thus, m′ = dimE V = 2m is even, and b is either symplectic (when E = F ) or anti-Hermitian (δ = −1) with Witt index m. If K is a field, then Hk is symplectic, metaplectic, or a quasi-split even unitary group. If K = k ⊕ k, then Hk = GLm′ (k). We use the notation as in Sec. 3.2. Now, the vector w0 ∈ Wm,ℓ = Wm,ℓ (F ) is isotropic. Recall, that, in this case, we choose, for simplicity, w0 = eℓ+1 , and denote ψℓ = ψℓ,eℓ+1 . This is a character of Nℓ (A), which we extend trivially to Nℓ+1 (A), still denoting the extension by ψℓ . Thus, at the place v, with notations as in the last section, we have the following. Assume, first, that K is a √ field. Denote by ψK,−1 the character ψv , when K = k, and when K = k[ ρ] is a quadratic extension, this is the character of K given by ψv ( 12 trK/k (x). Then     z y x   Nℓ+1 (k) = v =  I2(m−ℓ−1) y ′  ∈ h(V )(k)|z ∈ Zℓ+1 (k)   (3.60) z∗   ℓ P zi,i+1 . (ψv )ℓ (v) = ψK,−1 i=1

The stabilizer of (ψv )ℓ , viewed as a character of Nℓ (k), inside h(Wm,ℓ )k , is (with obvious identifications)     1y x  Lℓ (k) =  g y ′  ∈ h(Wm,ℓ )k .   1

ˆ ℓ ( see (3.12), (3.13).) Let ω −1 −1 = ω −1 be the Weil represenRecall the groups H ψv ,γv ψv f m,ℓ+1 (k)), in case H is symplectic, or metaplectic. If H is tation of HWm,ℓ+1 (k) ⋊Sp(W f unitary, then let ω −1 −1 be the Weil representation of HW (k) ⋊ Sp(Wm,ℓ+1 (k)), ψv ,γv

m,ℓ+1

restricted to HWm,ℓ+1 (k) ⋊h(Wm,ℓ+1 )k . See Sec. 1.2. Recall, that we fixed in Sec. 3.2 an isomorphism jℓ from Nℓ \Nℓ+1 to the Heisenberg group HWm,ℓ+1 . See, also, Sec. 1.2. We keep denoting it by jℓ , at the place v. In terms of (3.60), we take 1 (3.61) jℓ (v) = (yℓ+1 ; trK/k (xℓ+1,1 ). 2 Let π be a smooth representation of Hk , acting in a space Vπ . Consider the action of Nℓ+1 (k) on π ⊗ ωψv−1 ,γv−1 , where Nℓ+1 (k) acts on (the space of) ωψv−1 ,γv−1 through jℓ (so that Nℓ (k) acts trivially on ωψv−1 ,γv−1 ). Let J(ψv )ℓ (π ⊗ ωψv−1 ,γv−1 ) be the Jacquet module of π ⊗ ωψv−1 ,γv−1 , with respect to the unipotent group Nℓ+1 (k) ˆ ℓ (k). We will denote and its character (ψv )ℓ . This is a module for H

(3.62) F J(ψv )ℓ ,γv (π) = J(ψv )ℓ (π ⊗ ωψv−1 ,γv−1 ). √ Assume that K ∼ = k ⊕ k, so that E = F [ ρ] is a quadratic extension of F , and ρ = d2 is a square in k; we use the isomorphism i in (1.7). As in the previous

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∼ GL(X) ∼ section, with the same notation, Hk = = GL2m (k), by the isomorphism g 7→ r(g). As in (3.50), the elements of Nℓ+1 (k) have the following form, using the identification given by the isomorphism r,   z1 y1 x u =  I2(m−ℓ−1) y2  ∈ GL2m (k), (3.63) z2

where z1 , z2 ∈ Zℓ+1 (k). Similarly to (3.52), (3.58), we get that the character (ψv )ℓ of Nℓ+1 (k), extended trivially from Nℓ (k), is given, after a simple conjugation, by ! ℓ 1X ((z1 )i,i+1 + (z2 )i,i+1 ) . (3.64) (ψv )ℓ (u)) = ψv 2 i=1

When we consider (ψv )ℓ as a character of Nℓ (k), its stabilizer inside GL2(m−ℓ) (k), embedded as diag(Iℓ , GL2(m−ℓ) (k), Iℓ ), is the subgroup     1 y1 x  Lℓ (k) =  g y2  g ∈ GL2(m−ℓ−1) (k) .   1

Finally, as in the previous section, Wm,ℓ+1 (k) ∼ = Ym,ℓ=1 ⊕ Ym,ℓ+1 , and the isomorphism jℓ : Nℓ (k)\Nℓ+1 (k) ∼ = HYm,ℓ+1 ⊕Ym,ℓ+1 , to the Heisenberg group, is given through the following, where u ∈ Nℓ+1 (k) is in the form (3.63),

1 (3.65) jℓ (u) = ((y1 )ℓ+1 , (t y2 )1 J2(m−ℓ−1) ; xℓ+1,1 − (y1 )ℓ+1 · (y2 )1 ). 2 See (1.21), (1.22). Let ωψv−1 ,γv−1 be the Weil representation of HWm,ℓ+1 (k) ⋊ ∼ f m,ℓ+1 (k)), restricted to HW ⋊ Sp(W ⊕Y (k) ⋊ h(Wm,ℓ+1 )k = HY m,ℓ+1

m,ℓ+1

m,ℓ+1

GL2(m−ℓ−1) (k). See Sec. 1.3. Let π be a smooth representation of Hk ∼ = GL2m (k), acting in a space Vπ . As before, we have the action of Nℓ+1 (k) on π ⊗ ωψv−1 ,γv−1 , and we define, as in (3.62), the Jacquet module corresponding to π and the Fourier-Jacobi character (ψv )ℓ by F J(ψv )ℓ ,γv (π) = J(ψv )ℓ (π ⊗ ωψv−1 ,γv−1 ). ˆ ℓ (k) = h(Wm,ℓ+1 )k ∼ This is a module for H = GL(Ym,ℓ ) ∼ = GL2(m−ℓ−1) (k).

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Chapter 4

Some double coset decompositions

In this chapter, we describe in detail the double cosets of h(V ), modulo two maximal parabolic subgroups, over Fv . We write a set of representatives and compute their stabilizers. We already encountered a similar situation in Sec. 2.6, Sec. 2.7, where we used representatives which are Weyl elements of minimal length, described in terms of the root system. For the convenience of the reader, and also as part of our effort to make this text self-contained, we will describe this set of double cosets in an elementary way, using only linear algebra. Using the same methods, we will also consider the double cosets of h(Wm,ℓ ), over Fv , modulo a maximal parabolic subgroup on one side and the stabilizer group Lℓ,w0 on the other side (notation of (3.7), (3.11)). The material in this chapter will be needed for the study of the Jacquet modules J(ψv )ℓ,w0 (π), F J(ψv )ℓ ,γv (π) = J(ψv )ℓ (π ⊗ ωψv−1 ,γv−1 ), as in Sec. 3.7, Sec. 3.8, when we take π to be a representation induced from a maximal parabolic subgroup of H. We will analyze such Jacquet modules by Bruhat theory, and hence the need of the set of double cosets above. We will use the notation of Sec. 3.7, Sec. 3.8. Thus, v is a fixed place of F , k = Fv , K = Ev = k ⊗F E, Vk = k ⊗F V etc. We will not mention the place v, but rather the field k, and then distinguish the cases where K = k, a quadratic extension of k, or K ∼ = k ⊕ k, so that what we do here clearly applies to any field k (of characteristic different than 2). In this chapter we work in a general setup and we assume that the Witt index of our form is arbitrary.

4.1

The space Qj \h(V )k /Qℓ

1. The case where K is a field Assume that K is a field. Denote by m ˜ k the Witt index of Vk , that is the dimension, over K, of a maximal isotropic subspace of Vk ; we will continue to denote by b the form bv , over k. We use the same notation as in Sec. 3.1, Sec. 3.2, usually + dropping mention of the field k. We fix dual maximal isotropic subspaces V + = Vm ˜k 65

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− + and V − = Vm ˜ k in Vk , and also a maximal isotropic flag in V , as in (3.1), with + corresponding dual bases, over K, {e1 , ..., em ˜ k } and {e−1 , ..., e−m ˜ k } of V , and of
⊥ + − V − , respectively; V0 = (V + ⊕ V − )⊥ , and Wm,ℓ = Vℓ + Vℓ (m′ = dimK Vk , m′ ˜ k , let Qj be the (standard) parabolic subgroup of m = [ 2 ]). For 1 ≤ j ≤ m h(Vk ) = h(V )k , which preserves Vj+ . Consider the action, from the left, of h(Vk ) on the variety Yj of all j-dimensional isotropic K - subspaces of V , given by

X 7→ h−1 · X, for h ∈ h(Vk ) and X ∈ Yj . Denote by Oj the h(Vk ) - orbit of Vj+ . Since Qj is the stabilizer of Vj+ , inside h(Vk ), we may identify Qj \h(Vk ) with Oj . Note that, except the case where h(Vk ) is split even orthogonal, and j = m = m ˜ k , the orbit Oj is the full variety of j - dimensional isotropic subspaces of Vk , i.e. Oj = Yj . In case h(Vk ) is split even orthogonal, and j = m, there are two h(Vk ) - orbits in Ym , namely Ym = h(Vk )(Vm+ ) ∪ h(Vk )(+ Vm ) (see Sec. 2.1). Let 1 ≤ ℓ ≤ m ˜ k . In order to describe Qj \h(Vk )/Qℓ , we consider the action of Qℓ on Oj . Before we continue, let us make the following Definition: Let (Vk , b) be so that m′ = 2m and b is symmetric with m ˜ k = m. We say that two maximal isotropic subspaces Y , Y ′ of Vk are in the same orientation class, if there is h ∈ h(Vk ) (in particular, det(h) = 1) such that Y ′ = h · Y , i.e. Y and Y ′ lie in the same h(Vk )-orbit. Note that for Y , Y ′ , as in the definition, it is impossible to have also h′ in the full isometry group of (Vk , b), with det(h′ ) = −1, such that h′ Y = Y ′ . The following proposition is known, and appears, for example, for orthogonal groups, in [Ginzburg, Piatetski-Shapiro and Rallis (1997)], Prop. 3.1. It describes invariants, which characterize the Qℓ -orbits in Oj . Proposition 4.1. (1) Assume that h(Vk ) is not even orthogonal and split. Then two isotropic subspaces X, X ′ in Oj , are in the same Qℓ -orbit, if and only if dimK (X ∩ Vℓ+ ) = dimK (X ′ ∩ Vℓ+ ) dimK (X ∩ (Vℓ+ + Wm,ℓ )) = dimK (X ′ ∩ (Vℓ+ + Wm,ℓ )).

(4.1)

(2) Assume that h(Vk ) is even orthogonal and split. Then the assertion of (1) holds, unless ℓ, j < m, and ℓ + dimK (X ∩ (Vℓ+ + Wm,ℓ )) − dimK (X ∩ Vℓ+ ) = m.

(4.2)

In this case, we have to add one more condition to (4.1), which is that Vℓ+ + X ∩(Vℓ+ + Wm,ℓ ) and Vℓ+ + X ′ ∩(Vℓ+ + Wm,ℓ ) (which are now maximal isotropic subspaces of dimension m) are in the same orientation class.

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Some double coset decompositions

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Proof. Let us denote (Vℓ+ )⊥ = Vℓ+ + Wm,ℓ . The conditions are clearly necessary, for if X, X ′ ∈ Oj , and there is g ∈ Qℓ , such that X ′ = g −1 X, then since g −1 Vℓ+ = Vℓ+ and g −1 (Vℓ+ )⊥ = (Vℓ+ )⊥ , we get that X ∩ Vℓ+ = gX ′ ∩ Vℓ+ = g(X ′ ∩ Vℓ+ ), X ∩ (Vℓ+ )⊥ = gX ′ ∩ (Vℓ+ )⊥ = g(X ′ ∩ (Vℓ+ )⊥ ). In particular, we get the equalities of the dimensions in (4.1). Note, also, that , by the last equality, we get g(Vℓ+ + X ′ ∩ (Vℓ+ )⊥ ) = gVℓ+ + g(X ′ ∩ (Vℓ+ )⊥ ) = Vℓ+ + X ∩ (Vℓ+ )⊥ , and so, in case h(Vk ) is split even orthogonal, and the spaces Vℓ+ + X ∩ (Vℓ+ )⊥ , Vℓ+ + X ′ ∩ (Vℓ+ )⊥ are maximal isotropic (of dimension m), they are in the same orientation class, since det(g) = 1. Conversely, let X, X ′ ∈ Oj satisfy (4.1). Denote dimK (X ∩ Vℓ+ ) = dimK (X ′ ∩ + Vℓ ) = r, and dimK (X ∩ (Vℓ+ )⊥ ) = dimK (X ′ ∩ (Vℓ+ )⊥ ) = s. Of course, 0 ≤ r ≤ s ≤ j≤m ˜ k , and also r ≤ ℓ. Choose a K-basis {x1 , ..., xr } of the subspace X ∩ Vℓ+ . If r = 0, then regard the last set as empty. Consider the restriction of the form b to X ×Vℓ+ . It factors through X/X ∩(Vℓ+ )⊥ ×Vℓ+ . Let t = dimK X/X ∩(Vℓ+ )⊥ = j −s. If t > 0, then we can find t K-linearly independent vectors x(1) , ..., x(t) ∈ X, which form, via b, a K-linearly independent set of linear functionals on Vℓ+ . Thus, we can find a K-basis of Vℓ+ of the form {x1 , ..., xr , v1 , ..., vt , vr+t+1 , ..., vℓ },

(4.3)

such that b(x(i1 ) , vi2 ) = δi1 ,i2 , (i1 )

b(x

, vi2 ) = 0,

for

for

1 ≤ i1 , i2 ≤ t

1 ≤ i1 ≤ t,

r + t + 1 ≤ i2 ≤ ℓ.

Note, that since X is isotropic, we also have b(x(i1 ) , xi2 ) = 0,

for 1 ≤ i1 ≤ t,

1 ≤ i2 ≤ r.

Choose an extension {xr+1 , ..., xs } of {x1 , ..., xr } to a K-basis of X ∩ (Vℓ+ )⊥ . Then the set {x1 , ..., xs , x(1) , ..., x(t) }

(4.4)

is K-linearly independent, and forms a K-basis of X. Note that s+t = j = dimK X. We conclude that the set {v1 , ..., vt , vr+t+1 , ..., vℓ , x1 , ..., xs , x(1) , ..., x(t) }

(4.5)

is a K basis of the subspace Vℓ+ + X. Note that the first ℓ + s − r elements of (4.5) span the isotropic subspace Z + = Vℓ+ + X ∩ (Vℓ+ )⊥ , and, in particular, ℓ+s−r ≤m ˜ k.

(4.6)

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Note, also, that by (4.3), r + t ≤ ℓ, and hence

s − r ≥ j − ℓ.

(4.7)

′ {v1′ , ..., vt′ , vr+t+1 , ..., vℓ′ , x′1 , ..., x′s , (x′ )(1) , ..., (x′ )(t) },

(4.8)

′

Let us repeat the same construction for X , to obtain a set which is analogous to (4.5), with self-evident notation. In particular, the Gram matrices of the ordered sets (4.5) and (4.8) are the same. By Witt’s theorem, we can find g in the (full) isometry group of (Vk , b), which takes the set (4.5) to the set (4.8), so that g · vi = vi′ , g · xi = x′i , g · x(i) = (x′ )(i) , for all relevant indices. In particular, gVℓ+ = Vℓ+ and gX = X ′ (using that (4.4) is a basis of X, and the analogous subset of (4.8) is a basis of X ′ ). If h(Vk ) is not orthogonal, then this means that g ∈ Qℓ takes X to X ′ , and hence X, X ′ are in the same Qℓ -orbit of Oj , and the proposition is proved. Assume that h(Vk ) is orthogonal. Then it might be that det(g) = −1. If m′ is odd, and det(g) = −1, replace g by −g, and now −g ∈ Qℓ and (−g)X = X ′ . Assume now that h(Vk ) is even orthogonal and not split (over k), and det(g) = −1. Then we may correct g as follows. Let Z − be an ℓ+s−r - dimensional isotropic subspace of Vk , which is dual to Z + , with respect to b. We may choose Z − , so that it contains {x(1) , ..., x(t) }. By our assumption, (Z + + Z − )⊥ is a non-trivial subspace of Vk , on which the symmetric form b is non-degenerate. Hence we may choose g0 in the full isometry group of (Vk , b), which acts as the identity on Z + + Z − , and acts by determinant -1 on (Z + + Z −)⊥ . Since Z + = Vℓ+ + X ∩ (Vℓ+ )⊥ , and Z − contains {x(1) , ..., x(t) }, it follows that g0 X = X and g0 Vℓ+ = Vℓ+ . Now, we replace g by gg0 , which has determinant 1, lies in Qℓ and takes X to X ′ . The same argument works when h(Vk ) is even orthogonal and split, provided ℓ + s − r < m = m ˜ k (recall that dimK Z + = ℓ + s − r.) Finally, assume that h(Vk ) is split and even orthogonal, and that ℓ + s − r = m. If det(g) = −1, then we must also have ℓ, j < m. (Recall that if j = m, then X, X ′ are assumed to be in the same orientation class.) Denote Z ′+ = Vℓ+ + X ′ ∩ (Vℓ+ )⊥ . Now, we use the extra assumption in this case, namely that Z + , Z ′+ are in the same orientation class. (They are isotropic subspaces of dimension m.) Since, by our construction, gZ + = Z ′+ , we must have det(g) = 1. This completes the proof of Proposition 4.1.  2. The case where K = k ⊕ k

Assume that K ∼ = k ⊕ k. The set of double cosets, analogous to the set in Sec. 4.1 is that of Pj,m′ −j \GLm′ (k)/Pℓ,m′ −ℓ ,

where Pt,m′ −t is the standard parabolic subgroup of GLm′ (k), which corresponds to the partition (t, m′ − t) of m′ . If we consider the analogy as one coming from a place over which the unitary group splits and becomes a general linear group, then we should consider Pj,m′ −j \GLm′ (k)/Pℓ,m′ −2ℓ,ℓ .

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See (1.19). We will consider a little more general set up, as follows. Let ℓ˜ = (ℓ1 , ℓ2 , ℓ3 ) be a partition of a positive integer N . Let Pℓ1 ,ℓ2 ,ℓ3 = Pℓ˜ be the ˜ We standard parabolic subgroup of GLN (k), which corresponds to the partition ℓ. will consider the space of double cosets Pj,N −j \GLN (k)/Pℓ˜. We allow degenerate cases, where one member of the partition is zero. We realize Pj,N −j \GLN (k) as the variety Sj of j-dimensional subspaces X ⊂ k N . Let us denote the standard basis of k N by {ǫ1 , ..., ǫN }. Let, for i ≤ N , X (i) = Spank {ǫ1 , ..., ǫi }. The analogous characterization to that of Proposition 4.1 of the Pℓ˜-orbits in Sj is the following proposition. It is a simple special case of a more general proposition, describing the double cosets of the symmetric group SN modulo two subgroups corresponding to two general partitions. See [Zelevinsky (1981)], p. 170. Proposition 4.2. Let X, X ′ be two subspaces in Sj . Then they are in the same Pℓ˜-orbit, if and only if dimk (X (ℓ1 ) ∩ X) = dimk (X (ℓ1 ) ∩ X ′ ), dimk (X (ℓ1 +ℓ2 ) ∩ X) = dimk (X (ℓ1 +ℓ2 ) ∩ X ′ ).

(4.9)

Proof. The conditions of the proposition are clearly necessary. We prove the converse. Denote dimk (X (ℓ1 ) ∩ X) = r, dimk (X (ℓ1 +ℓ2 ) ∩ X) = s. Let A1 be a basis of X (ℓ1 ) ∩ X. Choose an extension, A2 , of A1 , to a basis, A1 ∪ A2 , of X (ℓ1 +ℓ2 ) ∩ X, and an extension A3 , of A1 , to a basis, A1 ∪ A3 , of X (ℓ1 ) . Then A1 ∪ A2 ∪ A3 is a basis of X (ℓ1 ) + X (ℓ1 +ℓ2 ) ∩ X. Choose an extension, A4 , of A1 ∪ A2 , to a basis, A1 ∪ A2 ∪ A4 , of X. Since X ∩ (X (ℓ1 ) + X (ℓ1 +ℓ2 ) ∩ X) = X (ℓ1 +ℓ2 ) ∩ X,

we get that ∪4i=1 Ai is a basis of X (ℓ1 ) + X (ℓ1 +ℓ2 ) ∩ X + X = X (ℓ1 ) + X. Finally, extend the basis A1 ∪ A2 ∪ A3 of X (ℓ1 ) + X (ℓ1 +ℓ2 ) ∩ X to a basis A1 ∪ A2 ∪ A3 ∪ A5 of X (ℓ1 +ℓ2 ) . Then ∪5i=1 Ai is a basis of X + X (ℓ1 +ℓ2 ) . Indeed, we have Note that

Hence

X (ℓ1 +ℓ2 ) ∩ (X (ℓ1 ) + X) = X (ℓ1 ) + X (ℓ1 +ℓ2 ) ∩ X. |A5 | = ℓ2 − (s − r), dimk (X (ℓ1 +ℓ2 ) + X) = ℓ1 + ℓ2 + j − s.

(4.10)

s − r ≤ ℓ2 , j − s ≤ ℓ3 .

(4.11)

Let us carry a similar construction of the sets A′i , for the subspace X ′ ; i ≤ 5. Note that, by (4.9) and (4.10), we have |Ai | = |A′i |, for i ≤ 5. Since ∪5i=1 Ai and ∪5i=1 A′i are linearly independent, we can find g ∈ GLN (k), such that gAi = A′i , for i ≤ 5. Since A1 ∪ A3 and A′1 ∪ A′3 are bases of X (ℓ1 ) , we get that gX (ℓ1 ) = X (ℓ1 ) . Since ∪46=i≤5 Ai , and ∪46=i≤5 A′i are bases of X (ℓ1 +ℓ2 ) , we get that gX (ℓ1 +ℓ2 ) = X (ℓ1 +ℓ2 ) , and hence g ∈ Pℓ˜. Since A1 ∪ A2 ∪ A4 and A′1 ∪ A′2 ∪ A′4 are bases of X and X ′ , respectively, we get that gX = X ′ . This shows that X and X ′ are in the same Pℓ˜-orbit. 

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4.2

A set of representatives for Qj \h(V )k /Qℓ

1. The case where K is a field and h(Vk ) is not even orthogonal and split Assume that K is a field and h(Vk ) is not even orthogonal and split. We keep the notation of the first part of Sec. 4.1. Here, Yj = Oj . By Proposition 4.1, the Qℓ -orbit of a j-dimensional isotropic subspace X of V is determined by the numbers r = dimK (X ∩ Vℓ+ ), and s = dimK (X ∩ (Vℓ+ )⊥ ), which satisfy, by (4.6), (4.7), 0≤r≤s≤j≤m ˜ k, j ≤ℓ+s−r ≤m ˜ k.

(4.12)

Conversely, each pair (r, s), satisfying (4.12), determines a Qℓ -orbit in Yj , namely the orbit of Xr,s = SpanK {e1 , ..., er , eℓ+1 , ..., eℓ+s−r , e−ℓ , e−(ℓ−1) , ..., e−(ℓ−t+1) }.

(4.13)

Recall that t = j − s. If r = 0, then {e1 , ..., er } does not appear in (4.13). Similarly, if s − r = 0, then {eℓ+1 , ..., eℓ+s−r } does not appear in (4.13), and if t = 0, then {e−ℓ , e−(ℓ−1) , ..., e−(ℓ−t+1) } does not appear in (4.13). Note that we cannot have 0 = r = s − r = t, since then 0 = t = j − s = j, while j ≥ 1. Since, by (4.12), r + t ≤ ℓ, it follows that Xr,s is a j-dimensional isotropic subspace of Vk . We clearly have Xr,s ∩ Vℓ+ = SpanK {e1 , ..., er }, Xr,s ∩ (Vℓ+ )⊥ = SpanK {e1 , ..., er , eℓ+1 , ..., eℓ+s−r },

so that dimK (Xr,s ∩ Vℓ+ ) = r, and dimK (Xr,s ∩ (Vℓ+ )⊥ ) = s. The spaces Xr,s form a set of representatives for the Qℓ -orbits in Yj . We may choose representatives + ǫr,s ∈ h(Vk ), such that ǫ−1 r,s Vj = Xr,s , as follows. Consider the matrix, ar,s defined by   Ir   0 0 Iℓ−r−t 0 0 0 0     0 0 0 0 0 It 0    I  0 0 0 0 0   s−r 0   −1 ar,s =  (4.14) 0 0 0 Im′ −2(ℓ+s−r) 0 0 0 .     0 0 0 0 0 0 Is−r     0 δI 0 0 0 0 0 t     0 0 0 0 Iℓ−r−t 0 0 Ir

Then ǫr,s is defined by

ǫr,s = ωbt ar,s ,

(4.15)

where ωb = Im′ , when b is symplectic, or δ-Hermitian; when h(Vk ) is odd orthogo0 nal, ωb = −Im′ ; when h(Vk ) is even orthogonal, take ωb = diag(Im ˜ k , ωb , Im ˜ k ), where

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ωb0 has determinant −1 and is an element of the orthogonal group of the anisotropic kernel, V0 = V0 (k), of b. For example, if m ˜ k = m− 1 and we take a basis of V0 (k), as in (3.2), then we can take ωb0 = diag(1, −1). We really should add to the notation of these representatives the indices ℓ, j and b. We will avoid doing this, in order to lighten our notation. When we mention ǫr,s , ℓ, j and b will always be clear. 2. The case where h(Vk ) is even orthogonal and split Assume that h(Vk ) is even orthogonal and split (i.e. m ˜ k = m). We use the same notations. 2a. The case ℓ + s − r < m Assume that ℓ + s − r < m. Then the considerations of the previous case are valid, and Xr,s from (4.13) is a representative in Oj , and ǫr,s from (4.15) represents the corresponding Qj − Qℓ -double coset. Here, we may take   01 ωb = diag(Im−1 , ωb0 , Im−1 ) = ωb0 = . 10 Next, by Proposition 4.1, if ℓ + s − r = m, and ℓ = m, or j = m, then we are still in the first case of the proposition, i.e. the Qℓ -orbit is still determined solely by (r, s). 2b. The case j = m Assume that j = m. Then, by (4.12), ℓ + s − r = m, and t = ℓ − r. If t is even, then we choose the representative (4.13), as before. Note that since, in this case, we are considering elements X ∈ Om , such that dimk (X ∩ Vm+ ) = r, then, if, also, ℓ = m, then we must have that m − r is even. Thus, if ℓ − r is odd, then ℓ < m, and, in this case, we choose the following representative from Oj Spank {e1 , ..., er , eℓ+1 , ..., em−1 , e−m , e−ℓ , ..., e−(r+1) } = ωb Xr,s . A representative, ǫr,s , of the corresponding double coset is given by ar,s ωb , where ar,s is given by (4.14) (note the order). 2c. The case j < m and ℓ = m Assume that j < m and ℓ = m. Then s = r and t = j − r. Here, a representative in Oj , is the space Xr,s = Xr,r defined in (4.13), and a representative, ǫr,s , of the corresponding Qj − Qm -double coset is given by (4.15). 2d. The case j, ℓ < m and ℓ + s − r = m Finally, assume that j, ℓ < m and ℓ + s − r = m. Let Xr,s be the space defined

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in (4.13). Then it corresponds to (r, s), as before, and Vℓ+ + Xr,s ∩ (Vℓ+ )⊥ = Vm+ . Let ǫr,s be the matrix defined in (4.15). Then it is a representative of the corresponding Qj − Qℓ -double coset. Note, that, since j < m, ωb Vj+ = Vj+ , and hence (ωbt ar,s )−1 Vj+ = Xr,s . A representative for the second case in Proposition 4.1(2) ˜ r,s = ωb Xr,s . This is a j-dimensional isotropic subspace of can be chosen to be X Vk , whose intersections with Vℓ+ and with (Vℓ+ )⊥ are of dimensions r and s, respec˜ r,s ∩ (V + )⊥ =+ Vm . A representative of the corresponding tively, and, also, Vℓ+ + X ℓ Qj − Qℓ -double coset is ǫ˜r,s = ωbt+1 ar,s ωb = ωb ǫr,s ωb . Indeed, we have + ˜ (˜ ǫr,s )−1 Vj+ = ωb a−1 r,s Vj = ωb Xr,s = Xr,s .

Here, again, we used the same notation ǫr,s , as for the other groups, and we didn’t add b, or ℓ, j to the notation. Since the group we are considering will always be clear, no confusion will arise. 3. The case K = k ⊕ k Here, we write the representatives for the more general set up of Proposition 4.2. The Pℓ˜-orbits in Sj are determined by the pairs (r, s), which satisfy (4.11)), and, of course, are such that 0 ≤ r ≤ ℓ1 , r ≤ s ≤ j < N . Conversely, such a pair determines a Pℓ˜-orbit. Here is a representative of this orbit. It is the subspace Xr,s ⊂ k N , where (we use the notations from Sec. 4.1(2)) Xr,s = Spank {ǫ1 , ..., ǫr , ǫℓ1 +1 , ..., ǫℓ1 +s−r , ǫℓ1 +ℓ2 +1 , ..., ǫℓ1 +ℓ2 +j−s }.

(4.16)

Note that dimk Xr,s = j, and that the intersections of this subspace with X (ℓ1 ) , and with X (ℓ1 +ℓ2 ) have the dimensions r, s, respectively. Here is a Weyl element, in GLN (k), w = wr,s , such that w−1 X (j) = Xr,s .   Ir 0 0 0 0 0 0 0 0 Iℓ1 −r 0 0       0 Is−r 0 0 0 0  −1 .  w = (4.17) 0 0 Iℓ2 −(s−r) 0   0 0    0 0 Ij−s 0 0 0  0

4.3

0

0

0

0

Iℓ3 −j+s

Stabilizers

We list here the various stabilizers of the representatives written in the previous section. Put (w)

Qℓ,j = Qℓ ∩ w−1 Qj w.

(4.18)

1. The case where K is a field and h(V ) is not even orthogonal and split

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For the representatives, ǫr,s , from Sec. 4.2(1), (ǫ

)

Qℓ,jr,s = {g ∈ h(V )|

gVℓ+ = Vℓ+

gXr,s = Xr,s },

and (ǫ

)

where Xr,s is defined in (4.13). The elements of Qℓ,jr,s have the form   r a x1 x2 y1 y2 y3 z1 z2 z3 ′ 0 b x3 0 y4 y5 0 z4 z ′  ℓ =ℓ−r−t 2  0 0 c 0 0 y 0 0 z ′  t 6  1  s−r d u v y6′ y5′ y3′      (r,s) gℓ,j =  0 e u′ 0 y4′ y2′  m”=m′ −2(ℓ+s−r)   ∗ ′  s−r 0 0 d 0 0 y1    ∗ ′ ′  t c x3 x2    ℓ′ =ℓ−r−t 0 b∗ x′1  0 0 a∗ r

(4.19)

We have



(r,s)

ǫr,s gℓ,j ǫ−1 r,s

a y1 δz1 0 d δy ′ 6   0 0 c∗     =      

x1 0 0 b 0 0

y2 u 0 y4 e 0

z2 y5′ δx′3 z4 y4′ b∗

x2 0 0 x3 0 0 c 0 0

y3 v 0 y5 u′ 0 y6 d∗ 0

ωbt z3 y3′   δx′2   z2′    y2′  ′  x1   z1′   y1′  a∗

r s−r t ℓ′ =ℓ−r−t m”=m′ −2(ℓ+s−r)

(4.20)

ℓ′ =ℓ−r−t t s−r r

Of course, there shouldn’t be any confusion between the block matrix ”b” and b in the subscript of ωb , the form, which defines the group h(Vk ). 2. The case where h(V ) is even orthogonal and split Here, in the cases, where ℓ + s − r < m, or where j < m, and ℓ = m, we get (ǫ ) exactly the same form of the stabilizer Qℓ,jr,s , as in (4.19), and the same conjugation formula, as in (4.20). Consider the case where j = m. Then ℓ + s − r = m, and t = ℓ − r. We saw, in this case, that ωbt Xr,s is a corresponding representative in Om . As we already noted, in Sec. 4.2(2b), if s = r, then t is even. Thus, ωbt Vℓ+ = Vℓ+ , and hence (r,s) the stabilizer, in Qℓ , consists of elements g, such that ωbt gωbt has the form gℓ,j in (4.19) (with the columns and rows indexed by ℓ′ and m” omitted). In this case, (r,s) t ǫr,s = ar,s ωbt , and so, to get the analog of (4.20), let g = (gℓ,j )ωb be an element of the stabilizer of ωbt Xr,s , in Qℓ . Then (r,s)

t

(r,s)

t ωb t −1 −1 ǫr,s gǫ−1 r,s = ar,s ωb (gℓ,j ) ωb ar,s = ar,s gℓ,j ar,s .

(4.21)

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t

If we write g = (gℓ,j )ωb in the form  a x2 y1 0 c 0   d    0  

then we get

y3 y6 v d∗

z1 0 y6′ 0 c∗ 0

ω t z3 b z1′    y3′   y1′   x′2  a∗

r t s−r

(4.22)

s−r t r

 r x2 y3 z3 ′ 0 v y3  s−r  0 0 x′2  t ǫr,s gǫ−1 (4.23) r,s ′ c y6 z1  t  0 d∗ y1′  s−r 0 0 a∗ r Assume now that ℓ, j < m and ℓ + s − r = m. Then the stabilizer of Xr,s in Qℓ is as in (4.19), and conjugation of its elements by ǫr,s is given by (4.20). Finally, we have the representatives ωb Xr,s . As in the last case, the stabilizer, in Qℓ , consists (r,s) (r,s) of all elements of the form g = (gℓ,j )ωb , as gℓ,j varies over the elements (4.19). ωb The corresponding representative is (ǫr,s ) . Conjugation of g by this element gives i h 

a y1 z1 0 d y ′  6   0 0 c∗ =   

(r,s)

ǫr.s gℓ,j ǫ−1 r,s in (4.20)).

ωb

, i.e. the matrix in (4.20), conjugated by ωbt+1 (and not by ωbt , as

3. The case K = k ⊕ k Assume that K = k ⊕ k. Consider the representative Xr,s from (4.16), and the representative, given in (4.17), w = wr,s of the corresponding Pj,N −j − Pℓ˜ - dou(w) = Pℓ˜ ∩ w−1 Pj,N −j w, in Pℓ˜, of Xr,s , consists of the ble coset. The stabilizer, Pℓ,j ˜ elements of the form   r a1 x y 1 y 2 z 1 z 2  0 a 0 y 0 z  ℓ −r 1 2 3 3    b1 u v1 v2  s−r  g= (4.24)   0 b2 0 v3  ℓ2 −(s−r)    c1 t  j−s 0 c2 ℓ3 −j+s and conjugation of g in (4.24), by w is carried out, as follows.  r a1 y 1 z 1 x y 2 z 2  0 b v 0 u v  s−r 1 1 2      0 0 c1 0 0 t  j−s −1 wgw =  (4.25)   a2 y3 z3  ℓ1 −r    0 b2 v3  ℓ2 −(s−r) 0 0 c2 ℓ3 −j+s

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The set Q\h(Wm,ℓ )k /Lℓ,w0

1. The case where K is a field and w0 is anisotropic Assume that K is a field and that w0 is an anisotropic vector in Wm,ℓ = Wm,ℓ (k), and consider its stabilizer Lℓ,w0 in h(Wm,ℓ ). (We assume that δ = 1.) Denote W = Wm,ℓ (k) and Lw0 = Lℓ,w0 (k). The setup is now general, namely we have a vector space W over k, equipped with a non-degenerate bilinear form b, which is symmetric, when K = k, or Hermitian, when [K : k] = 2; Lw0 is the stabilizer in h(W ) of the anisotropic vector w0 . Proposition 4.3. Let X ⊂ W be a non-trivial (totally) isotropic subspace, and let Q be the parabolic subgroup of h(W ), which preserves X. Then Q\h(W )/Lw0 contains at most three elements. Proof. Denote i = dimK X. Let us realize Q\h(W ) as the variety O of idimensional isotropic subspaces of W . If h(W ) is even orthogonal and split, and X is maximal, then we have to add that these isotropic subspaces are in the same orientation class of X. We consider the orbits of the action of Lw0 on O. Let Y1 , Y2 ∈ O. If Y1 , Y2 lie in the same Lw0 -orbit, then w0 ∈ Y1⊥ ⇔ w0 ∈ Y2⊥ .

This means that when we view w0 as a linear functional on the subspaces Y1 , Y2 (via the form b), then it is trivial on Y1 , if and only if it is trivial on Y2 . Conversely, assume that Y1 , Y2 ∈ O are such that w0 is either trivial on both Y1 , Y2 or non-trivial on both Y1 , Y2 . Suppose, first, that w0 ∈ Y1⊥ ∩Y2⊥ . Then, by Witt’s theorem, we can find g, in the full isometry group of W , such that gY1 = Y2 , and gw0 = w0 . Next, suppose that w0 ∈ / Y1⊥ and w0 ∈ / Y2⊥ . Then, since w0 defines a non-trivial linear (2) (2) (1) (1) functional on Y1 and Y2 , we can find bases {y1 , ..., yi } of Y1 , and {y1 , ..., yi } (1) (2) (1) of Y2 , such that b(yj , w0 ) = b(yj , w0 ) = 0, for 1 ≤ j ≤ i − 1, and b(yi , w0 ) = (2)

b(yi , w0 ) = 1. By Witt’s theorem, we can find g in the full isometry group of W , (2) (1) such that g · yj = yj and g · w0 = w0 , for 1 ≤ j ≤ i. In particular, gY1 = Y2 , and except in case h(W ) is orthogonal, we also have that g ∈ Lw0 ; the same holds when h(W ) is orthogonal, provided we know that det(g) = 1. Thus, if h(W ) is unitary, then there are at most two Lw0 -orbits in O, since an Lw0 -orbit is characterized by the triviality or non-triviality of w0 , as a linear functional on its elements. Let us denote by W itt(Z) the Witt index of a non-degenerate subspace Z ⊂ W . Assume now, that X is not maximal isotropic, that is dimK X < W itt(W ). We will show that Q\h(W )/Lw0 has exactly two elements, in all cases. Let us, first, show that, under this assumption, we must have W itt(w0⊥ ) ≥ dimK X.

(4.26)

If w0 ∈ X ⊥ , then (4.26) is clear. If w0 ∈ / X ⊥ , then, as before, we can decompose ⊥ X = X0 ⊕ Ku0 , such that w0 ∈ X0 and b(w0 , u0 ) = 1. Span{w0 , u0 } is a non-

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degenerate subspace of Witt index 1. Decompose

X0′

Span{w0 , u0 }⊥ = X0 ⊕ S0 ⊕ X0′ ,

(4.27)

where is an i − 1 dimensional isotropic subspace, dual to X0 , with respect to b, and S0 is a non-degenerate subspace, orthogonal to X0 + X0′ . We conclude that W itt(W ) = dimK X + W itt(S0 ), and, hence, by our assumption, W itt(S0 ) ≥ 1. From (4.27), we get that W itt(Span{w0 , u0 }⊥ ) ≥ dimK X, and, in particular, W itt(w0⊥ ≥ dimK X. Moreover, in this case, let s0 ∈ S0 be a nonzero isotropic vector. Then Y = X0 ⊕ Ks0 is an i dimensional isotropic subspace, such that w0 ∈ Y ⊥ . Thus, in this case, when h(W ) is unitary, there are exactly two orbits. In the first case (when w0 ∈ X ⊥ ), we can decompose ′

w0⊥ = X ⊕ Z0 ⊕ X ′ ,

(4.28)

where X is an isotropic subspace of dimension i, which is dual to X, with respect to b, and Z0 is a non-degenerate subspace, which is orthogonal to X + X ′ . Let us choose vectors x0 ∈ X and x′0 ∈ X ′ , such that trK/k b(x0 , x′0 ) = −b(w0 , w0 ). Then ˜ be an v0 = w0 + x0 + x′0 is isotropic, and satisfies b(v0 , w0 ) = b(w0 , w0 ) 6= 0. Let X ⊥ ˜ i-dimensional isotropic subspace of W , which contains v0 . Then w0 ∈ / X . Thus, when h(W ) is unitary, we have two orbits in the first case, as well. Assume that h(W ) is orthogonal. Then, by (4.26), there is an i-dimensional isotropic subspace U , such that w0 ∈ U ⊥ . Consider the decomposition w0⊥ = U ⊕ T0 ⊕ U ′ , analogous to (4.28). Since T0 is non-trivial, we can find h in the full orthogonal group of W , which acts as the identity on U + U ′ + kw0 , and on T0 , it acts by determinant -1. Let B ⊂ W be an i-dimensional isotropic subspace, such that w0 ∈ B ⊥ . We saw that there is g in the full orthogonal group of W , such that B = gU , and gw0 = w0 . If det(g) = −1, then ghw0 = w0 , ghU = B, and det(gh) = 1, and hence B is in the Lw0 -orbit of U . Let now U ⊂ W be an i-dimensional isotropic subspace, which is not orthogonal to w0 (we explained that such a space exists.) Then we can decompose U = U0 ⊕ ku0 , and obtain (4.27) above, with U replacing X. Since S0 is nontrivial (it even has a positive Witt index), there is an element h in the full orthogonal group of W , such that h is the identity on U0 + U0′ + Spank {w0 , u0 }, and, on S0 , it has determinant -1. Note that hU = U, h · w0 = w0 , and det(h) = −1. Now, we argue as before and conclude that all such subspaces U form one orbit under the action of Lw0 . This proves that there are exactly two orbits when X is not maximal isotropic. Assume that X is maximal isotropic, that is W itt(W ) = dimK X. As in the beginning of the proof, we either have W itt(w0⊥ ) = dimK X, or W itt(w0⊥ ) = dimK X − 1. Assume that W itt(w0⊥ ) = dimK X. Let U be a maximal isotropic subspace of W , such that w0 ∈ U ⊥ . Since w0⊥ is non-degenerate, we have a decomposition w0⊥ = U ⊕ Z0 ⊕ U ′ , analogous to (4.28). Note that Z0 may be trivial. Now, exactly as before (right after (4.28)), we can produce a maximal isotropic subspace ˜ , such that w0 ∈ ˜ ⊥ . Hence, in this case, if h(W ) is unitary, then we have U / U

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exactly two orbits. Assume that h(W ) is orthogonal. If Z0 is nontrivial, then there is an element h in the full orthogonal group of W , which is the identity on U + U ′ + kw0 , and, on Z0 , it has determinant -1. Then, as before, any maximal isotropic subspace of W , which is orthogonal to w0 , lies in the Lw0 -orbit of U . Let B be a maximal isotropic subspace of W, such that w0 ∈ / B ⊥ . Then we have the analogue of (4.27) Span{w0 , u0 }⊥ = B0 ⊕ S0 ⊕ B0′ , with B = B0 ⊕ ku0 , as before. Since we assume that Z0 is nontrivial, dimk W ≥ 2dimk X + 2, and we conclude that dimk S0 ≥ 2, and, in particular, S0 is non-trivial. Now, we argue as before, and ˜ , and, again, we got exactly two orbits. conclude that B lies in the Lw0 -orbit of U Assume, now, that Z0 is trivial, that is w0⊥ = U ⊕ U ′ . Thus h(W ) is split, odd orthogonal; dimk W = 2dimk X + 1. If B is a maximal isotropic subspace of W , which is orthogonal to w0 , then we have an analogous decomposition w0⊥ = B ⊕ B ′ , and now B is in the Lw0 -orbit of U means that B is in the same orientation class of U , when we identify Lw0 with the split special even orthogonal group of U ⊕ U ′ . Thus, there are two Lw0 -orbits of maximal isotropic subspaces which are orthogonal to w0 . The maximal isotropic subspaces which are not orthogonal to w0 form one Lw0 -orbit. The proof is exactly as before (we get that dimS0 = 1). Thus, in this case, we got exactly three orbits. Note that here dW = (−1)dimX b(w0 , w0 )(mod(k ∗ )2 ),

(4.29)

where dW is the discriminant of W . We have a similar condition, when h(W ) is a quasi-split, odd unitary group, for w0 to be such that W itt(w0⊥ ) = dimK X. Here, we need to replace (k ∗ )2 by the group of norms NK/k (K ∗ ). Finally, assume that X is maximal isotropic, and W itt(w0⊥ ) = dimK X − 1. This implies that w0 ∈ / U ⊥ , for any maximal isotropic subspace U . Thus, if h(W ) is unitary, all these subspaces form one Lw0 -orbit. The same is true when h(W ) is orthogonal. Indeed, if dimk W > 2dimk X, then we argue as before. If dimk W = 2dimk X, then h(W ) is a split even orthogonal group. Recall that now O is the set of all maximal isotropic subspaces which lie in the same orientation class of X, and so given U ∈ O, we already know that there is g in the full orthogonal group of W , which fixes w0 , and gX = U . Since X and U lie in the same orientation class, we must have det g = 1, and, so, U lies in the Lw0 -orbit of X. This completes the proof of Proposition 4.3.  The last proof gives, in detail, the number of orbits in each case. We summarize this now, still keeping the notation of the last proposition. Proposition 4.4. (1) Assume that dimK X < W itt(W ). Then the set Q\h(W )/Lw0 consists of two elements. (2) Assume that X is a maximal isotropic subspace of W , and that W itt(w0⊥ ) = dimK X. (a) If h(W ) is unitary, then Q\h(W )/Lw0 consists of two elements.

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(b) If h(W ) is orthogonal and dimk W ≥ 2dimk X + 2, then Q\h(W )/Lw0 consists of two elements. (c) If h(W ) is orthogonal and dimk W = 2dimk X + 1, then Q\h(W )/Lw0 consists of three elements. (3) Assume that X is a maximal isotropic subspace of W , and that W itt(w0⊥ ) = dimK X − 1. Then Q\h(W )/Lw0 consists of one element. (4) Assume that X is a maximal isotropic subspace of W , and dimK W = 2dimK X. Then W itt(w0⊥ ) = dimK X −1, and, in particular, Q\h(W )/Lw0 consists of one element. Let us write representatives for Q\h(W )/Lw0 . Write W = W + + V0 + W − , where W ± are maximal isotropic subspaces, say of dimension d, over K, which are in duality, with respect to b, and V0 = (W + + W − )⊥ . Choose a basis {x1 , ..., xd } of W + , and a dual basis {x−d , ..., x−1 } of W − . Choose an orthogonal basis B0 of V0 . We will write the elements of h(W ) as matrices with respect to the basis {x1 , ..., xd } ∪ B0 ∪ {x−d , ..., x−1 }.

Let w0 = xα = xd + (−1)dimK V0 +1 α2 x−d , where α ∈ k ∗ . Note that b(w0 , w0 ) = (−1)dimK V0 +1 α. Let X = Wi+ = SpanK {x1 , ..., xi }, i ≤ d. (1) Assume that i < d. Then, by Proposition 4.4, Q\h(W )/Lw0 consists of two elements. One representative is IW , and the other can be taken as follows   Ii  Id−i   .  (4.30) I V0     Id−i Ii

(2) Assume that i = d and W itt(w0⊥ ) = d − 1. Here, Q\h(W )/Lw0 consists of one element element. We choose IW as a representative. For example, this happens when V0 = 0, that is h(W ) is quasi-split even unitary, or split even orthogonal. + − Here, x⊥ α = Wd−1 + Kx−α + Wd−1 . (3) Assume that i = d and W itt(w0⊥ ) = d. ′

(i) Assume that [K : k] = 2 and (−1)m +1 α is represented by b on V0 , or ′ that K = k, (−1)m +1 α is represented by b on V0 , and dimk W ≥ 2d + 2. Then there is vα ∈ V0 , such that x−α + vα is isotropic. We know that Q\h(W )/Lw0 consists of two elements. One representative is IW , and the other can be taken to be   Id−1   1     −vα IV0 (4.31) .    ′ α ′ m   (−1) 2 vα 1 Id−1

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For example, assume that [K : k] = 2 and dimK W = 2d + 1, that is h(W ) is quasi-split, odd unitary, and assume that α = NK/k β is a norm. Then the second representative can be taken can be taken to be   Id−1   1     (4.32) −β 1   α ¯   −2 β 1 Id−1

(ii) Assume that K = k, dimk W = 2d + 1, and α = β 2 is a square in k ∗ . Then Q\h(W )/Lw0 consists of three elements. One representative is IW , and the two others can be taken as follows     Id−1 1  ; (4.33) γ± =  ±β 1  γ± Id−1 − α2 ∓β 1

2. The case where K = k ⊕ k (and w0 - anisotropic)

Assume that K = k ⊕ k. In light of the end of Sec. 3.7, the set of double cosets, analogous to the previous case is Pi,D−i \GLD (k)/L, where 0 < i < D, and    1 L= ∈ GLD (k) . g Proposition 4.5. The set Pi,D−i \GLD (k)/L has three elements. Proof. Denote by C the unipotent radical of P1,D−1 . Note that P1,D−1 = k ∗ L ⋉ C. Start with Pi,D−i \GLD /P1,D−1 , which has two elements: Here, hi =



ID−i

GLD (k) = Pi,D−i P1,D−1 ∪ Pi,D−i hi P1,D−1 .  Ii . It is clear that Pi,D−i P1,D−1 = Pi,D−i L. We have

Pi,D−i hi P1,D−1 = ∪ζ∈kD−1 Pi,D−i hi

 1

ζ ID−1





1

0

L = ∪z∈ki Pi,D−i hi  ID−i−1

For g ∈ GLi (k),       −1  1 0 z 1 1 0 zg g hi  ID−i−1 0   ID−i−1  = hi  ID−i−1 0  . ID−i g Ii Ii If z 6= 0, we can choose gz ∈ GLi (k), such that zgz = (0, ..., 0, 1). Thus, Pi,D−i hi P1,D−1 = Pi,D−i hi L∪ Pi,D−i hi u0 L,   1 0 1 where u0 =  ID−2 0. All in all, we got 1 GLD (k) = Pi,D−i L ∪ Pi,D−i hi L∪ Pi,D−i hi u0 L.

 z 0  L. Ii



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Remark: In the cases where w0 is isotropic, the group Lℓ,w0 is almost a parabolic subgroup of h(Wm,ℓ ) (up to a diagonal GL1 -subgroup). In this case the description of the set Q\h(Wm,ℓ )/Lℓ,w0 is a special case of Proposition 4.1, or Proposition 4.2.

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Jacquet modules corresponding to Gelfand - Graev characters of parabolically induced representations We study Jacquet modules corresponding to Gelfand-Graev characters, when apH(k ) plied to parabolic induction, that is J(ψv )ℓ,w0 (IndQj (Fv v ) τ ⊗ σ), in the notation of Sec. 3.7. We will obtain a Leibniz rule for these Jacquet modules, expressing them in terms of Bernstein-Zelevinsky derivatives of τ and similar Jacquet modules of σ, of Gelfand-Graev type. The decomposition of these Jacquet modules were obtained in [Ginzburg, Piatetski-Shapiro and Rallis (1997)] for orthogonal groups.

5.1

The case where K is a field

We keep notations and assumptions as in Sec. 3.7, only that we will drop the subscript v. Thus, for the finite place v, we re-denote the character ψv by ψ, the stabilizer group Lℓ,w0 ,v by Lℓ,w0 , and so on. We assume that K is a field, and that δ = 1 (in particular, (V, b) is not symplectic). When K is a quadratic extension √ of k, we assume that K = k[ ρ]. We will also abbreviate and drop the field k from the notation, when convenient. For example, V = Vk , Nℓ = Nℓ (k), but we will write GLn (k), rather than GLn . We will also write m ˜ =m ˜ k (in this section, the Witt index m ˜ is arbitrary.) Let Qj be the standard parabolic subgroup of H = h(V ), which preserves the isotropic subspace Vj+ . It has a Levi decomposition Qj = Dj ⋉ Uj and Dj ∼ = GLj (K) × h(Wm,j ). Consider a parabolically induced representation π = IndH Qj τ ⊗ σ, where τ and σ are smooth representations of GLj (K) = GL(Vj+ ) and h(Wm,j ) respectively. We want to analyze the Jacquet module Jψℓ,w0 (π), introduced in Sec. 3.7 (ℓ ≤ m, ˜ w0 ∈ Wm,ℓ is anisotropic.) Since Jψℓ,w0 (π) = Jψℓ,w0 (ResQℓ π), we first look at the restriction of π to Qℓ . By Bruhat theory [Silberger (1979)], this representation has a finite Qℓ - filtration, whose subquotients are parametrized by the elements of Qj \H/Qℓ , which we analyzed explicitly in Sec. 4.1. We will choose representatives w as in Sec. 4.2(1,2). If ℓ = 0, then Qℓ = h(V ), and we can skip the following and consider just the double cosets 81

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Qj \h(V )/L0,w0 , described in Sec. 4.4 (L0,w0 is the fixator of w0 inside h(V ).) The subquotient, corresponding to a representative w, is (w)

ρw = indQℓ(w) πℓ,j (τ, σ).

(5.1)

Qℓ,j

(w)

Here, ind refers to compact induction; for g ∈ Qℓ,j 1

(w)

πℓ,j (τ, σ)(g) = δ(g) 2 (τ ⊗ σ)(wgw−1 ), where δ(g) = δQℓ (g)δ −1(w) (g)δQj (wgw−1 ), Qℓ,j

(w)

and δQℓ , δQj , δQ(w) are the modular functions of the groups Qℓ , Qj , Qℓ,j , respecℓ,j

tively. Our goal now is to analyze Jψℓ,w0 (ρw ), for each w, as an Lℓ,w0 -module. Since Jψℓ,w0 (ρw ) = Jψℓ,w0 (ResRℓ,w0 (ρw )), our next step is to consider ResRℓ,w0 (ρw ). (w)

For this, again, by Bruhat theory, we need to look at the space Qℓ,j \Qℓ /Rℓ,w0 . (w)

Referring to the structure of Qℓ,j in Sec. 4.3(1)(2), it is clear, that we can pick representatives for the last set, of the form   ǫ η =  γ , (5.2) ǫ∗

where ǫ is a Weyl element in GLℓ (K), and γ is a representative for Q′w \h(Wm,ℓ )/Lℓ,w0 , where Q′w is the maximal parabolic subgroup of h(Wm,ℓ ), as follows. If w = ǫr,s , in (4.15), or as in Sec. 4.2(2a), then Q′w is the parabolic subgroup of h(Wm,ℓ ), which preserves the standard s−r dimensional isotropic subspace + Vℓ,s−r of Wm,ℓ . We denote ± Vℓ,s−r = SpanK {e±(ℓ+1) , ..., e±(ℓ+s−r) }.

(5.3)

See (4.19), Sec. 4.3(2). The remaining cases are those, where h(V ) is split, even orthogonal, and ℓ + s − r = m = j, or ℓ, j < m and ℓ + s − r = m. If j = m, and + w = ǫr,s , then Q′w is the parabolic subgroup, which preserves ωbt Vℓ,m−ℓ . If ℓ, j < m (and ℓ + s − r = m) then, when w = ǫr,s , Q′w is the parabolic subgroup, which + preserves Vℓ,m−ℓ , and when w = ωb ǫr,s ωb , Q′w is the parabolic subgroup, which + + preserves Vℓ,m−ℓ = ωb Vℓ,m−ℓ . Proposition 5.1. Let w be any representative, as above, corresponding to (r, s). If r > 0, then Jψℓ,w0 (ρw ) = 0.

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Proof. The representation ResRℓ,w0 ρw has a finite Rℓ,w0 -filtration, whose subquotients are Rℓ,w0

(w,η) (τ, σ), (w) π Rℓ,w0 ∩η −1 Qℓ,j η ℓ,j

ρw,η = ind

(w,η)

where η varies over the representatives (5.2), and πℓ,j −1

(w) Qℓ,j η

(w) πℓ,j (τ, σ)(ηhη −1 ),

η to the form of η, it is clear that

up to appropriate modular functions. Looking at (w)

(w)

(w)

(τ, σ) takes h ∈ Rℓ,w0 ∩ (w)

Rℓ,w0 ∩ η −1 Qℓ,j η = (Lℓ,w0 ∩ η −1 Mℓ,j η)(Nℓ ∩ η −1 Qℓ,j η), (w)

(w)

(w)

where Qℓ,j = Mℓ,j Uℓ,j , and Mℓ,j = Dℓ ∩ w−1 Qj w, (w)

(w)

(w)

Uℓ,j = Uℓ ∩ w−1 Qj w. Note

that Lℓ,w0 ∩ η −1 Mℓ,j η normalizes Nℓ ∩ η −1 Qℓ,j η. A function f in the space of ρw,η (w)

is supported in a set of the form (Lℓ,w0 ∩ η −1 Mℓ,j η)Nℓ Ω2 , where Ω2 is a compact subset of Lℓ,w0 , and f is determined by its restriction to Nℓ Ω2 . The restriction of (w) f to the last set is supported in a set of the form (Nℓ ∩ η −1 Qℓ,j η)Ω1 Ω2 , where Ω1 is a compact subset of Nℓ , and this restriction of f is determined by its values on Ω1 Ω2 . Consider Z −1 (u)ρ(u)f du, ψℓ,w 0

(5.4)

(i) Nℓ

(i)

where ρ(u) denotes right translation by u, and {Nℓ }∞ i=1 is an ascending sequence of compact open subgroups in Nℓ , whose union is all of Nℓ . We have to prove that (5.4) vanishes, for i large enough (depending on f ). By the comments above, it is enough to show that, for i large enough, Z −1 (u)f (nωu)du = 0, ∀n ∈ Nℓ , ω ∈ Ω2 . (5.5) ψℓ,w 0 (i)

Nℓ

(i)

(i)

There is a compact open subgroup L0 of Lℓ,w0 , such that ωNℓ ω −1 = Nℓ , for all ω ∈ L0 and i large. We may replace Ω2 by Ω2 L0 , since Ω2 ⊂ Ω2 L0 . Thus, we can write Ω2 as a finite union of cosets ωj L0 , 1 ≤ j ≤ N . The number of these cosets depends on f only. We may change variable, u 7→ a−1 ua, and since Lℓ,w0 fixes ψℓ,w0 , we get that it is enough to prove that for i′ sufficiently large, and 1 ≤ j ≤ N Z −1 (u)f (nωj ua)du = 0, ∀n ∈ Nℓ , a ∈ L0 . (5.6) ψℓ,w 0 (i′ )

Nℓ

We may assume that, given i′ large enough, there is i also sufficiently large, such (i′ ) −1 (i) (i′ ) (i) that ωj Nℓ ω0 ⊃ Nℓ . Writing ωj Nℓ ω0−1 as a union (finite) of cosets n′ Nℓ , with representatives n′ ∈ Nℓ , it is enough to show that for i sufficiently large and 1 ≤ j ≤ N, Z −1 (u)f (nuωj a)du = 0, ∀n ∈ Nℓ , a ∈ L0 . ψℓ,w 0 (i)

Nℓ

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Thus, it is enough to show that Z −1 (u)f (nuω)du = 0, ψℓ,w 0

∀n ∈ Nℓ , ω ∈ Ω2 .

(5.7)

(i)

Nℓ

(w)

It is enough to take, in (5.7), nu ∈ (Nℓ ∩η −1 Qℓ,j η)Ω1 . We may take i large enough, (i)

(i)

(i)

so that Ω1 Nℓ = Nℓ . Thus, it is enough to prove (5.7), for n ∈ Nℓ , and then, changing variable in u, it is enough to prove (5.7), for n = 1. Thus, we need to show that, for i large enough, Z −1 (u)f (uω)du = 0, ∀ω ∈ Ω2 . (5.8) ψℓ,w 0 (i)

Nℓ

(i)

(w)

We will show (5.8) by finding a subgroup J ⊂ Nℓ ∩ η −1 Qℓ,j η, such that, for i, large enough, we have (w,η) ψℓ,w0 6= 1, and πℓ,j (τ, σ) = 1. (5.9) J

J

(w)

Consider the various stabilizers Qℓ,j described in (4.19) and in Sec. 4.3(2). Let S be a simple root subgroup inside Nℓ ∩ Dℓ . Denote by S1 the projection of S to GLℓ (K). Assume, further, that S1 is a simple root subgroup in GLℓ (K). Note the form (5.2) of η. If (see (4.19))      Ir x1 x2 ǫS1 ǫ−1 ⊂ Zr,ℓ−r−t,t =  Iℓ−r−t x3  ∈ GLℓ (K) ,   It (w)

then it is clear that S ⊂ Nℓ ∩ η −1 Qℓ,j η and S satisfies the requirements of (5.9). (i) ∩ Nℓ ,

Then we may take J = S for i sufficiently large. (This applies to split, even orthogonal groups as well.) Note that the Weyl element ǫ, which appears in η is in the Weyl group of GLℓ , modulo the Weyl group of the standard Levi subgroup, Mr,ℓ−r−t,t , which corresponds to the partition (r, ℓ − r − t, t) of ℓ. According to the   Ir following lemma, we may assume that ǫ =  Iℓ−r−t  . It Lemma 5.1. Let ǫ be a permutation matrix in GLℓ (K), modulo (from the left) the Weyl group of Mr,ℓ−r−t,t. Assume that, for each simple (standard) root subgroup U of GLℓ (K), ǫU ǫ−1 does not lie inside Zr,ℓ−r−t,t . Then, modulo Mr,ℓ−r−t,t,   Ir (5.10) ǫ =  Iℓ−r−t  . It

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Proof. Let us denote by 1 ≤ j1 , ..., jℓ ≤ ℓ the columns, such that ǫi,ji = 1, 1 ≤ i ≤ ℓ. Multiplying ǫ from the left by permutation matrices in Mr,ℓ−r−t,t , we may assume that j1 < ... < jr , jr+1 < ... < jℓ−t , jℓ−t+1 < ... < jℓ . Let 1 ≤ i1 , ..., iℓ ≤ ℓ be the rows, such that ǫij ,j = 1. Denote, for 1 ≤ i 6= j ≤ ℓ and x ∈ K, zi,j (x) = Iℓ + xei,j , where ei,j is the ℓ × ℓ matrix, which has 1 at the coordinate (i, j), and zero elsewhere. Put i′1 = ij1 +1 , i.e. ǫi′1 ,j1 +1 = 1. Then (ǫ−1 )j1 +1,i′1 = 1. We have ǫzj1 ,j1 +1 (x)ǫ−1 = z1,i′1 (x). By our assumption, we must have i′1 ≤ r, and hence, we must have i′1 = 2, and so j2 = j1 + 1. Put i′2 = ij2 +1 , i.e. ǫi′2 ,j2 +1 = 1. Then (ǫ−1 )j1 +2,i′2 = 1. We have ǫzj2 ,j2 +1 (x)ǫ−1 = z2,i′2 (x). As before, we conclude that i′2 ≤ r, and hence i′2 = 3, and j3 = j1 +2. We continue in this way, and get that j2 = j1 +1, j3 = j1 +2, ..., jr = j1 +r−1. If j1 +r−1 < ℓ, then we can repeat the argument. Put i′r = ijr +1 , then since ǫzjr ,jr +1 (x)ǫ−1 = zr,i′r (x), we conclude, as before, that i′r ≤ r, and this is impossible. Thus j1 + r − 1 = ℓ, and so j1 = ℓ − r + 1, j2 = ℓ − r + 2, ..., jr = ℓ. A similar argument shows that jr+1 = t + 1, jr+2 = t + 2,...,jℓ−t = ℓ − r. Finally, by our assumptions, it follows that jℓ−t+1 = 1, jℓ−t+2 = 2,...,jℓ = t. This proves the lemma.  We continue with the proof of the proposition. Assume that η is such that ǫ has the form (5.10). Consider the following subgroup S of Nℓ , which consists of elements of the form   Iℓ−r
t′   Ir 0 y2 y3 ωb γ z3           Is−r   ∗   (5.11) Im” .      Is−r         Ir Iℓ−r

Here, γ is the middle block of η; t′ = 0, except in the following cases, where t′ = 1: h(V ) is split, even orthogonal, and either j = m and t is odd, or, j, ℓ < m, ℓ+s−r = (w) b m, and w = ǫω and Sec. 4.3(2), S ⊂ η −1 Uℓ,j η, and by (4.20), and r,s . By (4.19), (w,η) Sec. 4.3(2), πℓ,j (τ, σ) = 1. Note that, by definition of ψℓ,w0 , dimK V − 2ℓ ≥ 1, S and since w0 is anisotropic, ψℓ,w0 cannot be identically trivial on the elements of (i) the form (5.11). Now, we can take J = S ∩ Nℓ , for i sufficiently large, and (5.9) is verified. Note that this argument breaks down when r = 0. This completes the proof of Proposition 5.1. 

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It remains to analyze Jψℓ,w0 (ρw ), for w which corresponds to (r, s), where r = 0, s≤j≤m ˜ and (see (4.12)) j ≤ ℓ + s ≤ m. ˜ Note that now, the representatives η have the form (5.2), with ǫ of the form (5.10). We will re-denote ρw,η = ρw,γ,t , where γ appears in the middle block of η. We start with the contribution of γ = Im′ −2ℓ . In this case, denote η = ηt . The inducing subgroup, in the induced (w) representation ρw,Im′ −2ℓ ,t , i.e. Rℓ,w0 ∩ηt−1 Qℓ,j ηt , consists of the elements (see (4.19) and Sec. 4.3(2)) 

n1 0 0 0  n2 0 y 4   d u   h= 0 e   0 0  

y6 y5 v u′ d∗

0 z4 y5′ y4′ 0 n∗2

′

ωbt′ 0 0  y6′    0  0  0 n∗1

t ′

ℓ =ℓ−t s m”=m′ −2(ℓ+s)

(5.12)

s ℓ′ =ℓ−t t

Here, n1 ∈ Zt (K), n2 ∈ Zℓ−t (K), t is as in the end of the last proof, and   du v ′ ′  e u′  ωbt (w0 ) = ωbt (w0 ). d∗

(5.13)

ωbt+t′ n2 y 4 z 4 ⊗σ e y4′  n∗2

(5.14)

We identify ωb as an element of h(Wm,ℓ ), written as a matrix group. The action of (w,η ) πℓ,j t (τ, σ) on h, of the form (5.12), satisfying (5.13) is 1

δR2 ℓ,w (h)δ 0

− 21

(w) Rℓ,w0 ∩η −1 Qℓ,j η

1

(h)δQ2 j (whw−1 )τ



 ′

d y6 0 n∗1



Proposition 5.2. In the notation above, assume that w corresponds to r = 0, − j < ℓ + s, and assume that w0 is not orthogonal to Vℓ,s (see (5.3)). Then Jψℓ,w0 (ρw,Im′ −2ℓ ,t ) = 0. ′

m +1 α ∗ In particular, if w0 = yα = em ˜ and j < m. ˜ ˜ + (−1) ˜ , α ∈ k , ℓ + s = m, 2 e −m Then

Jψℓ,α (ρw,Im′ −2ℓ ,t ) = 0. Proof. As in the proof of Proposition 5.1, it is enough to find a subgroup S ⊂ (w) Nℓ ∩ ηt−1 Qℓ,j ηt , such that the requirements of (5.9) are satisfied. Let S be the subgroup of Nℓ , which consists of the elements (5.12), such that all the diagonal

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blocks are identity blocks, and all blocks off the diagonal are zero, except the blocks y5 , y5′ . Note, that ℓ − t = ℓ − j + s > 0. Also, the assumption on w0 implies that s > 0. Thus, S is nontrivial. The assumption on w0 also implies that ψℓ,α 6= 1, S (w,η ) and by (5.14), we see that πℓ,j t (τ, σ) = 1. When we take w0 = yα , this implies S

that ℓ < m ˜ (yα does not lie in V0 = Wm,m ˜ we see that s > 0. ˜ ). Since ℓ + s = m, If t = ℓ, then j = ℓ + s = m, ˜ contrary to our assumption. Now we can apply the argument above. This proves the proposition. 

Let τ (t) denote the Bernstein-Zelevinski derivative ([Bernstein and Zelevinsky (1976)]) of τ , along the subgroup    Is y ′ Zt = ∈ GLj (K) z ∈ Zt (K) 0 z

and corresponding to the character   −1 ′ Is y ψt = ψK,1 (z1,2 + z2,3 + ... + zt−1,t ). 0 z

τ (t) is the representation of GLs (K), acting in the Jacquet module JZt′ ,ψt′ (Vτ ), via the embedding d 7→ diag(d, It ) of GLs (K) in GLj (K). Consider, for a ∈ K ∗ , the following character of Zt′ ,   Is y −1 ” (z1,2 + z2,3 + ... + zt−1,t + ays,1 ). (5.15) = ψK,1 ψt,a 0 z Denote the corresponding Jacquet module JZt′ ,ψt,a ” (Vτ ) by τ(t),a . This is a representation of the mirabolic subgroup Ps−1,1 of GLs (K). We have the following easy lemma. Lemma 5.2. For all a, a′ ∈ K ∗ , τ(t),a and τ(t),a′ are isomorphic, as representations of Ps−1,1 . Proof.

The map T : Vτ → Vτ , given by a ′ Is T (v) = τ a

It



,

factors to a Ps−1,1 - isomorphism from JZt′ ,ψa” (Vτ ) to JZt′ ,ψ”′ (Vτ ).



a

We will, sometimes, denote by τ(t) anyone of the representations τ(t),a of Ps−1,1 . ′ Denote by Nℓ−t the unipotent radical of the (standard) parabolic subgroup of h(Wm,j ), which preserves the flag 0 ⊂ SpanK {ej+1 } ⊂ SpanK {ej+1 , ej+2 } ⊂ · · · ⊂ SpanK {ej+1 , ..., ej+ℓ−t }. This is the analog of Nℓ inside h(Wm,j ). If j = m ˜ (and then ℓ − t = 0), this is the identity subgroup. Let (w)

Q′s = Lℓ,w0 ∩ ηt−1 Qℓ,j ηt

(5.16)

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When w0 ∈ Wm,ℓ+s , or when h(V ) is split, even orthogonal, this is the (maximal) + ∩w0⊥ (see parabolic subgroup of Lℓ,w0 , which preserves the isotropic subspace ωbt Vℓ,s (5.3)). Otherwise, it is a proper subgroup of the last parabolic subgroup. In any case, it is identified with the subgroup of h(Wm,ℓ ), consisting of the elements (5.13). We will now distinguish two cases, according to whether ℓ < m, ˜ or ℓ = m. ˜ Assume, first, that ℓ < m. ˜ In this case, we may take w0 = yα . According to the last proposition, we may assume, in this case, that ℓ + s < m, ˜ or j = m. ˜ Assume that R (w,η ) ℓ + s < m. ˜ The following map, on ind ℓ,α −1 (w) −1 πℓ,j t (τ, σ), is well defined, T (f )(x) =

Rℓ,α ∩ηt Qℓ,j ηt

Z

(w)

Nℓ ∩ηt−1 Qℓ,j ηt \Nℓ

  −1 ′ (f (nx))ψℓ,α (n)dn. JZt′ ,ψt′ ⊗ Jψℓ−t,α Rℓ,α

(w,ηt ) π (τ, σ), (w) Rℓ,α ∩ηt−1 Qℓ,j ηt−1 ℓ,j

Here, f is a function in (the space of) ind

(5.17)

and x ∈

′ is applied to Lℓ,α (recall that we denote Rℓ,α = Rℓ,yα etc.). The functor Jψℓ−t,α t

(w)

σ ωb . To check that (5.17) is well defined, let n ∈ Nℓ ∩ ηt−1 Qℓ,j ηt be of the form (5.12), (with the block upper triangular matrix in (5.13) being the identity.) Then  ′ JZt′ ,ψt′ ⊗ Jψℓ−t,α (f (nx)) is equal to  JZt′ ,ψt′ This equals

   ωbt′      I y ′    n2 y 4 z 4 τ s 6 ⊗ σ ωbt  Im” y ′   (f (x)) . ′ ⊗ Jψℓ−t,α 4     0 n∗1 n∗2 

   ωbt′   n y z 2 4 4    I y′ ′  Im” y ′   JZ ′ ,ψ′ ⊗ Jψ′ (f (x)) ψt′ s 6∗ ψℓ−t,α 4 t t   ℓ−t,α 0 n1 n∗2   ′ = ψℓ,α (n) JZt′ ,ψt′ ⊗ Jψℓ−t,α (f (x)).

Note that the integral (5.17) converges absolutely, since the function n 7→ f (nx), (w) on Nℓ , has compact support, modulo Nℓ ∩ ηt−1 Qℓ,j ηt . Lemma 5.3. The map T defines an Lℓ,α -isomorphism Rℓ,α

(w,ηt ) π (τ, σ)) (w) Rℓ,α ∩ηt−1 Qℓ,j ηt ℓ,j

T ′ : Jψℓ,α (ρw,Im′ −2ℓ ,t ) = Jψℓ,α (ind L

1−t

∼ =

(5.18)

t

′ indQℓ,α | det ·|K2 τ (t) ⊗ Jψℓ−t,α (σ ωb ). ′ s

The action of the inducing representation of Q′s , on the right hand side of (5.18), is by (5.14). (Our assumptions in this lemma are that ℓ, ℓ + s < m ˜ and w0 = yα .)

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Proof. It is clear that T , in (5.17), factors through the Jacquet module on the left hand side of (5.18). Denote by T ′ the induced map on this Jacquet module. It is straight forward to check that, for q ′ ∈ Q′s , of the form ωbt′ du v q ′ =  e u′  , d∗ 

which is the projection of the element (5.12) to Q′s , and satisfies (5.13), we have 1−t

1

1

δQ2 j (wq ′ w−1 ) = δQ2 ′s (q ′ )| det(d)|K2

+ℓ

,

where we view q ′ back as an element of h(V ) (by letting it act as the identity, on Vℓ+ + Vℓ− ). Thus, T ′ (and also T ) takes values inside 1−t

L

t

′ indQℓ,α | det ·|K2 τ (t) ⊗ Jψℓ−t,α (σ ωb ). ′ s t

t

′ If t = ℓ, then Jψℓ−t,α (σ ωb ) = ResLℓ,α ∩h(Wm,j (σ b ). Let us prove that T ′ is injective. Assume that T (f ) = 0. We will use the notation of the proof of Proposition 5.1. Thus, our assumption is equivalent to T (f )(x) = 0, for all x ∈ Ω2 . For x ∈ Ω2 , the (w) integration dn in (5.17) takes place, modulo Nℓ ∩ ηt−1 Qℓ,j ηt , in Ω1 . (We may even

(w)

take Ω1 to lie in the subgroup of Nℓ , which is a complement of Nℓ ∩ ηt−1 Qℓ,j ηt ; take the elements (5.12) in Nℓ , “fill” the upper zero blocks, with general coordinates, replace n1 , n2 by identity matrices, and replace y4 , y4′ , ..., z4 with zero blocks.) Thus, for all x ∈ Ω2 ,     Z  ψ −1 (n)f (nx)dn = 0. ′ ′ JZt′ ,ψt′ ⊗ JNℓ−t ,ψℓ−t,α ℓ,α Ω1

This means (see again the calculation, right after (5.17)) that there is i0 , such that, for all i ≥ i0 and x ∈ Ω2   Z Z t −1 −1 ψℓ,α (u)(τ ⊗ σ ωb )(u)  ψℓ,α (n)f (nx)dn du = 0. (i)

Ω1

(w)

Nℓ ∩ηt−1 Qℓ,j ηt

Thus, for all i ≥ i0 and x ∈ Ω2 , Z

Z

−1 ψℓ,α (un)f (unx)dndu = 0.

(5.19)

(i) (w) Nℓ ∩ηt−1 Qℓ,j ηt Ω1

We want to show that f satisfies the property Z −1 ψℓ,α (n)ρ(n)f dn = 0, (i) Nℓ

(5.20)

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for i sufficiently large. Here, ρ denotes right translation in the representation R (w,η ) ind ℓ,α −1 (w) πℓ,j t (τ, σ). We have to show, that there is i′0 , such that, for Rℓ,α ∩ηt Qℓ,j ηt

all n0 ∈ Nℓ and x ∈ Ω2 ,

Z

−1 ψℓ,α (n)f (n0 xn)dn = 0,

(i′ )

Nℓ

for all i′ ≥ i′0 . Since Lℓ,α normalizes Nℓ and fixes ψℓ,α , we may replace, for x ∈ Ω2 , xn by nx, in the last integral. Indeed, as in Proposition 5.1, we may take a compact (i) open subgroup L0 of Lℓ,α , which normalizes Nℓ , for i large, and we may assume that Ω2 is a union of finitely many cosets ωj L0 , 1 ≤ j ≤ N (depending on f ). Then we first replace an by na, for a ∈ L0 , by a simple change of variable. Next, we may (i) (i) assume that i′ , i are large, and such that ωj Nℓ ωj−1 ⊃ Nℓ , for all 1 ≤ j ≤ N . (i)

(i)

Since we can decompose ωj Nℓ ωj−1 as a (finite) union of cosets of the form n′ Nℓ , with representatives n′ ∈ Nℓ , it is enough to show that, for i sufficiently large, Z −1 ψℓ,α (n)f (n0 nx)dn = 0, (i)

Nℓ

(i)

for all x ∈ Ω2 and n0 ∈ Nℓ . Let us also assume that i is so large, that Ω1 ⊂ Nℓ . (i) Then, in the last integral, we may integrate on n ∈ Nℓ , such that n0 n ∈ (Nℓ ∩ (i) −1 (w) −1 (w) ηt Qℓ,j ηt )Ω1 . This implies n0 ∈ (Nℓ ∩ ηt Qℓ,j ηt )Nℓ . Thus, we may assume that (i)

n0 ∈ Nℓ , and changing variable, n 7→ n−1 0 n, in the last integral, we may assume that n0 = 1. Thus, in order to prove (5.20), we have to show, that, for i sufficiently large, we have, for all x ∈ Ω2 , Z −1 ψℓ,α (n)f (nx)dn = 0. (i)

Nℓ

(i)

(w)

In the last integral, it is enough to integrate over (Nℓ ∩ ηt−1 Qℓ,j ηt )Ω1 . Now, we get what we want from (5.19). This proves the injectivity of T ′ . We show the surjectivity of T ′ . Let φ be a function in the space of L

1−t

t

′ (σ ωb ). Assume that φ is supported in Q′s Ω2 , where indQℓ,α | det ·|K2 τ (t) ⊗ Jψℓ−t,α ′ s ′ Ω2 is compact and open. Since JZt′ ,ψt′ ⊗ Jψℓ−t,α is surjective, from the space

(w,ηt )

of πℓ,j

t

′ (τ, σ) to the space of τ (t) ⊗ Jψℓ−t,α (σ ωb ), it is easy to see that we

L

(w,η )

can find a (smooth) function f in the space of indQℓ,α πℓ,j t (τ, σ), such that ′ s   ′ (f (x)) = φ(x), for all x ∈ Ω2 . Now, let us extend f to a funcJZt′ ,ψt′ ⊗ Jψℓ−t,α Rℓ,α

(w,ηt ) π (τ, σ), (w) Rℓ,α ∩ηt−1 Qℓ,j ηt ℓ,j

tion in the space of ind

so that, for all x ∈ Ω2 , the (w)

function on Nℓ , n 7→ f (nx), is supported inside (Nℓ ∩ Qℓ,j )Ω01 , where Ω01 is a sufficiently small compact open subgroup of Nℓ , and is such that f right invariant to x−1 Ω01 x, for all x ∈ Ω2 . (Recall that Ω2 is compact and open.) Now it is clear

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that T (f )(x) = c0 φ(x), for all x ∈ Ω2 . Here c0 is the measure of Ω01 , modulo (w) Nℓ ∩ ηt−1 Qℓ,j ηt . This implies that T ′ is surjective, and the lemma is proved.  Note that, since we assume, in the last lemma, that ℓ + s < m, ˜ Q′s is a parabolic subgroup of Lℓ,α , and the representation in the r.h.s. of (5.18) is parabolically induced from Q′s , i.e. the inducing representation acts trivially on the unipotent radical of Q′s . By Proposition 5.2 and the last lemma, it remains to prove the following lemma in order to complete the study of Jψℓ,α (ρw,Im′ −2ℓ ,t ). Lemma 5.4. Assume that ℓ < m ˜ and w0 = yα . Assume also that j = m ˜ = ℓ + s. Then, as Lℓ,α -modules, −ℓ

t L Jψℓ,α (ρw,Im′ −2ℓ ,t ) ∼ | det ·|K2 τ(ℓ) ⊗ σ ωb . = indQℓ,α ′ s

(5.21)

If m ˜ = m, then, of course σ is not there, and should be omitted from the statement. The proof of the lemma is exactly as that of Lemma 5.3, where we replace in ′ ′ the definition of T , in (5.17), JZt′ ,ψt′ ⊗ JNℓ−t by JZt′ ,ψ”t,a ⊗ ResL′0,α , where ,ψℓ−t,α ′ L0,α = Lℓ,α ∩ h(Wm,j ). Here,  α 1−t′ ′ a = (−1)m − . 2 See (5.15) and Lemma 5.2. Note that the inducing representation on the right hand side of (5.21), is not trivial on the unipotent radical of Q′s . Let us summarize. Proposition 5.3. Assume that ℓ < m ˜ and w0 = yα . Let the Weyl element w correspond to r = 0 and s ≤ j, such that j ≤ ℓ+s ≤ m. ˜ Consider the representation ρw,Im′ −2ℓ ,t . We have (1) If ℓ + s = m ˜ and j < m, ˜ then Jψℓ,α (ρw,Im′ −2ℓ ,t ) = 0. (2) If ℓ + s = m ˜ and j = m, ˜ then (see (5.16)) −ℓ

L

t

Jψℓ,α (ρw,Im′ −2ℓ ,t ) ∼ | det ·|K2 τ(ℓ) ⊗ σ ωb . = indQℓ,α ′ s (3) If ℓ + s < m, ˜ then L

1−t

t

′ Jψℓ,α (ρw,Im′ −2ℓ ,t ) ∼ | det ·|K2 τ (t) ⊗ Jψℓ−t,α (σ ωb ). = indQℓ,α ′ s

Note again, that in Case (3) of the last proposition, Q′s is a parabolic subgroup of Lℓ,α and the representation on the right hand side is parabolically induced. In Case (2), Q′s is a parabolic subgroup of Lℓ,α , only when h(V ) is split and even orthogonal, and in all cases, the inducing representation, in the right hand side is not trivial on the unipotent radical of Q′s . Assume that ℓ = m, ˜ and w0 ∈ Wm,ℓ = V0 . Of course, we also assume that V0 6= 0. Then we can repeat the same arguments as above. In this case, we must

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have s = 0, t = j. Thus, 1 ≤ j ≤ ℓ = m. ˜ The inducing subgroup, in the induced (w) representation ρw,Im′ −2ℓ ,t , i.e. Rℓ,w0 ∩ ηt−1 Qℓ,j ηt consists of the elements (see (5.12)   n1 0 0 0 0 t ′   n y z 0 ℓ =ℓ−t 2 4 4   ′ h= e y4′ 0    m”=m −2ℓ ∗   ′ n2 0 ℓ =ℓ−t n∗1 t with n1 ∈ Zt (K), n2 ∈ Zℓ−t (K), and e · w0 = w0 , thinking of e as an element of (w,η ) h(V0 ). The action of πℓ,j t (τ, σ) on h, of the form above is by ωt  n2 y 4 z 4 b τ (n∗1 ) ⊗ σ  e y4′  . n∗2 Now, Lemma 5.3 works exactly the same and shows that Jψℓ,w0 (ρw,Im′ −2ℓ ,t ) is isot morphic to τ (j) ⊗ Jψℓ−t,w0 (σ ωb ). Note that τ (j) is the Jacquet module of τ , with −1 respect to the Whittaker character corresponding to ψK,1 . Thus, if τ has no Whittaker functionals, we get zero, and if it does, we get a multiple dτ = dimτ (j) of t Jψℓ−t,w0 (σ ωb ). Therefore, Proposition 5.4. Assume that ℓ = m ˜ and w0 ∈ V0 . Let the Weyl element w correspond to r = 0 and (necessarily) s = 0. Consider the representation ρw,Im′ −2ℓ ,t . Then (j = t) t Jψℓ,w0 (ρw,Im′ −2ℓ ,t ) ∼ = dτ · Jψℓ−t,w0 (σ ωb ). Let us examine the Jacquet modules Jψℓ,w0 (ρw,γ,t), for γ 6= I. Of course, this implies that ℓ < m ˜ and s > 0. In particular, we may take here w0 = yα . Assume, first, that ℓ + s < m. ˜ This puts us in Case (1) of Proposition 4.4 (with W = Wm,ℓ + and X = Vℓ,s ). Thus, we may take (see (4.30))   0 Is Im−ℓ−s  0  ˜  . γ = γs =  (5.22) I V0     0 Im−ℓ−s ˜ Is 0 We will denote the corresponding matrix η (5.2) (with ǫ as in (5.10)) η = ηt,s . (w) −1 (w) −1 We have Rℓ,α ∩ ηs,t Qℓ,j ηs,t = ηs,t (Rℓ,ηs,t yα ∩ Qℓ,j )ηs,t . As in (5.12), this is the subgroup of elements h in h(V ), such that  ωt′ n2 0 0 y 4 y 5 0 z 4 b ℓ−t  n1 0 0 y 6 0 0  t    d u v y6′ y5′  s     −1 (5.23) ηs,t hηs,t = 0 e u′ 0 y4′  m”=m′ −2(ℓ+s)   ∗   0 0 d 0 0 s    n∗1 0  t n∗2 ℓ−t

master

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where n1 ∈ Zt (K), n2 ∈ Zℓ−t (K), and   du v ′ ′  e u′  ωbt ηs,t (yα ) = ωbt ηs,t (yα ). d∗ (w,ηs,t )

The action of πℓ,j

93

(5.24)

(τ, σ) on h, of the form (5.23), satisfying (5.24) is

ωbt+t′   ′ n y z 2 4 4 1 d y6 −1 −1 −1 δ 2 −1 (w) (h)δQ2 j (wηs,t hηs,t ⊗σ w )τ e y4′  Rℓ,α ∩ηs,t Qℓ,j ηs,t 0 n∗1 n∗2

(5.25)

Denote

(w)

(w)

−1 −1 Q′s,t = Lℓ,α ∩ ηs,t Qℓ,j ηs,t = ηs,t (Lℓ,ηs,t yα ∩ Qℓ,j )ηs,t .

(5.26)

This is a subgroup of the maximal parabolic subgroup of Lℓ,α , which preserves the ′ + isotropic subspace (ωbt ηs,t )−1 Vℓ,s ∩ yα⊥ . It is the projection of the elements (5.23), which satisfy (5.24), to h(Wm,ℓ ). Lemma 5.5. Assume that 0 < s ≤ j < ℓ + s < m. ˜ Then Jψℓ,α (ρw,γs ,t ) = 0. Proof. As in Proposition 5.2, we will find a subgroup S ⊂ Nℓ , which satisfies (5.9). Take S to be the subgroup of elements (5.23), with the matrix having identity ′ diagonal blocks, and all off-diagonal blocks, except y5 , y5 , being zero. Note that S ⊂ Nℓ and ψℓ,α 6= 1. In fact, ψℓ,α evaluated at such elements gives (with a J

′

′

convenient slight abuse of notation) ψK,1 ((−1)m +1 α2 (y5 ωbt )ℓ−t,1 ). By (5.25), we have π (w,ηt,s ) = 1.  S

Assume, next, that 0 < s ≤ j = ℓ + s < m. ˜ This forces ℓ < j < m. ˜ As in Lemma 5.3, we get ℓ

ℓ − L Jψℓ,α (ρw,γs ,t ) ∼ | det ·|K 2 τ(ℓ) ⊗ σ ωb . = indQℓ,α ′ s,ℓ

(5.27)

The action of the inducing representation of Q′s,η , in (5.27), is by (5.25). We will denote, in this case, Q′s,ℓ = Q′′s .

(5.28)

˜ This It remains to compute Jψℓ,w0 (ρw,γ,t ), for γ 6= I and 0 < s ≤ j ≤ ℓ + s = m. puts us in Case (2) of Proposition 4.4. In particular, we must have W itt(Ky−α + V0 ) = 1.

(5.29)

Indeed, this is equivalent to the condition that W itt(yα⊥ ∩ Wm,ℓ ) = m ˜ − ℓ. In this case, we can choose the representative γ to be of the form γ = diag(Is−1 , γα , Is−1 ),

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where γα is described in (4.31); it acts as follows. Let vα ∈ V0 be such that ′ b(vα , vα ) = −b(y−α , y−α ) = (−1)m +1 α. Then γα (em ˜ ) = y−α − vα ; γα (e−m ˜ ) = e −m ˜, and, for v ∈ V0 , γα (v) − v ∈ Ke−m ˜ . By Proposition 4.4, this is the only (nontrivial) representative to consider, except when h(V ) is split, odd orthogonal, in which case, there are two choices for vα above, and we get the two nontrivial representatives γα = γ± in (4.33). In this case, α = β 2 is a square, and the two choices of vα are ±βe0 . Denote the corresponding representative η by η = ηγα ,t . Note that ηγα ,t (yα ) = em ˜ − vα . = Rℓ,ηγα ,t yα . Thus, we Since ℓ < m, ˜ we have ηγα ,t Nℓ ηγ−1 = Nℓ , and ηγα ,t Rℓ,yα ηγ−1 α ,t α ,t have the same formulas as those of (5.23)– (5.25). Denote (w)

(w)

Qℓ,j ηγα ,t = ηγ−1 (Lℓ,ηγα ,t yα ∩ Qℓ,j )ηγα ,t . Q′α = Lℓ,α ∩ ηγ−1 α ,t α ,t

(5.30)

(w)

It is a parabolic subgroup of Lℓ,α . The Levi subgroup of Lℓ,em˜ +vα ∩Qℓ,j is naturally isomorphic to GLs (K)×h(vα⊥ ∩V0 ). In fact, ηγα ,t Q′α ηγ−1 is realized as the subgroup α ,t of elements, as in (5.24), such that d ∈ GLs (K), e(vα ) = vα and   0  ..  u(vα ) = (d − Is )  .  . 1

Remark 5.1. In case h(V ) is split odd orthogonal, the above is meaningful for both γ± and we will denote the two groups corresponding to Q′α by Q′s,± . Note that, for elements h ∈ Nℓ , of the form (5.23) (with η = ηγα ,t ), we have, if t < ℓ, ′

ψℓ,α (h) = ψ((y4 ωbt )ℓ−t · vα′ ), where we realize V0 as a column space and vα′ is the column vector corresponding to vα (as in (3.47)). As in Lemma 5.3, we get here that L

1−t

t

′ (σ ωb ). Jψℓ,α (ρw,γα ,t ) ∼ | det ·|K2 τ (t) ⊗ Jψℓ−t,v = indQℓ,α ′ α α

(5.31)

Finally, if t = ℓ, then ψℓ,yα is trivial on elements h ∈ Nℓ , of the form (5.23) (with η = ℓ t ′ ηγα ,t ). We get (5.31), with the usual interpretation Jψ0,v (σ ωb ) = Resh(vα⊥ ∩V0 ) (σ ωb ). α Let us summarize. Proposition 5.5. Let w correspond to r = 0 and s, such that 0 < s ≤ j ≤ ℓ+s ≤ m. ˜ Assume that γ 6= I (in (5.2), and choose γ as above). (1) If t < ℓ and ℓ + s < m, ˜ then (see (5.22)) Jψℓ,α (ρw,γs ,t ) = 0.

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(2) If t = ℓ and ℓ + s < m, ˜ then (see (5.28)) −ℓ

L

ℓ

ωb 2 . Jψℓ,α (ρw,γs ,t ) ∼ = indQℓ,α ′′ | det ·|K τ(ℓ) ⊗ σ s

(3) Assume that W itt(Ky−α + V0 ) = 1 and ℓ + s = m. ˜ Then (see (5.30)) 1−t

t L ′ Jψℓ,α (ρw,γα ,t ) ∼ | det ·|K2 τ (t) ⊗ Jψℓ−t,v (σ ωb ). = indQℓ,α ′ α α

Note Case (3) in the last proposition, when h(V ) is split odd orthogonal. Here, α t ′ must be a square β 2 , vα = ±βe0 , and Jψℓ−t,±βe (σ ωb ) are Jacquet modules with 0 respect to Whittaker characters. In this case, we will rewrite the isomorphism in the proposition as (see Remark 5.1) t 1−t L Jψℓ,α (ρw,γ± ,t ) ∼ | det ·| 2 τ (t) ⊗ JW h,± (σ ωb ). = indQℓ,α ′

(5.32)

s,±

t

In case t = ℓ, σ “is not there”, and JW h,± (σ ωb ) should be omitted. We summarize the last three propositions in the following Leibniz rule. Here, “≡” denotes isomorphism of representations, up to semi-simplification. Theorem 5.1. (1) Assume that 0 ≤ ℓ < m ˜ and 1 ≤ j < m. ˜ Then h(V )

Jψℓ,α (IndQj τ ⊗ σ) ≡ 1−t L t L ′ (σ ωb ) indQℓ,α | det ·|K2 τ (t) ⊗ Jψℓ−t,α ′ ℓ+j−m
j−t

0≤t≤j

 ℓ −ℓ L | det ·|K 2 τ(ℓ) ⊗ σ ωb L indQℓ,α ′′ j−ℓ 0, L

1 δh(V ),α

(

L

1+m−ℓ−j ˜ 2

indQℓ,α | det ·|K ′ α

0

ℓ<j otherwise ℓ+j−m ˜

˜ ′ τ (ℓ+j−m) ⊗ Jψm−j,v (σ ωb ˜ α

 L 1+m−ℓ−j t 2  τ (ℓ+j−m) ⊗ JW h,+ (σ ωb )⊕ indQℓ,α | det ·| ′   m−ℓ,+ L 2 1+m−ℓ−j t ℓ,α 2 δh(V ),α indL τ (ℓ+j−m) ⊗ JW h,− (σ ωb ) Q′m−ℓ,− | det ·|    0

)

0<m−ℓ≤j ˜ otherwise

0<m−ℓ≤j otherwise

1 Here, δh(V ),α = 0, unless W itt(Ky−α + V0 ) = 1 and h(V ) is not split odd 2 orthogonal, in which case it is 1, and δh(V ),α = 0, unless h(V ) is split odd orthogonal, and α is a square, in which case it is 1. See (5.16) for the definition of Q′j−t , (5.28) for the definition of Q′′j−ℓ , (5.30) for the definition of Qα , and Remark 5.1 for the definition of Q′m−ℓ,± .

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(2) Assume that 0 ≤ ℓ < m ˜ and j = m. ˜ Then h(V )

Jψℓ,α (IndQj τ ⊗ σ) ≡ −ℓ

L

ℓ

indQℓ,α | det ·|K2 τ(ℓ) ⊗ σ ωb ′ m−ℓ ˜

1−ℓ

L

L

ℓ

ℓ,α 1 (ℓ) 2 ′ δh(V ⊗ Jψ0,v (σ ωb ) ),α indQ′α | det ·|K τ α

 Lℓ,α 2 δh(V ),α indQ′

m−ℓ,+

| det ·|

1−ℓ 2

L

τ (ℓ) ⊕ indQℓ,α ′

L

m−ℓ,−

| det ·|

1−ℓ 2

(3) Assume that ℓ = m ˜ and w0 ∈ V0 . Then

 τ (ℓ) .

j

h(V )

Jψℓ,w0 (IndQj τ ⊗ σ) ≡ dτ · Jψℓ−j,w0 (σ ωb ). We end this section with several special cases of the last theorem. We keep the same notation. Theorem 5.2. Assume that ℓ < m ˜ and j < m, ˜ and that τ is a supercuspidal representation of GLj (K). (1) If ℓ ≥ j, then h(V )

Jψℓ,α (IndQj τ ⊗ σ) ≡  indLℓ,α | det ·| 12 τ ⊗ J ′ (σ), j L ψℓ,α K ωb Q′j ′ dτ Jψℓ−j,α (σ ) 0, L

L

1 δh(V ),α

2 δh(V ),α

(

(

j<m ˜ −ℓ otherwise

1

L

2 ′ | det ·|K indQℓ,α τ ⊗ Jψℓ,v (σ), ′ α

m−ℓ=j

α

0, otherwise 1 1 Lℓ,α L 2 τ ⊗ J (σ) ⊕ ind | det ·| 2 τ ⊗ JW h,− (σ), indQℓ,α | det ·| W h,+ ′ Q′

m−ℓ=j

0,

otherwise

j,−

j,+

(2) If ℓ < j, then h(V )

Jψℓ,α (IndQj τ ⊗ σ) ≡ −ℓ L indQℓ,α | det ·|K 2 τ(ℓ) ′′ j−ℓ

L

2 δh(V ),α

(

⊗σ

ωbℓ

L

1 δh(V ),α

( L 1 2 ′ | det ·|K indQℓ,α τ ⊗ Jψℓ,v (σ), ′ α

α

m−ℓ = j

0, otherwise 1 1 Lℓ,α L 2 2 | det ·| τ ⊗ J (σ) ⊕ ind τ ⊗ J (σ), m−ℓ=j | det ·| indQℓ,α W h,+ W h,− ′ Q′ j,−

j,+

0,

otherwise

Proof. We use Theorem 5.1(1). Since τ is supercuspidal, τ (t) = 0, unless t = 0 or t = j. If t = 0, in the first sum in Theorem 5.1(1), then we must have j < m ˜ − ℓ, 1

L

2 ′ τ ⊗ Jψℓ,α (σ). If t = j, in the first sum and then we get the summand indQℓ,α | det ·|K ′ j

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97 j

′ of Theorem 5.1(1), then we must have ℓ ≥ j, and we get the summand Jψℓ−j,α (σ ωb j (if ℓ = j, this is just σ ωb ). Of course, this summand appears dimτ (j) = dτ

Lℓ,α

times, that is the dimension of the space of ψ-Whittaker functionals of τ . Finally, the third and fourth summands in Theorem 5.1(1) contribute only when ℓ + j = m, ˜ and, then, we get 1 L Lℓ,α 1 2 ′ δh(V ),α indQ′α | det ·|K τ ⊗ Jψℓ,vα (σ)   1 1 Lℓ,α Lℓ,α 2 2τ ⊗ J 2τ ⊗ J δh(V ind | det ·| (σ) ⊕ ind (σ) . | det ·| ′ ′ W h,+ W h,− ),α Q Q j,+

j,−



1 Theorem 5.3. Assume that 0 < ℓ < m, ˜ j=m ˜ and that, in case δh(V ),α = 1 (resp. 2 δh(V ),α = 1), τ is a supercuspidal representation of GLm (K). Then −ℓ

L

h(V )

ℓ

Jψℓ,α (IndQj τ ⊗ σ) ∼ | det ·|K2 τ(ℓ) ⊗ σ ωb = indQℓ,α ′ m−ℓ ˜

2 Proof. This follows from Theorem 5.1(2), where, if δh(V ),α = 1 (resp. δh(V ),α = 1), the second (resp. the third) summand in Theorem 5.1(2) is zero, since, by assumption, τ is supercuspidal and 0 < ℓ < m. ˜ The fact that we have an isomorphism (and not just up to semisimplification) follows from the fact that, in the proofs of the propositions, which led to Theorem 5.1, the r.h.s. corresponds to one double coset. 

Next, we consider the case where j = m ˜ and τ is as follows. Let µ1 , ..., µi be characters of K ∗ . Let m ˜ = m1 + · · · + mi be a partition of m. ˜ Let Pm1 ,...,mi be the corresponding standard parabolic subgroup of GLm (K). Its Levi part is isomorphic ˜ to GLm1 (K) × · · · GLmi (K). We consider GL (K)

τ = IndPmm˜,...,m µ1 (det)(m1 ) ⊗ · · · ⊗ µi (det)(mi ), 1

i

(5.33)

where µj (det)(mj ) = µj (detGLmj ). Then the highest derivative of τ is τ (i) , and as a representation of GLm−i (K), we have ˜ | det ·|

1−i 2

GL ˜ (K) µ1 (det)(m1 − 1) ⊗ · · · ⊗ µi (det)(mi − 1). τ (i) ∼ = IndPmm−i 1 −1,...,mi −1

(5.34)

Here, if mi′ = 1, we omit µi′ (detGLm ′ −1 ), and we omit mi′ − 1 from the partition i (m1 − 1, ..., mi − 1) of m ˜ − i. This follows from [Bernstein and Zelevinsky (1977)], p.455, and the fact that µi′ (detGLm′ ), as a representation of GLmi′ (K), admits only i two non-trivial derivatives, namely the one with index 0, which is µi′ (detGLm′ ), and i the one with index 1, which is the restriction of µi′ (detGLm′ ) to GLmi′ −1 (K). i

Theorem 5.4. Let τ be a representation of GLm ˜ (K), of the form (5.33) and let σ be a smooth representation of h(V0 ).

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1 2 (1) Assume that 0 ≤ ℓ < m, ˜ and w0 = yα . Assume that δh(V ),α = δh(V ),α = 0, that is W itt(Ky−α + V0 ) = 0. Then h(V )

Jψℓ,α (IndQm˜ τ ⊗ σ) = 0, for all ℓ ≥ i, and for ℓ = i − 1,

h(V ) ∼ Jψi−1,α (IndQm˜ τ ⊗ σ) = Li−1,α ωbi−1 . indQ′ (m1 −1,...,mi −1) µ1 (det)(m1 − 1) ⊗ · · · ⊗ µi (det)(mi − 1) ⊗ σ m−i+1 ˜

(5.35) ′ Here, Q′m−i+1 (m − 1, ..., m − 1) is the intersection of Q (see (5.16)) 1 i ˜ m−i+1 ˜ with the parabolic subgroup of Li−1,α , which preserves the flag (see (5.3)) + + + Vi−1,m ⊂ Vi−1,m ⊂ · · · ⊂ Vi−1, . m−i ˜ 1 −1 1 +m2 −2

1 2 (2) Assume that 0 ≤ ℓ < m, ˜ and w0 = yα . Assume that one of δh(V ),α , or δh(V ),α is nonzero, that is W itt(Ky−α + V0 ) = 1. Then h(V )

Jψℓ,α (IndQm˜ τ ⊗ σ) = 0, for all ℓ > i, and for ℓ = i, h(V ) Jψi,α (IndQm˜ τ ⊗ σ) ∼ = i Li,α 1 ′ δh(V ),α ·IndQ(m1 −1,...,mi −1) µ1 (det)(m1 −1)⊗· · ·⊗µi (det)(mi −1)⊗Jψ0,v (σ ωb ) α L  Li,α 2 δh(V ),α IndQ+ (m1 −1,...,mi −1) µ1 (det)(m1 − 1) ⊗ · · · ⊗ µi (det)(mi − 1)⊕  L Ind+ i,α µ (det)(m − 1) ⊗ · · · ⊗ µ (det)(m − 1) . 1 1 i i Q(m1 −1,...,mi −1) (5.36) 1 = 1, Q(m − 1, ..., m − 1) is the parabolic subgroup of Here, in case δh(V 1 i ),α Li,α , which preserves the flag + + Vi,m ⊂ Vi,m ⊂ · · · ⊂ Vm+1 +···+mi−1 −i+1 ⊂ Vi,+m−i−1 ⊕ k(y−α − vα ), ˜ 1 −1 1 +m2 −2

2 2 and, in case δh(V ),α = 1, h(V ) is split, odd orthogonal, and α = β is a square, + Q (m1 − 1, ..., mi − 1) is the parabolic subgroup of Li,α , which preserves the flag

β2 e−m ), 2 and + Q(m1 − 1, ..., mi − 1) is the parabolic subgroup of Li,α , which preserves the flag + + + Vi,m ⊂ Vi,m ⊂ · · ·⊂ Vm+1 +···+mi−1 −i+1 ⊂ Vi,m−i−1 ⊕k(em−βe0− 1 −1 1 +m2 −2

+ + + Vi,m ⊂ Vi,m ⊂ · · ·⊂ Vm+1 +···+mi−1 −i+1 ⊂ Vi,m−i−1 ⊕k(em+βe0− 1 −1 1 +m2 −2

β2 e−m ). 2

(3) Assume that ℓ = m ˜ and w0 ∈ V0 . Then,

h(V )

Jψℓ,α (IndQm˜ τ ⊗ σ) = 0, for i < ℓ = m, ˜ and for i = ℓ (which means that m1 = · · · = mi = 1 and i = m), ˜ m ˜ h(V ) Jψi,α (IndQm˜ τ ⊗ σ) ∼ = Jψ0,w0 (σ ωb ).

(5.37)

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1 2 Proof. Assume, first, that 0 ≤ ℓ < m. ˜ Assume, that δh(V ),α = δh(V ),α = 0. By Theorem 5.1(2), we have ℓ

ℓ − L h(V ) | det ·|K 2 τ(ℓ) ⊗ σ ωb . Jψℓ,α (IndQm˜ τ ⊗ σ) ∼ = indQℓ,α ′ m−ℓ ˜

1 Note that τ(ℓ) is a smooth representation of the mirabolic subgroup, Pm−ℓ of ˜ GLm−ℓ (K). By [Bernstein and Zelevinsky (1977)], p. 452, we know that this repre˜ 1 sentation has a finite Pm−ℓ -filtration, whose subquotients are (compactly) induced ˜ from derivatives of τ(ℓ) . Note that (τ(ℓ) )(e) = τ (ℓ+e) . We conclude that if ℓ + 1 > i, h(V )

then τ(ℓ) = 0, and hence Jψℓ,α (IndQm˜ τ ⊗ σ) = 0. Let, now, ℓ = i − 1. Then, 1 since τ (i+e) = 0, for e > 0, then it follows, by looking at the above Pm−i+1 - filtra˜ (i) tion of τ(i−1) , that it has only one subquotient, namely τ , and we conclude that 1 τ(i−1) ∼ (K) (embedded inside Pm−i+1 ). From = τ (i) , as representations of GLm−i ˜ ˜ (5.34), we get that h(V ) Jψℓ,α (IndQm˜ τ ⊗ σ) ∼ = L

indQi−1,α ′

GL

m−(i−1) ˜

i−1

(K)

˜ IndPmm−i −1,...,m 1

i

µ (det)(m1 − 1) ⊗ · · · ⊗ µi (det)(mi − 1) ⊗ σ ωb −1 1



.

By transitivity of induction, we get (5.35). 1 2 Assume now that one of δh(V ),α , or δh(V ),α is nonzero. From the first part, we know that τ(ℓ) = 0, for all ℓ > i − 1. From Theorem 5.1(2), it follows, that, for ℓ > i − 1, we have h(V )

Jψℓ,α (IndQm˜ τ ⊗ σ) ≡ 1−ℓ

L

ℓ

ℓ,α 1 (ℓ) 2 ′ δh(V ⊗ Jψ0,v (σ ωb ) ),α indQ′α | det ·|K τ α

 Lℓ,α 2 δh(V ),α indQ′

m−ℓ,+

| det ·|

1−ℓ 2

L

τ (ℓ) ⊕ indQℓ,α ′

L

m−ℓ,−

| det ·|

1−ℓ 2

h(V )

 τ (ℓ) .

Since τ (ℓ) = 0, for all ℓ > i, it follows that Jψℓ,α (IndQj τ ⊗ σ) = 0, for all ℓ > i. Let ℓ = i. Then, from the last equality and from (5.34), we conclude (5.36). Assume, next, that ℓ = m ˜ (and w0 ∈ V0 ). Then by Theorem 5.1(3), h(V )

m ˜

Jψℓ,w0 (IndQm˜ τ ⊗ σ) ≡ dτ · Jψℓ−j,w0 (σ ωb ), where dτ is the dimension of the space of ψ-Whittaker functionals on τ . For τ of the form (5.33), we know that this is zero, unless m1 = · · · = mi = 1. This, also, forces i = m, ˜ and, in this case, dτ = 1. This proves (5.37). Finally, the fact that we have isomorphisms in the theorem (and not just isomorphisms, up to semisimplifications) follows from the fact that in the proof of the propositions, which led 1 to Theorem 5.1, the r.h.s. of (5.35), or of (5.36), when δh(V ),α = 1, or the r.h.s. of 2 (5.37), corresponds to one double coset, and the r.h.s. of (5.36), when δh(V ),α = 1, corresponds to two open double cosets. 

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We will need especially the following case: h(V ) is split orthogonal or quasi-split even unitary (in particular, m ˜ = m); i = i1 + i2 , m1 = · · · = mi1 = 2, mi1 +1 = · · · = mi = 1. In practice, we will have i2 = 0, 1, 2. In such a case, let τ be a representation of GLm (K), of the form (5.33). Note that, now, m = m ˜ = 2i1 + i2 . Thus, GL (K)

m µ1 (det)(2) ⊗ · · · ⊗ µi1 (det)(2) ⊗ µi1 +1 ⊗ · · · µi1 +i2 . τ = IndP2,...,2,1,...1

(5.38)

We get the following special case of Theorem 5.4. Theorem 5.5. Assume that h(V ) is split orthogonal, or quasi-split even unitary. Let τ be a representation of GLm (K), of the form (5.38) (m = 2i1 + i2 , i = i1 + i2 ). Assume that 0 ≤ ℓ < m and w0 = yα . 2 (1) Assume that δh(V ),α = 0. Then h(V )

Jψℓ,α (IndQm τ ) = 0, for all ℓ ≥ m − i1 , and for ℓ = m − i1 − 1 = i1 + i2 − 1, Lm−i1 −1,α

Jψm−i1 −1,α (IndQm τ ) ∼ = indB ′ h(V )

µ1 ⊗ · · · ⊗ µi1 .

(5.39)

Here, B ′ is the intersection of Q′i1 +1 with the Borel subgroup of Lm−i1 −1,α , which preserves the flag SpanK {em−i1 } ⊂ SpanK {em−i1 , em−i1 +1 } ⊂ · · · ⊂ SpanK {em−i1 , ..., em−1 }. 2 2 (2) Assume that δh(V ),α = 1, that is h(V ) is split odd orthogonal, and α = β is a square. Then h(V )

Jψℓ,α (IndQm τ ) = 0, for all ℓ > m − i1 , and for ℓ = m − i1 , h(V )

Lm−i1 ,α

Jψm−i1 ,α (IndQm τ ) ∼ = IndB ′

Lm−i1 ,α

µ1 ⊗· · ·⊗µi1 −1 ⊗µ−1 i1 . (5.40) Here, B ′ is the Borel subgroup of Lm−i1 ,α , which preserves the flag µ1 ⊗· · ·⊗µi1 ⊕IndB ′

Spank {em−i1 +1 } ⊂ 2 Spank {em−i1 +1 , em−i1 +2 } ⊂ · · ·⊂ Spank {em−i1 +1 , ..., em−1 , em −βe0 − β2 e−m }. Proof. Only (5.40) needs explanation. For this, note that the parabolic subgroups Q+ (m1 − 1, ..., mi − 1), + Q(m1 − 1, ..., mi − 1) of Li,α , in (5.36), are conjugate by the outer conjugation ωb , namely the one which sends eu to itself, for u = ±1, ..., ±m and e0 to −e0 . In this theorem, it is clear that these two parabolic subgroups coincide and equal to the Borel subgroup B ′ . The inducing data in the second summand of (the summand in) (5.36), in our case, are obtained by the outer conjugation, applied to the inducing data of the first summand. Thus, the character of B ′ , µ1 ⊗ · · · ⊗ µi1 −1 ⊗ µi1 , becomes after the outer conjugation, the character µ1 ⊗ · · · ⊗ µi1 −1 ⊗ µi−1 .  1

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We will encounter the setup of Theorem 5.5, when we consider an irreducible, generic, unramified and self-conjugate representation τ of GLm (K) (i.e. τˆ ∼ = τ ′, ′ where τ denotes composition of τ with the conjugation of K/k; in case K = k, this is the identity.) Let us also assume that K/k is unramified, when [K : k] = 2. Such τ is the full induction, to GLm (K), of an unramified character of the Borel subgroup, and such a character is of the form ξ1 ⊗ · · · ⊗ ξm , where the ξi -s are unramified characters of K ∗ . Since τ is self-conjugate and unramified, its central character ωτ , is quadratic, i.e. ωτ2 = 1. Let λ0 be the unique, non-trivial, unramified, quadratic character of K ∗ . Then we distinguish the following four cases, according to whether ωτ = 1, or ωτ = λ0 , and the parity of m. (i) If m = 2n is even and ωτ = 1, then (up to a permutation), {ξ1 , ..., ξm } has the form {µ1 , ..., µn , µ1−1 , ..., µ−1 n }.

(ii) If m = 2n is even and ωτ = λ0 , then (up to a permutation), {ξ1 , ..., ξm } has the form {µ1 , ..., µn−1 , 1, λ0 , µ1−1 , ..., µ−1 n−1 }.

(iii) If m = 2n + 1 is odd and ωτ = 1, then (up to a permutation), {ξ1 , ..., ξm } has the form −1 {µ1 , ..., µn , 1, µ−1 1 , ..., µn }.

(iv) If m = 2n + 1 is odd and ωτ = λ0 , then (up to a permutation), {ξ1 , ..., ξm } has the form −1 {µ1 , ..., µn , λ0 , µ−1 1 , ..., µn }.

h(V )

1

2 . We assume For such τ , let πτ denote the unramified constituent of IndQm τ | det ·|K that h(V ) is as in Theorem 5.5. In the following theorem, π1 ≺ π2 denotes the fact that π1 is a subquotient of π2 , up to semi-simplification. (In most of the cases in the 1 theorem below, the l.h.s. is a true subquotient of the r.h.s.). Since now δh(V ),α = 0, 2 we will re-denote δh(V ),α = δh(V . ),α

Theorem 5.6. Let h(V ) be as in Theorem 5.5. Let τ be an irreducible, generic, unramified representation of GLm (K). Assume that τ is self-conjugate. Assume, also, that K/k is unramified, when [K : k] = 2. Let 0 ≤ ℓ < m and w0 = yα . (1) Assume that ωτ = 1 and m = 2n is even. Assume, also, that δh(V ),α = 0. Then Jψℓ,α (πτ ) = 0, when ℓ ≥ n, and, for ℓ = n − 1 (using the notation of (i) above), L

′

µ1 ⊗ · · · ⊗ µn . Jψn−1,α (πτ ) ≺ indBn−1,α ′

Here, B is the intersection of which preserves the flag

Q′n+1

(5.41)

with the parabolic subgroup of Ln−1,α ,

SpanK {en } ⊂ SpanK {en , en+1 } ⊂ · · · ⊂ SpanK {en , ..., e2n−1 }.

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(2) Assume that ωτ = 1 and m = 2n is even. Assume, also, that δh(V ),α = 1. Then Jψℓ,α (πτ ) = 0, when ℓ ≥ n + 1, and for ℓ = n (notation of (i) above) L

L

µ1 ⊗ · · · ⊗ µn ⊕ IndBn,α µ1 ⊗ · · · ⊗ µ−1 Jψn,α (πτ ) ≺ IndBn,α ′ ′ n .

(5.42)

′

Here, B is the Borel subgroup of Ln,α , which preserves the flag (see (5.40)) Spank {en+1 } ⊂ Spank {en+1 , en+2 } ⊂ · · · ⊂ Spank {en+1 , ..., e2n−1 , e2n − βe0 −

β2 e−2n }. 2

(3) Assume that ωτ = λ0 and m = 2n is even. Assume, also, that δh(V ),α = 0. Then Jψℓ,α (πτ ) = 0, when ℓ ≥ n + 1, and for ℓ = n, we have (notation of (ii) above), L

′

µ1 ⊗ · · · ⊗ µn−1 ⊗ µ0 , Jψn,α (πτ ) ≺ IndBn,α ′

(5.43)

where B is the Borel subgroup of Ln,α , which preserves the flag SpanK {en+1 } ⊂ SpanK {en+1 , en+2 } ⊂ · · · ⊂ SpanK {en+1 , ..., e2n−1 }. In case, h(V ) is even orthogonal, µ0 should be omitted, and in case h(V ) is even unitary, or odd orthogonal, µ0 is the trivial character (of U1 (k) or of SO2 (k) respectively). (4) Assume that ωτ = λ0 , m = 2n is even, and δh(V ),α = 1. Then when ℓ ≥ n, Jψℓ,α (πτ ) = 0. (5) Assume that m = 2n + 1 is odd. Then Jψℓ,α (πτ ) = 0, for all ℓ ≥ n + 1. Let ℓ = n, and assume that ωτ = 1. Then (in the notation of (iii) above) L

′

µ1 ⊗ · · · ⊗ µn ⊗ µ0 , Jψn,α (πτ ) ≺ indBn,α ′

(5.44)

where B is the Borel subgroup of Ln,α , which preserves the flag SpanK {en+1 } ⊂ SpanK {en+1 , en+2 } ⊂ · · · ⊂ SpanK {en+1 , ..., e2n }. As before, µ0 should be omitted, when h(V ) is even orthogonal, and when h(V ) is even unitary, or odd orthogonal, µ0 is the trivial character (of U1 (k), or SO2 (k), respectively). (6) Assume that m = 2n + 1 and ωτ = λ0 . Then the assertion (5.44) is modified as follows (with the notation of (iv) above). For h(V ) even orthogonal, or odd orthogonal, such that δh(V ),α = 0, (5.44) holds, and for h(V ) odd orthogonal, such that δh(V ),α = 1, (5.44) holds with µ0 , taken as the restriction of λ0 to SO1,1 (k) ∼ = k ∗ . For h(V ) even unitary, (5.44) holds, after omitting µ0 . In this ′ case, B is a proper subgroup of the Borel subgroup (the torus is “missing” the center).

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Proof. Assume, first, that m = 2n and ωτ = 1, and assume, also, in case h(V ) is h(V ) even orthogonal, that n is even. Then πτ is the unramified constituent of IndQm τ ′ , GL

(K)

2n where τ ′ = IndP2,...,2 µ1 (detGL2 ) ⊗ · · · µn (detGL2 ). The reason for this is that πτ is the unramified constituent of the representation of h(V ), induced from the Borel subgroup and its character 1

1

1

1

−1 2 2 2 2 ⊗ · · · µ n | · |K ⊗ µ−1 µ 1 | · |K n | · |K ⊗ · · · ⊗ µ 1 | · | K .

There is a Weyl element of h(V ), which conjugates this character to 1

−1

1

−1

2 2 (µ1 | · |K ⊗ µ 1 | · |K 2 ) ⊗ · · · ⊗ µ n | · | K ⊗ (µn | · |K 2 ).

Here, we use the fact that n is even, when h(V ) is even orthogonal. Denote by τµi , the representation of GL2 (K), induced from the Borel subgroup and the character 1

−1

2 ⊗µi |·|K 2 . This representation admits the character µi (detGL2 ) as a quotient. µi |·|K Inducing by stages, it is clear that πτ is a constituent of the representation of h(V ), which is parabolically induced from τµ1 ⊗ · · · ⊗ τµn , and since πτ is unramified, we get that πτ is the unramified constituent of the representation of h(V ), which is parabolically induced from µ1 (detGL2 ) ⊗ · · · µn (detGL2 ), i.e. the representation h(V ) IndQm τ ′ . Now we can use Theorem 5.5, with m = 2n, i1 = n, i2 = 0. This proves parts (1), (2), except in case h(V ) is even orthogonal and n is odd. Assume, now, that this is the case. Since n− 1 is even, we can repeat the last argument to get that πτ is the unramified constituent of the representation of h(V ), parabolically induced 1

1

2 2 ⊗ µ−1 from µ1 (detGL2 ) ⊗ · · · µn−1 (detGL2 ) ⊗ µn | · |K n | · |K . We conclude that πτ is GL2n−2 (k) h(V ) ′ ′ the unramified constituent of IndQ2n−2 τ ⊗ σ, where τ = IndP2,...,2 µ1 (detGL2 ) ⊗ ∼ · · · µn−1 (detGL2 ), and σ is the representation of h(W2n,2n−2 ) = SO2,2 (k), which is the outer conjugation of the representation induced from the Siegel type parabolic subgroup of SO2,2 (k) and the character µn (detGL2 ). Now, let us use Theorem 5.2(1) with m = 2n and j = 2n − 2. Let ℓ > n. Then the second summand in Theorem ′ 5.2(1) is zero, since τ(ℓ) = 0. In the first summand, we have a summation over ℓ−2 < t ≤ ℓ, i.e. t = ℓ−1, ℓ. Since ℓ−1 > n−1, we conclude that τ ′(t) = 0, and hence Jψℓ,α (πτ ) = 0. Let ℓ = n, then in the summation of Theorem 5.2(1), t = n − 1, n ′ (note also that τ(n) = 0). Since τ ′(n) = 0, we need consider only t = n − 1, and then 2−n

we get | det ·| 2 τ ′(n−1) ⊗Jψ1,α (σ). We have Jψ1,α (σ) = 0, since, in this case, the last Jacquet module measures whether the representation σ of SO2,2 (k) is generic, and σ ′ is, clearly, non-generic. Note, also, that since τ(n) = 0, the second summand is zero, ′ = as well. This shows that Jψn,α (πτ ) = 0, as well. Let, now ℓ = n−1. Again, τ(n−1) 0, and, in the first summand of Theorem 5.2(1), we need consider only t = n−2, n−1. The second summand is zero. The case t = n − 2 yields 0, since Jψ1,α (σ) = 0, and 2−n 2−n L t = n − 1 gives IndQn−1,α | det ·| 2 τ ′(n−1) ⊗ ResSO2,1 (k) σ; | det ·| 2 τ ′(n−1) is the ′ n−1 representation of GLn−1 (k), parabolically induced from µ1 ⊗· · ·⊗µn−1 . Finally, it is easy to see that ResSO2,1 (k) σ has the same semi-simplification as the representation induced from the Borel subgroup and the character µn . This completes the proof of part (1) (part (2) is already proved).

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Assume that ωτ = λ0 and m = 2n. Reasoning as above, we see that πτ is the unramified constituent of the representation of h(V ), parabolically induced from ν

1

2 2 ⊗ | · |K , where τ ′ is the representation of GL2n−2 (K), parabolically τ ′ ⊗ λ0 | · |K induced from µ1 (detGL2 ) ⊗ · · · µn−1 (detGL2 ), and ν = 1, except in case h(V ) is even orthogonal, and n is odd, and then ν = −1. Assume, first, that h(V ) is even orthogonal. Then we use Theorem 5.5, with m = 2n, j = 2n − 2, i1 = n − 1, i2 = 2, and we get what we want. Assume, next, that h(V ) is even unitary, or odd h(V ) orthogonal. Then it is clear that πτ is the unramified constituent of IndQ2n−2 τ ′ ⊗ σ, where σ is the representation of h(W2n,2n−2 ) (U2,2 (k) or SO3,2 (k) respectively) induced from the parabolic subgroup, which preserves an isotropic line and the 1 character λ0 | · | 2 ⊗ 1. Let us use Theorem 5.1(1), with m = 2n, j = 2n − 2. Note ′ that σ is non-generic and τ(ℓ) = 0, for ℓ ≥ n− 1. As before, it is now easy to see that Jψℓ,α (πτ ) = 0, for ℓ ≥ n + 1, and, for ℓ = n, we have, assuming that δh(V ),α = 0, 2−n

L

Jψn,α (πτ ) ≺ IndQn,α | det ·|K2 τ ′(n−1) ⊗ Jψ1,α (σ). Apply Theorem 5.1(1) again to ′ n−1 get that Jψ1,α (σ) ∼ = 1, as representations of h(W2n,2n−1 ∩ yα⊥ ) (i.e. U1 (k), SO2 (k), respectively). This proves part (3). In order to prove part (4), assume that δh(V ),α = 1. Note that in the previous part, we could switch the order of the characters λ0 and 1, and replace σ by the representation of h(W2n,2n−2 ) = SO3,2 (k), induced from the 1 parabolic subgroup, which preserves an isotropic line, and the character | · | 2 ⊗ λ0 , where, now we view λ0 as a character of SO2,1 (k) = P GL2 (k), by composing it with the determinant, modulo squares (recall that λ0 is a quadratic character). SO (k) Thus, we get that Jψn,α (πτ ) is a subquotient of both IndB ′ n,n µ1 ⊗ · · · ⊗ µn−1 ⊗ 1 SO

(k)

and IndB ′ n,n µ1 ⊗ · · · ⊗ µn−1 ⊗ λ0 . If Jψn,α (πτ ) 6= 0, then we conclude that the ± ± ± sets {µ± 1 , ..., µn−1 , 1} and {µ1 , ..., µn−1 , λ0 } are equal, and this is impossible. This proves that Jψn,α (πτ ) = 0, which is part (4). Assume that m = 2n + 1 is odd. Reasoning as above, it is clear that πτ is h(V ) GL2n+1 (K) the unramified constituent of IndQm τ˜, where τ˜ = IndP2,...,2,1 µ1 (detGL2 ) ⊗ · · · ⊗ ν

2 µn (detGL2 ) ⊗ ωτ | · |K . Here, ν = 1, except when h(V ) is even orthogonal and n is odd, where ν = −1. Apply now Theorem 5.5, with m = 2n + 1, i1 = n, i2 = 1. We conclude, that if δh(V ),α = 0, then Jψℓ,α (πτ ) = 0, for ℓ ≥ n + 1, and if δh(V ),α = 1, the theorem guarantees that the last Jacquet module vanishes, for ℓ > n + 1. Let us show that it vanishes, for ℓ = n + 1, as well. In this case, h(V ) we have, as before, that πτ is the unramified constituent of IndQ2n τ ′ ⊗ σ, where

GL

(k)

2n µ1 (detGL2 ) ⊗ · · · µn (detGL2 ), and σ is the unramified constituent of τ ′ = IndP2,...,2 the representation of h(W2n+1,2n ) ∼ = SO2,1 (k), induced from the Borel subgroup and 1 the character ωτ | · | 2 , i.e. σ is the character ωτ , composed with the determinant (modulo squares) of P GL2 (k). Now, we use Theorem 5.5(1), with m = 2n + 1, j = 2n, ℓ = n + 1. The last summand there is zero, since σ is non-generic; the ′ second summand is zero, since τ(n+1) = 0, and the first summand is zero, since it ′(n+1) corresponds just to t = n + 1, and τ = 0. This completes the proof of the first part of part (5). Note that Theorem 5.5 also gives parts (5) and (6) in full, when

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105

h(V ) is even orthogonal, and when h(V ) is even unitary and ωτ = λ0 . Assume that h(V ) is even unitary, and ωτ = 1, or that h(V ) is odd orthogonal. Then we can repeat, in all these cases, the last argument, and realize πτ as the unramified h(V ) constituent of IndQ2n τ ′ ⊗ σ, as before; when h(V ) is even unitary, σ is the trivial character. Let us apply Theorem 5.5(1), again, now with ℓ = n, and we get what we want. This completes the proof of Theorem 5.6.  Let us list, in detail, all the cases treated in the last theorem; µ1 , ..., µn denote n unramified characters of K ∗ , and BG denotes the “standard” Borel subgroup of G. The groups h(V ) that we consider in this list are split orthogonal, or quasi-split even unitary. 1. Let K/k be quadratic and unramified, h(V ) = U4n (k) (quasi-split) GL

τ1 = IndBGL2n

(K)

2n (K)

GL

τ2 = IndBGL2n

(K)

2n (K)

−1 µ1 ⊗ · · · ⊗ µn ⊗ µ−1 n ⊗ · · · ⊗ µ1 , −1 µ1 ⊗ · · · ⊗ µn−1 ⊗ 1 ⊗ λ0 ⊗ µn−1 ⊗ · · · ⊗ µ−1 1 .

Lℓ,α ∼ = U2(2n−ℓ−1)+1 (k) (quasi-split), 0 ≤ ℓ ≤ 2n − 1. We have n ≤ ℓ ≤ 2n − 1.

Jψℓ,α (πτ1 ) = 0, First possible “occurrence”: ℓ = n − 1. U

(k)

µ1 ⊗ · · · ⊗ µn . Jψn−1,α (πτ1 ) ≺ indB2n+1 ′       b∗ ∗    ′ Here, B = 0 1 ∗  ∈ U2n+1 (k) b ∈ BGLn (K) , is almost the Borel subgroup of     0 0 b∗ U2n+1 (k). n + 1 ≤ ℓ ≤ 2n − 1.

Jψℓ,α (πτ2 ) = 0, First possible occurrence: ℓ = n. U

Jψn,α (πτ2 ) ≺ IndB2n−1 U

(k)

2n−1 (k)

µ1 ⊗ · · · ⊗ µn−1 ⊗ 1.

2. h(V ) = SO4n (k) (split), GL

τ1 = IndBGL2n

(k)

2n (k)

GL

τ2 = IndBGL2n

(k)

2n (k)

−1 µ1 ⊗ · · · ⊗ µn ⊗ µ−1 n ⊗ · · · ⊗ µ1 , −1 µ1 ⊗ · · · ⊗ µn−1 ⊗ 1 ⊗ λ0 ⊗ µ−1 n−1 ⊗ · · · ⊗ µ1 .

Lℓ,α ∼ = SO2(2n−ℓ−1)+1 (k) (split), 0 ≤ ℓ ≤ 2n − 1. We have Jψℓ,α (πτ1 ) = 0,

n ≤ ℓ ≤ 2n − 1.

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First possible occurrence: ℓ = n − 1. SO

Jψn−1,α (πτ1 ) ≺ IndBSO2n+1

(k)

2n+1 (k)

µ1 ⊗ · · · ⊗ µn .

We have n + 1 ≤ ℓ ≤ 2n − 1.

Jψℓ,α (πτ2 ) = 0, First possible occurrence: ℓ = n. SO

Jψn−1,α (πτ2 ) ≺ IndBSO2n−1

(k)

2n−1 (k)

µ1 ⊗ · · · ⊗ µn−1 .

3. h(V ) = SO4n+1 (split) GL

τ1 = IndBGL2n

(k)

2n (k)

GL

τ2 = IndBGL2n

(k)

2n (k)

−1 µ1 ⊗ · · · ⊗ µn ⊗ µ−1 n ⊗ · · · ⊗ µ1 , −1 µ1 ⊗ · · · ⊗ µn−1 ⊗ 1 ⊗ λ0 ⊗ µ−1 n−1 ⊗ · · · ⊗ µ1 .

For α not a square, Lℓ,α ∼ = SO2n−ℓ+1,2n−ℓ−1 (k) (quasi-split with discriminant (−1)ℓ α), 0 ≤ ℓ ≤ 2n. We have n ≤ ℓ ≤ 2n − 1.

Jψℓ,α (πτ1 ) = 0, First possible occurrence: ℓ = n − 1. SO

(k)

Jψn−1,α (πτ1 ) ≺ indB ′ n+2,n µ1 ⊗ · · · ⊗ µn .     b ∗ ∗  Here, B ′ = 0 I2 ∗  ∈ SOn+2,n (k) b ∈ BGLn (k) , is almost the Borel subgroup   0 0 b∗ of SOn+2,n (k). For α a square, Lℓ,α ∼ = SO2(2n−ℓ) (k) (split), 0 ≤ ℓ ≤ 2n. We have, Jψℓ,α (πτ1 ) = 0,

n + 1 ≤ ℓ ≤ 2n − 1,

Jψ2n,e0 (πτ1 ) = 0. First possible occurrence: ℓ = n. SO

Jψn,α (πτ1 ) ≺ IndBSO2n

(k)

2n (k)

SO

µ1 ⊗ · · · ⊗ µn ⊕ IndBSO2n

(k)

2n (k)

µ1 ⊗ · · · ⊗ µ−1 n .

We have Jψℓ,α (πτ2 ) = 0,

n + 1 ≤ ℓ ≤ 2n − 1,

Jψ2n,e0 (πτ2 ) = 0. First possible occurrence, for α not a square class: ℓ = n, and then SO

Jψn,α (πτ2 ) ≺ IndBSOn+1,n−1

(k)

n+1,n−1 (k)

For α a square, Jψn,α (πτ2 ) = 0.

µ1 ⊗ · · · ⊗ µn−1 ⊗ 1.

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It is easy to write the possible unramified parameters of Jψn−1,α (πτ2 ), using Theorem 5.1, but this will not be needed here. 4. h(V ) = U4n+2 (k) (quasi-split); K/k - quadratic, unramified. GL

τ1 = IndBGL2n+1

(K)

2n+1 (K)

GL

τ2 = IndBGL2n+1

(K)

2n+1 (K)

µ1 ⊗ · · · ⊗ µn ⊗ 1 ⊗ µn−1 ⊗ · · · ⊗ µ−1 1 , µ1 ⊗ · · · ⊗ µn ⊗ λ0 ⊗ µn−1 ⊗ · · · ⊗ µ−1 1 .

Lℓ,α ∼ = U2(2n−ℓ)+1 (k) (quasi-split), 0 ≤ ℓ ≤ 2n. We have, for i=1,2, n + 1 ≤ ℓ ≤ 2n.

Jψℓ,α (πτi ) = 0, First possible occurrence: ℓ = n U

Jψn,α (πτ1 ) ≺ IndB2n+1 U

(k)

2n+1 (k)

U

Jψn,α (πτ2 ) ≺ indB2n+1 ′

µ1 ⊗ · · · ⊗ µn ⊗ 1.

(k)

µ1 ⊗ · · · ⊗ µn .

B ′ is as in No. 1. 5. h(V ) = SO4n+2 (k) (split), GL

τ1 = IndBGL2n+1

(k)

2n+1 (k)

GL

τ2 = IndBGL2n+1

(k)

2n+1 (k)

−1 µ1 ⊗ · · · ⊗ µn ⊗ 1 ⊗ µ−1 n ⊗ · · · ⊗ µ1 , −1 µ1 ⊗ · · · ⊗ µn ⊗ λ0 ⊗ µ−1 n ⊗ · · · ⊗ µ1 .

Lℓ,α ∼ = SO2(2n−ℓ)+1 (k) (split), 0 ≤ ℓ ≤ 2n. We have, for i=1,2, Jψℓ,α (πτi ) = 0,

n + 1 ≤ ℓ ≤ 2n.

First possible occurrence: ℓ = n SO

Jψn,α (πτi ) ≺ IndBSO2n+1

(k)

2n+1 (k)

µ1 ⊗ · · · ⊗ µn .

6. h(V ) = SO4n+3 (k) (split), GL

τ1 = IndBGL2n+1

(k)

2n+1 (k)

GL

τ2 = IndBGL2n+1

(k)

2n+1 (k)

µ1 ⊗ · · · ⊗ µn ⊗ 1 ⊗ µn−1 ⊗ · · · ⊗ µ−1 1 , −1 µ1 ⊗ · · · ⊗ µn ⊗ λ0 ⊗ µ−1 n ⊗ · · · ⊗ µ1 .

For α not a square, Lℓ,α ∼ = SO2n−ℓ+2,2n−ℓ (k) (quasi-split with discriminant (−1)ℓ α), 0 ≤ ℓ ≤ 2n + 1. We have, for i = 1, 2, Jψℓ,α (πτi ) = 0,

n + 1 ≤ ℓ ≤ 2n.

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First possible occurrence: ℓ = n SO

Jψn,α (πτi ) ≺ IndBSOn+2,n

(k)

n+2,n (k)

µ1 ⊗ · · · ⊗ µn ⊗ 1.

For α a square class, Lℓ,α ∼ = SO2(2n−ℓ+1) (k) (split), 0 ≤ ℓ ≤ 2n + 1. We have, for i = 1, 2, Jψℓ,α (πτi ) = 0,

n + 1 ≤ ℓ ≤ 2n,

Jψ2n+1,e0 (πτi ) = 0. First possible occurrence: ℓ = n SO

Jψn,α (πτ1 ) ≺ IndBSO2n+2

(k)

2n+2 (k)

SO

Jψn,α (πτ2 ) ≺ IndBSO2n+2

(k)

2n+2 (k)

5.2

µ1 ⊗ · · · ⊗ µn ⊗ 1, µ1 ⊗ · · · ⊗ µn ⊗ λ0 .

The case K = k ⊕ k

In this section we carry out the work, analogous to that of the previous section for the split version of the unitary group, that is, for the general linear group GLm′ (k). We want to compute the Jacquet module Jψℓ,w0 ,v of a parabolically induced representation, from a maximal parabolic subgroup of GLm′ (k). By the end of Sec. 3.7, such a Jacquet module is isomorphic, in a natural sense, to the Jacquet module with respect to the character (ψ˜v )ℓ in (3.58). We will analyze such ˜ ℓ . See (3.59). As in the previous Jacquet modules as modules over the group L section we will drop the subscript v. We will work in a slightly more general setup, as in Sec. 4.1(2). GL (k) Consider π = IndPj,NN−j τ1 ⊗ τ2 (0 < j < N ), where τ1 , τ2 are smooth representations of GLj (k), GLN −j (k), respectively. Let ℓ˜ = (ℓ1 , ℓ2 , ℓ3 ) be a partition of N , where ℓ2 ≥ 1, and ℓ1 , ℓ3 ≥ 0. Let Pℓ˜ be the corresponding standard parabolic subgroup of GLN (k). Its unipotent radical is     z1 y1 x   Nℓ˜ = v =  Iℓ2 y2  ∈ GLN (k) z1 ∈ Zℓ1 (k), z2 ∈ Zℓ3 (k) , (5.45)   z2 and consider the character ψℓ˜ of Nℓ˜, defined analogously to (3.58). For v ∈ Nℓ˜, of the form (5.45), ψℓ˜(v) = ψZℓ1 (z1 )ψZℓ3 (z2 )ψ((y1 )ℓ1 ,1 + (y2 )1,1 ),

(5.46)

where, for z ∈ Zt (k), we denote ψZt (z) = ψ(z1,2 + z2,3 + · · · + zt−1,t ).

(5.47)

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Let Lℓ˜ be the stabilizer of ψℓ˜ inside GLℓ2 (k) (embedded in GLN (k) as   Iℓ1  ). Then GLℓ2 (k) Iℓ3

   1 Lℓ˜ = ∈ GLℓ2 (k) ∼ = GLℓ2 −1 (k). g

Put

Rℓ˜ = Nℓ˜Lℓ˜. We will analyze Jψℓ˜(π) as an Lℓ˜-module. Since Jψℓ˜(π) = Jψℓ˜(ResPℓ˜π), we first look at ResPℓ˜π. It has a finite Pℓ˜filtration, whose subquotients correspond to the representatives w = wr,s of Pj,N −j \GLN (k)/Pℓ˜, found in (4.17). The Pℓ˜-subquotient, which corresponds to w = wr,s is Pℓ˜

ρw = ind

(w) ℓ,j

P˜

(w)

πℓ,j ˜ (τ1 , τ2 ),

(5.48)

(w)

(w)

= Pℓ˜ ∩ w−1 Pj,N −j w, which takes where πℓ,j ˜ ˜ (τ1 , τ2 ) is the representation of Pℓ,j (w)

g ∈ Pℓ˜

1

to δ(g) 2 (τ1 ⊗ τ2 )(wgw−1 ); δ(g) = δPℓ˜(g)δ −1(w) δPj,N −j (wgw−1 ). Since P˜

ℓ,j

Jψℓ˜(ρw ) = Jψℓ˜(ResRℓ˜ρw ), we need to analyze ResRℓ˜ρw . By Bruhat theory, we have (w)

to consider Pℓ,j ˜ \Pℓ˜/Rℓ˜. Clearly, the representatives can be chosen to be of the form   ǫ1 η =  γ , (5.49) ǫ3 where ǫ1 , ǫ3 are in the Weyl groups of GLℓ1 (k), GLℓ3 (k), respectively, and γ is a representative of Ps−r,ℓ2 −(s−r) \GLℓ2 (k)/Lℓ˜. By Proposition 4.5, if 0 < s − r < ℓ2 , then Ps−r,ℓ2 −(s−r) \GLℓ2 (k)/Lℓ˜ has three elements. We get that ResRℓ˜ρw has a finite Rℓ˜-filtration, whose subquotients are Rℓ˜

(w,η)

ρw,η = ind

(w) η ℓ,j

Rℓ˜∩η −1 P ˜ (w)

(w,η)

πℓ,j ˜

(τ1 , τ2 ),

(w)

−1 ), up to an appro(τ1 , τ2 ) takes h ∈ Rℓ˜ ∩ η −1 Pℓ,j and πℓ,j ˜ (τ1 , τ2 )(ηhη ˜ η to πℓ,j ˜ priate modular function. Let w = wr,s be as in (4.17). As in Proposition 5.1, we (w) look for subgroups J ⊂ Nℓ˜ ∩ η −1 Pℓ,j ˜ η, which satisfy the analog of (5.9), namely (w,η) ψℓ˜ 6= 1 and πℓ,j (τ , τ ) (5.50) 1 2 = 1. ˜ J

J

Let J be a simple root subgroup inside Nℓ˜, which lies in the Levi part of Pℓ˜. Denote its natural projections to GLℓ1 (k) and GLℓ3 (k) by J1 , J3 respectively. If (recall the form (5.49) of η) one of the following holds    Ir x −1 ǫ1 J1 ǫ1 ⊂ Zr,ℓ1 −r = ∈ GLℓ1 (k) 0 Iℓ1 −r    Ij−s t ǫ3 J3 ǫ−1 ⊂ Z = (k) , ∈ GL ℓ3 j−s,ℓ3 −(j−s) 3 0 Iℓ3 −r

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then J ⊂ Nℓ˜ ∩ η −1 Pℓ,j ˜ η satisfies (5.50), and we conclude, as in Proposition 5.1, that Jψℓ˜(ρw,η ) = 0. By Lemma 5.1, we may now assume that     Ij−s Ir , ǫ3 = . (5.51) ǫ1 = Iℓ3 −(j−s) Iℓ1 −r Proposition 5.6. Assume that r > 0 and j − s < ℓ3 . Then Jψℓ˜(ρw ) = 0. Proof. We have to show that Jψℓ˜(ρw,η ) = 0, for all η of the form (5.49), with ǫ1 , ǫ3 of the form (5.51), and γ being one of the representatives from Proposition 4.5. Consider the subgroup J ⊂ Nℓ˜, which consists of elements of the form   Iℓ1 −r
  0 Ir 0 y2 γ       v2   −1 v= (5.52) Iℓ2 γ ,   0     Iℓ3 −j+s Ij−s where y2 ∈ Mr×(ℓ2 −s+r) (k), v2 ∈ M(s−r)×(ℓ3 −j+s) (k). It is clear that J ⊂ Nℓ˜ ∩ (w,η) −1 (w) (τ1 , τ2 ) = 1. We have, for v ∈ J, of the form (5.52) η Pℓ,j ˜ ˜ η and πℓ,j J   −1 v2 ]1,1 )). (5.53) ψℓ˜(v) = ψ(([(0, y2 )γ]r,1 + [γ 0 Let us examine each possible representative γ separately. Let γ = Iℓ2 . Here, (5.53) gives ψℓ˜(v) = ψ(([(0, y2 )]r,1 + (v2 )1,1 )).

(5.54)

Thus, (5.54) is not (identically) trivial when either s = r > 0, ors > r and j−s  < ℓ3 . Is−r Assume that (see the proof of Proposition 4.5) γ = hs−r = (in Iℓ2 −(s−r) particular, 0 < s − r < ℓ2 ). Here, (5.53) gives ψℓ˜(v) = ψ(((y2 )r,1 ).

(5.55)

This is not (identically) trivial when r > 0. Note that y2 has ℓ2 − s + r > 0 columns.   1 0 1 Finally, assume that γ = hs−r u0 = hs−r  Iℓ2 −2 0 (in particular, 0 < s−r < ℓ2 ). 1 Here, (5.53) gives ψℓ˜(v) = ψ(((y2 )r,1 − (v2 )s−r,1 )).

(5.56)

This is not (identically) trivial when either r > 0, or j − s < ℓ3 . We therefore see, that in all three cases (5.54)–(5.56), ψℓ˜ 6= 1, as long as r > 0 and j − s < ℓ3 . This J proves the proposition. 

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It remains to analyze Jψℓ˜(ρw ), when r = 0, or j − s = ℓ3 . Proposition 5.7. Let w correspond to r = 0 and j − s < ℓ3 (recall that s ≤ ℓ2 , j). Then when s = ℓ2 , Jψℓ˜(ρw ) = 0, and, for s < ℓ2 GLℓ −1 (k) Jψℓ˜(ρw ) ∼ = IndPs,ℓ 2−s−1 | det ·|

1−(j−s)+ℓ3 −ℓ1 2

2

(j−s)

τ1

⊗ | det ·|

j−s 2

Jψ(ℓ1 ,ℓ2 −s,ℓ3 −j+s) (τ2 ).

Proof. As in the last proposition, we have to examine Jψℓ˜(ρw,η ), for all η of the form (5.49), with ǫ1 = Iℓ1 , ǫ3 of the form (5.51), and γ being one of the representatives from Proposition 4.5. Using the subgroup (5.52) and (5.54)–(5.56), we get that Jψℓ˜(ρw,η ) = 0, when s > 0 and γ = Iℓ2 , and when 0 < s < ℓ2 and γ = hs u0 . Thus, for j − ℓ3 < s < ℓ2 and 0 < s ≤ j, we have that Jψℓ˜(ρw ) ∼ = Jψℓ˜(ρw,η ), (w)

where η is as above, with γ = hs . In this case, by (4.24), Rℓ˜ ∩ η −1 Pℓ,j ˜ η consists of elements of the form
  a2 0 y 3 hs z3 0 ℓ1 hs     b1 u v2 v1  s    h−1 s g= (5.57) v3 0  ℓ2 −s , 0 b2    c2 0  ℓ3 −j+s j−s 0 c1 where a2 ∈ Zℓ1 (k), c1 ∈ Zj−s (k), c2 ∈ Zℓ3 −j+s (k),     b1 0 u ′ b1 u =  1 0, 0 b2 b′2

(5.58)

Note that, for v ∈ Nℓ˜, of the form (5.57), we have   a2 ψℓ˜(v) = ψZℓ1 +ℓ3  c1  ψ((y3 )ℓ1 ,1 + (v3 )1,1 )). c2

(5.60)

where b1 ∈ GLs (k), b′2 ∈ GLℓ2 −s−1 (k). From (4.25), we have, for g of the form (5.57), satisfying (5.58),     a2 y 3 z 3 ℓ3 +1−(j−s) b v (w,η) 1 1 2 ⊗ τ2  b2 v3  . (5.59) τ1 (τ1 , τ2 )(g) = | det b1 | πℓ,j ˜ c1 c2

Now, as in Lemma 5.3, we show that GLℓ −1 (k) Jψℓ˜(ρw,η ) ∼ = IndPs,ℓ 2−s−1 | det ·| 2

1−(j−s)+ℓ3 −ℓ1 2

(j−s)

τ1

⊗ | det ·|

j−s 2

Jψ(ℓ1 ,ℓ2 −s,ℓ3 −j+s) (τ2 ). (5.61)

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R˜ Indeed, ρw,η ∼ = ind ℓ

T (f )(x) =

Z

(w,η)

(w) η ℓ,j

Rℓ˜∩η −1 P ˜

(w) η\Nℓ˜ ℓ,j

Nℓ˜∩η −1 P ˜

πℓ,j ˜ 

(τ1 , τ2 ), and consider, as in (5.17), the map

 −1 ′ JZj−s ,ψj−s ⊗ Jψ(ℓ1 ,ℓ2 −s,ℓ3 −j+s) (f (nx))ψℓ˜ (n)dn.

(5.62)    I x s ′ ′ Here, Zj−s = ∈ GLj (k) : z ∈ Zj−s (k) , and JZj−s ,ψj−s is the Jacquet z ′ functor along Zj−s , with respect to the character   I x ψj−s s = ψ(z1,2 + z2,3 + · · · + zj−s−1,j−s ). z This Jacquet functor defines the Bernstein-Zelevinski derivative of order j − s of τ1 , (j−s) τ1 . By (5.57)-(5.60), the integral (5.62) is well defined, and now, as in Lemma 5.3, we get the isomorphism (5.61). Assume that s = 0. Then we need to consider η as above, just with γ = Iℓ2 . In (w) this case, by (4.24), Rℓ˜ ∩ η −1 Pℓ,j ˜ η consists of the elements of the form 

 a2 0 y 3 z 3  b2 v3 0   g=  c2 0  c1

ℓ1 ℓ2 ℓ3 −j

,

(5.63)

j

where a2 ∈ Zℓ1 (k), c1 ∈ Zj (k), c2 ∈ Zℓ2 −j (k), and   1 , b′2 ∈ GLℓ2 −1 (k). b2 = b′2

(5.64)

From (4.25), we have, for g of the form (5.63), satisfying (5.64),   a2 y 3 z 3 ℓ1 (w,η) (τ1 , τ2 )(g) = | det b2 |− 2 τ1 (c1 ) ⊗ τ2  b2 v3  . πℓ,j ˜ c2

For v ∈ Nℓ˜ of the form (5.63), we have that ψℓ˜(v) is given exactly by the r.h.s. of (5.60), and now, exactly as before, we get the isomorphism, analogous to (5.61), with s = 0, i.e. j (j) Jψℓ˜(ρw,η ) ∼ = τ1 ⊗ | det ·| 2 Jψ(ℓ1 ,ℓ2 ,ℓ3 −j) (τ2 ).

(j)

Here, τ1 is the Jacquet module of τ1 , with respect to the Whittaker character (corresponding to ψ), and if it is non-zero, then we think of it as a representation of the trivial group. This completes the proof of the proposition.  Let us examine, now, the case where r > 0 and j − s = ℓ3 . Let η be of the form (5.49), with ǫ3 = Iℓ3 , ǫ1 of the form (5.51), and γ- one of the representatives from Proposition 4.5. Using the subgroup (5.52) and (5.54)-(5.56), we get that Jψℓ˜(ρw,η ) = 0, when s = r and γ = Iℓ2 , and when 0 < s − r < ℓ2 and γ =

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hs−r , hs−r u0 . Thus, for 0 < r < s = j − ℓ3 , s − r ≤ ℓ2 , and r ≤ ℓ1 , we have that Jψℓ˜(ρw ) ∼ = Jψℓ˜(ρw,η ), where η is as above, with γ = Iℓ2 . In this case, by (4.24), (w)

Rℓ˜ ∩ η −1 Pℓ,j ˜ η consists of the elements of the form   a2 0 0 y 3 0 ℓ1 −r  a1 y 1 y 2 z 1  r   g= b1 u v1    s−r ,  0 b2 0  ℓ2 −s+r c1 j−s

(5.65)

where a1 ∈ Zr (k), a2 ∈ Zℓ1 −r (k), c1 ∈ Zj−s (k), and     1 b1 u =  b′1 u′  , b′1 ∈ GLs−r−1 (k), b2 ∈ GLℓ2 −s+r (k). b2 b2

(5.66)

From (4.25), we have, for g of the form (5.65), satisfying (5.66),     a1 y 1 z 1 r+1 a2 y 3 (w,η) − 2   . τ1 (τ1 , τ2 )(g) = | det b2 | πℓ,j b1 v1 ⊗ τ2 ˜ b2 c1

(5.67)

For v ∈ Nℓ˜ of the form (5.65), we have that   a1 ψℓ˜(v) = ψZℓ1 +ℓ3  a2  ψ((y1 )r,1 + (v1 )1,1 ). c1

(5.68)

For a representation τ of GLn (k), denote by τ ∗ , the representation of GLn (k), acting in the space of τ , by τ ∗ (g) = τ (g ∗ ), where g ∗ = wℓ′ 1 −r t g −1 wℓ′ 1 −r −1 , and here wℓ′ 1 −r = diag(wℓ1 −r , wℓ2 −s+r ). Then, it is clear that the Jacquet module of τ2 , with   a2 y 3 , a2 ∈ Zℓ1 −r (k), and respect to the subgroup, which consists of n = Iℓ2 −s+r the character, which takes n to ψZℓ1 −r (a2 ) is isomorphic, as a GLℓ2 −s+r (k)-module, to [ℓ −r]

[(τ2∗ )(ℓ1 −r) ]∗ := τ2 1

,

(5.69)

where the outer * denotes composition with the map g 7→ wℓ2 −s+r ·t g −1 wℓ−1 on 2 −s+r GLℓ2 −s+r (k). Now, we repeat the arguments of Proposition 5.7, to conclude from (5.65)–(5.68) GLℓ −1 (k) Jψℓ˜(ρw,η ) ∼ = IndPj−ℓ2 −r−1,ℓ 3

2 +ℓ3 −j+r

This proves

| det ·|−

ℓ1 −r 2

Jψ(r,j−ℓ3 −r,ℓ3 ) (τ1 ) ⊗ | det |

ℓ3 −r−1 2

[ℓ −r]

τ2 1

.

Proposition 5.8. Let w correspond to r > 0 and j − s = ℓ3 . Then , for s = r, Jψℓ˜(ρw ) = 0,

and for s > r, GLℓ −1 (k) Jψℓ˜(ρw ) ∼ = IndPj−ℓ2 −r−1,ℓ 3

2 +ℓ3 −j+r

| det ·|−

ℓ1 −r 2

Jψ(r,j−ℓ3 −r,ℓ3 ) (τ1 ) ⊗ | det |

ℓ3 −r−1 2

[ℓ −r]

τ2 1

.

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

Finally, assume that r = 0 and ℓ3 = j − s. Thus, η =  −1

Iℓ1 γ Iℓ3

(w) Pℓ,j ˜ η



, and γ =

consists of the elements Iℓ2 , hs , hs u0 . Here, the subgroup Rℓ˜ ∩ η
  a2 0 y 3 γ 0 ℓ1  γ     s=j−ℓ b u v 3 1 1  g= , γ −1  0 b2 0  ℓ2 −s=ℓ2 +ℓ3 −j j−s=ℓ3 c1 where a2 ∈ Zℓ1 (k), c1 ∈ Zℓ3 (k), and  γ   b1 u 1 ∈ . 0 b2 GLℓ2 −1 (k)

(5.70)

(5.71)

(w,η)

(τ1 , τ2 ), on g, as above, is     a2 y 3 b1 v1 (w,η) , ⊗ τ2 (τ1 , τ2 )(g) = δγ (b1 , b2 )τ1 πℓ,j ˜ b2 c1

The action of πℓ,j ˜

where 1

δIℓ2 (b1 , b2 ) = | det b2 |− 2 , 1 δhs (b1 , b2 ) = | det b1 |− 2 , δhs u0 (b1 , b2 ) = 1. (w)

The restriction of ψℓ˜ to Nℓ˜ ∩ η −1 Pℓ,j ˜ η is given, in terms of (5.70), by     v a2 ψ([(0, y3 )γ]ℓ1 ,1 + [γ −1 1 ]1,1 ). ψℓ˜(v) = ψZℓ1 +ℓ3 0 c1

(5.72)

Let us write, in the detail, the second factor of this restriction, as well as (5.71), and then we can compute Jψℓ˜(ρw,η ), as in the last two propositions. Assume that γ = Iℓ2 . Then the second factor of (5.72) is ψ((v1 )1,1 ), ψ((y3 )ℓ1 ,1 ),

if ℓ3 < j if ℓ3 = j.

In (5.71), we have     1 b1 u =  b′1 u′  , 0 b2 b2

where b′1 ∈ GLj−ℓ3 −1 (k), b2 ∈ GLℓ2 +ℓ3 −j (k). Then we conclude ( GLℓ −1 (k) ℓ1 ℓ3 −1 [ℓ ] IndPj−ℓ2 −1,ℓ +ℓ −j | det ·|− 2 (τ1 )(ℓ3 ) ⊗ | det | 2 τ2 1 , 3 2 3 Jψℓ˜(ρw,η ) ∼ = ℓ3 (τ1 )(j) ⊗ | det ·|− 2 (τ2 )[ℓ1 ] ,

if if

ℓ3 < j

. ℓ3 = j (5.73)

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115

Here, (τ2 )[ℓ1 ] = [(τ2∗ )(ℓ1 ) ]∗

(5.74)

as a module over the mirabolic subgroup of GLN −j−ℓ1 (k). Note that (τ2 )[ℓ1 ] ∼ = J(ℓ−1−r,ℓ2 −s+r,0) (τ2 ) (and, similarly, (τ1 )(ℓ3 ) = J(0,ℓ2 ,ℓ3 ) (τ1 )). Assume that 0 < j − ℓ3 < ℓ2 and γ = hs . Then the second factor of (5.72) is ψ((y3 )ℓ1 ,1 ), and, in (5.71), we have b1 u 0 b2

!

  b1 0 u ′   =  1 0, b′2

where b1 ∈ GLj−ℓ3 (k), b′2 ∈ GLℓ2 +ℓ3 −j−1 (k). Then we get GL

(k)

ℓ −1 Jψℓ˜(ρw,η ) ∼ = IndPj−ℓ2 ,ℓ +ℓ 3

2

3 −j−1

| det ·|

1−ℓ1 2

ℓ3

(τ1 )(ℓ3 ) ⊗ | det | 2 τ2[ℓ1 ] .

(5.75)

Assume that 0 < j − ℓ3 < ℓ2 and γ = hs u0 . Then the second factor of (5.72) is ψ((y3 )ℓ1 ,1 − (v1 )s,1 ), and, in (5.71), we have b1 u 0 b2

!

 ′  b1 x 0 u ′    1 0 u” = , 1 u”  b′2

where b′1 ∈ GLj−ℓ3 −1 (k), b′2 ∈ GLℓ2 +ℓ3 −j−1 (k). Let

′ Pj−ℓ 3 −1,1,ℓ3 +ℓ3 −j−1

   ′ ′    b1 x u    ′ ′ =  1 u” ∈ GLℓ2 −1 (k) : b1 ∈ GLj−ℓ3 −1 (k), b2 ∈ GLℓ2 +ℓ3 −j−1 (k) .     b′2

This “almost” a parabolic subgroup. We get GLℓ −1 (k) Jψℓ˜(ρw,η ) ∼ = indP ′ 2

j−ℓ3 −1,1,ℓ2 +ℓ3 −j−1

ℓ1

ℓ3

| det ·|− 2 (τ1 )(ℓ3 ) ⊗ | det | 2 τ2[ℓ1 ] .

(5.76)

Note that this is not parabolic induction, in the sense that the unipotent radical of ′ Pj−ℓ does not act trivially. Let us summarize Propositions 5.6–5.8 3 −1,1,ℓ3 +ℓ3 −j−1 and (5.73)–(5.76) in the form of the following Leibniz rule.

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Theorem 5.7. Let ℓ˜ = (ℓ1 , ℓ2 , ℓ3 ) be a partition of N (ℓ2 ≥ 1). Then, up to semi-simplification, GL (k)

Jψ ˜(IndPj,NN−j τ1 ⊗ τ2 ) ≡ Lℓ 1−(j−s)+ℓ3 −ℓ1 j−s GL −1 (k) (j−s) 2 τ1 ⊗ | det ·| 2 Jψ(ℓ1 ,ℓ2 −s,ℓ3 −j+s) (τ2 ) IndPs,ℓℓ2−s−1 | det ·| j−ℓ3 <s<ℓ2

L

2

0≤s≤j

GL

0
(k)

ℓ2 −1 IndPj−ℓ −r−1,ℓ 3

2 +ℓ3 −j+r

| det ·|−

ℓ1 −r 2

Jψ(r,j−ℓ3 −r,ℓ3 ) (τ1 ) ⊗ | det |

ℓ3 −r−1 2

[ℓ −r]

τ2 1

j−ℓ2 −ℓ3 ≤r≤ℓ1

 ℓ3 −1 [ℓ ] ℓ1 GLℓ2 −1 (k)  IndPj−ℓ | det ·|− 2 (τ1 )(ℓ3 ) ⊗ | det | 2 τ2 1 ,   −1,ℓ2 +ℓ3 −j 3 L ℓ3 (j) τ1 ⊗ | det ·|− 2 τ2[ℓ1 ] ,   0, ( GLℓ −1 (k) 2 L IndPj−ℓ ,ℓ +ℓ 3

2

| det ·|

3 −j−1

0,

1−ℓ1 2

0 < j − ℓ3 ≤ ℓ2

if if

ℓ3 = j

otherwise

ℓ3

(τ1 )(ℓ3 ) ⊗ | det | 2 τ2[ℓ1 ] ,

if

0 < j − ℓ3 < ℓ2

otherwise

 ℓ3 ℓ1 GLℓ2 −1 (k) | det ·|− 2 (τ1 )(ℓ3 ) ⊗ | det | 2 τ2[ℓ1 ] , L indPj−ℓ ′ −1,1,ℓ +ℓ −j−1 3 2 3 0,

if

0 < j − ℓ3 < ℓ2

otherwise

Note that, for a smooth representation π of GLn (k), π[ℓ] can be considered as Jψ(ℓ,n−ℓ,0) (π). We will apply the last theorem, with ℓ˜ = (ℓ, 2(m − ℓ), ℓ), to representations of GL2m (k) of the following form. Let τ be an irreducible, unramified, generic repre1 GL2m (k) τ | det ·| 2 ⊗ sentation of GLm (k). Let πτ be the unramified constituent of IndPm,m 1

τ | det ·|− 2 . Write τ , as the representation induced from the standard Borel subgroup and the character µ1 ⊗ · · · ⊗ µm , where the µi are unramified characters of k ∗ . Then πτ is the unramified constituent of GL

IndBGL2m

1

(k)

2m (k)

1

1

1

(µ1 | · | 2 ⊗ µ1 | · |− 2 ) ⊗ · · · ⊗ (µm | · | 2 ⊗ µm | · |− 2 ).

Since the representation of GL2 (k), induced from the Borel subgroup, from the 1 1 character µi | · | 2 ⊗ µi | · |− 2 , admits the character µi (detGL2 ) as a quotient, we see that πτ is the unramified constituent of GL

(k)

2m µ1 (2) ⊗ · · · ⊗ µm (2), π(τ ) = IndP2,...,2

where µi (2) = µi (detGL2 ). We will use Theorem 5.7 to compute, for ℓ˜ above, Jψℓ˜(π(τ )), by pulling out, one character µj (detGL2 ), at a time. We will re-denote now Jψℓ,2(m−ℓ),ℓ (π(τ )) = Jψℓ (π(τ )). Theorem 5.8. We have, for all

m−1 2

< ℓ < m,

Jψℓ (π(τ )) = 0.

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Proof.

≤

Denote, for 1

GL2(m−i) (k) IndP2,...,2 µi+1 (detGL2 )

i

≤

m, τ1,i

=

117

µi (detGL2 ), and τ2,i

⊗ · · · µm (detGL2 ). Let us apply Theorem 5.7 τ1 = τ1,1 , τ2 = τ2,1 , N = 2m, j = 2, ℓ1 = ℓ3 = ℓ, and ℓ2 = 2(m − ℓ). that in Theorem 5.7, the second sum is empty, and the third and the fifth mands are zero, since (τ1,1 )(t) = 0, for 1 ≤ t ≤ 2. The fourth summand is

=

with Note sumzero,

(2)

unless ℓ = 1. In the first summand, we may take only s = 1, 2, since τ1,1 = 0. Thus, assuming that ℓ > 1, we get Jψℓ (π(τ )) ≡ L

GL

2−ℓ<s<2(m−ℓ)

(k)

2(m−ℓ)−1 IndPs,2(m−ℓ)−s−1 | det ·|

−1+s 2

(2−s)

τ1,1

⊗ | det ·|

2−s 2

Jψ(ℓ,2(m−ℓ)−s,ℓ−2+s) (τ2,1 ).

s=1,2

(5.77) Let ℓ1 = ℓ, ℓ2 = 2(m − ℓ) − s, ℓ3 = ℓ − 2 + s, and apply Theorem 5.7, with τ1 = τ1,2 and τ2 = τ2,2 , in order to compute the term Jψ(ℓ,2(m−ℓ)−s,ℓ−2+s) (τ2,1 ), in (5.76). Assume that ℓ3 , ℓ2 > 1. Then, we get, as in the first step, Jψ(ℓ,2(m−ℓ)−s,ℓ−2+s) (τ2,1 ) ≡ L

GL

2−ℓ3 <s<ℓ2

(k)

−1 IndPs,ℓℓ2−s−1 | det ·|

−1+s+ℓ3 −ℓ 2

2

(2−s)

τ1,2

⊗ | det ·|

2−s 2

Jψ(ℓ,ℓ2 −s,ℓ3 −2+s) (τ2,2 ).

s=1,2

(5.78) Apply Theorem 5.7 again, where we substitute ℓ instead of ℓ1 , ℓ2 − s instead of ℓ2 , ℓ3 − 2 + s (from (5.78)) instead of ℓ3 , τ1 = τ1,3 and τ2 = τ2,3 , and we continue, in this way, as long as the current ℓ3 , ℓ2 are larger than 1. (Note that if ℓ3 > 1 and ℓ2 = 2, we get zero.) We get L Jψℓ (π(τ )) ≡ s1 ,...,si =1,2

ℓ−2i+s1 +···+si =1 GL2(m−ℓ)−1 (k) (2−s ) IndPs ,...,s ,ℓ −1 | det ·|z1 τ1,1 1 1 i 2

(2−si )

⊗ | det ·|zi+1 Jψ(ℓ,ℓ2 ,1) (τ2,i ). (5.79) Here, the exponents zt depend on i, s1 , ..., si , and ℓ2 = 2(m−ℓ)−(s1 +· · ·+si ). Now consider Jψ(ℓ,ℓ2 ,1) (τ2,i ), in (5.79). Again, we use Theorem 5.7. Note that except the first and the fourth terms, the other terms are zero, and that in the first term, only the summand corresponding to s = 2 may have a nonzero contribution. We get, if ℓ2 > 2, ⊗ · · · ⊗ | det ·|zi τ1,i

Jψ(ℓ,ℓ2 ,1) (τ2,i ) ≡ GL

−1 IndP2,ℓℓ2−3

(k)

2

L

GL

| det ·|

−1 IndP1,ℓℓ2−2 2

(k)

2−ℓ 2

| det ·|

τ1,i+1 ⊗ Jψ(ℓ,ℓ2 −2,1) (τ2,i+1 ) 1−ℓ 2

1

(τ1,i+1 )(1) ⊗ | det | 2 (τ2,i+1 )[ℓ] .

(5.80)

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If ℓ2 = 2, omit the first term, and if ℓ2 = 1, we get zero. By a recursive application of (5.79), (5.80), we get that the semi-simplification of Jψℓ (π(τ )) is a direct sum of parabolically induced representations (to L2(m−ℓ)−1 (k)) from representations (on Levi subgroups) of the form (2−s1 )

(2−si′ )

(1)

⊗ | det ·|zi′ +1 τ1,i′ +1 ⊗ | det ·|zi′ +2 (τ2,i′ +1 )[ℓ] , (5.81) where i′ ≥ i, i, s1 , ..., si are as in (5.79), and st = 2, for i < t ≤ i′ (in case, i′ > i). Thus, | det ·|z1 τ1,1

⊗ · · · ⊗ | det ·|zi′ τ1,i′

ℓ = 2i − (s1 + · · · + si ) + 1 = 2i′ − (s1 + · · · + si′ ) + 1 ≤ 2i′ − i′ + 1 = i′ + 1. As explained in the proof of Theorem 5.4, in order that (τ2,i′ +1 )[ℓ] be nonzero, we must have that ℓ + 1 ≤ m − (i′ + 1). Thus, ℓ ≤ i′ + 1 ≤ m − ℓ − 1, and hence, ℓ ≤ theorem.

m−1 2 ,

contradicting our assumption. This completes the proof of the 

Finally, let us examine Jψℓ (π(τ )), at the “first occurrence” ℓm = [ m−1 2 ]. Theorem 5.9. For m = 2n + 1 odd, we have Jψn (πτ ) ≺ Jψn (π(τ )) ∼ = τ. Proof.

Let i′ be as in the last proof, with ℓ = n. Then, keeping the same notation, n = 2i′ − (s1 + · · · + si′ ) + 1 n + 1 ≤ 2n + 1 − (i′ + 1).

We conclude that s1 = · · · = si′ = 1 and i′ = n − 1. From (5.77)–(5.80), we get that n (1) GL2n+1 (k) (1) τ1,1 ⊗ · · · ⊗ τ1,n ⊗ | det ·| 2 (τ2,n )[n] Jψn (π(τ )) ∼ = IndP1,...,1,n+1 n GL2n+1 (k) ∼ µ1 ⊗ · · · ⊗ µn ⊗ | det ·| 2 (τ2,n )[n] . = IndP1,...,1,n+1 n n As in the proof of Theorem 5.4, we have | det ·| 2 (τ2,n )[n] ∼ = | det ·| 2 (τ2,n )[n+1] , as n n ∗ )(n+1) )∗ , which is GLn+1 (k)-modules, and then | det ·| 2 (τ2,n )[n+1] ∼ = (| det ·|− 2 (τ2,n isomorphic to

GL

(IndBGLn+1

(k)

n+1 (k)

GLn+1 (k) −1 ∗ ∼ µn+1 ⊗ · · · ⊗ µ−1 µn+1 ⊗ · · · ⊗ µ2n+1 , 2n+1 ) = IndBGL (k) n+1

as GLn+1 (k)-modules. Thus, GL

Jψn (π(τ )) ∼ = IndBGL2n+1

(k)

2n+1 (k)

µ1 ⊗ · · · µ2n+1 ∼ = τ.



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119

Consider now the case m = 2n. Here, ℓm = n − 1. Theorem 5.10. Let m = 2n be even. Then Jψn−1 (πτ ) ≺ L

GL

0≤t
L

IndEt 2n+1 GL

n−1≤t<2n

L

GL

(k)

IndEt 2n+1

indB ′ 2n+1

(k)

1

µ1 ⊗ · · · ⊗ µt ⊗ | det ·| 2 µt+1 (detGL2 ) ⊗ µt+2 ⊗ · · · ⊗ µ2n

(k)

1

µ1 ⊗ · · · ⊗ µt ⊗ | det ·|− 2 µt+1 (detGL2 ) ⊗ µt+2 ⊗ · · · ⊗ µ2n

µ1 ⊗ · · · ⊗ µ2n .

Here, Et is the standard parabolic subgroup of type (1, ..., 1, 2, 1, ..., 1), where 2 is located at the t+1 place. B ′ is the following subgroup of the standard Borel subgroup,     b1 x y  B ′ =  1 z  ∈ BGL2n+1 (k) b1 ∈ BGLn−1 (k) , b2 ∈ BGLn+1 (k) .   b2

The character µ1 ⊗ · · · ⊗ µ2n of B ′ takes an element with diagonal (x1 , ..., xn−1 , 1, xn , ..., x2n ) to µ1 (x1 ) · ... · µ2n (x2n ).

Proof. The proof is similar to the proof of the previous theorem. Using the same notation, we see that n − 1 = 2i′ − (s1 + · · · + si′ ) + 1 n ≤ 2n − (i′ + 1).

We conclude that i′ ≤ s1 + · · · + si′ ≤ i′ + 1, and hence s1 + · · · + si′ = i′ or s1 + · · · + si′ = i′ + 1. In the first case, we get that s1 = · · · = si′ = 1 and i′ = n − 2, and in the second case, i′ = n − 1 and all of the st are equal to 1, except one of them, which is equal to 2. As before, the first case will contribute to the semi-simplification of Jψn−1 (π(τ )) the representation of GL2n+1 (k), parabolically induced from µ1 ⊗ · · · ⊗ µn−1 ⊗ | det ·|

n−1 2

(τ2,n−1 )[n−1] .

For the second case, let st+1 = 2, while all the other sr are equal to 1. This will contribute (see (5.79)) the representation of GL2n+1 (k), parabolically induced from 1

µ1 ⊗ · · · ⊗ | det ·| 2 µt+1 (detGL2 ) ⊗ µt+2 ⊗ · · · ⊗ µn−1 ⊗ n−2 | det ·| 2 Jψ(n−1,n+2,1) (τ2,n−1 ). Apply now (5.80), and note that Jψ(n−1,n,1) (τ2,n ) = 0 (due to the fact that (τ2,r )[n−1] = 0, for r > n). Thus, the last representation is the representation of GL2n+1 (k), parabolically induced from 1

µ1 ⊗ · · · ⊗ | det ·| 2 µt+1 (detGL2 ) ⊗ µt+2 ⊗ · · · ⊗ µn ⊗ | det ·| n−1 2

n−1 2

(τ2,n )[n−1] .

As in the proof of Theorem 5.9, the representation | det ·| (τ2,n )[n−1] is isomorphic, as a representation of GLn (k), to the representation, induced from the Borel

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subgroup and the character µn+1 ⊗ · · · ⊗ µ2n . To end the proof, it remains to n−1 consider | det ·| 2 (τ2,n−1 )[n−1] , as a representation of GLn+2 (k). Note that this 1 representation is originally a representation of the “mirabolic-like” subgroup P1,n+2 of GLn+3 (k). As in the proof of Theorem 5.4, this representation has a finite 1 P1,n+2 -filtration, whose subquotients are compactly induced from (τ2,n−1 )[n−1+e] , n−1 e ≥ 1. When e = 1, we get the representation | det ·| 2 (τ2,n−1 )[n] , which is trivial 1 on the unipotent radical of P1,n+2 , and, as a representation of GLn+2 (k), it is isomorphic, up to semi-simplification, to the sum of the representations, parabolically induced from the tensor product of the characters, µn , ..., µ2n of k ∗ , except that 1 one of them, µt+1 , should be replaced by | det ·|− 2 µt+1 (detGL2 ). Note, that for 1 e > 2, (τ2,n−1 )[n−1+e] = 0. For e = 2, we get the representation of P1,n+2 , com[n+1] pactly induced from (τ2,n−1 ) . Note that the inducing representation acts by 1 a non-trivial character on the unipotent radical of P1,n+2 , and as a representation 1 1 of P1,n+1 , it acts trivially on the unipotent radical of P1,n+1 , and, on GLn+1 (k), it is the representation τ (n), induced from the Borel subgroup, from the character 1 µn ⊗ · · · ⊗ µ2n . Hence the restriction to GLn+2 (k), of this representation of P1,n+2 1 is the representation, compactly induced from the representation τ (n) of P1,n+1 1 (extended trivially to the unipotent radical of P1,n+1 ). This completes the proof of the theorem. 

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Chapter 6

Jacquet modules corresponding to Fourier - Jacobi characters of parabolically induced representations We carry out a similar study, as in the previous chapter, of Jacquet modules corresponding to Fourier-Jacobi characters, when applied to parabolic induction. We will obtain similar Leibniz rules for these Jacquet modules. 6.1

The case where K is a field

We keep notations and assumptions as in Sec. 3.8, and as in the previous chapter, we will drop the subscript v. We assume that K is a field, and that δ = −1, that is (V, b) is symplectic, or anti-Hermitiean of dimension 2m, over K, with Witt index √ m. When K is a quadratic extension of k, we assume that K = k[ ρ], and, as before, we will drop k from the notation, when convenient. The group in question H = H(V ) (now V = Vk ) is either a symplectic group, a metaplectic group, or a quasi-split even unitary group. Recall that ψK,−1 denotes the character ψ ◦ trK/k . Let π be the representation of H, which is parabolically induced from Qj and the (smooth) representation τ ⊗ σ of GL(Vj+ ) × H(Wm,j ). In case H is not metaplectic, Qj denotes, as before, the standard parabolic subgroup of H = h(V ), which preserves the isotropic subspace Vj+ . It has a Levi decomposition Qj = Dj ⋉ Uj , Dj ∼ = GLj (K) × h(Wm,j ), and π is the parabolically induced representation π = IndH Qj τ ⊗ σ.

In case H is metaplectic, we denote by Qj the inverse image, in H, of the parabolic subgroup of h(V ), which preserves Vj+ , and π = IndH Qj (τ ⊗ σ)µψ , where

 a (τ ⊗ σ)µψ  g

a

∗





 , ǫ = (det a, x(g))γψ (det a)τ (a) ⊗ σ(g, ǫ).

(6.1)

f 2(m−j) (k); ( , ) is the Hilbert symbol on k × k, and x(g) g ∈ Sp f is the x-function, which enters in the definition of the Rao cocycle on Sp 2(m−j) (k).

Here, a ∈ GLj (k),

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See Sec. 1.2 and [Rao (1993)]. We will still say that π is parabolically induced from τ ⊗ σ. Recall the unipotent radical Nℓ and its character ψℓ . In this chapter, we assume that 0 ≤ ℓ ≤ m − 1. Recall, also, that we extend ψℓ trivially to Nℓ+1 . We want to compute Jψℓ (π ⊗ ωψ−1 ,γ −1 ). In case H is metaplectic, Nℓ+1 splits in H, and we identify Nℓ+1 with the subgroup {(v, 1) : v ∈ Nℓ+1 }. ∼ The center Cℓ = k of the Heisenberg group HWm,ℓ+1 ∼ = Nℓ \Nℓ+1 , acts through the 0 Weil representation ωψ−1 ,γ −1 , by the character ψ −1 . Put Nℓ+1 = Nℓ Cℓ , and let ψℓ0 0 denote the extension of ψℓ , from Nℓ to Nℓ+1 , by ψ on Cℓ . For this, we identify Cℓ with k, as     Iℓ         1 0 z        z ∈ k . Cℓ = c(z) =  I 0          1      Iℓ

We let ψ(c(z)) = ψ(z). We denote by Jψℓ0 (π), the Jacquet module of π with respect 0 to (Nℓ+1 , ψℓ0 ). We clearly have Jψℓ (π ⊗ ωψ−1 ,γ −1 ) ∼ = JHWm,ℓ+1 /Cℓ (Jψℓ0 (π) ⊗ ωψ−1 ,γ −1 ). Denote ¯ ℓ = H(Wm,ℓ+1 ) ⋉ HW L . m,ℓ+1

¯ ℓ . This will run parallel Our main task will be to compute Jψℓ0 (π), as a module over L to Sec. 5.1. Recall that for ℓ = m − 1, we define H(Wm,ℓ+1 ) = {1}, HWm,ℓ+1 = Cℓ and ωψ−1 ,γ −1 = ψ −1 , so that Jψℓ0 (π) becomes the Jacquet module, with respect to the Whittaker character defined by ψK,−1 , along Nm . Then we know that this Jacquet module is non-trivial if and only if τ and σ are generic (with respect to ψK,−1 ). As in Sec. 5.1, since Jψℓ0 (π) = Jψℓ0 (ResQℓ π), we first consider the restriction of π to Qℓ . This restriction has a finite Qℓ - filtration, whose subquotients are the representations ρw , as in (5.1), where w ∈ Qj \H/Qℓ . We choose representatives w = ǫr,s as in Sec. 4.2. Let ¯ ℓ = H(Wm,ℓ+1 )Nℓ+1 . R Then Jψℓ0 (ρw ) = Jψℓ0 (ResR¯ ℓ ρw ), and as in Sec. 5.1, we analyze ResR¯ ℓ ρw , by Bruhat (w) ¯ ℓ . We choose, for the last space, the repretheory, and so, we consider Qℓ,j \Qℓ /R sentatives η, of the form (5.2), where ǫ is a Weyl element of GLℓ (K), and γ, in (5.2), ¯ ℓ . In the metaplectic case, we replace η by is a representative for Q′w \H(Wm,ℓ )/L (η, 1), and so on. For w = ǫr,s = ar,s (see (4.15)), Q′w is the parabolic subgroup of + H(Wm,ℓ ), which preserves the isotropic subspace (5.3) Vℓ,s−r of Wm,ℓ . The analog of Proposition 4.3 is the following proposition, which is a special case of Proposition 4.1. See the remark at the end of Chapter 4.

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¯ℓ Proposition 6.1. Assume that s − r > 0. If s − r < m − ℓ, then Q′w \H(Wm,ℓ )/L ′ ¯ consists of three elements, and if s − r = m − ℓ, then Qw \H(Wm,ℓ )/Lℓ consists of two elements. ¯ ℓ by Proof. The set of double cosets above does not change, when we replace L the parabolic subgroup of H(Wm,ℓ ), which preserves the line through eℓ+1 . Also, in the metaplectic case, this set is the same one as in the linear (symplectic) case. Thus, we may apply Proposition 4.1 with s − r replacing j and 1 replacing ℓ. By (4.12), we get that these double cosets are parametrized by pairs (r′ , s′ ), such that r′ = 0, 1 and 0 ≤ r ′ ≤ s′ ≤ s − r ≤ m − ℓ s − r ≤ 1 + s′ − r′ ≤ m − ℓ. Let s − r < m − ℓ. Then, when r′ = 0, we get that s′ = s − r − 1, or s′ = s − r, and when r′ = 1, we get that s′ = s − r. This gives the three pairs (1, s − r), (0, s − r), (0, s − r − 1). If s − r = m − ℓ, then 1 + s′ − r′ = m − ℓ = s − r, and we get two pairs: (1, s − r), (0, s − r − 1).  The representatives above can be chosen as in (4.14). We write them in matrix form inside h(Wm,ℓ ), and apply the standard modifications in the metaplectic group case. For the pair (1, s − r), we choose the representative γ1 = Im−2ℓ . For the pair (0, s − r − 1), we choose   0 Is−r−1 0 0 0 0 0 0 0 −1    γ3 = 0 (6.2) 0 I2(m−ℓ−s+r) 0 0 . 1 0 0 0 0 0 0 0 Is−r−1 0 If s − r < m − ℓ, we choose, for the pair (0, s − r),  Is−r 1  γ2 =  I2(m−ℓ−s+r−1)   Let us prove now the analog of Proposition 5.1.



Is−r

  .  1

(6.3)

Proposition 6.2. Let w = ǫr,s = ar,s (in (4.15)). Assume that r > 0. Then Jψℓ0 (ρw ) = 0. Proof. We repeat the proof of Proposition 5.1, word for word, until the end of Lemma 5.1. (Recall that in case H is metaplectic, we identify Nm with the subgroup {(x, 1)|x ∈ Nm }. Note also, that if g ∈ h(V ), x ∈ Nm ⊂ h(V ) are such that

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gxg −1 lies in the Siegel parabolic subgroup Qm of h(V ), then (g, 1)(x, 1)(g, 1)−1 = (gxg −1 , 1), inside H.) Let us consider the subgroup S of Nℓ , defined by (5.11), for ′ γ = γ2 , γ3 (of course, now ωbt = I). In the notation of (5.11), we have e′r (0, y2 , y3 )γ2 = ((y2 )r,1 , ... ) e′r (0, y2 , y3 )γ3 = ((y3 )r,1 , ... ), where e′r = (0, ..., 0, 1) ∈ k r . Recall that here, y2 ∈ Mr×2(m−ℓ−s+r)(K), y3 ∈ Mr×(s−r) (K). The character ψℓ0 , applied to the element (5.11), is ψK,−1 ((y2 )r,1 ), in case γ = γ2 , and ψK,−1 ((y3 )r,1 ), in case γ = γ3 . Thus, the restriction of ψℓ0 to (w) w,η) S is non-trivial, while, by (4.19), (4.20), S ⊂ η −1 Uℓ,j η, and πℓ,j (τ, σ) is trivial on S. We conclude, from (5.9), that Jψℓ0 (ρw,η ) = 0, for η of the form (5.2), with γ = γ2 , γ3 . Note again, that these arguments apply to the metaplectic case, with the obvious modifications. It remains to show that Jψℓ0 (ρw,η ) = 0, for η of the form (5.2), with γ = γ1 = Im−2ℓ . In this case, let S = Cℓ , the center of the Heisenberg group HWm,ℓ+1 . It is clear that the elements of Cℓ commute with η, and, of course, w,η)

ψℓ0 is not trivial on Cℓ , while, from (4.19), (4.20), πℓ,j (τ, σ) is trivial on Cℓ . By (5.9), we get that Jψℓ0 (ρw,η ) = 0. 

It remains to analyze Jψℓ0 (ρw,η ), for w = ǫ0,s , where 0 ≤ s ≤ j, j ≤ ℓ + s ≤ m, and η has the form (5.2), with ǫ of the form (5.10). As in Sec. 5.1, we re-denote ρw,η = ρw,γ,t , where γ = γi , 1 ≤ i ≤ 3, appears in the middle block of η := ηt (t = j − s). Note that the argument in the end of the last proof, shows also that Jψℓ0 (ρw,I,t ) = 0, except in case s = 0, and then we must have ℓ ≥ j. In this case, we see, using (w) (4.19), in case H is linear, that Rℓ ∩ η −1 Qℓ,j η consists of the elements   n1  n2 y 4 z 4    , h= e y4′   ∗  n2  n∗1

where n1 ∈ Zj (K), n2 ∈ Zℓ−j (K), and e ∈ h(Wm,ℓ+1 )HWm,ℓ+1 . The action of (w,η ) πℓ,j t (τ, σ) on such h is by   n2 y 4 z 4 τ (n∗1 ) ⊗ σ  e y4′  . n∗2

In case H is metaplectic, we apply the obvious modifications. As in Lemma 5.3, we ¯ ℓ , Jψ0 (ρw,I,t ) ∼ 0 get, in this case, that, as modules over L (σ). Thus, = τ (j) ⊗ Jψℓ−j ℓ Proposition 6.3. Let w = ǫ0,s , where s ≤ j ≤ ℓ + s ≤ m. (1) If s > 0, then Jψℓ0 (ρw,I,t ) = 0.

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(2) If s = 0 (i.e. t = j), then 0 Jψℓ0 (ρw,I,t ) ∼ (σ). = τ (j) ⊗ Jψℓ−j

Assume next, that 0 < s < m − ℓ, and let w = ǫ0,s , and η of the form (5.2), with ǫ as in (5.10), and γ = γ2 . Let us analyze first the case where H is linear. In this (w) case, the subgroup Rℓ ∩ η −1 Qℓ,j η consists of the elements of the form (5.23), with ′ ωbt = I, ηs,t replaced by our present η, and instead of (5.24), we require that u1 = 0 (w,η) and e ∈ Lℓ+s , where u1 is the first column of u. The action of πℓ,j (τ, σ) on such elements is given by (5.25), except that there y6′ should be replaced by −y6′ . Note that if h of the form (5.23), lies in Nℓ0 , then ψℓ0 (h) = ψZt (n1 )ψZℓ−t (n2 )ψK,−1 ((y4 )ℓ−t,1 )ψ(e1,2(m−ℓ−s) ). As in Lemma 5.3, we consider the map, on ρw,η , given by Z 0 )(f (nx))ψℓ0 (n−1 )dn. T (f )(x) = (JZt′ ,ψt′ ⊗ Jψℓ−t (w)

N 0 ∩η −1 Qℓ,j η\Nℓ0

Here, x ∈ Lℓ , and in Lemma 5.3, we need to replace ψK,1 by ψK,−1 . The map T factors through Jψℓ0 (ρw,η ), and it is straightforward to check that it takes values 1−t

(t) 2 ℓ 0 ⊗ Jψℓ−t (σ). Here, inside indL Ps′ | det ·|K τ

   10 y c z         d u v c′      ′ ′ ′ ′ ′  ∈ h(Wm,ℓ d ∈ GLs (K), e ∈ h(Wm,ℓ+s+1 ) . Ps =  e u y         d∗ 0      1

(6.4)

1−t

0 (σ) acts on an element in Ps′ , of the form The representation | det ·|K2 τ (t) ⊗ Jψℓ−t (6.4), by   1 y z 1−t 0 (σ)  e′ y ′  . | det(d)|K2 τ (t) (d) ⊗ Jψℓ−t 1

As in Lemma 5.3, we get that T defines an isomorphism, over Lℓ , 1−t

(t) 2 ℓ 0 ⊗ Jψℓ−t (σ). Jψℓ0 (ρw,η ) ∼ = indL Ps′ | det ·|K τ

(6.5)

¯ ℓ, In case H is metaplectic (6.5) remains the same, except that Lℓ is replaced by L ′ Ps is defined as the inverse image, in H(Wm,ℓ ), of (6.4), and in (6.5), we should replace τ (t) by µψ τ (t) (see (6.1)). In order to see this, we have to check that (w, 1)(η, 1)g(η, 1)−1 (w, 1)−1 = (wηgη −1 w−1 , 1), (w)

for all g ∈ Rℓ ∩ η −1 Qℓ,j η. Indeed, let us start with (η, 1)g(η, 1)−1 = (ηgη −1 , c(η, g)c(ηg, η −1 )c(η, η −1 )),

(6.6)

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where c denotes the normalized Rao cocycle. See [Rao (1993)]. (See also [Jiang and Soudry (2007)], for a convenient summary). We have (since η ∈ Qm ) c(η, g) = (x(η), x(g)) c(ηg, η −1 ) = (x(η), x(g))(x(η), x(η −1 )) c(η, η −1 ) = (x(η), x(η −1 )). In the last three equalities, (, ) denotes the Hilbert symbol of k. Now it is clear that (η, 1)g(η, 1)−1 = (ηgη −1 , 1). Next, consider (w, 1)(ηgη −1 , 1)(w, 1)−1 = (wηgη −1 w−1 , c(w, g η )c(wg η , w−1 )c(w, w−1 )). Let us write ηgη −1 = g η in the form (5.23). Then we can express g η = ndˆe˜′ = dˆe˜′ n′ , where n, n′ are upper unipotent, dˆ is the block diagonal matrix, whose blocks are Iℓ , d, I2(m−ℓ−s) , d∗ , Iℓ (d ∈ GLs (k)) and e˜′ is the block diagonal matrix, whose blocks are Iℓ+s+1 , e′ , Iℓ+s+1 (e′ ∈ h(Wm,ℓ+s+1 )). Note that dˆ and e˜′ commute. Since n′ is upper unipotent and since dˆ ∈ Qm , we have ˆ = (det d, x(e˜′ ))(det d, x(we˜′ ))c(w, e˜′ ). c(w, g η ) = c(w, e˜′ d)

We have c(wg η , w−1 ) = c(wg η w−1 w, w−1 ), and for g as above, we see, from (4.20), that we can write wg η w−1 = ndˇe˜′ , where n is upper unipotent and dˇ is the block diagonal matrix, with blocks d, I2(m−s) , d∗ . Thus, we have c(wg η , w−1 ) = (det d, x(e˜′ w))(det d, x(e˜′ ))c(e˜′ w, w−1 ), and hence (since w and e˜′ commute), we get that c(w, g η )c(wg η , w−1 )c(w, w−1 ) = c(w, e˜′ )c(e˜′ w, w−1 )c(w, w−1 ). The last expression is identically 1 on h(Wm,ℓ+s+1 ) = Sp(Wm,ℓ+s+1 ). Indeed, conf m,ℓ+s+1 ) (embedded inside Sp(V f ) by {(e˜′ , ǫ)|e′ ∈ sider the conjugation in Sp(W Sp(Wm,ℓ+s+1 ), ǫ = ±1}) by (w, 1), (w, 1)(e˜′ , ǫ)(w, 1)−1 = (e˜′ , c(w, e˜′ )c(we˜′ , w−1 )c(w, w−1 )ǫ).

f m,ℓ+s+1 ), we get that Since this conjugation defines an automorphism of Sp(W ′ −1 −1 ′ ′ ˜ ˜ χ(e ) = c(w, e )c(we , w )c(w, w ) is a character of Sp(Wm,ℓ+s+1 ), and hence is trivial. This proves (6.6). Let µ′ψ be the following character of Ps′ . In the linear case, µ′ψ is the trivial character. In the metaplectic case (our convention is that we keep denoting by Ps′ the inverse image of (6.4))    10y c z   d u v c′      ′   µψ   (6.7) e u′ y ′   , ǫ = (det d, x(e))γψ (det d)ǫ.  d∗ 0   1

We proved

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Proposition 6.4. Let 0 < s < m − ℓ, s ≤ j ≤ ℓ + s, w = ǫ0,s and   Iℓ−t  I   t     η= .   γ2    It  Iℓ−t

Then

1−t

¯ℓ ′ (t) 2 0 Jψℓ0 (ρw,η ) ∼ ⊗ Jψℓ−t (σ). = indL Ps′ µψ | det ·|K τ

Finally, let s > 0, w = ǫ0,s and η as in the last proposition, with γ3 replacing γ2 . See (w) (6.2). Again, we start with the linear case. As before, the subgroup Rℓ ∩ η −1 Qℓ,j η ′ consists of the elements of the form (5.23) (with ωbt = I), and instead of (5.24), we  ′  d x require that (in the notation of (5.23)) d has the form , the last row of u is 0 1 (w,η)

zero, and the last row and the first column of v are zero. The action of πℓ,j (τ, σ) on such elements is given by (5.25), except that there y6′ should be replaced by −y6′ . Note that if h of the form (5.23), lies in Nℓ0 , then it lies in Nℓ and ψℓ (h) = ψZt (n1 )ψZℓ−t (n2 )ψK,−1 ((y5 )ℓ−t,1 ). (w)

Assume that t < ℓ. Then consider the subgroup S ⊂ Nℓ ∩η −1 Qℓ,j η, where n1 , n2 are identity blocks, and y4 , y6 are zero. Then, for h ∈ S, we have ψℓ (h) = ψ((y5 )ℓ−t,1 ), (w,η) and hence ψℓ is nontrivial on S, while πℓ,j (τ, σ) is trivial on S. This proves that, for t < ℓ, we have JNℓ ,ψℓ (ρw,η ) = 0, and, hence also, Jψℓ0 (ρw,η ) = 0. This is clearly valid in the metaplectic case, as well. Assume now, that t = ℓ. Thus, j = ℓ + s, and, in particular, ℓ < j. In this case, if h of the form (5.23), lies in Nℓ0 , (then it lies in Nℓ ) ψℓ (h) = ψZt (n1 )ψZℓ−t (n2 )ψK,−1 ((y6 )t,1 ). The same calculation, as before, now gives ℓ

− L JNℓ ,ψℓ (ρw,η ) ∼ = indPsℓ′′ | det ·|K 2 τ(ℓ) ⊗ σ.

Here  100       d u ′′ Ps =  e       

x′ v u′ d∗

  0     x    0 ∈ h(Wm,ℓ ) d ∈ GLs−1 (K), e ∈ h(Wm,ℓ+s ) .    0   1

(6.8)

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The action of | det ·|K 2 τ(ℓ) ⊗ σ on an element of the form (6.8) is by   d −x −ℓ | det(d)|K 2 τ(ℓ) ⊗ σ(e). 0 1 As in the previous case, we check in the same way that, in the metaplectic case we ¯ ℓ and τ(ℓ) ⊗ σ by just need to apply the obvious modifications, replacing Lℓ by L ′ ′ µψ (τ(ℓ) ⊗ σ); µψ is defined in a similar way to (6.7) (and we keep this notation). Now we can summarize Proposition 6.5. Let 0 < s ≤ j ≤ ℓ + s, w = ǫ0,s and   Iℓ−t  I   t     η= .   γ3    It  Iℓ−t

(1) If t < ℓ, then

JNℓ ,ψℓ (ρw,η ) = 0, and hence Jψℓ0 (ρw,η ) = 0. (2) If t = ℓ, then ¯

−ℓ

L JNℓ ,ψℓ (ρw,η ) ∼ = indPsℓ′′ µ′ψ | det ·|K 2 τ(ℓ) ⊗ σ.

We record the last three propositions as the following Leibniz rule. Theorem 6.1. Let π be the representation of H, which is parabolically induced from Qj and the representation τ ⊗ σ of GL(Vj+ ) × H(Wm,j ) = GLj (K) × H(Wm,j ) (in the metaplectic case, we induce from µψ (τ ⊗ σ)). Then Jψℓ0 (π) ≡ L

1−t

¯

ℓ+j−m
(t) ′ 2 ℓ 0 ⊗ Jψℓ−t (σ) indL P ′ µψ | det ·|K τ j−t

(6.9)

0≤t≤j

 ¯ℓ − 2ℓ ′ L JCℓ ,ψ (indL ′′ µψ | det ·|K τ(ℓ) ⊗ σ), Pj−ℓ 0,

if

ℓ<j

otherwise.

When j = m (i.e. π is parabolically induced from τ , on Qm ) (6.9) means ¯

ℓ Jψℓ0 (π) ∼ = JCℓ ,ψ (indL P ′′

m−ℓ

−ℓ

µ′ψ | det ·|K 2 τ(ℓ) ).

(6.10)

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Consider now F Jψℓ ,γ (π) = JCℓ \HWm,ℓ+1 (Jψℓ0 (π) ⊗ ωψ−1 ,γ −1 ), for π as in the last theorem. In the next proposition, we treat the summands in the first term of (6.9). ˆ ℓ , defined in (3.12), (3.13). Recall the group H Proposition 6.6. Let 0 ≤ t ≤ j be such that ℓ + j − m < t ≤ ℓ. Then 1−t

¯ℓ 0 F Jψ1 ,γ (indL µ′ψ | det ·|K2 τ (t) ⊗ Jψℓ−t (σ)) ∼ = ′ Pj−t   1−t ˆℓ H −1 ′ IndQj−t γ µψ−1 | det ·|K2 τ (t) ⊗ F Jψℓ−t ,γ (σ) .

(6.11)

ˆ ℓ , preserving the Here, we still denote by Qj−t the standard parabolic subgroup of H standard j − t dimensional isotropic subspace (of Wm,ℓ+1 ). In (6.11), µ′ψ on the left hand side, is trivial except in case H(Wm,ℓ ) (the “Levi ¯ ℓ ) is metaplectic, and then it is given by (6.1). On the right hand side part” of L ′ ˆ ℓ is metaplectic, and then it is given by (6.1), with µψ−1 is trivial except in case H −1 ˆ ℓ is (linear) symplectic, ψ replacing ψ. Note that when H(Wm,ℓ ) is metaplectic, H ˆ ℓ is metaplectic. and that when H(Wm,ℓ ) is (linear) symplectic, H Proof.

Define the map ¯

1−t

ℓ 0 (σ)) ⊗ ωψ−1 ,γ −1 7→ T : indL (µ′ψ | det ·|K2 τ (t) ⊗ Jψℓ−t ′ Pj−t   1−t ˆℓ (t) −1 ′ 2 τ ⊗ F J (σ) , γ µ | det ·| IndH −1 ψ ,γ ℓ−t K Qj−t ψ

by

T (f ⊗ φ)(h) = Z

Ys Hℓ+s+1 \Hℓ+1

(1τ (t) ⊗ JCℓ+s \Hℓ+s+1 )(f (nh) ⊗ (ωψ−1 ,γ −1 (nh)φ(0s , ·)))dn.

(6.12)

Here, we used the following notation: s = j − t, 0s is the zero vector in K s , Hr = HWm,r (r ≤ m). Let us write the elements of Hℓ+1 in matrix form (inside h(Wm,ℓ )) as follows.   1 a b c z ′  Is 0 0 c    u(a, b, c; z) =  IWm,ℓ+s+1 0 b′  ,  Is a′  1

where a, c ∈ K s , b ∈ K 2(m−ℓ−s−1) , z ∈ k. Then

Hℓ+1 = Xs Ys Hℓ+s+1 , where Xs = {u(a, 0, 0; 0)|a ∈ K s } Ys = {u(0, 0, c; 0)|c ∈ K s },

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and we identify Hℓ+s+1 = {u(0, b, 0; z)|b ∈ K 2(m−ℓ−s−1) , z ∈ k}.

The function, on K m−ℓ−s−1 , ξ 7→ ωψ−1 ,γ −1 (nh)φ(0s , ξ) is in S(K m−ℓ−s−1 ). For fixed n, h, we regard f (nh) ⊗ (ωψ−1 ,γ −1 (nh)φ(0s , ·)), as an element in 1−t 0 (σ) ⊗ ωψ′ −1 ,γ −1 ), where ωψ′ −1 ,γ −1 is the Weil representaµ′ψ | det ·| 2 τ (t) ⊗ (Jψℓ−t ¯ ℓ+s = H(Wm,ℓ+s+1 )HW tion of L , and 1τ (t) ⊗ JCℓ+s \Hℓ+s+1 denotes the m,ℓ+s+1 application of JCℓ+s \Hℓ+s+1 to the second factor. Clearly, T factors through 1−t

¯

ℓ 0 F Jψ1 ,γ (indL µ′ψ | det ·|K2 τ (t) ⊗Jψℓ−t (σ)). Note that T (f ⊗φ)(h) is invariant to left ′ Pj−t translations by the elements of the unipotent radical of the parabolic subgroup Qs ˆ ℓ . This follows from the following easy facts (see (1.5)). Let ϕ ∈ S(K m−ℓ−1 ), of H then   Is x  Im−ℓ−s−1   ωψ−1 ,γ −1  ′  ϕ(0s , ξ) = ϕ(0s , ξ)  Im−ℓ−s−1 x Is



 ωψ−1 ,γ −1  

Is

0 Im−ℓ−s−1

u 0 Im−ℓ−s−1

 v u′   ϕ(0s , ξ) = ϕ(0s , ξ). 0 Is

Similarly, it is straightforward to check that   d T (f ⊗ φ)( I2(m−ℓ−s−1)  h) = d∗ 1

1−t

δQ2 s (d)γ −1 µ′ψ−1 (det(d))| det(d)|K2 (τ (t) (d) ⊗ id)T (f ⊗ φ)(h) 

T (f ⊗ φ)(

Is e′ Is



 h) =

0 (id ⊗ JCℓ+s \Hℓ+s+1 )(Jψℓ−t (σ)(e′ ) ⊗ ωψ′ −1 ,γ −1 (e′ ))T (f ⊗ φ)(h).

Here, the meaning of µ′ψ−1 is as in the remark right after the statement of the proposition. This shows that the image of T is in   1−t ˆℓ −1 ′ µψ−1 | det ·|K2 τ (t) ⊗ F Jψℓ−t ,γ (σ) . IndH Qj−t γ ¯

1−t

ℓ We keep denoting by T the map it induces on F Jψ1 ,γ (indL µ′ | det ·|K2 τ (t) ⊗ ′ Pj−t Pψ 0 Jψℓ−t (σ)). Let us show that T is injective. For this, assume that i T (fi ⊗ φi ) = 0. We need to show that there is an integer r0 > 0, such that for all r ≥ r0 , XZ ρ(n)fi ⊗ ωψ−1 ,γ −1 (n)φi dn = 0. (6.13)

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Here, (Cℓ \Hℓ+1 )(P −r ) = Cℓ (P −r )\Hℓ+1 (P −r ); ρ denotes right translation. The meaning of (6.13) is, due to the Iwasawa decomposition in H(Wm,ℓ+1 ), that for all ω ∈ H(Wm,ℓ+1 )(O), n′ ∈ Xs , XZ fi (n′ ωn) ⊗ ωψ−1 ,γ −1 (n)φi dn = 0, (Cℓ \Hℓ+1 )(P −r )

i

for all r ≥ r0 . Since ωψ−1 ,γ −1 (n′ ω) is invertible, this is the same as XZ fi (n′ ωn) ⊗ ωψ−1 ,γ −1 (n′ ωn)φi dn = 0, (Cℓ \Hℓ+1 )(P −r )

i

for all ω ∈ H(Wm,ℓ+1 )(O), n′ ∈ Xs , r ≥ r0 . Since conjugation by ω preserves (Cℓ \Hℓ+1 )(P −r ), we may replace, in the last integrals, n′ ωn by n′ nω. Since the functions n 7→ fi (nω) have compact support in Hℓ+1 , modulo Ys Hℓ+s+1 , uniformly in ω ∈ H(Wm,ℓ+1 )(O), we get, for r0 large enough, that n′ must lie Xs (P −r ) (or otherwise we get zero). Changing variable in the integrals, n 7→ n′−1 n, we may assume that n′ = 1. Thus, we need to show that XZ fi (nω) ⊗ ωψ−1 ,γ −1 (nω)φi dn = 0, i

(Cℓ \Hℓ+1 )(P −r )

for all ω ∈ H(Wm,ℓ+1 )(O) and r large enough. Write Z fi (nω) ⊗ ωψ−1 ,γ −1 (nω)φi dn = Z

(Cℓ \Hℓ+1 )(P −r )

(CYs \Hℓ+1 )(P −r )

fi (nω) ⊗

Z

Ys (P −r )

ωψ−1 ,γ −1 (ynω)φi dydn.

Again, since n 7→ fi (nω) has compact support in Hℓ+1 , modulo Ys Hℓ+s+1 , uniformly in ω ∈ H(Wm,ℓ+1 )(O), there is r1 such that for all r ≥ r1 (and all i) n in the last integrand maybe taken of the form n′ n′′ , where n′ ∈ Xs (P −r1 ) and n′′ ∈ Hℓ+s+1 (P −r ). The function on K s , x 7→ ωψ−1 ,γ −1 (n′ n′′ ω)φi (x, ·) vanishes outside a compact set, which depends only on r1 , for all r ≥ r1 . Indeed, ωψ−1 ,γ −1 (ω)φi takes finitely many values, as ω varies in H(Wm,ℓ+1 )(O). Each one of these finite values can be written as a sum of functions of the form ϕ1 ⊗ ϕ2 , where ϕ1 ∈ S(K s ), ϕ2 ∈ S(K m−ℓ−s−1 ). We have ωψ−1 ,γ −1 (n′ n′′ )(ϕ1 ⊗ ϕ2 )(x, ξ) = ϕ1 (x + a)ωψ′ −1 ,γ −1 (n′′ )ϕ2 (ξ),

where n′ = u(a, 0, 0; 0). Thus, there is r0 ≥ r1 , such that for all r ≥ r0 , x ∈ K s , ξ ∈ K m−ℓ−s−1 , Z ωψ−1 ,γ −1 (yn′ n′′ ω)φi (x, ξ)dy = cr χr (x)ωψ−1 ,γ −1 (n′ n′′ ω)φi (0s , ξ), Ys (P −r )

where χr is the characteristic function of a certain small neighborhood of 0 in K s (depending on r and ψ); cr is the measure of Ys (P −r ). Thus, it is enough to show that, for r ≥ r0 , we have XZ fi (nω) ⊗ ωψ−1 ,γ −1 (nω)φi (0s , ·)dn = 0, i

(Cℓ Ys \Hℓ+1 )(P −r )

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P for all ω ∈ H(Wm,ℓ+1 )(O). By our assumption that i T (fi ⊗ φi ) = 0, we have XZ (id ⊗ JC\Hℓ+s+1 )(fi (nω) ⊗ (ωψ−1 ,γ −1 (nω)φi (0s , ·)))dn = 0, i

Ys Hℓ+s+1 \Hℓ+1

for all ω ∈ H(Wm,ℓ+1 )(O). Note, again that fi (nω) has a compact support modulo Ys Hℓ+s+1 , and, in particular, fi (nω) ⊗ (ωψ−1 ,γ −1 (nω)φi (0s , ·)) takes finitely many (vector) values. Thus, there is an integer r2 > 0, such that, for all r ≥ r2 , and all ω ∈ H(Wm,ℓ+1 )(O), Z XZ fi (unω) ⊗ ωψ−1 ,γ −1 (unω)φi (0s , ·)dudn i

(Ys Hℓ+s+1 \Hℓ+1 )(P −r )

(Cℓ+s \Hℓ+s+1 )(P −r )

is zero. The integrals in the last sum can be rewritten as Z fi (nω) ⊗ ωψ−1 ,γ −1 (nω)φi (0s , ·)dn, (Cℓ Ys \Hℓ+1 )(P −r )

and we get what we want (with r0 = r2 ). To see the surjectivity of T , we note that 1−t

¯ℓ (t) ′ 2 ⊗ Jψℓ0 (σ)) ⊗ ωψ−1 ,γ −1 ∼ indL = P ′ (µψ | det ·|K τ

1−t ¯ℓ ′ (t) 2 0 −1 −1 indL (µ | det ·| τ ⊗ J (σ) ⊗ ω ′ ψ ,γ ψℓ ψ K P

ℓ−t

′ Pℓ−t

ℓ−t

Consider the projection defined on ωψ−1 ,γ −1

′ Pℓ−t

).

, by φ 7→ φ(0s , ·). Composing 1−t

¯

′ (t) 2 ℓ this with the last isomorphism, we get a surjection to indL ⊗ P ′ (µψ −1 | det ·|K τ ℓ−t

(Jψℓ0 (σ) ⊗ ωψ′ −1 ,γ −1 )), and now the surjectivity of T follows. This completes the proof of the proposition. 

We can repeat the last proof, when we consider the contribution ofthe second   dx d term of (6.9) to F Jψℓ ,γ (π), provided we know that τ(ℓ) = τ(ℓ) (d ∈ 01 1 ∼ GLj−ℓ−1 (K)). In this case, τ(ℓ) = τ (ℓ+1) . We get, in the notation of GLj−ℓ−1 (K)

Theorem 6.1,

Proposition 6.7. Assume that ℓ < j, and that τ(ℓ) d ∈ GLj−ℓ−1 (K) and x ∈ K j−ℓ−1 . Then



   dx d = τ(ℓ) , for all 01 1

¯ℓ ℓ ′ ∼ 2 F Jψ1 ,γ (indL P ′′ µψ | det ·| τ(ℓ) ⊗ σ) = j−ℓ

−1 ′ ℓ IndH µψ−1 τ (ℓ+1) ⊗ (σ ⊗ ωψ′ −1 ,γ −1 )). Qj−ℓ−1 (γ

Here, ωψ′ −1 ,γ −1 is the Weil representation of H(Wm,j ).

Let us apply, as in Sec. 5.1, Theorem 6.1 and the last two propositions to several special cases. We keep the notation of Theorem 6.1. Theorem 6.2. Let j < m. Assume that that τ is supercuspidal.

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(1) If ℓ < j < m − ℓ, then Jψℓ0 (π) ≡ L ¯ℓ ′ 1 ¯ℓ ′ − 2ℓ 2 indL JCℓ ,ψ (indL τ(ℓ) ⊗ σ). P ′ µψ | det ·| τ ⊗ Jψℓ0 (σ) P ′′ µψ | det ·| j

j−ℓ

(2) If ℓ < j and m − ℓ ≤ j, then

¯ℓ ′ − 2ℓ τ(ℓ) ⊗ σ). Jψℓ0 (π) ∼ = JCℓ ,ψ (indL P ′′ µψ | det ·| j−ℓ

(3) If m − ℓ ≤ j ≤ ℓ, then 0 Jψℓ0 (π) ∼ (σ). = dim(τ (j) )Jψℓ−j

(4) If j ≤ ℓ and j < m − ℓ, then ¯

1

ℓ ′ 2 Jψℓ0 (π) ≡ indL P ′ µψ | det ·| τ ⊗ Jψℓ0 (σ) j

Proof. t = j.

M

0 dim(τ (j) )Jψℓ−j (σ).

The proof is immediate from Theorem 6.1, since τ (t) = 0, unless t = 0, or 

Consider now a representation τ of the form (5.33). Theorem 6.3. Let µ1 , ..., µi be characters of K ∗ , and consider GL (K)

τ = IndPmm,...,m µ1 (detGLm1 ) ⊗ · · · ⊗ µi (detGLmi ). 1

i

Then, for all ℓ ≥ i, ′ Jψℓ0 (IndH Qm µψ τ ) = 0,

and for ℓ = i − 1,

′ ∼ 0 Jψi−1 (IndH Qm µψ τ ) = ¯ L

JCi−1 ,ψ (indP i−1 ′′

m−i+1 (m1 −1,...,mi −1)

µ′ψ µ1 (detGLm1 −1 ) ⊗ · · · ⊗ µi (detGLmi −1 )),

′′ ′′ where Pm−i+1 (m1 − 1, ..., mi − 1) is the subgroup of Pm−i+1 , obtained by replacing the reductive part, GLm−i (K), by Pm1 −1,...,mi −1 . Moreover, ′ ∼ F Jψi−1 ,γ (IndH Qm µψ τ ) = ˆ H

i−1 IndQm −1,...,m 1

i −1

γ −1 µ′ψ−1 µ1 (detGLm1 −1 ) ⊗ · · · ⊗ µi (detGLmi −1 ),

ˆ i−1 , whose Levi part is where Qm1 −1,...,mi −1 is the standard parabolic subgroup of H isomorphic to GLm1 −1 (K) × · · · × GLmi −1 (K). Proof. The proof is exactly as that of Theorem 5.4, where we use Theorem 6.1 for the case j = m, i.e. (6.10). The second follows from Proposition 6.7, since  part Im−i x τ(i−1) acts trivially on the subgroup . Indeed, the highest derivative 0 1 of τ is of index i. (See (5.34).) 

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Consider representations τ of GLm (K), as in Theorem 5.6. Let πτ be the unramified 1

′ 2 constituent of IndH Qm µψ τ | det ·|K . When H is metaplectic, we will denote it by πτ,ψ . We will assume that the residual characteristic of k is odd, and that ψ is normalized. When [K : k] = 2, we will assume that K/k is an unramified extension. We write λ0 (x) = (x, ǫ), where ǫ is a non-square unit, and ( , ) is the Hilbert symbol of K. (2),+ f 2 (k), with the unramified piece of the Weil representation of SL Denote by ωψǫ f 2 (k), induced respect to ψǫ . This is a subrepresentation of the representation of SL 1

from the Borel subgroup and γψ λ0 | · | 2 . Then we have the following analogue of Theorem 5.6. Theorem 6.4. Let τ be an irreducible, generic, unramified representation of GLm (K). Assume that τ is self-conjugate. (1) Assume that ωτ = 1 and m = 2n is even. Then Jψℓ0 (πτ ) = 0, for all ℓ ≥ n, and for ℓ = n − 1,

ˆ H

F Jψn−1 ,γ (πτ ) ≺ IndBnn−1 γ −1 µ′ψ−1 (µ1 ⊗ · · · µn ),

(6.14)

ˆ n−1 . where Bn is the standard Borel subgroup of H (2) Assume that ωτ = λ0 and m = 2n is even. Let H be symplectic. Then Jψℓ0 (πτ ) = 0, for all ℓ ≥ n + 1, and for ℓ = n,

f Sp

2n−2 F Jψn (πτ ) ≺ IndBn−1

(k)

µψ−1 (µ1 ⊗ · · · µn−1 ),

(6.15)

f 2n−2 (k) is the (inverse image of the) standard Borel subgroup. where Bn−1 ⊂ Sp (3) Assume that ωτ = λ0 and m = 2n is even. Let H be (even) unitary. Then Jψℓ0 (πτ ) = 0,

for all ℓ ≥ n, and for ℓ = n − 1, U

F Jψn−1 ,γ (πτ ) ≺ IndB2n ′

(k) −1

n−1

γ

(2)

(µ1 ⊗ · · · ⊗ µn−1 ) ⊗ ωψ−1 ,γ −1 ,

(6.16)

′ where Bn−1 is the parabolic subgroup, which preserves the flag

SpanK {en+1 } ⊂ SpanK {en+1 , en+2 } ⊂ · · · ⊂ SpanK {en+1 , ..., e2n−1 };

(2) ωψ−1 ,γ −1

denotes the Weil representation of U2 (k). In particular, if π is an irreducible, unramified representation of U2n (k), which is a subquotient of F Jψn−1 ,γ (πτ ), then U

π ≺ IndB2n n

(k) −1

γ

(µ1 ⊗ · · · µn−1 ⊗ 1),

where Bn ⊂ U2n (k) is the standard Borel subgroup.

(6.17)

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(4) Assume that ωτ = λ0 and m = 2n is even. Let H be metaplectic. Then Jψℓ0 (πτ,ψ ) = 0, for all ℓ ≥ n, and for ℓ = n − 1, Sp

F Jψn−1 (πτ,ψ ) ≺ IndB ′ 2n

(k)

n−1

(2),+

µ1 ⊗ · · · ⊗ µn−1 ⊗ (ωψǫ

(2)

⊗ ωψ−1 ),

(6.18)

′ where Bn−1 is the parabolic subgroup, which preserves the flag

Spank {en+1 } ⊂ Spank {en+1 , en+2 } ⊂ · · · ⊂ Spank {en+1 , ..., e2n−1 }. Moreover, assume that π is an irreducible, unramified representation of Sp2n (k), which is a subquotient of F Jψn−1 ,γ (πτ,ψ ), then Sp

π ≺ IndBn2n

(k)

µ1 ⊗ · · · µn−1 ⊗ λ0 .

(6.19)

(5) Assume that ωτ = 1 and m = 2n + 1 is odd. Let H be symplectic. Then Jψℓ0 (πτ ) = 0, for all ℓ ≥ n + 1, and for ℓ = n, f Sp

F Jψn (πτ ) ≺ IndBn2n µψ−1 (µ1 ⊗ · · · ⊗ µn ).

(6.20)

(6) Assume that ωτ = 1 and m = 2n + 1 is odd. Let H be unitary. Then Jψℓ0 (πτ ) = 0, for all ℓ ≥ n, and for ℓ = n − 1, U

F Jψn−1 (πτ ) ≺ IndB2n+2 ′ n−1

(k) −1

γ

(2)

(µ1 ⊗ · · · ⊗ µn ) ⊗ ωψ−1 ,γ −1 .

(6.21)

In particular, if π is an irreducible, unramified representation of U2n+2 (k), which is a subquotient of F Jψn−1 ,γ (πτ ), then U

π ≺ IndB2n+2 n+1

(k) −1

γ

(µ1 ⊗ · · · µn ⊗ 1),

(6.22)

(7) Assume that ωτ = 1 and m = 2n + 1 is odd. Let H be metaplectic. Then Jψℓ0 (πτ,ψ ) = 0, for all ℓ ≥ n + 1, and for ℓ = n,

Sp

F Jψn (πτ,ψ ) ≺ IndBn2n

(k)

µ1 ⊗ · · · ⊗ µn .

(6.23)

(8) Assume that ωτ = λ0 and m = 2n + 1 is odd. Let H be symplectic. Then Jψℓ0 (πτ ) = 0, for all ℓ ≥ n + 1, and for ℓ = n, f Sp

F Jψn , (πτ ) ≺ IndBn2n

(k)

µψ−1 (µ1 ⊗ · · · ⊗ µn ).

(6.24)

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(9) Assume that ωτ = λ0 and m = 2n + 1 is odd. Let H be metaplectic. Then Jψℓ0 (πτ,ψ ) = 0, for all ℓ ≥ n, and for ℓ = n − 1, Sp

F Jψn−1 (πτ,ψ ) ≺ IndBn′2n+2

(k)

(2),+

µ1 ⊗ · · · ⊗ µn ⊗ (ωψǫ

(2)

⊗ ωψ−1 ),

(6.25)

and if π is an irreducible, unramified representation of Sp2n+2 (k), which is a subquotient of F Jψn−1 (πτ,ψ ), then Sp

2n+2 π ≺ IndBn+1

(k)

µ1 ⊗ · · · µn ⊗ λ0 .

(6.26)

(10) Assume that ωτ = λ0 and m = 2n + 1 is odd. Let H be unitary. Then Jψℓ0 (πτ ) = 0, for all ℓ ≥ n + 1, and for ℓ = n,

U

F Jψn (πτ ) ≺ IndB2n n

(k) −1

γ

(µ1 ⊗ · · · ⊗ µn ).

(6.27)

Proof. The proof is similar to that of Theorem 5.6. Assume that ωτ = 1 and m = 2n. Then, as in the proof of Theorem 5.6, πτ is the unramified constituent of ′ ′ ′ IndH Q2n µψ τ , where τ is the representation of GL2n (K), parabolically induced from µ1 (detGL2 ) ⊗ · · · ⊗ µn (detGL2 ). Now use Theorem 6.3. This proves part (1). Assume that ωτ = λ0 and m = 2n. Denote now by τ ′ the representation of GL2n−2 (K), parabolically induced from µ1 (detGL2 ) ⊗ · · · ⊗ µn−1 (detGL2 ). Assume Sp (k) first, that H is symplectic . Then πτ is the unramified constituent of IndQm4n τ , 1 1 where τ is the representation of GL2n (k), parabolically induced from τ ′ ⊗λ0 |·| 2 ⊗|·| 2 . Now use Theorem 6.3, with i = n + 1, m1 = · · · = mn−1 = 2, mn = mn+1 = 1, µn = 1 1 λ0 | · | 2 , µn+1 = | · | 2 . This proves part (2). Assume next, that H is unitary. Then U (k) ′ τ ⊗ σ, where σ is the representation πτ is the unramified constituent of IndQ4n 2n−2 1

2 ⊗ 1U2 . Let us use Theorem 6.1, with of U4 (k) parabolically induced from λ0 | · |K ′ j = 2n − 2. Note that τ(ℓ) = 0, for ℓ ≥ n − 1. In the first sum of (6.9), we have t = ℓ − 1, ℓ. Note that Jψ10 (σ) = 0, since σ does not have any Whittaker model. Thus, only t = ℓ contributes to the first sum. Finally, note that τ ′(t) = 0, for t ≥ n. This proves that Jψℓ0 (πτ ) = 0, for ℓ ≥ n. For ℓ = n − 1, as we just explained, (6.9) gives just the term which corresponds to t = ℓ, and by Proposition 6.6, we get that

U

2−n

(k)

γ −1 | det ·|K2 τ ′(n−1) ⊗ F Jψ0 ,γ (σ). F Jψn−1 ,γ (πτ ) ≺ IndQ2n n−1 In order to compute F Jψ0 ,γ (σ), we use Theorem 6.1, with ℓ = 0, j = 1, m = 2, 1 τ = λ0 | · | 2 , and σ = 1U2 . In this case, Jψ00 (σ) = 0. (The character ψ0 is trivial, and the character ψ00 is the character ψ of the center C0 ∼ = F of the Heisenberg group in question; in our cases, C0 is the one parameter unipotent subgroup corresponding to the highest root.) In this case, (6.9) reads as HW

2,1 JC0 ,ψ (indU2 (k)

·U2 (k)

1U2 ).

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Now, we apply Proposition 6.7. This will give (6.16), and (6.17) follows, since the (2) unramified piece of ωψ−1 ,γ −1 is the unramified constituent of the representation of U2 (k), induced from the character γ −1 of the Borel subgroup. This proves part (3). Finally, assume that H is metaplectic. Then we reason as in the last case, where f (k) Sp f 4 (k) induced from πτ,ψ ≺ Ind 4n µψ τ ′ ⊗ σ, and σ is the representation of Sp 1

Q2n−2 (2),+

µψ | · | 2 ⊗ ωψǫ . Note, that since ǫ is not a square, σ does not have a ψ-Whittaker model, i.e. Jψ10 (σ) = 0. Now, we are exactly in the situation of the last case, and we get the first assertion of part (4) and (6.18). To get (6.19), we compute the Jacquet (2) (2),+ module of ωψǫ ⊗ωψ−1 (along N1 ⊂ SL2 (k)). This is done as follows. First, consider (2)

the restriction of ωψ−1 to the Borel subgroup B1 . It has two constituents: a one 1 dimensional quotient, which is the character (of k˜∗ ) γψ−1 | · | 2 , and one irreducible

subrepresentation, wψ , which is the (genuine) compactly induced representation, (2),+ does not have a ψ-Whittaker model, we get that from ψ −1 on N1 . Since ωψǫ (2),+

JN1 (ωψǫ

⊗ wψ ) = 0, and hence (2),+

JN1 (ωψǫ

(2),+

(2)

⊗ ωψ−1 ) ∼ = JN1 (ωψǫ

1 ) ⊗ γψ−1 | · | 2 ∼ = λ0 | · |.

This proves part (4). Assume that ωτ = 1 and m = 2n + 1. Let H be symplectic or metaplectic. Then πτ (resp. πτ,ψ ) is the unramified constituent of the representation parabolically 1 induced from µ1 (detGL2 ) ⊗ · · · ⊗ µn (detGL2 ) ⊗ | · | 2 . Now parts (5), (7) follow in a straightforward way from Theorem 6.3. In case H is unitary, we use the fact that U (k) ′ GL2n (K) µ1 (detGL2 ) ⊗ · · · ⊗ τ ⊗ 1U2 , where τ ′ = IndP2,...,2 πτ is a constituent of IndQ4n+2 2n µn (detGL2 ), and we reason as in part (3). This proves part (6). Parts (8), (10) follow directly from Theorem 6.3, with i = n + 1, m1 = · · · = 1

2 . Finally, part (9) is proved similarly to part mn = 2, mn+1 = 1, µn+1 = λ0 | · |K (4). This completes the proof of the theorem. 

6.2

The case K = k ⊕ k

In this section, we compute the Jacquet module corresponding to Fourier - Jacobi models, in case K = k ⊕ k, when applied to parabolically induced representations. This is the split version of the previous section for the unitary group. We will work in a set-up similar to that of Sec. 5.2. We consider a parabolic induction GL (k) π = IndPj,NN−j τ1 ⊗ τ2 , where τ1 and τ2 are smooth representations of GLj (k) and GLN −j (k) respectively. Let ℓ˜ = (ℓ1 , ℓ2 , ℓ3 ) be a partition of N . We assume that

ℓ2 ≥ 2 and ℓ1 , ℓ2 ≥ 0. Let Pℓ˜ be the corresponding standard parabolic subgroup of GLN (k). Let Nℓ˜ be the unipotent subgroup (5.45), and consider its character ψℓ˜, defined as follows. Let v ∈ Nℓ˜ be written as in (5.45). Then ψℓ′˜(v) = (ψ 12 )Zℓ1 (z1 )(ψ 12 )Zℓ3 (z2 )ψ 12 (((y1 )ℓ1 ,1 + (y2 )ℓ2 ,1 )).

(6.28)

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This is the appropriate generalization of the restriction of the character (3.64) to Nℓ (k) = N(ℓ,2(m−ℓ),ℓ)(k). Note that this character is different than the one in (5.46), and hence the different notation. The stabilizer of ψℓ˜ inside GLℓ2 (k) is     1 y1 x  L′ℓ˜ =  g y2  ∈ GLℓ2 (k) .   1

We identify L′ℓ˜ as the subgroup

 I   ℓ1    1 y1 x   g y2      1  

         ∈ GLN (k)        

Iℓ3

∼ GLℓ −2 (k)H2(ℓ −2) , where H2(ℓ −2) is the Heisenberg of GLN (k). Note that L′ℓ˜ = 2 2 2 group corresponding to the 2(ℓ2 − 2)–dimensional symplectic space over k. See (1.21), (3.65). Denote     1 0 z   C = c(z) =  Iℓ2 −2 0 |z ∈ k   1

the center of of the Heisenberg group H2(ℓ2 −2) . Denote its corresponding image in GLN (k) by Cℓ˜ and identify C and Cℓ˜, when convenient. Let Nℓ˜0 = Nℓ˜Cℓ˜, and extend ψℓ′˜ to a character ψℓ0˜ , of Nℓ˜0 , by letting act as ψ, on Cℓ˜, namely ψ(c(z)) = ψ(z). We will compute F Jψ′˜,γ (π) = JH2(ℓ2 −2) /C (JN ˜0 ,ψ0˜ (π) ⊗ ωψ−1 ,γ −1 ), ℓ

ℓ

ℓ

with respect to (ψ −1 , γ −1 ). We where ωψ−1 ,γ −1 is the Weil representation of will use the realization given in (1.23). Note that when ℓ2 = 2,    1z L′ℓ˜ = |z ∈ k , 1 L′ℓ˜,

and we define ωψ−1 ,γ −1 , in this case, to be the character ψ −1 (z). As in the previous section, we compute first Jψ0˜ (π) = JN ˜0 ,ψ0˜ (π). Note that ℓ ℓ ℓ when ℓ2 = 2, Jψ0˜ (π) is the Jacquet module of π with respect to a Whittaker charℓ acter, and hence it is non-trivial if and only if τ1 , τ2 have non-trivial Whittaker functionals. Since Jψ0˜ (π) = Jψ0˜ (ResPℓ˜(π)), we consider the finite Pℓ˜-filtration of ℓ ℓ ResPℓ˜(π), whose subquotients are the representations ρw in (5.48) and the representatives w = wr,s of Pj,N −j \GLN (k)/Qℓ˜ are given in (4.17).

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Let Rℓ˜ = L′ℓ˜Nℓ˜. Then Jψ0˜ (ρw ) = Jψ0˜ (ResRℓ˜ρw ), and to analyze ResRℓ˜ρw , we con(w)

ℓ

ℓ

sider Pℓ,j ˜ \Pℓ˜/Rℓ˜. The representatives η can be chosen as in (5.49), where γ (in (5.49)) is a representative of Ps−r,ℓ2 −(s−r) \GLℓ2 (k)/L′ℓ˜ = Ps−r,ℓ2 −(s−r) \GLℓ2 (k)/P1,ℓ2 −2,1 . This is a special case of Proposition 4.2. Of course, if s − r = 0, or s − r = ℓ2 , then the last set is a singleton, and we choose γ = γ1 = Iℓ2 . Assume that 0 < s − r < ℓ2 . Then the double cosets are parametrized by pairs ′ ′ (r , s ), such that r′ = 0, 1 and 0 ≤ s′ − r ′ ≤ ℓ 2 − 2 0 ≤ s − r − s′ ≤ 1. See (4.17). Thus, we get, for 2 ≤ s − r ≤ ℓ2 − 2, four representatives. These correspond to the pairs (1, s − r), (1, s − r − 1), (0, s − r), (0, s − r − 1). For s − r = 1, we get three representatives. They correspond to (0, s − r), (0, s − r − 1), (1, s − r), and for s − r = ℓ2 − 1, we get three representatives. They correspond to (0, s − r − 1), (1, s − r), (1, s − r − 1). We choose the following representatives, written in matrix form. For (1, s − r), 1 ≤ s − r ≤ ℓ2 − 1, we choose γ1 = Iℓ2 . For (1, s − r − 1), 2 ≤ s − r ≤ ℓ − 2 − 1, we choose   Is−r−1 γ2 =  1 . Iℓ2 −(s−r) For (0, s − r − 1), 1 ≤ s − r ≤ ℓ2 − 1, we choose  0 Is−r−1 0 0 0 0 γ3 =  1 0 0 0 0 Iℓ2 −(s−r)−1

 0 1 . 0

0

For (0, s − r), 1 ≤ s − r ≤ ℓ2 − 2, we choose   0 Is−r 0 . γ4 = 1 0 0 0 0 Iℓ2 −(s−r)−1

Exactly as in Sec. 5.2, and using similar notations, we may assume that, for w = wr,s , of the form (4.17), η, of the form (5.49), with γ = γ1 , ..., γ4 , is such that ǫ1 , ǫ3 are of the form (5.51). Proposition 5.6 is valid here as well, i.e. for w = wr,s , such that r > 0 and j − s < ℓ3 , Jψ0˜ (ρw ) = 0. ℓ

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To prove this, we have to show that Jψ0˜ (ρw,η ) = 0, for η as above. Assume, first, ℓ (w,η) that 0 < s − r < ℓ2 . Let γ = γ1 . From (4.25), it is clear that πℓ,j = 1, while, ˜ Cℓ˜ of course, ψℓ0˜ 6= 1. Let now γ = γi , i = 2, 3, 4. Consider the subgroup J ⊂ Nℓ˜ Cℓ˜ (w,η) (w) of elements v, of the form (5.52). Then J ⊂ Nℓ˜ ∩ η −1 Qℓ,j = 1. We ˜ ˜ η, and πℓ,j J

have

  1 ′ 0 −1 v2 ψℓ˜ (v) = ψℓ˜(v) = ψ( ([(0, y2 )γ]r,1 + [γ ]ℓ2 ,1 )). 0 2

(6.29)

Here, y2 ∈ Mr×(ℓ2 −(s−r)) (k), v2 ∈ Ms−r×(ℓ3 −j+s) (k). For γ2 , (6.29) equals ψ( 12 (v2 )s−r,1 ), which is non-trivial, unless j − s = ℓ3 (here, we have s − r ≥ 2). For γ3 , (6.29) equals ψ( 12 ((y2 )r,1 + (v2 )s−r,1 )), and this is non-trivial, unless r = 0 and j − s = ℓ3 . For γ4 , (6.29) equals ψ( 12 (y2 )r,1 ), which is non-trivial, unless r = 0. (Here, s − r ≤ ℓ2 − 2.) Finally, if s − r = 0, ℓ2 , then we have to consider only γ1 . Here, we consider the last subgroup J once more, and note that (6.29) equals, in the first case, ψ( 12 (y2 )r,1 ), which is non-trivial, unless r = 0, and equals, in the second case, ψ( 21 (v2 )ℓ2 ,1 ), which is non-trivial unless j − s = ℓ3 . It remains to analyze Jψ0˜ (ρw ), in case r = 0, or j − s = ℓ3 . The next proposition ℓ is analogous to Proposition 5.7; the details of the proof are very similar, and so we omit them. Proposition 6.8. Let w correspond to r = 0 and j − s < ℓ3 . (Recall that s ≤ ℓ2 .) Then when s = ℓ2 − 1, ℓ2 , Jψ0˜ (ρw ) = 0, ℓ

and when s ≤ ℓ2 − 2,

′

L˜ Jψ0˜ (ρw ) ∼ = indPsℓ′ | det ·|

1−(j−s)+ℓ3 −ℓ1 2

ℓ

where

(j−s)

τ1

⊗ | det ·|

j−s 2

0 Jψ(ℓ

1 ,ℓ2 −s,ℓ3 −j+s)

(τ2 ),

   1 0 x z        b u u 1 1 2 ′ ′   Ps =  ∈ L ∈ GL (k), b ∈ GL (k) , b ˜ 1 s ℓ −s−2 2 2 ℓ   b2 y      1

and the inducing representation acts on an element of Ps′ by | det(b1 )|

1−(j−s)+ℓ3 −ℓ1 2

(j−s) τ1 (b1 )

⊗ | det(b2 )|

j−s 2

  1 x z 0 Jψ(ℓ (τ2 )  b2 y . 1 ,ℓ2 −s,ℓ3 −j+s) 1

Similarly, we get the following analogue of Proposition 5.8.

Proposition 6.9. Let w correspond to r > 0 and j −s = ℓ3 . Then when s = r, r+1, Jψ0˜ (ρw ) = 0, ℓ

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and when 2 ≤ s − r ≤ ℓ2 , ′

L˜ Jψ0˜ (ρw ) ∼ = indQℓ′

ℓ2 −s+r

ℓ

| det ·|−

ℓ1 −r 2

0 Jψ(r,j−ℓ

3 −r,ℓ3 )

(τ1 ) ⊗ | det ·|

ℓ3 −r−1 2

[ℓ −r]

τ2 1

,

where    1 x u1 z        b u y 1 2 ′   Qℓ2 −s+r  ∈ L ∈ GL (k), b ∈ GL (k) , b ˜ 1 s 2 ℓ −s−2 2 ℓ   b2 0      1

and the inducing representation acts on an  1 x ℓ1 −r  b1 0 (τ ) | det(b1 )| 2 Jψ(r,j−ℓ 1 −r,ℓ ) 3

3

[ℓ −r]

See (5.69) for the definition of τ2 1

element of Q′ℓ2 −s+r by  z ℓ3 −r−1 [ℓ −r] y  ⊗ | det(b2 )| 2 τ2 1 (b2 ). 1

. 

Finally, assume that r = 0 and j − s = ℓ3 . Thus, η = 

Iℓ1 γ



. Again, the

Iℓ3 details are similar to those which appear right after Proposition 5.8. The subgroup (w) looks the same, except that instead of (5.71), we have Rℓ˜ ∩ η −1 Qℓ,j ˜  γ b1 u ∈ Lℓ˜. 0 b2 (w)

The restriction of ψℓ0˜ to Nℓ˜0 ∩ η −1 Qℓ,j is given, using the notation of (5.72) by ˜ ψℓ0˜ (v)

= (ψ 12 )Zℓ1 +ℓ3

 a2

c1



   γ ! 1 b1 u −1 v1 ψ( ([(0, y3 )γ]ℓ1 ,1 + [γ ]ℓ2 ,1 ))ψ . 0 0 b2 1,ℓ 2 2

We get the following. Proposition 6.10. Let w correspond to r = 0 and j − s = ℓ3 . Then (1) Let γ = γ1 . Then, for 0 < s < ℓ2 , Jψ0˜ (ρw,η ) = 0, ℓ

for s = 0 (i.e. j = ℓ3 ), ℓ3

(j)

Jψ0˜ (ρw,η ) ∼ = τ1 ⊗ | det ·| 2 JC,ψ ((τ2 )[ℓ1 ] ), ℓ

(see (5.74) for the definition of (τ2 )[ℓ1 ] ) and for s = ℓ2 (i.e. j = ℓ2 + ℓ3 ), ℓ

1 [ℓ ] Jψ0˜ (ρw,η ) ∼ = | det ·|− 2 JC,ψ ((τ1 )(ℓ3 ) ) ⊗ τ2 1 . ℓ

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(2) Let γ = γ2 (and so 2 ≤ s ≤ ℓ2 − 1). Then ′

L˜ Jψ0˜ (ρw,η ) ∼ = indQℓ′

ℓ1

ℓ2 −s

ℓ

| det ·|− 2 JC,ψ ((τ1 )(ℓ3 ) ) ⊗ | det ·|

ℓ3 −1 2

[ℓ ]

τ2 1 .

  1 x u1 z  b u y  1 2  The inducing representation acts on   ∈ Q′ℓ2 −s ,  b2 0  1 (b1 ∈ GLs−2 (k), b2 ∈ GLℓ2 −s (k)) by   1 x z ℓ1 ℓ3 −1   [ℓ ] | det(b1 )|− 2 JC,ψ ((τ1 )(ℓ3 ) )  b1 y  ⊗ | det(b2 )| 2 τ2 1 (b2 ). 1

(3) Let γ = γ3 (1 ≤ s ≤ ℓ2 − 1). Then  L′  ℓ1 ℓ3 ˜ Jψ0˜ (ρw,η ) ∼ = JC,ψ indPsℓ′′ | det ·|− 2 (τ1 )(ℓ3 ) ⊗ | det ·| 2 (τ2 )[ℓ1 ] . ℓ

   1 0 b 0          b u a 1   ′′ Here, Ps =   |b1 ∈ GLs−1 (k), b2 ∈ GLℓ2 −s−1 (k) .    b2 0        1 The inducing representation acts on Ps′′ by ! ℓ3 ℓ1 b a 1 ⊗ | det(b2 )| 2 (τ2 )[ℓ1 ] | det(b1 )|− 2 (τ1 )(ℓ3 ) 1

1 b b2

!

.

(4) Let γ = γ4 (and so 1 ≤ s ≤ ℓ2 − 2). Then ′

L˜ Jψ0˜ (ρw,η ) ∼ = indPsℓ′ | det ·| ℓ

1−ℓ1 2

ℓ3

(τ1 )(ℓ3 ) ⊗ | det ·| 2 JC,ψ ((τ2 )[ℓ1 ] ). 

1 0 x z



 b u u   1 1 2 Here, the inducing representation acts on   ∈ Ps′ (b1 ∈ GLs (k),  b2 y  1

b2 ∈ GLℓ2 −s−2 (k)) by | det(b1 )|

1−ℓ1 2



1 x z



ℓ3   (τ1 )(ℓ3 ) (b1 ) ⊗ | det(b2 )| 2 JC,ψ ((τ2 )[ℓ1 ] )  b2 y  . 1

We summarize the last three propositions in the following Leibniz rule.

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Theorem 6.5. Let ℓ˜ = (ℓ1 , ℓ2 , ℓ3 ) be a partition of N with ℓ2 ≥ 2. Then, up to semi-simplification, GL (k)

Jψ0˜ (IndPj,NN−j τ1 ⊗ τ2 ) ≡ ℓ

L

L′

j−ℓ3 <s≤ℓ2 −2

L

indPsℓ˜′ | det ·|

1−(j−s)+ℓ3 −ℓ1 2

(j−s)

τ1

⊗ | det ·|

j−s 2

0 Jψ(ℓ

1 ,ℓ2 −s,ℓ3 −j+s)

(τ2 )

0≤s≤j L′

0
indQℓ˜′

ℓ2 +ℓ3 −j+r

| det ·|−

ℓ1 −r 2

0 Jψ(r,j−ℓ

3 −r,ℓ3 )

(τ1 ) ⊗ | det |

ℓ3 −r−1 2

[ℓ −r]

τ2 1

j−ℓ2 −ℓ3 ≤r≤ℓ1

 L′˜ L ind ℓ′ Qℓ

0,

ℓ1

2 +ℓ3 −j

| det ·|− 2 JC,ψ ((τ1 )(ℓ3 ) ) ⊗ | det |

ℓ3 −1 2

[ℓ ]

τ2 1 ,

if

2 ≤ j − ℓ3 ≤ ℓ2

otherwise

 ℓ3 1−ℓ1 L′˜ L indP ℓ′ | det ·| 2 (τ1 )(ℓ3 ) ⊗ | det | 2 JC,ψ (τ2[ℓ1 ] ), j−ℓ3 0,

0 ≤ j − ℓ3 ≤ ℓ2 − 2

if

otherwise

  L′  ℓ1 ℓ3 ˜ L JC,ψ indP ℓ′′ | det ·|− 2 (τ1 )(ℓ3 ) ⊗ | det | 2 τ2[ℓ1 ] , j−ℓ3 0,

if

0 < j − ℓ3 < ℓ2

otherwise.

0 Note that we can consider, in the last theorem JC,ψ ((τ1 )(ℓ3 ) ) as Jψ(0,j−ℓ (τ1 ), and 3 ,ℓ3 ) 0 JC,ψ−1 ((τ2 )[ℓ1 ] ) as Jψ(ℓ (τ ). Then we can rewrite the assertion of the last 2 1 ,ℓ2 +ℓ3 −j,0) theorem as

GL (k)

Jψℓ˜(IndPj,NN−j τ1 ⊗ τ2 ) ≡ L L

L′

j−ℓ3 ≤s≤ℓ2 −2

indPsℓ˜′ | det ·|

1−(j−s)+ℓ3 −ℓ1 2

(j−s)

τ1

⊗ | det ·|

j−s 2

0 Jψ(ℓ

1 ,ℓ2 −s,ℓ3 −j+s)

(τ2 )

0≤s≤j L′

indQℓ˜′

0≤r<j−ℓ3 −1

ℓ2 +ℓ3 −j+r

| det ·|−

ℓ1 −r 2

0 Jψ(r,j−ℓ

3 −r,ℓ3 )

(τ1 ) ⊗ | det |

ℓ3 −r−1 2

[ℓ −r]

τ2 1

j−ℓ2 −ℓ3 ≤r≤ℓ1

 L J

C,ψ

0,



L′

indP ℓ˜′′

j−ℓ3

 ℓ3 ℓ1 | det ·|− 2 (τ1 )(ℓ3 ) ⊗ | det | 2 τ2[ℓ1 ] ,

if

0 < j − ℓ3 < ℓ2

otherwise.

(6.30) Recall that our goal is to describe F Jψ′˜,γ (π). We have the following proposition, ℓ where, for simplicity, we keep the notation of the last theorem.

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Proposition 6.11. (1) We have, for j − ℓ3 ≤ s ≤ ℓ2 − 2, 0 ≤ s ≤ j,  L′ (j−s) 0 JH2(ℓ2 −2) /C (indPsℓ˜′ | det ·|α1 τ1 ⊗ | det ·|α2 Jψ(ℓ

1 ,ℓ2 −s,ℓ3

GLℓ −2 (k) (j−s) ∼ ′ ⊗ | det ·|α2 F Jψ(ℓ = IndPs,ℓ22−s−2 γ −1 | det ·|α1 τ1

(τ2 )) ⊗ ωψ−1 ,γ −1 −j+s)

1 ,ℓ2 −s,ℓ3 −j+s)



,γ (τ2 ).

(2) We have, for 0 ≤ r < j − ℓ3 − 1, j − ℓ3 − ℓ2 ≤ r ≤ ℓ1 , JH2(ℓ

2



−2) /C

L′

(indQℓ˜′

ℓ2 +ℓ3 −j+r

GLℓ −2 (k) ∼ = IndPj−ℓ2 −r−2,ℓ 3

Proof.

[ℓ1 −r]

| det ·|α1 Jψ0

(r,j−ℓ3 −r,ℓ3 )

2 +ℓ3 −j+r

(τ1 ) ⊗ | det ·|α2 τ2

| det ·|α1 F Jψ′

(r,j−ℓ3 −r,ℓ3 )

,γ (τ1 )

) ⊗ ωψ−1 ,γ −1 [ℓ1 −r]

⊗ γ −1 | det ·|α2 τ2



.

Define L

(j−s)

T : (indPsℓ˜′ | det ·|α1 τ1 GL

(k)

0 ⊗ | det ·|α2 Jψ(ℓ

(j−s)

−2 7→ IndPs,ℓℓ2−s−2 γ −1 | det ·|α1 τ1 2

by T (f ⊗ φ)(g) =

Z

H′s \H2(ℓ2 −2)

1 ,ℓ2 −s,ℓ3 −j+s)

′ ⊗ | det ·|α2 F Jψ(ℓ

(τ2 )) ⊗ ωψ−1 ,γ −1

1 ,ℓ2 −s,ℓ3 −j+s)

,γ (τ2 )

(1 ⊗ JH2(ℓ2 −s−2) /C )[f (ng) ⊗ (ωψ−1 ,γ −1 (ng)φ(0s , ·))]dn.

Here, Hs′ = Ps′ ∩ H2(ℓ2 −2) . The function ξ 7→ ωψ−1 ,γ −1 (ng)φ(0s , ξ) is in S(k ℓ2 −s−2 )(ℓ −s−2)

2 the space of ωψ−1 ,γ −1 , the Weil representation of GLℓ2 −s−2 (k) (with respect to −1 −1 ψ , γ ). For fixed n,g, we regard f (ng) ⊗ ωψ−1 ,γ −1 (ng)φ(0s , ·) as an element

(ℓ −s−2)

(j−s)

2 0 in γ −1 | det ·|α1 τ1 ⊗ | det ·|α2 Jψ(ℓ (τ2 ) ⊗ ωψ−1 ,γ −1 . Note that the 1 ,ℓ2 −s,ℓ3 −j+s) integrand is left Hs′ - invariant. Since g ∈ GLℓ2 −2 (k) normalizes H2(ℓ−2) , it is clear by definition that T factors through the Jacquet module along H2(ℓ−2) /C. It is easy to check that T (f ⊗ φ)(g)is invariant under left translations by the unipotent  b1 , where b1 ∈ GLs (k) and b2 ∈ GLℓ2 −s−2 (k). radical of Ps,ℓ2 −s−2 . Let b = b2 Then it is easy to check that 1

T (f ⊗ φ)(bg) = δP2s,ℓ

2 −s−2

(j−s)

(τ1

(b)γ −1 (det(b1 ))| det(b1 )|α1 | det(b2 )|α2

′ (b1 ) ⊗ F Jψ(ℓ

1 ,ℓ2 −s,ℓ3 −j+s),γ

(τ2 )(b2 ))(T (f ⊗ φ)(g)).

The details of the rest of the proof, establishing that T is injective on the Jacquet module along H2(ℓ−2) /C, and that T is surjective, are similar to those of Proposition 6.6, and we omit them. This proves part (1). The proof of part (2) is similar with T (f ⊗ φ)(g) = Z Z ωψ−1 ,γ −1 (ng)φ(·, x)dx]dn, (JH2(j−ℓ3 −r−2) /C ⊗ 1)[f (ng) ⊗ H′′ r \H2(ℓ2 −2)

kℓ2 +ℓ3 −j+r

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where Hr′′ = H2(ℓ2 −2) ∩Q′ℓ2 +ℓ3 −j+r . Recall also, that ℓ2 +ℓ3 −j+r = ℓ2 −(j−ℓ3 −r) ≤ ℓ2 − 2. Note that the inner dx integration is the evaluation at 0ℓ2 +ℓ3 −j+r of the partial Fourier transform of x 7→ ωψ−1 ,γ −1 (ng)φ(·, x).  As in Sec. 5.2, we apply the results above to the unramified constituent πτ of 1 1 GL2m (k) τ | det ·| 2 ⊗ τ | det ·|− 2 , where τ is an irreducible, generic, unramified repIndPm,m resentation of GLm (k). Write τ as the representation induced from the standard Borel subgroup and its character µ1 ⊗ · · · ⊗ µm , where µi are unramified characters of k ∗ . Then, as in Sec. 5.2, πτ is the unramified constituent of the representation π(τ ), which is the representation induced from P2,...,2 and its character µ1 (detGL2 ) ⊗ · · · ⊗ µm (detGL2 ). m−2 2

Theorem 6.6. We have, for all

< ℓ < m,

F Jψℓ ,γ (π(τ )) = 0. ′ Recall that F Jψℓ ,γ (π(τ )) = F Jψ(ℓ,2(m−ℓ),ℓ) ,γ (π(τ )). We will use the no-

Proof.

GL

2(m−i) tation of Theorem 5.8. Thus, τ1,i = µi (detGL2 ), τ2,i = IndP2,...,2 µi+1 (detGL2 ) ⊗ · · · ⊗ µm (detGL2 ). Let us apply Theorem 6.5 (the version in (6.30)) and Proposition 6.11 with τ1 = τ1,1 , τ2 = τ2,1 , N = 2m, j = 2, ℓ1 = ℓ3 = ℓ, ℓ2 = 2(m − ℓ). Note that the second sum in (6.30) is empty (since ℓ ≥ 1). The third summand in (6.30) is zero since (τ1,1 )(1) = 0. We get L F Jψℓ ,γ (π(τ )) ≡

2−ℓ≤s≤2(m−ℓ−1) s=1,2

GL

(k)

2(m−ℓ−1) IndPs,2(m−ℓ−1)−s γ −1 | det ·|

−1+s 2

(2−s)

τ1,1

⊗ | det ·|

2−s 2

′ F Jψ(ℓ,2(m−ℓ)−s,ℓ−2+s) ,γ (τ2,1 ).

Substitute in Theorem 6.5, (6.30), Proposition 6.11, ℓ1 = ℓ, ℓ2 = 2(m − ℓ) − s, ′ ℓ3 = ℓ − 2 + s, τ1 = τ1,2 and τ2 = τ2,2 , to compute F Jψ(ℓ,2(m−ℓ)−s,ℓ−2+s) ,γ (τ2,1 ). ˜ Assume that ℓ2 ≥ 2, ℓ3 ≥ 1. Then we get, as above, for ℓ = (ℓ1 , ℓ2 , ℓ3 ), L F Jψℓ˜,γ (τ2,1 ) ≡ 2−ℓ3 ≤s≤ℓ2 −2 s=1,2

GL

(k)

−2 IndPs,ℓℓ2−2−s γ −1 | det ·|

−1+s+ℓ3 −ℓ1 2

2

(2−s)

τ1,2

⊗ | det ·|

2−s 2

′ F Jψ(ℓ,ℓ

2 −s,ℓ3 −2+s)

,γ (τ2,2 ).

Continue with ℓ1 = ℓ and the new ℓ2 , ℓ3 being ℓ2 − s, ℓ3 − 2 + s, τ1 = τ1,3 , τ2 = τ2,3 , ′ and so on, as long as ℓ2 ≥ 2 and ℓ3 ≥ 1. Note that F Jψ(ℓ,2,ℓ ,γ (τ2,i ) = 0, if ℓ3 > 1 3) (and ℓ + 2 + ℓ3 = 2(m − i)). We get (as in (5.79)) L F Jψℓ ,γ (π(τ )) ≡ s1 ,...,si =1,2

ℓ−2i+s1 +···+si =1

GL

(k)

(2−s1 )

IndPs 2(m−ℓ−1) γ −1 | det ·|z1 τ1,1 ,...,s ,ℓ −2 1

i

2

(2−si )

⊗ · · · ⊗ γ −1 | det ·|zi τ1,i

′ ⊗ F Jψ(ℓ,ℓ

2 ,1)

,γ (τ2,i ).

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Here, ℓ2 = 2(m − ℓ) − (s1 + · · · + si ), and the exponents zt depend on i, s1 , ..., si . ′ Consider F Jψ(ℓ,ℓ ,γ (τ2,i ). As above, we get ,1) 2

′ F Jψ(ℓ,ℓ

2 ,1)

,γ (τ2,i )

GL

−2 ≡ IndP2,ℓℓ2−4 2

L

GL

−2 IndP1,ℓℓ2−3

(k) −1

γ

(k) −1

| det ·|

2−ℓ 2

′ τ1,i+1 ⊗ F Jψ(ℓ,ℓ

1−ℓ 2

2 −2,1)

,γ (τ2,i+1 )

(1)

′ τ1,i+1 ⊗ F Jψ(ℓ,ℓ ,γ (τ2,i+1 ), 2 −1,0)   (2(m−i−2)−ℓ) ′ . where F Jψ(ℓ,ℓ ,γ (τ2,i+1 ) = JH2(2(m−i−2)−ℓ) /C JC,ψ ((τ2,i+1 )[ℓ] ⊗ ωψ −1 ,γ −1 −1,0) 2 If ℓ2 = 2, we get zero. If ℓ − 2 = 3, omit the first term. Now, consider F Jψ(ℓ,ℓ2 −2,1) ,γ (τ2,i+1 ) (if ℓ2 ≥ 4), and as in (5.81), we get that the semi-simplification of F Jψℓ ,γ (π(τ )) is the direct sum of representations of GL2(m−ℓ−1) (k), parabolically induced from representations of the form 2

(2−s1 )

γ −1 | det ·|z1 τ1,1

γ

| det ·|

(2−si′ )

⊗ · · · ⊗ γ −1 | det ·|zi′ τ1,i′ ′ ⊗| det ·|zi′ +2 F Jψ(ℓ,ℓ

2 ,0)

(1)

⊗ γ −1 | det ·|zi′ +1 τ1,i′ +1

,γ (τ2,i′ +1 ),

where now, i′ ≥ i, for i as above, and then s1 , ..., si remain as above, and st = 2, for i < t ≤ i′ (if i′ > i); also, ℓ2 = 2(m − ℓ) − (s1 + · · · + si′ ). We have ℓ = 2i − (s1 + · · · + si ) + 1 = ℓ = 2i′ − (s1 + · · · + si′ ) + 1 ≤ 2i′ − i′ + 1 = i′ + 1.

We explained in the proof of Theorem 5.4, that in order that (τ2,i′ +1 )[ℓ] be nonzero, we must have that ℓ+1 ≤ m−(i′ +1). The same explanation (which actually appears also in the next proof) shows that in order that JC,ψ ((τ2,i′ +1 )[ℓ] ) be nonzero, we must have the stronger inequality ℓ + 2 ≤ m − (i′ + 1). Thus, and hence ℓ ≤

m−2 2 .

ℓ ≤ i′ + 1 ≤ m − ℓ − 2,

This proves the theorem.



Let us examine F Jψℓ ,γ (π(τ )) at the “first occurrence” ℓm = [ m−2 2 ].

Theorem 6.7. For m even, we have F Jψℓm ,γ (π(τ )) ∼ = F Jψℓm ,γ (πτ ) ∼ = γ −1 ⊗ τ. Proof. Write m = 2n. Then ℓm = n − 1. We keep the notation of the last proof, with ℓ = n − 1. Then we have n − 1 = 2i′ − (s1 + · · · + si′ ) + 1

n + 1 ≤ 2n − (i′ + 1).

Hence, we must have s1 = · · · = si′ = 1 and i′ = n − 2, and we get F Jψ ,γ (π(τ )) ∼ = n−1

GL

(k)

GL

(k)

(1)

(1)

2n γ −1 τ1,1 ⊗ · · · ⊗ γ −1 τ1,n−1 ⊗ | det ·| IndP1,...,1,n+1 2n γ −1 µ1 ⊗ · · · ⊗ γ −1 µn−1 ⊗ | det ·| IndP1,...,1,n+1

n−1 2

n−1 2

∼ ′ F Jψ(n−1,n+3,0) ,γ (τ2,n−1 ) =

′ F Jψ(n−1,n+3,0) ,γ (τ2,n−1 ).

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1 The representation (τ2,n−1 )[n−1] is a module over P1,n+2 , the subgroup of P1,n+2 , ∗ ∗ where, we replace k by 1, in the Levi subgroup k ×GLn+2 (k). Since the application of JC,ψ to the Jacquet module of (τ2,n−1 )[n−1] , with respect to the trivial character 1 of the unipotent radical of P1,n+2 , is zero, we have P1

1

ψ ⊗ | det ·| 2 (τ2,n−1 )[n] ), JC,ψ ((τ2,n−1 )[n−1] ) ∼ = JC,ψ (indP1,n+2 1,1 1,1,n+1

where 1,1 P1,1,n+1

     1zx =  1 u ∈ GLn+3 (k) ,   g

1,1 to representation takes an element of P1,1,n+1   1 1u ψ(z)| det(g)| 2 (τ2,n−1 )[n] . Since the highest derivative of τ2,n−1 is of index g 1 n + 1, we get that (τ2,n−1 )[n] is isomorphic, as a representation of P1,n+1 , to [n+1] 1 (τ2,n−1 ) (extended trivially, from GLn+1 (k) to P1,n+1 ). Thus,

and

the

inducing

1

1 P ψ ⊗ | det ·| 2 (τ2,n−1 )[n+1] ). JC,ψ ((τ2,n−1 )[n−1] ) ∼ = JC,ψ (indP1,n+2 1,1 1,1,n+1

It will be more convenient to use the following isomorphic realization, which results from a conjugation by an appropriate Weyl element, 1

1 P ψ ′ ⊗ | det ·| 2 (τ2,n−1 )[n+1] ), JC,ψ ((τ2,n−1 )[n−1] ) ∼ = JC,ψ (indP¯1,n+2 1,1 1,n+1,1

where 1,1 P¯1,n+1,1

    1vz  =  g 0 ∈ GLn+3 (k) ,   u1 1

and the inducing representation acts by ψ(z)| det(g)| 2 (τ2,n−1 )[n+1] (g). P1

ψ′ ⊗ Consider the map t, which takes a function f in the space of indP¯1,n+2 1,1 1,n+1,1

1

| det ·| 2 (τ2,n−1 )[n+1] to ResP 1,1

1,n+1,1

1,1 P1,n+1,1

Clearly, ResP 1,1

1,n+1,1

P 1,1

f , where

    1xz  =  g y  ∈ GLn+3 (k) .   1 1

f ∈ indQ1,n+1,1 ψ ′ ⊗ | det ·| 2 (τ2,n−1 )[n+1] , where ′ n+1

Q′n+1

  1x =  g 

  z  0 ∈ GLn+3 (k) .  1

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1,1 It is easy to see that t induces an isomorphism of P1,n+1,1 -modules 1,1

1

ResP 1,1

1,n+1,1

1 1 P1,n+1,1 ′ P ψ ⊗| det ·| 2 (τ2,n−1 )[n+1] . JC,ψ (ind ¯1,n+2 ψ ′ ⊗| det ·| 2 (τ2,n−1 )[n+1] ) ∼ = indQ′ 1,1

P1,n+1,1

n+1

Now, we are at the situation of Proposition 6.11(2), and we get, with a similar proof, that P

1,1

1

1

(n+1)

−1 [n+1] ψ ′ ⊗ | det ·| 2 (τ2,n−1 )[n+1] ⊗ ωψ−1 ,γ −1 ) ∼ JH2(n+1) /C (indQ1,n+1,1 , = γ ⊗ | det ·| 2 (τ2,n−1 ) ′ n+1

as GLn+1 (k)-modules. This shows that ∼ −1 ⊗ | det ·| 12 (τ2,n−1 )[n+1] , ′ F Jψ(n−1,n+3,0) ,γ (τ2,n−1 ) = γ as GLn+1 (k)-modules. Thus, n GL2n (k) γ −1 µ1 ⊗ · · · ⊗ γ −1 µn−1 ⊗ γ −1 | det ·| 2 (τ2,n−1 )[n+1] . F Jψn−1 ,γ (π(τ )) ∼ = IndP1,...,1,n+1 n GL (k) As in the proof of Theorem 5.9, we have | det ·| 2 (τ2,n−1 )[n+1] ∼ = IndBGLn+1 (k) µn ⊗ n+1 · · · µ2n , and so

GL (k) F Jψn−1 ,γ (π(τ )) ∼ = γ −1 ⊗ τ. = IndBGL2n (k) γ −1 µ1 ⊗ · · · γ −1 µ2n ∼ 2n

This proves the theorem.



Theorem 6.8. Assume that m = 2n + 1 is odd, (so that ℓm = n − 1). Then F Jψℓm ,γ (πτ ) ≺ M 1 GL (k) γ −1 ⊗ IndEt 2n+2 µ1 ⊗ · · · ⊗ | det ·| 2 µt+1 (detGL2 ) ⊗ · · · ⊗ µ2n+1 0≤t
M

GL

n−1≤t≤2n

γ −1 ⊗ IndEt 2n+2

(k)

1

µ1 ⊗ · · · ⊗ | det ·|− 2 µt+1 (detGL2 ) ⊗ · · · ⊗ µ2n+1 M

GL

indB ′ 2n+2

(k) −1

γ

µ1 ⊗ · · · ⊗ γ −1 µ2n+1 .

Here, Et is the standard parabolic subgroup of GL2n+2 (k) of type (1, ..., 1, 2, 1, ..., 1), where 2 appears at the (t + 1)-th place;     b1 x y  B ′ =  1 z  ∈ GL2n+2 (k) b1 ∈ BGLn−1 (k) , b2 ∈ BGLn+2 (k) ,   b2

and γ −1 µ1 ⊗ · · · ⊗ γ −1 µ2n+1 takes a matrix in B ′ , whose diagonal is diag(x1 , ..., xn−1 , 1, xn , ..., x2n+1 ) to γ −1 (x1 · · · x2n+1 )µ1 (x1 ) · · · µ2n+1 (x2n+1 ). Proof.

As in the last proof, and with similar notation, we have n − 1 = 2i′ − (s1 + · · · + si′ ) + 1

n + 1 ≤ 2n + 1 − (i′ + 1).

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Thus, s1 +· · ·+si′ = i′ , i′ +1. In the first case, we get s1 = · · · si′ = 1, and i′ = n−2. In the second case, we get that i′ = n − 1 and all sr are equal to 1, except one of them, which is equal to 2. Consider the first case, where i′ = n − 2. Then, as in the last proof, we get the following contribution to the semi-simplification of F Jψn−1 ,γ (π(τ )): GL

(k)

2n+2 γ −1 µ1 ⊗ ⊗γ −1 µn−1 ⊗ | det ·| IndP1,...,1,n+3

n−1 2

′ F Jψ(n−1,n+5,0) ,γ (τ2,n−1 ).

Again, as in the last proof,

1

1 P ψ ⊗ | det ·| 2 (τ2,n−1 )[n] ). JC,ψ ((τ2,n−1 )[n−1] ) ∼ = JC,ψ (indP1,n+4 1,1 1,1,n+3

1 P1,n+3 -filtration

Now, consider the of (τ2,n−1 )[n] . It has two pieces. One piece is 1 (τ2,n−1 )[n+1] , regarded as a representation of P1,n+3 , where the unipotent radical acts trivially. The second piece is 1

1

1,1,n+2

1,1,n+2

1 1 P P ψ ⊗ | det ·| 2 (τ2,n−1 )[n+1] ∼ ψ ⊗ | det ·| 2 (τ2,n−1 )[n+2] , indP1,n+3 = indP1,n+3 1,1 1,1

1 where (τ2,n−1 )[n+2] acts trivially on the unipotent radical of P1,n+2 . Again, this follows from the fact that the highest derivative of τ2,n−1 is of index n + 2. For the first piece, we repeat the arguments of the last proof, and get the contribution (to the semi-simplification of F Jψn−1 ,γ (π(τ ))) GL

n

(k)

2n+2 IndP1,...,1,n+3 γ −1 µ1 ⊗ · · · ⊗ γ −1 µn−1 ⊗ γ −1 | det ·| 2 (τ2,n−1 )[n+1] ≡

M

GL

n−1≤t≤2n

γ −1 ⊗ IndEt 2n+2

(k)

1

µ1 ⊗ · · · ⊗ | det ·|− 2 µt+1 (detGL2 ) ⊗ · · · ⊗ µ2n+1 .

Similarly, the second piece will contribute to F Jψn−1 ,γ (π(τ )), the representation GL (k) indB ′ 2n+2 γ −1 µ1 ⊗ · · · ⊗ γ −1 µ2n+1 . Consider the second case, where i′ = n− 1. This gives the following contribution to F Jψn−1 ,γ (π(τ )): M 1 GL2n+2 (k) −1 IndP1,...,2,...,1 γ µ1 ⊗ · · · ⊗ γ −1 | det ·| 2 µt+1 (detGL2 ) ⊗ · · · ⊗ γ −1 µn 0≤t
⊗| det ·|

Again,

n−1 2

′ F Jψ(n−1,n+3,0) ,γ (τ2,n ).

1

1 P ψ ⊗ | det ·| 2 (τ2,n )[n] ) JC,ψ ((τ2,n )[n−1] ) ∼ = JC,ψ (indP1,n+2 1,1 1,1,n+1

1 P1,n+2

JC,ψ (indP 1,1

1,1,n+1

1

ψ ⊗ | det ·| 2 (τ2,n )[n+1] ).

As before, we get that | det ·|

n−1 2

∼ −1 ⊗ IndGLn+1 (k) µn+1 ⊗ · · · ⊗ µ2n+1 , ′ F Jψ(n−1,n+3,0) ,γ (τ2,n ) = γ BGL (k) n+1

and we get the contribution M 1 GL (k) γ −1 ⊗ IndEt 2n+2 µ1 ⊗ · · · | det ·| 2 µt+1 (detGL2 ) ⊗ · · · ⊗ µ2n+1 . 0≤t
This proves the theorem.



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We finished the local preparatory work for our plan, outlined in Sec. 3.6. In the last chapter and this one we obtained general Leibniz rules for Jacquet modules corresponding to Gelfand-Graev characters or Fourier-Jacobi characters. For the sake of obtaining this property in general and for future reference, we did not restrict ourselves just to the cases which correspond to functoriality and to our descent map. We now proceed to the global part of the plan in Sec. 3.6.

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Chapter 7

The tower property

We proceed to the global part of the plan outlined in Sec. 3.6. The assumptions and notations in this chapter are those of Sec. 3.1, Sec. 3.2. Thus, to recall some of these, our classical group h(V ), over the number field F , is split, or quasi-split. We denote the corresponding group over A by HA , and in case h(V ) is symplectic, we allow HA to be the metaplectic group of h(V )A (see Sec. 2.1). We denote by m ˜ the Witt index of V (and the corresponding form b) over F , and we denote m = [ 21 dimE V ]. In this chapter we consider smooth automorphic functions of moderate growth f on HA and develop general formulas for constant terms, along unipotent radicals ∗ of parabolic subgroups of the groups (3.31), Hℓ,w of Fourier coefficient of Gelfand0 Graev type (resp. of Fourier-Jacobi type) of f . We will apply these formulas to cuspidal representations of HA and also to the residual representations Eτ¯ of Chapter 2, and thereby prove that the descents σψ,α (¯ τ ), σψ,γ (¯ τ ) defined in Sec. 3.6 are ∗ cuspidal representations of the group Hℓ∗m ,yα (A) (resp. Hm (A) = Hℓ∗m ,eℓm +1 (A); see (3.40), (3.41)), in the sense that all constant terms, along unipotent radicals of parabolic subgroups, are zero. This implies, in particular, that the elements of σψ,α (¯ τ ), or σψ,γ (¯ τ ) are rapidly decreasing (and, certainly, square integrable). Indeed these elements are smooth and of uniform moderate growth, and now we may use Corollary I.2.12 in [Moeglin and Waldspurger (1995)]. We will show that the various constant terms of elements of the descents above are expressed in terms of elements in the spaces σψℓ,α (¯ τ ) of (3.40) and σψℓ ,γ (¯ τ ) of (3.33), (3.34) are zero, for ℓ > ℓm , where ℓm is given by (3.36), (3.38). The vanishing of the last spaces will follow from the theorems of the previous chapter. We will therefore get a tower property for these spaces; at the first index, counting backwards, ℓ = m, m − 1, ..., which later will be shown to be ℓm , where the space, corresponding to ℓ, is nonzero, we get a cuspidal representation. This resembles the Rallis tower property in the theory of the theta correspondence. The tower property of the descents above was first observed by the authors in [Ginzburg, Rallis and Soudry (1999)]. The full proof for the case where H is symplectic appeared in loc. cit. and in [Ginzburg, Rallis and Soudry I (1999)], [Ginzburg, Rallis and Soudry (2002)]. It was announced, in general, in [Soudry (2005)]. The fact that the descents above are nontrivial will be proved in the next two chapters. 151

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A general lemma on “exchanging roots”

The following lemma will be used repeatedly. We call it the lemma on exchanging roots, because it allows to replace, in an integration along a unipotent group, computing a Fourier coefficient, certain one parameter subgroups (root subgroups in most cases) by others, in the sense that there is a simple identity relating the two integrations, which, in particular, implies that the first Fourier coefficient is non-trivial, if and only if the second Fourier coefficient is nontrivial. The setup is as follows. Let C ⊂ h(V ) be an F - subgroup of a maximal unipotent subgroup of h(V ), and let ψC be a non-trivial character of CF \CA . Assume that there are two unipotent F - subgroups X, Y , such that the following conditions are satisfied. (1) X and Y normalize C. (2) X ∩ C and Y ∩ C are normal in X and Y , respectively, and X ∩ C\X and Y ∩ C\Y are abelian. (3) XA and YA preserve ψC (when acting by conjugation). (4) ψC is trivial on (X ∩ C)A and on (Y ∩ C)A . (5) [X, Y ] ⊂ C. (6) The pairing (X ∩ C)A \XA × (Y ∩ C)A \YA −→ C∗ , given by (x, y) 7→ ψC ([x, y]), is multiplicative in each coordinate, non-degenerate, and identifies (Y ∩C)F \YF with the dual of XF (X ∩ C)A \XA and (X ∩ C)F \XF with the dual of YF (Y ∩ C)A \YA . Note that the first five conditions imply that, for each y ∈ YA , ψC ([x, y]) is a character of XA , which is trivial on (X ∩C)A . (Similarly, when we fix x, we get a character of (Y ∩ C)A \YA .) Indeed, for x1 , x2 ∈ XA , we have [x1 x2 , y] = x1 [x2 , y]x−1 1 [x1 , y]. Hence ψC ([x1 x2 , y]) = ψC (x1 [x2 , y]x−1 1 )ψC ([x1 , y]) = ψC ([x1 , y])ψC ([x2 , y]). Since y preserves ψC , we have ψC ([y, c]) = ψC (ycy −1 )ψC (c−1 ) = ψC (c)ψC (c−1 ) = 1, and so, ψC ([y, c]) = 1, for all c ∈ CA . We represent the setup above by the following diagram, A B = CY

ր

տ

տ

ր

C

D = CX

(7.1)

Here, A = BX = DY = CXY . Extend the character ψC to a character ψB , of BF \BA , and to a character ψD of DF \DA , by making it trivial on YA and on XA .

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The conditions above can be adapted to metaplectic groups HA . In this case, we let A′ , B ′ , C ′ , D′ , X ′ , Y ′ ⊂ h(V ) be F - subgroups of the corresponding symplectic group, satisfying the conditions above. The subgroups C ′ , X ′ , Y ′ , being unipotent, split uniquely, when we take Adele points (and also when we take points over a local field). Let CA , XA , YA ⊂ HA be the corresponding splittings. Then our assumptions above are for the groups CA , XA , YA , i.e. XA and YA normalize CA etc. Note that the splitting of the F -rational points of an F -subgroup R is R(F ) × 1. Finally, the character ψC is by definition ψC ′ , that is, for (c, ǫ(c)) ∈ CA , ψC ((c, ǫ(c))) = ψC ′ (c). Lemma 7.1. Let f be a smooth automorphic function on HA . Assume that f is of uniform moderate growth. (See [Moeglin and Waldspurger (1995)], I.2.3.) Then Z Z Z −1 −1 f (uy)ψD (u)dudy. (7.2) f (v)ψB (v)dv = (Y ∩C)A \YA DF \DA

BF \BA

The r.h.s. of (7.2) converges in the sense Z Z −1 f (uyh)ψD (u)du (Y ∩C)A \YA

DF \DA

dy < ∞,

and this convergence is uniform as h varies in compact subsets of HA . Proof.

By our assumptions and notation, we have Z Z Z −1 −1 f (cy)ψC (c)dcdy. f (v)ψB (v)dv = BF \BA

(7.3)

YF (Y ∩C)A \YA CF \CA

For fixed h ∈ HA , consider the following function on (X ∩ C)A \XA , Z −1 f (cxh)ψC (c)dc. x 7−→ φh (x) = CF \CA

Since X normalizes C and preserves ψC , this function is left- invariant by XF , and we view it as a function on the compact abelian group XF (X ∩ C)A \XA . By our assumptions, a character of this group has the form ψC ([x, y ′ ]), for some y ′ ∈ (Y ∩ C)F \YF . Let us write the Fourier expansion of φh and evaluate it at 1: Z X −1 φh (x)ψC ([x, y ′ ])dx. (7.4) φh (1) = y ′ ∈(Y ∩C)F \YF X (X∩C) \X F A A

This Fourier expansion converges absolutely and uniformly in compact subsets of HA . This follows from our assumption that f is smooth and of uniform moderate growth. Thus, for y ∈ YA , Z Z Z X −1 −1 f (cxy)ψC (c[x, y ′ ])dcdx. f (cy)ψC (c)dc = CF \CA

y ′ ∈(Y ∩C)F \YF X (X∩C) \X C \C F F A A A

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Change variable in the dc-integration, c 7→ c[x, y ′ ]−1 . Note that, by our assumptions, [x, y ′ ] ∈ CA . We get Z Z X −1 f (c[y ′ , x]xy)ψC (c)dcdx. y ′ ∈(Y ∩C)F \YF X (X∩C) \X C \C F F A A A

Now, note that, in the last integral, we have f (c[y ′ , x]xy) = f (cy ′ xy ′−1 y) = f (y ′−1 cy ′ xy ′−1 y). We used the fact that f is automrphic. Change variable in the dc-integration c 7→ y ′ cy ′−1 . This operation preserves CF \CA and ψC . This gives (changing y ′−1 to y ′ ) Z Z X −1 f (cxy ′ y)ψC (c)dcdx = y ′ ∈(Y ∩C)F \YF X (X∩C) \X C \C F F A A A

Z

X

−1 f (uy ′ y)ψD (u)du.

y ′ ∈(Y ∩C)F \YF D \D F A

We proved

Z

−1 f (cy)ψC (c)dc

=

X

Z

−1 f (uy ′ y)ψD (u)du.

(7.5)

y ′ ∈(Y ∩C)F \YF D \D F A

CF \CA

Substitute this in (7.3), and we get (7.2). For this, and for the statement about uniform absolute convergence, made at the end of the lemma, we use the uniform absolute convergence, in compact subsets of HA , of the Fourier expansion (7.4). This proves the lemma.  Corollary 7.1. Let f be a smooth automorphic function on HA . Assume that f is of uniform moderate growth. Then Z −1 (v)dv ≡ 0, ∀a ∈ A(A), f (va)ψB BF \BA

if and only if

Z

−1 (u)du ≡ 0, f (ua)ψD

∀a ∈ A(A).

DF \DA

Proof.

Let a ∈ A(A) be such that Z −1 f (va)ψB (v)dv 6= 0. BF \BA

Then by (7.2), with f replaced by the right a translate of f , there is y ∈ YA , such that Z −1 f (uya)ψD (u)du 6= 0, DF \DA

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this being an inner integral of a nonzero double integral. Conversely, let a ∈ A(A) be such that Z −1 f (ua)ψD (u)du 6= 0. DF \DA

Replacing f by its right translate by a, we may assume that a = 1. Then there is a Schwartz function ϕ′ ∈ S((Y ∩ C)A \YA ), such that Z Z −1 ′ f (uy)ψD (u)dudy 6= 0. ϕ (y) DF \DA

(Y ∩C)A \YA

By our assumptions, we may write ϕ′ as the Fourier transform ϕˆ of a Schwartz function ϕ ∈ S((X ∩ C)A \XA ), using the pairing ψC ([x, y]). The last integral is then equal to Z Z Z −1 −1 f (uy)ψD (u)dudy = ϕ(x)ψC ([x, y])dx DF \DA

(Y ∩C)A \YA (X∩C)A \XA

Z

Z

Z

ϕ(x)

−1 f (uy)ψD (u[x, y])dudydx.

(7.6)

DF \DA

(X∩C)A \XA (Y ∩C)A \YA

The order change of the dx, dy integrations is justified, since, from the last lemma, it follows that Z Z Z −1 f (uy)ψD (u)du dydx < ∞. ϕ(x) DF \DA

(X∩C)A \XA (Y ∩C)A \YA

Change variable, in (7.6), u 7→ u[x, y]−1 x. Recall that the restriction of ψD to CA is ψC , [x, y] ∈ CA , ψD is trivial on XA , and that D = CX. We get Z Z Z −1 f (u[y, x]xy)ψD (u)dudydx = ϕ(x) 0 6= DF \DA

(X∩C)A \XA (Y ∩C)A \YA

Z

Z

ϕ(x)

−1 f (uyx)ψD (u)dudydx =

DF \DA

(X∩C)A \XA (Y ∩C)A \YA

Z

Z

ϕ(x)

(X∩C)A \XA

Z

−1 f (vx)ψB (v)dv.

BF \BA

We used (7.2). Thus, there exists x ∈ XA , such that Z −1 f (vx)ψB (v)dv 6= 0. BF \BA

This proves the corollary.



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Using similar ideas as in the last corollary, we can explain better the convergence in Lemma 7.1. Corollary 7.2. Let f be a smooth automorphic function on HA . Assume that f is of uniform moderate growth. Then there exist smooth automorphic functions on HA , of uniform moderate growth, f1 , ..., fr , and Schwartz functions φ1 , ..., φr ∈ S((Y ∩ C)A \YA ), such that, for all y ∈ (Y ∩ C)A \YA , Z Z r X −1 −1 fi (uy)ψD (u)du. (7.7) φi (y) f (uy)ψD (u)du = i=1

DF \DA

DF \DA

Proof. By definition, there is a positive integer n, such that, for all elements Z in the complexified Lie algebra of the product of H(Fv ), over the archimedean places, we have that vZ,n (f ) = suph∈HA khknF |Z ∗ f (h)| is finite. Here, we fix a norm on HA , as in [Moeglin and Waldspurger (1995)], I.2.2, and define, for h ∈ HA , khkF = infγ∈HF kγhk. Denote the space of such smooth automorphic functions on HA , by Aumg,n (H). This is a Fr´echet space (with the topology defined by the seminorms vX,n , as X varies) on which HA acts smoothly. See [Casselman (1989)], [Kudla Rallis and Soudry (1992)], Lemma 2.4. By the lemma of Dixmier-Malliavin (see [Dixmier and Malliavin (1978)]), there exist f1 , ..., fr ∈ Aumg,n (H) and ϕ1 , ..., ϕr ∈ Cc∞ (XA ), such that r X ϕi ∗ fi , f= i=1

where

ϕi ∗ fi (h) =

Z

ϕi (x)fi (hx)dx.

XA

We regard here Aumg,n (H) as a module over A(A). We have Z Z Z −1 −1 ϕi (x)fi (u[y, x]xy)ψD (u)dxdu. ϕi ∗ fi (uy)ψD (u)du = DF \DA XA

DF \DA

We may switch the order of integration and then change variable u 7→ u[x, y]x−1 . We Zget Z Z −1 −1 −1 fi (uy)ψD (u)du, (u)du = φi (y) fi (uy)ψD ϕi (x)ψD ([x, y])dx XA

DF \DA

DF \DA

where φi is the function in S((Y ∩ C)A \YA ), which is the Fourier transform of the function ϕ′i ∈ Cc∞ ((X ∩ C)A \XA ) defined by Z ϕi (x′ x)dx′ . ϕ′i (x) = (X∩C)A

This proves the corollary.



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7.2

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A formula for constant terms of Gelfand-Graev coefficients

In this section, assumptions are as in Sec. 3.1. Thus, the form b which defines H = h(V ) is symmetric, or Hermitian, and H is either orthogonal, split or quasisplit, over F , or H is unitary, quasi-split over F . Let f be a smooth automorphic function of uniform moderate growth on HA . Let 0 ≤ ℓ ≤ m, and consider the Fourier coefficient of Gelfand-Graev type f ψℓ,α , given by (3.8) (with w0 = yα , see ∗ (3.10)). We restrict this Fourier coefficient to Hℓ,α (A) = Lℓ,yα (A). For ℓ = 0, this is ⊥ ∗ just the restriction of f to h(yα )A . We know that f ψℓ,α is automorphic on Hℓ,α (A). See (3.9). Assume that ℓ < m. ˜ Denote by m ˜ ℓ,α the Witt index of the restriction of the form b to Wm,ℓ ∩ yα⊥ . Of course, m ˜ ℓ,α ≤ m ˜ − ℓ. For 1 ≤ p ≤ m ˜ ℓ,α , consider ∗ ∗ the (standard) maximal parabolic subgroup Qp of Hℓ,α , which preserves the totally isotropic subspace Vℓ,p,+ = SpanE {eℓ+1 , ..., eℓ+p } ∩ yα⊥ ⊂ Wm,ℓ ∩ yα⊥ . Denote by Up∗ the unipotent radical of Q∗p . The main result of this section is a general formula for the constant term of f ψℓ,α along Up∗ , Z f ψℓ,α (u)du. cp (f ψℓ,α ) = Up∗ (F )\Up∗ (A)

We will use the matrix form of our groups, as before. For any 1 ≤ j ≤ m, ˜ we identify GL(Vj+ )A with GLj (AE ) by   γ γˆ =  Im′ −2j  . γ∗ Denote ( )  ∧ Ip+ℓ−i ∗ i Uℓ+p = z ∈ ResE/F Zi · Uℓ+p ⊂ Nℓ+p , 1 ≤ i ≤ p + ℓ − 1 z L = Lp,ℓ

L(i)

( ∧ ) Ip = ∗ Iℓ

   λ1  ∧  Ip  ..  = ∈ L |λ =  .  ,   λ Iℓ λℓ

λj = 0,

( ) ∧ Ip Li = ∈ L |λℓ−i = · · · = λℓ = 0 , λ Iℓ β = βp,ℓ =



Iℓ

Ip

∧

∀j 6= ℓ − i

    

,

i = 0, ..., ℓ − 1,

i = 0, ..., ℓ − 1,

. (7.8)

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Finally, recall that Qp = Dp ⋉Up is the Levi decomposition of the parabolic subgroup Qp ⊂ H. We denote by f Up the constant term of f along the unipotent radical Up . Theorem 7.1. Let f be a smooth automorphic function, of uniform moderate growth on HA , say f ∈ Aumg,n (H). Let 0 ≤ ℓ < m ˜ and 1 ≤ p ≤ m ˜ ℓ,α . Assume, also, that when m′ = 2m + 1 is odd, and m ˜ ℓ,α = m − ℓ, we have that p < m − ℓ. Then cp (f ψℓ,α ) = Z

X

Z

−1 f (uˆ γ λβ)ψℓ+p,α (u)dudλ +

1 γ∈Pp−1,1 (E)\GLp (E)LA U ℓ (F )\U ℓ (A) ℓ+p ℓ+p

Z

(f Up )ψℓ,α (λβ)dλ.

LA

(7.9)

We denote (f Up )ψℓ,α (g) = ((ρ(g)f )Up )ψℓ,α (1), where ρ(g) is the right translation by g. Here, (f Up )ψℓ,α is the Gelfand-Graev coef1 ficient, with respect to ψℓ,α of the restriction of f Up to H(Wm,p )A ⊂ Dp (A); Pp−1,1 is the mirabolic subgroup of GLp . Both dλ-integrations converge as follows. Let 0 ≤ i ≤ ℓ − 1. Then in the second term of (7.9) Z Z Z (f Up )ψℓ,α (λ(i) · · · λ(0) β)dλ(0) · · · dλ(i−1) dλ(i) < ∞; ··· (i)

LA

(i−1)

LA

(0)

LA

in the first term, we have, for each γ ∈ GLp (E), Z Z (i)

LA

···

(i−1)

LA

Z

Z

ℓ (0) U ℓ (F )\Uℓ+p (A) ℓ+p

LA

and

Z

X

−1 (u)dudλ(0) · · · dλ(i−1) dλ(i) < ∞, f (uˆ γ λ(i) · · · λ(0) β)ψℓ+p,α

Z

1 γ∈Pp−1,1 (E)\GLp (E) LA U ℓ (F )\U ℓ (A) ℓ+p ℓ+p

−1 f (uˆ γ λβ)ψℓ+p,α (u)dudλ < ∞.

Moreover, there is ϕ ∈ Aumg,n (H), in W (f ), the closed cyclic module generated by 1 (AE ), f , such that, for all g ∈ Pp−1 Z g ), g λβ)dλ = (ϕUp )ψℓ,α (ˆ (f Up )ψℓ,α (ˆ LA

and Z

Z

LA U ℓ (F )\U ℓ (A) ℓ+p ℓ+p

−1 f (uˆ gλβ)ψℓ+p,α (u)dudλ =

Z

ℓ ℓ Uℓ+p (F )\Uℓ+p (A)

−1 ϕ(uˆ g )ψℓ+p,α (u)du.

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Proof. We start by performing a conjugation by β in the integration defining cp (f ψℓ,α ). Note that β ∈ HF . Z Z −1 f (βvrβ −1 β)ψℓ,α (v)dvdr. (7.10) cp (f ψℓ,α ) = Up∗ (F )\Up∗ (A) Nℓ (F )\Nℓ (A)

The elements βvrβ −1 , in (7.10), have the form   Ip 0 x d y u z a e d′    −1 βvrβ =  Im′ −2(ℓ+p) a′ x′    = s(z; u, a, d, e; x, y), ∗  z 0

(7.11)

u′ Ip

(ℓ+p)

(ℓ+p)

with z ∈ Zℓ (AE ) and x · yα = 0, where yα is the vector yα , regarded as a m′ −2(ℓ+p) ′ column vector in E . Note that m − 2(ℓ + p) 6= 1, since, otherwise, we must have that m′ = 2m + 1 is odd (and then m ˜ = m) and p = m − ℓ, contrary to our assumption. We will treat this case separately later. Note that (ℓ+p)

βUp∗ β −1 = {s(Iℓ ; 0, 0, 0, 0; x, y) ∈ H|x · yα = 0}, βNℓ β −1 = {s(z; u, a, d, e; 0, 0) ∈ H|z ∈ ResE/F Zℓ }, (ℓ+p) ψℓ,α (v) = ψE,1 (z1,2 + z2,3 + · · · + zℓ−1,ℓ + aℓ · yα ).

(7.12)

Denote S = βUp∗ Nℓ β −1 . This is the subgroup of elements of the form (7.11), which (ℓ+p)

lie in H, and satisfy x · yα = 0. Denote by ψS the character of SA , given by the third formula in (7.12). It is obtained by, first, extending ψℓ,α to Up∗ (A)Nℓ (A), so that it is trivial on Up∗ (A), and then let, for s ∈ SA , ψS (s) = ψℓ,α (β −1 sβ). Let Se be the subgroup of S, which consists of the elements of the form (7.11), with u = 0, z = Iℓ . Note that L = Lp,ℓ is the subgroup of elements in S, which are lower triangular. In this proof, we will identify ResE/F Zℓ as a subgroup of S. Thus, Z Z Z (7.13) f (sλzβ)ψS−1 (sz)dsdλdz. cp (f ψℓ,α ) = eF \S eA Zℓ (E)\Zℓ (AE ) LF \LA S

Note that ψS is trivial on LA . Let J = {s(Iℓ ; 0, 0, 0, 0; x, y) ∈ H} and J0 = J ∩ S = βUp∗ β −1 . Then J0 \J e Y = is naturally identified with E p , and hence it is abelian. The groups C = S, (0) (0) e e L , X = J, B = SL , D = SJ satisfy the assumptions (1)-(6) represented in (7.1). Assumptions (1)-(4) are immediately checked, and for the remaining two, it is easy to see that the commutator of an element of Y and an element of X, written as above, has the form   0   s(Iℓ ; 0,  ...  , ∗, ∗; 0, 0), u·x

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e and on adele points, application of ψC gives and hence lies in C = S, (ℓ+p) ψE,1 (uℓ xyα ). Here, all the rows of u, except the last one, uℓ , are zero. e = Uℓ+p , and ψD = ψℓ+p,α . Note that D = SJ By Lemma 7.1, we conclude that Z Z Z −1 f (sλh)ψS (s)dsdλ = (0)

(0)

LF \LA

eF \S eA S

DA

(0)

LA

Z

−1 f (uλh)ψℓ+p,α (u)dudλ.

Uℓ+p (F )\Uℓ+p (A)

(7.14)

Note that by the lemma, we also know that Z Z −1 f (uλh)ψℓ+p,α (u)du dλ < ∞, (0)

LA

(7.15)

Uℓ+p (F )\Uℓ+p (A)

and this convergence is uniform in compact subsets of HA . We conclude that Z Z Z −1 ψℓ,α f (uλzβ)ψℓ+p,α (uz)dudλdz. cp (f )= Zℓ (E)\Zℓ (AE ) L0 (F )\LA Uℓ+p (F )\Uℓ+p (A)

(7.16)

Note that ResE/F Zℓ normalizes L0 . Write L0 = L(1) L1 , Write also, Zℓ = Z (1) ResE/F Zℓ−1 , where Z (1) =

  Iℓ−1 ∗ , 1

ResE/F Zℓ−1 =

   z ∈ ResE/F Zℓ . 1

Recall that we identify ResE/F Zℓ with {s(z; 0, 0, 0, 0; 0, 0)|z ∈ ResE/F Zℓ }. Rewrite (7.16) as Z Z Z −1 I1 (f, ψ)(λzλ(0) β)ψℓ+p,α (z)dλdzdλ(0) , cp (f ψℓ,α ) = (0)

LA

Zℓ−1 (E)\Zℓ−1 (AE ) L1 (F )\L1 (A)

(7.17)

where, for h ∈ HA I1 (f, ψ)(h) = Z (1)

(1)

LF \LA

Z

Z

−1 f (uz (1) λ(1) h)ψℓ+p,α (uz (1) )dudz (1) dλ(1) .

Z (1) (E)\Z (1) (AE ) Uℓ+p (F )\Uℓ+p (A)

(7.18)

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Note that, by Corollary 7.2 and its proof, there is f0 ∈ W (f ), such that, for all z ∈ Zℓ (AE ), λ0 ∈ L0 (A), Z Z Z −1 −1 f0 (uλ0 z)ψℓ+p,α (u)du, f (uλ0 zλβ)ψℓ+p,α (u)dudλ = (0)

LA

Uℓ+p (F )\Uℓ+p (A)

Uℓ+p (F )\Uℓ+p (A)

(7.19)

and then, it follows, from (7.14), (7.16), that Z Z −1 I1 (f0 , ψ)(λz)ψℓ+p,α (z)dλdz. cp (f ψℓ,α ) = Zℓ−1 (E)\Zℓ−1 (AE ) L1 (F )\L1 (A)



(7.20)

∧ 0 x Consider the groups C = Uℓ+p Z (1) , Y = L(1) , X = { Iℓ−1 0 }, ψC = 1 ψℓ+p,α . It is easy to verify that we have the set-up (7.1). By Lemma 7.1, we get

Ip

CA

I1 (f, ψ)(h) =

Z

(1)

LA

Z

−1 f (uλ(1) h)ψℓ+p,α (u)dudλ(1) .

(7.21)

1 1 Uℓ+p (F )\Uℓ+p (A)

1 . We conclude from (7.17), (7.21), Note that, in this case, D in Corollary 7.2 is Uℓ+p that

cp (f ψℓ,α ) = Z Z (1,0)

LA

Z

Z

−1 f (uλzλ′ β)ψℓ+p,α (uz)dudλdzdλ′ ,

1 1 Zℓ−1 (E)\Zℓ−1 (AE ) L1 (F )\L1 (A) Uℓ+p (F )\Uℓ+p (A)

(7.22) where L(1,0) = L(1) L(0) , and the integration dλ′ is a repeated integration dλ(0) dλ(1) ; the dλ(1) -integration is absolutely convergent. Repeating the last argument for f0 , and using Corollary 7.2 and its proof, as in (7.19), (7.20), we get that there is f1 ∈ W (f ), such that Z Z Z −1 ψℓ,α cp (f )= f1 (uλz)ψℓ+p,α (uz)dudλdz. 1 1 Zℓ−1 (E)\Zℓ−1 (AE ) L1 (F )\L1 (A) Uℓ+p (F )\Uℓ+p (A)

Denote, for 1 ≤ i ≤ ℓ − 1, ResE/F Zℓ−i =

 z

Ii



 ∈ ResE/F Zℓ ,

embedded in H, via (7.12). Assume, by induction, that, for 1 ≤ i ≤ ℓ − 2,

(7.23)

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cp (f ψℓ,α ) = Z (i,...,0)

LA

Z

Z

Z

−1 f (uλzλ′ β)ψℓ+p,α (uz)dudλdzdλ′ ,

i i Zℓ−i (E)\Zℓ−i (AE ) Li (F )\Li (A) Uℓ+p (F )\Uℓ+p (A)

(7.24) where L(i,...,0) = L(i) · ... · L(1) · L(0) , and dλ′ is a repeated integration dλ(i) · · · dλ(0) . Each integration dλ(j) is absolutely convergent, for j = 0, 1, ..., i. Also, assume, by induction, that there is fi ∈ W (f ), such that Z Z Z −1 fi (uλz)ψℓ+p,α (uz)dudλdz. cp (f ψℓ,α ) = i i Zℓ−i (E)\Zℓ−i (AE ) Li (F )\Li (A) Uℓ+p (F )\Uℓ+p (A)

(7.25)

Write where

Li = L(i+1) Li+1 , Z (i+1)

ResE/F Zℓ−i = Z (i+1) ResE/F Zℓ−i−1 ,

   Iℓ−i−1 x 0  =  1 0  ⊂ ResE/F Zℓ .   Ii

This is a subgroup of ResE/F Zℓ , embedded in H, via (7.12). Rewrite (7.24) as cp (f ψℓ,α ) = Z Z Z −1 Ii+1 (f, ψ)(λzλ′ β)ψℓ+p,α (z)dλdzdλ′ , (i,...,0)

LA

Zℓ−i−1 (E)\Zℓ−i−1 (AE ) Li+1 (F )\Li+1 (A)

where, for h ∈ HA Ii+1 (f, ψ)(h) =

Z

(i+1)

LF

(i+1)

\LA

Z

(7.26) Z

f (uz (i+1) λ(i+1) h)

i i Z (i+1) (E)\Z (i+1) (AE ) Uℓ+p (F )\Uℓ+p (A)

(7.27)

−1 ψℓ+p,α (uz (i+1) )dudz (i+1) dλ(i+1) . i Consider the groups C = Uℓ+p Z (i+1) , Y = L(i+1) ,  ∧    Ip 0 x        I 0 ℓ−i−1   X=  ,  1         Ii

and the character ψC = ψℓ+p,α By Lemma 7.1, we get Z Ii+1 (f, ψ)(h) =

(i+1) LA

. It is easy to verify that we have the set-up (7.1).

CA

Z

i+1 i+1 Uℓ+p (F )\Uℓ+p (A)

−1 f (uλ(i+1) h)ψℓ+p,α (u)dudλ(i+1) .

(7.28)

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We conclude from (7.27), (7.28), that cp (f ψℓ,α ) = Z (i+1,...,0)

LA

Z

Z

Z

−1 f (uλzλ′ β)ψℓ+p,α (uz)

Zℓ−i−1 (E)\Zℓ−i−1 (AE ) Li+1 (F )\Li+1 (A) U i+1 (F )\U i+1 (A) ℓ+p

ℓ+p

(7.29) dudλdzdλ′ ,

where the integration dλ′ is a repeated integration dλ(0) ...dλ(i+1) ; the dλ(i+1) integration is absolutely convergent. Repeating the last argument for (7.25), and using Corollary 7.2 and its proof, as in (7.19), (7.20), we get that there is fi+1 ∈ W (f ), such that cp (f ψℓ,α ) = Z

Z

Z

−1 fi+1 (uλz)ψℓ+p,α (uz)dudλdz.

Zℓ−i−1 (E)\Zℓ−i−1 (AE ) Li+1 (F )\Li+1 (A) U i+1 (F )\U i+1 (A) ℓ+p

ℓ+p

(7.30) Consider (7.29), for i = ℓ − 2. This is Z Z cp (f ψℓ,α ) =

−1 f (uλβ)ψℓ+p,α (u)dudλ,

(7.31)

LA U ℓ−1 (F )\U ℓ−1 (A) ℓ+p ℓ+p

where the integration dλ is a repeated integration, and convergence is as above. 1 (AE ) (the When deriving (7.30), it is easy to see that we also get, for all g ∈ Pp−1,1 mirabolic subgroup of GLp ) Z Z Z −1 −1 f (uˆ gλβ)ψℓ+p,α (u)dudλ = fℓ−1 (uˆ g )ψℓ+p,α (u)du. LA U ℓ−1 (F )\U ℓ−1 (A) ℓ+p ℓ+p

ℓ−1 ℓ−1 Uℓ+p (F )\Uℓ+p (A)

(7.32) Denote ϕ = fℓ−1 . Let x ∈ ApE be a column vector. Define  ∧ I x nℓ (x) = p ∈ Nℓ+p (A). 1 Consider the function φ(x) =

Z

−1 ϕ(unℓ (x))ψℓ+p,α (u)du.

ℓ−1 ℓ−1 Uℓ+p (F )\Uℓ+p (A)

This is a smooth function on E p \ApE . Write its Fourier expansion, and evaluate at x = 0. A general character of nℓ (x), trivial on E-rational elements, has the

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form ψE,1 (t η · x), where η ∈ E p is a column vector. The Fourier coefficient of φ, corresponding to the trivial character, i.e. η = 0, is Z Z −1 ϕ(unℓ (x))ψℓ+p,α (u)dudx = (ϕUp )ψℓ,α (1). ℓ−1 ℓ−1 E p \Ap E Uℓ+p (F )\Uℓ+p (A)

1 Next, let η 6= 0, and write t η = (0, ..., 1)γ, with γ ∈ Pp−1,1 (E)\GLp (E). Then the Fourier coefficient of φ, corresponding to η, is Z Z −1 ϕ(unℓ (x))ψℓ+p,α (unℓ (γx))dudx = ℓ−1 ℓ−1 E p \Ap E Uℓ+p (F )\Uℓ+p (A)

Z

Z

−1 ϕ(ˆ γ uˆ γ −1 nℓ (γx)ˆ γ )ψℓ+p,α (unℓ (γx))dudx.

ℓ−1 ℓ−1 E p \Ap E Uℓ+p (F )\Uℓ+p (A)

Here, we used the fact that ϕ is automorphic, and hence, ϕ(unℓ (x)) = ϕ(ˆ γ unℓ (x)) = ϕ(ˆ γ uˆ γ −1 nℓ (γx)ˆ γ ). Changing variables u 7→ γˆ −1 uˆ γ, Z

x 7→ γ −1 x, we get −1 ϕ(uˆ γ )ψℓ+p,α (u)du.

ℓ ℓ Uℓ+p (F )\Uℓ+p (A)

From (7.31), (7.32), we get X

cp (f ψℓ,α ) =

Z

1 γ∈Pp−1,1 (E)\GLp (E)U ℓ

−1 ϕ(uˆ γ )ψℓ+p,α (u)du + (ϕUp )ψℓ,α (1).

ℓ ℓ+p (F )\Uℓ+p (A)

(7.33) Finally, we have Z

−1 ϕ(uˆ γ )ψℓ+p,α (u)du =

ℓ ℓ Uℓ+p (F )\Uℓ+p (A)

Z

Z

Z

−1 f (unℓ (x)ˆ γ λβ)ψℓ+p,α (unℓ (x))dudλdx =

LA U ℓ−1 (F )\U ℓ−1 (A) E p \Ap E ℓ+p ℓ+p

Z

Z

−1 f (uˆ γ λβ)ψℓ+p,α (u)dudλ.

ℓ ℓ LA Uℓ+p (F )\Uℓ+p (A)

In order to see this, we have to justify the switch of integration order, in the second line, of dx and dλ(ℓ−1) , dx and dλ(ℓ−2) , up to dx and dλ(0) . Indeed, this follows

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by tracing back, at each step, the construction, via Corollary 7.2, of the functions ϕ = fℓ−1 , fℓ−2 , ..., f0 . The function ϕ has the form X Z ϕ(h) = φi (λ)fℓ−2,i (hλ)dλ, i

(ℓ−1)

LA

(ℓ−1)

for certain finite number of φi ∈ S(LA ), fℓ−2,i ∈ W (f ). Moreover, each φi is the Fourier transform of a function φ′i ∈ Cc∞ (XA ), where   ∧    Ip 0 x  X =  1 0 ∈ Nℓ+p .     1 We have (and now the switch of integration is justified) Z −1 ϕ(uˆ γ )ψℓ+p,α (u)du = ℓ ℓ Uℓ+p (F )\Uℓ+p (A)

X Z i

Z

(ℓ−1)

LA

(ℓ−1)

LA

Z

−1 φi (λ)fℓ−2,i (uˆ γ λ)ψℓ+p,α (u)dudλ =

ℓ ℓ Uℓ+p (F )\Uℓ+p (A)

Z

ℓ ℓ Uℓ+p (F )\Uℓ+p (A)

Z

(ℓ−1)

LA

  XZ −1  φ′i (x)fℓ−2,i (uˆ γ λx)dx ψℓ+p,α (u)dudλ = i X A

Z

−1 fℓ−2 (uˆ γ λ)ψℓ+p,α (u)dudλ.

ℓ ℓ Uℓ+p (F )\Uℓ+p (A)

−1 Here, we used that ψℓ+p,α ([x, λ]) = ψℓ+p,α ([x−1 , λ]), γˆ [x, λ] = [x, λ]ˆ γ , γˆ xˆ γ −1 ∈ −1 ℓ Uℓ+p (A), and ψℓ+p,α (ˆ γ xˆ γ ) = 1. Now write fℓ−2 as a finite sum of convolutions, (ℓ−2)

along LA , of Schwartz functions and elements of W (f ), and so on. We get, by induction, that Z Z −1 fℓ−2 (uˆ γ λ)ψℓ+p,α (u)dudλ = (ℓ−1)

LA

Z

ℓ ℓ Uℓ+p (F )\Uℓ+p (A)

Z

−1 f (uˆ γ λβ)ψℓ+p,α (u)dudλ.

ℓ ℓ LA Uℓ+p (F )\Uℓ+p (A)

Similarly, we get, for any g ∈ GLp (AE ), Z g λβ)dλ. g ) = (f Up )ψℓ,α (ˆ (ϕUp )ψℓ,α (ˆ LA

Substituting back in (7.33), we get (7.9) and the theorem.



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Let us continue the series of Fourier expansions of the last proof, along the first p − 1 coordinates of column p, then along the first p − 2 coordinates of column p − 1, and so on, exactly as the Shalika expansion in [Shalika, J. (1974)], p. 190. The only difference being that we get also contributions from Fourier coefficients of trivial characters. Thus, in the notation of the end of the last proof, consider, for fixed γ ∈ GLp (E), the following function on E p−1 \Ap−1 E , Z −1 ϕ(unℓ+1 (x)ˆ γ )ψℓ+p,α (u)du, φ(x) = ℓ ℓ Uℓ+p (F )\Uℓ+p (A)

 ∧ Ip−1 x ∈ Nℓ+p (A). Then, exactly as before, when we wrote 1 the Fourier expansion in nℓ (∗), we get Z X −1 φ(0) = γ ); ϕ(uγˆ′ γˆ)ψℓ+p,α (u)du + (ϕUp−1 )ψℓ+1,α (ˆ where nℓ+1 (x) =

1 γ ′ ∈Pp−2,1 (E)\GLp−1 (E) ℓ+1 ℓ+1 Uℓ+p (F )\Uℓ+p (A)

we denote (ϕUp−1 )ψℓ+1,α (g) = ((ρ(g)ϕ)Up−1 )ψℓ+1,α (1). We conclude from (7.33) and the last theorem that Z X cp (f ψℓ,α ) =

−1 ϕ(uˆ γ )ψℓ+p,α (u)du

1,1 (E)\GLp (E)U ℓ+1 (F )\U ℓ+1 (A) γ∈Pp−2,1,1 ℓ+p

X

+

γ) + (ϕUp−1 )ψℓ+1,α (ˆ

1 γ∈Pp−1,1 (E)\GLp (E)

where, for 1 ≤ i ≤ p, 1,...,1 Pp−i,1,...,1 (E)

ℓ+p

Z

(f Up )ψℓ,α (λβ)dλ,

LA

  gx = ∈ GLp (E) | z

 z ∈ Zi (E) .

Here, 1, ..., 1 is repeated i times. As in the end of the last proof, we get Z Z X −1 cp (f ψℓ,α ) = f (uˆ γ λβ)ψℓ+p,α (u)dudλ 1,1 (E)\GLp (E)LA U ℓ+1 (F )\U ℓ+1 (A) γ∈Pp−2,1,1 ℓ+p ℓ+p

+

X

Z

1 γ∈Pp−1,1 (E)\GLp (E)LA

γ λβ)dλ + (f Up−1 )ψℓ+1,α (ˆ

Z

(f Up )ψℓ,α (λβ)dλ.

LA

We continue and write the Fourier expansion along nℓ+i (x) = ℓ+p−1 Nℓ+p (A), x ∈ E p−i \Ap−i E , for i = 2, ..., p − 1. Note that Uℓ+p 1,...,1 that for i = p − 1, Pp−i,1,...,1 = Zp . This proves

 ∧ Ip−i x ∈ 1 = Nℓ+p , and

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Theorem 7.2. Let f be a smooth automorphic function, of uniform moderate growth on HA . Let 0 ≤ ℓ < m ˜ and 1 ≤ p ≤ m ˜ ℓ,α . Assume, also, that when ′ m = 2m + 1 is odd, and m ˜ ℓ,α = m − ℓ, we have that p < m − ℓ. Then cp (f ψℓ,α ) = X

Z

f

=

p X

γ∈Zp (E)\GLp (E)L

ψℓ+p,α

(ˆ γ λβ)dλ+

p−1 X i=0

A

X

Z

γ λβ)dλ (f Up−i )ψℓ+i,α (ˆ

1,...,1 (E)\GLp (E)LA γ∈Pp−i,1...,1

Z

X

(7.34)

γ λβ)dλ. (f Up−i )ψℓ+i,α (ˆ

i=0 γ∈P 1,...,1 p−i,1,...,1 (E)\GLp (E)LA

The convergence in (7.34) is as in Theorem 7.1. As a corollary, we get Theorem 7.3. With assumptions and notations as in Theorem 7.2, assume that for all 0 ≤ i ≤ p − 1, (f Up−i )ψℓ+i,α = 0.

Then cp (f

ψℓ,α

Z

X

)=

γ∈Zp (E)\GLp (E)L

f ψℓ+p,α (ˆ γ λβ)dλ.

(7.35)

A

In particular, (7.35) is valid when f is cuspidal. Finally, let us consider the case where m′ = dimE V = 2m + 1 is odd, and m ˜ α,ℓ = m − ℓ i.e. H is (split) odd orthogonal, or (quasi-split) odd unitary and α = NE/F α′ , i.e. α is a square, α′2 in F , in case E = F , or is a norm of an element ∗ ∼ α′ ∈ E, when [E : F ] = 2, i.e. Hℓ,α = SO2(m−ℓ) is split even orthogonal, in the first ∗ ∼ case, or Hℓ,α = U2(m−ℓ) is quasi-split unitary, in the second case. As explained in Chapter 3, right after (3.10), we may replace here yα by y(α′ ) = α′ e0 . The point is that, in this case y(α′ ) and yα lie in the same h(Wm,ℓ ) orbit (for any 0 ≤ ℓ < m), and hence the groups h(Wm,ℓ ∩ yα⊥ ) and h(Wm,ℓ ∩ y( α′ )⊥ ) are conjugate within h(Wm,ℓ ). The choice y(α′ ) is very convenient, since the proofs of the last three theorems are valid word for word, for this case, provided we replace everywhere, in the proofs, the characters ψℓ+i,α = ψℓ+i,yα by the characters ψℓ+i,y(α′ ) . For example, we get (7.34). More precisely, for 0 ≤ ℓ < m and 1 ≤ p ≤ m − ℓ, Z X cp (f ψℓ,y(α′ ) ) = f ψℓ+p,y(α′ ) (ˆ γ λβ)dλ+ γ∈Zp (E)\GLp (E)L

p−1 X

X

Z

i=0 γ∈P 1,...,1 p−i,1...,1 (E)\GLp (E)LA

A

γ λβ)dλ (f Up−i )ψℓ+i,y(α′ ) (ˆ

(7.36)

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=

p X i=0

Z

X

γ λβ)dλ. (f Up−i )ψℓ+i,y(α′ ) (ˆ

1,...,1 (E)\GLp (E)LA γ∈Pp−i,1,...,1

Now, we conclude Theorem 7.3 for this case Theorem 7.4. Assume that H is (split) odd orthogonal, or (quasi-split) odd unitary (i.e. m′ = 2m + 1 and m ˜ = m) and that α = NE/F α′ (i.e. m ˜ ℓ,α = m − ℓ, for all 0 ≤ ℓ < m). With remaining assumptions and notations as in Theorem 7.2, assume that for all 0 ≤ i ≤ p − 1, (f Up−i )ψℓ+i,y(α′ ) = 0. Then cp (f

ψℓ,y(α′ )

)=

X

Z

γ∈Zp (E)\GLp (E)L

f ψℓ+p,y(α′ ) (ˆ γ λβ)dλ.

(7.37)

A

In particular, (7.37) is valid when f is cuspidal. In case H is split odd orthogonal and α = α′2 , there is one more constant term to ∗ ∼ consider, since Hℓ,α = SO2(m−ℓ) has one more standard maximal parabolic that we did not consider, namely (we keep the choice y(α′ )) the parabolic subgroup which preserves the isotropic subspace +

Vℓ,m−ℓ = Span {eℓ+1 , ..., em−1 , e−m } .

Denote its unipotent radical by ∗ Um−ℓ . We need to consider the constant term of f ψℓ,y(α′ ) with respect to ∗ Um−ℓ . Denote this corresponding constant term by c′m−ℓ (f ψℓ,y(α′ ) ). Let  Im−1  1  ω′ = −  1   1

 Im−1

   ∈ SO2m+1 (F ).  

Then it is clear that ′

c′m−ℓ (f ψℓ,y(α′ ) ) = cm−ℓ ((f ω )ψℓ,y(α′ ) ), ′

(7.38)

where f ω denotes the conjugation of f by ω ′ . We conclude that (7.36) holds for ′ c′m−ℓ (f ψℓ,y(α′ ) ), where we replace in the r.h.s. f by f ω , and since f is automorphic, we may just replace f by its right ω ′ -translate, and similarly for Theorem 7.4, in this case.

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Global Gelfand-Graev models for cuspidal representations

Theorem 7.3 has a nice application to cuspidal representations of HA , namely they always have global Gelfand-Graev models, with respect to some ψℓ,α and a cuspidal ∗ representation of Hℓ,α (A) = Lℓ,yα (A). We keep the same assumptions as in the previous section. Let π be an irreducible automorphic cuspidal representation of HA . Let 0 ≤ ℓ ≤ m, ˜ and consider ψℓ,α the Gelfand-Graev coefficients f of the cusp forms f in the space of π, when re∗ ∗ stricted to Hℓ,α (A). These are smooth automorphic functions on Hℓ,α (A), which are clearly of uniform moderate growth. (Cuspidality is not needed in order to get this property.) Denote by σψℓ,α (π) the space of all these functions. Note that σψ0,α (π) ∗ is the space of all restrictions to H0,α (A) = h(yα⊥ )A . Of course, σψ0,α (π) 6= 0. Let ℓ0 = ℓπ,α,ψ be the maximal ℓ, 0 ≤ ℓ ≤ m, ˜ such that σψℓ,α (π) 6= 0. Then, by the cuspidality of π, (7.35), (7.37) and (7.38) imply that σψℓ0 ,α (π) is a cuspidal representation of Hℓ∗0 ,α (A). Of course, this last statement is meaningless, when Hℓ∗0 ,α is trivial or abelian . Moreover, since (7.35) is obtained by a sequence of Fourier expansions of cp (f ψℓ,α ), it is clear that for ℓ < ℓ0 , σψℓ,α (π) is noncuspidal (and in particular, non-trivial). This is the tower property of the spaces σψℓ,α (π), ℓ = m, ˜ ..., ℓ0 , ..., 1, 0: for the first index ℓ0 , counting from m ˜ to 0, such that σψℓ0 ,α (π) 6= 0, the representation σψℓ0 ,α (π) (of Hℓ∗0 ,α (A)) is cuspidal, and for ℓ < ℓ0 , σψℓ,α (π) is not cuspidal. We sometimes refer to the index ℓ0 as “the first occurrence” for π, when ψ, α are understood. Since σψℓ0 ,α (π) is a direct sum of irreducible representations, there is an irreducible, automorphic, cuspidal representation σ of Hℓ∗0 ,α (A), which has a non-trivial L2 - pairing with σψℓ0 ,α (π), i.e. there are f ∈ Vπ , ξ ∈ Vσ , such that Z f ψℓ0 ,α (g)ξ(g)dg 6= 0. Hℓ∗

0 ,α

(F )\Hℓ∗

0 ,α

(A)

We say that π has a global Gelfand-Graev model with respect to (ψℓ0 ,α , σ). We summarize this in Theorem 7.5. Let π be an irreducible, automorphic, cuspidal representation of HA . Then there is an index 0 ≤ ℓ0 ≤ m, ˜ such that (1) σψℓ,α (π) = 0, for all ℓ0 < ℓ ≤ m. ˜ (2) σψℓ0 ,α (π) is a non-trivial cuspidal representation. (3) σψℓ,α (π) is non-cuspidal, for all 0 ≤ ℓ < ℓ0 . In particular, there is an irreducible, automorphic, cuspidal representation σ of Hℓ∗0 ,α (A), such that π has a global Gelfand-Graev model with respect to (ψℓ0 ,α , σ).

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The general case: H is neither split nor quasi-split

As we remarked at the end of Sec. 3.1, the notions of Fourier coefficients of GelfandGraev type make sense, without restricting the Witt index m, ˜ and so we may consider our group H = h(V ), for arbitrary symmetric or Hermitiean non-degenerate form b, and consider Gelfand-Graev coefficients with respect to (Nℓ , ψℓ,w0 ), for 0≤ℓ≤m ˜ and anisotropic w0 ∈ Wm,ℓ . Assume that H is neither split, nor quasisplit over F . Our notations and theorems in the previous section make sense for any Witt index m ˜ ≤ m, and so are the proofs, with the following slight modification. In the formulations of Theorems 7.1, 7.2, we take 1 ≤ p ≤ m ˜ ℓ,α . If p = m−ℓ ˜ is possible, then this means that m ˜ ℓ,α = m ˜ − ℓ, and therefore α is represented by a vector, say ∗ wα , in Wm,m ˜ = V0 , and then the stabilizer group Hℓ,α is conjugate, within h(Wm,ℓ ) ⊥ to h(Wm,ℓ ∩ wα ). In this case, we will consider the coefficients f ψℓ,wα instead of f ψℓ,α . Now, the theorems, and the formula (7.34) are valid in the general, with the same notation, and the proofs are the same word for word. For the case where α is represented by a vector in Wm,m ˜ = V0 , we just need to replace in (7.34), and in the proofs, the characters ψℓ+i,α by the characters ψℓ+i,wα . In particular, we get Theorem 7.3 and Theorem 7.5 with the same exact formulation. Therefore, for an irreducible, automorphic, cuspidal representation π of HA , π satisfies the tower property, with respect to the Gelfand-Graev coefficients ψℓ,α ; 0 ≤ ℓ ≤ m, ˜ and π has ∗ ˜ a global Gelfand-Graev model with respect to certain (Hℓ0 ,α , σ), where 0 ≤ ℓ0 ≤ m and an irreducible, automorphic, cuspidal representation σ of Hℓ∗0 ,α (A). 7.5

Global Gelfand-Graev models for the residual representations Eτ¯

In this section, we show that the conditions of Theorem 7.3 are satisfied for most of our residual representations Eτ¯ , from  Chapter 2, when we take ℓ = ℓm given  0 . In this section, since we consider the by (3.36), that is ℓm = m−1+dimV 2 residual representations Eτ¯ , we take (V, b) is as in Sec. 2.1, i.e. Assumption 2.1 holds. (Thus, now, we assume that m ˜ = m, so that when H is orthogonal, it is split, and when H is unitary, it is quasi-split, and, also, in this case, it is in an even number of variables.) Recall that involved in the notation of the residual representation Eτ¯ on HA are, in the notations of Chapter 2, τ1 , ..., τr , which are pairwise different irreducible, unitary, automorphic representations of GL(Vϕ+ (1))A ∼ = GLmr (AE ), respectively, and each τi = GLm1 (AE ),...,GL(Vϕ+ (r))A ∼ (2)

is cuspidal whenever mi = dimE Vϕ+ (i) > 1, and is such that LS (τi , αϕ,i , s) has a pole at s = 1. These are the assumptions of Theorem 2.1, which asserts the existence of the residue representation Eτ¯ of HA . Now, we consider the representations σψℓ,α (¯ τ ) = σψℓ,α (Eτ¯ ). See (3.40). The following proposition implies that, for f ∈ Eτ¯ , the summands in (7.35) are zero, when ℓ ≥ ℓm . Once we we prove this, the verification of of the conditions of Theorem 7.3, for ℓ = ℓm , will show that all

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constant terms, along unipotent radicals, of σψℓm ,α (¯ τ ) are zero. Proposition 7.1. We have σψℓ,α (¯ τ ) = 0, for all ℓ > ℓm . Proof. Write Eτ¯ ∼ = ⊗v πv . (By Theorem 2.1, Eτ¯ is irreducible.) It is enough to show that there is a finite place v, such that J(ψv )ℓ,α (πv ) = 0, for all ℓ > ℓm . For this, take a finite place, such that πv is unramified, and if [E : F ] = 2, assume that (2) Ev is an unramified quadratic extension of Fv . Our assumption that LS (τi , αφ,i , s) (2)

has a pole at s = 1, implies that τi is self-conjugate. Moreover, in case αφ,i = ∧2 , or (2) αφ,i = Asai, we know that ωτi = 1 (1 ≤ i ≤ r). See [Jacquet and Shalika (1990)], A∗ [Shahidi, F. (1981)], [Jacquet and Rallis (1997)], [Flicker (1988)], [Flicker (1991)]. Let τv be the representation of GLm (Ev ), parabolically induced from τ1,v ⊗· · ·⊗τr,v . Then τv is irreducible, generic, unramified, self-conjugate, and πv is the unramified 1 v 2 constituent πτv of IndH Qm,v τv | det ·| . Now, we can apply Theorem 5.6. See also the list in the end of Sec. 5.1. There we proved that J(ψv )ℓ,α (πτv ) = 0, for all ℓ > ℓm . In detail, when H = U4n , SO4n , m = 2n and ℓm = n − 1; see Theorem 5.6(1). When H = SO4n+1 , m = 2n and ℓm = n; see Theorem 5.6(2,4), for the case where α is a square, and Theorem 5.6(1,3), for the case where α is not a square. When H = U4n+2 , SO4n+3 , m = 2n + 1 and ℓm = n; see Theorem 5.6(5). Note that Theorem 5.6 and the list following it contain many cases that are not relevant for the proof of this proposition. Indeed, when H is even unitary or even orthogonal, we know that ωτi = 1, and when H = SO2m , we know that m must be even A∗

(since Ress=1 LS (τi , ∧2 , s) 6= 0 implies that mi is even. See [Jacquet and Shalika (1990)]). 

Theorem 7.6. With notations and assumptions as above, assume that τ1 , ..., τr are not characters, i.e. m1 , ..., mr ≥ 2, in case H = U4n , or H = SO4n+3 . Then the residual representation Eτ¯ of HA satisfies the assumptions of Theorem 7.3, when ℓ = ℓm . This, together with Proposition 7.1, implies (by (7.35)) that σℓm ,α (¯ τ ) is cuspidal in the sense that all constant terms of all elements of σℓm ,α (¯ τ ), along all unipotent radicals of Hℓ∗m ,α , are zero. Proof. Let 1 ≤ p ≤ m − ℓm and 0 ≤ i ≤ p − 1. We want to show that, for all f in the space of Eτ¯ , (f Up−i )ψℓm +i,α = 0. Assume that the constant term f Up−i is not identically zero. Then, by Corollary 2.1, we get that the restriction of this constant term to H(Wm,p−i )A can be expressed as a sum of residual Eisenstein series Eτ ′ of the same form, i.e. obtained as

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multi-residues at (1, ..., 1) from inducing data (see (2.61), (2.63)) τj1 | det |Ej1

s −1 · · · τjt | det |Ejt 2

− 21

⊗

(j1 , ..., jt depend on the Weyl element w ∈ W (Mϕ , Dp−i ), such that bw = 0, in (2.63)). Now, we use Proposition 7.1 for Eτ ′ , with m − (p − i) replacing m. We need to verify that, for 0 ≤ i ≤ p − 1, ℓm + i > ℓm−(p−i) , i.e.

a − (p − i) a ], [ ]+i>[ 2 2 where a = m − 1 + dimF V0 . If a = 2n is even, then [

(7.39)

a − (p − i) p−i 1 a ] = [n − ] ≤ [n − ] = n − 1 < n + i = [ ] + i. 2 2 2 2

If a = 2n + 1 is odd, then the l.h.s. of (7.39) is n + i, and so we may get equality only when i = 0, and then we must have p = 1. Thus, (7.39) is true, except when i = 0, p = 1 and a = 2n + 1 is odd. But if f U1 is not identically zero, then one of the τi is a character of GL1 (AE ) (see (2.24)), which is self-conjugate. Also, we have m + dimF V0 = 2n + 2. If V0 = 0 and H is (even) unitary, then H = U4(n+1) , and this is one of the cases we excluded in our assumptions. H cannot be even orthogonal, since, in this case, our assumption that all exterior square L-functions of τ1 , ..., τr have poles at s = 1, imply in particular that m1 , ..., mr are all even. The remaining case is H = SO2m+1 , and then dimF V0 = 1, and m = 2n + 1, i.e. H = SO4n+3 . This is the second case we excluded in our assumptions. This proves the theorem.  Remark: We do not know yet that σψℓm ,α (¯ τ ) is nonzero. In fact, it may happen that this space is zero. Indeed, consider the case H = SO4n+1 . Assume that the product of central characters of the representations τi , i = 1, ..., r, is trivial, and assume that α is not a square. Then there are infinitely many places v, where α is not a square in Fv . By Theorem 5.6(1), for such a finite place, where data are unramified, we get that the corresponding Jacquet module J(ψv )n,α (πτv ) is zero, and τ ) = 0. The issue of non-vanishing of σℓm ,α (¯ τ ) is the subject of the hence σψn,α (¯ next two chapters. 7.6

A formula for constant terms of Fourier-Jacobi coefficients

In this section, we develop a formula analogous to (7.9), for Fourier-Jacobi coefficients. We assume that (V, b) is as in Sec. 3.2; V is even dimensional (m′ = 2m), the form b is either symplectic, or anti-Hermitian (i.e. δ = −1) with Witt index m. Thus, h(V ) is either symplectic, or quasi-split even unitary. Let f be a smooth automorphic function of uniform moderate growth on HA . Let −1 ≤ ℓ < m, and consider the Fourier-Jacobi coefficient F Jψℓ ,γ (f, φ), given by (3.14), (3.15). We

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ˆ ℓ (A) (see (3.12), (3.13)). When ℓ = m − 1, this coefrestrict this coefficient to H ˆ m−1 = {1}. Thus, we will assume that ficient is the ψ-Whittaker coefficient and H ˆ ℓ (A) (see (3.16)). For ℓ < m − 1. We know that F Jψℓ ,γ (f, φ) is automorphic on H ˆ ˆ ℓ, 1 ≤ p ≤ m − ℓ − 1, let Qp be the (standard) maximal parabolic subgroup of H which preserves the totally isotropic subspace (see (5.3)) + Vℓ+1,p = SpanE {eℓ+2 , ..., eℓ+p+1 }.

ˆp the unipotent radical of Q ˆ p . We will re-denote, Denote by U f ψℓ,γ;φ = F Jψℓ ,γ (f, φ). Consider the constant term cp (f

ψℓ,γ;φ

)=

Z

(7.40)

f ψℓ,γ;φ (u)du.

ˆp (E)\U ˆ p (A) U

We will use the same notation as in Sec. 7.2, (7.8), with(ℓ + 1 replacing ℓ, whenever ∧ )  Ip , β = βℓ+1,p ℓ is not specified in the notation. Thus, L = Lℓ+1,p = ∗ Iℓ+1 etc. Note that in case of a metaplectic group, all these subgroups have a simple splitting, namely of the form (x, 1), when x varies in each of these subgroups. We will therefore identify x with (x, 1). Theorem 7.7. Let f be a smooth automorphic function of uniform moderate growth on HA , say f ∈ Aumg,n (H). Let −1 ≤ ℓ ≤ m − 2 and 1 ≤ p ≤ m − ℓ − 1. Let p m−ℓ−p−1 + + φ1 ∈ S(Vℓ+1,p (A)) ∼ (A)) ∼ ). Put = S(AE ) and φ2 ∈ S(Vℓ+p+1,m−ℓ−p−1 = S(AE φ = φ1 ⊗ φ2 . Then cp (f ψℓ,γ;φ ) = X

Z

1 η∈Pp−1,1 (E)\GLp (E)LA

(m−ℓ−p−1),φ2

θψ−1 ,γ −1

Z

φ1 (i(λ))

−1 f (uˆ ηλβ)ψℓ+p (u)

(7.41)

ℓ+1 ℓ+1 Uℓ+1+p (F )\Uℓ+1+p (A)

(jℓ+p (u))dudλ +

Z

φ1 (i(λ))(f Up )ψℓ,γ;φ2 (λβ)dλ.

LA

∧  Ip Here, for λ = ∈ LA , i(λ) = yℓ+1 - the last row of y. As before, we y Iℓ+1 denote (f Up )ψℓ,γ;φ2 (g) = ((ρ(g)f )Up )ψℓ,γ;φ2 (1), where ρ(g) denotes the right translation by g. In case ℓ = −1, the group L is trivial, and φ1 (i(λ)) should be replaced by φ1 (0p ). Note that Up0 = Up and βp,0 = I. The convergence in (7.41) is as in Theorem 7.1.

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Finally, as in Theorem 7.1, there exists ϕ ∈ Aumg,n (H), lying in W (f ), such that, 1 for all g ∈ Pp−1 (AE ), Z g ), g λβ)dλ = (ϕUp )ψℓ,γ;φ2 (ˆ φ1 (i(λ))(f Up )ψℓ,γ;φ2 (ˆ LA

and

Z

Z

φ1 (i(λ))

LA

(m−ℓ−p−1),φ2

−1 f (uˆ gλβ)ψℓ+p (u)θψ−1 ,γ −1

(jℓ+p (u))dudλ

ℓ+1 ℓ+1 Uℓ+1+p (F )\Uℓ+1+p (A)

=

Z

(m−ℓ−p−1),φ2

−1 ϕ(uˆ g )ψℓ+p (u)θψ−1 ,γ −1

(jℓ+p (u))du.

ℓ+1 ℓ+1 Uℓ+1+p (F )\Uℓ+1+p (A)

Proof. As in the proof of Theorem 7.1, we perform conjugation by β in the integration defining cp (f ψℓ,γ;φ ) Z Z (m−ℓ−1),φ f (βvrβ −1 β)ψℓ−1 (v)θψ−1 ,γ −1 (jℓ (v)r)dvdr. cp (f ψℓ,γ;φ ) = ˆ p (E)\U ˆp (A) Nℓ+1 (F )\Nℓ+1 (A) U

(7.42) In case of a metaplectic group, β is identified with (β, 1), and, similarly, an upper unipotent matrix u is identified with (u, 1). Change variable βvrβ −1 7→ u and ˆp Nℓ+1 β −1 . The elements of S have the form integrate over SF \SA , where S = β U ′ (7.11) with m = 2m and ℓ + 1 replacing ℓ,   Ip 0 x d y u z a e d′    −1  βvrβ =  (7.43) I2(m−(ℓ+1+p)) a′ x′   = s(z; u, a, d, e; x, y),  z∗ 0  u′ Ip

with z ∈ ResE/F Zℓ+1 . Note that in case of a metaplectic group, the corresponding elements are of the form (x, 1), where x is of the form (7.43). Thus, we can keep identifying x with (x, 1). We will continue doing this till the end of this section without further comments. Note also that ˆp β −1 = {s(Iℓ+1 ; 0, 0, 0, 0; x, y) ∈ H}, βU βNℓ+1 β −1 ψℓ (v) jℓ (v) w

= {s(z; u, a, d, e; 0, 0) ∈ H|z ∈ ResE/F Zℓ+1 }, = ψE,−1 (z1,2 + z2,3 + · · · + zℓ,ℓ+1 ), = (uℓ+1 , aℓ+1 , d′ℓ+1 ; eℓ+1,1 − 21 b(w, w)); = (uℓ+1 , aℓ+1 , d′ℓ+1 ) ∈ Wm,ℓ+1 .

(7.44)

Denote by U ′ the subgroup of elements (7.43) with z = Iℓ+1 , u = 0, x = 0, y = 0, ˆp β −1 . Note that U ′′ U ′ = Uℓ+p+1 . We identify, as in the proof and denote U ′′ = β U of Theorem 7.1, ResE/F Zℓ+1 , via (7.43), as a subgroup of H. Then

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cp (f ψℓ,γ;φ ) =

Z

Z

Z

175

−1 f (uλzβ)ψℓ+p (z)

(7.45)

Zℓ+1 (E)\Zℓ+1 (AE ) LF \LA Uℓ+p+1 F \Uℓ+p+1 A (m−ℓ−1),φ

θψ−1 ,γ −1

(jℓ (β −1 uλβ))dudλdz.

(m−ℓ−1)

Recall that ωψ−1 ,γ −1 denotes the Weil representation of H(Wm,ℓ+1 )A . Here, and in the sequel, we extend jℓ to standard unipotent subgroups of H(Wm,ℓ+1 )A as the identity. We have (see (1.4), (1.5)) X (m−ℓ−1) φ −1 θψ uλβ)) = ωψ−1 ,γ −1 (jℓ (β −1 uλβ)φ(η, t)) = −1 ,γ −1 (j(β + η∈Vℓ+1,p (F ) + t∈Vℓ+p+1,m−ℓ−p−1 (F )

X

(m−ℓ−1)

ωψ−1 ,γ −1 ((η, 02(m−ℓ−p−1) , 0p ; 0)jℓ (β −1 uλβ))φ(0, t).

+ η∈Vℓ+1,p (F ) + t∈Vℓ+p+1,m−ℓ−p−1 (F )

Let us write the du integration along Uℓ+p+1 as du′′ du′ , where we first integrate along U ′ and then along U ′′ . By (1.4), (1.5), for fixed η, u′′ , λ, the function (on UA′ ) X (m−ℓ−1) u′ 7→ ωψ−1 ,γ −1 ((η, 02(m−ℓ−p−1) , 0p ; 0)jℓ (β −1 u′ u′′ λβ))φ(0, t) + t∈Vℓ+p+1,m−ℓ−p−1 (F )

is left UF′ -invariant. We have, using the notation of the proof of Theorem 7.1, Z

(m−ℓ−1),φ

f (u′ u′′ λzβ)θψ−1 ,γ −1

(jℓ (β −1 u′ u′′ λβ))du′ =

Z

X

f (η (0) u′ u′′ λzβ)

+ ′ \U ′ η∈V UF ℓ+1,p (F ) A

′ \U ′ UF A

X

(m−ℓ−1)

ωψ−1 ,γ −1 ((η, 0, 0p ; 0)jℓ (β −1 u′ u′′ λβ))φ(0, t)du′ ,

+ t∈Vℓ+p+1,m−ℓ−p−1 (F )

and this is equal to X

Z

f (u′ η (0) u′′ λzβ)

(0) ′ η (0) ∈LF UF \UA′

X

(m−ℓ−1)

ωψ−1 ,γ −1 (jℓ (β −1 u′ η (0) u′′ λβ))φ(0, t)du′ .

(7.46)

+ t∈Vℓ+p+1,m−ℓ−p−1 (F )

∧  Ip Here, η (0) =  o Iℓ  . We used the fact that f is automorphic. Note also η 0 1 that (η, 02(m−ℓ−p−1) , 0p ; 0) = jℓ (β −1 η (0) β). Next, we changed variable u′ 7→

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(η (0) )−1 u′ η (0) . Finally, the switch of order of the du′ -integration and the summation over η is easy to justify. Indeed, the series X X |ωψ−1 ,γ −1 ((η, 0, 0p ; 0)h)φ(0, t)| + + η∈Vℓ+1,p (F ) t∈Vℓ+p+1,m−ℓ−p−1 (F )

converges uniformly, as h varies in compact subsets. Similarly, it is easy to see that when we integrate (7.46) along U ′′ , we can switch the order of summation over (0) η (0) ∈ LF and the du′′ - integration, and then it is easy to see that we get Z Z X f (u′ u′′ η (0) λzβ) (0)

′′ \U ′′ U ′ \U ′ η (0) ∈LF UF A F A

X

(m−ℓ−1)

ωψ−1 ,γ −1 (jℓ (β −1 u′ u′′ η (0) λβ))φ(0, t)du′ du′′ ,

+ t∈Vℓ+p+1,m−ℓ−p−1 (F )

that is X

Z

(0)

f (uη (0) λzβ)

η (0) ∈LF Uℓ+p+1 (F )\Uℓ+p+1 (A)

X

(m−ℓ−1)

ωψ−1 ,γ −1 (jℓ (β −1 uη (0) λβ))φ(0, t)du.

+ t∈Vℓ+p+1,m−ℓ−p−1 (F )

Thus, for z ∈ Zℓ+1 (AE ) and λ0 ∈ L0 (A), Z Z (m−ℓ−1),φ f (uλ(0) λ0 zβ)θψ−1 ,γ −1 (jℓ (β −1 uλ(0) β))dudλ(0) = (0)

(0)

LF \LA

Uℓ+p+1 F \Uℓ+p+1 A

Z

(0)

LA

X

Z

f (uλ(0) λ0 zβ)

Uℓ+p+1 (F )\Uℓ+p+1 (A) (m−ℓ−1)

ωψ−1 ,γ −1 (jℓ (β −1 uλ(0) β))φ(0, t)dudλ(0) .

(7.47)

+ t∈Vℓ+p+1,m−ℓ−p−1 (F )

Let us take φ = φ1 ⊗ φ2 , as in the statement of the theorem. Write, for u, λ(0) , λ0 in (7.47), u = u′ u′′ , u′ ∈ UA′ , u′′ ∈ UA′′ , and λ = λ(0) λ0 . Recall the definition of i(λ) in the statement of the theorem. Note that i(λ) = i(λ(0) ), and jℓ (β −1 λβ) = + jℓ (β −1 λ(0) β). Then we have, for t ∈ Vℓ+p+1,m−ℓ−p−1 (A) (m−ℓ−1)

(m−ℓ−p−1)

ωψ−1 ,γ −1 (jℓ (β −1 uλβ))φ(0, t) = φ1 (i(λ))ωψ−1 ,γ −1

(jℓ+p (u))φ2 (t).

(7.48)

To prove (7.48), write u′ = s(Iℓ+1 , 0, a, d, e; 0, 0), u′′ = s(Iℓ+1 , 0, 0, 0, 0; x, y), as in (7.43). We have (see (7.44)) jℓ (β −1 u′ β) = (0p , aℓ+1 , d′ℓ+1 ; e′ ),

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where e′ = eℓ+1,1 − 12 b(aℓ+1 , aℓ+1 ). Note that

(0p , aℓ+1 , d′ℓ+1 ; e′ ) = (0p , 02(m−ℓ−p−1) , d′ℓ+1 ; 0)(0p , aℓ+1 , 0p ; e′ ).

By (1.4), (1.5), we then have (m−ℓ−1)

ωψ−1 ,γ −1 ((0, aℓ+1 , d′ℓ+1 ; e′ )jℓ (β −1 u′′ λβ))φ(0, t) = ψE,−1 (2b((0p , t, 0m−ℓ−1 ), (02(m−ℓ−1)−p , d′ℓ+1 )) (m−ℓ−1)

ωψ−1 ,γ −1 ((0p , aℓ+1 , 0p ; e′ )jℓ (β −1 u′′ λβ))φ(0, t) (m−ℓ−1)

= ωψ−1 ,γ −1 ((0p , aℓ+1 , 0p ; e′ )jℓ (β −1 u′′ λβ))φ(0, t). Note that (0p , aℓ+1 , 0p ; e′ ) and jℓ (β −1 u′′ β) commute. Next, by (1.4), (1.5), we have, writing x = (x1 , x2 ), with x1 , x2 ∈ Mp,m−ℓ−p−1 (AE ), (m−ℓ−1)

ωψ−1 ,γ −1 (jℓ (β −1 u′′ β)(0p , aℓ+1 , 0p ; e′ )jℓ (β −1 λβ))φ(0, t) 

 Ip 0 x2 y ′    Im∗ 0 x′2  (m−ℓ−1)   (0p , aℓ+1 , 0p ; e′ )jℓ (β −1 λβ))φ (0, t) Ip x1 = ωψ−1 ,γ −1 (  Im∗ Im∗ 0  Ip

  x2 y ′ ))) = ψE,−1 (b((0p , t, 0m∗ , 0p ), (0p , 0m∗ , (0p , t) 0 x′2 (m−ℓ−1)

ωψ−1 ,γ −1 ((0p , aℓ+1 , 0p ; e′ )jℓ (β −1 λβ))φ(0, t) (m−ℓ−1)

= ωψ−1 ,γ −1 ((0p , aℓ+1 , 0p ; e′ )jℓ (β −1 λβ))φ(0, t). Here, we put, for short, y ′ = y − x1 x′2 , m∗ = m− ℓ − p− 1. Note that (0p , aℓ+1 , 0p ; e′ ) and jℓ (β −1 λβ) commute, and that (aℓ+1 ; e′ ) = jℓ+p (u′ ) = jℓ+p (u′ u′′ ) = jℓ+p (u). Finally, by (1.4), (1.5), we have (m−ℓ−1)

ωψ−1 ,γ −1 (j(β −1 λβ)(0p , aℓ+1 , 0p ; e′ )φ(0, t) = (m∗ )

(m∗ )

φ1 (i(λ))ωψ−1 ,γ −1 ((aℓ+1 ; e′ ))φ2 (t) = φ1 (i(λ))ωψ−1 ,γ −1 (jℓ+p (u))φ2 (t). This proves (7.48). We conclude that (7.47) is equal to Z Z (m−ℓ−p−1),φ2 f (uλ0 λ(0) zβ)θψ−1 ,γ −1 (jℓ+p (u))dudλ(0) . φ1 (i(λ(0) )) (0)

LA

Uℓ+p+1 (F )\Uℓ+p+1 (A)

(7.49)

Substituting this in (7.45), we get cp (f

ψℓ,γ;φ

)=

Z

Z

Z

Zℓ+1 (E)\Zℓ+1 (AE ) L0 (F )\L0 (A) L(0) A

φ1 (i(λ(0) ))

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Z

(m−ℓ−p−1),φ2

−1 f (uλ0 λ(0) zβ)ψℓ+p (z)θψ−1 ,γ −1

(jℓ+p (u))dudλ(0) dλ0 dz.

Uℓ+p+1 (F )\Uℓ+p+1 (A)

(7.50) We have in (7.50) λ(0) z = z[z −1 , λ(0) ]λ(0) . Note that z and λ0 commute and that [z −1 , λ(0) ] ∈ L0 (A). Now, it is clear that we can switch the order of the dλ(0) , dλ0 integrations in (7.50), and then we change variable λ0 [z −1 , λ(0) ] 7→ λ0 . We get Z Z Z ψℓ,γ;φ cp (f )= φ1 (i(λ(0) )) Zℓ+1 (E)\Zℓ+1 (AE ) L0 (F )\L0 (A) L(0) A

Z

(m−ℓ−p−1),φ2

−1 f (uzλ0 λ(0) β)ψℓ+p (z)θψ−1 ,γ −1

(jℓ+p (u))dudλ(0) dλ0 dz.

Uℓ+p+1 (F )\Uℓ+p+1 (A)

(7.51) We switched the order of the dλ(0) , dλ0 -integrations again. This is easily justifiable, due to the presence of the Schwartz function φ1 . In fact, we can carry out the dλ(0) at any stage we want in (7.51). When we perform it first, let Z φ1 (i(λ(0) ))f (hλ(0) β)dλ(0) . f ′ (h) = (0)

LA

Then f ′ ∈ W (f ) and we get from (7.51) Z Z cp (f ψℓ,γ;φ ) =

Z

f ′ (uzλ0 )

Zℓ+1 (E)\Zℓ+1 (AE ) L0 (F )\L0 (A) Uℓ+p+1 (F )\Uℓ+p+1 (A) (m−ℓ−p−1),φ2

−1 ψℓ+p (z)θψ−1 ,γ −1

(jℓ+p (u))dudλ0 dz.

When we perform the dλ(0) -integration last, we get Z Z (0) ψℓ,γ;φ φ1 (i(λ )) cp (f )= (0)

LA

Z

(7.52) Z

Zℓ+1 (E)\Zℓ+1 (AE ) L0 (F )\L0 (A) (m−ℓ−p−1),φ2

−1 f (uzλ0 λ(0) β)ψℓ+p (z)θψ−1 ,γ −1

(jℓ+p (u))dudλ0 dλ(0) dz.

Uℓ+p+1 (F )\Uℓ+p+1 (A)

(7.53) Now, we are exactly at the same situation as in (7.17), and we perform the same series of Fourier expansions along the first p coordinates of column number p+ ℓ + 1, and then columns p + ℓ, p + ℓ − 1, ..., p + 2. We obtain the analogue of (7.31). cp (f ψℓ,γ;φ ) =

Z

LA

φ1 (i(λ))

Z

ℓ ℓ Uℓ+p+1 (F )\Uℓ+p+1 (A)

(m−ℓ−p−1),φ2

−1 f (uλβ)ψℓ+p (u)θψ−1 ,γ −1

(jℓ+p (u))dudλ. (7.54)

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Note that jℓ+p (u) remains unchanged when u ∈ Uℓ+p+1 (A) is multiplied by an element in Zℓ+p+1 (AE ). Exactly as in (7.32), and using (7.52), there is ϕ ∈ W (f ), 1 such that, for all g ∈ Pp−1,1 (AE ), Z Z (m−ℓ−p−1),φ2 −1 (jℓ+p (u))dudλ f (uˆ gλβ)ψℓ+p (u)θψ−1 ,γ −1 φ1 (i(λ)) LA

ℓ ℓ Uℓ+p+1 (F )\Uℓ+p+1 (A)

=

Z

(m−ℓ−p−1),φ2

−1 ϕ(uˆ g )ψℓ+p (u)θψ−1 ,γ −1

(jℓ+p (u))du.

(7.55)

ℓ ℓ Uℓ+p+1 (F )\Uℓ+p+1 (A)

Now we continue as in (7.33) and get (7.41), as well as the two statements computing the terms of (7.41) in terms of ϕ (which means that the integration along LA is “redundant”). This completes the proof of the theorem.  We continue exactly as we did after the proof of Theorem 7.1 and get the analogue of Theorem 7.2 Theorem 7.8. With notations and assumptions as in Theorem 7.7, we have Z p X X ψℓ,γ;φ cp (f )= φ1 (i(λ))(f Up−k )ψℓ+k,γ;φ2 (ˆ η λβ)dλ. (7.56) k=0 η∈P 1,...,1

p−k,1,...,1 (E)\GLp (E)LA

The convergence in (7.56) is as in Theorems 7.1, 7.2. As a corollary, we get the analogue of Theorem 7.3 Theorem 7.9. Let notations and assumptions be as in Theorem 7.7. Assume that for all 0 ≤ k ≤ p − 1, (f Up−k )ψℓ+k,γ;φ2 = 0. Then cp (f ψℓ,γ;φ ) =

X

Z

η∈Zp (E)\GLp (E)L

φ1 (i(λ))f ψℓ+p,γ;φ2 (ˆ η λβ)dλ.

(7.57)

A

In particular, (7.57) is valid when f is cuspidal. 7.7

Global Fourier-Jacobi models for cuspidal representations

As in the case of Gelfand-Graev coefficients, Theorem 7.9 has a nice application to cuspidal representations of HA , namely they always have global Fourier-Jacobi ˆ ℓ (A). models, with respect to some ψℓ,γ and a cuspidal representation of H We keep the assumptions and notations of the previous section. We also fix the function γ, which enters in the definition of the Weil representation of unitary groups. Let π be an automorphic representation of HA , realized as a subspace Vπ

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of smooth automorphic functions of moderate growth on HA . Let 0 ≤ ℓ < m, and consider the Fourier-Jacobi coefficients f ψℓ,γ;φ , as f varies over Vπ , and φ varies + ˆ ℓ (A). These are in S(Vℓ+1,m−ℓ−1 (A). We consider the restriction of f ψℓ,γ;φ to H smooth automorphic functions. Consider the resulting space of all these functions, ˆ ℓ (A). Let C be the center of the Heisenberg group σψℓ ,γ (π), as a module over H HWm,1 = N1 ,     1 0 z   C = c(z) =  I2m−2 0 |z ∈ F . (7.58)   1 The character ψ defines a character of CA , still denoted by ψ, ψ(c(z)) = ψ(z). We denote by f C,ψ the ψ-Fourier coefficient of f (in Vπ ) along C, Z f (ch)ψ −1 (c)dc. f C,ψ (h) = CF \CA

Proposition 7.2. We have f C,ψ = 0, for all f ∈ Vπ , if and only if f ψ0,γ;φ = 0, for + all f ∈ Vπ and all φ ∈ S(V1,m−1 (A)). Proof.

ˆ 0 (A), By definition (see (3.14)), for h ∈ H Z (m−1),φ f (vh′ )θψ−1 ,γ −1 (j0 (v)h′′ )dv f ψ0,γ;φ (h) = N1 (F )\N1 (A)

=

Z

(m−1),φ

f C,ψ (vh′ )θψ−1 ,γ −1 (j0 (v)h′′ )dv.

N1 (F )CA \N1 (A)

Thus, if f C,ψ = 0, then f ψ0,γ;φ = 0. Conversely, assume that f ψ0,γ;φ ≡ 0 identically. Let us write N1 = N1+ N1− C, where     1 a 0 0         I 0 0 m−1 +  ∈ h(V ) N 1 = a+ =  ′  Im−1 a       1 N1−

   

   1 0 a 0    ′  I 0 a m−1 −   = a = ∈ h(V ) .   Im−1 0      1

Note that for v = a+ e− c(z) ∈ N1 , j0 (v) = (a, 0m−1 ; z)(0m−1 , e; 0), and then, using (1.4), (1.5), we have Z Z X (m−1) f ψ0,γ;φ (1) = f C,ψ (a+ e− ) ωψ−1 ,γ −1 (e− )φ(x+a)da+ de− N1− (F )\N1− (A) N1+ (F )\N1+ (A)

+ x∈V1,m−1

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Z

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(m−1)

f C,ψ (a+ e− )ωψ−1 ,γ −1 (e− )φ(a)dade.

F m−1 \Am−1 Am−1

We have (m−1)

−1 (2b((a, 0m−1 ), (0m−1 , e)))φ(a) ωψ−1 ,γ −1 (e− )φ(a) = ψE,−1

Put, for short b((a, 0m−1 ), (0m−1 , e)) = (a|e). This a non-degenerate form on E m−1 × E m−1 , linear in the first variable, and conjugate-linear in the second variable. We have a+ e− = e− a+ c(trE/F ((a|e))), (7.59) ψE,−1 (2(a|e)) = ψ(trE/F ((a|e)). Hence, continuing the last calculation, we get (the switch of integration order being easy to justify) Z Z ψ0,γ;φ f (1) = f C,ψ (e− a+ )φ(a)deda. (7.60) Am−1 F m−1 \(Am−1

Thus, if f ψ0,γ;φ (1) = 0, for all f ∈ Vπ and all φ ∈ S(Am−1 , then E Z f C,ψ (e− a+ )de = 0,

(7.61)

F m−1 \Am−1

for all f ∈ Vπ and all a ∈ Am−1 . Take now a ∈ E m−1 . Using (7.59) again, we get E that for all f ∈ Vπ , Z −1 f C,ψ (e− )ψE,−1 ((a|e))de ≡ 0, F m−1 \Am−1

for all a ∈ E m−1 . Here, we used the fact that f C,ψ is N1+ (F )- left invariant. Thus, the smooth function e 7→ f C,ψ (e− ), on E m−1 \Am−1 , has zero Fourier coefficients E with respect all characters of E m−1 \Am−1 . This implies that f C,ψ = 0, for all E f ∈ Vπ . 

Note that CN1− is a group over F and ce− 7→ ψ(c) is a character of CN1− (A) (c ∈ CA , e− ∈ N1− (A)). We keep denoting this character by ψ. Put, for f in Vπ , Z − f (ch)ψ −1 (u)du. f N1 C,ψ (h) = (N1− C)F \(N1− C)A

We can rewrite (7.60) as f

ψ0,γ;φ

(h) =

Z

−

f CN1

,ψ

(a+ )ωψ−1 ,γ −1 φ(a)da.

(7.62)

Am−1 −

Then we conclude that f CN1

,ψ

= 0. We record this as −

Proposition 7.3. We have f CN1 ,ψ = 0, for all f ∈ Vπ , if and only if f ψ0,γ;φ = 0, + for all f ∈ Vπ and all φ ∈ S(V1,m−1 (A)).

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Lemma 7.2. Assume that f C,ψ = 0, for all nontrivial characters ψ of F \A. Then f is constant. Proof. Writing the Fourier expansion of the function c 7→ f (ch), along CF \CA (for fixed h), we conclude that f is left CA - invariant. Since f is automorphic it is also left HF - invariant, and hence it is invariant under all left translations by the elements of the group generated by HF and CA . Since f is smooth, we conclude that f is constant on a dense subset of HA , and so f is constant. Note, for example, that when H is symplectic, the group generated by HF and CA is HA .  We conclude from Proposition 7.2 and Lemma 7.2 Corollary 7.3. Assume that the automorphic representation π of HA is not a multiple of the trivial representation. Then there is a nontrivial character ψ of F \A, such that σψ0 ,γ (π) 6= 0. Let now π be an irreducible, automorphic, cuspidal representation of HA . Recall that when H is metaplectic we assume that π is genuine. By the last corollary, there is a nontrivial character ψ of F \A, such that σψ0 ,γ (π) 6= 0. Let ℓ0 = ℓπ,ψ be the maximal ℓ, 0 ≤ ℓ < m, such that σψℓ ,γ (π) 6= 0. Since π is cuspidal, Theorem ˆ ℓ0 (A). Of 7.9 implies that σψℓ0 ,γ (π) is a (nontrivial) cuspidal representation of H course, this statement is meaningless when ℓ0 = m − 1, since then the last group is trivial (but then, as we explained before, π has nontrivial ψ-Whittaker coefficients). Theorem 7.9 also implies that, for ℓ < ℓ0 , σψℓ ,γ (π) is not cuspidal. Thus, the spaces σψℓ ,γ (π), ℓ = m−1, m−2, ..., ℓ0, ..., 1, 0 satisfy the tower property: for the first index ℓ0 , counting from m − 1 backwards, such that σψℓ0 ,γ (π) 6= 0, σψℓ0 ,γ (π) is cuspidal ˆ ℓ0 (A)), and for ℓ < ℓ0 , σψ ,γ (π) is not cuspidal. Since σψ ,γ (π) is a direct (on H ℓ ℓ0 sum of irreducible representations, there is an irreducible, automorphic, cuspidal ˆ ℓ0 (A), which has a nontrivial L2 -pairing with σψ ,γ (π), i.e. representation σ of H ℓ0 there are f ∈ Vπ , φ ∈ S(Vℓ+ (A)), ξ ∈ Vσ , such that 0 +1,m−ℓ0 −1 Z ¯ f ψℓ0 ,γ;φ (g)ξ(g)dg 6= 0. h(Wm,ℓ0 +1 )F \h(Wm,ℓ0 +1 )A

We say that π has a global Fourier-Jacobi model with respect to (ψℓ0 , γ, σ) . Summarizing Theorem 7.10. Let π be an irreducible, automorphic, cuspidal representation of HA . Then there is 0 ≤ ℓ0 < m, such that (1) σψℓ ,γ (π) = 0, for all ℓ0 < ℓ < m. (2) σψℓ0 ,γ (π) is a nontrivial cuspidal representation. (3) σψℓ ,γ (π) is non-cuspidal, for all 0 ≤ ℓ < ℓ0 . In particular, there is an irreducible, automorphic, cuspidal representation σ of ˆ ℓ0 (A), such that π has a global Fourier-Jacobi model with respect to (ψℓ0 , γ, σ). H

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Global Fourier-Jacobi models for the residual representations Eτ¯

We keep the assumptions and notations of the previous two sections. As in Sec. 7.5, the conditions of Theorem 7.9 are satisfied for most of our residual representations Eτ¯ , from Chapter 2, when we take ℓ = ℓm given by (3.38), that is ℓm = [ m−2 2 ], except . In this section ψ and γ in case H is metaplectic and m is odd, and then ℓm = m−1 2 are fixed. Consider the representations σψℓ ,γ (¯ τ ) = σψℓ ,γ (Eτ¯ ). See (3.33) and (3.34). In case H is metaplectic, we consider σψℓ ,γ (¯ τ ) = σψℓ ,γ (Eτ¯,ψ ). Recall that, in the notations of Chapter 2, τ1 , ..., τr , are pairwise different irreducible, unitary, automorphic representations of GL(Vϕ+ (1))A ∼ = GLmr (AE ), = GLm1 (AE ),...,GL(Vϕ+ (r))A ∼ + respectively, and each τi is cuspidal whenever mi = dimE Vϕ (i) > 1, and is such (1)

(2)

that LS (τi , αϕ,i , s) has a pole at s = 1, and LS (τi , αϕ,i , 21 ) 6= 0. Theorem 2.1, asserts the existence of the residue representation Eτ¯ of HA . We first prove the analogue of Proposition 7.1. Proposition 7.4. We have σψℓ ,γ (¯ τ ) = 0, for all ℓ > ℓm . Proof. The proof is similar to the proof of Proposition 7.1 and follows from (2) Theorem 6.4. Note again that the assumption that LS (τi , αϕ,i , s) has a pole at (2)

(2)

s = 1 implies that τi is self-conjugate. Moreover, in case αϕ,i = ∧2 , or αϕ,i = Asai, ωτi = 1. Let v be a finite place of F , such that τ1,v , ..., τr,v are unramified, and if A∗

[E : F ] = 2, then Ev is an unramified quadratic extension of Fv . Assume also that v is odd and that ψv is normalized (this is needed in case H is metaplectic). Let τv be the representation of GLm (Ev ), parabolically induced from τ1,v ⊗· · ·⊗τr,v . Then τv is an irreducible, unramified, generic, self-conjugate representation of GLm (Ev ). 1

′ v 2 Let πτv be the unramified constituent of IndH Qm,v µψv τv | det |Ev . As in the proof of Proposition 7.1, it is enough to show that J(ψv )0ℓ (πτv ) = 0. This follows from Theorem 6.4. In detail, when H = U4n , Sp4n , m = 2n and ℓm = n − 1 (note that f 4n , m = 2n and ℓm = n − 1; see ωτν = 1); see Theorem 6.4(1). When H = Sp Theorem 6.4(1),(4). When H = U4n+2 , m = 2n + 1 and ℓm = n − 1; see Theorem f 4n+2 , m = 2n + 1, ℓm = n; see Theorem 6.4(7),(9). 6.4(6). When H = Sp 

f 4n+2 Remark: Note the special case in the last proof corresponding to H = Sp and ωτv 6= 1. Then Theorem 6.4(9) tells us, more precisely, that J(ψv )0ℓ (πτv ) = 0, for ℓ > n − 1. In the proof of Proposition 7.4, we proved a little more. Let ψℓ0 be the character of Nℓ0 (A) (see Sec. 6.1), defined by ψℓ0 (v) = ψE,−1 (v1,2 + · · · vℓ,ℓ+1 + vℓ+1,2m−ℓ ).

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Denote by f ψℓ the Fourier coefficient Z 0 f ψℓ (h) =

f (vh)ψℓ0 (v −1 )dv.

(7.63)

Nℓ0 (F )\Nℓ0 (A)

Then we proved Proposition 7.5. For any automorphic form f in the residual representation Eτ¯ and ℓ > ℓm , 0

f ψℓ = 0. We end this chapter with the analogue of Theorem 7.6. Theorem 7.11. With notations and assumptions as above, assume that τ1 , ..., τr f 4n . Then are not characters, i.e. m1 , ..., mr ≥ 2, when H = U4n+2 , or H = Sp the residual representation Eτ¯ of HA satisfies the assumptions of Theorem 7.9 when ℓ = ℓm . This together with Proposition 7.4 implies that σψℓm ,γ (¯ τ ) is cuspidal in the sense that all constant terms of all elements of σψℓm ,γ (¯ τ ), along unipotent radicals ˆ ℓm are zero. of H Proof.

Let 1 ≤ p ≤ m − ℓm − 1 and 0 ≤ k ≤ p − 1. Then we want to show that (f Up−k )ψℓm +k ,γ;φ = 0,

identically, for all f in the space of Eτ¯ and all φ ∈ S(Vℓ+ (A)) ∼ = m +p+1,m−ℓm −p−1 m−ℓm −p−1 S(AE ). Exactly as in the proof of Theorem 7.6, using the last proposition, it remains to verify that, for 0 ≤ k ≤ p − 1, ℓm + k > ℓm−(p−k) .

(7.64)

Assume that H is either unitary or symplectic. Then (7.64) means that m − 2 − (p − k) m−2 ]+k > [ ]. 2 2 When m is even this inequality holds. When m = 2n + 1 is odd, the l.h.s. is n − 1 + k, and then (7.64) holds, except when k = 0, p = 1, where we get an equality. In this case, we must have H = U4n+2 . (H = Sp4n+2 is impossible, since here the assumption that LS (τi , ∧2 , s) have poles at s = 1 implies that all mi are even, and hence m is even.) Now, since f U1 is not identically zero on Eτ¯ , we get (see (2.24)) that one of the τi is a character. This is one of the cases we excluded in our assumptions. Assume that H is metaplectic. Let δm = 1, for m odd and δm = 0, for m even. Then (7.64) means that [

m − 2 − (p − k) + δm−(p−k) m − 2 + δm ]+k > [ ]. 2 2 Assume that m = 2n + 1 is odd. Then the l.h.s. is n + k and the r.h.s. is δm−(p−k) − 1 − (p − k) δm−(p−k) − 2 n+[ ]≤n+[ ] = n − 1 < n + k. 2 2 [

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Thus, (7.64) holds in this case. Assume that m = 2n is even. Then the l.h.s. is δ −(p−k) n − 1 + k, and the r.h.s. is n − 1 + [ 2n−(p−k) ]. Thus, (7.64) holds, except 2 in case p = 1, k = 0, where we get an equality. (Note that δ2n−1 = 1.) Thus, f 4n , and as before, we get that one of the τi is a character. This is the other H = Sp case that we excluded in the assumptions of the theorem. 

Remark: The remark after Proposition 7.4 shows that it may happen that f σψℓm ,γ (¯ τ ) = 0, namely take H = Sp 4n+2 and ωτ¯ 6= 1. See Theorem 6.4(9). We have to remember that in this case the character ψ appears also in the induction data of Eτ¯ = Eτ¯,ψ in the form of µψ . Note that Eτ¯,ψ = Eτ¯χα ,ψα , for all α ∈ F ∗ , where χα denotes the quadratic character χα (x) = (x, α), given by the product of all local Hilbert symbols. In this case, ωτ¯χα = ωτ¯ χα . In the next two chapters we deal with the issue of non-vanishing of our descent representations.

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Non-vanishing of the descent I

The main goal of the next two chapters is to prove that the descents σψ,α (¯ τ ), σψ,γ (¯ τ) are nontrivial, provided some simple compatibility conditions are satisfied; these will be precisely formulated in our theorems. See the statements of the main theorem Theorem 3.1. We keep the same notation as in Theorem 3.1, and as in the last chapter. Thus, in this chapter, HA is one of the groups in Chapter 2, so that the residue representation Eτ¯ of HA exists. Thus, HA is the group of Adele points of one of the following: a split orthogonal group, a quasi-split even unitary group, a symplectic group, or HA is a metaplectic group. In detail, HA is one of the following groups:

f (A), Sp f SO4n (A), SO4n+1 (A), SO4n+3 (A), U4n (A), U4n+2 (A), Sp4n (A), Sp 4n 4n+2 (A). Recall, again, that in the even orthogonal case, we assume that all representations τi , of GLmi (A), are such that LS (τi , ∧2 , s) has a pole at s = 1, and hence each mi must be even. This is why we do not consider SO4n+2 , and, for the same reason, we do not consider Sp4n+2 . The rough outline of the proof that the descents are nontrivial is an explicit relation between the Whittaker coefficient of the descent above (viewed ∗ ∗ as an automorphic representation of Hα,m (A), resp. Hm (A); see (3.40), (3.41)) and the Fourier coefficient of the residual representation Eτ¯ , corresponding to the partition (m, m, m′ − 2m) of m′ = dimE V . This will be done in the next chapter. In turn (in this chapter) we write a precise relation between this last coefficient and the Whittaker coefficient of the constant term of Eτ¯ , along Um (viewed as an automorphic representation of the Levi subgroup, isomorphic to GLm (AE )). This last Fourier coefficient is clearly non-trivial. From this, we conclude that the Whittaker coefficient of the descent above is nontrivial, and hence the descent itself is nontrivial; moreover it is globally generic, and (by the last chapter) cuspidal. The main ideas of the proofs in this and the next chapter appeared in [Ginzburg, Rallis and Soudry I (1999)] and in [Ginzburg, Rallis and Soudry (2002)] where the case h(V ) = Sp4n was treated. 187

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The Fourier coefficient corresponding to the partition (m, m, m′ − 2m)

We introduce the unipotent subgroup of h(V ) and its character, both corresponding to the partition (m, m, m′ − 2m). Here we use the conventions of [Ginzburg (2006)]. Let hm (t) be the one parameter subgroup in the diagonal subgroup of h(V ), given by hm (t) = diag(tm−1 , tm−1 , tm−3 , tm−3 , ..., t3−m , t3−m , t1−m , t1−m ). Explicitly, for m = 2n even and h(V ) symplectic, even orthogonal, or (even) unitary, h2n (t) = diag(t2n−1 , t2n−1 , ..., t, t, t−1 , t−1 , ..., t1−2n , t1−2n ), and for h(V ) odd orthogonal, h2n (t) = diag(t2n−1 , t2n−1 , ..., t, t, 1, t−1 , t−1 , ..., t1−2n , t1−2n ). For m = 2n + 1 odd and h(V ) symplectic, or (even) unitary h2n+1 (t) = diag(t2n , t2n , ..., t2 , t2 , 1, 1, t−2 , t−2 , ..., t−2n , t−2n ), and for h(V ) odd orthogonal, h2n+1 (t) = diag(t2n , t2n , ..., t2 , t2 , 1, 1, 1, t−2, t−2 , ..., t−2n , t−2n ). Recall that Nm denotes the standard maximal unipotent subgroup of h(V ). Consider the action by conjugation of hm (t) on Nm . On each root subgroup, corresponding to a positive root α inside Nm , we have hm (t)xα (r)hm (t)−1 = xα (tiα r), where iα is a non-negative integer which depends on α. Let Sm ⊂ Nm be the F - subgroup of Nm generated by all the root subgroups, corresponding to positive roots α, such that iα ≥ 2. The explicit matrix form of these groups is as follows. Let m = 2n be even and h(V ) symplectic, even orthogonal, or (even) unitary. Then    I2 x1          I2 x2 ⋆            .   .     .              x n−1          I y   2 ∈ h(V ) , (8.1) Sm =       I2 x′n−1             I2            .   .     .        ′       I x 2 1     I2

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and for h(V ) odd orthogonal (SO4n+1 ),    I2 x1          I2 x2 ⋆            .   .     .              x n−1            I 0 y   2    Sm =  1 0  ∈ h(V ) = SO4n+1 .          I2 x′n−1             I  2            .   .     .        ′       I x 2  1    I2

(8.2)

Let m = 2n + 1 be odd and h(V ) symplectic, (even) unitary, or odd orthogonal. Then    I2 x1        I2 x2   ⋆            .   .     .              x n−1            I y 2      ′ ′ Sm =  (8.3) Im −4n y  ∈ h(V ) .          I2 x′n−1             I2             .   .     .        ′       I x 2   1   I2

Note that in the last case, m′ − 4n = 2, when h(V ) is symplectic, or unitary, and m′ − 4n = 3, when h(V ) = SO4n+3 is odd orthogonal. Sm is a subgroup of the unipotent radical of the parabolic subgroup Pϕm , which stabilizes the flag ϕm : V2+ ⊂ V4+ ⊂ · · · ⊂ V2[+m ] . 2

See (2.3). The Levi part Mϕm of Pϕm is isomorphic to (see (2.6)) [m 2 ]

Y

i=1

m

GL(Vϕ+m (i)) × h(Wm,2[ m2 ] ) = (ResE/F GL2 )[ 2 ] × h(Wm,2[ m2 ] ).

Except the case where h(V ) = SO4n+1 , Sm is equal to the unipotent radical of Pϕm . Note that, for m even, Wm,2[ m2 ] = 0, when h(V ) is not odd orthogonal, and Wm,2[ m2 ] = V0 , when h(V ) is odd orthogonal. When m is odd, Wm,2[ m2 ] = SpanE {em , e−m }, when h(V ) is not odd orthogonal, and Wm,2[ m2 ] = SpanE {em , e0 , e−m }, when h(V ) is odd orthogonal.

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Consider the following character ψ (m) of Sm (A). In the notation of (8.1)-(8.3), ψ takes an element of Sm (A) to (m)

ψE,δ (tr(x1 ) + · · · tr(xn−1 ) + tr(ya)),

[m 2 ].

(8.4)

The matrix a is defined as follows. where n = When h(V ) = Sp4n , U4n , 1 a = I2 . 2 When h(V ) = SO4n , SO4n+1 ,   1 1 a= . −1 2 When h(V ) = SO4n+3 ,

 10 a = 0 0  . 01 

When h(V ) = Sp4n+2 , U4n+2 ,

a=



1 δ



.

The characterψ (m) istrivial on Sm (F ). Note that when m is even, y in (8.4) has x ∗ the form y = , and its contribution to the character (8.4) is ∗ −δ¯ x ψE,δ (tr(ya) = ψE,δ (x).

The stabilizer R(m) of ψ (m) inside Mϕm is described as follows. Assume that m = 2n is even. Then R(m) consists of ( diag(g, ..., g, g ∗ , ..., g ∗ ), h(V ) = SO4n , Sp4n , U4n ∧ g = ∗ ∗ diag(g, ..., g, 1, g , ..., g ), h(V ) = SO4n+1 where g ∈ ResE/F GL2 is repeated n times and satisfies where w2 =



t g¯w2 ag = w2 a,  1 . Thus, R(m) is the isometry group of w2 a, diagonally embedded

1 in Mϕm . In case h(V ) = Sp4n , g lies in the (split) group O2 ; in case h(V ) = SO4n , g lies in SL2 , and in case h(V ) = U4n , g lies in the group U2 , written with respect to δ = 1. Assume that m = 2n + 1 is odd. Then R(m) consists of g ∧ = diag(g, ..., g, g ′ , g ∗ , ..., g ∗ ), where g ∈ ResE/F GL2 is repeated n times and satisfies the following. If h(V ) is symplectic, or unitary (i.e. h(V ) = Sp4n+2 , U4n+2 ), then g ∈ h(W2n+1,n ) = SL2 , U2 and     1 1 g′ = g . δ δ

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If h(V ) = SO4n+3 , then g lies in the split orthogonal group O2 , and g ′ (now δ = 1) is the matrix in h(W2n+1,n ) = O3 which fixes e0 and acts on Span{e2n+1 , e−(2n+1) } by g. For a smooth automorphic function f on HA , we consider the following Fourier coefficient of f , attached to the partition (m, m, m′ − 2m), Z f (v ω h)ψ (m) (v −1 )dv, (8.5) Jm,ψ (f )(h) = Sm (F )\Sm (A)

[m]

where ω = ωb 2 ; see (4.15) and Sec. 4.2. Thus, ω = I, except in the following cases: h(V ) = SO4n and n is odd, and in this case ω is the outer flip in O4n , as in Sec. 4.2; h(V ) = SO2m+1 , and ω = −I2m+1 . In the last case, v ω = v in (8.5), so that we may also replace there ω by I2m+1 . (The reason for this slight modification will me made clear later in this chapter.) As usual, in case H is metaplectic, we identify Sm (A) with the subgroup Sm (A) × 1 of HA . Jm,ψ (f ) is left (R(m) )ω (F ) - invariant. One of the main theorems of this chapter is that for our residual representations Eτ¯ , Jm,ψ (Eτ¯ ) is nonzero. This will follow from Propositions 7.1, 7.4 and the fact that the Eisenstein series on GLm (AE ) induced from τ1 ⊗ · · · ⊗ τr (on Pm1 ,...,mr (AE )) is globally generic. In order to formulate the precise relation between the Whittaker coefficient of the last Eisenstein series on GLm (AE ) and Jm,ψ , we need more notations. Assume that dimE V = m′ = 2m is even. For an m × m matrix, we denote     I I x . , u¯m (x) = m um (x) = m x Im Im Let X = {x ∈ ResE/F Mm×m | um (x) ∈ h(V )}, X0 = {x ∈ X |

x is nilpotent and upper triangular} .

Put um (X0 ) = {um (x) | x ∈ X0 } ,

u¯m (X0 ) = {¯ um (x) | x ∈ X0 } .

We regard X0 , um (X0 ), u¯m (X0 ) as algebraic groups over F . Assume that dimV = m′ = 2m + 1 is odd. In this case, h(V ) is odd orthogonal. Here, we denote, for an m × m matrix x,     Im 0 x Im cm (x) =  1 0  , c¯m (x) =  0 1  , x 0 Im Im and define X , X0 ⊂ Mm×m as before, with cm (x) replacing um (x). Thus, here, X consists of all m × m matrices x, such that xwm is anti-symmetric. (Recall that

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wm denotes the permutation m × m matrix with 1 along the anti-diagonal.) Denote ¯m ) as also a general element of the unipotent radical Um (resp. U      Im v x Im resp. u um (v, x) =  ¯m (v, x) =  v ′ 1  , 1 v′  , Im x v1

where v ′ = −t vwm , and t (xwm ) = −xwm − v t v. Note that 1 1 um (v, x) = cm (x + v t vwm )um (v, − v t vwm ), 2 2 and similarly for u¯m . Denote cm (X0 ) = {cm (x) | x ∈ X0 },

c¯m (X0 ) = {¯ cm (x) | x ∈ X0 }

Finally, define Ym = {¯ um (e, x) ∈ h(V )| ei = 0, i > [  γ This is a group. Recall our notation γˆ =  Im′ −2t

m−1 ], x ∈ X0 }. 2  ∗

, for γ ∈ GLt (AE ); t ≤ m.

γ Let us state now the main result of this chapter; for notational reasons we break it into two theorems. Theorem 8.1. f (A), and (1) Let HA be one of the groups SO4n (A), U4n (A), U4n+2 (A), Sp4n (A), Sp 4n consider our residual representation Eτ¯ on HA . Then for any automorphic form f in Eτ¯ , Z Z ′ (z −1 )dzdy. (8.6) z yαm )ψZ f Um (ˆ Jm,ψ (f )(1) = m u ¯ m (X0 )A Zm (E)\Zm (AE )

(2) Let HA = SO4n+3 (A). Then, for any automorphic form f in Eτ¯ , Z Z ′ (z −1 )dzdy. z yαm )ψZ f Um (ˆ Jm,ψ (f )(1) = m

(8.7)

Ym (A) Zm (F )\Zm (A)

′ Here, αm is a certain Weyl element in h(V ), and ψZ is a certain Whittaker m ′ character of Zm (AE ) (trivial on Zm (E)). Both αm and ψZ will be specified m Um in the proof. Finally, f denotes the constant term of f along Um .

(3) The Fourier coefficient Jm,ψ (f )(1) is not identically zero on Eτ¯ . f 4n (A), the double cover splits over u¯m (X0 )A and we identify this In case HA = Sp group  with itssplitting inside HA ; it is obtained by conjugation of um (X0 )A × 1 Im , 1). Similarly, Zˆm (A) is identified with Zˆm (A) × 1 and αm with by ( −Im (αm , 1).

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f 4n+2 (A) (and m = 2n, 2n + 1 respecAssume now that HA = SO4n+1 (A), Sp tively). Denote, for t ∈ A,   1t x(t) = , 1 in (t) =

(

x(t)∧ diag(I2n , x(−t), I2n ) ∈ Sp4n+2 (A)

x(t)∧ ∈ SO4n+1 (A)

.

Note that in (t) ∈ R(m) (A), i.e. in (t) preserves the character ψ (m) of Sm (A). In f case HA = Sp 4n+2 (A), we identify in (t) with (in (t), 1). Finally, denote, for b ∈ A,   1 0 b 0 − 12 b2  100 0    1 0 −b  jn (b) = diag(I2(n−1) ,   , I2(n−1) ) ∈ SO4n+1 (A).   1 0  1

Note that jn (b) normalizes Sm (A) (and for b ∈ F , it normalizes Sm (F )) and preserves the character ψ (m) . Note also that in (t) and jn (b) commute. We have Theorem 8.2. f 4n+2 (A), (m = 2n + 1) and consider our residual representation (1) Let HA = Sp Eτ¯ on HA . Then for any automorphic form f in Eτ¯ , Z Z Z ′ (z −1 )dzdy. (8.8) z yαm )ψZ f Um (ˆ Jm,ψ (f )(in (t))dt = m F \A

u ¯ m (X0 )A Zm (F )\Zm (A)

(2) Let HA = SO4n+1 (A) (m = 2n). Then, for any automorphic form f in Eτ¯ , Z Z Z ′ (z −1 )dzdy. z yαm )ψZ f Um (ˆ Jm,ψ (f )(jn (b)in (t))dbdt = m [F \A]2

Ym (A) Zm (F )\Zm (A)

(8.9) ′ Here, αm is a certain Weyl element in h(V ), and ψZ is a certain Whittaker m ′ character of Zm (AE ) (trivial on Zm (E)). Both αm and ψZ will be specified m in the proof. ′ (f )(1). Then (3) Denote the Fourier coefficient on the l.h.s. of (8.8), (8.9) by Jm,ψ ′ Jm,ψ (f )(1) is not identically zero on Eτ¯ .

Denote also, for H = SO4n , in (t) = x(t)∧ = diag(x(t), ..., x(t), x(−t), ..., x(−t)); for t¯ = −t and H = U4n ,

in (t) = x(t)∧ = diag(x(t), ..., x(t), x(−t¯), ..., x(−t¯));

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for t¯ = −δt, H = U4n+2 ,

in (t) = x(t)∧ = diag(x(t), ..., x(t), x(δt), x(−t¯), ..., x(−t¯)).

In all three cases x(t) is repeated n times. Note again that in (t) ∈ R(m) (A). We will prove Theorem 8.3. (1) Let HA = SO4n (A). Then for any automorphic form f in Eτ¯ , Jm,ψ (f )(in (t)ω ) = Jm,ψ (f )(1),

for all t ∈ A.

(2) Let HA = U4n (A). Then for any automorphic form f in Eτ¯ , Jm,ψ (f )(in (t)) = Jm,ψ (f )(1),

for all t ∈ AE ,

such that t¯ = −t.

(3) Let HA = U4n+2 (A). Then for any automorphic form f in Eτ¯ , Jm,ψ (f )(in (t)) = Jm,ψ (f )(1), 8.2

for all t ∈ AE ,

such that t¯ = −δt.

Conjugation of Sm by the element αm

We will rewrite the integral Jm,ψ (f )(h) in terms of a conjugated subgroup αm Sm α−1 m , for a certain Weyl element αm ∈ h(V ). Assume that m = 2n is even. Let βm be the following element in the isometry group of (V, b). Its action on the standard basis of V is as follows. βm (ei ) = e2i−1 , 1≤i≤n βm (en+i ) = e−2n+2i−2 , 1 ≤ i ≤ n βm (e−n−i ) = δe2n−2i+2 , 1 ≤ i ≤ n βm (e−i ) = e−2i+1 , 1 ≤ i ≤ n.

−1 In case h(V ) = SO4n+1 , we also require that βm (e0 ) = e0 . The matrix form of β2n , in case dimE V = 4n, is as follows. Consider the following n × 2n matrices.     1 01  01    01         ′ ′′ .. 01 βn,2n = =  , βn,2n . .     ..     . 1

10

Then



−1 β2n

01

′ βn,2n 0n×2n



    0  ′  n×2n βn,2n    = .  ′′  δβn,2n 0n×2n      ′′ 0n×2n βn,2n

(8.10)

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−1 When h(V ) = SO4n+1 , β2n is obtained by the last matrix, following the embedding     a0b ab 7→ 0 1 0 cd c0d

of O4n inside O4n+1 . Assume that m = 2n + 1. Then βm is the element in the isometry group of (V, b) satisfying βm (ei ) = e2i−1 , 1≤i≤n+1 βm (en+1+i ) = e−2n+2i−2 , 1≤i≤n βm (e−n−1−i ) = δe2n−2i+2 , 1≤i≤n βm (e−i ) = e−2i+1 , 1 ≤ i ≤ n + 1.

In case h(V ) = SO4n+3 , we also require that βm (e0 ) = e0 . The matrix form of −1 β2n+1 , in case dimE V = 4n + 2, is the following.   ′ βn,2n 0 0 0n×2n  0 10 0      ′   0n×2n 0 0 βn,2n −1 β2n+1 =  ′′ (8.11) . δβn,2n 0 0 0n×2n     0 01 0  ′′ 0n×2n 0 0 βn,2n

−1 When h(V ) = SO4n+3 , β2n+1 is obtained, as above, by taking the last matrix with δ = 1 (it lies in O4n+2 ), and then applying the embedding O4n+2 ֒→ O4n+3 . When h(V ) = SO4n , SO4n+1 , det(β2n ) = (−1)n , and also when h(V ) = SO4n+3 , det(β2n+1 ) = (−1)n . Define (see (4.15) and Sec. 4.2) [m]

−1 αm = βm ωb 2 .

We have Jm,ψ (f )(h) =

Z

f (αm v ω h)ψ (m) (v −1 )dv =

Sm (F )\Sm (A)

Z

(m) −1 f (αm v ω α−1 (v )dv = m αm h)ψ

Z

−1 −1 f (βm vβm αm h)ψ (m) (v −1 )dv =

Sm (F )\Sm (A)

Sm (F )\Sm (A)

Z

−1 −1 βm Sm (F )βm \βm Sm (A)βm

−1 −1 f (vαm h)ψ (m) (βm v βm )dv.

(8.12)

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−1 We used the fact that αm is rational and f is automorphic. Denote S m = βm Sm βm . This subgroup is described as follows. Assume first that h(V ) is not odd orthogonal. Then ( )   z x z, z ′ ∈ ResE/F Zm ; −1 βm Sm βm = v = ∈ h(V ) x, y are nilpotent and upper triangular, y z′

= um (X0 )(ResE/F Zm )∧ u ¯m (X0 ).

(8.13) In case HA is metaplectic, we have to carry out the conjugation −1 (βm , 1)(Sm (A), 1)(βm , 1), which is the subgroup of elements of the form (v, ǫ(v)), where v ∈ h(V )A is of the form (8.13) and ǫ(v) is the appropriate splitting arising from the conjugation by βm . Similarly, this subgroup is equal to (um (X0 )A × 1)(Zˆm (A) × 1)¯ um (X0 )A , where now, u ¯m (X0 )A is identified with the corresponding subgroup of −1 (βm , 1)(Sm (A), 1)(βm , 1). Note that u¯m (X0 )A , as any other unipotent subgroup, has a unique splitting. We keep denoting the conjugated Adele group (in the metaplectic case) by S m (A). The character ψ (m) becomes, after the conjugation above, the character ψS m of S m (A), which is trivial on um (X0 )A , u ¯m (X0 )A , and on Zˆm (AE ), it is the following Whittaker character: ′ (z) = ψE,δ (z1,2 + z2,3 + · · · + z[ m2 ],[ m2 ]+1 − z¯[ m2 ]+1,[ m2 ]+2 − · · · − z¯m−1,m ). (8.14) ψZ m

Note that

αm in (t)ω α−1 m

  um (δtIm ),        !     tI n   , u2n    −tIn     ! =  tI  n c2n  ,   −tIn        !     δtIn   ,  u2n tI

h(V ) = U4n+2 , Sp4n+2

h(V ) = SO4n (8.15) h(V ) = SO4n+1

h(V ) = U4n .

n

f 4n+2 (A) (m = 2n + 1). In this case, we consider also Let HA = Sp Z Jψ (in (t)h)dt. Jψ′ (f )(h) = F \A

(8.16)

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As before, Jψ′ (f )(h) ′ m

=

Z

197

−1 f (vαm h)ψ(S ′ )m (v)dv,

′ m (S ′ )m F \(S )A

−1 where (S ) = S um (Xc ); Xc = {tIm }. Here, αm = βm and by (8.15), −1 βm in (t)βm = um (−tIm ). Denote X0′ = X0 ⊕ Xc . Then, as before, um (X0 )A , (8.17) (S ′ )m (A) = um (X0′ )A Zˆm (A)¯ where we identify this subgroup with its splitting in HA , as before. The character ψ(S ′ )m is extended from the character ψS m to this group, so that it is trivial on um (Xc )) over A. Assume that H is odd orthogonal. Then    z, z ′ ∈ Z ; x, y are nilpotent and upper triangular, z u x   m S m = v = e′ 1 u′  ∈ h(V ) ] < i ≤ m, ui = ei = 0, for all [ m−1 2   y e z′ u′i = e′i = 0, for all 1 ≤ i ≤ [ m 2]+1 m

= u′m cm (X0 )Zˆm c¯m (X0 )u˜′ m ,

(8.18) where u′m is the subset which consists of the matrices um (v, − 21 v t vwm ), such that ′ ˜′ vi = 0, for all [ m−1 2 ] < i ≤ m. Note that um cm (X0 ) is a group. Finally, u m is the m−1 1 t subset of matrices u¯m (v, − 2 v vwm ), such that vi = 0, for all [ 2 ] < i ≤ m. Note −1 again, that u˜′ m c¯m (X0 ) is a group. The character ψ (m) , conjugated by βm becomes m ′ the character ψS m of S (A), which is trivial on um (A), u˜′ m (A), cm (X0 )A , c¯m (X0 )A and on Zˆm (A), it is the Whittaker character given by (8.14) (with δ = 1). Assume that h(V ) = SO4n+1 . In this case, we consider also Z ′ Jψ (jn (b)in (t)h)dbdt. (8.19) Jψ (f )(h) = Conjugating by αm , as before, Jψ′ (f )(h) = ′ m

m

[F \A]2

Z

−1 f (vαm h)ψ(S ′ )m (v)dv.

′ m (S ′ )m F \(S )A ·αm [Image(jn ×in )]α−1 m .

Here,  (S ) =S tIn ), and cm ( −tIn

αm jn (b)α−1 m

−1 By (8.15), αm in (t)α−1 m = βm in (t)βm =

 In−1  1 0 b 0 − 21 b2   In 0 0 0   = 1 0 −b   In 0   1



In−1

     .    

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Let Xa =

 tIn

−tIn

 . Denote

X0′ = X0 ⊕ Xa ,

and let vm be the subset of matrices of the form um (x, − 12 xt xwm ), where xi = 0, ′ for [ m+1 2 ] = n + 1 ≤ i ≤ 2n = m. Note that vm cm (X0 ) is a group. As before, (S ′ )m = vm cm (X0′ )Zˆm c¯m (X0 )˜ u′m .

(8.20)

The character ψ(S ′ )m is extended to this group, from the character ψS m , so that it is trivial on vm , cm (Xa ), over A. 8.3

Exchanging the roots y1,2 and x1,1 (dimE V = 2m , m > 2)

This “practical” title refers to an application of Lemma 7.1 to the integral Jm,ψ (f )(h) (using the matrix notation (8.13)) in order to exchange the coordinate roots above. This section is a warm up towards what is coming, and is intended as a specific illustration of the technic of proof. Assume that h(V ) is not odd orthogonal, and that m > 2. We now set up the necessary data for Lemma 7.1. We introduce the groups A, B, C, D, X, Y of this lemma. Consider Y 1,2 = {y ∈ X0 | yi,j = 0, for all (i, j) 6= (1, 2), (m − 1, m)}. Symbolically, in matrix form   0t     0      Y 1,2 = y 1,2 (t) =         

Put

  ··· 0 0     ··· 0 0      .. .. ∈ Res M .  m×m E/F . .      ¯  0 −δ t   0

Y 1,2 = u¯m (Y 1,2 ). The following subspace is a direct complement of Y 1,2 in X0 , Y1,2 = {y ∈ X0 |y1,2 = 0 (= ym−1,m )}. Put Y1,2 = u ¯m (Y1,2 ). In case of a metaplectic group, we identify, as usual, Y 1,2 (A) and Y1,2 (A) with their splitting (in the unique splitting of u ¯m (XA )). Next, let X 1,1 = {x ∈ X | xi,j = 0, for all (i, j) 6= (1, 1), (m, m)}. In matrix form, X 1,1 = {x1,1 (t) = diag(t, 0, · · · , −δ t¯) ∈ ResE/F Mm×m }.

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Put X 1,1 = um (X 1,1 ). Let C 1,2 = um (X0 )(ResE/F Zm )∧ Y1,2 , B 1,2 = C 1,2 Y 1,2 , D1,2 = C 1,2 X 1,1 , A1,2 = B 1,2 X 1,1 = D1,2 Y 1,2 = C 1,2 X 1,1 Y 1,2 . It is easy to check that C 1,2 is a group. In case of a metaplectic group, C 1,2 (A) −1 is a subgroup of (βm , 1)(Sm (A), 1)(βm , 1), and its elements have the form (c, ǫ(c)), 1,2 where c ∈ C (A), and for c = xˆ z y, where x ∈ Um (X0 (A)), z ∈ Zm (AE ), y ∈ Y1,2 (A), ǫ(c) = ǫ(y). Here, (¯ u, ǫ(¯ u)), as u ¯ varies in u ¯m (X (A)), describes the splitting of u ¯m (X (A)). The groups X = X 1,1 , Y = Y 1,2 , C = C 1,2 satisfy the requirements of Lemma 7.1, with respect to the character ψC 1,2 , which is the restriction of ψS m to C 1,2 (A), and is given via (8.14). Let us explain this. Note that X and Y are abelian and intersect C trivially. This takes care of the conditions (2), (4) for the set-up (7.1). Let x1,1 ∈ X 1,1 , y 1,2 ∈ Y 1,2 and z ∈ ResE/F Zm . Then by a straightforward multiplication, [x1,1 , zˆ] ∈ um (X0 ),

[y 1,2 , zˆ] ∈ Y1,2 .

From this we get that the elements of X and of Y conjugate the elements of (ResE/F Zm )∧ into C 1,2 , and also that, over A, this conjugation preserves ψC 1,2 . Next, note that for um (x1,1 (t)) ∈ X 1,1 , u ¯m (y 1,2 (s)) ∈ Y 1,2 , ∧  1 ts   1   ∈ (ResE/F Zm )∧ . [um (x1,1 (t)), u¯m (y 1,2 (s))] =  ..   .  1

In particular, when we take t, s ∈ AE , the application of the character (8.14) to this commutator gives ψE,δ (ts). This verifies the conditions (5) and (6) in the set-up (7.1). A similar simple commutator computation shows that the elements of X (resp. Y ) conjugate the elements of Y1,2 (resp. um (X0 )) into C 1,2 , and when we take Adele points, the character ψC 1,2 is trivial on such commutators. This verifies conditions (1) and (3), ( of course X 1,1 commutes element-wise with um (X0 ) and Y 1,2 commutes element-wise with Y1,2 ). Now we can apply Lemma 7.1 and Corollary 7.1. Note that, so far, we did not use the fact that our automorphic forms are in the residual representation Eτ¯ . We get Lemma 8.1. Let f be a smooth automorphic function on HA . Assume that f is of uniform moderate growth. Then

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(1) Jm,ψ (f )(h) =

Z

Z

−1 f (vyαm h)ψD 1,2 (v)dvdy,

(8.21)

1,2 YA1,2 DF \DA1,2

1,1 −1 where ψD 1,2 is the extension of ψC 1,2 by the trivial character on XA .

(2) Z

YA1,2

Z

−1 f (vyαm h)ψD 1,2 (v)dv

1,2 DF \DA1,2

dy < ∞.

This convergence is uniform in h varying in compact subsets of HA . (3) Jm,ψ (f ) 6= 0, if and only if there is h ∈ HA , such that Z −1 f (vh)ψD 1,2 (v)dv 6= 0. 1,2 DF \DA1,2

Consider the elements in (t), which lie in the stabilizer R(m) of ψ (m) , as in Theorem 8.3. By (8.15), αm in (t)ω α−1 = um (tJ) := xt , where J = m diag(In , −In ), diag(δIn , In ), δI2n+1 , depending on the group. From the last lemma we get Corollary 8.1. Assume that h(V ) = SO4n , U4n , U4n+2 . In the notation of the last lemma, Z Z −1 Jm,ψ (f )(in (t)ω h) = f (vxt yαm h)ψD (8.22) 1,2 (v)dvdy, 1,2 YA1,2 DF \DA1,2

for all t ∈ AE , such that in (t) ∈ R(m) (A). Proof. Then

Assume, for simplicity, that n > 1. Let y = u¯m (Y ), where Y ∈ Y 1,2 . [y, xt ] = (Im − tJY )∧ um (t2 Y ).

This is an element of D1,2 (and even of S m ). Indeed, um (t2 Y ) ∈ um (X0 ) and Im − tJY ∈ ResE/F Zm . It is easy to verify that, for t as above, and Y ∈ YA1,2 , ′ ψZ (Im − tJY ) = 1, and so, in the integrand of (8.21), we have m f (vyαm in (t)ω h) = f (v[y, xt ]xt yαm h), and now change variable v[y, xt ] 7→ v, and we get the corollary.



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8.4

201

First induction step: exchanging the roots yi,j and xj−1,i , for ]; dimE V = 2m 1 ≤ i < j ≤ [ m+1 2

The idea is to continue the process started in Lemma 8.1 (we assume that h(V ) is not odd orthogonal) and “fatten” the integration along D1,2 , by “adding” the integration along the coordinate x2,2 , and this can be done by “taking out” the coordinate y2,3 from Y1,2 . Next, we do the similar root exchange of y1,3 and x2,1 . Now, we continue with the fourth column of y (in the notation of (8.13)) and exchange y3,4 and x3,3 , y2,4 and x3,2 , y1,4 and x3,1 , and “so on”. Of course, in each step we need to verify the set-up (7.1). Before we formalize all of this we introduce more notation. Denote by ea,b the matrix of the standard basis of the m × m matrices, which has 1 in the coordinate (a, b), and zero in all other coordinates. Define for 1 ≤ i < j ≤ [ m+1 2 ], Y i,j = {tei,j − δ t¯em+1−j,m+1−i },

Yi,j = {y ∈ X0 | yi′ ,j ′ = 0, ∀ i′ , j ′ < j, and yi′ ,j = 0, ∀ i′ ≥ i}, Y i,j = u ¯m (Y i,j ), We have the relation Yi,j ⊕ Y i,j =

(

Yi,j = u ¯m (Yi,j ).

Yi+1,j ,

Y1,j−1 ,

The elements of Yi,j have the form   0 y1 y2 ∗ ∗  0 y3 ∗ ∗    Y = y0 y3′ y2′     0 y1′  0

i<j−1

i = j − 1.

(8.23)

i−1 j−i+1 m−2j .

(8.24)

1 j−1

Here, Y ∈ X0 (in particular y0 is nilpotent and upper triangular); the first block column in Y has j − 1 columns, the second has one column, the third has m − 2j columns, the fourth has j − i + 1 columns and the last has i − 1 columns. Define next, for 1 ≤ s ≤ r ≤ [ m−1 2 ], X r,s = {ter,s − δ t¯em+1−s,m+1−r },

L Lr ′ ′ ′ Xr,s = X0 ⊕ ( s′ ≤r′
We have the relation Xr,s =

(

Xr,s = um (Xr,s ).

Xr,s+1 ⊕ X r,s , Xr−1,1 ⊕ X

r,s

,

s≤r−1 s = r.

(8.25)

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The elements of Xr,s have the form  x1 x2 ∗ ∗  0 x3 ∗ ∗  X= x0 ∗   x′3 0

 ∗ r−1  ∗ 1 ∗  m−2r . x′2  r−s+1 x′1 s−1

(8.26)

Here, X ∈ X and x0 is nilpotent and upper triangular. If r = s − 1, simply omit x2 , x′2 , x3 , x′3 . The block columns of X, from left to right have s − 1, r − s + 1, m − 2r, 1, r − 1 columns, respectively. Assume that HA is metaplectic and m is odd. Define ′ Xr,s = Xr,s ⊕ Xc , ′ ′ Xr,s = um (Xr,s ).

The following lemma is easy to prove. Lemma 8.2. The group (ResE/F Zm )∧ normalizes (over F ) each one of the groups ′ above Xr,s , Yi,j , Xr,s . Denote, for short, Let, for 1 ≤ i < j ≤ [ m+1 2 ], ( C i,j =

Zm = (ResE/F Zm )∧ . Xj−1,i+1 Zm Yi,j ,

Xj−2,1 Zm Yj−1,j ,

i+1≤j−1

i=j−1

,

(8.27)

m−1 where X0,1 = X0 . Note that since i < j ≤ [ m+1 2 ], we have i ≤ j − 1 ≤ [ 2 ]. Assume that HA is metaplectic and m is odd. Then we also define (C ′ )i,j as in ′ (8.27), replacing Xj−1,i+1 (resp. Xj−2,1 ) by Xj−1,i+1 (resp. Xj−2,1 ). Define

B i,j = C i,j Y i,j Di,j = C i,j X j−1,i Ai,j = C i,j Y i,j X j−1,i . When HA is metaplectic and m is odd, we also define (B ′ )i,j , (D′ )i,j , (A′ )i,j , replacing C i,j by (C ′ )i,j . Note the relation ( X j−1,i+1 C i+1,j , i≤j−2 i,j i,j C Y = (8.28) j−2,1 1,j−1 X C , i = j − 1. Lemma 8.3. Assume that dimE V = 2m. Let 1 ≤ i < j ≤ [ m+1 2 ]. Then (1) C i,j is a group. (2) Define for c = uz¯ v ∈ C i,j (A), where z ∈ Zm (A), v¯ ∈ Yi,j (A), and u ∈ Xj−1,i+1 (A) when i + 1 ≤ j − 1 (resp. u ∈ Xj−2,1 (A) when i = j − 1), ′ (z). ψC i,j (c) = ψS m (z¯ v ) = ψS m (z) = ψZ m

See (8.14). Then ψC i,j is a character of C i,j (A).

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(3) A = Ai,j , B = B i,j , C = C i,j , D = Di,j , X = X j−1,i , Y = Y i,j satisfy the requirements of Lemma 7.1 with respect to the character ψC i,j . (4) Assume that HA is metaplectic and m is odd. Then the assertions above are valid, when we replace Ai,j , ..., Di,j by (A′ )i,j , ..., (D′ )i,j (and X = X j−1,i , Y = Y i,j remain the same). The interpretation of these statements in case of metaplectic groups is as usual and will be made precise, once more, in the proof. Proof. Assume first, for simplicity of notations, that H is not metaplectic. By Lemma 8.2, Zm normalizes Xj−1,i+1 (resp. Xj−2,1 ) and Yi,j . In order to show (1), it remains to prove that for all u ∈ Xj−1,i+1 (resp. u ∈ Xj−2,1 ) and v¯ ∈ Yi,j , we have v¯u ∈ C i,j . Let X ∈ Xj−1,i+1 (resp. Xj−2,1 ) and Y ∈ Yi,j . Assume that u = um (X), v¯ = u ¯m (Y ) respectively. We have   I X u¯m (Y )um (X) = m . (8.29) Y Im + Y X Let us solve

  z + az ∗ b az ∗ u ¯m (Y )um (X) = um (a)ˆ z u¯m (b) = , z ∗b z∗

(8.30)

for a, b of the same form as X, Y and z ∈ Zm . Solving the last equation, we first get that z ∗ = Im +Y X. For Y, X of the form (8.24), (8.26) respectively, we see that Y X is upper nilpotent, such that all its coordinates in the second upper diagonal are also zero. Hence Im + Y X is upper unipotent, with a zero second upper diagonal. Thus, when we take z with coordinates in Fν or with Adelic coordinates, any Whittaker character is trivial on z. Solving for a, b, we must have a = X(Im + Y X)−1 ,

b = (Im + Y X)−1 Y.

We have to show that a lies in Xj−i,i+1 (resp. Xj−2,1 ) and b lies in Yi,j . Let us explain this for a. The case for b is similar. Since a is obtained from X by right multiplication by an upper unipotent matrix, it is enough to show that a ∈ X , i.e. t (wm a) = −δwm a. This is equivalent to ¯ t Y¯ )−1 wm X = wm X(Im + Y X)−1 . (Im +t X

¯ t Y¯ )−1 wm = (Im +t (Xwm )t (wm Y ))−1 = (Im + XY )−1 , it remains Since wm (Im +t X to verify that X(Im + Y X) = (Im + XY )X, which is clear. Finally, z, a, b solve (8.30), if we verify that z + az ∗ b = Im . We have t (Im + Y X)−1 wm = (Im + XY )−1 . z = wm

Thus, we need to verify that Im = (Im + XY )−1 + X(Im + Y X)−1 Y,

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which is equivalent to XY = (Im + XY )X(Im + Y X)−1 Y. This follows simply by using the nilpotence of Y X. Say (Y X)N +1 = 0, then we have (Im + Y X)−1 =

N X

(−)k (Y X)k .

k=0

This completes the proof of (1). In this part we also proved that any Whittaker character is trivial on z = (Im + Y X)∗ above, say in Adele coordinates. This, together with Lemma 8.2, immediately imply (2). Let us prove (3). We start with the property that Y i,j normalizes C i,j . First, Y i,j commutes element-wise with Yi,j . Next, let Y ∈ Y i,j and z ∈ Zm . Then u¯m (Y )ˆ z u¯m (Y )−1 = zˆu¯m ((z ∗ )−1 Y z − Y ).

(8.31)

By (8.23) and Lemma 8.2, it is clear that (z ∗ )−1 Y z − Y ∈ Yi+1,j when i + 1 < j (resp. in Y1,j−1 , when i + 1 = j). All we have to note now is that (z ∗ )−1 Y z − Y has zero coordinate in Y i,j . This follows by a simple matrix multiplication, and we conclude that (z ∗ )−1 Y z − Y ∈ Yi,j . This, together with (8.31), also shows that Y i,j conjugates Zm into C i,j , preserving the character ψC i,j , when we take Adele coordinates. Now, consider the conjugation of Xj−1,i+1 (resp. Xj−2,1 ) by Y i,j . Let Y ∈ Y i,j and X ∈ Xj−1,i+1 (resp. Xj−2,1 ). As in (8.29), we have u ¯m (Y )um (X)¯ um (Y )−1 = um (X(z ∗ )−1 )ˆ zu ¯m ((z ∗ )−1 Y − Y ),

(8.32)

where z ∗ = Im + Y X ∈ Zm . As in the proof related to b in (8.30), (z ∗ )−1 Y − Y ∈ Yi,j . Also, Y X is upper nilpotent with its second diagonal being zero, so that any Whittaker character is trivial on z. Similarly, X(z ∗ )−1 ∈ Xj−1,i+1 (resp. Xj−2,1 ). This completes the proof that Y i,j normalizes C i,j and preserves ψC i,j over A. In particular, B i,j = C i,j Y i,j is a group and we can extend the character ψC i,j to a character ψB i,j of B i,j (A) by making it trivial on Y i,j (A). The proof that X j−1,i normalizes C i,j and preserves ψC i,j is carried out similarly. For this, we just need the analogs of (8.31), (8.32). For z ∈ Zm , X ∈ X j−1,i , um (X)ˆ z um (X)−1 = zˆum (z −1 Xz ∗ − X),

(8.33)

and for X ∈ X j−1,i , Y ∈ Yi,j , um (X)¯ um (Y )um (X)−1 = um (X − X(z ∗ )−1 )ˆ zu ¯m ((z ∗ )−1 Y ),

(8.34)

where z ∗ = Im −Y X ∈ Zm . We will extend ψC i,j to Di,j (A), by the trivial character of X j−1,i (A), and denote this extension by ψDi,j . Note that by the relations (8.28), we conclude the relations Di,j = B i−1,j , D1,j = B j,j+1

if

i≥2

(8.35)

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Finally, let X = aej−1,i − δ¯ aem+1−i,m+2,j ∈ X j−1,i and Y i,j ¯ δ bem+1−j,m+1−i ∈ Y . (Recall that 1 ≤ i < j ≤ [ m+1 2 ]). Then

= bei,j −

[um (X), u ¯m (Y )] = (Im + abej−1,j )∧ .

Thus, [X

j−1,i

,Y

i,j

] ⊂ Zm ⊂ C

i,j

(8.36)

, and when a, b ∈ AE , we have from (5.34),

ψC i,j ([um (X), u ¯m (Y )]) = ψE,δ (ab). This is a general character of Y i,j (F )\Y i,j (A) as a varies in X j−1,i (F ), and it is a general character of X j−1,i (F )\X j−1,i (A), as b varies in Y i,j (F ). This completes the proof of the lemma in case H is not metaplectic. Assume now that H is metaplectic. We first consider assertions (1)-(3). We will employ an inductive argument to show that the cocycle behaves well in the considerations above. Recall that the elements of u ¯m (XA ) are now written as (y, ǫ(y)) according to its unique splitting. We order the pairs (i, j), (i′ , j ′ ) satisfying 1≤i<j≤[

m+1 ], 2

1 ≤ i′ < j ′ ≤ [

m+1 ], 2

by (i′ , j ′ ) < (i, j),

if and only if

j ′ < j,

or

j′ = j

and

i′ > i.

(8.37)

For this part of the proof, in order to distinguish between subgroups of h(V )A and their splittings in the metaplectic cover, we will denote by Y i,j (A) the subgroup in h(V )A and by Y˜ i,j (A) its corresponding splitting in the metaplectic group, and similarly for the other subgroups, such as C i,j , etc. We have already seen that C˜ 1,2 (A) is a group, and that its elements have the form (xˆ z y, ǫ(y)), where x ∈ um (X0 (A)), z ∈ Zm (A), y ∈ Y1,2 (A). Assume, by induction, that C˜ i,j (A) is a group, and that its elements have the form (xˆ z y, ǫ(y)), where x, z, y are in the corresponding factors in (8.27). Now, examine the argument above, where we proved that X j−1,i normalizes C i,j . Fix x in X j−1,i . Then for c in C i,j (A), we have, in the metaplectic group, (x, 1)(c, ǫ(c))(x, 1)−1 = (xcx−1 , ǫ(c)), as follows easily by the properties of the Rao cocycle. Thus, for fixed x, both ǫ(xcx−1 ) and ǫ(c) give a splitting of C i,j (A), and hence their quotient is a character of C i,j (A), which takes values in {±1}. Such a character must be trivial. Thus, for all x, c as above, ǫ(xcx−1 ) = ǫ(c). ˜ j−1,i (A) (which is X j−1,i (A) × 1) normalizes C˜ i,j (A) (fixing This shows that X ˜ i,j (A) = ψC i,j , as before), and we conclude that the elements of the group D i,j j−1,i ˜ ˜ C (A)X (A) have the form (xc, ǫ(c)), with x, c as above. Indeed, we have, using the Rao cocycle, (x, 1)(c, ǫ(c)) = (xc, ǫ(c)).

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˜ i,j (A) have the form (x′ zˆy, ǫ(y)), where x′ ∈ Xj−1,i , z ∈ Thus, the elements of D Zm (A) and y ∈ Yi,j (A). Assume that i > 1. Then when we take y ∈ Yi−1,j (A), we get the subgroup of elements (x′ zˆy, ǫ(y)), where x′ ∈ Xj−1,i , z ∈ Zm (A) and y ∈ Yi−1,j (A). This is C˜ i−1,j (A), and hence C˜ i−1,j (A) is a group. Assume that i = 1. Then when we take y ∈ Yj,j+1 (A), we get the subgroup of elements (x′ zˆy, ǫ(y)), where x′ ∈ Xj−1,i , z ∈ Zm (A) and y ∈ Yj,j+1 (A). This is C˜ j,j+1 (A), and hence C˜ j,j+1 (A) is a group. This completes the proof of (1), by induction in case of a metaplectic group. Note that our arguments make sense and are valid when we replace the Adeles by a local field. The rest of the proof follows similarly. This proves assertions (1)-(3), when HA is metaplectic. ˜ It remains to prove (4). Thus, assume that HA = Sp 4n+2 (A) (m = 2n + 1). The proof is the same as above, almost word for word. Let us add only one comment. In the proof of (1)-(3), we encounter (e.g. in (8.30), (8.32)) the matrices Y X, where ′ ′ Y ∈ Yi,j , Y i,j and X ∈ Xj−1,i+1 (resp. in Xj−2,1 ). Note, first, that Y X is upper nilpotent, so that Im + Y X ∈ Zm , and similarly with XY . In (1)-(3), we also had that the second upper diagonal of XY is zero, and hence concluded that any Whittaker character on Im +XY is trivial. Here, the second upper diagonal may not be zero, due to the presence of Xc . Write then X = tIm + X1 , where X1 ∈ Xj−1,i+1 (resp. in Xj−2,1 ). Then the second upper diagonal of X1 Y is zero, so that (over A), ′ by the definition of ψZ (8.14), m n X ′ ′ (I + tY ) = ψ( t(Yk,k+1 − Y2n+1−k,2n+2−k )) = 1. (I + XY ) = ψ ψZ m Zm m m k=1

Indeed, Yk,k+1 = Y2n+1−k,2n+2−k . Having made this comment, the proof now is exactly the same as that of (1)-(3).  From Lemma 7.1 and Corollary 7.1, we conclude Lemma 8.4. Assume that dimE V = 2m. Let f be a smooth automorphic function on HA . Assume that f is of uniform moderate growth. Then, for 1 ≤ i < j ≤ [ m+1 2 ], (1)

Z

−1 f (vh)ψB i,j (v)dv

i,j \BAi,j BF

(2)

=

Z

Z

−1 f (vyh)ψD i,j (v)dvdy.

i,j \DAi,j YAi,j DF

Z

YAi,j

Z

i,j DF \DAi,j

−1 f (vyh)ψD i,j (v)dv dy < ∞.

This convergence is uniform in h varying in compact subsets of HA . (3) There is h ∈ HA , such that Z −1 f (vh)ψB i,j (v)dv 6= 0, i,j BF \BAi,j

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if and only if there is h ∈ HA , such that Z −1 f (vh)ψD i,j (v)dv 6= 0. i,j \DAi,j DF

(4) The same assertions hold, when HA is metaplectic, m is odd, and we replace B i,j , Di,j by (B ′ )i,j , (D′ )i,j , respectively. Corollary 8.2. Assume that h(V ) = SO4n , U4n , U4n+2 . Let xt be as in Corollary 8.1. Then, in the notation of the last lemma, Z Z Z −1 −1 f (vxt h)ψB i,j (v)dv = f (vxt yh)ψD i,j (v)dvdy. i,j \BAi,j BF

i,j \DAi,j YAi,j DF

Proof. The proof is as that of Corollary 8.1, with Y i,j replacing Y 1,2 . Let us write it now in more detail. We use the same notation. Let y = u ¯m (Y ), Y ∈ Y i,j . Then a straightforward multiplication gives   Im − tJY t2 JY J [y, xt ] = . −tY JY Im + tY J + t2 Y JY J Assume that m = 2n. Then 1 ≤ i < j ≤ n. Write Y = aei,j − δ¯ aem+1−j,m+1−i and J = diag(d1 , ..., dm ). Then Y JY = (aei,j − δ¯ aem+1−j,m+1−i )(adi ei,j − δ¯ adm+1−j em+1−j,m+1−i ) = 0,

since j 6= i, m + 1 − j and i 6= m + 1 − i. Thus,   Im − tJY t2 JY J [y, xt ] = . 0 Im + tY J

This element lies in Di,j (and even in S m ). With Adele coordinates, we have ′ (Im − tJY ) = 1. ψDi,j ([y, xt ]) = ψZ m

In order to see this, we have to check just one case, namely j = i + 1 and i < n. ′ Let h(V ) = SO4n . Then J = diag(In , −In ), and, by the definition of ψZ , (8.14) m ′ ′ (Im − taei,i+1 − tae2n−i,2n−i+1 ) = 1. (Im − tJY ) = ψZ ψZ m m

Let h(V ) = U4n . Then J = diag(δIn , In ), and

−1 ′ ′ a) = 1, (Im − δtaei,i+1 + tδ¯ ae2n−i,2n−i+1 ) = ψE,δ (δta + δt¯ (Im − tJY ) = ψZ ψZ m m since t¯ = −t. Assume that m = 2n + 1, so that h(V ) = U4n+2 . In this case, J = δI2n+1 , and 1 ≤ i < j ≤ n + 1. If j ≤ n, then all the considerations above are valid. Namely, Y JY = δY 2 = 0, [y, xt ] ∈ S m ⊂ Di,j , and we have, for j = i + 1, −1 ′ ′ (Im − δtaei,i+1 + tae2n−i,2n−i+1 ) = ψE,δ (δta + t¯a) = 1, (Im − tJY ) = ψZ ψZ m

m

since t¯ = −δt. Assume that j = n + 1. Then Y JY = δY 2 = −δa¯ aei,2n+2−i . Since 2n + 2 − i ≥ n + 2, u ¯m (−δtY 2 ) ∈ Yi,j . It is easy to verify that [y, xt ] = u ¯m (−δtY 2 )(Im − δtY )∧ um (t2 (Y + tY 2 )).

Thus, [y, xt ] ∈ Di,j , and, exactly as before,

′ (Im − δtY ) = 1. ψDi,j ([y, xt ]) = ψZ m

Now, we finish the proof as in Corollary 8.1.



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At this point, we go back to the initial step in Lemma 8.1, where (i, j) = (1, 2), and proceed by induction, following the order (8.37), using repeatedly Lemma 8.4 and the relation (8.35), which allows us to switch from Di,j to B i−1,j when i > 1 and from D1,j to B j,j+1 . Define, for 1 ≤ i < j ≤ [ m+1 2 ], ∗ Yi,j = Y 1,2 Y 2,3 Y 1,3 · · · Y j−1,j Y i,j . ∗ m Put Ym∗ = Y1,[ = D1,[ m+1 and D ] 2

1,[ m+1 2 ]

let (D′ )m = (D′ )

m+1 2 ]

. Similarly, for HA metaplectic, with m odd,

. Then we conclude

Proposition 8.1. Assume that dimE V = 2m. Let f be a smooth automorphic function on HA . Assume that f is of uniform moderate growth. Then (1) Jm,ψ (f )(h) =

Z

Z

−1 f (vyαm h)ψD m (v)dvdy.

∗ (A) D m \D m Ym F A

The convergence of this integral is as in Lemma 8.4. Moreover, Jm,ψ (f ) 6= 0, if and only if there is h ∈ HA , such that Z −1 f (vαm h)ψD m (v)dv 6= 0. m DF \DAm

(2) Assume that HA is metaplectic and m is odd. Then Z Z −1 ′ f (vyαm h)ψ(D Jψ (f )(h) = ′ )m (v)dvdy. ∗ (A) (D ′ )m \(D ′ )m Ym F A

The convergence of this integral is as above. Moreover, Jψ′ (f ) 6= 0, if and only if there is h ∈ HA , such that Z −1 f (vαm h)ψ(D ′ )m (v)dv 6= 0. ′ m (D′ )m F \(D )A

Similarly, we conclude from Corollaries 8.1, 8.2, Corollary 8.3. Assume that h(V ) = SO4n , U4n , U4n+2 . Let xt be as in Corollary 8.1. Then, in the notation of the last proposition, Z Z −1 f (vxt yαm h)ψD Jm,ψ (f )(in (t)ω h) = m (v)dvdy. ∗ (A) D m \D m Ym F A

Let us specify the form of Dm and the character ψDm . Assume that m = 2n is even. Then Dm = X n−1,1 C 1,n = um (Xn−1,1 )Zm u ¯m (Y1,n ),

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and

Xn−1,1

    x1 x2 x3 x4′   0 x5 x3   ∈ X =    0 x′2    x′1

x1 ∈ ResE/F Mn−1

Note that when h(V ) = SO4n , x5 above is zero.

Y1,n =

 0n×n

y 0n×n



∈X



.

      

.

(8.38)

(8.39)

For an element of Dm (A), a = xˆ z y, written as above, ′ (z). ψDm (a) = ψZ m

See (8.14). Assume that m = 2n + 1 is odd. Then Dm = X n,1 C 1,n+1 = um (Xn,1 )Zm u ¯m (Y1,n+1 ), and

Xn,1

   x1 x2 x3 =  0 x′2  ∈ X  ′ x1 Y1,n+1

  0n×n 0 =  0 

x1 ∈ ResE/F Mn   y  0 ∈X . 

  

.

(8.40)

(8.41)

0n×n

For an element of Dm (A), a = xˆ z y, written as above, ′ (z). ψDm (a) = ψZ m

Similarly, when HA is metaplectic and m is odd, note that ′ (D′ )m = um (Xn,1 )Zm u¯m (Y1,n+1 ),

and ′ Xn,1

   x1 x2 x3 =  t x′2  ∈ X  x′1

x1 ∈ Mn

  

.

(8.42)

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First induction step: odd orthogonal groups

The analog of Proposition 8.1 in this case is obtained similarly. For 1 ≤ i < j ≤ i,j [ m+1 2 ], we define Y , Yi,j ⊂ X exactly as in the last section, and then we define Y i,j = c¯m (Y i,j ), Yi,j = c¯m (Yi,j ).

r,s , Xr,s ⊂ X exactly as in the last Similarly, for 1 ≤ s ≤ r ≤ [ m−1 2 ], we define X section, and then we define

X r,s = cm (X r,s ), Xr,s = cm (Xr,s ). ′ ′ ′ When h(V ) = SO4n+1 , we also define Xr,s = Xr,s ⊕Xa , Xr,s = cm (Xr,s ). See (8.20). m+1 i,j Next, as in (8.27), define, for 1 ≤ i < j ≤ [ 2 ], the groups C . ( Xj−1,i+1 u′m Zm u˜′ m Yi,j , i+1≤j−1 i,j . C = ′ ′ ˜ i=j−1 Xj−2,1 um Zm u m Yj−1,j ,

See (8.18). Note that u′m Xr,s and u˜′ m Yi,j are groups, for r, s and i, j as above. When h(V ) = SO4n+1 , define ( ′ Xj−1,i+1 vm Zm u˜′ m Yi,j , i+1≤j−1 ′ i,j (C ) = . i=j−1 X′ vm Zm u˜′ m Yj−1,j , j−2,1

′ vm Xr,s

Again, are groups. Now, define B i,j , Di,j , Ai,j as in the last subsection, and similarly, define (B ′ )i,j , (D′ )i,j , (A′ )i,j , when h(V ) = SO4n+1 (replacing C i,j by (C ′ )i,j ). We get the analog of Lemma 8.3 Lemma 8.5. Assume that h(V ) is odd orthogonal. Let 1 ≤ i < j ≤ [ m+1 2 ]. Then

(1) C i,j is a group. (2) Define for c = uz¯ v ∈ C i,j (A), where z ∈ Zm (A), v¯ ∈ Yi,j u˜′ m , and u ∈ ′ Xj−1,i+1 um when i + 1 ≤ j − 1 (resp. u ∈ Xj−2,1 u′m when i = j − 1), ′ (z). ψC i,j (c) = ψS m (z¯ v ) = ψS m (z) = ψZ m

See (8.14). Then ψC i,j is a character of C i,j (A). (3) A = Ai,j , B = B i,j , C = C i,j , D = Di,j , X = X j−1,i , Y = Y i,j satisfy the requirements of Lemma 7.1 with respect to the character ψC i,j . (4) Assume that h(V ) = SO4n+1 . Then the assertions above are valid when we replace Ai,j , ..., Di,j by (A′ )i,j , ..., (D′ )i,j , (and X = X j−1,i , Y = Y i,j remain the same). Proof.

Denote ∗ i,j

(C )

=

(

Xj−1,i+1 Zm Yi,j , Xj−2,1 Zm Yi,j ,

i+1<j−1 i=j−1

.

By the proof of Lemma 8.3 for SO2m , it follows that (C ∗ )i,j is a group and the restriction of ψC i,j to (C ∗ )i,j (A) is a character. For this, we use the natural embedding of SO2m inside SO2m+1 . This observation and the following conjugation

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formulae easily show that C i,j is a group. Let c¯m (Y ) ∈ Yi,j , um (v, − 12 v t vwm ) ∈ u′m , u ¯m (e, − 21 et ewm ) ∈ u˜′ m , and cm (X) ∈ Xj−1,i+1 (resp. in Xj−2,1 ). When ′ h(V ) = SO4n+1 , we also allow um (v, − 21 v t vwm ) ∈ vm and cm (X) ∈ Xj−1,i+1 (resp. ′ in Xj−2,1 ). Put 1 vm (v) = um (v, − v t vwm ), 2

1 u˜′ m (e) = u ¯m (e, − et ewm ). 2

Then u˜′ m (e)cm (X)u˜′ m (e)−1 = um (−Xe, X(Im + 21 et ewm X))(Im − 12 Xet ewm )∧ , vm (v)−1 c¯m (Y )vm (v) = (Im − 12 v t vwm Y )∧ u ¯m (Y v, (Im + 12 Y v t vwm )Y ), u˜′ m (e)vm (v)u˜′ m (e)−1 = (Im + v t ewm )∧ vm (v). (8.43) ′ Note that ψZm is trivial on (Im − 12 Xet ewm )∧ , (Im − 12 v t vwm Y )∧ , and (Im + v t ewm )∧ , when Adele coordinates are taken. Indeed, Xet ewm , v t vwm Y, v t ewm are upper nilpotent and also their second upper diagonal is zero. When h(V ) = SO4n+1 (m = 2n), we still have to show that c¯m (Y )cm (X) ∈ (C ′ )i,j , for Y ∈ Yi,j , X ∈ Xa . As in (8.30), this follows from c¯m (Y )cm (X) = cm (X(Im + Y X)−1 )[(Im + XY )−1 ]∧ c¯m ((Im + Y X)−1 Y ). (8.44) ′ Finally, over A, ψZ (Im + XY ) = 1. For this, write X = diag(tIm , −tIm ). Then m n n−1 X X ′ (I + XY ) = ψ( tY − (−t)Y2n−k,2n+1−k ). ψZ m k,k+1 m k=1

k=1

Since Yn,n+1 = 0, and Y2n−k,2n+1−k = −Yk,k+1 , we get n−1 X

ψ(

(t(Y2n−k,2n+1−k + Yk,k+1 ))) = 1.

k=1

This proves (1) and (2) for C i,j and also for (C ′ )i,j . Let us prove (3). Again, from the proof of Lemma 8.3, we know that X j−1,i , Y i,j satisfy the requirements of Lemma 7.1, with respect to (C ∗ )i,j and the restriction of ψC i,j to (C ∗ )i,j (A). Finally, note that the formulae (8.43) are valid for X ∈ X j−1,i , Y ∈ Y i,j , as well. For such X, Y and u′m (v), u˜′ m (e), as above, we get cm (X)−1 u˜′ m (e)cm (X) = um (−Xe, 12 Xet ewm X)(Im − 12 Xet ewm )∧ u˜′ m (e), (8.45) ¯m (Y v, 21 Y v t vwm Y ). c¯m (Y )vm (v)¯ cm (Y )−1 = vm (v)(Im − 21 v t vwm Y )∧ u This shows that Y i,j and X j−1,i conjugate u′m and u˜′ m into C i,j , and also, in case h(V ) = SO4n+1 , they conjugate vm (and u˜′m ) into (C ′ )i,j . Again, the matrices Xet ewm , v t vwm Y are upper nilpotent and their second upper diagonal is zero, so ′ that ψZm is trivial on (Im − 21 Xet ewm )∧ , (Im − 21 v t vwm Y )∧ , when we take Adele coordinates. Note also that X j−1,i commutes with the elements of u′m (in case h(V ) = SO4n+1 , it commutes also with vm ) and similarly with Y i,j and u˜′ m . Finally, when h(V ) = SO4n+1 , Y ∈ Y i,j , X ∈ Xa , we see from (8.44), which is still valid, ′ that c¯m (Y )cm (X)¯ cm (Y )−1 ∈ (C ′ )i,j , and over A, ψZ is trivial on Im + XY . This m completes the proof of (3) and of (4), as well. 

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Now we proceed exactly as in the previous section. We first obtain Lemma 8.4. Lemma 8.6. Assume that h(V ) is odd orthogonal. Let f be a smooth automorphic function on HA . Assume that f is of uniform moderate growth. Then, for 1 ≤ i < j ≤ [ m+1 2 ], (1) Z

−1 f (vh)ψB i,j (v)dv

i,j \BAi,j BF

=

Z

Z

−1 f (vyh)ψD i,j (v)dvdy.

i,j \DAi,j YAi,j DF

(2) Z

YAi,j

Z

i,j DF \DAi,j



−1 f (vyh)ψD i,j (v)dv dy

< ∞.

This convergence is uniform in h varying in compact subsets of HA . (3) There is h ∈ HA , such that Z −1 f (vh)ψB i,j (v)dv 6= 0, i,j BF \BAi,j

if and only if there is h ∈ HA , such that Z −1 f (vh)ψD i,j (v)dv 6= 0. i,j DF \DAi,j

(4) Assume that h(V ) = SO4n+1 . Then the same assertions hold when we replace B i,j , Di,j by (B ′ )i,j , (D′ )i,j . Finally, we obtain, with analogous notation, Proposition 8.1. Proposition 8.2. Assume that h(V ) is odd orthogonal. Let f be a smooth automorphic function on HA . Assume that f is of uniform moderate growth. Then (1) Jψ (f )(h) =

Z

Z

−1 f (vyαm h)ψD m (v)dvdy.

∗ (A) D m \D m Ym F A

The convergence of this integral is as in Lemma 8.6. Moreover, Jψ (f ) 6= 0, if and only if there is h ∈ HA , such that Z −1 f (vαm h)ψD m (v)dv 6= 0. m \D m DF A

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(2) Assume that h(V ) = SO4n+1 . Then Z Z ′ Jψ (f )(h) =

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213

−1 f (vyαm h)ψ(D ′ )m (v)dvdy,

∗ (A) (D ′ )m \(D ′ )m Ym F A

and Jψ′ (f ) 6= 0, if and only if there is h ∈ HA , such that Z −1 f (vαm h)ψ(D ′ )m (v)dv 6= 0. ′ m (D′ )m F \(D )A

The form of the elements of Dm is as follows. Assume that m = 2n is even. Then Dm = X n−1,1 C 1,n = u′m Xn−1,1 Zm Y1,n u˜′ m . Also, ′ (D′ )m = X n−1,1 (C ′ )1,n = vm Xn−1,1 Zm Y1,n u˜′ m .

The elements of u′m Xn−1,1 have the form  In−1 v x1  I 0 0 2   In−1 0 0   1 0    In−1  

The elements of

′ vm Xn−1,1

              

x2 0 0 0

x3 x′2 x′1 v′

I2 In−1

have the form

In−1 1 1

0 In−1

v x1 b 0 0 0 0 0 1 0 In−1

x2 t 0 0 0

x3 0 −t 0 −b



      ∈ SO4n+1 .    

x4 x′3 x′2 x′1 v′

1 1 In−1

The elements of Y1,n u˜′ m have the form   In   0 In    ∈ SO4n+1 ,  0 e′ 1    0 y e In  0 0 0 0 In

(8.46)



        ∈ SO4n+1 .      

en = 0.

Assume that m = 2n + 1 is odd. Then

Dm = X n,1 C 1,n+1 = u′m Xn,1 Zm Y1,n+1 u˜′ m .

(8.47)

(8.48)

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The elements of u′m Xn,1 have the form  In v x1 x2  1 0 0 0   I  n 0 0 0  1 0 0    In   1

 x3 x′2   x′1    v ′  ∈ SO4n+3 .     In

The elements of Y1,n+1 u˜′ m have the form   In   1     In     ′  ∈ SO4n+3 . 0 0 e 1     0 0 y e In   0 0 0 0 1  0 0 0 0 In

(8.49)

(8.50)

In all cases, for an element a ∈ Dm (A) (resp. (D′ )m ) of the form uˆ z v¯, as above, ¯m (A) and z ∈ Zm (A), where u ∈ Um (A), v¯ ∈ U ′ (z). ψDm (a) = ψZm

Assume that h(V ) = SO4n+1 (m = 2n). We can proceed from Proposition 8.2, and keep applying Lemma 7.1, so that row n, in (8.47), is “filled”, from column 2n + 2 till column 3n. For this, define, for 1 ≤ i ≤ n − 1, C i,n+1 B i,n+1 Di,n+1 Ai,n+1

= vm Xn,i+1 Zm Yi,n+1 u ˜′m i,n+1 i,n+1 =C Y = C i,n+1 X n,i = C i,n+1 Y i,n+1 X n,i .

Here, as usual, Y i,n+1 = c¯m (F (ei,n+1 − en,2n+1−i )),

X n,i = cm (F (en,i − e2n+1−i,n+1 )),

(as algebraic groups over F ), and Yi,n+1 = {¯ cm (y) ∈ Y1,n |yi′ ,n+1 = 0, ∀i ≤ i′ < n} ′ X n,n−1 X n,n−2 · · · X n,i+1 . Xn,i+1 = Xn−1,1

′ Note that Xn−1,1 = Xn−1,1 X n,n , and hence

(D′ )1,n = B n−1,n+1 ; also, for 2 ≤ i ≤ n − 1,

Di,n+1 = B i−1,n+1 .

We leave it to the reader to show that the assertions of Lemma 8.5 are valid for (i, n + 1). In fact, the formulae (8.43), (8.44), (8.45) are still valid, in this case, as

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well as (8.36). (All this will be done more generally, one more time later in this chapter, and we carry this step here for the sake of uniformity of notation.) Now, we conclude Lemma 8.6 for (i, n + 1) and obtain the analog of Proposition 8.2, Proposition 8.3. Assume that h(V ) = SO4n+1 . Let f be a smooth automorphic function on HA . Assume that f is of uniform moderate growth. Then Z Z −1 ′ f (vyαm h)ψD Jψ (f )(h) = 1,n+1 (v)dvdy, ∗ Y1,n+1 (A) D1,n+1 \D1,n+1 F

∗ Y1,n+1

∗ Y1,n Y n−1,n+1

where = is h ∈ HA , such that

Z

···Y

A

1,n+1

. Moreover, Jψ′ (f ) 6= 0, if and only if there

−1 f (vαm h)ψD 1,n+1 (v)dv 6= 0.

1,n+1 DF \DA1,n+1

m Denote [ m = D1,n+1 , and so Proposition 8.2, applied to 2 ] = n. When m is odd, D SO4n+3 , has the same form as Proposition 8.3 (for SO4n+1 ). When m = 2n is even, D1,n+1 = vm Xn,1 Zm Y1,n+1 u˜′m . The elements of vm Xn,1 have the form (compare with (8.47) and (8.49))   In 0 v x1 x2  In 0 0 x′1     (8.51) 1 0 v′   ∈ SO4n+1 .   In 0  In

The elements of Y1,n+1 u˜′ m have the form   In+1   0 In−1   ′  ∈ SO4n+1 .  0 e 1     0 y e In−1 0 0 0 0 In+1 8.6

(8.52)

Second induction step: exchanging the roots yi,j and xj−1,i , for i + j ≤ m + 1, j > [ m+1 ] (dimE V = 2m) 2

Assume that m′ = dimE V = 2m. Thus, h(V ) is not odd orthogonal. Denote n′ = nm = [

m+1 ]. 2

Our plan is to go on and take roots off Y1,n′ and add them to Xn′ −1,1 in the same order as before. In this part, we will have to use the fact that our automorphic

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forms f lie in the residual representations Eτ¯ . (We did not use this fact, up to this point, in this chapter.) Let j > n′ and i + j ≤ m + 1. Define Y i,j = {tei,j − δ t¯em+1−j,m+1−i }, Yi,j = {y ∈ Y1,n′ | yi′ ,j ′ = 0, j ′ < j, and yi′ ,j = 0, i ≤ i′ ≤ m + 1 − j}, Y i,j = u ¯m (Y i,j ), Yi,j = u ¯m (Yi,j ). Note that when i + j = m + 1, Y m+1−j,j = {tem+1−j,j | t¯ = −δt}.

Thus, if E = F , then Y m+1−j,j = 0, when δ = 1 (this is the case when h(V ) = SO4n ). If E = F and δ = −1, then the F -points of Y m+1−j,j are F em+1−j,j (this is the case when h(V ) is symplectic. If [E; F ] = 2, then the F -points of Y m+1−j,j √ are F em+1−j,j , when δ = −1, or F ρem+1−j,j , when δ = 1 (this is the case when h(V ) is unitary). The elements of Yi,j have the form (8.39), or (8.41), such that   0 y1 y2 y3 0 0 y4 y2′   y= (8.53) 0 0 0 y ′  , 1 0 0 0 0

where y1 appears in column j − n′ of y and it has i − 1 coordinates; (y1′ is in row m + 1 − j of y and is “dual” to y1 ). We have the relation ( Yi+1,j , i+j <m+1 Yi,j ⊕ Y i,j = (8.54) Y1,j−1 , i+j =m+1 Next, define, for r ≥ n′ , r + s ≤ m + 1, X r,s = {ter,s − δ t¯em+1−s,m+1−r }, L Lm+1−r ′ ′ ′ Xr,s = Xn′ −1,1 ⊕ ( X r ,s ) ⊕ ( r≥s′ =s X r,s ), ′ ′ ′ s ,n ≤r
r ′ +s′ ≤m+1

X r,s = um (X r,s ), Xr,s = um (Xr,s ). Note, again, that when r + s = m + 1, X m+1−s,s can be zero, F em+1−s,s , or √ F ρem+1−s,s , according to the group at hand. The elements of Xr,s have the form   x1 x2 x3 x4    0 x5 x6 x′3  a b  ∈ X, x= (8.55) ′  0 0 x′ x′  , xa 5 2 0 0 0 x′1 where x is an (m + 1 − n′ ) × (m + 1 − n′ ) matrix, x1 is an (r − n′ ) × (s − 1) matrix, and x5 is a row vector of m + 1 − (r + s) coordinates (x′5 is its “dual” column). Finally, let C i,j B i,j Di,j Ai,j

= Xj−1,i+1 Zm Yi,j = C i,j Y i,j = C i,j X j−1,i = C i,j Y i,j X j−1,i .

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Note that the assumptions j > n′ and i + j ≤ m + 1 imply that j − 1 ≥ n′ and ′ (i + 1) + (j − 1) ≤ m + 1. Note also the following relations (D1,n was defined in the previous sections) ′

′

′

D1,n = B n ,n +1 , m even Di,j = B i−1,j , i≥2 D1,j X j,m+1−j = B m−j,j+1 , D1,j X j,m−j = B m−j−1,j+1 , h(V ) even orthogonal

(8.56)

We have first the following easy lemma. Lemma 8.7. The group Zm normalizes (over F ) each one of the groups above Xr,s , Yi,j . Next, we have the analog of Lemma 8.3 Lemma 8.8. Assume that dimE V = 2m. Let n′ = [ m+1 2 ] < j ≤ m and i + j ≤ m + 1. Then (1) C i,j is a group. (2) Define for c = uz¯ v ∈ C i,j (A), where z ∈ Zm (A), v¯ ∈ Yi,j , and u ∈ Xj−1,i+1 , ′ (z). ψC i,j (c) = ψS m (z¯ v ) = ψS m (z) = ψZ m

See (8.14). Then ψC i,j is a character of C i,j (A). (3) The groups X j−1,i and Y i,j normalize C i,j and preserve the character ψC i,j . (4) Assume that i + j < m + 1 when h(V ) is not symplectic. Then A = Ai,j , B = B i,j , C = C i,j , D = Di,j , X = X j−1,i , Y = Y i,j satisfy the requirements of Lemma 7.1 with respect to the character ψC i,j . Proof. The proof is the same as that of Lemma 8.3. Note that for Y ∈ Yi,j as in (8.53) and X ∈ Xj−1,i+1 as in (8.55), Y X is upper nilpotent and its second upper diagonal is also zero (similarly with XY ). Now we can use equation (8.29) and its solution (8.30). Hence any Whittaker character is trivial on (Im + Y X)∗ , when we take Adele coordinates. The arguments following (8.30) used just the facts that X, Y ∈ X and Y X is nilpotent. This proves (1) and (2). The proof that Y i,j normalizes C i,j and preserves ψC i,j is exactly as in Lemma 8.3, using (8.31), (8.32), and similarly for X j−1,i . This proves (3). Finally, the general form of formula (8.36) is as follows. Let X = aej−1,i − δ¯ aem+1−i,m+2−j and Y = bei,j − δ¯bem+1−j,m+1−i . Then   z u [um (X), u ¯m (Y )] = , (8.57) 0 z∗ where z = Im + abej−1,j − δa¯bδm+1−j,i ej−1,m+1−i ,

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and u = a¯ aδm+1−j,i (δb − ¯b). Thus, when i + j < m + 1, we get (8.36) again (up to ±1). Assume that i + j = m + 1. Then (8.57) becomes [um (X), u¯m (Y )] = um (a¯ a(δb − ¯b)ej−1,m+2−j )(Im + a(b − δ¯b)ej−1,j )∧ . (8.58)

Of course, this element lies in C m+1−j,j , and when we take Adele coordinates, we get ψC m+1−j,j ([um (X), u ¯m (Y )]) = ψE,δ (±a(b − δ¯b)). (8.59)

When h(V ) is orthogonal, this character is, of course, trivial. (In this case, Y m+1−j,j is trivial.) When h(V ) is unitary, this is not a general character of a as b varies in E. Indeed, as b varies in E, b − δ¯b varies over the elements t in E, which satisfy t¯ = −δt. Finally, when h(V ) is symplectic, we get the character ψ(2ab), and so the requirements of Lemma 7.1 are satisfied in this case as well. This proves (4). When HA is metaplectic, the proof of the lemma follows exactly as in Lemma 8.3, using induction on (i, j), where we order the pairs (i′ , j ′ ), (i, j), satisfying n′ < j, j ′ ≤ m, i + j, i′ + j ′ ≤ m + 1, in the same way we did in (8.37). This proves the lemma.  Extend the character ψC i,j to B i,j (A) and to Di,j (A), by making it trivial on Y i,j (A) and on X j−1,i (A), respectively. Denote the corresponding extensions by ψB i,j and ψDi,j . From Lemma 7.1 and Corollary 7.1 we obtain, as in the two previous sub-sections

Lemma 8.9. Assume that dimE V = 2m. Let f be a smooth automorphic function on HA . Assume that f is of uniform moderate growth. Assume that n′ = [ m+1 2 ] < j ≤ m and i+j < m+1, and in case h(V ) is symplectic, we also allow i+j ≤ m+1. Then (1)

Z

−1 f (vh)ψB i,j (v)dv =

i,j BF \BAi,j

(2)

Z

Z

−1 f (vyh)ψD i,j (v)dvdy.

i,j YAi,j DF \DAi,j

Z

YAi,j

Z

i,j \DAi,j DF



−1 f (vyh)ψD i,j (v)dv dy

< ∞.

This convergence is uniform in h varying in compact subsets of HA . (3) There is h ∈ HA , such that Z −1 f (vh)ψB i,j (v)dv 6= 0, i,j BF \BAi,j

if and only if there is h ∈ HA , such that Z −1 f (vh)ψD i,j (v)dv 6= 0. i,j \DAi,j DF

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Our next goal is to deal with the pairs (i, j), such that n′ < j ≤ m and i+j = m+1, in case h(V ) is not symplectic. Note again that when h(V ) is orthogonal, Y m+1−j,j is trivial, while X j−1,m+1−j is a root group, and when h(V ) is unitary, Y m+1−j,j is a root group of dimension 1 over F , while X j−1,m+1−j is a root group of dimension 2 over F . Thus, we cannot “exchange” these two root groups as in the last lemma. In case h(V ) is unitary, we will be able to “exchange” Y m+1−j,j with “half” of X j−1,m+1−j . Assume that h(V ) is unitary. Define X0j−1,m+1−j = {um (tej−1,m+1−j − δ t¯ej,m+2−j )|t¯ = t}, D0m+1−j,j = C m+1−j,j X0j−1,m+1−j Am+1−j,j = C m+1−j,j X0j−1,m+1−j Y m+1−j,j . 0 Since X0j−1,m+1−j ⊂ X j−1,m+1−j , it follows from Lemma 8.8 that X0j−1,m+1−j normalizes C m+1−j,j and preserves ψC m+1−j,j . Consider now the commutator of an element um (X) = um (tej−1,m+1−j − δ t¯ej,m+2−j ) of X0j−1,m+1−j (A) and an element u ¯m (Y ) = u¯m (dem+1−j,j ) of Y m+1−j,j (A). Here, t¯ = t and d¯ = −δd. By (8.59) (with a = t, d = b − δ¯b), ψ([um (X), u ¯m (Y )]) = ψE,δ (td). √ Note that when δ = 1, d has the form x ρ, where x ∈ A, and when δ = −1, d ∈ A. Thus, ψE,δ (td) = ψ(tx). This shows that the groups Am+1−j,j , B m+1−j,j , C m+1−j,j , D0m+1−j,j , X0j−1,m+1−j , Y m+1−j,j satisfy the re0 quirements of Lemma 7.1. Extend the character ψC m+1−j,j to B m+1−j,j (A) and to D0m+1−j,j (A), by making it trivial on X0j−1,m+1−j (A) and on Y m+1−j,j (A), respectively. Denote the corresponding extensions by ψB m+1−j and ψDm+1−j,j . As 0 before, we conclude Lemma 8.10. Assume that h(V ) is (even) unitary. Let f be a smooth automorphic function on HA . Assume that f is of uniform moderate growth. Assume that n′ < j ≤ m . Then (1)

Z

−1 f (vh)ψB m+1−j,j (v)dv =

m+1−j,j BF \BAm+1−j,j

Z

Z

−1 f (vyh)ψD m+1−j,j (v)dvdy. 0

YAm+1−j,j D0m+1−j,j (F )\D0m+1−j,j (A)

(2) Z

YAm+1−j,j



Z

D0m+1−j,j (F )\D0m+1−j,j (A)



−1 f (vyh)ψD m+1−j,j (v)dv dy 0

< ∞,

This convergence is uniform in h varying in compact subsets of HA .

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(3) There is h ∈ HA , such that Z

−1 f (vh)ψB 2n+1−j,j (v)dv 6= 0,

m+1−j,j \BAm+1−j,j BF

if and only if there is h ∈ZHA , such that −1 f (vh)ψD m+1−j,j (v)dv 6= 0. 0

D0m+1−j,j (F )\D0m+1−j,j (A)

Corollary 8.4. Assume that h(V ) = U4n . Let f be a smooth automorphic function on HA . Assume that f is of uniform moderate growth. Then for all t ∈ AE , such that t¯ = −t, Z −1 f (vxt h)ψB n,n+1 (v)dv =

n,n+1 \BAn,n+1 BF

Z

Z

−1 f (vxt yh)ψD n,n+1 (v)dvdy. 0

YAn,n+1 D0n,n+1 (F )\D0n,n+1 (A)

Proof.

Let Y = aen,n+1 , a ¯ = −δa, and y = u ¯m (Y ) ∈ Y n,n+1 . Then   I − δtaen,n+1 δt2 en,n+1 . [y, xt ] = m 0 Im + taen,n+1

In particular, [y, xt ] ∈ D0n,n+1 . We have −1 ′ (Im − δtaen,n+1 ) = ψE,δ (δta) = 1, ψDn,n+1 ([y, xt ]) = ψZ m 0

since δta = ta. Now we finish the proof as in Corollary 8.1.



In the following two theorems we will need, for the first time, that our automorphic forms lie in the residual representation Eτ¯ . Note that the analog of the following theorem in case h(V ) is symplectic appears already in Lemma 8.9. Theorem 8.4. Assume that dimE V = 2m and n′ = [ m+1 2 ] < j ≤ m. Let f be an automorphic form lying in the residual representation Eτ¯ . (1) Assume that h(V ) is orthogonal. Then the following identity is satisfied for all h ∈ HA ,Z Z −1 −1 f (vh)ψB (v)dv = f (vh)ψD m+1−j,j m+1−j,j (v)dv. m+1−j,j BF \BAm+1−j,j

m+1−j,j DF \DAm+1−j,j

(2) Assume that h(V ) is unitary. Then the following identity is satisfied for all h ∈ HA , Z −1 f (vh)ψD m+1−j,j (v)dv = 0

D0m+1−j,j (F )\D0m+1−j,j (A)

Z

m+1−j,j \DAm+1−j,j DF

−1 f (vh)ψD m+1−j,j (v)dv.

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Proof. Assume that h(V ) is orthogonal. Then B m−j+1,j = C m−j+1,j and Dm−j+1,j = C m−j+1,j X j−1,m+1−j . The elements of XAj−1,m+1−j have the form t ∈ A.

x(t) = um (t(ej−1,m+1−j − ej,m+2−j )), Assume that h(V ) is unitary. Let

aej,m+2−j )|¯ a = −a}. X1j−1,m+1−j = {um (aej−1,m+1−j − δ¯

Write the elements of X1j−1,m+1−j (A) as √ x(t) = um (t ρ(ej−1,m+1−j + δej,m+2−j )),

t ∈ A.

Note that X j−1,m+1−j = X0j−1,m+1−j X1j−1,m+1−j and Dm+1−j,j = m+1−j,j j−1,m+1−j D0 X1 . Denote by R the group B m−j+1,j , in case h(V ) is orthogonal, and the group D0m+1−j,j , in case h(V ) is unitary. We follow the beginning of the proof of Lemma 7.1 (see (7.4)). For fixed h ∈ HA , consider the following function on A, Z −1 f (rx(t)h)ψR (r)dr. t 7→ φh (t) = RF \RA

We know from Lemma 8.8 that x(t) normalizes R and preserves ψR , when t ∈ A. Thus, φh is a function on F \A. Let us write its Fourier expansion and evaluate it at zero: X Z φh (t)ψ −1 (λt)dt. φh (0) = λ∈F F \A

Thus, what we need to prove is that all the Fourier coefficients, with respect to nontrivial characters, of φh are zero. So, let λ ∈ F ∗ . Then we want to show that the Fourier coefficient Z Z −1 λ f (rx(t)h)ψR (r)ψ −1 (λt)drdt φh = F \A RF \RA

is zero. We know that the set of all rx(t) in the integrand is a group R′ (algebraic over F ). Note that R′ = Xj−1,m+1−j Zm Ym−j+1,j ,

(8.60)

and ψR′ ,λ (rx(t)) = ψR (r)ψ(λt) is a character of RA′ , which is trivial on RF′ . In terms of (8.60), ′ (z)ψλ (um (x)), ψR′ ,λ (um (x)ˆ zu ¯m (y)) = ψZ m

where ψλ (um (x)) =

(

ψ(λxj−1,m+1−j ), λx √ ψ( 12 trE/F ( j−1,m+1−j ρ

h(V ) orthogonal )),

h(V ) unitary

.

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Thus, we may rewrite φλh

=

Z

−1 f (rh)ψR ′ ,λ (r)dr.

R′F \R′A

Denote Z m,j =

 z

Zj =

e Im−j+1



 |z ∈ ResE/F Zj−1 ,

Z m,j = (Z m,j )∧ ,

   Ij−1 |z ∈ ResE/F Zm+1−j , z

Z j = (Z j )∧ .

Define Bj = Xj−1,m+1−j Z m,j Ym−j+1,j . Using equation (8.30) and its solution, it is easy to see that Bj is a subgroup and that Z j normalizes Bj and R′ = Bj Z j . Denote by ψBj ,λ the restriction of ψR′ ,λ to Bj (A), and denote the restriction to Z j (A) by ψ(j) . Then Z Z −1 −1 λ f (vzh)ψB (v)ψ(j) (z)dvdz, φh = j ,λ j ZF \ZAj Bj (F )\Bj (A)

and so, it is enough to prove that for all h ∈ HA , the Fourier coefficient Z −1 ψ;j,λ f (vh)ψB (v)dv f (h) = j ,λ Bj (F )\Bj (A)

∗

is zero (for all λ ∈ F ). Now we can use Lemma 7.1 repeatedly, as before. This is done as follows. Define, for 1 ≤ i < m + 1 − j, Cji = Xj−1,i+1 Z m,j Yi,j , Bji = Cji Y i,j Dji = Cji X j−1,i .

Note that Bjm−j = Bj ,

Dji = Bji−1 ,

i ≥ 2.

(8.61) Cji

Using equation (8.30) and its solution, it is easy to see that is a group. Let c = um (x)z u ¯(y) be an element of Cji (A) (according to the decomposition above). Define ψCji (c) = ψBj ,λ (um (x)z). Then ψCji is a character of Cji . Again, this follows as in the proof of Lemma 8.3. For this, (and for showing that Cji is a group), the main point is that for X ∈ Xj−1,i+1   I a and Y ∈ Yi,j , the matrix z = (Im + Y X)∗ has the form z = j−1 , where Im+1−j aj−1,1 = 0, and in particular z ∈ Z m,j , with its second upper diagonal being zero.

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Now, it is easy to see that X ′ = X(Im + Y X)−1 , which lies in Xj−1,i+1 (see the proof of Lemma 8.3) is such that, when we take Adele coordinates, ψCji (um (X ′ )) = ψCji (um (X)). Examine now the conjugation by Y i,j . As in (8.31), it is clear that Y i,j conjugates Z m,j into Cji and preserves ψCji , when we take Adele coordinates. Similarly, when we conjugate Xj−1,i+1 by Y i,j . This is as in (8.32). Indeed, let Y ∈ Y i,j and X ∈ Xj−1,i+1 . Then

u ¯m (Y )um (X)¯ um (Y )−1 = um (X(z ∗ )−1 )ˆ zu ¯m ((z ∗ )−1 Y − Y ), (8.62)   I a where z = (Im +Y X)∗ . Then z ∗ has the form m+1−j , where am+1−j,1 = 0. Ij−1 In particular, z ∈ Z m,j . As in the proof of Lemma 8.3, (z ∗ )−1 Y −Y ∈ Yi,j . Looking ′ at the form of z, it is clear that when we take Adele coordinates, ψZ (z) = 1 and m ∗ −1 i,j ψλ (um (X(z ) ) = ψλ (um (X)). Of course, Y commutes element-wise with Yi,j , and so we showed that Y i,j acts on Cji by conjugation and preserves ψCji , when we take Adele coordinates. Next, consider conjugation by X j−1,i . That X j−1,i conjugates Z m,j into Cji , preserving ψCji , when we take Adele coordinates, follows from (8.33). Indeed, let X ∈ X j−1,i and z ∈ Z m,j . Then um (X)ˆ z um (X)−1 = zˆum (z −1 Xz ∗ − X).

It is easy to see that z −1 Xz ∗ − X ∈ Xj−1,i+1 . Since z ∈ Z m,j , we also see that (z −1 Xz ∗)j−1,m+1−j = (X)j−1,m+1−j = 0 and hence ψCji (um (z −1 Xz ∗ − X)) = 1 (in Adele coordinates). Here, it is crucial that we take Z m,j rather than all of Zm . Using (8.34), it is easy to see, as in (8.62), that X j−1,i conjugates Yi,j into Cji and that ψCji (xyx−1 ) = 1, for all x ∈ XAj−1,i , y ∈ Yi,j (A). Finally, formula (8.36) holds, when we consider [X j−1,i , Y i,j ], and this is explained in (8.57) and what follows. Now we can apply Lemma 7.1 with A = Aij , ..., D = Dji , X = X j−1,i , Y = Y i,j . We conclude that Z Z Z −1 −1 f (vyh)ψD (8.63) f (vh)ψB i (v)dv = i (v)dvdy, j

j

YAi,j Dji (F )\Dji (A)

Bji (F )\Bji (A)

and

Z

YAi,j

Z

Dji (F )\Dji (A)

−1 f (vyh)ψD (v)dv dy < ∞. i j

Here, ψBji and ψDji are the extensions of ψCji by the trivial character to Bji (A), Dji (A), respectively. Note that ψB m−j = ψBj ,λ . Apply (8.63) for i = j

m − j, m − j − 1, ..., 1, and using (8.61), we get Z Z −1 ψ;j,λ f (vh)ψBj (v)dv = f (h) = Bj (F )\Bj (A)

Z

−1 f (vyh)ψD 1 (v)dvdy, j

Yj (A) Dj1 (F )\Dj1 (A)

(8.64)

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where Yj = Y m−j,j Y m−j−1,j · · · Y 1,j . The convergence of the r.h.s. of (8.64) is in the sense above, repeated for i = m − j, ..., 1. Recall that Dj1 = Xj−1,1 Z m,j Y1,j . Note that Xj−1,1 Z m,j = Nj−1 , and Y1,j normalizes Nj−1 and preserves ψDj1 , restricted to Nj−1 , when we take Adele coordinates. Thus, Z Z Z −1 ′ ψ;j,λ (8.65) f (vy ′ yh)ψD f (h) = 1 (v)dvdy dy. j

Yj (A) Y1,j (F )\Y1,j (A) Nj−1 (F )\Nj−1 (A)

The inner dv-integration is a Gelfand-Graev coefficient of f corresponding to a character, which is conjugate to ψj−1,wλ , where wλ = ej − t(λ)e−j ; t(λ) = λ, when δ = 1, and t(λ) = √λρ , when δ = −1. (See (3.5).) Note that wλ is h(Wm,j−1 )conjugate to yαλ (see (3.10)) where αλ = ±2t(λ). Thus, there is γλ ∈ Mj−1 , such that Z −1 ψj−1,αλ (γλ h), f (vh)ψD 1 (v)dv = f j

Nj−1 (F )\Nj−1 (A)

and so (8.65) reads as f ψ;j,λ (h) =

Z

Z

f ψj−1,αλ (γλ y ′ yh)dy ′ dy.

Yj (A) Y1,j (F )\Y1,j (A)

By Proposition 7.1, we know that, for f lying in Eτ¯ , f ψj−1,αλ = 0. Indeed, j − 1 > m−1 n′ − 1 = [ m+1 2 ] − 1 = [ 2 ] = ℓm . This is the first time in this chapter that we use the fact that f is in our residual representation. This proves that f ψ;j,λ = 0, for all λ ∈ F ∗ . We remark here, that in Proposition 7.1 unitary groups were assumed to √ be with respect to a Hermitian form, but conjugation by the matrix diag(Im , ρIm ) takes a unitary group h(V ) with respect to a Hermitian form to a unitary group with respect to an anti-Hermitian form and vice versa, and defines an F -isomorphism between the two groups. Thus, our use of Proposition 7.1, in case of a unitary group with respect to an anti-Hermitian form is justified. The proof of Theorem 8.4 is now complete.  Theorem 8.5. Assume that dimE V = 2m, and h(V ) is either symplectic or unitary. Let f be an automorphic form lying in the residual representation Eτ¯ . Denote n = [m 2 ], and let n < j − 1 ≤ m. In case HA is metaplectic and m = 2n + 1 is odd, we require that n + 1 < j − 1 ≤ m. Then, for all h ∈ HA , Z Z −1 −1 f (vh)ψD f (vh)ψB 1,j−1 (v)dv = m+1−j,j (v)dv. 1,j−1 DF \DA1,j−1

m+1−j,j BF \BAm+1−j,j

Proof. For simplicity of notation, we assume that HA is not metaplectic. The slight modifications for the metaplectic case are exactly as in the end of the proof of Lemma 8.3. Recall that D1,j−1 = Xj−2,1 Zm Y1,j−1 and that D1,j−1 X j−1,m+2−j = B m+1−j,j . (Note that in case h(V ) is orthogonal, X j−1,m+2−j = 1, and hence the identity asserted in this theorem is trivial in this case.) We proceed as in the proof of

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Theorem 8.4. The elements of X j−1,m+2−j have the form um (tej−1,m+2−j ), where t¯ = −δt. Denote ( um (tej−1,m+2−j ), δ = −1 x(t) = √ um (t ρej−1,m+2−j ), δ=1 Note that X j−1,m+2−j normalizes D1,j−1 . Indeed, X j−1,m+2−j commutes elementwise with Y1,j and with Xj−1,1 , and by (8.33), it conjugates Zm into D1,j , preserving ψD1,j , when we take Adele coordinates. Consider, for fixed h ∈ HA , the following function on F \A, Z −1 f (vx(t)h)ψD φh (t) = 1,j−1 (v)dv. 1,j−1 \DA1,j−1 DF

Write its Fourier expansion, and evaluate it at zero: X Z φh (t)ψ −1 (λt)dt. φh (0) = λ∈F F \A

As in the proof of Theorem 8.4, we need to show that all the Fourier coefficients, with respect to nontrivial characters, of φh are zero. Let λ ∈ F ∗ . We need to show that the Fourier coefficient Z Z −1 −1 f (vx(t)h)ψD (λt)dvdt φλh = 1,j−1 (v)ψ F \A D1,j−1 \D1,j−1 F

A

is zero. By the remarks above, we can rewrite this Fourier coefficient as Z −1 f (vh)ψB φλh = m+1−j,j ,λ (v)dv, m+1−j,j BF \BAm+1−j,j

where ψB m+1−j,j ,λ is the character of B m+1−j,j (A) defined as follows. Write v ∈ B m+1−j,j (A) in the form v = ux(t), where u ∈ D1,j−1 (A) and t ∈ A. Then ψB m+1−j,j ,λ (v) = ψD1,j−1 (u)ψ(λt). Decompose, as in the proof of Theorem 8.4, Zm = Z m,j−1 Z j−1 . It is easy to see that Bj = Xj−1,m+2−j Z m,j−1 Y1,j−1 is a group, and for a corresponding v = xˆ zy ∈ Bj (A), ψBj ,λ (v) = ψB m+1−j,j ,λ (xˆ z ) defines a character of Bj (A). Also, B m+1−j,j = Bj Z j−1 , and Z j−1 normalizes Bj and preserves ψBj ,λ , when we take Adele points. For this, just note that Z j−1 commutes element-wise with X j−1,m+2−j , and of, course, it normalizes Y1,j−1 and Xj−1,1 . We have 0 B m+1−j,j = Bj Z j−1 = Xj−1,m+2−j Z m,j−1 Y1,j−1 Z j−1 = Nj−2 Y1,j−1 Z j−1 ,

where 0 Nj−2 = Xj−1,m+2−j Z m,j−1 = Nj−2 X j−1,m+2−j

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0 is the group defined in the beginning of Sec. 6.1. Denote by ψNj−2 ,λ the restriction 0 of ψBj ,λ to Nj−2 (A), and denote by ψ(j−1) the restriction of ψD1,j−1 to Z j−1 (A). Then Z Z

φλh =

f

ψN 0

j−2

,λ

−1 (yzh)ψ(j−1) (z)dydz,

j−1 ZF \ZAj−1 Y1,j−1 (F )\Y1,j−1 (A)

where f

ψN 0

j−2

,λ

Z

(h) =

−1 f (vh)ψN 0

j−2 ,λ

(v)dv.

0 0 Nj−2 \Nj−2 (A)

Clearly, we can find a diagonal element of Mj−2 (F ), γλ , such that it conjugates 0 0 ψNj−2 ,λ to (ψλ )j−2 , where ψλ (a) = ψ(λa), and then f

ψN 0

j−2

,λ

0

(h) = f (ψλ )j−2 (γλ h). 0

See (7.63). By Proposition 7.5, we see that f (ψλ )j−2 = 0. Indeed, assume first that m−2 HA is not metaplectic, with m odd. Then j − 2 > n − 1 = [ m 2 ] − 1 = [ 2 ] = ℓm . In case HA is metaplectic and m = 2n + 1 is odd, ℓm = m−1 = n, and by our 2 assumption, in this case, j − 2 > n = ℓm . This shows that φλh = 0, for all λ ∈ F ∗ and all h ∈ HA . Finally, the same remark as in the end of the proof of Theorem 8.4 explains that even though we formulated Proposition 7.5 for unitary groups with respect to anti-Hermitian forms, it can be used for unitary groups with respect to Hermitian forms as well. This proves Theorem 8.5. 

8.7

Completion of the proof of Theorems 8.1, 8.2; dimE V = 2m

Now we can complete the proof of Theorems 8.1, 8.2 when dimE V = 2m. Assume, first, that HA is not metaplectic, with m odd. Define, for i + j ≤ m + 1 and j > n′ = [ m+1 2 ], ′

′

′

∗ Yi,j = Ym∗ Y m−n ,n +1 · · · Y 1,n +1 · · · Y m+1−j,j · · · Y i,j .

We will prove, by induction, that for all n′ ≤ k ≤ m, and all h ∈ HA , Z Z −1 f (vyαm h)ψD Jψ (f )(h) = 1,k (v)dvdy. ∗ (A) Y1,k

1,k \DA1,k DF

(8.66)

For k = n′ , this is Proposition 8.1. Denote n = [ m 2 ]. We first prove (8.66), for k = n + 1. If m = 2n + 1 is odd, then n + 1 = n′ , and there is nothing to prove. Assume that m = 2n is even (and hence n′ = n). Recall that D1,n = B n,n+1 (see (8.56)). By Lemma 8.9, for (i, j) = (n, n + 1) and h(V ) symplectic, Theorem 8.4, for j = n + 1 and h(V ) orthogonal, and Lemma 8.10, Theorem 8.4, for j = n + 1 and h(V ) unitary, we have, using (8.66), for k = n, Z Z −1 Jψ (f )(h) = f (vyαm h)ψD n,n+1 (v)dvdy. ∗ Yn,n+1 (A) Dn,n+1 \Dn,n+1 F

A

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Recall now that Di+1,n+1 = B i,n+1 , for i ≥ 1 (see (8.56)) and apply Lemma 8.9 repeatedly, for j = n + 1 and i = n − 1, n − 2, ..., 1 (in this order). We get (8.66) for k = n + 1, when m is even as well. Assume, by induction, that (8.66) holds for k = j − 1 > n. By Theorem 8.5, for h(V ) symplectic or unitary, we get Z Z −1 Jψ (f )(h) = f (vyαm h)ψB m+1−j,j (v)dvdy. ∗ Y1,j−1 (A) B m+1−j,j \B m+1−j,j F

A

1,j−1

Note that when h(V ) is orthogonal, D = B m+1−j,j , so that the last equality is just a rewriting of the induction assumption. By Lemma 8.9, for i = m + 1 − j and h(V ) symplectic, Theorem 8.4, for h(V ) orthogonal, and Lemma 8.10, Theorem 8.4, for h(V ) unitary, we get Z Z −1 Jψ (f )(h) = f (vyαm h)ψD m+1−j,j (v)dvdy. ∗ Ym+1−j,j (A) Dm+1−j,j \Dm+1−j,j F

A

Now use the relations (8.56) Di+1,j = B i,j , for i ≥ 1, and apply Lemma 8.9 repeatedly , for i = m−j, m−j −1, ..., 1. This will prove (8.66), for k = j. We conclude, by ∗ induction, that (8.66) holds for k = m. Note that Y1,m (A) = u ¯m (X0 (A)), Y1,m = 1, and when h(V ) is orthogonal Xm−1,1 = Um , while for h(V ) unitary, or symplectic, Xm−1,1 X m,1 = Um . Note also that D1,m = Xm−1,1 Zm . Thus, Z Z Z ′ (z −1 )dvdzdy. f (vzyαm h)ψZ Jψ (f )(h) = m u ¯ m (X0 )A Xm−1,1 (F )\Xm−1,1 (A) Zm (E)\Zm (AE )

This proves (8.6) when h(V ) is orthogonal. When h(V ) is symplectic or unitary, we need to apply Theorem 8.5 one more time, namely for j = m + 1, to get that Z Z ′ (z −1 )dvdz = f (vzh)ψZ m Xm−1,1 (F )\Xm−1,1 (A) Zm (E)\Zm (AE )

Z

Z

′ (z −1 )dudz. f (uzh)ψZ m

Um (F )\Um (A) Zm (E)\Zm (A)

Note again that when h(V ) is orthogonal this equality is tautological. The last integral is equal to Z ′ (z −1 )dz. (8.67) z h)ψZ f Um (ˆ 2n Zm (E)\Zm (AE )

This proves (8.6) when h(V ) is symplectic, or unitary. Finally, Proposition 8.1, Lemma 8.9, Lemma 8.10, Theorem 8.4 and Theorem 8.5 also prove that there is h ∈ HA , such that Jψ (f )(h) 6= 0, if and only if there is h ∈ HA , such that (8.67) is nonzero, and this is clear, since f ∈ Eτ¯ and the Eisenstein series on GLm (AE ), induced from τ¯ is globally generic (even at any point). Thus, there is f in Eτ¯ , such that Jψ (f )(1) 6= 0.

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Finally, assume that HA is metaplectic and m = 2n + 1 is odd. By Proposition 8.1, Z Z −1 ′ f (vyαm h)ψ(D Jψ (f )(h) = ′ )m (v)dvdy. ∗ (A) (D ′ )m \(D ′ )m Ym F A

Note that (D′ )m = D1,n+1 X n+1,n+1 = B n,n+2 . Thus, Z Z −1 ′ Jψ (f )(h) = f (vyαm h)ψB n,n+2 (v)dvdy. ∗ (A) Ym B n,n+2 \B n,n+2 F

A

Use the relations (8.56) Di+1,n+2 = B i,n+2 , for i ≥ 1, and apply Lemma 8.9 repeatedly , for i = n, n − 1, ..., 1 and j = n + 2. We get Z Z −1 ′ Jψ (f )(h) = f (vyαm h)ψD 1,n+2 (v)dvdy. ∗ Y1,n+2 D1,n+2 \D1,n+2 F

A

Now, we can prove, by induction, the analog of (8.66). For n + 2 ≤ k ≤ m, Z Z −1 f (vyαm h)ψD (8.68) Jψ′ (f )(h) = 1,k (v)dvdy. ∗ (A) Y1,k D1,k \D1,k F

A

We just proved this equality when k = n + 2. The rest of the proof, by induction, is exactly as above, word for word, since now we can use Theorem 8.5, which required, in this case, that j − 1 ≥ n + 2. This completes the proof of Theorems 8.1, 8.2 in case dimE V = 2m. 8.8

Completion of the proof of Theorem 8.3

Let h(V ) = SO4n . By Corollary 8.3, the fact that D1,n = B n,n+1 , and Theorem 8.4 for j = n + 1, we have for all t ∈ A, Z Z −1 ω Jm,ψ (f )(in (t) h) = f (vxt yαm h)ψD 1,n (v)dvdy Z

=

∗ (A) Ym D1,n \D1,n

Z

F

A

−1 f (vxt yαm h)ψD n,n+1 (v)dvdy.

∗ (A) Ym Dn,n+1 \Dn,n+1

Since xt ∈

F

A

DAn,n+1 ,

we get, changing variable vxt 7→ v, Z Z −1 Jm,ψ (f )(in (t)ω h) = f (vyαm h)ψD n,n+1 (v)dvdy = Jm,ψ (f )(h). ∗ (A) Ym Dn,n+1 \Dn,n+1 F

A

Assume that h(V ) = U4n . By Corollary 8.3, the fact that D1,n = B n,n+1 , and Corollary 8.4 for j = n + 1, we have, for all t ∈ AE , such that t¯ = −t, Z Z −1 f (vxt yαm h)ψD Jm,ψ (f )(in (t)h) = n,n+1 (v)dvdy. 0

∗ )(A) (Y n,n+1 Ym Dn,n+1 (F )\Dn,n+1 (A) 0

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By Theorem 8.4, we get

Z

Jm,ψ (f )(in (t)h) =

Z

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229

−1 f (vxt yαm h)ψD n,n+1 (v)dvdy.

∗ )(A) (Y n,n+1 Ym Dn,n+1 \Dn,n+1 F

A

DAn,n+1 .

Now we finish, as before, since xt ∈ Assume that h(V ) = U4n+2 . By Corollary 8.3 and Theorem 8.5 for j = n + 2, we get, for all t ∈ AE , such that t¯ = −δt, Z Z −1 Jm,ψ (f )(in (t)h) = f (vxt yαm h)ψD 1,n+1 (v)dvdy ∗ (A) Ym D1,n+1 \D1,n+1

=

Z

F

Z

A

−1 f (vxt yαm h)ψB n,n+2 (v)dvdy.

∗ (A) Ym B n,n+2 \B n,n+2 F

A

BAn,n+2 ,

Since xt ∈ we can change variables vxt 7→ v, as before, and thus complete the proof of Theorem 8.3.

8.9

Second induction step: odd orthogonal groups

Assume that h(V ) is odd orthogonal. Denote n = [ m 2 ]. For n + 2 ≤ j ≤ m, i + j ≤ m + 1, define Y i,j = {t(ei,j − em+1−j,m+1−i )}, Yi,j = {y ∈ Y1,n+1 |yi′ ,j ′ = 0, j ′ < j, and yi′ ,j = 0, i ≤ i′ ≤ m + 1 − j}, Y i,j = c¯m (Y i,j ,

)Yi,j = c¯m (Yi,j ).

Note that Y m+1−j,j = 1. Let 1 Y 0,j = {¯ um (tξm+1−j , − t2 em+1−j,j )}, 2 where ξ1 , ..., ξm is the standard basis of column vectors in m coordinates. This is a (negative) root group. Consider the subset 1 u ˜′ (j) = {¯ um (e, − et ewm )|ei = 0, i > m − j}. 2 Then Y1,j−1 u˜′ (j), Yi,j u ˜′ (j) are subgroups. The elements of Yi,j u ˜′ (j) have the form   Ij−1   Im+1−j   ′  ∈ SO2m+1 ,  (8.69) e 1     Y e Im+1−j Ij−1 where em+1−j = 0 and yi′ ,j = 0, for i ≤ i′ . Define, next, for r ≥ n+1, r +s ≤ m+1, X r,s = {t(er,s − em+1−s,m+1−r) }, L Lm+1−r ′ ′ ′ Xr,s = Xn,1 ⊕ ( ′ X r ,s ) ⊕ ( s′ =s X r,s ), ′ s ,n+1≤r
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X r,s = um (X r,s ), Xr,s = um (Xr,s ), 1 X r,0 = {um (tξr , − t2 er,m+1−r )}. 2 This is a (positive) root group. Consider the subset 1 v(r) = {um (v, − v t vwm )|vi = 0, i > r}. 2 Then v(r)Xr−1,1 , v(r)Xr,s are groups. The elements of v(r)Xr,s have the form   Ir 0 v x1 x2  Im−r 0 0 x′1     1 0 v′  (8.70)  ∈ SO2m+1 ,    Im−r Ir

where (x1 )r,s′ = 0, for 1 ≤ s′ < s. Define, for n + 2 ≤ j ≤ m, i + j ≤ m + 1 C 0,j B 0,j D0,j A0,j

= v(j − 2)Xj−2,1 Zm Y1,j−1 u ˜′ (j), 0,j 0,j =C Y , = C 0,j X j−1,0 , = C 0,j Y 0,j X j−1,0 .

C i,j B i,j Di,j Ai,j

= v(j − 1)Xj−1,i+1 Zm Yi,j u ˜′ (j), i,j i,j =C Y , = C i,j X j−1,i , = C i,j Y i,j X j−1,i .

Lemma 8.11. Let n + 2 ≤ j ≤ m, i + j ≤ m + 1, i ≥ 0. Then (1) C i,j is a group. (2) Write c ∈ C i,j (A) in the form c = u1 zˆu ¯2 , according to the definition above, ¯ where u1 ∈ Um (A), u ¯2 ∈ Um (A) and z ∈ Zm (A). Define ′ (z). ψC i,j (c) = ψZ m

Then ψC i,j is a character of C i,j (A). (3) Assume that i+j < m+1. Then A = Ai,j , B = B i,j , C = C i,j , D = Di,j , X = X j−1,i , Y = Y i,j satisfy the requirements of Lemma 7.1, with respect to ψC i,j . Proof. We start with i = 0. In this case, we use formulae (8.43), (8.44) which are still valid, where in loc.cit. we take vm (v) to be a general element of v(j − 2) and u˜′m (e) to be a general element of u˜′ (j). Note that, for X ∈ Xj−2,1 and Y ∈ Y1,j−1 , the matrices (in (8.43), (8.44)) Xet ewm , v t ewm , XY are upper nilpotent and their second upper diagonal is zero. Also, cm (Y ) and vm (v) commute. This proves (1), (2). Denote v(a) = um (aξj−1 , − 21 t2 ej−1,m+2−j ) and u ˜(b) = u ¯m (bξm+1−j , − 12 b2 em+1−j,j ). The formulae (8.43) are still valid when

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X ∈ Xj−2,1 and Y ∈ Y1,j−1 , and when vm (v) is replaced by v(a), or a general element of v(j − 2) and u˜′m (e) is replaced by u ˜(b), or a general element of u ˜′ (j). In particular, the third formula in (8.43) shows [v(a), u˜(b)] = (Im − abej−1,j )∧ ;

′ ([v(a), u˜(b)]) = ψ(ab). ψZ m

(8.71)

Finally, v(a) conjugates zˆ ∈ Zm into v(j − 2)Xj−2,1 zˆ, and u˜(b) conjugates zˆ into zˆY1,j−1 u˜(j). This is seen by the following formulae um (v, 21 vv ′ )ˆ z um (v, 12 vv ′ )−1 = um (v − zv, 21 (zvv ′ (z ∗ )−1 − 2vv ′ (z ∗ )−1 + vv ′ )), 1 ′ zu ¯m (e, 21 ee′ )−1 = zˆu ¯m ((z ∗ )−1 e − e, 21 ((z ∗ )−1 ee′ z − 2(z ∗ )−1 ee′ + ee′ )). u ¯m (e, 2 ee )ˆ (8.72) Here, v ′ = −t vwm . Thus, we proved that X j−1,0 and Y 0,j act on C 0,j by conjugation, and preserve ψC 0,j (over A). This proves (3). Let 1 ≤ i ≤ m+1−j. We refer again to (8.43), (8.44), which are still valid, when X ∈ Xj−1,i+1 , Y ∈ Yi,j , vm (v) replaced by a general element of v(j − 1), and u ˜′m (e) ′ t t replaced by a general element of u˜ (j). Note again that Xe ewm , v ewm , XY are upper nilpotent, their second upper diagonal is zero, and that cm (Y ) and vm (v) commute. This proves (1), (2). Let X ∈ X j−1,i . The first formula of (8.45) is ˜′ (j). This formula shows that valid , with u˜′m (e) replaced by a general element of u j−1,i ′ i,j X conjugates u ˜ (j) into C . Note also that Xet ewm is upper nilpotent and its second upper diagonal is zero. The second formula of (8.45) is valid when we take Y ∈ Y i,j and replace vm (v) by a general element of v(j − 1). We see that c¯m (Y ) commutes with the elements of v(j − 1). The fact that X j−1,i (resp. Y i,j ) conjugates Yi,j (resp. Xj−1,i+1 ) into C i,j , preserving ψC i,j , over A, follows exactly as in Lemma 8.8. Finally, let i + j < m + 1, X j−1,i = cm (a(ej−1,i − em+1−i,m+2−j )), Y i,j = c¯m (b(ei,j − em+1−i,m+1−j )). As in (8.57), [X j−1,i , Y i,j ] = (Im + abej−1,j )∧ , and so, over A, ′ ([X j−1,i , Y i,j ]) = ψ −1 (ab). ψZ m

This proves (3).



We conclude, with similar notations, the analog of Lemma 8.9. Lemma 8.12. Assume that h(V ) is odd orthogonal (dimF V = 2m + 1, [ m 2 ] = n). Let f be a smooth automorphic function on HA . Assume that f is of uniform moderate growth. Assume that n + 2 ≤ j ≤ m and 0 ≤ i < m + 1 − j. Then (1) Z

i,j \BAi,j BF

−1 f (vh)ψB i,j (v)dv =

Z

Z

i,j \DAi,j YAi,j DF

−1 f (vyh)ψD i,j (v)dvdy.

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(2)

Z

YAi,j

Z

i,j DF \DAi,j



−1 f (vyh)ψD i,j (v)dv dy

< ∞.

This convergence is uniform in h varying in compact subsets of HA . (3) There is h ∈ HA , such that Z −1 f (vh)ψB i,j (v)dv 6= 0, i,j \BAi,j BF

if and only if there is h ∈ HA , such that Z −1 f (vh)ψD i,j (v)dv 6= 0. i,j DF \DAi,j

The analog of Theorem 8.4(1) is valid also. Theorem 8.6. Assume that h(V ) is odd orthogonal. Let f be an automorphic form lying in the residual representation Eτ¯ . Let n + 2 ≤ j ≤ m. Then the following identity is satisfied for all h ∈ HA , Z Z −1 −1 f (vh)ψB m+1−j,j (v)dv = f (vh)ψD m+1−j,j (v)dv. m+1−j,j BF \BAm+1−j,j

m+1−j,j DF \DAm+1−j,j

Proof. The proof is the same as that of Theorem 8.4(1), with the obvious modifications. Thus, the group Bj in loc. cit. should be Bj = v(j − ′ 1)Xj−1,m+1−j Z m,j Ym+1−j,j u˜′ (j) and the character ψBj ,λ of Bj (A) is ψZ on m m,j ′ Z (A); it is trivial on the Adele points of Ym+1−j,j u˜ (j) , v(j − 1), and on Xj−1,m+1−j , it is given by cm (x) 7→ ψ(λxj−1,m+1−j ). We need to show that the Fourier coefficients Z f (vh)ψB j ,λ (v)−1 (v)dv,

f ψ;j,λ (h) =

Bj (F )\Bj (A)

are identically zero, for all λ ∈ F ∗ . For the repeated application of Lemma 7.1, define, for 1 ≤ i < m + 1 − j, the groups Cji , etc., where now Cji = v(j − 1)Xj−1,i+1 Zm,j Yi,j u˜′ (j); the relations (8.61) hold. The details of the proof that these groups satisfy the requirements of Lemma 7.1 are the same as in the proof of Theorem 8.4 and Lemma 8.11. We conclude, as in loc. cit. that Z Z f ψj−1,αλ (γλ y ′ yh)dy ′ dy, f ψ;j,λ (h) = Yj (A) Y1,j (F )\Y1,j (A)

where γλ is a certain element in Mj−1 (F ) and f ψj−1,αλ is the Gelfand-Graev coefficient of f with respect to ψj−1,αλ , and αλ = ±2λ. By Proposition 7.1, we know that, for f lying in Eτ¯ , f ψj−1,αλ = 0. Indeed, j − 1 > n = [ m 2 ] = ℓm . This proves the theorem. 

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8.10

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∗ Assume that h(V ) is odd orthogonal (dimF V = 2m + 1, [ m 2 = n). Let Jψ (f ) = ∗ ′ Jψ (f ), when m is odd, and Jψ (f ) = Jψ (f ), when m is even. Then by Propositions 8.2, 8.3, Z Z −1 Jψ∗ (f )(h) = f (vyαm h)ψD 1,n+1 (v)dvdy. ∗ Y1,n+1 (A) D1,n+1 \D1,n+1 F

A

Let us prove by induction that, for all n + 1 ≤ k ≤ m and all h ∈ HA , Z Z −1 f (vyαm h)ψD Jψ∗ (f )(h) = 1,k (v)dvdy.

(8.73)

∗ (A) Y1,k D1,k \D1,k F

A

Here, we define, as before, for 0 ≤ i ≤ m + 1 − j and j > n, ∗ ∗ Yi,j = Y1,n+1 Y 0,n+2 Y m−n−1,n+2 · · · Y 1,n+2 · · · Y 0,j Y m+1−j,j · · · Y i,j .

Assume that (8.73) holds, for k = j − 1 ≥ n + 1. By, construction, D1,j−1 = B 0,j . From Lemma 8.12, for (0, j), Z Z −1 ∗ Jψ (f )(h) = f (vyαm h)ψD 0,j (v)dvdy. ∗ (A) Y0,j D0,j \D0,j F

By definition, D

0,j

m+1−j,j

=B Z ∗ Jψ (f )(h) =

A

. Theorem 8.6 implies now Z −1 f (vyαm h)ψD m+1−j,j (v)dvdy.

∗ (A) Y0,j Dm+1−j,j \Dm+1−j,j F

A

By construction, Di+1,j = B i,j , for i ≥ 1. Apply Lemma 8.12 repeatedly, for i = m − j, m − j − 1, ..., 1. This proves (8.73), for k = j. By induction, (8.73) holds ∗ for k = m. Note that Y1,m = Ym and D1,m = v(m − 1)Xm−1,1 Zm . This is the unipotent radical     z a b  D1,m =  I3 a′  ∈ SO2m+1 |z ∈ Zm−1 .   z∗

Denote

fD

1,m

,ψD1,m

Z

(h) =

−1 f (vh)ψD 1,m (v)dv.

D1,m (F )\D1,m (A)

Then Jψ∗ (f )(h) =

Z

Ym (A)

fD

1,m

,ψD1,m

(yαm h)dy.

(8.74)

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The following argument is a special case of the proof of Theorem 8.6. Consider the (simple) root group x(t) = um (tξm , − 21 t2 em,1 ). Define φh (t) = f D

1,m

,ψD1,m

(x(t)h).

This is a function on F \A. Write its Fourier expansion, and evaluate it at zero. Then X Z 1,m D1,m ,ψD1,m f D ,ψD1,m (x(t)h)ψ −1 (λt)dt. f (h) = λ∈F F \A

Clearly, the Fourier coefficients of φh , with respect to nontrivial characters are Whittaker coefficients of f (Gelfand-Graev coefficients with respect to ψm,λ ), and hence are identically zero, since f lies in the residual representation Eτ¯ . Thus, only the trivial coefficient remains, and we get Z Z 1,m 1,m ′ (z −1 )dz. z h)ψZ f Um (ˆ f D ,ψD1,m (x(t)h)dt = f D ,ψD1,m (h) = m F \A

Zm (F )\Zm (A)

Substitute this in (8.74) and we get the identities (8.7), (8.9). Finally, Propositions 8.2, 8.3, Lemma 8.12, Theorem 8.6 and the last identity imply that there is h ∈ HA , such that Jψ∗ (f )(h) 6= 0, if and only if there is h ∈ HA , such that Z ′ (z −1 )dz 6= 0, z h)ψZ f Um (ˆ m Zm (F )\Zm (A)

and this is true (same argument as in case dimE V even). Thus, there is f , such that Jψ∗ (f ) 6= 0. This completes the proofs of Theorems 8.1, 8.2.

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Non-vanishing of the descent II

We will use the theorems of the last chapter, Theorems 8.1, 8.2, and prove that the descents σψ,α (¯ τ ), σψ,γ (¯ τ ) are nontrivial, provided that there are some compatibility relations between the central character of τ¯, ψ and α. In most cases, we will even find explicit identities, which relate a Whittaker coefficient of the descent with the coefficient Jm,ψ . We do not know how to find such identities when HA = f 4n+2 (A), SO4n+1 (A) (the “symmetric square” cases). Here, the proof will be by Sp contradiction. We keep the notation of the last chapter. 9.1

f 4n+2 (A) The case HA = Sp

f 4n+2 (A) (m = 2n + 1). By our assumptions, the representations Let HA = Sp τ1 , ..., τr are self dual. Recall that (since m is odd) we may assume that ωτ¯ , the product of their central characters is trivial. Otherwise, ωτ¯ = χα is a quadratic character and we replace τi by τi ⊗ χα . See (3.35). Thus, we assume that ωτ¯ = 1. This is a normalized form of the compatibility between ωτ¯ and ψ. Theorem 9.1. Let, for 1 ≤ i ≤ r, τi be an irreducible, automorphic, representation of GLmi (A), which is cuspidal when mi > 1, and is such that LS (τi , sym2 , s) has a pole at s = 1. Assume that m = m1 + · · · mr = 2n + 1 is odd, that ωτ¯ = 1, and that τ1 , ..., τr are pairwise different. Then the descent σψ (¯ τ ) = σψ,en+1 (Eτ¯,ψ ) is nontrivial. Moreover, this is an automorphic, cuspidal representation of Sp2n (A), such that every one of its irreducible summands lifts locally, at almost all places v, to GL

(Fv ) τ1,v r (Fv )

τv = IndPm2n+1 ,...,m 1

Proof.

⊗ · · · ⊗ τr,v .

Let f be an automorphic form in the residual representation Eτ¯,ψ . Denote Z −1 D1,n+1 ,ψ f (h) = f (vαm h)ψD 1,n+1 (v)dv. 1,n+1 DF \DA1,n+1

See Proposition 8.1. Recall that D1,n+1 = Xn,1 Zm Y1,n+1 . We know that the one parameter group um (tIm ) normalizes D1,n+1 and preserves the character ψD1,n+1 235

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over A. In fact, it is clear that um (tIm ) conjugates Xn,1 Zm into itself, and by formula (8.34), which is still valid, um (tIm ) conjugates Y1,n+1 into D1,n+1 . Next, (D′ )1,n+1 is a group and ψ(D′ )1,n+1 is a character of (D′ )1,n+1 (A), which is trivial on um (tIm ). Recall that αm in (t)α−1 m = um (−tIm ). Consider the Fourier expansion of the function on F \A, t 7→ f D

1,n+1

,ψ

(in (−t)h)

(for fixed h) and evaluate at zero. Note that the Fourier coefficient of this function, with respect to the trivial character is Z −1 (D′ )1,n+1 ,ψ f (h) = f (vαm h)ψ(D ′ )1,n+1 (v)dv. 1,n+1 1,n+1 (D′ )F \(D′ )A

Thus, fD X Z

λ∈F ∗

F \A

1,n+1

Z

,ψ

′ 1,n+1

(h) = f (D )

,ψ

(h)+

−1 −1 f (vum (tIm )αm h)ψD (λt)dvdt. 1,n+1 (v)ψ

(9.1)

1,n+1 DF \DA1,n+1

Note that (D′ )1,n+1 is the set (which is a group over F ) of elements vum (tIm ), where v varies in D1,n+1 , and also the function ψ(D′ )1,n+1 ,λ (vum (tIm )) = ψ(λt) is a character of (D′ )1,n+1 (A). As in the proof of Theorem 8.5, write (D′ )1,n+1 = Z n+1 Y1,n+1 Nn0 , where (according to the notation of Theorems 8.4, 8.5), Z n+1 is the subgroup  ∧ In of elements in Zm of the form . Note that Z n+1 normalizes Y1,n+1 Nn0 z and Y1,n+1 normalizes Nn0 . Thus, we can rewrite the summand in (9.1), which corresponds to λ, as Z Z ψ −1 f Nn0 ,λ (yzαm h)ψ(n+1) (z)dydz, n+1 ZF \ZAn+1 Y1,n+1 (F )\Y1,n+1 (A)

′ where, as in Theorem 8.5, ψ(n+1) is the restriction of ψZ , thought of as a character m ψ n+1 of Zm (A), to Z (A); ψNn0 ,λ is the restriction of ψ(D′ )1,n+1 ,λ to Nn0 (A) and f Nn0 ,λ is the Fourier coefficient of f , along Nn0 , with respect to ψNn0 ,λ . By Theorem 6.4(9), ψ

it follows that if λ is not a square, then, for all f , f Nn0 ,λ = 0. Indeed, as in ψ the proof of Proposition 7.4, if there is f , such that f Nn0 ,λ 6= 0, then at almost all finite places, the Jacquet module J(ψvλ )0n (πτv ,ψv ) is nonzero. Here, πτv ,ψv is the f Sp

(Fv )

1

spherical constituent of IndQm4n+2 µψv τv |det | 2 . Note that (for almost all v) (Fv ) πτv ,ψv = πτv ⊗χλ,v ,ψvλ , where χλ,v (x) = (x, λv ) (local Hilbert symbol). By Theorem

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6.4(9), we conclude that the central character of τv ⊗ χλ,v is trivial, for almost all finite v, and since ωτv = 1, we get that λ is a square in Fv∗ , for almost all finite v. This implies that λ is a square in F ∗ . For such λ, write λ = α2 , with α ∈ F ∗ . We can find a diagonal element in γα ∈ Mn (F ), which conjugates ψNn0 ,λ to ψn0 , so that f

ψN 0 ,λ n

0

(h) = f ψn (γα h).

We get the identity f

D1,n+1 ,ψ

(h) = f

(D′ )1,n+1 ,ψ

(h) +

X

α∈{±1}\F ∗

Z Z

0

−1 (z)dydz. f ψn (γα yzαm h)ψ(n+1)

(9.2) Here, the dy and dz integrations are over Y1,n+1 (F )\Y1,n+1 (A) and ZFn+1 \ZAn+1 respectively. 0 Assume, by contradiction, that σψ (¯ τ ) = 0. By Proposition 7.2, f ψn = 0, for all f ∈ Eτ¯ , and now (9.2) becomes fD

1,n+1

,ψ

′ 1,n+1

= f (D )

By Proposition 8.1, Z Z D1,n+1 ,ψ f (yαm h)dy = Jm,ψ (f )(h) =

.

′ 1,n+1

f (D )

,ψ

′ (yαm h)dy = Jm,ψ (f )(h),

∗ (A) Ym

∗ (A) Ym

and hence Jm,ψ (f ) = that, for all t ∈ A

,ψ

′ (f ), Jm,ψ

′ , this implies for all f ∈ Eτ¯ . By the definition of Jm,ψ

Jm,ψ (f )(in (t)h) = Jm,ψ (f )(h).

Thus, Jm,ψ (f ) is left invariant by in (A). Recall that we identify in (t) with (in (t), 1),   1t and that in (t) is the embedding of in the stabilizer R(m) of the character 1 ψ (m) (8.4) of the group Sm (A) (8.3). Recall, also, that R(m) consists of all g ∧ = diag(g, ..., g, g δ , g ∗ , ..., g ∗ ), where g ∈ SL2 and g δ is the outer conjugation of g by diag(1, −1). Thus, Jm,ψ (f ) is also left invariant by all (g ∧ , 1), where g ∈ SL2 (F ). The subgroup generated by f 2 (A), embedded as the subgroup of all (g ∧ , ǫ), these elements and in (A) × 1 is SL f 2 (A) invariant, where g ∈ SL2 (A) and ǫ = ±1. We conclude that Jm,ψ (f ) is left SL and, in particular, it is not genuine, i.e. Jm,ψ (f )((1, ǫ)h) = Jm,ψ (f )(h). Since f is itself genuine, Jm,ψ (f )(1, ǫ)h) = ǫJm,ψ (f )(h). This implies that Jm,ψ (f ) = 0, for all f ∈ Eτ¯ , and we get a contradiction to Theorem 8.2. This proves that the descent σψ (¯ τ ) is nontrivial. Theorem 7.11 implies that this descent is cuspidal, and so it decomposes into a direct sum of genuine, irreducible, automorphic, cuspidal representations of Sp2n (A). Pick such an irreducible summand σ. Then, in the notation above, at almost all finite places v, σv is an unramified constituent of F J(ψv )n (πτv ,ψv ). By Theorem 6.4(7), σv is uniquely determined as the unramified representation of Sp2n (Fv ), whose standard lift to GL2n+1 (Fv ) is τv . This completes the proof of the theorem. 

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The case H = SO4n+1

Let H = h(V ) = SO4n+1 (m′ = 2m+1, m = 2n). Since the representations τ1 , ..., τr are self-dual, the product of their central characters ωτ¯ is a quadratic character χατ , where ατ ∈ F ∗ , and χατ (x) = (x, ατ ) (global Hilbert symbol). We will consider τ ). Recall that this is a representation of SO2n (A), when ωτ¯ is the descent σψ,ατ (¯ trivial, and it is a representation of SOn+2,n−2 (A), corresponding to χατ , when ωτ¯ is nontrivial. In Chapter 3, we denoted either one of these orthogonal groups in 2n variables, by Hα∗τ ,n . See (3.40). We will denote, for short, α = ατ , and re-denote, for clarity, as in (3.42), SO2n,α = Hα∗τ ,n . The proof that the descent σψ,α (¯ τ ) is nontrivial resembles the proof in the previous section. Theorem 9.2. Let, for 1 ≤ i ≤ r, τi be irreducible, automorphic, representation of GLmi (A), cuspidal when mi > 1, such that LS (τi , sym2 , s) has a pole at s = 1, and m = m1 + · · · mr = 2n is even. Assume that ωτ¯ = χα , and that τ1 , ..., τr are pairwise different. Then the descent σψ,α (¯ τ ) is nontrivial. Moreover, this is an automorphic, cuspidal representation of SO2n,α (A), such that every one of its irreducible summands lifts locally, at almost all places v, to GL

(F )

v τv = IndPm2n,...,m 1

r (Fv )

τ1,v ⊗ · · · ⊗ τr,v .

Proof. Let f be an automorphic form in the residual representation Eτ¯ . Consider ˜′m of (D′ )1,n . See Sec. 8.5. Let ψ(D∗ )1,n the subgroup (D∗ )1,n = vm Xn−1,1 Zˆm Y1,n u denote the restriction of the character ψ(D′ )1,n to (D∗ )1,n (A), and consider Z ∗ 1,n −1 f (D ) ,ψ (h) = f (vαm h)ψ(D ∗ )1,n (v)dv. ∗ 1,n (D∗ )1,n F \(D )A

 tIn We know that the one parameter subgroup c(t) = cm (

 ) normalizes

−tIn (D∗ )1,n and preserves ψ(D∗ )1,n over A. This can be seen by using the formulae in the proof of Lemma 8.5. Note that the group generated by (D∗ )1,n and the image of c is (D′ )1,n , and that αm in (t)α−1 m = c(t). Consider the Fourier expansion of the following function on F \A, t 7→ f (D

∗ 1,n

)

,ψ

(c(t)h),

and evaluate at zero. The Fourier coefficient of this function, with respect to the trivial character is Z ′ 1,n −1 f (vαm h)ψ(D f (D ) ,ψ (h) = ′ )1,n (v)dv. ′ 1,n (D′ )1,n F \(D )A

Thus, f (D

∗ 1,n

)

,ψ

′ 1,n

(h) = f (D )

,ψ

(h)+

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X Z

λ∈F ∗

F \A

Z

239

−1 −1 f (vc(t)αm h)ψ(D (λt)dvdt. ∗ )1,n (v)ψ

(9.3)

∗ 1,n (D∗ )1,n F \(D )A

The function ψ(D∗ )1,n (v)ψ(λt) defines a character ψ(D′ )1,n ,λ of (D′ )1,n A . Exactly as in the proof of Proposition 8.3, and the proof of Theorem 8.4, we can apply repeatedly Lemma 7.1 and, in the summand corresponding to λ in (9.3), exchange the roots yn−1,n+1 and xn,n−1 , then yn−2,n+1 and xn,n−2 ,..., and finally y1,n+1 and xn,1 . Let Y n+1 = Y n−1,n+1 Y n−2,n+1 · · · Y 1,n+1 . Then the Fourier coefficient, corresponding to λ, in (9.3), is equal to Z Z −1 f (vyαm h)ψD 1,n+1 ,λ (v)dvdy, 1,n+1 YAn+1 DF \DA1,n+1

where ψD1,n+1 ,λ is the extension of ψ(D′ )1,n ,λ to D1,n+1 (A) by the trivial character on Xn,i (A), i = n − 1, ..., 2, 1. Write D1,n+1 = Z n+1 (Y1,n+1 u˜′m )Nn , where, as in the previous chapter, in the notation of the proof of Theorem 8.4, Z n+1 is the subgroup  ∧ In ′ . Then the of matrices of Zm . Denote the subgroup Y1,n+1 u˜′m by Y1,n+1 z last integral is equal to Z Z Z −1 f ψn,λ (y ′ zyαm h)ψ(n+1) (z)dy ′ dzdy. ′

′

n+1 YAn+1 ZF \ZAn+1 Y1,n+1 (F )\Y1,n+1 (A)

Here, f ψn,λ is the Gelfand-Graev coefficient of f , with respect to ψn,λ , and ψ(n+1) ′ is the restriction of ψZ , thought of as a character of Zm (A), to Z n+1 (A). By m Theorem 5.6(1,4), if λ is not in the square class of α, then f ψn,λ = 0, for all f ∈ Eτ¯ . Indeed, if f ψn,λ is not identically zero, then, for almost all finite places v, the Jacquet module Jψn,λ (πτv ) is nontrivial, where πτv is the unramified constituent of SO

(F )

1

v τv | det | 2 . By Theorem 5.6(1,4), λα−1 is a square in Fv∗ , for almost IndQm4n+1 (Fv ) all finite v, and hence λα−1 is a square in F ∗ . For such λ, write λ = η 2 α, with η ∈ F ∗ . Then we can find an element γη ∈ Mn (F ) (Mn is the Levi subgroup of Pn = Mn Nn ) which conjugates ψn,λ into ψn,α , so that

f ψn,λ (h) = f ψn,α (γη h). Thus, the Fourier expansion (9.3) has the form Z Z Z X ∗ 1,n ′ 1,n −1 (z)dy ′ dzdy. f (D ) ,ψ (h) = f (D ) ,ψ (h)+ f ψn,α (γη y ′ zyαm h)ψ(n+1) η∈{±1}\F ∗

′

′ ′ Y1,n+1 (F )\Y1,n+1 (A),

Here, the dy , dz, dy integrations are over YAn+1 respectively. Assume, by contradiction, that σψ,α (¯ τ ) = 0. Then (9.4) becomes f (D

∗ 1,n

)

,ψ

′ 1,n

= f (D )

,ψ

.

(9.4) and

ZFn+1 \ZAn+1 ,

(9.5)

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Consider Jm,ψ (f )(h), which is the Fourier coefficient of f along the group Sm (8.2), 1 with respect to the character ψ (m) (8.4). Write Sm = Sm X 2n−1,1 , where 1 Sm = {v ∈ Sm |v2n−1,2n+2 = 0},

X 2n−1,1 = {cm (t(e2n−1,1 − e2n,2 ))}.

1 Clearly, X 2n−1,1 normalizes Sm and, over A, preserves the restriction of ψ (m) to 1 Sm (A). Let Z 1 f (vh)ψ (m) (v −1 )dv. Jm,ψ (f )(h) = 1 (F )\S 1 (A) Sm m

Then Jm,ψ (f )(h) =

Z

1 Jm,ψ (f )(xh)ψ (m) (x−1 )dx.

(9.6)

2n−1,1 XF \XA2n−1,1

1 The group Sm is the unipotent radical of the standard parabolic subgroup of SO4n+1 , whose Levi part is isomorphic to GL2n−1 × SO5 . The stabilizer, in the 1 Adele points of this Levi part, of the restriction of ψ (m) to Sm (A) is isomorphic to SL2 (A) × HA , as follows

i(h)i(g) = diag(I2n−2 , h, I2n−2 )diag(g, ..., g, 1, g ∗ , ..., g ∗ ), where g ∈ SL2 (A) is repeated n times and h ∈ HA ; H ⊂ SO5 is the following standard unipotent radical, which is the Heisenberg group, corresponding to the two dimensional symplectic space,    1 0 b t − 12 bc − 21 b2         1 c − 1 c2 −t − 1 bc  2 2    . H = h(c, b; t) =  1 −c −b         1 0     1 Our notation already provides an isomorphism with the Heisenberg group, namely 1 h(c1 , b1 ; t1 )h(c2 , b2 ; t2 ) = h(c1 + c2 , b1 + b2 ; t1 + t2 + (c1 b2 − c2 b1 )). 2 Denote h1 (c) = h(c, 0; 0), h2 (c) = h(0, c; 0), x(t) = h(0, 0; t). Then 1 h(c, b; t) = h1 (c)h2 (b)x(t − bc). 2 Note that i(h2 (b)) = jn (b) and X 2n−1,1 is the group of all i(I2 , x(t)). Let ωψ−1 be the f 2 (A)HA (corresponding to ψ −1 ). It acts on the Schwartz Weil representation of SL f 2 (A), the space S(A). See (1.4). For φ ∈ S(A), consider, for h ∈ HA , (g, ǫ) ∈ SL corresponding theta series X φ θψ ωψ−1 (h(g, ǫ))φ(ξ). −1 (h(g, ǫ)) = ξ∈F

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f 2 (A), Define, for g˜ = (g, ǫ) ∈ SL φ Jm,ψ (f )(˜ g) =

Z

master

241

φ 1 Jm,ψ (f )(i(h)i(g))θψ (h˜ g )dh.

HF \HA

f 2 (A). This is a smooth, genuine, automorphic function, of moderate growth, on SL We will show that it is not identically zero, and that (under the assumption that the descent is zero) is constant, and in particular, not genuine. This will be the contradiction. Write in the last integral h = x(t)h1 (c)h2 (b), and carry out the dh-integration φ in the order dtdbdc as t, b, c vary in F \A. Since θψ g) = −1 (x(t)h1 (c)h2 (b)˜ φ ψ −1 (t)θψ g ), we get from (9.6), −1 (h1 (c)h2 (b)˜ Z Z φ φ Jm,ψ (i(h1 (c)h2 (b))i(g))θψ g)dbdc. Jm,ψ (f )(˜ g) = −1 (h1 (c)h2 (b)˜ F \A F \A

φ Substitute the series expression of θψ The r.h.s of the last equality becomes −1 . (using (1.4)) Z Z X Jm,ψ (i(h1 (c + ξ)h2 (b))i(g))ψ(−b(c + ξ))ωψ−1 (˜ g )φ(c + ξ)dbdc. F \A F \A ξ∈F

We used the left HF invariance of Jm,ψ (f )(i(h)i(g)). Since

Jm,ψ (i(h1 (c + ξ)h2 (b))i(g))ψ(−b(c + ξ)) = Jm,ψ (i(h1 (c + ξ)h2 (b))x(−b(c + ξ))i(g)) = Jm,ψ (i(h2 (b)h1 (c + ξ))i(g)),

we get φ Jm,ψ (f )(˜ g) =

Z

Z X

Jm,ψ (i(h2 (b)h1 (c + ξ))i(g))ωψ−1 (˜ g )φ(c + ξ)dbdc.

F \A F \A ξ∈F

Since ωψ−1 (˜ g )φ is a Schwartz function, we have the following convergence, for fixed g, Z Z |Jm,ψ (i(h2 (b)h1 (c)i(g))ωψ−1 (˜ g )φ(c)|dbdc < ∞. A F \A

Hence we can interchange the summation over ξ and the db-integration, and then collapse the summation and the dc-integration. We get Z Z φ Jm,ψ (i(h2 (b)h1 (c))i(g))db)ωψ−1 (˜ g)φ(c)dc. Jm,ψ (f )(˜ g) = ( A F \A

φ Jm,ψ (f )(˜ g)

We conclude that is identically zero (for all φ, f, g), if and only if the inner db-integral is identically zero (for all f, c, g), which is the same as Z Jm,ψ (jn (b))db = 0, F \A

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′ for all f . The last integral is an inner integral of the Fourier coefficient Jm,ψ (f )(1), φ g ) is not which is not identically zero, by Theorem 8.2. This proves that Jm,ψ (f )(˜ identically zero. By Proposition 8.2(1), Z Jm,ψ (i(h2 (b)h1 (c)i(g))db = F \A

Z

Z

Z

−1 f (vyvm (b)αm i(h1 (c))i(g))ψD 1,n (v)dvdydb.

∗ (A) F \A Ym D1,n \D1,n F

A

um (bξn , − 21 b2 en,n+1 ) 2n

(ξn is the n-th vector in the standard basis of Here, vm (b) = the column space F ). From (8.43), we can write yvm (b) = v ′ (y, b)vm (b)y, where v ′ (y, b) ∈ D1,n , and, over A, ψD1,n (v ′ (y, b)) = 1. By (the convergence statement in) Proposition 8.2, it is easy to see that, after changing variable vv ′ (y, b) 7→ v, we can switch the order of the dy, db integrations. Since the group of all vvm (b), v ∈ D1,n is (D∗ )1,n , we get that the last integral is equal to Z Z −1 f (vyαm i(h1 (c))i(g))ψ(D ∗ )1,n (v)dvdy. ∗ (A) Ym (D∗ )1,n \(D∗ )1,n F

This shows that φ Jm,ψ (f )(˜ g)

=

Z

A

Z

f (D

∗ 1,n

)

,ψ

(yαm i(h1 (c))i(g))ωψ−1 (h1 (c)˜ g )φ(0)dydc.

∗ (A) A Ym

By (9.5) and Proposition 8.2(2), Z φ ′ (f )(i(h1 (c))i(g))ωψ−1 (h1 (c)˜ Jm,ψ (f )(˜ g ) = Jm,ψ g )φ(0)dc.

(9.7)

A

  1t ′ ) commutes with i(h1 (c)), and that, by definition, Jm,ψ (f ) Note that in (t) = i( 1 is left in (A) invariant. Similarly,     1t 1t ωψ−1 (h1 (c)( , 1)˜ g)φ(0) = ωψ−1 (( , 1)h1 (c)˜ g )φ(0) = ωψ−1 (h1 (c)˜ g )φ(0). 1 1 These observations and (9.7) imply that, for all t ∈ A,   1t φ φ Jm,ψ (f )(( , 1)˜ g) = Jm,ψ (f )(˜ g ), 1 φ and since Jm,ψ (f ) is automorphic, it is left invariant by SL2 (F )×1. We conclude, as f 2 (A), in the end of the proof of Theorem 9.1, that J φ (f ) is left invariant under SL m,ψ

φ and hence constant. In particular, it is not genuine. However, by definition, Jm,ψ (f ) φ is genuine, and hence Jm,ψ (f ) must be identically zero. This is a contradiction. The proof is now complete. 

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243

Whittaker coefficients of the descent corresponding to GelfandGraev coefficients: the unipotent group and its character; h(V ) 6= SO4n+1 .

Let h(V ) be one of the groups SO4n , SO4n+3 , U4n , U4n+2 . Recall that ℓm = ∗ E V0 [ m−1+dim ], and Hα,m = Lℓm ,α = h(Wm,ℓm ∩ yα⊥ ) (see (3.36), (3.40), (3.42)). 2 Our next goal is to write an explicit formula for certain Whittaker coefficients of the descent σψ,α (¯ τ ). This formula will relate in an explicit way the Whittaker coefficient with the coefficient Jm,ψ of the previous chapter, and finally, we will conclude that the descent is nontrivial. ∗ Let us write the “standard” maximal unipotent subgroup of Hα,m and its ′ Whittaker characters. Assume that V0 = 0, i.e. m = 2m is even and h(V ) = SO4n , U4n , U4n+2 . Note that yα = em − α2 e−m . We choose {eℓm +1 , ..., em−1 , y−α , e1−m , ..., e−ℓm −1 } as a basis for Wm,ℓm ∩ yα⊥ . Its Gram matrix is   wm−ℓm −1 .  α wm−ℓm −1

∗ A standard upper unipotent matrix, with respect to this basis, in Hα,m has the form   z 2zx y  1 2x′′  , z∗ ′

t

xw

m −1 where z ∈ ResE/F Zm−ℓm −1 and x′′ = xα = − m−ℓ . Its image in h(V ), with α respect to the standard basis of V , is   z z(x, 2x α )  y′  2x   α  I2 u = diag(Iℓm ,  (9.8) ′  , Iℓm ).  x z∗

∗ Denote by NHα,m the group of elements of the form (9.8). This is the “standard” ∗ maximal unipotent subgroup of Hα,m . We will consider the following Whittaker ∗ characters of NHα,m (A),

ψNHα,m ∗ ,λ (u) = ψZℓm (z)ψE,1 (λxm−ℓm −1 ),

(9.9)

where λ ∈ F ∗ . Assume that V0 6= 0. Then h(V ) = SO4n+3 (m′ = 2m + 1 = 4n + 3) and ℓm = n. Here, yα = em + α2 e−m . In this case, we will take {en+1 , ..., e2n , e0 , y−α , e−2n , ..., e−n−1 }

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as a basis for Wm,ℓm ∩ yα⊥ . Its Gram matrix is   wn   1 0  .   0 −α wn

∗ A standard upper unipotent matrix, with respect to this basis, in Hα,m has the form   z zx1 2zx2 y  1 0 2x′1  ,   1 − α2 x′2  z∗

where z ∈ Zn and x′ = −t xwn . Its image in h(V ), with respect to the standard basis of V , is   y z z(x2 , x1 , − α2 x2 )  2 ′  − α x2      I3 u = diag(In ,  (9.10) x′1   , In ).   x′ 2

z∗

Note that when α = η 2 is a square, we can use the basis 1 1 1 {en+1 , ..., e2n , e0 + y−α , (e0 − y−α ), e−2n , ..., e−n−1 } η 2 η

of Wm,ℓm ∩ yα⊥ . Its Gram matrix is w2n+2 . The image in SO4n+3 of a standard ∗ unipotent matrix (with respect to this basis) in Hα,m = SO2n+2 is still of the form ∗ (9.10). Denote by NHα,m the group of elements of the form (9.10). This is the ∗ “standard” maximal unipotent subgroup of Hα,m . We will consider the following ∗ Whittaker characters of NHα,m (A), ψNHα,m ∗ ,λ,µ (u) = ψZn (z)ψ(λ(x1 )n + µ(x2 )n ),

(9.11)

where λ, µ ∈ F are not both zero. Denote anyone of the characters (9.9), (9.11) by ∗ and denote by ψNℓm ,α the character of Nℓm ,α (A) ψNHα,m . Let Nℓm ,α = Nℓm NHα,m ∗ defined by ψNℓm ,α (vu) = ψℓm ,α (v)ψNHα,m (u), ∗

∗ (A). v ∈ Nℓm (A), u ∈ NHα,m

For a smooth automorphic function f on HA , let Z −1 f ψℓm ,α (uh)ψN Wm,ψ (f )(h) = H∗

(u)du.

(9.12)

α,m

NHα,m ∗ (F )\NHα,m ∗ (A)

This is the ψNHα,m -Whittaker coefficient of f ψℓm ,α . Directly, in terms of f , we have ∗ Z −1 f (vh)ψN (v)dv. (9.13) Wm,ψ (f )(h) = ℓm ,α Nℓm ,α (F )\Nℓm ,α (A)

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9.4

245

Conjugation by the element η ˆm

As in the previous chapter, when we started the computation of Jm,ψ (f )(h), we will −1 rewrite the integral Wm,ψ (f )(h) in terms of a conjugated subgroup ηˆm Nℓm ,α ηˆm , for a certain Weyl element ηm in GLm (F ). Its action on the standard basis {e1 , ..., em } of Vm+ is as follows. Assume that m = 2n is even. Then ηm (ei ) = e2i , 1≤i≤n−1 ηm (en−1+i ) = e2i−1 , 1≤i≤n . ηm (e2n ) = e2n If we think of ei , 1 ≤ i ≤ m as column vectors in F m , then the columns of the matrix ηm are ηm = (e2 , e4 , ..., e2n−2 , e1 , e3 , ..., e2n−1 , e2n ). Assume that m = 2n + 1 is odd. Then

Thus,

ηm (ei ) = e2i , 1≤i≤n . ηm (en+i ) = e2i−1 , 1 ≤ i ≤ n + 1 ηm = (e2 , e4 , ..., e2n , e1 , e3 , ..., e2n+1 ).

We have Wm,ψ (f )(h) =

Z

−1 f (ˆ ηm vh)ψN ℓ

m ,α

(v)dv

Nℓm ,α (F )\Nℓm ,α (A)

=

Z

−1 f (v ηˆm h)ψN (v)dv, m,α

(9.14)

Nm,α (F )\Nm,α (A) −1 ηm v ηˆm ). where Nm,α = and ψNm,α (v) = ψNℓm ,α (ˆ Assume that m = 2n is even, so that h(V ) = SO4n , U4n , and ℓm = n − 1. Denote   T Y C v(T, Y, C) =  I2 Y ′  ∈ h(V ). (9.15) T∗ −1 ηˆm Nℓm ,α ηˆm ,

Then Nm,α consists of all (unipotent) matrices of the form v(T, Y, C), such that Y has the form   x1 α2 x1  a   1 b1   x  2  2 α x2    a b2  , Y = (9.16)  .2 ..   .   .  .   an−1 bn−1  xn α2 xn

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and T has the form 

  T = 

1 v1 .. . vn−1

 u1 . . . un−1 T1,1 . . . T1,n−1   , .. ..  . . Tn−1,1 . . . Tn−1,n−1

(9.17)

where Ti,j are all 2×2 upper triangular matrices, such that Ti,i are (upper) unipotent and, for i > j, Ti,j are (upper) nilpotent. Finally the vectors in the first row have   ∗ the form ui = (0, ∗), and the vectors in the first column have the form . Thus 0 T looks like this   1 0 ∗ 0 ∗ ... 0 ∗ ∗ 1 ∗ ∗ ∗ . . . ∗ ∗   0 0 1 0 ∗ . . . 0 ∗     ∗ 0 ∗ 1 ∗ . . . ∗ ∗   0 0 0 0 1 . . . 0 ∗ .   ..     .   ∗ 0 ∗ 0 ∗ . . . 1 ∗ 0 0 0 0 0 ... 0 1 For v = v(T, Y, C) ∈ Nm,α (A) of the form above, α bn−1 + λxn ). 2 (9.18) Assume that m = 2n + 1 is odd, so that h(V ) = U4n+2 , SO4n+3 , and ℓm = n. Denote again ψNm,α (v) = ψE,1 ((u1 )2 + tr(T1,2 + T2,3 + · · · + Tn−2,n−1 ) + an−1 −



v(T, Y, C) = 

T

Y I2+dimV0

 C Y ′  ∈ h(V ). T∗

(9.19)

Then Nm,α consists of all (unipotent) matrices of the form v(T, Y, C), where, in case V0 = 0 (h(V ) = U4n+2 ), Y has the form  x1 α2 x1 a b   1 1  x 2 x   2 α 2   a b  Y =  .2 .2  ,  . ..    .   2 xn α xn  an b n 

(9.20)

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and in case dimV0 = 1 (h(V ) = SO4n+3 ), Y has the form   x1 y1 − α2 x1 a c   1 1 b1  x y − 2 x   2 2 α 2   a c b2  ; Y =  .2 .2   . .   . .    xn yn − α2 xn  an cn b n

T has the form

 T1,1 . . . T1,n  ..  , T =  ... .  Tn,1 . . . Tn,n 

master

247

(9.21)

(9.22)

where Ti,j are all 2×2 lower triangular matrices, such that Ti,i are (lower) unipotent and, for i > j, Ti,j are (lower) nilpotent. Thus T looks like this   1 0 ∗ 0 ... ∗ 0 ∗ 1 ∗ ∗ . . . ∗ ∗   0 0 1 0 . . . ∗ 0     ∗ 0 ∗ 1 . . . ∗ ∗ .   ..     .   0 0 0 0 . . . 1 0 ∗ 0 ∗ 0 ... ∗ 1

For v = v(T, Y, C) ∈ Nm,α (A), of the form above, ψNm,α (v) is given as follows. Assume that h(V ) = U4n+2 . Then α ψNm,α (v) = ψE,1 (tr(T1,2 + T2,3 + · · · + Tn−1,n ) + an − bn + λxn ). (9.23) 2 Assume that h(V ) = SO4n+3 . Then α ψNm,α (v) = ψ(tr(T1,2 + T2,3 + · · · + Tn−1,n ) + an + bn + λxn + µyn ). (9.24) 2

9.5

Exchanging roots: h(V ) = SO4n , U4n

Assume that h(V ) = SO4n , U4n . As in the previous chapter we are now going to apply repeatedly Lemma 7.1 to the integral (9.14). The first step is to use the nontrivial coordinates of the row T2n−2 of T in (9.17) in order to “fill in” Y in (9.16) the rows (xj , − α2 xj ), j = 1, 2, ..., n in rows 1, 3, 5, ..., 2n − 1, 2n of Y . Let 1 ≤ j ≤ n. Denote by C2n−2,2j−1 the subset of matrices which have the form v(T, Y, C) as in (9.15), so that T has the form (9.17), with vn−1 = 0, Tn−1,1 = · · · Tn−1,j−2 = 0, (Tn−1,j−1 )1,2 = 0,

(9.25)

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and Y has the form Y = has the form



 Y1 , where Y1 is any matrix of size (2j − 2) × 2 and Y2 Y2 

xj aj .. .

2 α xj



 bj     ..    . . Y2 =    xn−1 α2 xn−1    an−1 bn−1  2 xn α xn

(9.26)

If j = 1, then (9.25) is just (vn−1 )1,1 = 0. This is a group over F . Indeed, consider −1 2n−2,2j−1 ηˆm C ηˆm . This consists of the elements of the form v(z, u, e) ∈ h(V ), with     z1 x1 x2 u1 v1 (9.27) z =  z2 x3  , u = u2 v2  , z3 u3 α2 u3

where z1 ∈ ResE/F Zn−1 , z2 ∈ ResE/F Zj , z3 ∈ ResE/F Zn−j , u1 , v1 (resp. u2 , v2 ) are (n − 1)- dimensional (resp. j-dimensional) column vectors , and u3 is an (n − j)dimensional column vector. Finally, the last row of x1 is zero. This description −1 2n−2,2j−1 makes it clear that ηˆm C ηˆm is a group (over F ), and hence so is C2n−2,2j−1 . Similarly, the function ψC2n−2,2j−1 , defined on C2n−2,2j−1 (A), in terms of the coordinates above, by α ψE,1 ((u1 )2 + tr(T1,2 + T2,3 + · · · + Tn−2,n−1 ) + an−1 − bn−1 + λxn ), (9.28) 2 is a character of C2n−2,2j−1 (A), trivial on C2n−2,2j−1 (F ). In terms of the coordi−1 2n−2,2j−1 nates of ηˆm C0 ηˆm , this character becomes   z1 x1 x2 α (9.29) ψZ2n−1 ,1  z2 x3  ψE,1 ((u1 )n−1 − (v1 )n−1 + λ(u3 )n−j ), 2 z3 where ψZ2n−1 ,1 denotes the standard Whittaker character, corresponding to ψE,1 . Define Y2n−2,2j−1 = {(I2n + te2n−2,2j−1 )∧ }, X2j−1;2n,2n+1 = {(I2n + te2j−1,2n )∧ · um (− α2 te2j−1,1 + α2 t¯e2n,2n+2−2j ))}.

The proof that C2n−2,2j−1 is a group shows that C2n−2,2j−1 Y2n−2,2j−1 is an F -group, and moreover, that Y2n−2,2j−1 normalizes C2n−2,2j−1 over F , and Y2n−2,2j−1 (A) preserves ψC2n−2,2j−1 . Let us show now that X2j−1;2n,2n+1 has the same property. We can write an element of X2j−1;2n,2n+1 in the form x = v(I2n−1 , tvj , ∗), where vj is the (2n − 1) × 2 matrix all of whose coordinates are zero except the (2j − 1)-th row, which is (1, − α2 ). Conjugation by x of an element in C2n−2,2j−1 gives us xv(T, Y, C)x−1 = v(T, Y + tvj − tT vj , ∗).

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Assume, for simplicity of notation, that j > 1. Using the form (9.17), (9.25), (9.26), and denoting   0 rk Tk,j−1 = ; k = j, ..., n − 1, 0 0 we have  ′ Y1 vj − T vj = , Y2′ where Y1′ is a (2j − 2) × 2 matrix (easily found) and   0 0   −r 2 j  α rj    0   0 ′  Y2 =  . ..  ,  .. .    −rn−1 α2 rn−1  0 0 with rn−1 = 0. Comparing with (9.26), we see that conjugation by x preserves C2n−2,2j−1 . Moreover, when we take Adele coordinates, the character ψC2n−2,2j−1 , given by (9.28), becomes after conjugation, 2 α ψE,1 ((u1 )2 +tr(T1,2 +T2,3 +· · ·+Tn−2,n−1 )+(an−1 −trn−1 )− (bn−1 + trn−1 )+λxn ). 2 α Since rn−1 = 0, we see that the character is preserved. Finally, in order to apply Lemma 7.1, it remains to examine [X2j−1;2n,2n+1 , Y2n−2,2j−1 ]. Let 2t¯ 2t x = (I2n + te2j−1,2n )∧ um (− e2j−1,1 + e2n,2n+2−2j ), y = (I2n + se2n−2,2j−1 )∧ . α α (9.30) Then 2st 2stt¯ 2¯ stt¯ 2st e2n−2,1− e2n,3+ e2n−2,2n+2−2j− e2j−1,3 ). [x, y] = (I2n−ste2n−2,2n )∧ um ( α α α α (9.31) Thus, [x, y] ∈ C2n−2,2j−1 and, by (9.28), we have, for s, t ∈ AE , ψC2n−2,2j−1 ([x, y]) = ψE,1 (−2st).

We have now verified all conditions of Lemma 7.1. Let B2n−2,2j−1 = C2n−2,2j−1 Y2n−2,2j−1 D2n−2,2j−1 = C2n−2,2j−1 X2j−1;2n,2n+1 . Note that D2n−2,2j−1 = B2n−2,2j+1 , B2n−2,1 = Nm,α .

(9.32)

Extend ψC2n−2,2j−1 to B2n−2,2j−1 (A) and to D2n−2,2j−1 (A), by making it trivial on Y2n−2,2j−1 (A) and on X2j−1;2n,2n+1 (A), respectively. Denote the extended characters by ψB2n−2,2j−1 , ψD2n−2,2j−1 . We conclude Lemma 9.1. Let h(V ) = SO4n , U4n , and let f be a smooth automorphic function of uniform moderate growth on HA . Then, for all h ∈ HA ,

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(1)

Z

Z

−1 f (vh)ψB 2n−2,2j−1 (v)dv =

B2n−2,2j−1 (F )\B2n−2,2j−1 (A)

Z

−1 f (vyh)ψD 2n−2,2j−1 (v)dvdy.

Y 2n−2,2j−1 (A) D2n−2,2j−1 (F )\D2n−2,2j−1 (A)

(2) The convergence of the r.h.s. is such that Z Z −1 f (vyh)ψD 2n−2,2j−1 (v)dv dy < ∞. Y 2n−2,2j−1 (A) D2n−2,2j−1 (F )\D2n−2,2j−1 (A)

(3) As h varies,

Z

−1 f (vh)ψB 2n−2,2j−1 (v)dv ≡ 0,

Z

−1 f (vh)ψD 2n−2,2j−1 (v)dv ≡ 0.

B2n−2,2j−1 (F )\B2n−2,2j−1 (A)

if and only if

D2n−2,2j−1 (F )\D2n−2,2j−1 (A)

Let Y2n−2 =

n Y

Y2n−2,2j−1 .

(9.33)

j=1

Using Lemma 9.1, and (9.32), we obtain Proposition 9.1. Let h(V ) = SO4n , U4n and let f be a smooth automorphic function of uniform moderate growth on HA . Then Z Z −1 f (vy ηˆm h)ψD Wm,ψ (f )(h) = 2n−2,2n−1 (v)dvdy, YA2n−2 D2n−2,2n−1 \D2n−2,2n−1 F A

where the dy-integration converges absolutely as a repeated integration, according to (9.33), starting with j = 1. Finally, Wm,ψ (f ) 6= 0, if and only if there is h ∈ HA , such that Z −1 f (v ηˆm h)ψD 2n−2,2n−1 (v)dv 6= 0. D2n−2,2n−1 \D2n−2,2n−1 F A

The group D2n−2,2n−1 consists of all elements v(T, Y, C) ∈ h(V ), (9.15), such that T is of the form (9.17), with vn−1 = 0, Tn−1,1 = · · · Tn−1,n−2 = 0, Tn−1,n−1 = I2 ,

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(and Y is arbitrary). The character ψD2n−2,2n−1 is given by α λ α ψE,1 ((u1 )2 +tr(T1,2 +· · · tr(Tn−2,n−1 )+Y2n−2,1 − Y2n−2,2 + (Y2n−1,1 + Y2n−1,2 )). 2 2 2 (9.34) We continue with the process of exchanging roots, as follows. We exchange the coordinates in columns 1, 3, ..., 2n − 3 of row 2n − 4 of T in v(T, Y, C) ∈ D2n−2,2n−1 with the coordinates in rows 1, 3, ..., 2n−3 of column 2n−2. Then we exchange the coordinates in columns 1, 3, ..., 2n−5 of row 2n−6 with those in rows 1, 3, ..., 2n−5 of column 2n − 4, and so on. In precise terms, let, for 1 ≤ i ≤ n − 1, Ti be the group of matrices of the form   1 u1 . . . ui−1 ui  v1 T1,1 . . . T1,i−1 T1,i     .. ..  , (9.35)  . .    vi−1 Ti−1,1 . . . Ti−1,i−1 Ti−1,i  0 0 ... 0 I2 where the Tr,s are 2 × 2 upper triangular matrices, such that, for r > s, Tr,s are (upper) nilpotent, and Tr,r are (upper) unipotent. The column vectors v1 , ..., vi−1 have zero in their second coordinate, and the row vectors u1 , ..., ui have zero in their first coordinate. This group is a conjugate of the following subgroup of the standard maximal unipotent subgroup of GL2i+1 by the Weyl element η2i+1 ,    zx | z ∈ ResE/F Zi , ζ ∈ ResE/F Zi+1 , xi = 0 . ζ This conjugation also takes a standard Whittaker character, restricted to the Adele points of this subgroup to the character ψTi ψE,1 ((u1 )2 + tr(T1,2 + T2,3 + · · · + Ti−1,i )) (9.36) of the Adele points of Ti . For 1 ≤ j ≤ i, let Ti′′ (j) be the subgroup of Ti , where we further impose the conditions vi−1 = 0, (Ti−1,1 )1,2 = · · · = (Ti−1,j−1 )1,2 = 0. Next, we denote by Ti′ (j) the group of matrices defined like the elements of Ti , where we impose the conditions which define Ti′′ (j), but also relax the conditions on ui , T1,i , ..., Tj−2,i and let them be arbitrary. Thus, Ti′ (j) = Ti′′ (j)Ti∗ (j), where Ti∗ (j) is the subgroup of matrices   I2j−3 y  I2i−2j+2 0   ,  1 

1 and y is a column vector, all of whose even coordinates are zero. For j = 1, Ti∗ (1) = I. It is easy to check that Ti′ (j) is a group (over F ), and that formula (9.36) defines a character ψTi′ (j) of Ti′ (j)(A). Finally, let    I2 ∗ ∗ ∗          I2 ∗ ∗  T Z   ′ Ti (j) = ∈ ResE/F GL2n−1 T ∈ Ti (j), U =   . (9.37) . . . ∗   U        I2

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This is a group (over F ) and the character ψTi′ (j) extends to the character ψTi (j) of Ti (j)(A) by   I2i+1 7→ ψE,1 (tr(U1,2 + · · · Un−i−2,n−i−1 )), U

where we write U as a matrix of 2 × 2 blocks. Now we can define the groups (C, X, Y ) which satisfy Lemma 7.1. For h(V ) = U4n , define C2i−2,2j−1 = {v(T, Y, C) ∈ h(V ) | T ∈ Ti (j)}, ψC2i−2,2j−1 (v(T, Y, C)) = ψTi (j) (T )ψE,1 (Y2n−2,1 − α2 Y2n−2,2 + λ2 (Y2n−1,1 + α2 Y2n−1,2 )), Y2i−2,2j−1 = {(I2n + te2i−2,2j−1 )∧ } X2j−1,2i = {(I2n + te2j−1,2i )∧ }.

(9.38) Since Ti (j) is a group over F , it is clear that C2i−2,2j−1 is a group over F , and it is also clear that ψC2i−2,2j−1 is a character of C2i−2,2j−1 (A) (trivial on the F -points). Now, it is a simple routine to check that the conditions of Lemma 7.1 are verified. Denote, as usual, B2i−2,2j−1 = C2i−2,2j−1 Y2i−2,2j−1 ,

D2i−2,2j−1 = C2i−2,2j−1 X2j−1,2i ,

and extend ψC2i−2,2j−1 to characters ψB2i−2,2j−1 , ψD2i−2,2j−1 of B2i−2,2j−1 (A), D2i−2,2j−1 (A), by the trivial characters of Y2i−2,2j−1 (A), X2j−1,2i (A), respectively. Note that D2i−2,2j−1 = B2i−2,2j+1 , for j < i, (9.39) D2i−2,2i−1 = B2i−4,1 We conclude, as before, the assertions of Lemma 9.1 replacing B2n−2,2j−1 by B2i−2,2j−1 ,D2n−2,2j−1 by D2i−2,2j−1 and Y2n−2,2j−1 by Y2i−2,2j−1 . Thus, for a smooth automorphic function of uniform moderate growth on HA , f , we have, for all h ∈ HA , Z −1 f (vh)ψB 2i−2,2j−1 (v)dv = Z

B2i−2,2j−1 (F )\B2i−2,2j−1 (A)

Z

−1 f (vyh)ψD 2i−2,2j−1 (v)dvdy.

Y 2i−2,2j−1 (A) D2i−2,2j−1 (F )\D2i−2,2j−1 (A)

The convergence of the r.h.s. is interpreted as in Lemma 9.1. We have Z −1 f (vh)ψB 2n−2,2j−1 (v)dv ≡ 0, B2n−2,2j−1 (F )\B2n−2,2j−1 (A)

if and only if

Z

D2n−2,2j−1 (F )\D2n−2,2j−1 (A)

−1 f (vh)ψD 2n−2,2j−1 (v)dv ≡ 0.

(9.40)

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Let i Q

Y2i−2 =

Y2i−2,2j−1

j=1

Y′ =

n−1 Q

Y2i

i=1

−1 ′ Ym = ηˆm Y ηˆm ′ Sm = D2,3 .

By Proposition 9.1, (9.39) and (9.40), we conclude, re-denoting Wm,ψ (f ) also by Wm,ψ;α,λ (f ), in order not to forget the dependence on α, λ, Theorem 9.3. Let h(V ) = SO4n , U4n and let f be a smooth automorphic function of uniform moderate growth on HA . Then Z Z f (v ηˆm yh)ψS−1 Wm,ψ (f )(h) = Wm,ψ;α,λ (f ) = ′ (v)dvdy, m ′ (F )\S ′ (A) Ym (A) Sm m

where the dy-integration converges absolutely as a repeated integration, as before. Finally, Wm,ψ (f ) 6= 0, if and only if there is h ∈ HA , such that Z f (v ηˆm h)ψS−1 ′ (v)dv 6= 0. m ′ (F )\S ′ (A) Sm m

′ ′ , ψSm (m = 2n) is as follows. The explicit form of Ym , Sm

Ym

′ Sm

  ∗∗      ∧  ∗ ∗   In−1 Y 0   =  In 0 Y =     1  ∗ ∗    ∗∗

   1 u ∗ ··· ∗ ∗      I2 x1 ∗         I ∗  2   = v( , ..    .  ∗         I x 2 n−2   I2

0 ∗ .. . ∗ ∗

 0 ··· 0 0     0 · · · 0 0   ..  .     ∗ · · · ∗ 0   ∗ ··· ∗ ∗

        

∗ ∗ ∗ .. . ∗

xn−1

        



   u1 = 0   , ∗) ∈ h(V ) x1 , ..., xn−1 ∈ ResE/F M2×2           

′ (v) = ψE,1 (u2 + tr(x1 + · · · + xn−2 ) + tr(xn−1 ψSm



1 − α2

′ Note that Sm resembles Sm in shape, but not quite.

λ 2 αλ 4

 ))

(9.41)

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9.6

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Nonvanishing of the Whittaker coefficient of the descent corresponding to Gelfand-Graev coefficients: h(V ) = SO4n , U4n

Denote

Z

′

f Sm ,ψ (h) =

f (vh)ψS−1 ′ (v)dv. m

′ (F )\S ′ (A) Sm m

′ ′ . Thus, The group of all (I2n + te1,2 )∧ normalizes Sm and, over A, preserves ψSm we can consider the Fourier expansion, along E\AE of the function ′

t 7→ f Sm ,ψ ((I2n + te1,2 )∧ h).

The Fourier coefficient, corresponding to the character ψE,1 (µt), µ ∈ E is Z

Z

′

−1 f Sm ,ψ ((I2n + te1,2 )∧ h)ψE,1 (µt)dt =

˜

Sm ,ψ,α,λ,µ f (vh)ψS−1 (h), ˜m (v)dv := f

˜m (F )\S ˜m (A) S

E\AE

where ′ S˜m = {v(I2n + te1,2 )∧ | v ∈ Sm }, ∧ ′ (v)ψE,1 (µt) = ψ ˜ ψS˜m (v(I2n + te1,2 )∧ ) = ψSm Sm ,α,λ,µ (v(I2n + te1,2 ) ).

The group S˜m consists of the elements as in (9.41), without the restriction u1 = 0. The character ψS˜m has the form as in (9.41), only that u2 is replaced by µu1 + u2 . Theorem 9.4. Assume that h(V ) = SO4n , U4n . Let λ = ˜

˜

2

2 α,

µ = − α1 , and denote

1

f Sm ,ψ,α = f Sm ,ψ,α, α ,− α . ˜

(1) The Fourier coefficient f Sm ,ψ,α is not identically zero, as f varies in the space of the residual representation Eτ¯ . Moreover, we have an explicit formula, of the form Z ˜m ,ψ,α ω S Jm,ψ (f )(yγα )dy, (f ) (1) = ′ (A) Yn

where Yn′ is a certain unipotent subgroup of h(V ) and γα is a certain element in h(V ), both specified in the proof. (Recall that ω = 1 except when h(V ) = SO4n and n is odd. See (8.5).) (2) The descent σψ,α (¯ τ ) is non-trivial and globally generic (on SO2n+1 (A) and U2n+1 (A) respectively). Assume that when h(V ) = U4n , m1 , ..., mr ≥ 2 (reτ ) is also cuspidal. Let call that τi is on GLmi (AE ), for 1 ≤ i ≤ r). Then σψ,α (¯ σ be an irreducible summand of σψ,α (¯ τ ). (3) Assume that h(V ) = SO4n . Then σ lifts locally, at almost all places v, to GL

(F )

v τv = IndPm2n,...,m 1

r (Fv )

τ1,v ⊗ · · · ⊗ τr,v .

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(4) Assume that h(V ) = U4n and that σ is globally generic. Then there is an automorphic character ǫ of A∗E , lifted from U1 (A), (i.e. has the form ǫ(x) = ǫ′ ( xx¯ ), where ǫ′ is an automorphic character of U1 (A)), such that, at almost all places v, σv lifts to GL

(Ev ) τ1,v r ,1 (Ev )

IndPm2n+1 ,...,m 1



⊗ · · · ⊗ τr,v ⊗ ǫv .

 1 λ2 Proof. Let b = , and γ = diag(1, b, ..., b, I2 , b∗ , ..., b∗ , 1). Then − α2 αλ 4 γ S˜m γ −1 = S˜m , and, writing v ∈ S˜m (A) as in (9.41), with u = (u1 , u2 ), we have   µ ψS˜m ,α,λ,µ (v) = ψE,1 (ub + tr(x1 + · · · + xn−1 )). 1     µ 0 λ 1 2 = Choose λ = α , µ = − 2 = − α , so that b . For this choice, put γ = γ(α) 1 1 and ψS˜m ,α,λ,µ (γ(α)−1 vγ(α)) = ψS˜m (v) = ψE,1 (u2 + tr(x1 + · · · xn−1 )). We have

Z

˜

f Sm ,ψ,α (h) =

f (γ(α)vh)ψS−1 ˜ ,α, 2 ,− 1 (v)dv = m

α

α

˜m (F )\S ˜m (A) S

Z

f (vγ(α)h)ψS−1 ˜ (v)dv. m

˜m (F )\S ˜m (A) S ˜ The passage from S˜m to Sm and from f Sm ,ψ,α to Jm,ψ (f ) will be done, once again, by a series of exchanges of roots. Assume that h(V ) = SO4n . Define, for 1 ≤ i ≤ n − 1,

Y2i−1,2i = {(Im + ae2i−1,2i )∧ } X2i,2i+1 = {(Im + be2i,2i+1 )∧ }

Let C2i−1,2i be the subgroup of the elements in SO4n of the form      ∗ 1 u ∗ ··· ∗ ∗      ∗   ∗    I2 x1        ..    ..   .    .           I2 xi−1 ∗ ∗ ··· ∗   ∗             v I2 xi ∗ ∗   ,  ∗  , ∗ ,       zi+1 xi+1 ∗   ∗           ..    ..     . ∗   .          zn−2 xn−2   ∗    zn−1 xn−1

(9.42)

(9.43)

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such that x1 , ..., xn−1 ∈ M2×2 , (xi−1 )2,1 = · · · = (xn−1 )2,1 , ! 1∗ zi+1 , ..., zn−1 = . 01

(9.44)

It is easy to see that C2i−1,2i is indeed a group over F . For v ∈ C2i−1,2i (A), written as in (9.43), define ψC2i−1,2i (v) = ψ(u2 + tr(x1 + · · · xn−1 )). 2i−1,2i

(9.45) 2i−1,2i

This is a character of C (A). To see this, note that for v ∈ C (A), such that, in the notation of (9.43), zi+1 = · · · zn−1 = I2 , (xi−1 )2,1 = · · · = (xn−1 )2,1 = t, and for z = diag(I2i+1 , zi+1 , ..., zn−1 , 1)∧ ∈ C2i−1,2i (A); zi+1 =

1 ki+1 0 1

!

, ..., zn−1 =

1 kn−1 0

1

!

,

conjugation of v by z results in −1 −1 −1 (u, x1 , ..., xn−1 ) 7→ (u, x1 , ..., xi−1 , xi zi+1 , zi+1 xi+1 zi+2 , ..., zn−2 xn−2 zn−1 , zn−1 xn−1 ),

and so −1 −1 −1 tr(xi zi+1 + zi+1 xi+1 zi+2 + · · · + zn−2 xn−2 zn−1 + zn−1 xn−1 ) = tr(xi + · · · xn−1 ) + t(−ki+1 + (ki+1 − ki+2 ) + · · · (kn−2 − kn−1 ) + kn−1 ) = tr(xi + · · · xn−1 ).

Similar verifications show that conjugation by Y2i−1,2i and by X2i,2i+1 preserve C2i−1,2i and ψC2i−1,2i , over A. Let y = (Im + ae2i−1,2i )∧ , x = (Im + be2i,2i+1 )∧ . Then [x, y] = (Im − abe2i−1,2i+1 )∧ ∈ C2i−1,2i , and for a, b ∈ A,

ψC2i−1,2i ([x, y]) = ψ −1 (ab).

Thus, we verified all the conditions of Lemma 7.1. Let B2i−1,2i = C2i−1,2i Y2i−1,2i , D2i−1,2i = C2i−1,2i X2i,2i+1 , and, as usual, extend ψC2i−1,2i to characters ψB2i−1,2i , ψD2i−1,2i of B2i−1,2i (A) and D2i−1,2i (A) respectively, by making the character trivial on Y2i−1,2i (A) and on X2i,2i+1 (A), respectively. Note that when i = 1, the last two conditions of (9.44) are   1∗ u1 = (x1 )2,1 = · · · = (xn−1 )2,1 ; z2 , ..., zn−1 = . 01

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We have ˜

= B2n−3,2n−1 D = B2i−3,2i−2 ; i = n − 1, ..., 3, 2 1,2 D = {vin (t) | v ∈ Sm , t ∈ Ga } ψD1,2 (vin (t)) = ψSm (v), v ∈ Sm (A), t ∈ A. Sm 2i−1,2i

Denote

Yn′′ = Y2n−3,2n−2 Y2n−5,2n−4 · · · Y1,2 .

We conclude, as usual, by a repeated application of Lemma 7.1 and Corollary 7.1 that Z Z Z ˜m ,ψ,α S f (vin (t)yγ(α)h)ψS−1 (v)dvdtdy. (9.46) (h) = f m ′′ (A) F \A S (F )\S (A) Yn m m

˜

Moreover, f Sm ,ψ,α is nonzero, if and only if the Fourier coefficient Z Z (v)dvdt f (vin (t)h)ψS−1 m F \A Sm (F )\Sm (A)

is nonzero (as a function of h). When n is even, we recognize the inner dvdt integral of the r.h.s. of (9.46) as Z Jm,ψ (f )(in (t)yγ(α)h)dt = Jm,ψ (f )(yγ(α)h). F \A

We used Theorem 8.3. Hence, when n is even, Z ˜m ,ψ,α S Jm,ψ (f )(yγ(α)h)dy. (h) = f ′′ (A) Yn

Assume that n is odd. Recall that in this case ω = diag(I2n−1 ,



ω Denote f ω (h) = f (hω ). Then, since Nn−1 = Nn−1 , Z −1 f (v ω hω )ψn−1,α (v)dv = (f ω )ψn−1,α (h) =

1

 1

, I2n−1 ).

Nn−1 (F )\Nn−1 (A)

Z

−1 f (vhω )ψn−1,α (v ω )dv.

Nn−1 (F )\Nn−1 (A)



 1  1   , I2n−2 ). Then ω ′ ∈ SO4n (F ), and since N ω′ = Let ω ′ = diag(I2n−2 ,  n−1  1  1 Nn−1 , we get Z ′ −1 f (v ω ω ′ hω )ψn−1,α (v ω )dv = (f ω )ψn−1,α (h) = Nn−1 (F )\Nn−1 (A)

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Z

′

−1 f (vω ′ hω )ψn−1,α (v ω ω )dv.

Nn−1 (F )\Nn−1 (A)

Since ψn−1,α (v

ω′ ω

) = ψn−1,α (v), we get (f ω )ψn−1,α (h) = f ψn−1,α (ω ′ hω ),

and hence Wm,ψ;α, α2 (f ω )(h) = Wm,ψ;α, α2 (f )(ω ′ hω ).

(9.47)

ω

Applying (9.46) to f , we get, using Theorem 8.3, (for n odd or even) Z Z Z ˜ (v)dvdtdy f (v ω in (t)ω (yγ(α)h)ω )ψS−1 (f ω )Sm ,ψ,α (h) = m ′′ (A) F \A S (F )\S (A) Yn m m

=

Z

Z

Z

Jm,ψ (f )(y ω γ(α)ω hω )dy.

Jm,ψ (f )(in (t)ω y ω γ(α)ω hω )dtdy

′′ (A) F \A Yn

=

′′ (A) Yn

Let

Yn′

=

(Yn′′ )ω .

(Thus, for n even Yn′ = Yn′′ .) Then Z ˜m ,ψ,α ω S Jm,ψ (f )(yγα hω )dy, (h) = (f )

(9.48)

′ (A) Yn

˜

and (f ω )Sm ,ψ,α 6= 0 if and only if Jm,ψ (f ) 6= 0. In (9.48), γα = γ(α)ω . By Theorem 8.1, we know that as f varies in the residual representation Eτ¯ , Jm,ψ (f )(1) is not ˜ identically zero. We conclude that (f ω )Sm ,ψ,α (1) is not identically zero on Eτ¯ , and ′ hence, (f ω )Sm ,ψ is not identically zero on Eτ¯ (see the beginning of this section). By Theorem 9.3 Wm,ψ;α, α2 (f ω ) is not identically zero on Eτ¯ , and hence, using (9.47), for n odd, that Wm,ψ;α, α2 (f )(1) is not identically zero on Eτ¯ . Thus, the Whittaker coefficient of the descent σψ,α (¯ τ ), with respect to λ = α2 (and hence with respect ∗ to any λ ∈ F ) is non-trivial. In particular, the descent σψ,α (¯ τ ) is nontrivial and globally generic. By Theorem 7.6, we also know that it is cuspidal (on SO2n+1 (A)). Finally, since the exterior square L-functions of the representations τi all have a pole at s = 1, the central characters ωτi are all trivial. By Theorem 5.6(1), we get that each irreducible summand of σψ,α (¯ τ ) lifts almost everywhere to τv (on GL2n (Fv )). This proves the theorem, for SO4n . Assume that h(V ) = U4n . The proof is similar, but needs some modifications. Let Y02n−1,2n = {(Im + ae2n−1,2n )∧ | a ¯ = a} X2n,2n+1 = {um (be2n,1 ) | ¯b = −b} C2n−1,2n = {v ∈ S˜m | v¯2n−1,2n = −v2n−1,2n }.

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It is clear that C2n−1,2n is an F -subgroup of S˜m . Let ψC2n−1,2n be the restriction of the character ψS˜m to C2n−1,2n (A). It is also clear that Y02n−1,2n normalizes C2n−1,2n , and its Adele points preserve ψC2n−1,2n . So does X2n,2n+1 . To see this, write v ∈ C2n−1,2n in the form      ∗ 1 u ∗ ··· ∗     I2 x1 ∗    ∗         . . .. v = v   ,  ..  , ∗ ,       I2 xn−2   ∗   I2

xn−1

where x1 , ..., xn−1 ∈ ResE/F M2×2 and (¯ xn−1 )2,1 = −(xn−1 )2,1 . When we conjugate  ′ ′ x1 x2 ¯ v by um (be2n,1 ), b = −b, u, x1 , ..., xn−2 remain unchanged and xn−1 = x′3 x′4 becomes     ′ ′ 1 −b x1 x2 − bx′1 . xn−1 = 0 1 x′3 x′4 − bx′3

This an element of C2n−1,2n , and when we take Adele coordinates, the application of ψC2n−1,2n to the last conjugation is −1 ψE,1 (u2 + tr(x1 + · · · xn−1 ))ψE,1 (bx′3 ).

But ψE,1 (bx′3 ) = 1, since bx′3 = (−b)(−x′3 ) = bx′3 . Similarly, we see that for y = (Im + ae2n−1,2n )∧ and x = um (be2n,1 ), with a ¯ = a, ¯b = −b, [x, y] = um (−abe2n−1,1 − abe2n,2 ) ∈ C2n−1,2n ,

and in Adele coordinates, ψC2n−1,2n ([x, y]) = ψE,1 (−ab). Note that ab = −ab, and so, we verified all the requirements of Lemma 7.1. Let D2n−1,2n = C2n−1,2n X2n,2n+1 and extend ψC2n−1,2n to a character of D2n−1,2n (A) by making it trivial on X2n,2n+1 (A). Since S˜m = C2n−1,2n Y02n−1,2n , we conclude that Z Z ˜ −1 f (vyγα h)ψD f Sm ,ψ,α (h) = 2n−1,2n (v)dvdy, 2n−1,2n \D2n−1,2n Y02n−1,2n (A) DF A

˜

and that, with our usual notation, f Sm ,ψ,α 6= 0, if and only if there is h ∈ U4n (A), such that Z −1 f (vγα h)ψD 2n−1,2n (v)dv 6= 0. D2n−1,2n \D2n−1,2n F A

Here γα = γ(α). From this point on, we continue as in the case of SO4n . Define, for i = n − 1, n − 2, ..., 1 Y2i−1,2i = {(Im + ae2i−1,2i )∧ },

X2i,2i+1 = {(Im + be2i,2i+1 )∧ },

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and let C2i−1,2i be the group of all elements of the form v = v ′ um (xe2n,1 ), where x ¯ = −x, and v ′ has the form (9.43), such that (xi−1 )2,1 = · · · = (xn−1 )2,1 = t, where t¯ = −t. Consider the character of C2n−1,2n (A), given, in the notation above, and that of (9.43) by ψC2i−1,2i (v) = ψC2i−1,2i (v ′ ) = ψE,1 (u2 + tr(x1 + · · · + xn−1 )). The requirements of Lemma 7.1 are satisfied for C2i−1,2i , Y2i−1,2i , X2i,2i+1 , ψC2i−1,2i , and we apply this lemma and its corollary repeatedly. Note that, with the similar notation as in the SO4n case, D1,2 = {vin (t) | v ∈ Sm , t¯ = −t}.

Let Yn′ = Y02n−1,2n Y2n−3,2n−2 · · · Y1,2 . Then we get, using Theorem 8.3, Z Z Z √ ˜ (v)dvdtdy f (vin (t ρ)yγα h)ψS−1 f Sm ,ψ,α (h) = m ′ (A) F \A S (F )\S (A) Yn m m

=

Z

Z

√ Jm,ψ (f )(in (t ρ)yγα h)dtdy

′ (A) F \A Yn

=

Z

Jm,ψ (f )(yγα h)dy.

′ (A) Yn

Exactly as in the SO4n case, we conclude that the Whittaker coefficient of σψ,α (¯ τ) is nontrivial (of course, now the parity of n is irrelevant). Note that in order to apply Theorem 7.6 and get the cuspidality of σψ,α (¯ τ ), we need the assumption that m1 , ..., mr ≥ 2. Finally, we analyze the unramified parameters, at almost all places of any given irreducible generic summand σ of σψ,α (¯ τ ). Since we assume that the Asai L-functions of the representations τi all have a pole at s = 1, we know that their central characters are trivial on A∗ , and hence the relevant case of Theorem 5.6 is (1). Let v be a place where Ev is a quadratic unramified extension of Fv , and GL (E ) τi,v are unramified. Let τv = IndPm2n,...,mv (Ev ) τ1,v ⊗ · · · τr,v and assume that τv is r 1 induced from the standard Borel subgroup and the unramified character µ1 ⊗· · · µn ⊗ µn−1 ⊗ · · · ⊗ µ−1 1 . (We also know that the representations τi are self-conjugate.) By U (Fv ) Theorem 5.6(1), σv is a constituent of the representation indB2n+1 µ1 ⊗ · · · ⊗ µn , ′ where     b∗ ∗  B ′ =  1 ∗  ∈ U2n+1 (Fv ) | b ∈ BGL2n (Ev ) .   b∗ See the summary, right after Theorem 5.6. Since σv is irreducible and unramified, it follows that σv is the unramified constituent of the representation of U2n+1 (Fv ), induced from the character µ1 ⊗ · · · ⊗ µn ⊗ 1 of the Borel subgroup. Thus, the GL (Ev ) τ1,v ⊗ · · · τr,v ⊗ 1. Assume that local lift of σv to GL2n+1 (Ev ) is IndPm2n+1 1 ,...,mr ,1 (Ev ) v is a place where Ev = Fv ⊕ Fv , and the representations τi,v of GLmi (Ev ) are

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unramified. In the isomorphism rv of U4n (Fv ) with GL4n (Fv ) (see (1.15)), the parabolic subgroup Q2n (Fv ) becomes the standard parabolic subgroup P2n,2n (Fv ) 1 and the representation τv | det ·| 2 of GL2n (Ev ) becomes a representation of the form 1 1 τ ′ | det ·| 2 ⊗τ ′ | det ·|− 2 , of GL2n (Fv )×GL2n (Fv ), where τ ′ is an unramified (generic) representation of GL2n (Fv ). Again, we used the fact that the representations τi are self conjugate. Assume that τ ′ is induced from the Borel subgroup from the unramified character µ1 ⊗ · · · ⊗ µ2n . By Theorem 5.10, σv , which is now an irreducible, unramified, generic representation of GL2n+1 (Fv ), is a constituent of a certain direct sum of representations. Looking at this sum, we see that all its summands, except the last one are not generic. Thus σv is a constituent of the last summand GL (F ) in Theorem 5.10, which is indB ′ 2n+1 v µ1 ⊗ · · · ⊗ µ2n . This is the place where we use the genericity of σ. Since σv is irreducible and unramified (it has a central character and hence) there is an unramified character ǫ′v of Fv∗ , such that σv is the unramified constituent of the representation of GL2n+1 (Fv ), induced from the Borel subgroup and the character µ1 ⊗ · · · ⊗ µ2n ⊗ ǫ′v . It is easy to see that ǫ′v = ωσv ωτ−1 ′ . Since the central characters of the representations τi are trivial on A∗ , there is an automorphic character η of U1 (A), such that ωτ (x) = η( xx¯ ), for x ∈ A∗E . For the split places v as above, ηv = ωτ ′ and hence ǫ′v = ωσv ηv−1 . Thus, ǫ′ = ωσ η −1 is an automorphic character of U1 (A). Let ǫ be the automorphic character of A∗E given by ǫ(x) = ǫ′ ( xx¯ ). Then, at almost all places v of F , σv lifts to the representation of GL (Ev ) τ1 ⊗ · · · ⊗ τr ⊗ ǫ. The proof of Theorem 9.4 is now complete.  IndPm2n+1 ,...,m ,1 (Ev ) 1

r

Remark 1. Assume that h(V ) = U4n . The fact that the irreducible summands of σψ,α (¯ τ ) are not uniquely determined almost everywhere by τ1 , ..., τr , as is the case when h(V ) = SO4n , is not a surprise. Note that the relation between σ and GL (AE ) τ1 ⊗ · · · ⊗ τr is not a case of functoriality. We will see later τ = IndPm2n,...,m r (AE ) 1 that the descent construction, using Fourier-Jacobi coefficients gives irreducible, automorphic, cuspidal and globally generic representations σ ′ of U2n (A), which lift to τ almost everywhere. Let ǫ′ be an automorphic character of U1 (A). Then, by a theta lift to U2n+1 (A) one can lift σ ′ ⊗ ǫ′−1 to an irreducible, automorphic, cuspidal and globally generic representation σǫ′ of U2n+1 (A) and then σ = σǫ′ ⊗ ǫ′ is an irreducible, automorphic, cuspidal and globally generic, which lifts almost everywhere to GL

(AE ) τ1 r ,1 (AE )

IndPm2n+1 ,...,m 1

⊗ · · · ⊗ τr ⊗ ǫ,

where ǫ(x) = ǫ′ ( xx¯ ). Remark 2. In the next chapter, we will prove that all irreducible summands of the descent representation σψ,α (¯ τ ) are globally generic. Thus, the extra assumption on σ, in the last theorem, when h(V ) = U4n becomes unnecessary.

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Nonvanishing of the Whittaker coefficient of the descent corresponding to Gelfand-Graev coefficients: h(V ) = U4n+2 , SO4n+3

Assume that h(V ) = U4n+2 , SO4n+3 . Once again, we will apply repeatedly Lemma 7.1 to the integral (9.14). The first step is to use the nontrivial coordinates of the row T2n of T in (9.22) in order to “fill in” Y in (9.20) the rows (xj , − α2 xj ), j = 1, 2, ..., n in rows 1, 3, 5, ..., 2n − 1, 2n of Y , and similarly, the rows (xj , 0, α2 xj ), j = 1, 2, ..., n in Y in (9.21). Let 1 ≤ j ≤ n. Denote by C2n,2j−1 the subset of matrices which have the form v(T, Y, C) as in (9.19), so that T has the form (9.22), with Tn,1 = · · · Tn,j−1 = 0, (Tn,j )2,1 = 0, (9.49)   Y1 , where Y1 is any matrix of size (2j − 2) × (2 + dimV0 ) and Y has the form Y = Y2 and Y2 is such that its first and last columns are of the form   xj (−1)dimV0 α2 xj   aj bj     .. .. .  . .   xn (−1)dimV0 2 xn  α

an

bn

As in the previous case, this is a group over F , and, similarly, the function ψC2n,2j−1 , defined on C2n,2j−1 (A), in terms of the coordinates above, by α ψE,1 ((u1 )2 +tr(T1,2 +T2,3 +· · ·+Tn−1,n )+an +(−1)dimV0 +1 bn +λxn +δdimV0 ,1 µyn ), 2 (9.50) is a character of C2n,2j−1 (A), trivial on C2n,2j−1 (F ). Define Y2n,2j−1 = {(I2n+1 + ye2n,2j−1 )∧ }, X2j−1;2n+1,2n+2+dimV0 =

2 2 ¯e2n+1,2n+3−2j ))}. {(I2n+1 + xe2j−1,2n+1 )∧ · u′m ((−1)dimV0 (− xe2j−1,1 + x α α Here, u′m = um , when dimV0 = 0 (i.e. h(V ) = U4n+2 ) and u′m = cm , when dimV0 = 1. The groups C2n,2j−1 , Y2n−2,2j−1 , X2j−1;2n+1,2n+2+dimV0 satisfy the requirements of Lemma 7.1. The proof of this is very similar to the previous case (the groups h(V ) = SO4n , U4n ). For example, for x ∈ X2j−1;2n+1,2n+2+dimV0 and y ∈ Y2n−2,2j−1 , written as above, [x, y] = v(I2n , −xyv2n , ∗), where v2n is the 2n×(2+dimV0 ) matrix all of whose coordinates are zero except the first and last coordinates of its last row, which are 1 and (−1)dimV0 +1 α2 , respectively.

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Thus, [x, y] ∈ C2n,2j−1 and, in Adele coordinates, ψC2n,2j−1 ([x, y]) = ψE,1 (−2xy). Let B2n,2j−1 = C2n,2j−1 Y2n−2,2j−1 and D2n,2j−1 = C2n,2j−1 X2j−1;2n+1,2n+2+dimV0 . Note that D2n,2j−1 = B2n,2j+1 , B2n,1 = Nm,α .

Extend ψC2n,2j−1 to B2n,2j−1 (A) and to D2n,2j−1 (A), by making it trivial on Y2n,2j−1 (A) and on X2j−1;2n+1,2n+2+dimV0 (A) respectively. Denote the extended characters by ψB2n,2j−1 , ψD2n,2j−1 . Denote Y2n =

n Y

Y2n,2j−1 .

j=1

We conclude, as in Lemma 9.1 and Proposition 9.1, Proposition 9.2. Let h(V ) = U4n+2 , SO4n+3 and let f be a smooth automorphic function of uniform moderate growth on HA . Then Z Z −1 f (vy ηˆm h)ψD Wm,ψ (f )(h) = 2n,2n−1 (v)dvdy, 2n,2n−1 2n,2n−1 YA2n DF \DA

where the dy-integration converges absolutely as a repeated integration (as in Proposition 9.1. Wm,ψ (f ) 6= 0, if and only if there is h ∈ HA , such that Z −1 f (v ηˆm h)ψD 2n,2n−1 (v)dv 6= 0. 2n,2n−1 2n,2n−1 \DA DF

The group D2n,2n−1 consists of all elements v(T, Y, C) ∈ h(V ) (9.19), such that T is of the form (9.22), with Tn,1 = · · · Tn,n−1 = 0, Tn,n = I2 , (and Y is arbitrary). The character ψD2n,2n−1 is given by ψE,1 (tr(T1,2 + · · · tr(Tn−1,n ))ψE,1 (Y2n,1 + (−1)dimV0 +1 α2 Y2n,2+dimV0 )· ψE,1 ( λ2 (Y2n−1,1 + (−1)dimV0 α2 Y2n−1,2+dimV0 ) + δdimV0 ,1 µY2n−1,2 ).

(9.51)

We continue with the process of exchanging roots, as in the previous case. We exchange the coordinates in columns 1, 3, ..., 2n−3 of row 2n−2 of T in v(T, Y, C) ∈ D2n,2n−1 with the coordinates in rows 1, 3, ..., 2n − 3 of column 2n. Then we exchange the coordinates in columns 1, 3, ..., 2n − 5 of row 2n − 4 with those in rows 1, 3, ..., 2n− 5 of column 2n− 2, and so on. In precise terms, let, for 1 ≤ i ≤ n, Ti be the group of matrices of the form   T1,1 . . . T1,i−1 T1,i  .. ..   . .  (9.52) ,  Ti−1,1 . . . Ti−1,i−1 Ti−1,i  0

...

0

I2

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where the Tr,s are 2 × 2 lower triangular matrices, such that, for r > s, Tr,s are (lower) nilpotent, and Tr,r are (lower) unipotent. Denote by ψTi the following character of Ti (A) ψE,1 (tr(T1,2 + T2,3 + · · · + Ti−1,i )).

(9.53)

For 1 ≤ j ≤ i − 1, let

Ti′′ (j) = {T ∈ Ti |(Ti−1,1 )2,1 = · · · = (Ti−1,j )2,1 = 0} ,

   y  I2j−3  Ti∗ (j) =  I2i−2j+2 0 |y2 = y4 = · · · = y2j−4 = 0 ; Ti∗ (1) = I,   1 Ti′ (j) = Ti′′ (j)Ti∗ (j).

It is easy to see that Ti′ (j) is a group (over F ), and that formula (9.53) defines a character ψTi′ (j) of Ti′ (j)(A). As in (9.37), let    I2 ∗ ∗ ∗         I2 ∗ ∗   T Z   ′ Ti (j) = ∈ ResE/F GL2n T ∈ Ti (j), U =  . (9.54)  ..  U  . ∗        I2

This is a group (over F ) and the character ψTi′ (j) extends to the character ψTi (j) of Ti (j)(A) by   I2i 7→ ψE,1 (tr(U1,2 + · · · Un−i−1,n−i )), U

where we write U as a matrix of 2 × 2 blocks. Now we can define the groups (C, X, Y ) which satisfy Lemma 7.1. In fact, they are defined exactly as in (9.38). C2i−2,2j−1 = {v(T, Y, C) ∈ h(V ) | T ∈ Ti (j)}, Y2i−2,2j−1 = {(I2n+1 + te2i−2,2j−1 )∧ }

(9.55)

X2j−1,2i = {(I2n+1 + te2j−1,2i )∧ }. Let ψC2i−2,2j−1 be the following character of C2i−2,2j−1 (A) (trivial on the F -points) ψC2i−2,2j−1 (v(T, Y, C)) = ψTi (j) (T )ψE,1 (Y2n,1 + (−1)dimV0 +1 α2 Y2n,2+dimV0 )· ψ( λ2 (Y2n−1,1 + (−1)dimV0 α2 Y2n−1,2+dimV0 ) + δdimV0 ,1 µY2n−1,2 ).

(9.56) It is now a simple routine to check that the conditions of Lemma 7.1 are verified. Denote, as usual, B2i−2,2j−1 = C2i−2,2j−1 Y2i−2,2j−1 ,

D2i−2,2j−1 = C2i−2,2j−1 X2j−1,2i ,

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and extend ψC2i−2,2j−1 to characters ψB2i−2,2j−1 , ψD2i−2,2j−1 of B2i−2,2j−1 (A), D2i−2,2j−1 (A), by the trivial characters of Y2i−2,2j−1 (A), X2j−1,2i (A), respectively. Note that D2n,2n−1 = B2n−2,1 , D2i−2,2j−1 = B2i−2,2j+1 , D2i−2,2i−1 = B2i−4,1

for j < i − 1,

We conclude, as before, the analog of (9.40), for a smooth automorphic function of uniform moderate growth on HA , f . Let Y2i−2 =

i−1 Q

Y2i−2,2j−1

j=1

Y′ =

n Q

Y2i

i=1 −1 ′ Ym = ηˆm Y ηˆm .

By Proposition 9.2, we conclude, exactly as in the previous section, the analog of Theorem 9.3. Proposition 9.3. Let h(V ) = U4n+2 , SO4n+3 and let f be a smooth automorphic function of uniform moderate growth on HA . Then Z Z −1 f (v ηˆm yh)ψD Wm,ψ (f )(h) = 2,1 (v)dvdy, Ym (A) D2,1 (F )\D2,1 (A)

where the dy-integration converges absolutely as a repeated integration, as before. Finally, Wm,ψ (f ) 6= 0, if and only if there is h ∈ HA , such that Z −1 f (v ηˆm h)ψD 2,1 (v)dv 6= 0. D2,1 (F )\D2,1 (A)

Note that (see (8.3)) D2,1 = Sm . Write an element of D2,1 = Sm in the form v(T, Y, C), as in (9.19), with   I2 x1 . . . ∗    I2 ∗  y1       . .. T = , Y =  ...  , ∗     I2 xn−1  yn I2

where y1 , ..., yn are 2 × (2 + dimV0 ) matrices. For v ∈ D2,1 (A) written as above, ψD2,1 (v) = ψE,1 (tr(x1 + · · · + xn−1 ) + tr(yn A)),

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where

A=

          

λ 2 λα 4

1 − α2

!

,

h(V ) = U4n+2

  λ  1     2    µ 0  , h(V ) = SO4n+3        −λα α 4

(9.57)

2

Assume that h(V ) = U4n+2 . Let

1 1 γα,λ = diag(2A, ..., 2A, I2 , A∗ , ..., A∗ ) ∈ h(V ). 2 2

(9.58)

Here, 2A is repeated n times. Then, for v ∈ D2,1 (A) = Sm (A), (see (8.4)) −1 ψ (m) (v) = ψD2,1 (γα,λ vγα,λ ).

(9.59)

By Proposition 9.3, using its notation, we get, re-denoting Wm,ψ (f ) = Wm,ψ;α,λ (f ), Z Z −1 −1 f (γα,λ vγα,λ γα,λ ηˆm yh)ψD Wm,ψ;α,λ (f )(h) = 2,1 (v)dvdy Ym (A) D2,1 (F )\D2,1 (A)

=

Z

Z

f (vγα,λ ηˆm yh)ψ (m) (v −1 )dvdy

(9.60)

Ym (A) Sm (F )\Sm (A)

=

Z

Jm,ψ (f )(γα,λ ηˆm yh)dy.

Ym (A)

See (8.5). By Proposition 9.3, we conclude that Wm,ψ;α,λ (f ) 6= 0, if and only if there is h ∈ HA , such that Jm,ψ (f )(h) 6= 0. Note that this equivalence, as well as the identity (9.60), are valid for any automorphic form f on HA . Thus, if we let f vary in the residual representation Eτ¯ , we get, by Theorem 8.1 that Wm,ψ;α,λ is nontrivial on Eτ¯ and hence the descent σψ,α (¯ τ ) is nontrivial and globally generic. Note that (9.60) and (8.6) give a nice identity which relates the Whittaker coefficient of f ψn,α , ′ with respect to the character (9.9) and the Whittaker coefficient with respect ψZ2n+1 U2n+1 . We summarize this in the following theorem (on τ¯) of the constant term f Theorem 9.5. Assume that h(V ) = U4n+2 . (1) The descent σψ,α (¯ τ ) is non-trivial and globally generic (on U2n+1 (A)). We have the following identity for f ∈ Eτ¯ (in the notation of (9.60) and (8.6)), Z Z Z ′ (z −1 )dzdudy, z uγm,λ,α y)ψZ f Um (ˆ Wm,ψ;α,λ (f )(1) = m Ym (A) u ¯(X0 )A Zm (E)\Zm (AE )

where γm,λ,α = αm γλ,α ηˆm (m = 2n + 1).

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(2) The descent σψ,α (¯ τ ) is a cuspidal representation of U2n+1 (A). Each irreducible summand of σψ,α (¯ τ ), lifts, at almost all places v, to GL

(Ev ) τ1,v r (Ev )

τv = IndPm2n+1 ,...,m 1

⊗ · · · ⊗ τr,v .

The cuspidality of σψ,α (¯ τ ) follows from Theorem 7.6, and the assertion on the lift to GL2n+1 (AE ), almost everywhere, follows from Theorem 5.6(5)(see (4) in the summary following Theorem 5.6) and Theorem 5.9. Note again that by our assumptions, the central characters ωτi are trivial on A∗ . Assume that h(V ) = SO4n+3 . Consider the matrix A in (9.57). Choose t ∈ F ∗ , such that t2 α 6= ±1, and for such t, take tα µ= 2 λ. (9.61) t α+1 ′ Then it is a simple exercise to see that, for such a choice, we can find γλ,µ,α ∈ ′′ h(Wm,m ) = SO3 (F ), and γλ,µ,α ∈ GL2 (F ), such that   10 ′ ′′ (γλ,µ,α )−1 0 0 γλ,µ,α = A. 01

Define

′′ ′′ ′ ′′ ′′ γα,λ,µ = diag(γλ,µ,α , ..., γλ,µ,α , γλ,µ,α , (γλ,µ,α )∗ , ..., (γλ,µ,α )∗ ) ∈ SO4n+3 (F ).

′′ Here, γλ,µ,α is repeated n times. Exactly as in (9.59), we get for v ∈ D2,1 (A) = Sm (A), (see (8.4)) −1 ψ (m) (v) = ψD2,1 (γα,λ,µ vγα,λ,µ ).

By Proposition 9.3, using its notation, we get, as in (9.60) re-denoting Wm,ψ (f ) = Wm,ψ;α,λ,µ (f ), Z Z f (vγα,λ,µ ηˆm yh)ψ (m) (v −1 )dvdy Wm,ψ;α,λ,µ (f )(h) = Ym (A) Sm (F )\Sm (A)

=

Z

(9.62)

Jm,ψ (f )(γα,λ,µ ηˆm yh)dy.

Ym (A)

See (8.5). By Proposition 9.3 and Theorem 8.1, we conclude, as before, that (for µ as in (9.61)) Wm,ψ;α,λ (f ) 6= 0, as f varies in the residual representation Eτ¯ . Hence the descent σψ,α (¯ τ ) is nontrivial and globally generic with respect to the Whittaker character corresponding to (λ, µ), as above. Again, (9.62) and (8.7) give a nice identity which relates the Whittaker coefficient of f ψn,α , with respect to the ′ character (9.11) and the Whittaker coefficient with respect ψZ2n+1 (on τ¯) of the U2n+1 . Recall that when α is not a square, the descent σψ,α (¯ τ ) is a constant term f representation of SOn+2,n (A) (quasi-split with discriminant (−1)n α), and when α is a square, σψ,α (¯ τ ) is a representation of SO2n+2 (A) (split). We denote now both

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these groups by SO2n+2,α . Theorem 7.6 tells us that in order to get the cuspidality of σψ,α (¯ τ ), we must add the requirements that m1 , ..., mr ≥ 2, i.e. that no one of the representations τi is a character of A∗ . Let σ be any irreducible summand of σψ,α (¯ τ ). Then Theorem 5.6(5)(6) (see also (6) in the summary which follows the theorem) shows that σv lifts almost everywhere to the following representation of GL2n+2 (Fv ): GL

(F )

v IndPm2n+2 τ ⊗ · · · ⊗ τr,v ⊗ ωτ¯v χα,v , ,...,mr ,1 1,v 1

where ωτ¯v is the product of all central characters of the representations τi , at v, and χα,v is the quadratic character corresponding to α at v. We summarize this in the following theorem Theorem 9.6. Assume that h(V ) = SO4n+3 . (1) The descent σψ,α (¯ τ ) is non-trivial and a globally generic representation of SO2n+2,α (A). We have the following identity for f ∈ Eτ¯ , for µ as in (9.61), in the notation of (9.62) and (8.7)), Z Z Z ′ (z −1 )dzdudy, z uγm,α,λ,µ y)ψZ f Um (ˆ Wm,ψ;α,λ,µ (f )(1) = m Ym (A) Ym (A) Zm (F )\Zm (A)

where γm,α,λ,µ = αm γα,λ,µ ηˆm (m = 2n + 1). (2) Assume that m1 , ..., mr ≥ 2. Then the descent σψ,α (¯ τ ) is a cuspidal representation of SO2n+2,α (A). Each irreducible summand of σψ,α (¯ τ ), lifts, at almost all places v, to GL

(F )

v IndPm2n+2 τ ⊗ · · · ⊗ τr,v ⊗ ωτ¯v χα,v . ,...,mr ,1 1,v 1

Remark: The descent in the last theorem is not a case of functoriality, as it assoGL (A) τ1 ⊗· · ·⊗τr of GL2n+1 (A), irciates to the self-dual representation τ = IndPm2n+1 1 ,...,mr (A) reducible, automorphic, cuspidal (and globally generic) representations σ ⊂ σψ,α (¯ τ) of SO2n+2,α (A). However, we can use the descent to Sp2n (A) (Theorem 9.1) to obtain irreducible, automorphic, cuspidal (and globally generic) representations σ ′ of Sp2n (A), which lift almost everywhere to τ ⊗ ωτ . It is possible to prove that σ lifts by the theta correspondence to an automorphic, cuspidal (and globally generic) representation of Sp2n (A), all of whose irreducible summands are nearly equivalent to σ ′ (and even have a nontrivial L2 -pairing with the descent σψ (¯ τ ) to Sp2n (A)). 9.8

The Whittaker coefficient of the descent corresponding to f Fourier-Jacobi coefficients: HA = 6 Sp 4n+2 (A)

Let h(V ) be one of the groups Sp4n , U4n , U4n+2 , where now the form b is anti-symmetric, or anti-Hermitian, respectively. We consider the Adele groups f 4n (A), U4n (A), U4n+2 (A). Recall that now ℓm = [ m−2 ] = n−1 (m = 2n Sp4n (A), Sp 2

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∗ ∗ or m = 2n+ 1), and Hm = Hn−1 is the group corresponding to h(Wm,n ) (see (3.38), (3.41)). In detail,  f 2n (A),  Sp HA = Sp4n (A)    Sp (A), f (A) HA = Sp 2n 4n ∗ Hm (A) = (9.63)  U (A), H = U (A))  2n A 4n    U2n+2 (A), HA = U4n+2 (A)

The elements of the descent σψ,γ (¯ τ ) are given by Fourier-Jacobi coefficients f ψn−1,γ;φ + (see (7.40)), as f varies in the residual representation Eτ¯ and φ ∈ S(Wm,n (A)). By ψ

−

Proposition 7.3, σψ,γ (¯ τ ) 6= 0, if and only if f Nn is not identically zero, for f ∈ Eτ¯ , ψN − where f n is the following Fourier coefficient Z ψ − −1 f (vh)ψN (9.64) f Nn (h) = − (v)dv, n

− − Nn (F )\Nn (A)

where Nn− = Nn−1 C ′ , and denoting n′ = [ m+1 2 ],     1 0 e z        In′ 0 e′  ′ ′   , , I ) ∈ h(V ) C = c (e, z) = diag(In−1 ,  n−1 In′ 0        1

and ψNn− is the character of Nn− (A), which is ψn−1 on Nn−1 (A) and on CA′ is given by ψNn− (c′ (e, z)) = ψ(z). C ′ is (isomorphic under the isomorphism j of − (3.60) to) the maximal abelian subgroup Wm,n ⊕ F of the Heisenberg group HWm,n , ± ± where Wm,n = Vn−1,n′ are the standard maximal dual isotropic subspaces of Wm,n . ∗ + See (5.3). Clearly, the Siegel parabolic subgroup of Hm (corresponding to Wm,n ) − normalizes Nn and its Adele points preserve ψNn− , and in particular, this holds for ∗ the standard maximal unipotent subgroup of Hm ,     z y ∗ = ′ NHm u = diag(In , , I ) ∈ h(V ) |z ∈ Res Z . (9.65) n E/F n z∗ ∗ (A), Let λ ∈ F ∗ and consider the following Whittaker character of NHm

ψNHm ∗ ,λ (u) = ψE,−1 (z1,2 + · · · + zn−1,n + λyn′ ,1 ).

The Whittaker coefficient of f Wψ,λ (f

ψN − n

ψN −

)(h) =

n

, with respect to ψNHm ∗ ,λ is Z ψ − −1 f Nn (uh)ψN (u)du. H ∗ ,λ

(9.66)

(9.67)

m

NHm ∗ (F )\NH ∗ (A) m

Similarly, denote by Wψ,λ (f ψn−1,γ;φ ) the Whittaker coefficient of f ψn−1,γ;φ , with respect to ψNHm ∗ ,λ , Z −1 f ψn−1 ,γ,φ (uh)ψN (u)du. (9.68) Wψ,λ (f ψn−1,γ;φ )(h) = H ∗ ,λ m

NHm ∗ (F )\NH ∗ (A) m

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Denote



 1 a 0 0  In′ 0 0   , In−1 ) ∈ h(V ). a+ = diag(In−1 ,   In′ a′  1

Then, by (7.62), we have

f ψn−1,γ;φ (h) =

Z

f

ψN − n

(a+ h)ωψ−1 ,γ −1 (a+ h)φ(0)da.

′

An E

It is then easy to see that Wψ,λ (f ψn−1,γ;φ )(h) =

Z

Z

f

ψ

− Nn

−1 (ua+ h)ωψ−1 ,γ −1 (ua+ h)φ(0)ψN H∗

m

,λ (u)duda

′ N ∗ (F )\N ∗ (A) Hm Hm An E

=

Z

Wψ,λ (f

ψ

− Nn

)(a+ h)ωψ−1 ,γ −1 (h)φ(a)da.

′

An E

We record this formula, which is valid (together with the definitions in (9.64), (9.67), (9.68)) for any smooth automorphic function on HA . Proposition 9.4. For any smooth automorphic function f on HA , we have (in the above notation; n′ = [ m+1 2 ]) Z ψ − Wψ,λ (f ψn−1,γ;φ )(h) = Wψ,λ (f Nn )(a+ h)ωψ−1 ,γ −1 (h)φ(a)da. ′

An E

This proposition reduces the computation of the Whittaker coefficient of f ψn−1,γ;φ , with respect to ψNHm ∗ ,λ to the computation of the Whittaker coefficient of f

ψN − n

, with respect to ψNHm ∗ ,λ . Rewrite

Wψ,λ (f

ψN − n

)(h) = Wm,ψ,λ (f )(h) =

Z

−1 f (vh)ψB (v)dv, ′ m ,λ

(9.69)

′ (F )\B ′ (A) Bm m

′ − ′ ∗ N ′ ,λ is the character of B where Bm = NHm n . This is an F -subgroup; ψBm m (A), − ∗ (A). We will compute this integral and which is ψNn− on Nn (A) and ψNHm ∗ ,λ on NHm show that it is nontrivial on the residual representation Eτ¯ . We need to separate the cases above according to the parity of m.

9.9

The nonvanishing of the Whittaker coefficient of the descent corresponding to Fourier-Jacobi coefficients: f (A), U4n (A) HA = Sp4n (A), Sp 4n

f 4n (A), U4n (A). We will compute and obtain a nice Assume that HA = Sp4n (A), Sp identity for (9.69). As in the previous cases, we will rewrite the last integral in terms

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′ of a conjugated subgroup κ ˆ m Bm κ ˆ −1 m , for a Weyl element κm in GLm (F ) (similar to ηm ). Its action on the standard basis {e1 , ..., em } of Vm+ is as follows (m = 2n).

κm (ei ) = e2i , 1≤i≤n κm (en+i ) = e2i−1 , 1 ≤ i ≤ n If we think of ei , 1 ≤ i ≤ m as column vectors in F m , then the columns of the matrix κm are κm = (e2 , e4 , ..., e2n , e1 , e3 , ..., e2n−1 ). As in (9.14), we have Wm,ψ,λ (f )(h) =

Z

−1 f (vˆ κm h)ψB (v)dv, m ,λ

(9.70)

Bm (F )\Bm (A) ′ ′ ,λ (ˆ where Bm = κ ˆ m Bm κ ˆ −1 κ−1 κm ). The elements of Bm m , and ψBm ,λ (v) = ψBm m vˆ have the following form   T Y ∈ h(V ), T∗

where T has the form  T1,1 . . . T1,n−1 T1,n  .. ..   . .  ,  Tn−1,1 . . . Tn−1,n−1 Tn−1,n  0 ... 0 I2

(9.71)

ψE,−1 (tr(T1,2 + T2,3 + · · · + Tn−1,n ) + λY2n−1,2 + Y2n,1 ).

(9.72)



and the Tr,s are 2 × 2 lower triangular matrices, such that, for r > s, Tr,s are (lower) nilpotent, and Tr,r are (lower) unipotent. Note that T is exactly of the form (9.52), (with i = n). The character ψBm ,λ is given by

Now, we proceed exactly as we did in the previous case (of SO4n+3 , U4n+2 ), starting from (9.52), until we reach the exact analog of Proposition 9.3, which we formulate, with the following (analogous) notation. Put Y2i−2,2j−1 = {(I2n + te2i−2,2j−1 )∧ }, Y2i−2 =

i−1 Q

Y2i−2,2j−1

j=1

Y′ =

n−1 Q

Y2i

i=1

′ Ym = κ ˆ −1 ˆm. m Y κ

2 ≤ i ≤ n; 1 ≤ j ≤ i − 1,

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Recall the group Sm (see (8.3)). Write an element of Sm as T has the form,

  T Y ∈ h(V ), where T∗

  I2 x1 . . . ∗  I2 ∗      .. T =  . ∗    I2 xn−1  I2

For v ∈ Sm (A) written as above, let

   01 Y Y ψSm ,λ (v) = ψE,−1 (tr(x1 + · · · + xn−1 ) + tr( 2n−1,1 ¯2n−1,2 )). Y2n,1 Y2n−1,1 λ0

f 4n (A), U4n (A) and let f be a smooth Proposition 9.5. Let HA = Sp4n (A), Sp automorphic function of uniform moderate growth on HA . Then Z Z f (vˆ κm yh)ψS−1 (v)dvdy, Wm,ψ,λ (f )(h) = m ,λ Ym (A) Sm (F )\Sm (A)

where the dy-integration converges absolutely as a repeated integration. Finally, Wm,ψ,λ (f ) 6= 0, if and only if there is h ∈ HA , such that Z f (vˆ κm h)ψS−1 (v)dv 6= 0. m ,λ Sm (F )\Sm (A)

Choose now λ = −1. Let A =



 1 1 and −1 1

ε = diag(A, ..., A, A∗ , ..., A∗ ), where A is repeated n times. Then, for v ∈ Sm (A),

ψ (m) (v) = ψSm ,−1 (ε−1 vε).

See (8.4). By Proposition 9.5, we get Z Z Wm,ψ,−1 (f )(h) =

(v)dvdy f (ε−1 vεˆ κm yh)ψS−1 m ,−1

Ym (A) Sm (F )\Sm (A)

=

Z

Z

f (vεˆ κm yh)ψ (m) (v −1 )dvdy

(9.73)

Ym (A) Sm (F )\Sm (A)

=

Z

Jm,ψ (f )(εˆ κm yh)dy.

Ym (A)

See (8.5). By Proposition 9.5, we conclude that Wm,ψ,−1 (f ) 6= 0, if and only if there is h ∈ HA , such that Jm,ψ (f )(h) 6= 0. Thus, if we let f vary in the residual

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f 4n (A)), we get, by Theorem 8.1 that representation Eτ¯ (Eτ¯,ψ in case HA = Sp Wm,ψ,−1 is nontrivial on Eτ¯ and hence the descent σψ,γ (¯ τ ) is nontrivial and globally generic, with respect to the Whittaker character (9.66) ψNHm ∗ ,−1 . Summarizing f 4n (A), U4n (A). Theorem 9.7. Let HA = Sp4n (A), Sp

∗ f 2n (A), (1) The descent σψ,γ (¯ τ ) is non-trivial and globally generic (on Hm (A) = Sp Sp2n (A), U2n (A), respectively) with respect to the Whittaker character (9.66) ψNHm We have the following identity for f ∈ Eτ¯ (Eτ¯,ψ in case HA = ∗ ,−1 . f Sp4n (A)); we use the notation of (9.73) and (8.6). Z Z Z ′ (z −1 )dzdudy, z uγm y)ψZ f Um (ˆ Wm,ψ,−1 (f )(1) = m Ym (A) u ¯ (X0 )A Zm (E)\Zm (AE )

where γm = αm εˆ κm (m = 2n). Moreover, in the notation of Proposition 9.4, we have the following identity, computing the ψNHm ∗ ,−1 -Whittaker coefficient of ψn−1,γ;φ (the Fourier-Jacobi coefficient) f Wψ,−1 (f ψn−1 ,γ,φ )(1) = Z

Z

Z

Z

′ (z −1 )ωψ−1 ,γ −1 (h)φ(a)dzdudyda. z uγm ya+ )ψZ f Um (ˆ m

An ¯(X0 )A Zm (E)\Zm (AE ) E Ym (A) u

f (A), m1 , ..., mr ≥ 2 (i.e. the representations τi (2) Assume that in case HA = Sp 4n ∗ are not characters of A ). Then the descent σψ,γ (¯ τ ) is a cuspidal representation ∗ τ ), lifts, at almost all places v, of Hm (A). Each irreducible summand of σψ,γ (¯ to the representation τv , as follows. f 2n (A)) lifts almost (i) Assume that HA = Sp4n (A). Then each summand (on Sp everywhere (with respect to ψv−1 ) to GL

(F )

v τv = IndPm2n,...,m 1

r (Fv )

τ1,v ⊗ · · · ⊗ τr,v .

f 4n (A). Then each summand (on Sp2n (A)) lifts almost (ii) Assume that HA = Sp everywhere to GL

(Fv ) τ1,v r ,1 (Fv )

τv = IndPm2n+1 ,...,m 1

⊗ · · · ⊗ τr,v ⊗ ωτ¯v .

(iii) Assume that HA = U4n (A). Then each summand (on U2n (A)) lifts almost everywhere (with respect to γv ) to GL

(E )

τv = γv−1 ⊗ IndPm2n,...,mv 1

r (Ev )

τ1,v ⊗ · · · ⊗ τr,v .

The cuspidality of σψ,γ (¯ τ ) follows from Theorem 7.11, and the assertion on the lifts, almost everywhere, follows from Theorem 6.4(1), (4) and Theorem 6.7. Note again that by our assumptions, the central characters ωτi are trivial on A∗ in case HA = Sp4n (A), U4n (A).

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f 4n (A) is not a case of Remark: The descent in the last theorem, when HA = Sp GL (A) functoriality, as it associates to the representation τ = IndPm2n,...,m (A) τ1 ⊗ · · · ⊗ τr r 1 of GL2n (A) (where the symmetric square L-functions of the τi all have poles at s = 1) irreducible, automorphic, cuspidal (and globally generic) representations σ ⊂ σψ,γ (¯ τ ) of Sp2n (A). However, we can use the descent to SO2n,α (A) (Theorem 9.2) to obtain irreducible, automorphic, cuspidal (and globally generic) representations σ ′ of SO2n,α (A), (ωτ¯ = χα ) which lift almost everywhere to τ . It is possible to prove that σ lifts by the theta correspondence to an automorphic, cuspidal (and globally generic) representation of SO2n,α (A), all of whose irreducible summands are nearly equivalent to σ ′ (and even have a nontrivial L2 -pairing with the descent σψ,α (¯ τ ) to SO2n,α (A)). 9.10

Nonvanishing of the Whittaker coefficient of the descent corresponding to Fourier-Jacobi coefficients: h(V ) = U4n+2

Assume that h(V ) = U4n+2 . We want to carry out a similar computation for the Whittaker coefficient in (9.69). Consider the following element κm in GLm (F ) (m = 2n + 1). Its action on the standard basis {e1 , ..., em } of Vm+ is as follows. κm (ei ) = e2i , 1≤i≤n κm (en+i ) = e2i−1 , 1 ≤ i ≤ n + 1

If we think of ei , 1 ≤ i ≤ m as column vectors in F m , then the columns of the matrix κm are κm = (e2 , e4 , ..., e2n , e1 , e3 , ..., e2n+1 ). As in (9.70), we have

Z

Wm,ψ,λ (f )(h) =

−1 f (vˆ κm h)ψB (v)dv, m ,λ

(9.74)

Bm (F )\Bm (A) ′ κ ˆ m Bm κ ˆ −1 m ,

where Bm = have the following form

where T has the form



′ ,λ (ˆ κm ). The elements of Bm and ψBm ,λ (v) = ψBm κ−1 m vˆ

 1 v1 .. .

T Y T∗



∈ h(V ),

u1 . . . un−1 T1,1 . . . T1,n−1

     vn−1 Tn−1,1 . . . Tn−1,n−1 0 0 ... 0

 un T1,n   ..  , .   Tn−1,n  I2

(9.75)

where the Tr,s are 2 × 2 upper triangular matrices, such that, for r > s, Tr,s are (upper) nilpotent, and Tr,r are (upper) unipotent. The column vectors v1 , ..., vi−1

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have zero in their second coordinate, and the row vectors u1 , ..., ui have zero in their first coordinate. Note that this is the same form as (9.35) (with i = n). The character ψBm ,λ is given by ψE,−1 ((u1 )2 + tr(T1,2 + T2,3 + · · · + Tn−1,n ) + Y2n,2 + λY2n+1,1 ).

(9.76)

Now, we proceed exactly as we did in the case (of SO4n , U4n ), starting from (9.35), until we reach the exact analog of Theorem 9.3, which we formulate as the next proposition, with the following (analogous) notation. Put Y2i−2,2j−1 = {(I2n+1 + te2i−2,2j−1 )∧ }, Y2i−2 =

i Q

2 ≤ i ≤ n; 1 ≤ j ≤ i,

Y2i−2,2j−1

j=1

Y′ =

n−1 Q

Y2i

i=1

′ Ym = κ ˆ−1 ˆm. m Y κ

Let

′ Sm

form

  T Y ∈ h(V ), where now T has the be the subgroup of elements v = T∗ 

 1 u ∗ ··· ∗ ∗  I x ∗   2 1    I2 ∗     ..   .  ∗     I2 xn−1  I2

;

u1 = 0

(9.77)

′ For v ∈ Sm (A), written as above, let

   0λ Y2n,1 Y2n,2 ′ ,λ (v) = ψE,−1 (u2 + tr(x1 + · · · + xn−1 ) + tr( ψSm )). (9.78) Y2n+1,1 Y¯2n,1 10

Proposition 9.6. Let HA = U4n+2 (A) and let f be a smooth automorphic function of uniform moderate growth on HA . Then Z Z f (vˆ κm yh)ψS−1 Wm,ψ,λ (f )(h) = ′ ,λ (v)dvdy, m

′ (F )\S ′ (A) Ym (A) Sm m

where the dy-integration converges absolutely as a repeated integration. Finally, Wm,ψ,λ (f ) 6= 0, if and only if there is h ∈ HA , such that Z f (vˆ κm h)ψS−1 ′ ,λ (v)dv 6= 0. m

′ (F )\S ′ (A) Sm m

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Now, we continue exactly as in Theorem 9.4. Denote Z ′ f (vh)ψS−1 f Sm ,λ (h) = ′ ,λ (v)dv. m

′ (F )\S ′ (A) Sm m

′ ′ ,λ . Thus, The group of all (I2n+1 + te1,2 )∧ normalizes Sm and, over A, preserves ψSm we can consider the Fourier expansion, along E\AE of the function ′

t 7→ f Sm ,λ ((I2n+1 + te1,2 )∧ h). The Fourier coefficient, corresponding to the character ψE,−1 (µt), µ ∈ E is Z ′ −1 f Sm ,λ ((I2n+1 + te1,2 )∧ h)ψE,−1 (µt)dt = E\AE

Z

˜

Sm ,λ,µ f (vh)ψS−1 (h), ˜ ,λ,µ (v)dv := f m

˜m (F )\S ˜m (A) S

where ′ S˜m = {v(I2n+1 + te1,2 )∧ | v ∈ Sm }, ∧ ′ ψS˜m ,λ,µ (v(I2n+1 + te1,2 ) ) = ψSm ,λ (v)ψE,−1 (µt).

Note that S˜m consists of the elements as in (9.77), without the restriction u1 = 0. The character ψS˜m ,λ,µ has the form as in (9.78), only that u2 is replaced by µu1 +u2 .   1 1 Let b = . Put −1 1 1 γ = diag(2, b, ..., b, b∗, ..., b∗ , ). 2 ˜

Choose λ = µ = −1. We will compute the coefficient f Sm ,−1,−1 , and show that it is nontrivial on Eτ¯ . For v ∈ S˜m (A), as above, we have ψS˜m ,−1,−1 (γ −1 vγ) = ψE,−1 (u2 + tr(x1 + · · · xn−1 ) − Y2n,1 ) := ψS˜m (v), ˜

˜

and so, denoting f Sm ,ψ = f Sm ,−1,−1 , we have Z ˜ f (vγh)ψS−1 f Sm ,ψ (h) = ˜ (v)dv. m

˜m (F )\S ˜m (A) S

From this point on, we continue as in the case of SO4n , starting at (9.42). We want ˜ to pass from S˜m to Sm and from f Sm to Jm,ψ (f ). Define, for 1 ≤ i ≤ n, Y2i−1,2i = {(Im + ae2i−1,2i )∧ } X2i,2i+1 = {(Im + be2i,2i+1 )∧ }

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  T Y in U4n+2 where T is of the form T∗  ··· ∗ ∗ ∗      ∗ ∗ ··· ∗    xi ∗ ∗ ,  zi+1 xi+1 ∗    .. . ∗   zn−1 xn−1 

Let C2i−1,2i be the set of the elements v = 

1 u ∗  I2 x1   ..  .   I2 xi−1   I2        

such that

zn

x1 , ..., xn−1 ∈ ResE/F M2×2 , (xi−1 )2,1 = · · · = (xn−1 )2,1 = −Y2n+1,1 ,   1∗ zi+1 , ..., zn−1 = . 01

Note that Y¯2n+1,1 = Y2n+1,1 . C2i−1,2i is an F -subgroup of h(V ), and ψC2i−1,2i (v) = ψE,−1 (u2 + tr(x1 + · · · + xn−1 ) − Y2n,1 ) defines a character of C2i−1,2i (A). Exactly as in the SO4n -case, we verify the requirements of Lemma 7.1, and we continue until we reach (9.46). Denote Yn′ = Y2n−1,2n Y2n−3,2n−2 · · · Y1,2 . We conclude, by a repeated application of Lemma 7.1 and Corollary 7.1 that Z Z Z ˜ (v)dvdtdy. (9.79) f (vin (t)yγh)ψS−1 f Sm ,ψ (h) = m ′ (A) F \A S (F )\S (A) Yn m m

˜

Moreover, f Sm ,ψ is nonzero, if and only if the Fourier coefficient Z Z (v)dvdt f (vin (t)h)ψS−1 m F \A Sm (F )\Sm (A)

is nonzero (as a function of h). Now, the inner dvdt integral of the r.h.s. of (9.79) is Z Jm,ψ (f )(in (t)yγh)dt = Jm,ψ (f )(yγh). F \A

We used Theorem 8.3. Hence, ˜

f Sm ,ψ (h) =

Z

′ (A) Yn

Jm,ψ (f )(yγh)dy,

(9.80)

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and f Sm ,ψ 6= 0, if and only if Jm,ψ (f ) 6= 0. By Theorem 8.1, we get that as f varies in the residual representation Eτ¯ , Jm,ψ (f )(1) is not identically zero, and we ˜ conclude that f Sm ,ψ (1) is not identically zero on Eτ¯ , and hence that the ψNHm ∗ ,−1 Whittaker coefficient of the descent σψ,γ (¯ τ ) is non-trivial. In particular, the descent σψ,γ (¯ τ ) is nontrivial and is globally ψNHm ∗ ,−1 - generic. Now we finish as in the U4n case of Theorem 9.4. We apply Theorem 7.11 and get the cuspidality of σψ,γ (¯ τ ), adding the assumption that m1 , ..., mr ≥ 2. Finally, we analyze the unramified parameters at almost all places of any given irreducible generic summand σ of σψ,γ (¯ τ ). Recall again that since the Asai L-functions of the representations τi all have a pole at s = 1, their central characters are trivial on A∗ . Let v be a place where Ev is a quadratic unramified extension of Fv , and τi,v are unramified. GL (Ev ) τ1,v ⊗ · · · τr,v and assume that τv is induced from the Let τv = IndPm2n+1 ,...,m (Ev ) 1

r

standard Borel subgroup and the unramified character µ1 ⊗ · · · µn ⊗ 1 ⊗ µ−1 n ⊗···⊗ µ−1 . (We also know that the representations τ are self-conjugate.) By Theorem i 1 U (F ) 6.4(6), σv is a constituent of the representation IndB2n+2 v γv−1 (µ1 ⊗ · · · ⊗ µn ⊗ 1). GL (Ev ) τ1,v ⊗ · · · τr,v ⊗ 1. Thus, the local lift of σv to GL2n+2 (Ev ) is γv−1 ⊗ IndPm2n+2 1 ,...,mr ,1 (Ev ) Assume that v is a place where Ev = Fv ⊕ Fv , and the representations τi,v of GLmi (Ev ) are unramified. As in the proof of the U4n -case in Theorem 9.4, the 1 representation τv | det ·| 2 of GL2n+1 (Ev ) = GL2n+1 (Fv ) × GL2n+1 (Fv ) becomes a 1 1 representation of the form τ ′ | det ·| 2 ⊗τ ′ | det ·|− 2 , where τ ′ is an unramified (generic) representation of GL2n+1 (Fv ). Again, we used the fact that the representations τi are self conjugate. The character γv of Ev∗ = Fv∗ × Fv∗ becomes of the form γ1 × γ1−1 . Assume that τ ′ is induced from the Borel subgroup from the unramified character µ1 ⊗ · · · ⊗ µ2n+1 . By Theorem 6.8, σv , which is now an irreducible, unramified, generic representation of GL2n+2 (Fv ), is a constituent of a certain direct sum of representations. Looking at this sum, we see that all its summands, except the last one are not generic. Thus σv is a constituent of the last summand in Theorem 6.8, GL (F ) which is indB ′ 2n+2 v γ1−1 µ1 ⊗ · · · ⊗ γ1−1 µ2n+1 . (This is the place where we use the genericity of σ.) Since σv is irreducible and unramified, we conclude that there is an unramified character ǫ′v of Fv∗ , such that σv is the unramified constituent of the representation of GL2n+2 (Fv ), induced from the Borel subgroup and the character γ1−1 µ1 ⊗ · · · ⊗ γ1−1 µ2n+1 ⊗ γ1−1 ǫ′v . It is easy to see that ǫ′v = ωσv γ12n+2 ωτ−1 ′ . Since the central characters of the representations τi are trivial on A∗ , there is an automorphic character η of U1 (A), such that ωτ (x) = η( xx¯ ), for x ∈ A∗E . For the split places v as above, ηv = ωτ ′ and hence ǫ′v = ωσv γ12n+2 ηv−1 . Thus, ǫ′ = ωσ γ 2n+2 η −1 is an automorphic character of U1 (A). Let ǫ be the automorphic character of A∗E given by ǫ(x) = ǫ′ ( xx¯ ). Then, at almost all places v of F , σv lifts to the representation GL (Ev ) τ1 ⊗· · ·⊗τr ⊗ǫ. We summarize this in the following theorem. γv−1 ⊗IndPm2n+2 1 ,...,mr ,1 (Ev ) The remarks similar to those which follow Theorem 9.4 are applicable here, as well.

Theorem 9.8. Let h(V ) = U4n+2 (the form b being anti-Hermitian).

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(1) The descent σψ,γ (¯ τ ) is non-trivial and globally generic (on U2n+2 (A), respectively) with respect to the Whittaker character (9.66) ψNHm ∗ ,−1 . (2) Assume that m1 , ..., mr ≥ 2 (i.e. the representations τi are not characters of A∗ ). Then the descent σψ,γ (¯ τ ) is a cuspidal representation of U2n+2 (A). Let σ be an irreducible, globally generic summand of σψ,γ (¯ τ ). Then there is an ∗ automorphic character ǫ of AE , lifted from U1 (A), such that σ lifts locally, at almost all places v, (with respect to γv ) to the representation GL

(Ev ) τ1,v r ,1 (Ev )

γv−1 ⊗ IndPm2n+2 ,...,m 1

⊗ · · · ⊗ τr,v ⊗ ǫv .

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Chapter 10

Global genericity of the descent and global integrals of Rankin-Selberg type and of Shimura type The main result proved in this chapter is that all irreducible summands of the descent (σψ,α (¯ τ ) or σψ,γ (¯ τ )) are globally generic representations of GA , where GA ∗ ∗ is one of the groups Hα,m (A) or Hm (A); see (3.40), (3.41). In the previous chapter, we proved that an appropriate Whittaker Fourier coefficient is nontrivial on the f 4n+2 (A), SO4n+1 (A). Thus, in these cases, we already descent, in case HA 6= Sp know that at least one irreducible summand of the descent is globally generic. We do not know, in general whether the descent is irreducible or not, except in the cases HA = Sp4n (A), SO4n (A) (see [Jiang and Soudry (2007)], [Ginzburg, Jiang, and Soudry (2011)]), where we know that the descent is irreducible, and so, in these cases, the main result in this chapter already follows, since then the descent has just one irreducible summand. In case HA = SO4n+1 (A), (5.42) indicates that the descent is probably reducible. The proof that each irreducible summand of the descent is globally generic is achieved by viewing the L2 -product between cusp forms in the descent and cusp forms which lie in an irreducible summand of the descent as residues of the similar pairing where we replace the descent by the corresponding coefficient (Gelfand-Graev or Fourier-Jacobi) of the Eisenstein series E(h, fτ¯,¯s ) (see Chapter 2). Thus, global integrals of Rankin-Selberg type or of Shimura type come up “naturally”. The given datum for such integrals is a pair of irreducible, automorphic representations σ, τ of GA and GLm (AE ), respectively; σ is cuspidal and τ may be cuspidal or realized by Eisenstein series at a point of holomorphy. We reviewed these integrals in Sec. 3.4. In this chapter we show that if such integrals are not identically zero, then σ must be globally generic. In fact, we show that the global integrals unfold to Eulerian integrals, that is integrals, which can be written as a product of local “similar” integrals for decomposable data. The integrand in these Eulerian integrals contains, as a factor, a Whittaker coefficient of cusp forms in σ. Thus, the main work of this chapter is to introduce these global integrals for G×ResE/F GLm and show their unfolding to Eulerian integrals. These integrals were considered by many authors for the various cases. See [Jacquet and Shalika I (1981)], [Jacquet and Shalika II (1981)], [Jacquet Piatetski-Shapiro and Shalika (1983)], [Gelbart and Piatetski-Shapiro (1987)], [Ginzburg (1990)], [Soudry (1993)], [Tamir (1991)], [Ginzburg, Rallis and Soudry (1998)], [Watanabe (2000)],[Ben-Artzi and 281

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Soudry (2009)], [Ben-Artzi and Soudry (2011)], [Kaplan (2011)]. In these works, it is also proved that the integrals represent the standard partial L-function LS (σ ×τ, s). We will not show this, as this is beyond the scope of this book. Finally, in life, our journey really went in the opposite direction; it is through our work on the global integrals that we reached the idea of the descent construction. In the last two sections of this chapter we consider similar global integrals, with the roles of cuspidal representations and Eisenstein series reversed. The results (already reviewed in Sec. 3.5) are entirely similar and appear in the references cited above. 10.1

Statement of the theorems

Consider an irreducible summand π of the descent σψ,α (¯ τ ) or σψ,γ (¯ τ ), and let π be realized in a given irreducible subspace Vπ . The descent is an automorphic, cuspidal ∗ ∗ representation of Hα,m (A), or Hm (A), respectively (see (3.40), (3.41)). Let π ¯ be the representation by right translations in the space Vπ¯ of all complex conjugates of the cusp forms in Vπ . Then there are ξπ¯ ∈ Vπ¯ and f in the (space of the) descent, such that Z ξπ¯ (g)f (g)dg 6= 0. (10.1) G′F \G′A

∗ Here G′ = Hα,m , in case (V, b) is quadratic or Hermitian, and G′ = h(Wm,ℓm +1 ), ∗ , in case (V, b) is symplectic or anti-Hermitian. Note that in the last case, G′ = Hm ′ ∗ except when HA is symplectic, where Hm (A) is the double cover of GA . See (3.19), (3.23). The l.h.s. of (10.1) is the L2 -pairing between elements of the descent and elements of π and hence, it is not identically zero. Since the elements f of the descent are Gelfand-Graev, or Fourier-Jacobi coefficients applied to the elements of the residual representation Eτ¯ , we get that the following meromorphic functions have a pole at s¯ = (1, ..., 1), in the sense of Theorem 2.1, and in particular are nontrivial (as data vary). Case of Gelfand-Graev coefficients: Z

L(ξπ¯ , fτ¯,¯s ) =

ξπ¯ (g)E ψℓm ,α (g, fτ¯,¯s )dg;

G′F \G′A

Case of Fourier-Jacobi coefficients: L(ξπ¯ , fτ¯,¯s , φ) =

Z

ξπ¯ (g)E ψℓm ;γ,φ (g, fτ¯,¯s )dg.

G′F \G′A

We will show that the non-triviality of these functions forces π ¯ and hence π to be globally generic. This will prove that all irreducible summands of the descent are globally generic. Let us state this as a theorem, for the record. Theorem 10.1. Each irreducible summand of the descent σψ,α (¯ τ ), or σψ,γ (¯ τ ) is ∗ ∗ a globally generic (automorphic, cuspidal) representation of Hα,m (A), or Hm (A), respectively.

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In the course of proof, we will also specify a Whittaker character, ψ ′ , with respect to which these summands are globally generic. This theorem has a very nice corollary. Theorem 10.2. The descent σψ,α (¯ τ ), or σψ,γ (¯ τ ) is a multiplicity free representa∗ ∗ tion of Hα,m (A), or Hm (A), respectively. Proof. Let π1 and π2 be two isomorphic, irreducible subrepresentations of the descent, acting in subspaces Vπ1 , Vπ2 , respectively. These are automorphic cuspidal representations. Note that the descent, being cuspidal, decomposes into a direct sum of irreducible subspaces. By Theorem 10.1, we know that π1 and π2 are globally generic with respect to ψ ′ . By uniqueness up to scalars of ψ ′ -Whittaker functionals (at each place), we can choose an isomorphism of representations T : Vπ1 7→ Vπ2 , such that T (ϕ) − ϕ has a zero ψ ′ -Whittaker coefficient, identically over all cusp forms ϕ in Vπ1 . This forces T to be the identity map, since, otherwise, the space V ′ = {T (ϕ) − ϕ | ϕ ∈ Vπ1 }

is a nontrivial subspace of the descent, invariant to right translations, and affords an irreducible subrepresentation π ′ (isomorphic to π1 ) of the descent. By construction, the ψ ′ -Whittaker coefficient is zero on V ′ , that is, π ′ is not globally ψ ′ -generic. This contradicts Theorem 10.1, and hence T (ϕ) − ϕ = 0, for all ϕ ∈ Vσ1 . This means  that Vπ1 = Vπ2 . For the proof of Theorem 10.1, we will place ourselves in a more general set-up; we replace ℓm by any 0 ≤ ℓ ≤ m, and take τ1 , ..., τr to be any irreducible, automorphic, cuspidal representations of GLm1 (AE ), ..., GLmr (AE ), respectively. (We allow mi = 1 and now we make no assumptions on poles at s = 1 of symmetric square, exterior square or Asai L-functions.) Assumption 2.1 on (V, b) is still enforced. Finally, we let π be any irreducible, automorphic, cuspidal representation ˆ ℓ , respectively. See (3.10), (3.13). We of the group GA , where now, G = Lℓ,yα , H will denote σ = π ¯ . Note that according to the case at hand ℓ = m or ℓ = m − 1 corresponds to a Whittaker coefficient of E(g, fτ¯,¯s ), and then the integrals above are the starting point of the Langlands-Shahidi method. Although what follows takes care of this case as well, it will be convenient to assume that ℓ < m, and even more we assume that G is nontrivial so that we can consider the representations σ on GA . It will also be convenient to view E(g, fτ¯,¯s ) as an Eisenstein series corresponding to parabolic induction from Qm . Assume that HA is not metaplectic. For s¯ = (s1 , ..., sr ), re-denote sr = s, si = s′i + s (1 ≤ i < r) and let τ denote the automorphic representation of GLm (AE ) realized by the Eisenstein series corresponding to GL (A )

E IndPmm,...,m 1

s′1 , ..., s′r−1 ,

r (AE )

s′

s′

τ1 | det ·|E1 ⊗ · · · ⊗ τr−1 | det ·|Er−1 ⊗ τr .

(10.2)

We fix where the Eisenstein series (on GLm (AE )) is holomorphic, and suppress them from our notation. (Note that for the descent set-up, the point of interest is s′1 = · · · = s′r−1 = 0, and then since we assume that the τi are selfconjugate and pairwise inequivalent, the Eisenstein series above on GLm (AE ) is

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indeed holomorphic at this point.) When HA is metaplectic, we modify τ to be a f m (A) as in the remark in the end of Sec. 2.6. Consider now, representation of GL when HA is not metaplectic, s− 12

A ρτ,s = IndH Qm (A) τ | det ·|E

,

and for HA metaplectic, s− 12

A ρτ,s = IndH Qm (A) µψ τ | det ·|E

.

Here, we still denote by Qm (A) the inverse image in the metaplectic group of the Siegel parabolic subgroup of h(V )A = Sp(V )A . We now state the theorems. Theorem 10.3. Assume that (V, b) is either quadratic or even Hermitian. Let 0 ≤ ℓ < m be an integer and α ∈ F ∗ . Denote G = Lℓ,α (see (3.10)). Let σ be an irreducible, automorphic, cuspidal representation of GA , and let τ be an automorphic representation of GLm (AE ), of the form above. Consider the meromorphic functions Z ϕσ (g)E ψℓ,α (g, fτ,s )dg, (10.3) L(ϕσ , fτ,s ) = GF \GA

as ϕσ varies in the space of σ and fτ,s varies in the space of smooth holomorphic sections in ρτ,s . Assume that L(ϕσ , fτ,s ) is not identically zero (as a meromorphic function and as data vary). Then σ is globally generic, with respect to a certain “standard” Whittaker character (to be specified in the proof ). Moreover, for Re(s) large enough, we have an identity of the form Z Z −1 Zm ,ψ ψ (βℓ,α ug)ψℓ,α (u)dudg, fτ,s Wϕσ (g) L(ϕσ , fτ,s ) = NG (A)\GA

−1 Nℓ (A)∩βℓ,α Qm (A)βℓ,α \Nℓ (A)

(10.4) where Wϕψσ is the corresponding Whittaker function of ϕσ , NG is a “standard” maximal unipotent subgroup of G, βℓ,α is the product of a certain Weyl element and a certain (rational) diagonal element in h(V ); the super index (Zm , ψ) marks the application to fτ,s of the Whittaker coefficient along Zm (E)\Zm (AE ) with respect −1 −1 to the character ψZ (z1,2 + · · · + zm−1,m ), (z) = ψE,1 m Z Zm ,ψ fτ,s (ˆ z h)ψZm (z)dz. fτ,s (h) = Zm (E)\Zm (AE )

Theorem 10.4. Assume that (V, b) is either symplectic or even anti-Hermitian. Let ˆ ℓ (A) (see (3.12)). 0 ≤ ℓ < m− 1 be an integer. Denote G′ = h(Wm,ℓ+1 ) and GA = H Let σ be an irreducible, automorphic, cuspidal representation of GA , assumed to be genuine, when GA is metaplectic. Let τ be an automorphic representation of GLm (AE ), of the form above. Consider the meromorphic functions (see (3.20)) Z ϕσ (g)F Jψφℓ ,γ (E(g, fτ,s ))dg, (10.5) L(ϕσ , fτ,s , φ) = G′F \G′A

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as ϕσ varies in the space of σ, fτ,s varies in the space of smooth holomorphic + sections in ρτ,s , and φ varies in the Schwartz space of Vℓ+1,m−ℓ−1 (A) (see (5.3)). Assume that L(ϕσ , fτ,s , φ) is not identically zero (as a meromorphic function and as data vary). Then σ is globally generic, with respect to the “standard” Whittaker character determined by ψ (to be specified in the proof ). Moreover, for Re(s) large enough, we have an identity of the form L(ϕσ , fτ,s , φ) = Z Z Zm ,ψ (βℓ ug)ψℓ−1 (u)ωψ−1 ,γ −1 (j(u)g)φ(em )dudg, Wϕψσ (g) fτ,s

(10.6)

NG (A)\G′A

where the du-integration is carried over Nℓ+1 (A) ∩ βℓ−1 Nℓ+1 (A)βℓ \Nℓ+1 (A); Wϕψσ is the corresponding Whittaker function of ϕσ ; NG is a “standard” maximal unipotent (m−ℓ−1) subgroup of G; βℓ is a certain Weyl element in h(V ); ωψ−1 ,γ −1 = ωψ−1 ,γ −1 is f m,ℓ+1 )A , in the Weil representation of the semidirect product of HW and Sp(W m,ℓ+1

case h(V ) is symplectic, or h(Wm,ℓ+1 )A , in case h(V ) is unitary. In (10.6), we used, for simplicity, the same letter g to denote in the symplectic cases, both an element of a metaplectic group and its projection to the symplectic group. Finally, the super index (Zm , ψ) marks the application to fτ,s of the Whittaker coefficient along Zm (E)\Zm (AE ) with respect to the character

ψE,−1 (z1,2 +· · ·+zm−ℓ−2,m−ℓ−1 −2zm−ℓ−1,m−ℓ −zm−ℓ,m−ℓ+1 −· · ·−zm−1,m ). (10.7) 10.2

Proof of Theorem 10.3

The proof is an analog of the study carried out in Chapter 5 of the Jacquet module s− 12 with respect to (Nℓ , ψℓ,α ) of IndH at one finite place. The integral Qm τ | det ·|E (10.3) converges absolutely, whenever E(g, fτ,s ) is holomorphic, due to the rapid decrease of ϕσ and the moderate growth of E(g, fτ,s ). For Re(s) large enough, we may write E(h, fτ,s ) as an absolutely convergent series, and then, for h ∈ HA , Z X −1 fτ,s (δuh)ψℓ,α (u)du. (10.8) E ψℓ,α (h, fτ,s ) = Nℓ (F )\Nℓ (A) δ∈Qm (F )\h(V )

Let us factor the inner sum through Qm \h(V )/Qℓ . We already wrote down a set of representatives in (4.14), (4.15) (with j = m and ℓ + s − r = m). These are  ∧   Iℓ−r Ir  , Ir )ω ℓ−r ; 0 ≤ r ≤ ℓ ǫr =  Im′ −2ℓ 0 Im−ℓ  diag(Ir ,  b Iℓ−r Iℓ−r 0

(Recall that m′ = dimE V .) Thus, Z ℓ X E ψℓ,α (h, fτ,s ) =

X

(r) r=0 Nℓ (F )\Nℓ (A) δ∈QF \Qℓ (F )

−1 fτ,s (ǫr δuh)ψℓ,α (u)du,

(10.9)

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where Q(r) = Qℓ ∩ǫ−1 r Qm ǫr . This subgroup is described in (4.19), (4.22). From that description, we can write Q(r) = M (r) ⋉ U (r) , where M (r) , U (r) can be read from ℓ−r (4.19), (4.22). For example, (M (r) )ωb is the subgroup consisting of the following elements     du v  ∗ ′ ax  c x ′  diag( , ) ∈ h(V ), e u , c a∗ d∗

where a ∈ ResE/F GLr , c ∈ ResE/F GLℓ−r , d ∈ ResE/F GLm−ℓ . If m′ = 2m is even, omit the row and column of e, and otherwise, e = 1. Now factor the inner sum in (10.9) through Q(r) \Qℓ /Rℓ,α , where Rℓ,α = Lℓ,α Nℓ = GNℓ . By (5.2) and Proposition 4.4, the representatives for these double cosets can be taken to be of the form η = diag(ǫ, γ, ǫ∗ ), where ǫ is a representative in the quotient of Weyl groups WGLr ×GLℓ−r \WGLℓ , and γ = Im′ −2ℓ , unless h(V ) is odd orthogonal and α = t2 is a square, in which case γ can also be γ± , where   1   γ± = diag(Im−ℓ−1 ,  ±t 1  , Im−ℓ−1 ). t2 − 2 ∓t 1

See (4.33). Now, we rewrite (10.9) as E ψℓ,α (h, fτ,s ) = ℓ X

Z

X

X

−1 fτ,s (ǫr ηδuh)ψℓ,α (u)du.

(r) r=0 η∈Q(r) \Q (F )/R (F ) Nℓ (F )\Nℓ (A) δ∈Rℓ,α (F )∩η −1 QF η\Rℓ,α (F ) ℓ ℓ,α F

(10.10)

Since Rℓ,α ∩ η −1 Q(r) η = (G ∩ η −1 M (r) η)(Nℓ ∩ η −1 Q(r) η), we have a natural decomposition Rℓ,α ∩ η −1 Q(r) η\Rℓ,α ∼ = G ∩ η −1 M (r) η\G × Nℓ ∩ η −1 Q(r) η\Nℓ . We can write the du-integration in (10.10), for fixed r, η, as Z X X −1 fτ,s (ǫr ηδ1 δ2 uh)ψℓ,α (u)du, Nℓ (F )\Nℓ (A)

(r)

δ1 ∈Nℓ (F )∩η −1 QF η\Nℓ (F )

(r)

where the first summation is over δ2 ∈ GF ∩ η −1 MF η\GF . We can switch this sum with the du-integration and get Z X −1 fτ,s (ǫr ηuδh)ψℓ,α (u)du. (10.11) (r)

δ∈GF ∩η −1 MF η\GF

(r)

Nℓ (F )∩η −1 QF η\Nℓ (A)

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Here we used the absolute convergence for Re(s) large enough (that is when we −1 replace fτ,s (ǫr ηδ1 δ2 uh)ψℓ,α (u) by |fτ,s (ǫr ηδ1 δ2 uh)|). For fixed δ in the last sum, we factor the du-integration as Z Z −1 ′ fτ,s (ǫr ηu′ uδh)ψℓ,α (u u)du′ du. (r)

(r)

Nℓ (A)∩η −1 QA η\Nℓ (A)

(r)

Nℓ (F )∩η −1 QF η\Nℓ (A)∩η −1 QA η

(10.12) By the proof of Proposition 5.1, if r > 0, then there is an F -subgroup J ⊂ Nℓ ∩ η −1 Q(r) η, such that ψℓ,α is nontrivial on JA and ǫr ηJη −1 ǫ−1 r ⊂ Um (the unipotent radical of Qm ). Thus, (10.12) is zero for all r > 0, and in (10.10) only r = 0 contributes. Then η has the form diag(Iℓ , γ, Iℓ ), for γ as above. Assume that if h(V ) is odd orthogonal, then α is not a square. In this case, η = I and (10.10) becomes Z X −1 E ψℓ,α (h, fτ,s ) = fτ,s (ǫ0 uδh)ψℓ,α (u)du. (10.13) (0)

δ∈GF ∩MF \GF

(0)

Nℓ (F )∩QF \Nℓ (A)

Substitute (10.13) in (10.3). We get that for Re(s) ≫ 0 Z Z −1 fτ,s (ǫ0 ug)ψℓ,α (u)dudg. ϕσ (g) L(ϕσ , fτ,s ) =

(10.14)

(0) Nℓ (F )∩QF \Nℓ (A)

(0) GF ∩MF \GA

(0) It is straightforward to verify that the elements ǫ0 uǫ−1 form 0 , where u ∈ Nℓ ∩ Q ′ ∧ the subgroup (ResE/F Zℓ ) , where     Im−ℓ x ′ Zℓ = z = ∈ Zm , ζ

and that, for z ∈ Zℓ′ (AE ), written as above,

′ α −1 −1 ψℓ,α (ǫ0 zˆǫ0 ) = ψE,1 (ζ1,2 + · · · ζℓ−1,ℓ + (−1)m +1 xm−ℓ,1 ). 2 Let ψZℓ′ ,α be the following character of Zℓ′ (AE ): ′ α ψZℓ′ ,α (z) = ψE,1 ((−1)m +1 zm−ℓ,m−ℓ+1 + zm−ℓ+1,m−ℓ+2 + · · · + zm−1,m ). 2 Denote Z

Z ′ ,ψ,α

fτ,sℓ

fτ,s (ˆ z h)ψZℓ′ ,α (z)dz.

(h) =

Zℓ′ (E)\Zℓ′ (AE )

Then

Z

(0)

′

−1 ′ fτ,s (ǫ0 u′ ug)ψℓ,α (u )du′ = f Zℓ ,ψ,α (ǫ0 ug). (0)

Nℓ (F )∩QF \Nℓ (A)∩QA

(0)

Thus, when we factor the du-integration in (10.14) through Nℓ (A) ∩ QA , we get Z Z Z ′ ,ψ,α −1 ϕσ (g) fτ,sℓ (ǫ0 ug)ψℓ,α (u)dudg. (10.15) L(ϕσ , fτ,s ) = (0)

GF ∩MF \GA

(0)

Nℓ (A)∩QA \Nℓ (A)

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In (5.16), we denoted G ∩ M (0) by Q′m−ℓ . This is very close to the parabolic + subgroup QG,m−ℓ , which preserves the isotropic subspace Vℓ,m−ℓ ∩ yα⊥ of Wm,ℓ ∩ yα⊥ . ′ Let QG,m−ℓ = MG,m−ℓ ⋉ UG,m−ℓ be the Levi decomposition. Let Mm−ℓ be the subgroup of the g ∈ MG,m−ℓ , which are trivial on the subspace spanned by y−α ′ and V0 . Then Q′m−ℓ = Mm−ℓ UG,m−ℓ . Note that when h(V ) is even orthogonal, ′ Mm−ℓ = MG,m−ℓ , and when h(V ) is even unitary, or odd orthogonal, MG,m−ℓ = ′ Mm−ℓ × h(Span{y−α , V0 }). In coordinates, when dimE V = 2m,     d v α2 v u        1 0 2 v′  ′ α  , I ) ∈ h(V ) , (10.16) Qm−ℓ = q(d, v, u) = diag(Iℓ ,  ℓ    1 v′      d∗ and when dimF V = 2m + 1,     d v x − α2 v u         1 0 0 − 2 v′     α   ′ ′  , Iℓ ) ∈ h(V ) . Qm−ℓ = q(d, v, x, u) = diag(Iℓ ,  1 0 x          1 v′      ∗ d

(10.17)

In both cases,

′ Mm−ℓ

Let

  ∧    Iℓ  =  d  d ∈ ResE/F GLm−ℓ−1 .     1

CG,m−ℓ = {u ∈ UG,m−ℓ | uem = em }. Note that ǫ0 CG,m−ℓ ǫ−1 0 ⊂ Um . Since CG,m−ℓ normalizes Nℓ , and (commutes with) Nℓ ∩ Q(0) , and, of course preserves ψℓ,α , over A, we get from (10.15) L(ϕσ , fτ,s ) =

Z

Q′m−ℓ (F )CG,m−ℓ (A)\GA

C

ϕσ G,m−ℓ (g)

Z

Z ′ ,ψ,α

fτ,sℓ

−1 (ǫ0 ug)ψℓ,α (u)dudg,

(0)

Nℓ (A)∩QA \Nℓ (A)

(10.18) where is the constant term of ϕσ along CG,m−ℓ . Now, we use Shalika’s C expansion [Shalika, J. (1974)] to conclude that ϕσ G,m−ℓ is a sum of Whittaker ′ functions of ϕσ . Indeed, CG,m−ℓ \Qm−ℓ is isomorphic to the mirabolic subgroup of ResE/F GLm−ℓ    dv 1 Pm−ℓ = ∈ ResE/F GLm−ℓ , 1 C ϕσ G,m−ℓ

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    dv dv , or CG,m−ℓ q(d, v, x, u) 7→ , respectively 1 1 (using the notation of (10.16), (10.17)). Consider, for fixed g, the function C 1 q 7→ ϕσ G,m−ℓ (qg) on CG,m−ℓ (A)\Q′m−ℓ (A), identified with Pm−ℓ (A). This is a 1 cuspidal function on Pm−ℓ (A). The Shalika expansion, which converges absolutely and uniformly in g varying in compact subsets, tells us now that by CG,m−ℓ q(d, v, u) 7→

 ∧  Iℓ   C Wϕψσ  d  g  ϕσ G,m−ℓ (g) = d∈Zm−ℓ−1 (E)\GLm−ℓ−1 (E) 1 X

X

Wϕψσ (qg),

(10.19)

q∈NG (F )\Q′m−ℓ (F )

where Wϕψσ is the Whittaker function of ϕσ , Wϕψσ (g) =

Z

−1 ϕσ (vg)ψN (v)dv, G

NG (F )\NG (A)

with respect to the standard Whittaker character ψNG of the Adele points of the maximal unipotent subgroup NG of G, as follows (using the notation of (10.16), (10.17)). When dimE V = 2m, ψNG (q(z, v, u)) = ψE,1 (z1,2 + · · · zm−ℓ−2,m−ℓ−1 + vm−ℓ−1 ), and when dimF V = 2m + 1, ψNG (q(z, v, x, u)) = ψ(z1,2 + · · · zm−ℓ−2,m−ℓ−1 + vm−ℓ−1 ). Here z ∈ Zm−ℓ−1 (AE ). From (10.18), (10.19), we conclude L(ϕσ , fτ,s ) =

Z

NG (F )CG,m−ℓ (A)\GA

Wϕψσ (g)

Z

Z ′ ,ψ,α

fτ,sℓ

−1 (ǫ0 ug)ψℓ,α (u)dudg.

(0)

Nℓ (A)∩QA \Nℓ (A)

(10.20) Finally, factor the dg-integration in (10.20), through NG (F )\NG (A). Note that  ∧ zv −1 both ǫ0 q(z, v, u)ǫ−1 lie in  1  · Um , except when h(V ) is 0 , ǫ0 q(z, v, x, u)ǫ0 Iℓ  2 ∧ z αv  · Um . ∈  1 even orthogonal and ℓ is odd, in which case ǫ0 q(z, v, u)ǫ−1 0 Iℓ

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Thus,

Z

L(ϕσ , fτ,s ) =

Z

Wϕψσ (g)

ψNG (u′ )

NG (F )CG,m−ℓ (A)\NG (A)

NG (A)\GA

Z

Zℓ′ ,ψ,α

fτ,s

−1 (ǫ0 u′ ug)ψℓ,α (u)dudu′ dg

(0)

Nℓ (A)∩QA \Nℓ (A)

Z

=

Z

Wϕψσ (g)

−1 Zm ,ψ,α fτ,s (ǫ0 ug)ψℓ,α (u)dudg,

(0)

NG (A)\GA

Nℓ (A)∩QA \Nℓ (A)

where

Z

Zm ,ψ,α fτ,s (h) =

(10.21)

fτ,s (ˆ z h)ψZm ,α (z)dz,

Zm (E)\Zm (AE )

the application of the Whittaker coefficient along the standard maximal unipotent subgroup of the Levi part of Qm , with respect to the following character. Except when h(V ) is even orthogonal and ℓ is odd, ψZm ,α (z) is given by ′ α ψE,1 (z1,2 +· · ·+zm−ℓ−1,m−ℓ +(−1)m +1 zm−ℓ,m−ℓ+1 +zm−ℓ+1,m−ℓ+2 +· · ·+zm−1,m ), 2 (10.22) and when h(V ) is even orthogonal and ℓ is odd, ψZm ,α (z) is given by α (10.23) ψE,1 (z1,2 + · · · + zm−ℓ−2,m−ℓ−1 + (zm−ℓ−1,m−ℓ − zm−ℓ,m−ℓ+1))· 2 ψE,1 (zm−ℓ+1,m−ℓ+2 + · · · + zm−1,m ). Fix dα , a diagonal element in GLm (F ), such that ψZm ,α (dα zd−1 α ) = ψZm (z) = ψE,1 (z1,2 + · · · + zm−1,m ). For example, in case of (10.22), ′ 2 dα = diag((−1)m +1 Im−ℓ , Iℓ ), α

and in case of (10.23) 2 2 dα = diag(−( )2 Im−ℓ−1 , − , Iℓ ). α α Denote Zm ,ψ fτ,s (h) =

Z

fτ,s (ˆ z h)ψZm (z)dz.

Zm (E)\Zm (AE )

Then Zm ,ψ,α Zm ,ψ ˆ fτ,s (h) = fτ,s (dα h).

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Denote now βℓ,α = dˆα ǫ0 . Note that −1 Nℓ ∩ Q(0) = Nℓ ∩ Qℓ ∩ ǫ−1 0 Qm ǫ0 = Nℓ ∩ βℓ,α Qm βℓ,α .

Now (10.21) reads as Z Wϕψσ (g) L(ϕσ , fτ,s ) =

Z

−1 Zm ,ψ fτ,s (βℓ,α ug)ψℓ,α (u)dudg.

−1 Nℓ (A)∩βℓ,α Qm (A)βℓ,α \Nℓ (A)

NG (A)\GA

(10.24) This proves (10.4), except when h(V ) is odd orthogonal and α = t2 is a square. Assume now that this is the case. Then we have already seen that in (10.10) only r = 0 contributes, and then η is one of the three elements diag(Iℓ , γ, Iℓ ), where γ = I2(m−ℓ)+1 , γ+ , γ− . Re-denote these elements by η1 , η2 , η3 , respectively. Then (10.10) becomes Z 3 X X −1 fτ,s (ǫ0 ηi δuh)ψℓ,α (u)du. E ψℓ,α (h, fτ,s ) = i=1

(0)

−1 Nℓ (F )\Nℓ (A) δ∈Rℓ,α (F )∩ηi QF ηi \Rℓ,α (F )

(10.25) The contribution of η1 = I to (10.25) is exactly as the r.h.s. of (10.13), and then the contribution of η1 to L(ϕσ , fτ,s ) is exactly as the r.h.s. of (10.24). Thus, it is enough to show that the contributions of η2 , η3 to L(ϕσ , fτ,s ) are zero. Let η be one of the elements η2 , η3 . We continue as in (10.11), with r = 0. Since Nℓ ∩ η −1 Q(0) η = η −1 (Nℓ ∩ Q(0) )η, we have

Z

−1 fτ,s (ǫ0 ηuh)ψℓ,α (u)du = (0)

Nℓ (F )∩η −1 QF η\Nℓ (A)

Z

Z

(0)

(0)

−1 −1 fτ,s (ǫ0 vηuh)ψℓ,α (η vηu)dvdu. (0)

Nℓ (A)∩η −1 QA η\Nℓ (A) Nℓ (F )∩QF \Nℓ (A)∩QA

The group Nℓ ∩ Q(0) consists of all elements in h(V ) = SO2m+1 of the form   z 0 0 y 0  Im−ℓ 0 0 y ′    v= 1 0 0   , z ∈ Zℓ  Im−ℓ 0  z∗

(0)

and, for v ∈ Nℓ (A) ∩ QA ,

ψℓ,α (η −1 vη) = ψ(z1,2 + · · · + zℓ−1,ℓ ).

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Note that this is independent of y. Now we continue as we did right after (10.14). (0) The elements ǫ0 vǫ−1 form the subgroup Zℓ′ , and for 0 , for v ∈ Nℓ ∩ Q  ∧ Im−ℓ x −1 z = ǫ0 vǫ0 = , ζ we have ψℓ,α (η −1 vη) = ψ −1 (ζ1,2 + · · · + zℓ−1,ℓ ) = ψ −1 (zm−ℓ+1,m−ℓ+2 + · · · + zm−1,m ). 0 Denote this character of Zℓ′ by ψZ ′ , and let ℓ

0 ψZ ′ ℓ

fτ,s (h) =

Z

0 fτ,s (ˆ z h)ψZ ′ (z)dz. ℓ

Zℓ′ (E)\Zℓ′ (AE )

Then

Z

(0)

−1 fτ,s (ǫ0 vηug)ψℓ,α (v)dv = f

0 ψZ ′

ℓ

(ǫ0 ηug).

(0)

Nℓ (F )∩QF \Nℓ (A)∩QA

Note that this Fourier coefficient contains as inner integration a constant term integration along the unipotent radical of the standard parabolic subgroup of GLm , which corresponds to (m − ℓ, ℓ), and so, if τ is cuspidal, then this Fourier coefficient is identically zero. However, we allow τ to be non-cuspidal (when r > 1). We conclude that Z −1 fτ,s (ǫ0 ηug)ψℓ,α (u)du = (0)

Nℓ (F )∩η −1 QF η\Nℓ (A)

Z

f

0 ψZ ′

ℓ

−1 (ǫ0 ηug)ψℓ,α (u)du.

(10.26)

(0) Nℓ (A)∩η −1 QA η\Nℓ (A)

Next, G ∩ η −1 M (0) η = η −1 (Lℓ,ηyα ∩ M (0) )η. This is the group denoted Q′m−ℓ,η in Chapter 5 (right after (5.25)). There we described this group.     d u(d) v   Lℓ,η± yα ∩ M (0) = diag(Iℓ ,  1 u′ (d) , Iℓ ) ∈ SO2m+1 ,   d∗ where for d ∈ GLm−ℓ ,



d1,m−ℓ .. .



    u(d) = ∓t−1  .  dm−ℓ−1,m−ℓ  dm−ℓ,m−ℓ − 1

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Note that u(Im−ℓ ) = 0. Lℓ,η± yα ∩M (0) is a parabolic subgroup of Lℓ,±ηyα . It preserves the isotropic subspace + Vℓ,m−ℓ . Its unipotent radical is     Im−ℓ 0 v   ′ Um−ℓ = diag(Iℓ ,  1 0  , Iℓ ) ∈ SO2m+1 .   Im−ℓ −1 ′ ′ Note that ǫ0 η(η −1 Um−ℓ η)η −1 ǫ−1 0 = ǫ0 Um−ℓ ǫ0 ⊂ Um , and hence the function u 7→ ψ0 ′

′ f Zℓ (ǫ0 ηug) is invariant under left translations by elements of η −1 Um−ℓ (A)η. Thus, in the contribution of η to L(ϕσ , fτ,s ), when we factor the dg-integration through ′′ ′′ ′′ ′ Um−ℓ (F )\Um−ℓ (A), where Um−ℓ = η −1 Um−ℓ η, we get (similarly to (10.18)) Z Z ψ0 ′ U ′′ −1 ϕσ m−ℓ (g) f Zℓ (ǫ0 ηug)ψℓ,α (u)dudg, (0)

(0)

′′ (GF ∩η −1 MF η)Um−ℓ (A)\GA

Nℓ (A)∩η −1 QA η\Nℓ (A)

U ′′

where ϕσ m−ℓ is the constant term of ϕσ along the unipotent radical U ′′ . Since σ is cuspidal, this constant term is zero, and hence the last integral is (identically) zero. This completes the proof of Theorem 10.3. 10.3

Proof of Theorem 10.4

As in the proof of Theorem 10.3, the proof of this theorem runs in analogy to s− 12 ), at the study carried out in Chapter 6 of the modules F Jψℓ,γ (IndH Qm µψ τ | det ·|E one finite place. The integral (10.5) converges absolutely, whenever E(g, fτ,s ) is holomorphic, due to the rapid decrease of ϕσ and the moderate growth of E(g, fτ,s ) (m−ℓ−1),φ φ ˆ ℓ (A) (switching and of the theta series θψ . We have, for h ∈ H −1 ,γ −1 = θψ −1 ,γ −1 to the notation E ψℓ;γ,φ (h, fτ,s ) = F Jψφℓ,γ (E(h, fτ,s ))) Z φ ′′ ψℓ;γ,φ E(vh′ , fτ,s )ψℓ−1 (v)θψ E (h, fτ,s ) = −1 ,γ −1 (j(v)h )dv = Nℓ+1 (F )\Nℓ+1 (A)

Z

Z

φ ′′ E(uvh′ , fτ,s )ψℓ−1 (u)θψ (10.27) −1 ,γ −1 (vh )dudv

HWm,ℓ+1 (F )\HWm,ℓ+1 (A) Nℓ (F )\Nℓ (A)

See (3.20) and (3.60). We identify the Heisenberg group HWm,ℓ+1 as a subgroup of Nℓ+1 . As in (10.9), we have, for Re(s) large enough, E ψℓ;γ ,φ (h, fτ,s ) = ℓ X

Z

X

(r) r=0 Nℓ+1 (F )\Nℓ+1 (A) δ∈QF \Qℓ (F )

φ ′′ fτ,s (ǫr δuh′ )ψℓ−1 (u)θψ −1 ,γ −1 (j(u)h )du,

(10.28)

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where



ǫr = 

∧

Ir



I2(m−ℓ) 0 Im−ℓ  diag(Ir ,  Iℓ−r 0 Iℓ−r

−Iℓ−r



 , Ir ),

and Q(r) = Qℓ ∩ ǫ−1 r Qm ǫr . See (4.14), with j = m, ℓ + s − r = m, δ = −1. Let Rℓ = h(Wm,ℓ+1 )Nℓ+1 , and factor the sum in (10.28) through Q(r) \Qℓ /Rℓ . As in Sec. 6.1, the representatives have the form η = diag(ǫ, γi , ǫ∗ ), where ǫ is a representative in the quotient of Weyl groups WGLr ×GLℓ−r \WGLℓ , and, by Proposition 6.1, γi , i = 1, 3, is one of the two representatives in Q′ǫr \h(Wm,ℓ )/h(Wm,ℓ+1 ); Q′ǫr is the (Siegel) parabolic subgroup of h(Wm,ℓ ), which preserves the isotropic subspace + Vℓ,m−ℓ . Thus, as in (10.10), we get E

ψℓ;γ,φ

(h, fτ,s ) =

ℓ X

X

Z

r=0 η∈Q(r) \Q (F )/R (F ) Nℓ+1 (F )\Nℓ+1 (A) ℓ ℓ F

X

φ ′′ fτ,s (ǫr ηδuh′ )ψℓ−1 (u)θψ −1 ,γ −1 (j(u)h )du.

(10.29)

(r) δ∈Rℓ (F )∩η −1 QF η\Rℓ (F )

As in (10.11) and (10.12), and using the proof of Proposition 6.2, we see that the contribution of r > 0 to (10.29) is zero. Similarly, the argument of the proof of Proposition 6.3(1), shows that, when r = 0, the contribution of η = I to (10.29) is also zero. Thus, the double sum in (10.29) reduces to one term, namely the term corresponding to r = 0 and    ∧ 1 Im−ℓ−1  η = η0 = diag(Iℓ , I2(m−ℓ−1)  , Iℓ ). 1 −1

We have

′ Rℓ ∩ η0−1 Q(0) η0 = (Nℓ+1 ∩ η0−1 Q(0) η0 ) ⋊ (h(Wm,ℓ+1 ) ∩ η0−1 Q(0) η0 ) := Nℓ+1 Q′m−ℓ ,

where ′ Nℓ+1

and

   z 0 y 0       ′ I 0 y m−ℓ−1   =  ∈ h(V ) z ∈ ResE/F Zℓ+1 ,    Im−ℓ−1 0     z∗

Q′m−ℓ =



q ′ (d, v) = diag(Iℓ+1 ,

   d v , I ) ∈ h(V ) . ℓ+1 d∗

(10.30)

(10.31)

Note that Q′m−ℓ is a (Siegel) parabolic subgroup of h(Wm,ℓ+1 ), with Levi decom′ ′ ′ position Mm−ℓ ⋉ Um−ℓ ; Mm−ℓ is naturally identified with ResE/F GLm−ℓ−1 . Thus, (10.29) becomes

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E ψℓ;γ,φ (fτ,s , h) = X

Z

δ∈Q′m−ℓ (F )\h(Wm,ℓ+1 )(F )N ′

φ fτ,s (ǫ0 η0 uδh)ψℓ−1 (u)θψ −1 ,γ −1 (j(u)δh)du.

ℓ+1 (F )\Nℓ+1 (A)

(10.32) For simplicity of notation, we use, once again, the same letter h to denote (in case of symplectic groups) both an element in a metaplectic group and its projection to the symplectic group. We also used the fact that, for δ ∈ h(Wm,ℓ+1 (F )), φ φ φ −1 θψ )δh). −1 ,γ −1 (j(u)h) = θψ −1 ,γ −1 (δj(u)h) = θψ −1 ,γ −1 (j(δuδ

Substitute (10.32) in (10.5). We get (for Re(s) large) L(ϕσ , fτ,s , φ) = Z ϕσ (g) Q′m−ℓ (F )\G′A

Z

φ fτ,s (ǫ′0 ug)ψℓ−1 (u)θψ −1 ,γ −1 (j(u)g)dudg,

(10.33)

′ Nℓ+1 (F )\Nℓ+1 (A)

where ǫ′0 = ǫ0 η0 . Write X

φ θψ −1 ,γ −1 (j(u)g) =

ωψ−1 ,γ −1 (j(u)g)φ(x) = ωψ−1 ,γ −1 (j(u)g)φ(0)

+ x∈Wm,ℓ+1 (F )

+

X

ωψ−1 ,γ −1 (δ ′ j(u)g)φ(em ),

(10.34)

′ δ ′ ∈PF1 \Mm−ℓ (F )

′ (identified with ResE/F GLm−ℓ−1 ). where P 1 is the mirabolic subgroup of Mm−ℓ The contribution of ωψ−1 ,γ −1 (j(u)g)φ(0) to (10.33) is zero. For this, note that the ′ function ωψ−1 ,γ −1 (j(u)g)φ(0) is left Nℓ+1 (A)-invariant. The reason is that for u′ ′ of the form (10.30), j(u ) = (0, yℓ+1 ; 0), and now use (1.5). The contribution to (10.33) becomes Z Z fτ,s (ǫ′0 ug)ψℓ−1 (u)ωψ−1 ,γ −1 (j(u)g)φ(0)dudg. ϕσ (g) Q′m−ℓ (F )\G′A

′ Nℓ+1 (A)\Nℓ+1 (A)

′ ′ ′ Factor the dg-integration through Um−ℓ (F )\Um−ℓ (A). Since the elements of Um−ℓ ′ commute with Nℓ+1 , normalize Nℓ+1 , and, over A, preserve ψℓ , the last integral is equal to Z Z Z fτ,s (ǫ′0 vug)ψℓ−1 (u) ϕσ (vg) ′ ′ ′ ′ Mm−ℓ (F )Um−ℓ (A)\G′A Um−ℓ (F )\Um−ℓ (A)

′ Nℓ+1 (A)\Nℓ+1 (A)

ωψ−1 ,γ −1 (vj(u)g)φ(0)dudvdg. ′ By (1.5), for v ∈ Um−ℓ (A),

ωψ−1 ,γ −1 (vj(u)g)φ(0) = ωψ−1 ,γ −1 (j(u)g)φ(0).

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′ Also, ǫ′0 Um−ℓ (ǫ′0 )−1 ⊂ Um , and hence fτ,s (ǫ′0 vug) = fτ,s (ǫ′0 ug). Therefore the last integral is equal to Z Z U′ ϕσ m−ℓ (g) fτ,s (ǫ′0 ug)ψℓ−1 (u)ωψ−1 ,γ −1 (j(u)g)φ(0)dudg, ′ ′ Mm−ℓ (F )Um−ℓ (A)\G′A

U′

′ where the inner integration is over Nℓ+1 (A)\Nℓ+1 (A), and ϕσ m−ℓ is the constant ′ term of ϕσ along the unipotent radical Um−ℓ , and hence it is identically zero, as well as the last integral. Thus, when we substitute (10.34) in (10.33), we may ignore the first term in (10.34), and now the r.h.s. of (10.33) becomes Z Z X ωψ−1 ,γ −1 (δj(u)g)φ(em )dudg, ϕσ (g) fτ,s (ǫ′0 ug)ψℓ−1 (u) Q′m−ℓ (F )\G′A

′ where the du-integration is along Nℓ+1 (F )\Nℓ+1 (A) and the summation is over 1 ′ δ ∈ PF \Mm−ℓ (F ). We can switch the order of the inner summation and duintegration and get Z Z fτ,s (ǫ′0 ug)ψℓ−1 (u)ωψ−1 ,γ −1 (j(u)g)φ(em )dudg. ϕσ (g) ′ PF1 Um−ℓ (F )\G′A

′ Nℓ+1 (F )\Nℓ+1 (A)

Here we used the fact that, fτ,s (ǫ′0 δ −1 uδg) = fτ,s (ǫ′0 uδg). it now in the form  ζ        

′ (F ), ǫ′0 δ(ǫ′0 )−1 ∈ Qm (F ), and hence for δ ∈ Mm−ℓ ′ (A) (see (10.30)). Write Let u′ be an element of Nℓ+1

z1 1

0 0 Im−ℓ−1

z2 x 0 Im−ℓ−1

where ζ ∈ Zℓ (AE ). Then by (1.5),

0 0 x′ 0 1

 0 0   z2′  , 0  z1′  ζ∗

ωψ−1 ,γ −1 (j(u′ u)g)φ(em ) = ωψ−1 ,γ −1 ((0, x; 0)j(u)g)φ(em ) −1 (2x1 )ωψ−1 ,γ −1 (j(u)g)φ(em ). = ψE,−1

Note also that −1 ψℓ−1 (u′ ) = ψZ ℓ+1

  ζ z1 −1 = ψE,−1 (ζ1,2 + · · · ζℓ−1,ℓ + (z1 )ℓ ). 1

Next, we have ǫ′0 u′ (ǫ′0 )−1



∧ Im−ℓ−1 x′ −z2′ = 1 z1′  . ζ∗

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Thus, when we factor, in the last integration, the inner du-integration through ′ ′ Nℓ+1 (F )\Nℓ+1 (A), we get Z fτ,s (ǫ′0 ug)ψℓ−1 (u)ωψ−1 ,γ −1 (j(u)g)φ(em )du = ′ Nℓ+1 (F )\Nℓ+1 (A)

Z

Z′

,ψ

fτ,sℓ+1 (ǫ′0 ug)ψℓ−1 (u)ωψ−1 ,γ −1 (j(u)g)φ(em )du,

′ Nℓ+1 (A)\Nℓ+1 (A)

′ where Zℓ+1 =



   Im−ℓ−1 v ∈ ResE/F Zm , ζ Z ′ Zℓ+1 ,ψ ′ fτ,s (ˆ z h)ψZℓ+1 (z)dz, fτ,s (h) =

z=

′ ′ Zℓ+1 (E)\Zℓ+1 (AE )

′ ′ and, for z ∈ Zℓ+1 (AE ), as above, ψZℓ+1 (z) = ψZℓ+1 (ζ)ψE,−1 (2vm−ℓ−1 ). We conclude that (for Re(s) large)

L(ϕσ , fτ,s , φ) = Z ϕσ (g)

Z

Z′

,ψ

fτ,sℓ+1 (ǫ′0 ug)ψℓ−1 (u)ωψ−1 ,γ −1 (j(u)g)φ(em )dudg.

′ Nℓ+1 (A)\Nℓ+1 (A)

′ PF1 Um−ℓ (F )\G′A

(10.35) ′ ′ (A), and since the elements (F )\Um−ℓ Now, factor the dg-integration through Um−ℓ ′ ′ , the last normalize Nℓ+1 , preserve ψℓ , over A, and commute with Nℓ+1 of Um−ℓ integral becomes Z Z ϕσ (vg) ′ ′ ′ (F )\Um−ℓ (A) PF1 Um−ℓ (A)\G′A Um−ℓ

Z

Z′

,ψ

fτ,sℓ+1 (ǫ′0 vug)ψℓ−1 (u)ωψ−1 ,γ −1 (vj(u)g)φ(em )dudvdg.

′ Nℓ+1 (A)\Nℓ+1 (A)

′ ∈ Um , for v ∈ Um−ℓ , and since (by (1.5)) Since ǫ′0 vǫ′−1 0   I y −1 ωψ−1 ,γ −1 ( m−ℓ−1 j(u)g)φ(em ) = ψE,−1 (ym−ℓ−1,1 )ωψ−1 ,γ −1 (j(u)g)φ(em ), Im−ℓ−1

we get L(ϕσ , fτ,s , φ) = Z

′ PF1 Um−ℓ (A)\G′A

U′ ,ψ ϕσ m−ℓ (g)

Z

Z′

,ψ

fτ,sℓ+1 (ǫ′0 ug)ψℓ−1 (u)ωψ−1 ,γ −1 (j(u)g)φ(em )dudg, (10.36)

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′ where the du-integration is over Nℓ+1 (A)\Nℓ+1 (A), and Z ′ Um−ℓ ,ψ −1 ϕσ (g) = ϕσ (q ′ (Im−ℓ−1 , y)g)ψE,−1 (ym−ℓ−1,1 )dy.

′ ′ Here, the integration is over q ′ (Im−ℓ−1 , y) ∈ Um−ℓ (F )\Um−ℓ (A). As in (10.19), the   ′ du U ,ψ Shalika expansion is valid for the function ϕσ m−ℓ (q ′ ( , 0)g), as a function on 1 PF1 \PA1 , for fixed g, and we get that X U′ ,ψ (10.37) ϕσ m−ℓ (g) = Wϕψσ (γg), ˜m−ℓ−1 (F )\P 1 γ∈Z F

where Z˜m−ℓ−1 = {q (z, 0)|z ∈ ResE/F Zm−ℓ−1 }, and Wϕψσ is the Whittaker function of ϕσ , Z ′

−1 (v)dv, ϕσ (vg)ψN G

Wϕψσ (g) =

NG (F )\NG (A)

with respect to the standard Whittaker character ψNG of the Adele points of the maximal unipotent subgroup NG of G, as follows ψNG (q ′ (z, y)) = ψE,−1 (z1,2 + · · · + zm−ℓ−2,m−ℓ−1 + ym−ℓ−1,1 ), z ∈ Zm−ℓ−1 (AE ). We conclude that L(ϕσ , fτ,s , φ) = Z

Wϕψσ (g)

′ NG (F )Um−ℓ (A)\G′A

Z

Z′

,ψ

fτ,sℓ+1 (ǫ′0 ug)ψℓ−1 (u)ωψ−1 ,γ −1 (j(u)g)φ(em )dudg,

′ (A)\Nℓ+1 (A). Factor the dg-integration where the du-integration is over Nℓ+1 ′ throughZNG (A)\GA . Then it is easy to see that becomes Z Z the last integral ′ Z ,ψ fτ,sℓ+1 (ǫ′0 q ′ (z, 0)ug) Wϕψσ (g) ′ Nℓ+1 (A)\Nℓ+1 (A) Zm−ℓ−1 (E)\Zm−ℓ−1 (AE )

NG (A)\G′A

ψNG (q ′ (z, 0))ψℓ−1 (u)ωψ−1 ,γ −1 (j(u)g)φ(em )dzdudg. ∗ ′ −1 Since ǫ′0 q ′ (z, 0)(ǫ Z 0 ) = diag(z, I2ℓ+2 , z ), we get that Z′

,ψ

fτ,sℓ+1 (ǫ′0 q ′ (z, 0)ug)ψNG (q ′ (z, 0))dz = f Zm ,ψ (ǫ′0 ug),

Zm−ℓ−1 (E)\Zm−ℓ−1 (AE )

where f Zm ,ψ (h) is the Fourier coefficient Z

fτ,s (ˆ z h)·

Zm (E)\Zm (AE )

−1 (z1,2 + · · · zm−ℓ−2,m−ℓ−1 − 2zm−ℓ−1,m−ℓ − zm−ℓ,m−ℓ+1 − · · · − zm−1,m )dz. ψE,−1 Using this, and re-denoting ǫ′0 = βℓ , we finally get, for Re(s) large enough, that L(ϕσ Z , fτ,s , φ) is equal to Z

Wϕψσ (g)

NG (A)\G′A

Zm ,ψ fτ,s (βℓ ug)ψℓ−1 (u)ωψ−1 ,γ −1 (j(u)g)φ(em )dudg.

′ Nℓ+1 (A)\Nℓ+1 (A)

This completes the proof of Theorem 10.4.

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10.4

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A family of dual global integrals I

We will now consider the global integrals obtained from the ones introduced above by reversing the roles of cuspidal representations and Eisenstein series. Thus, this time, we pair Gelfand-Graev, or Fourier-Jacobi coefficients of cuspidal representations of HA with Eisenstein series on GA , induced from a “Siegel type” parabolic subgroup. This time, we assume that (V, b) is as in Sec. 3.5. In this section, we consider the case where (V, b) is quadratic or odd Hermitian. Recall, that by our assumption, HA = h(V )A is either split orthogonal, quasi-split even orthogonal, or odd quasi-split unitary. Let σ be an irreducible, automorphic, ψ cuspidal representation of HA . We will consider Gelfand-Graev coefficients ϕσ ℓ,w0 on the cusp forms of σ, as in Sec. 3.1, but for specific w0 , as follows. (Recall our ′ notations: m′ = dimE V0 ; m ˜ is the Witt index of b; m = [ m2 ].) Let 0 ≤ ℓ < m, ˜ and when (V, b) is even quadratic, we assume that ℓ < m ˜ − 1. If m′ is odd, then w0 = e0 , and if m′ is even, and m ˜ = m, then w0 = em − 12 e−m , and , (2) finally, if m′ is even, and m ˜ = m−1, then w0 = e0 (see (3.2)). In the last two cases (1) define w0′ = em + 21 e−m or w0′ = e0 , respectively. Denote G = h(Wm,ℓ ∩ w0⊥ ) = Lℓ,w0 . Note that G is, respectively, a quasi-split even unitary group, or a split orthogonal group (in all cases). Write G as a matrix group relative to the basis {eℓ+1 , ..., e[ m′ −1 ] , e−[ m′ −1 ] , ..., 2 2 e−ℓ−1 }, when m′ is odd, and in case m′ is even, we need to add the vector w0′ in the middle. Let BG be the subgroup of G which preserves the maximal isotropic flag, corresponding to the last basis; this is the standard Borel subgroup of G. Let BG = TG ⋉ NG be its Levi decomposition. The elements of TG are realized as diagonal matrices, and the elements of NG are realized as upper unipotent matrices. As usual, we will identify G as a subgroup of H = h(V ). Consider the “Siegel type” parabolic subgroup QG of G, stabilizing the maximal isotropic subspace WG = SpanE {eℓ+1 , ..., e[ m′ −1 ] } of Wm,ℓ ∩ w0⊥ . Write its Levi 2 decomposition as QG = MG UG ; the Levi part MG is isomorphic to GL(WG ), which is ResE/F GLm−ℓ , when m′ is odd, and it is GLm−ℓ−1 , when h(V ) is even orthogonal. Let τ be an automorphic representation of MG (A) of the form described in (10.2). Consider, for a complex number s, the parabolic induction 1

s− 2 A , ρτ,s = IndG QG (A) τ | det ·|

and let E(g, fτ,s ) denote an Eisenstein series on GA , corresponding to a smooth holomorphic section fτ,s in ρτ,s . In case G = SO1,1 , τ is just a character of GL1 (A) = A∗ and ρτ,s is the character  x

x−1



1

7→ τ (x)|x|s− 2 ,

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and there is no Eisenstein series. For an irreducible, automorphic, cuspidal representation σ of HA , define Z ψ ϕσ ℓ,w0 (g)E(g, fτ,s )dg. (10.38) L(ϕσ , fτ,s ) = GF \GA

We will denote, as usual, for h ∈ GLk , 1 ≤ k ≤ m,   h ˆ =  Im′ −2k  . h h∗

Lemma 10.1. The integral (10.38) converges absolutely and uniformly in vertical strips in C, away from poles of the Eisenstein series. Hence it it is a meromorphic function in the plane. Proof. The proof is similar to the one in [Jacquet and Shalika (1990)], p. 179. We thank Steve Miller for pointing out an error in a similar lemma in [Ben-Artzi and Soudry (2009)], p. 21. Let Sc = ΩAc KG be a Siegel domain for GA . Here, Ω is a compact subset of BG (A), KG is an appropriate standard maximal compact subgroup of GA , and, for a positive constant (small enough) c, Ac ⊂ TG (A) is ′ described as follows. Denote, for short, p = dimE WG = [ m 2−1 ] − ℓ. Let β1 ,...,βp be the simple roots of G relative to (TG , BG ). Embed the positive real numbers diagonally in the Archimedean part of A∗ , and at the finite part of A∗ , just put 1. Denote the image of this embedding by R+,∆ . Then Ac is the subset of all elements in TG (A), of the form −1 ∧ m(t) = diag(Iℓ , t1 , t2 , ..., t−1 2 , t1 , Iℓ ) = diag(Iℓ , t)

where t = diag(t1 , ..., tp ), ti ∈ R+,∆ , and

βi (m(t)) ≥ c,

i = 1, ..., p.

ti For example, if h(V ) is odd orthogonal, then βi (m(t)) = ti+1 , for i < p, and βp (m(t)) = tp−1 tp . For g = ωm(t)k ∈ ΩAc KG , consider Z ψ −1 (v)dv. ϕσ (vωm(t)k)ψℓ,w ϕσ ℓ,w0 (g) = 0 Nℓ (F )\Nℓ (A)

Write v ∈ Nℓ (A) in the form



z

y

v(z, y, x) =  Im′ −2ℓ

 x y′  , z∗

z ∈ Zℓ (AE ).

In the last integral, we may take v in a compact set Yℓ ⊂ Nℓ (A). We may then write vg = v(Iℓ , y1 , x1 )m(t)ω ′ kv(z, 0, 0), where v(Iℓ , y1 , x1 ) ∈ Nℓ (A) lies in a compact set, and ω ′ = m(t)−1 ωm(t). Since m(t) ∈ Ac , ω ′ lies in a compact set. Let us consider, first, the case where h(V ) is not odd orthogonal. In this case,we also have v(Iℓ , y1 , x1 )m(t) = v(Iℓ , (y2 , 0), 0)m(t)v(Iℓ , (0, y3 ), x2 ),

(10.39)

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where y2 ∈ Mℓ×p (AE ) and v(Iℓ , (0, y3 ), x2 ) ∈ Nℓ (A) lies in a compact set. Thus, vg = v(Iℓ , (y2 , 0), 0)m(t)r, where r lies in a fixed compact set C ⊂ h(VA′ ); this compact set C depends only on c. Then, in the last integral,  ∧ Iℓ y2 t vg = r, (10.40) t where r lies in the compact set C. We now prove that, for all N ≥ 1, there is cN > 0, such that ˆ ≤ cN min(| det(h)|N , | det(h)|−N ), |ϕσ (hr)|

(10.41) ′

for all h ∈ GLℓ+p (AE ) and r ∈ C. Note that ℓ + p = m, when m is odd, and ℓ + p = m − 1, when h(V ) is even orthogonal, so that when h(V ) is even orthogonal   h ˆ =  I2  . h h∗

The proof of (10.41) is the same as the proof of (8) in [Jacquet and Shalika II (1981)], p. 799. Indeed, replacing C by a larger compact set Ω0 , we may assume that h = a = diag(a1 , ..., aℓ+p ) is diagonal. Put ai , i = 1, ..., ℓ + p − 1; αℓ+p (a) = aℓ+p . αi (a) = ai+1 We may identify αi , 1 ≤ ℓ + p, with standard simple roots of h(V ). These are all the simple roots except in case h(V ) is split even orthogonal. In this case let αm be the remaining simple root (with respect to the standard Borel subgroup). For a, as above, det(a) = α1 (a)α2 (a)2 · ... · αℓ+p (a)ℓ+p . Put for any diagonal t = diag(t1 , ..., tm ), Λ(tˆ) = α1 (tˆ)α2 (tˆ)2 · ... · αm (tˆ)m . Note, that for a as above, Λ(ˆ a) = det(a).

There is an element w in the Weyl group of h(V ), such that wˆ aw−1 = bˆa , where ba = diag(b1 , ..., bℓ+p ) is diagonal and satisfies |αi (bˆa )| ≥ 1, i = 1, ..., m.

(10.42)

It is easy to see that there is a positive integer N0 , such that for any diagonal d, satisfying (10.42), and for any Weyl element w′ of h(V ), we have ˆ ′−1 )| ≤ κ(d) ˆ N0 , ˆ −N0 ≤ |Λ(w′ dw κ(d)

(10.43)

ˆ = max(|αi (d)| ˆ : 1 ≤ i ≤ m). where κ(d) Let N ≥ 1. Since ϕσ is rapidly decreasing, we get, due to (10.42), that there is cN > 0, such that, for all diagonal a ∈ GLℓ+p (AE ), and all r ∈ Ω0 , |ϕσ (ˆ ar)| = |ϕσ (bˆa wr)| ≤ cN κ(bˆa )−N0 N .

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From (10.43), we get a)|N , |Λ(ˆ a)|−N ) = |ϕσ (ˆ ar)| ≤ cN min(|Λ(w−1 bˆa w)|N , |Λ(w−1 bˆa w)|−N ) = cN min(|Λ(ˆ cN min(| det(a)|N , | det(a)|−N ). This proves (10.41). Now, getting back to (10.40), we conclude, that given N ≥ 1, there is cN > 0, such that |ϕσ (vg)| ≤ cN min(det(t)N , det(t)−N ),

(10.44)

for all v ∈ Yℓ and g = ωm(t)k ∈ Sc , as above. Now, (10.44) can be modified, by changing the constant to |ϕσ (vg)| ≤ c′N det(t)−N ,

(10.45)

and using the moderate growth of E(g, fτ,s ), we see that the integral (10.38) converges absolutely and uniformly when Re(s) is bounded and is away from poles. Finally, when h(V ) is odd orthogonal, we make the following modification in the beginning of the proof. When we wrote that vg = v(Iℓ , y1 , x1 )m(t)ω ′ kv(z, 0, 0), with ω ′ in a compact set (depending on c), we could also, at will, use a conjugation by   1 γ0 = diag(Im−1 ,  −1  , Im−1 ), 1 namely,

ϕσ (vg) = ϕσ (v(Iℓ , y1 , x1 )γ0 m(t)γ0 γ0 ω ′ kv(z, 0, 0)). Of course, we may assume that v(Iℓ , y1 , x1 )γ0 = v(Iℓ , y˜, x1 ) remains in the compact set Yℓ . The effect of the conjugation of γ0 on m(t) is just the flip of tp and t−1 p . So, we may assume that m(t) ∈ Ac and also tp > c, and now, we get the decomposition as in (10.39). The proof of (10.41), and hence that of (10.44) are, of course, valid here. Now we get the estimate (10.45), due to the extra assumption that tp is bounded below by c, and we finish as before. This completes the proof of the lemma.  Let us prove now the analogue of Theorem 10.3. Theorem 10.5. Assume that (V, b) is either quadratic or odd Hermitian. Let 0 ≤ ℓ<m ˜ be an integer and, when h(V ) is even orthogonal, assume also that ℓ < m− ˜ 1. Let G = h(Wm,ℓ ∩ w0⊥ ) = Lℓ,w0 . Let σ be an irreducible, automorphic, cuspidal representation of HA , and let τ be an automorphic representation of GLp (AE ) (p = dimE WG ), of the form (10.2) (with p replacing m). Consider the meromorphic functions (10.38) Z ψ ϕσ ℓ,w0 (g)E(g, fτ,s )dg, L(ϕσ , fτ,s ) = GF \GA

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as ϕσ varies in the space of σ and fτ,s varies in the space of smooth holomorphic sections in ρτ,s . Assume that L(ϕσ , fτ,s ) is not identically zero (as a meromorphic function and as data vary). Then σ is globally generic, with respect to a certain “standard” Whittaker character (to be specified in the proof ). Moreover, for Re(s) large enough, we have an identity of the form Z Z Zp ,ψ (g)dλdg, (10.46) Wϕψσ (λδℓ g)fτ,s L(ϕσ , fτ,s ) = NG (A)\GA LA

where Wϕψσ is the corresponding Whittaker function of ϕσ , ( ) ∧ ∧  Ip Ip L= ∈ h(V ) ; δℓ = ∈ h(V ); λ Iℓ Iℓ Z ,ψ

and fτ,sp

is as in Theorem 10.3.

Proof. For Re(s) large enough, we can write E(g, fτ,s ) as an absolutely convergent series, and then Z ψ (10.47) (ϕσ ℓ,w0 )UG (g)fτ,s (g)dg. L(ϕσ , fτ,s ) = MG (F )UG (A)\GA

By Theorem 7.3, ψ

(ϕσ ℓ,w0 )UG (g) =

X

Z

γ∈Zp (E)\GLp (E)L

γ λδℓ g)dλ. Wϕψσ (ˆ

A

ψ ϕσ ℓ+p,w0 .

This is the (standard) Whittaker function of ϕσ , with respect Here Wϕψσ = to ψ; L and δℓ are specified in the statement of the theorem. Hence, for Re(s) large enough, Z Z X γ λδℓ g)fτ,s (g)dλdg. (10.48) Wϕψσ (ˆ L(ϕσ , fτ,s ) = MG (F )UG (A)\GA γ∈Zp (E)\GLp (E)LA

Note that, for γ ∈ GLp (E), γˆ normalizes LA , and δℓ−1 γˆ δℓ is a general element of MG (F ). We then get Z Z (10.49) Wϕψσ (λδℓ g)fτ,s (g)dλdg. L(ϕσ , fτ,s ) = Zp (E)UG (A)\GA LA

Here, Zp (E) is regarded as a subgroup of MG . Finally, factor the dg-integration in (10.49) through Zp (E)\Zp (AE ), to get Z Z Zp ,ψ (g)dλdg, Wϕψσ (λδℓ g)fτ,s L(ϕσ , fτ,s ) = NG (A)\GA LA

Z ,ψ fτ,sp (g)

where theorem.

is exactly as in Theorem 10.3 (with p replacing m). This proves the 

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10.5

A family of dual global integrals II

Assume that (V, b) is symplectic , or even anti-Hermitian of dimension m′ = 2m over E (we take δ = −1). Consider, as usual, the corresponding groups HA = h(V )A , and also, when b is symplectic, we consider the double cover HA of h(V )A . Let σ be an irreducible, automorphic, cuspidal representation of HA (genuine, when HA is metaplectic). Consider Fourier-Jacobi coefficients of cusp forms ϕσ in σ. As in the previous section, we will pair these coefficients with Eisenstein series induced from the Siegel parabolic subgroup. More precisely, let ℓ ≤ m − 1. Consider the Fourier+ Jacobi coefficients F Jψφℓ ,γ (ϕσ ), where φ is a Schwartz function on Vℓ+1,m−ℓ−1 (A). ˆ See (3.14). This is an automorphic function on GA = Hℓ (A). See (3.12), (3.13). Denote also G′ = h(Wm,ℓ+1 ). As in the previous section, write G′ as a matrix group relative to the basis {eℓ+2 , ..., em , e−m , ..., e−ℓ−2 }. Let BG be the standard Borel subgroup of G′ , which preserves the maximal isotropic flag, corresponding to the last basis, and let BG = TG ⋉ NG be its Levi decomposition. The elements of TG are realized as diagonal matrices, and the elements of NG are realized as upper unipotent matrices. As usual, we will identify G′ as a subgroup of h(V ). Consider the “Siegel type” parabolic subgroup QG of G′ , stabilizing the maximal + isotropic subspace WG = SpanE {eℓ+2 , ..., em } = Vℓ+1,m−ℓ−1 of Wm,ℓ+1 . Write its Levi decomposition as QG = MG UG ; the Levi part MG is isomorphic to GL(WG ), which is ResE/F GLm−ℓ−1 . Let τ be an automorphic representation of MG (A) of the form described in (10.2). Consider, for a complex number s, the parabolic induction 1

s− 2 A , ρτ,s = IndG QG (A) τ | det ·|

in case GA is not metaplectic, and in case GA is metaplectic, consider 1

s− 2 A ρτ,s = IndG . QG (A) µψ τ | det ·|

Let E(g, fτ,s ) denote an Eisenstein series on GA , corresponding to a smooth holomorphic section fτ,s in ρτ,s . Define Z F Jψφℓ ,γ (ϕσ )(g ′ )E(fτ,s , g ′ )dg. (10.50) L(ϕσ , fτ,s , φ) = G′F \G′A

Here, if GA is metaplectic, then g ′ ∈ GA projects to g ∈ G′A , and if GA is symplectic, then g ′ = g ∈ GA . Recall the following extreme cases (see Sec. 3.2). When ℓ = m−1, G = 1 and the last integral is nothing but the standard Whittaker coefficient of ϕσ . Thus, we will assume that ℓ < m − 1. The possibility ℓ = −1 makes sense here. In this case, G′ = h(V ), and φ ′′ F Jψφℓ ,γ (ϕσ )(g) = ϕσ (g ′ )θψ −1 ,γ −1 (g ),

where g ′ = g ′′ = g, unless G′A is symplectic, and when it is, if HA is symplectic, then g = g ′′ projects to g ′ , and if HA is metaplectic, then g ′ = g ′′ projects to g. The analogue of Lemma 10.1 is valid for the integral (10.50), with exactly the

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same proof. Thus, L(ϕσ , fτ,s , φ) is a meromorphic function in C. Now, we have the analogue of Theorem 10.5 Theorem 10.6. Assume that (V, b) is either symplectic or even anti-Hermitian. Let −1 ≤ ℓ < m − 1 be an integer. Let G′ = h(Wm,ℓ+1 ). Let σ be an irreducible, automorphic, cuspidal representation of HA (genuine, when HA is metaplectic), and let τ be an automorphic representation of GLp (AE ) (p = dimE WG = m − ℓ − 1), of the form (10.2) (with p replacing m). Consider the meromorphic functions (10.50) Z F Jψφℓ ,γ (ϕσ )(g ′ )E(fτ,s , g ′ )dg, L(ϕσ , fτ,s , φ) = G′F \G′A

as ϕσ varies in the space of σ, fτ,s varies in the space of smooth holomorphic sections in ρτ,s , and φ varies in S(WG (A)). Assume that L(ϕσ , fτ,s , φ) is not identically zero (as a meromorphic function and as data vary). Then σ is globally generic, with respect to a certain “standard” Whittaker character (to be specified in the proof ). Moreover, for Re(s) large enough, we have an identity of the form (where, for simplicity, we use the same letter g to denote in the symplectic cases, both an element of a metaplectic group and its projection to the symplectic group) Z Z Zp ,ψ (g)dλdg, (10.51) Wϕψσ (λδℓ′ g)ωψ−1 ,γ −1 (g)φ(i(λ))fτ,s L(ϕσ , fτ,s ) = NG (A)\G′A LA

where Wϕψσ is the corresponding Whittaker function of ϕσ , ( ) ∧ ∧ ∧   Ip Ip Ip ′ L= ∈ h(V ) ; i = λℓ+1 ; δℓ = ∈ h(V ); λ Iℓ+1 λ Iℓ+1 Iℓ+1 Z ,ψ

and fτ,sp proof.)

is as in Theorem 10.3. (The case ℓ = −1 is specified at the end of the

Proof. Assume that Re(s) is large enough, so that we can replace E(g, fτ,s ) by an absolutely convergent series. Then Z (F Jψφℓ ,γ (ϕσ ))UG (g ′ )fτ,s (g ′ )dg. L(ϕσ , fτ,s , φ) = MG (F )UG (A)\G′A

By Theorem 7.9, we have (F Jψφℓ ,γ (ϕσ ))UG (g ′ ) =

X

Z

η∈Zp (E)\GLp (E)L

ψ

ωψ−1 ,γ −1 (˜ g )φ1 (i(λ))ϕσ ℓ+p,γ;φ2 (ˆ η λδℓ′ g ′′ )dλ,

A

(10.52) ψ 2 ψ (ϕ ) where in this case φ1 = φ (φ2 does not exist) and ϕσ ℓ+p,γ;φ2 = F Jψφℓ+p σ = Wϕσ ,γ is a standard Whittaker function of ϕσ ; L and δℓ′ are specified in the statement of f m,ℓ+1 )A and, the theorem. Finally, in (10.52), when G′ is symplectic, g˜ is in Sp(W ′ ′′ ′′ according to the case, either g = g˜ and projects to g , or g = g˜ and projects to

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g ′ . Note that for η ∈ GLp (E), ηˆ normalizes L, and (δ ′ )−1 ˆδℓ′ is a general element ℓ η of MG (F ). We conclude that, for Re(s) sufficiently large, Z Z Wϕψσ (λδℓ′ g)ωψ−1 ,γ −1 (g)φ(i(λ))fτ,s (g)dλdg. L(ϕσ , fτ,s , φ) = Zp (E)UG (A)\G′A LA

Here, we used that, for η ∈ GLp (E), ωψ−1 ,γ −1 (g)φ(i(λη)) = ωψ−1 ,γ −1 (g)φ(i(λ)η) = ωψ−1 ,γ −1 (ˆ η g)φ(i(λ)). Finally, it is easy to check that factoring the dg-integration through Zp (E)\Zp (AE ) gives Z Z Zp ,ψ (g)dλdg. Wϕψσ (λδℓ′ g)ωψ−1 ,γ −1 (g)φ(i(λ))fτ,s NG (A)\G′A LA

Note the case ℓ = −1 (see Theorem 7.7). Here, Z φ ϕσ (g)θψ L(ϕσ , fτ,s , φ) = −1 ,γ −1 (g)E(g, fτ,s )dg. G′F \G′A

Formula (10.52) reads as φ UG (g) = (ϕσ θψ −1 ,γ −1 )

X

η g). ωψ−1 ,γ −1 (g)φ(em )Wϕψσ (ˆ

(10.53)

η∈Zm (E)\GLm (E)

Thus, for Re(s) large enough, we get Z Zm ,ψ (g)dg. Wϕψσ (g)ωψ−1 ,γ −1 (g)φ(em )fτ,s L(ϕσ , fτ,s , φ) = NG (A)\G′A

10.6

(10.54) 

L-functions

The global integrals in the last four theorems represent (for suitable data), except when σ is on an even unitary group, the partial L-functions LS (σ × τ, s), where S is a finite set of places of F , containing those at infinity, outside of which σ and τ are unramified; when σ is on an even unitary group, we have to replace τ by τ ⊗ γ −1 ; also when σ is on a metaplectic group, the L-function depends on ψ, and then we take S, such that ψ is normalized outside S. We will write the precise statements, but will not prove them, as these proofs lie outside the scope of this book. The references cited in the beginning of this chapter contain the proofs. By uniqueness of Whittaker models, for a cusp form ϕσ , which corresponds to a decomposable vector, we can write its Whittaker function as a product of local Whittaker functions Y Wϕψσ = Wvψv , v

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where Wvψv is a local Whittaker function in the corresponding Whittaker model of σv , and for v outside S, Wvψv is spherical, such that its value at the identity element is 1. Denote this unique function in the Whittaker model of σv by Wσ0v . Similarly, assume that fτ,s is a decomposable section, and then when we apply to it the GL-Whittaker coefficient (on τ ) we can write in a similar way, say in Theorems 10.3, 10.4 Y Zm ,ψ fτ,s (h) = fτv ,s (hv ; Im ), v

where fτv ,s is a holomorphic section in ρτv ,s , taking values in the local Whittaker model of τv with respect to the character (at v) indicated in the theorems above; for fixed hv , we denote the corresponding Whittaker function in the Whittaker model of τv by x 7→ fτv ,s (hv ; x). For all places v outside S, fτv ,s is spherical and the function x 7→ fτv ,s (I; x) is the unique spherical and normalized Whittaker function in the corresponding Whittaker model of τv . Denote this unique section by fτ0v ,s . In the cases where we took Fourier-Jacobi coefficients, we assume that the Schwartz function φ is decomposable as a product of local Schwartz functions Y φ= φv , v

where outside S, φv is the characteristic function φ0v of the corresponding standard lattice. It follows from the last four theorems that, for Re(s) large enough, we have in Theorems 10.3, 10.5, Y L(ϕσ , fτ,s ) = Lv (Wvψv , fτv ,s ); v

and in Theorems 10.4, 10.6

L(ϕσ , fτ,s , φ) =

Y v

Lv (Wvψv , fτv ,s , φv ).

Here, the local factors Lv are read from the theorems above. More precisely, in the case of Theorem 10.3, we have, in the (analogous) notation of (10.4), (σv and τv are representations of GFv and GLm (Ev ), respectively) Lv (Wvψv , fτv ,s ) = Z Wvψv (g) NG (Fv )\GFv

Z

−1 Nℓ (Fv )∩βℓ,α Qm (Fv )βℓ,α \Nℓ (Fv )

fτv ,s (βℓ,α ug; Im )(ψv )−1 ℓ,α (u)dudg;

(10.55) in the case of Theorem 10.5, we have, in the notation of (10.46), (σv and τv are representations of HFv and GLp (Ev ), respectively) Z Z (10.56) Wvψv (λδℓ g)fτv ,s (g, Ip )dλdg; Lv (Wvψv , fτv ,s ) = NG (Fv )\GFv LFv

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in the case of Theorem 10.4, we have, in the notation of (10.6), (σv and τv are representations of GFv and GLm (Ev ), respectively) Lv (Wvψv , fτv ,s , φv ) = Z Z ψv Wv (g) fτv ,s (βℓ ug, Im )(ψv )−1 ℓ (u)ωψv−1 ,γv−1 (j(u)g)φv (em )dudg, NG (Fv )\G′Fv

(10.57) the du-integration being taken along Nℓ+1 (Fv ) ∩ βℓ−1 Nℓ+1 (Fv )βℓ \Nℓ+1 (Fv ); in the case of Theorem 10.6, we have, in the notation of (10.51), (σv and τv are representations of HFv and GLp (Ev ), respectively) Z Z Wvψv (λδℓ′ g)ωψv−1 ,γv−1 (g)φv (i(λ))fτv ,s (g, Ip )dλdg. Lv (Wvψv , fτv ,s , φv ) = NG (Fv )\G′Fv LFv

(10.58) Each local integral Lv (Wvψv , fτv ,s ), Lv (Wvψv , fτv ,s , φv ), converges absolutely in a fixed right half plane (independent of the vectors), and continues to a meromorphic function in the whole plane, rational in qv−s , when v is finite. When v is finite, we can find data Wvψv , fτv ,s (resp. φv ), such that Lv (Wvψv , fτv ,s ) (resp. Lv (Wvψv , fτv ,s , φv )) is identically 1 (as a function of s). When v is infinite, given s0 ∈ C, we can find data, Wvψv , fτv ,s (resp. φv ), such that Lv (Wvψv , fτv ,s ) (resp. Lv (Wvψv , fτv ,s , φv )) is holomorphic and nonzero in a neighbourhood of s0 . Assume that v is outside S. Then in case of (10.55) (resp. (10.56)), when G (resp. H) is odd orthogonal, L(σv × τv , s) ; L(τv , ∧2 , 2s)

(10.59)

L(σv × τv , s) ; L(τv , sym2 , 2s)

(10.60)

L(σv × τv , s) . L(τv , Asai, 2s)

(10.61)

Lv (Wσ0v , fτ0v ,s ) = when G (resp. H) is even orthogonal, Lv (Wσ0v , fτ0v ,s ) = when G (resp. H) is odd unitary, Lv (Wσ0v , fτ0v ,s ) =

In case of (10.57) (resp. (10.58)), when G (resp. H) is symplectic, L(σv × τv , s) ; L(τv , sym2 , 2s)

(10.62)

Lψv (σv × τv , s) ; L(τv , s + 12 )L(τv , ∧2 , 2s)

(10.63)

Lv (Wσ0v , fτ0v ,s , φ0v ) = when G (resp. H) is metaplectic, Lv (Wσ0v , fτ0v ,s , φ0v ) =

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when G (resp. H) is even unitary, Lv (Wσ0v , fτ0v ,s , φ0v ) =

L(σv × (τv ⊗ γv−1 ), s) . L(τv , Asai, 2s)

(10.64)

For v ∈ S, which is finite, let us choose data, such that the local integral is identically 1. Let Y L∞ (ϕσ , fτ,s ) = Lv (Wvψv , fτv ,s ), v∈S∞

and respectively

L∞ (ϕσ , fτ,s , φ) =

Y

v∈S∞

Lv (Wvψv , fτv ,s , φv ).

Then (for global decomposable data corresponding to the local data above at each place, we have in cases of (10.59) LS (σ × τ, s) ; LS (τ, ∧2 , 2s)

(10.65)

LS (σ × τ, s) ; LS (τ, sym2 , 2s)

(10.66)

LS (σ × τ, s) ; LS (τ, Asai, 2s)

(10.67)

L(ϕσ , fτ,s ) = L∞ (ϕσ , fτ,s ) in cases of (10.60) L(ϕσ , fτ,s ) = L∞ (ϕσ , fτ,s ) in cases of (10.61) L(ϕσ , fτ,s ) = L∞ (ϕσ , fτ,s ) in cases of (10.62)

L(ϕσ , fτ,s , φ) = L∞ (ϕσ , fτ,s , φ)

LS (σ × τ, s) ; LS (τ, sym2 , 2s)

(10.68)

in cases of (10.63) L(ϕσ , fτ,s , φ) = L∞ (ϕσ , fτ,s , φ) in cases of (10.64)

LSψ (σ × τ, s)

LS (τ, s + 12 )LS (τ, ∧2 , 2s)

L(ϕσ , fτ,s , φ) = L∞ (ϕσ , fτ,s , φ)

;

LS (σ × (τ ⊗ γ −1 ), s) . LS (τ, Asai, 2s)

(10.69)

(10.70)

Borrowing the notation of Sec. 2.3, the L-functions in the denominators in (10.65)–(10.70) are LS (τ, α(1) , s + 12 )LS (τ, α(2) , 2s), where, except in case (10.69), LS (τ, α(1) , z) = 1, and in case (10.69),LS (τ, α(1) , z) = LS (τ, z). Therefore, we can summarize (10.65)-(10.70) by Theorem 10.7. In the notation above, we can find data, at each place of F , such that LS (σ × τ, s) ; (10.71) L(ϕσ , fτ,s ) = L∞ (ϕσ , fτ,s ) S L (τ, α(2) , 2s)

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L(ϕσ , fτ,s , φ) = L∞ (ϕσ , fτ,s , φ)

LS (σ × (τ ⊗ γ −1 ), s) . + 21 )LS (τ, α(2) , 2s)

LS (τ, α(1) , s

(10.72)

Recall that γ = 1 except in case σ is a representation of an even unitary group. The meromorphic functions L∞ (ϕσ , fτ,s ), L∞ (ϕσ , fτ,s , φ) can be chosen to be holomorphic and nonzero in a neighbourhood of a pre-given point s0 ∈ C. Finally, in case σ is a representation of a metaplectic group, LS (σ × τ, s) should be replaced by LSψ (σ × τ, s). The last theorem allows us to control the poles of LS (σ × τ, s), at Re(s) ≥ 1, when τ is unitary, cuspidal. In the following theorem (AF )+ ∞ denotes the image of the diagonal embedding, at all Archimedean places of F , of the positive real ∗ numbers. We view (AF )+ ∞ as a subgroup of AE , by letting the coordinates at all finite places to be 1. Let us also denote by nτ the number defined by saying that τ is a representation of GLnτ (AE ). Theorem 10.8. Let notation be as in Theorem 10.7. Assume that nτ ≥ 2, and that τ is unitary and cuspidal. Assume that (ωτ ) = 1. Then LS (σ × τ, s) (resp. (AF )+ ∞

LSψ (σ × τ, s), when σ is a representation of a metaplectic group) is holomorphic at Re(s) > 1. If this L- function has a pole at s0 , such that Re(s0 ) = 1, then s0 = 1, the pole is simple, and LS (τ, α(1) , s − 21 )LS (τ ⊗ γ, α(2) , 2s − 1) has a pole at s = 1. Note, again, that LS (τ, α(1) , s) = 1, except in case σ is a representation of a metaplectic group, in which case LS (τ, α(1) , s) = LS (τ, s). Note also, that, except in case σ is a representation of an even unitary group, γ = 1. Proof. Assume that s0 , with Re(s0 ) > 1, is a pole of LS (σ × τ, s) (resp. LSψ (σ × τ, s)). Let us apply Theorem 10.7, with τ ⊗ γ replacing τ . Since τ is unitary, the denominators in (10.71), (10.72) are holomorphic and nonzero at s = s0 . Indeed, in this case, Re(2s0 ) > 2 > 1, and LS (τ, ∧2 s0 ), LS (τ, sym2 , 2s0 ), LS (τ, Asai, 2s0 ), LS (τ ⊗ γ, Asai, 2s0), LS (τ, s0 + 12 ) are given by convergent infinite products. We conclude that the global integral on the left hand side of (10.71), or (10.72), respectively, has a pole at s = s0 , as data vary. By the definition of this global integral, the Eisenstein series, E(·, fτ ⊗γ,s ), which appears in it, must have a pole at s = s0 , and so it must be simple. We conclude that the constant term of this Eisenstein series, along the unipotent radical of the corresponding parabolic subgroup, has a simple pole at s = s0 (as fτ ⊗γ,s varies). This constant term, evaluated at the identity has the form fτ ⊗γ,s (I) + Ms (fτ ⊗γ,s )(I), where Ms is the corresponding intertwining operator. Thus, Ms (fτ ⊗γ,s )(I) must have a pole at s = s0 . We may assume that fτ ⊗γ,s is decomposable and corresponds N to v fτv ⊗γv ,s . Let S ′ be a finite set of places of F , including those at infinity, such that outside S ′ , τv , γv and fτv ⊗γv ,s = fτ0v ⊗γv ,s are unramified. We conclude that

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311

N the product of local intertwining operators v Mv,s (fτv ⊗γv ,s ) has a pole at s = s0 . We know that (with an appropriate normalization), for v outside S ′ , Mv,s (fτ0v ⊗γv ,s ) is a multiple of fτ0v∗ ⊗γv ,1−s by the factor

Hence N v Mv,s (fτv ⊗γv ,s ) = MS ′ ,s (fτ ⊗γ,s )

L(τv , α(1) , s − 21 )L(τv ⊗ γv , α(2) , 2s − 1) . L(τv , α(1) , s + 21 )L(τv ⊗ γv , α(2) , 2s)

′

′

LS (τ, α(1) , s − 12 )LS (τ ⊗ γ, α(2) , 2s − 1) S ′ ,0 f ∗ , LS ′ (τ, α(1) , s + 21 )LS ′ (τ ⊗ γ, α(2) , 2s) τv ⊗γv ,1−s

(10.73)

′

,0 where MS ′ ,s (fτ ⊗γ,s) = ⊗v∈S ′ Mv,s (fτv ⊗γv ,s ), and fτSv∗ ⊗γ is the product over all v ,1−s ′ 0 places outside S of fτv∗ ⊗γv ,1−s . Since τ is unitary and Re(2s0 − 1) > 1, we conclude, as before, that the ratio of partial L-functions in (10.73) is holomorphic at s = s0 . Finally, it follows from Lemma 2.4 in [Kim (1999)] that each factor Mv,s (fτv ⊗γv ,s ) is holomorphic at Re(s) ≥ 12 , and, in particular at s0 . Note that the standard module conjecture needed in loc. cit. is needed here just for ResE/F GLm , and hence is valid. This shows that Ms (fτ ⊗γ,s (I) is holomorphic at Re(s) > 1, and we reach a contradiction. Thus, we proved that LS (σ × τ, s) is holomorphic at Re(s) > 1. Assume that Re(s0 ) = 1 and that s0 is a pole of LS (σ × τ, s). We run the same argument. Note again that since τ is unitary, the denominators in (10.71), (10.72), as well as the denominator in (10.73), are holomorphic and nonzero at s = s0 . Similarly, since Re(s0 ) = 1 > 12 , by [Kim (1999)], as above, each local intertwining operator Mv,s (fτv ⊗γv ,s ) is holomorphic at s0 . We conclude that ′ ′ LS (τ, α(1) , s − 12 )LS (τ ⊗ γ, α(2) , 2s − 1) has a pole at s = s0 . This means that ′ ′ LS (τ ⊗ γ, α(2) , 2s − 1) has a pole at s0 and LS (τ, α(1) , s0 − 21 ) 6= 0. By our assumption on the central character of τ , we must have s0 = 1. Indeed, we obtain that, ac′ ′ ′ cording to the case at hand, LS (τ ×τ, 2s−1) = LS (τ, ∧2 , 2s−1)LS (τ, sym2 , 2s−1), ′ ′ ′ or LS (τ × τ ′ , 2s − 1) = LS (τ, Asai, 2s − 1)LS (τ ⊗ γ, Asai, 2s − 1) has a pole at s = s0 . Here, we use Shahidi’s result ([Shahidi, F. (1981)]) that the L-functions ′ ′ LS (τ, α(2) , z) do not vanish at any point on the line Re(z) = 1. Thus, LS (τ × τ ′ , z) has a pole at z = 2s0 − 1. This forces, by [Jacquet and Shalika II (1981)], that | det ·|2s0 −2 ⊗ τ = τ ∗ . By our assumption on ωτ , we get that 2s0 − 2 = 0, i.e. s0 = 1. This completes the proof. 

Proposition 10.1. Assume that nτ = 1. Then the assertions of Theorem 10.7 are the same, with exactly the same proof, except when σ is a representation of an odd orthogonal group, or a metaplectic group. When σ is a representation of an odd orthogonal group, LS (σ × τ, s) is entire. f (A), LS (σ × τ, s) may When σ is a representation of a metaplectic group, Sp 2n ψ have a pole at Re(s) > 1, and then it must be at s = 32 . In this case, we must have τ = 1, and σ is obtained by the theta correspondence from an irreducible, automorphic, cuspidal, generic representation of SO2n−1 (A).

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Proof. Note, that in the proposition we assume that τ is a unitary character of E ∗ \A∗E , which is trivial on (AF )+ ∞ . When σ is a representation of SO2n+1 (A), the global integral (10.38) is integrated along SO1,1 (F )\SO1,1 (A) ∼ = F ∗ \A∗ , and now there is no Eisenstein series in the integrand, but rather the character   1 x 7→ τ (x)|x|s− 2 , −1 x and so, the global integral (10.38) is entire. Let us denote this global integral by L(ϕσ , τ, s). In this case, the denominator in (10.59) is trivial, and (10.71) reads as L(ϕσ , τ, s) = L∞ (ϕπ , τ, s)LS (σ × τ, s).

By Theorem 10.7, we conclude that LS (σ × τ, s) is entire. Assume now that σ is a representation (irreducible, genuine, automorphic, cusf (A). Repeating the argument of the proof of Theorem 10.8, pidal, generic) of Sp ′

2n

′

′

we see that now LS (τ, s − 12 )LS (τ, ∧2 , 2s − 1) = LS (τ, s − 12 ) may have a pole at a point s0 , with Re(s0 ) > 1. The only way that this can happen is when τ = 1, and s0 − 12 = 1, that is s0 = 32 . Assume then that LSψ (σ, s) has a pole at s = 23 . Consider its theta correspondence π (with respect to ψ) to SO2n+1 (A). By [Furusawa (1995)], it is globally generic. If π is cuspidal, then it is irreducible (see [Jiang and Soudry (2007)]). Since LS (π, s) = LSψ (σ, s), we get that LS (π, s) has a pole at s = 32 , and this contradicts what we just proved. Thus, π is not cuspidal, and, from the tower property of the theta correspondence, we conclude that σ has a nontrivial theta lift π ′ to SO2n−1 (A). By [Jiang and Soudry (2003)], Cor. 2.2, π ′ is cuspidal. The proof that π ′ is generic is in [Jiang and Soudry (2007)] (p. 745, second paragraph). We note that this proof uses only the results on theta correspondence proved in that reference (and no other results concerning the descent construction). This completes the proof. 

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Chapter 11

Langlands (weak) functorial lift and descent

In this chapter we will describe the image of the weak functorial lift from irreducible, automorphic, cuspidal, globally generic representations of GA to GLm (AE ), where m is the degree of the standard representation of L G0 . The group G is a classical group, f 2n (A), split, or quasi-split over F , or a metaplectic group (meaning that GA = Sp f 2n (Fv ), L G0 = Sp2n (C), m = 2n). When G is not metaplectic, the exisGFv = Sp tence of the weak functorial lift above was proved by use of the converse theorem ([Cogdell and Piatetski-Shapiro (1994)]) in [Cogdell, Kim, Piatetski-Shapiro and Shahidi (2004)], [Kim and Krishnamurthy (2005)], [Cogdell, Piatetski-Shapiro and f 2n (A) is metaplectic, this weak lift (with respect Shahidi (2011)]. When GA = Sp to ψ) can be carried out by using the theta correspondence (with respect to ψ) to f 2n (A) to automorphic representations lift cuspidal generic representations σ of Sp of SO2n+1 (A); these are known to be globally generic. If the theta lift of σ is not cuspidal, then the theta lift of σ to SO2n−1 (A) is cuspidal and generic. See Proposition 10.1. Using these lifts, it is easy to obtain (by [Cogdell, Kim, Piatetski-Shapiro and Shahidi (2004)]) the lift of σ (with respect to ψ) to GL2n (A). The main result of this chapter is that, except the case where G is metaplectic, where the following needs a slight modification, the image of the weak lift above consists exactly of all irreducible, automorphic representations of GLm (AE ), which are parabolic inductions (isobaric sums) τ1 × τ2 × · · · × τr ,

(11.1)

of irreducible, automorphic, unitary, self-conjugate, pairwise inequivalent representations τi , of GLmi (AE ), which are cuspidal, when mi > 1, and are such that all of them have the property that the same type of L-function (LS (τi , Λ2 , 2s − 1); LS (τi , Λ2 , 2s − 1)LS (τi , s − 12 ); LS (τi , sym2 , 2s − 1); LS (τi′ , Asai, 2s − 1); LS (τi′ ⊗ γ, Asai, 2s − 1)) has a pole at s = 1. The corresponding type of L-function, of course, depends on G (odd orthogonal; metaplectic; even orthogonal or symplectic; odd unitary; even unitary-respectively). Finally, there is a simple condition on the product of central characters of the τi . That the image of the weak lift is contained in the set of the representations (11.1) as above will follow by using L-functions for pairs of irreducible, automorphic, 313

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cuspidal, generic representations of GA and GLk (AE ) and the global integrals of the previous chapter. The fact that every representation of the form (11.1) as above is in the image of the weak lift was already proved in Chapter 9.

11.1

The cuspidal part of the weak lift

Let σ be an irreducible, automorphic, cuspidal and globally generic representation of GA . If GA is metaplectic, we also assume that σ is genuine. Let m be the degree of the standard representation of L G0 . By [Cogdell, Kim, Piatetski-Shapiro and Shahidi (2004)], [Kim and Krishnamurthy (2005)], [Cogdell, Piatetski-Shapiro and Shahidi (2011)], we know that σ has a weak functorial lift to an irreducible automorphic representation τ of GLm (AE ). This means that for all places v outside a finite set of places S, containing the Archimedean places of F , σv and τv are unramified, and σv lifts locally to τv . (In loc. cit. it is also proved that σv lifts to τv f 2n (A) is metaplectic, we need to first at the Archimedean places v.) When GA = Sp fix a nontrivial character ψ of F \A, and then require that σv lifts to τv with respect to ψv . As we remarked in the previous section, this can be done by first taking the theta correspondence of σ, with respect to ψ to SO2n+1 (A), call it π, and if π is not cuspidal, then consider the theta lift of σ, π ′ , to SO2n−1 (A). Now use [Cogdell, Kim, Piatetski-Shapiro and Shahidi (2004)]. In the first case, π is irreducible, cuspidal and generic, and therefore has a weak lift τ to GL2n (A). In the second case, as explained in the proof of Proposition 10.1, π ′ is irreducible, cuspidal and generic. Let the functorial weak lift of π ′ to GL2n−2 (A) be t. Then standard calculations of the unramified parameters of the local theta lift of σv to SO2n+1 (Fv ), for v outside S, show that σv lifts, with respect to ψv , to the unramified constituent of 1 1 | · | 2 × t v × | · |− 2 that is σv lifts, with respect to ψv , to 1GL2 (Fv ) × tv . (11.2) Here, × denotes parabolic induction to GL2n (Fv ). Having cleared out this case, let us treat the remaining ones. Let v be a place outside S. Since σv lifts to τv , it is clear that, except in case G is a quasi-split even orthogonal group (over F ), the restriction of the central character of τv to Fv∗ is trivial, i.e. ωτv = 1. In case G is a quasi-split even Fv∗

orthogonal group, let χG be the quadratic character of F ∗ \A∗ , associated to minus the discriminant of the two dimensional anisotropic kernel of the symmetric form preserved by G. Then, for v outside S, ωτv = χG,v . We conclude that when G is quasi-split even orthogonal ω τ = χG , (11.3) and otherwise, ωτ = 1. (11.4) A∗

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It is also easy to see that, for a place v outside S, τv∗ ∼ = τv .

(11.5)

Recall that τv∗ = τˆv′ , where τv′ (g) = τv (¯ g ), and g¯ denotes the application of the automorphism x 7→ x ¯ to the coordinates of g. In this section, we assume that τ is cuspidal. We will consider the general case in the following section. By the strong multiplicity one and multiplicity one theorems for GLm (AE ) (see [Shalika, J. (1974)], [Jacquet and Shalika I (1981)]), [Jacquet and Shalika II (1981)], we conclude from (11.5) that τ ∗ = τ.

(11.6)

LS (σ × τˆ, s) = LS (τ × τˆ, s),

(11.7)

By definition, we have

unless GA is metaplectic, in which case, we have LSψ (σ × τˆ, s) = LS (τ × τˆ, s).

(11.8)

Note that (11.6) (and also (11.3), (11.4)) implies that τ is unitary. In this case, it is well known that LS (τ × τˆ, s) has a pole at s = 1. See [Jacquet and Shalika I (1981)]. We conclude from (11.7) (resp. (11.8)) that LS (σ × τˆ, s) ( resp. LSψ (σ × τˆ, s)) has a pole at s = 1. Let us represent the partial L-function LS (σ × τˆ, s), or LSψ (σ × τˆ, s), by the global integrals from the previous chapter. Theorem 10.8 and Prop. 10.1 imply that LS (τ, α(1) , s − 12 )LS (τ ′ ⊗ γ, α(2) , 2s − 1) has a pole at s = 1. We also conclude that the global integrals L(ϕσ , fτ ′ ,s ), L(ϕσ , fτ ′ ⊗γ,s , φ), in Theorem 10.8 have nontrivial residues at s = 1, as data vary. The residue at s = 1 of such integrals is equal to the L2 -pairing between σ ¯ and the descent of τ . Thus, σ ¯ must have a nontrivial L2 -pairing with the descent of τ ′ ⊗ γ. Summarizing Theorem 11.1. Let σ be an irreducible, automorphic, cuspidal and globally generic representation of GA (genuine when GA is metaplectic). Let m be the degree of the standard representation of L G0 . Assume that the weak functorial lift of σ to GLm (AE ) is an irreducible, cuspidal representation τ . Then we have the following. (1) The representation τ is self-conjugate, that is τ ∗ = τ . (2) The restriction of ωτ to A∗ is trivial, except in case G is even orthogonal and quasi-split, where ωτ = χG . (3) The partial L-function LS (τ, α(1) , s− 12 )LS (τ ′ ⊗ γ, α(2) , 2s− 1) has a simple pole at s = 1. (4) The representation σ ¯ has a nontrivial L2 -pairing with the descent of τ ′ ⊗ γ ′ ′ (σψ,α (τ ), or σψ,γ (τ ⊗γ), with ψ chosen in a compatible way with the Whittaker character, with respect to which σ is globally generic). Recall, once again, that τ ′ = τ , unless G is unitary, and that γ = 1, unless G is unitary, in an even number of variables. We represent all cases of the theorem in

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the following table, where m = N2n , and LS (τ, βH , s) = LS (τ, α(1) , s − 12 )LS (τ ′ ⊗ γ, α(2) , 2s − 1). For later needs, we use in the table the index k; in cases (1), (2), (4), (5) Nk = k and in cases (3), (6), Nk = k + 1.

LS (τ, βH , s) 1) L (τ, ∧2 , 2s − 1) LS (τ, sym2 , 2s − 1) 2) LS (τ ′ , Asai, 2s − 1) 3) 4) LS (τ, s − 12 )LS (τ, ∧2 , 2s − 1) LS (τ ′ ⊗ γ, Asai, 2s − 1) 5) 6) LS (τ, sym2 , 2s − 1) (11.9) A special case of what we did in the first nine chapters of this book is to start with a cuspidal τ on GLm (AE ), such that LS (τ, βH , s) has a pole at s = 1 and consider the 1 parabolic induction of the representation τ ′ ⊗ γ| det ·|s− 2 , from the “Siegel type” parabolic subgroup to HA . We formed the corresponding Eisenstein series of HA , and considered its residue at s = 1. We studied the automorphic representation of GA obtained by applying to the residual representation Gelfand-Graev coefficients, or Fourier-Jacobi coefficients, stabilized by G. We got the descent of τ ′ ⊗ γ to GA . Every irreducible summand σ of the complex conjugate of this descent is globally generic and lifts weakly to τ . Thus, with the last theorem, we see that the cuspidal part of the image of the weak functorial lift from irreducible, automorphic, cuspidal and globally generic representations of GA to GLm (AE ) consists precisely of the irreducible, automorphic, cuspidal representations τ of GLm (AE ), such that LS (τ, βH , s) has a pole at s = 1 ( it can be shown that this implies that τ is self-conjugate and its central character satisfies (11.4)). Moreover, the descent is cannonic in the sense that it has a nontrivial L2 -pairing with the complex conjugate of any σ as in the theorem, which lifts to τ . We remark that in Cases (3) and (5) of the last table, we have, respectively, LS (τ ′ , Asai, 2s − 1) = LS (τ, Asai, 2s − 1) and LS (τ ′ ⊗ γ, Asai, 2s − 1) = LS (τ ⊗ γ, Asai, 2s − 1). So in (3) of the last theorem we may simply replace τ ′ by τ (this is relevant just for unitary groups). In (4) of the last theorem, σ has a nontrivial L2 -pairing with the complex conjugate of the descent of τ ′ ⊗ γ. This is easily seen to be the descent of τ ⊗ γ −1 with respect to (ψ −1 , γ −1 ), namely σψ−1 ,α (τ ), or σψ−1 ,γ −1 (τ ⊗ γ −1 ). G SO2n+1 SO2n U2n+1 f 2n Sp U2n Sp2n

11.2

ResE/F GLNk GLk GLk ResE/F GLk+1 GLk ResE/F GLk GLk+1

H = HG,k SO2k SO2k+1 U2k+2 Sp2k U2k f 2k+2 Sp

S

The image of the weak lift

Assume that the weak lift of σ to GLm (AE ) is an irreducible, automorphic representation τ (necessarily isomorphic to its conjugate), which is a constituent of a

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parabolic induction δ1 | det ·|z1 × · · · × δj | det ·|zj × τ1 × · · · × τr × δj∗ | det ·|−zj × · · · × δ1∗ | det ·|−z1 (11.10)

where Re(z1 ) ≤ · · · ≤ Re(zj ) ≤ 0; the representations δi , τk are irreducible, automorphic and unitary, with central characters which are trivial on (AF )+ ∞ , and τi = τi∗ , for 1 ≤ i ≤ ℓ. If δi (resp. τk ) is on GLa (AE ), a > 1, we assume it is cuspidal. Consider LS (σ × δˆ1 , s) (if G is metaplectic, consider LSψ (σ × δˆ1 , s)). As usual, we assume that S is finite, contains the places at infinity, and outside S, σ, δ1 (and ψ) are unramified. We have LS (σ × δˆ1 , s) =

j Y

i=1

LS (δi × δˆ1 , s + zi )LS (δi∗ × δˆ1 , s − zi )

r Y

i=1

LS (τi × δˆ1 , s). (11.11)

This product has a pole at s = 1 − z1 . (It comes from LS (δ1 × δˆ1 , s + z1 ). Note that Re(1 − z1 ), Re(1 − z1 ± zi ) ≥ 1, so that the other factors in the product do not cancel this pole.) Thus, LS (σ × δˆ1 , s) has a pole at s = 1 − z1 . (Similarly, when G is metaplectic, LSψ (σ × δˆ1 , s) has a pole at s = 1 − z1 ). By Theorem 10.8 and Proposition 10.1, except in case G is metaplectic, and δ1 = 1, we have z1 = 0 and LS (δˆ1 , βHG,k′ , s) has a pole at s = 1. Here, δ1 is on GLk (AE ), and k ′ = k in all cases of Table (11.9), except cases (3) and (6), where k ′ = k − 1. Note that since LS (δˆ1 , βHG,k′ , s) has a pole at s = 1, we must have δ1 = δ1∗ . (For example, in case of a unitary group, and η = δˆ1 , LS (η × η ′ , s) = LS (η, Asai, s)LS (η ⊗ γ, Asai, s),

and since one of the factors on the r.h.s. has a pole at s = 1, and, as is well known, by Shahidi’s theorem [Shahidi, F. (1981)], both of them do not vanish at s = 1, LS (η × η ′ , s) has a pole at s = 1, which implies that ηˆ′ = η, i.e. η ∗ = η). We conclude from (11.11) that LS (σ × δˆ1 , s) has a pole at s = 1, which is at least a double pole. This is impossible by Theorem 10.8 and Proposition 10.1. It follows that (11.10) has the form τ1 × · · · × τr ,

(11.12) S

and repeating the last argument, we conclude that L (τi , βHG,m′ , s) has a pole at i s = 1, for i = 1, ..., r, and also that τi 6= τj , for 1 ≤ i 6= j ≤ r. This also implies that the representation (11.12) is irreducible (in each place this is a parabolic induction to GLm from an irreducible, unitary representation of the Levi part, and hence this is irreducible). Here, τi is on GLmi (AE ). Consider the product of partial L-functions LS (σ × τ1′ , s1 ) · ... · LS (σ × τr′ , sr ),

(11.13)

where s1 , ..., sr are complex variables. In case GA is metaplectic, we replace each factor LS (σ × τi , si ) by LSψ (σ × τi , si ). By what we just proved, this product has a simple pole at (1, ..., 1), in the sense that the function (s1 − 1) · ... · (sr − 1)LS (σ × τ1′ , s1 ) · ... · LS (σ × τr′ , sr )

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is holomorphic and nonzero at (1, ..., 1). The product (11.13) can be represented by the global integrals of the previous chapter. Indeed, let τ (s1 , ..., sr−1 ) be the automorphic representation of GLm (AE ) realized by Eisenstein series corresponding to parabolic induction from ′ τ1′ | det ·|s1 −1 ⊗ · · · ⊗ τr−1 | det ·|sr−1 −1 ⊗ τr′ .

or L(ϕσ , Now consider the global integrals L(ϕσ , fτ (s1 ,...,sr−1 ),sr ), fτ (s1 ,...,sr−1 )⊗γ,sr , φ) and use Theorem 10.7. We conclude that these integrals have a pole at (s1 , ..., sr ) = (1, ..., 1), and, as before, we conclude that the representation σ ¯ has a nontrivial L2 -pairing with the descent σψ,α (τ¯′ ), or σψ,γ (τ¯′ ⊗ γ), with ψ chosen in a compatible way with the Whittaker character, with respect to which σ is globally generic. Here, we used the notation of Sec. 3.6 with τ¯′ meaning that we replaced τi by τi′ , and, similarly, τ¯′ ⊗ γ signifies that we take τi′ ⊗ γ instead of τi . Note that the multi-residue at (1, ..., 1) of the Eisenstein series E(fτ (s1 ,...,sr−1 )⊗γ , sr ) f 2n (A) is metaplectic, we is exactly Eτ¯′ ⊗γ of Chapter 2. Finally, in case GA = Sp see from Proposition 10.1, that it is possible to have δ1 = 1, and z1 = − 21 , and in this case σ is a (ψ) theta lift from an irreducible, automorphic, cuspidal, generic representation σ ′ of SO2n−1 (A), so that by the previous section (see (11.2)), the lift of σ to GL2n (A) has the form 1GL2 × τ1 × · · · × τr , where τi are as before, each one with its exterior square L-function having a pole at s = 1. Of course, if n = 1, then r = 0. In [Jiang and Soudry (2007)], it is proved, that (when n ≥ 2) LS (σ ′ , 12 ) = 0, and so, at least one of the representations τi is such that the central value of its L-function is zero. This proves Theorem 11.2. Let σ be an irreducible, automorphic, cuspidal and globally generic representation of GA (genuine when GA is metaplectic). Let m be the degree of the standard representation of L G0 . Assume that σ lifts weakly to an irreducible automorphic representation τ of GLm (AE ). Then, except in case G is metaplectic, τ has the form τ1 × · · · × τr ,

where for 1 ≤ i ≤ r, we have the following.

(1) The representation τi is an irreducible, automorphic, unitary representation of GLmi (AE ), and is cuspidal in case mi > 1. (2) Each representation τi is self-conjugate, that is τi∗ = τi . Q (3) The restriction of ωτ = ℓi=1 ωτi to A∗ is trivial, except in case G is even orthogonal and quasi-split, where ωτ = χG . (4) The partial L-function LS (τi , βHG,m′ , s) has a simple pole at s = 1. Here m′i = i mi in all cases of Table (11.9) except cases (3) and (6), where m′i = mi − 1. (5) The representations τi are pairwise different; τi 6= τj , for all 1 ≤ i 6= j ≤ r. (6) The complex conjugate of σ has a nontrivial L2 -pairing with the descent of τ ′ ⊗γ (σψ,α (τ¯′ ), or σψ,γ (τ¯′ ⊗γ), with ψ chosen in a compatible way with the Whittaker character, with respect to which σ is globally generic.

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f 2n (A) is metaplectic, either τ has the form above and satisfies When GA = Sp (1)–(6), or τ has the form 1GL2 × τ1 × · · · × τr ,

where, if r ≥ 1, then t = τ1 × · · ·× τr is the weak functorial lift of an irreducible, automorphic, cuspidal, globally generic representation σ ′ of (split) SO2n−1 (A) to GL2n−2 (A), and hence (t, σ ′ ) satisfy the properties (1)–(6) above. Moreover, LS (σ ′ , 12 ) = 0. In this case, σ has a nontrivial ψ- theta lift to a cuspidal, globally generic representation of SO2n−1 (A), and we can take σ ′ to be this theta lift. The heart of this work lies in the first nine chapters. There we went in the opposite direction. What we did was to take an irreducible, automorphic representation of GLm (AE ), of the form τ1 × · · · × τr , satisfying the first five properties listed in the last theorem, and considered the parabolic induction to HA , from 1

1

(τ1′ | det ·|s1 − 2 ⊗ · · · τr′ | det ·|sr − 2 ) ⊗ γ. We formed the corresponding Eisenstein series of HA , and considered its multiresidue at (1, ..., 1). We studied the automorphic representation of GA obtained by applying to the residual representation Gelfand-Graev coefficients, or Fourier-Jacobi coefficients, stabilized by G. We got the descent of τ ′ ⊗ γ to GA , and we proved that each irreducible summand σ of the descent is globally generic and lifts weakly to τ ′ . Finally, the same remark we made at the end of the previous section applies here. Thus in (6) of the last theorem, we have that σ has a nontrivial L2 -pairing with σψ−1 ,α (¯ τ ), or σψ−1 ,γ −1 (¯ τ ⊗ γ −1 ). In the following sections we will present applications to generalized endoscopy, base change and automorphic induction. 11.3

On generalized endoscopy

Theorem 11.2 tells us the structure of the functorial lift to GLm (AE ) of the generic cuspidal σ on GA . If the lift is not cuspidal, then the isobaric summands τi in the theorem are themselves in the image of the functorial lift from generic cuspidal representations σi lying in the descent of τi to the appropriate classical groups Gi (A). Let us list the various cases. 1. G = SO2n+1 Let σ be an irreducible, automorphic, cuspidal and globally generic representation of SO2n+1 (A). Assume that the lift of σ to GL2n (A) is not cuspidal. By Theorem 11.2 the lift of σ has the form τ = τ1 × · · · × τr , where the representations τi are pairwise inequivalent, and each τi is an irreducible, automorphic, cuspidal representation of GLmi (A), such that LS (τi , ∧2 , s) has a pole at s = 1. In particular, each

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mi = 2ni is even. Thus, we can apply descent to each τi . By Theorem 9.4, applied to τi , and Theorem 10.1, the descent of τi to SO2ni +1 (A) provides us with an irreducible, automorphic, cuspidal, globally generic representation σi of SO2ni +1 (A), which lifts to τi . Therefore σ is a (generalized) endoscopic weak lift of σ1 ⊗ · · · ⊗ σr on SO2n1 +1 (A) × · · · × SO2nr +1 (A). This lift corresponds to the L-group map Sp2n1 (C) × · · · × Sp2nr (C) ֒→ Sp2n (C) obtained by the direct sum embedding of symplectic spaces C2n1 ⊕ · · · ⊕ C2nr ֒→ C2n . Conversely, let σ1 , ..., σr be irreducible, automorphic, cuspidal, generic representations of SO2n1 +1 (A), ..., SO2nr +1 (A) respectively. Put n = n1 + · · · + nr . Consider the lifts τi of σi to GL2ni (A), τi = τi1 × · · · × τiki , as in Theorem 11.2, i = 1, ..., r. i . Clearly, if Ci ∩ Ci′ = ∅ for all 1 ≤ i 6= i′ ≤ r, then Denote Ci = {τij }kj=1 i τij lies in the image of the lift from SO2n+1 (A), and hence, τ = ×ri=1 τi = ×ri=1 ×kj=1 by Theorems 9.4, 10.1, the descent of τ to SO2n+1 (A) provides us with an irreducible, automorphic, cuspidal, general representation σ of SO2n+1 (A), which is a generalized endoscopic weak lift of σ1 ⊗ · · · ⊗ σr . Summarizing Theorem 11.3. Let σ be an irreducible, automorphic, cuspidal, generic representation of SO2n+1 (A). Assume that the lift of σ to GL2n (A) is not cuspidal. Then there exist irreducible, automorphic, cuspidal, generic representations σ1 , ..., σr of SO2n1 +1 (A), ..., SO2nr +1 (A), respectively, n1 + · · · + nr = n, such that σ is a generalized endoscopic weak lift of σ1 ⊗ · · · ⊗ σr . Conversely, let σ1 , σ2 , ..., σr be irreducible, automorphic, cuspidal, generic representations of SO2n1 +1 (A), ..., SO2nr +1 (A) respectively, n1 + · · · + nr = n. Consider the sets {Ci }ri=1 as above. If Ci ∩ Cj = ∅ for all 1 ≤ i 6= j ≤ r, then there is an irreducible, automorphic, cuspidal, generic representation σ of SO2n+1 (A), which is a generalized endoscopic weak lift of σ1 ⊗ · · · ⊗ σr . Otherwise, cuspidal data for SO2n+1 (A) can be specified, so that σ1 ⊗ · · · ⊗ σr lifts to a constituent of the corresponding parabolically induced representation. The last case of the theorem can be explained as follows. If the sets Ci are not pairwise disjoint, then we can rewrite the representation (up to semi-simplification) i τ = ×ri=1 τi = ×ri=1 ×kj=1 τij in the form η1 × · · · ηj × π1 × · · · × πk × ηj × · · · × η1 , where η1 , ..., ηj are irreducible, cuspidal, not necessarily pairwise disjoint; π1 , ..., πk are irreducible, cuspidal and pairwise disjoint (some of them may be equal to one of the ηi ). Of course, all ηi , πt are such that there exterior square L-function has a pole at s = 1. Let πi be a representation of GL2ti (A), and let t1 + · · · + tk = ℓ. Then the descent of π1 × · · · × πk to SO2ℓ+1 (A) provides us with an irreducible, automorphic, cuspidal, generic representation σ0 of SO2ℓ+1 (A), which lifts to π1 × · · · × πk . Now,

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it is clear that σ1 ⊗ · · · ⊗ σr has a weak functorial lift to any constituent of the parabolic induction (to SO2n+1 (A)) corresponding to η1 ⊗ · · · ηj ⊗ σ0 . Here is a simple, nice example. Let σ1 , ..., σn be pairwise different irreducible, automorphic, cuspidal representations of PGL2 (A). Then there is an irreducible, automorphic, cuspidal, generic representation σ of SO2n+1 (A), which is a generalized endoscopic weak lift of σ1 ⊗ · · · ⊗ σn . The descent from GL2n (A) to SO2n+1 (A), as in Theorem 9.4 is actually irreducible. This was proved by Jiang and Soudry. See [Jiang and Soudry (2003)], [Jiang and Soudry (2004)]. There, it is also proved that irreducible, automorphic, cuspidal, generic representations of SO2n+1 (A) are rigid, in the sense that if two such representations are isomorphic almost everywhere, then they are isomorphic. Thus, the cuspidal representations σ and endoscopic data σ1 , ..., σr in the last theorem are all determined uniquely, up to isomorphism. f 2. G = Sp 2n

Let σ be an irreducible, genuine, automorphic, cuspidal and globally ψ-generic f (A). Assume that the ψ- lift of σ to GL2n (A) is not cuspidal. representation of Sp 2n By Theorem 11.2, either σ lifts, by the theta correspondence, with respect to ψ, to an irreducible, cuspidal, generic representation σ ′ of SO2n−1 (A), or the lift of σ to GL2n (A) has the form τ = τ1 × · · · × τr , where the representations τi are pairwise inequivalent, and each τi is an irreducible, automorphic, cuspidal representation of GL2ni (A), such that LS (τi , ∧2 , s) has a pole at s = 1, and LS (τi , 12 ) 6= 0. So, we can apply descent to each τi . By Theorem 9.7, applied to τi , and Theorems 10.1, f 2n (A) provides us with an irreducible, automorphic, 10.4, the descent of τi to Sp i f 2n (A), which lifts to τi , with respect to cuspidal, ψ-generic representation σi of Sp i ψ. Therefore σ is a weak functorial lift, with respect to ψ, of σ1 ⊗ · · · ⊗ σr on f 2n (A). This lift corresponds, as in the previous case, to the f 2n (A) × · · · × Sp Sp r 1 L-group map Sp2n1 (C) × · · · × Sp2nr (C) ֒→ Sp2n (C)

given by the direct sum embedding. We will call such a lift a generalized endoscopic weak lift. Conversely, let σ1 , ..., σr be irreducible, genuine, automorphic, cuspidal, f 2n (A), respectively. Put n = n1 + · · ·+ f 2n (A),..., Sp ψ-generic representations of Sp r 1 nr . Consider the lifts, with respect to ψ, τi of σi to GL2ni (A), τi = τi1 × · · · × τiki , i = 1, ..., r. Now, by Theorem 11.2, we have to distinguish two cases, according to whether the ψ-theta lift of σi to SO2ni −1 (A) is nontrivial or trivial. Assume that this theta lift is nontrivial for 1 ≤ i ≤ r′ , and trivial for r′ + 1 ≤ i ≤ r. Denote, for 1 ≤ i ≤ r′ , by σi′ , the ψ-theta lift of σi to SO2ni −1 (A) (by [Jiang and Soudry (2007)], it is irreducible). If ni = 1, we ignore σi′ . By Theorem 11.2, LS (σi′ , 12 ) = 0. The lift of σi′ to GL2ni −2 (A) has the form ti = τi1 × · · · × τiki , so that the ψ-lift

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i . σi to GL2ni (A) is τi = 1GL2 × ti . Denote by Ci the set of representations {τij }kj=1 2 ′ S Note that, for each 1 ≤ i ≤ r and 1 ≤ j ≤ ki , L (τij , ∧ , s) has a pole at s = 1, and there exists 1 ≤ t ≤ ki , such that LS (τit , 21 ) = 0. For r′ + 1 ≤ i ≤ r, τi has the form τi1 × · · · × τiki , where, for each 1 ≤ j ≤ ki , LS (τij , ∧2 , 2s − 1)LS (τi , s − 21 ) has i . In the following theorem, we state a pole at s = 1. Denote, here, Ci = {τij }kj=1 ′ ′ the two basic cases r = 0 and r = r. The general case can be easily deduced from these two.

Theorem 11.4. 1. Let σ be an irreducible, genuine, automorphic, cuspidal, ψf 2n (A). Assume that the ψ- lift of σ to GL2n (A) generic representation of Sp is not cuspidal. Then either σ has a nontrivial ψ-theta lift to SO2n−1 (A), as in Theorem 11.2, or there exist irreducible, genuine, automorphic, cuspidal, ψf 2n (A), respecf 2n (A), ..., Sp f 2n (A), Sp generic representations σ1 , σ2 , ..., σr of Sp r 2 1 tively, n1 + · · · + nr = n, such that σ is a generalized endoscopic weak lift, with respect to ψ, of σ1 ⊗ · · · ⊗ σr . 2. Conversely, let σ1 , σ2 , ..., σr be irreducible, genuine, automorphic, cuspidal, ψf 2n (A) respectively, n1 + · · · + nr = n. f 2n (A), ..., Sp generic representations of Sp r 1 r Consider the sets {Ci }i=1 as above. 2.a Assume that r′ = 0. If Ci ∩ Cj = ∅, for all 1 ≤ i 6= j ≤ r, then there is an f (A), irreducible, genuine, automorphic, cuspidal, ψ-generic representation σ of Sp 2n which is a generalized endoscopic weak lift, with respect to ψ, of σ1 ⊗ · · · ⊗ σr . f 2n (A) can be If the sets Ci are not pairwise disjoint, then cuspidal data for Sp specified, so that σ1 ⊗ · · · ⊗ σr lifts to a constituent of the corresponding parabolically induced representation. 2.b Assume that r′ = r. If Ci ∩ Cj = ∅, for all 1 ≤ i 6= j ≤ r, then there is an irreducible, genuine, automorphic, cuspidal, ψ-generic representation σ of f 2(n−r+1) (A), such that σ1 ⊗ · · · ⊗ σr lifts weakly, with respect to ψ, to (any conSp f 2n (A) from stituent of ) the parabolic induction to Sp 1

1

(| · | 2 ⊗ · · · ⊗ | · | 2 ⊗ σ)µψ ,

(11.14)

1 2

where the character | · | is taken r − 1 times. f 2(n−r+1 (A) can If the sets Ci are not pairwise disjoint, then cuspidal data for Sp be specified, so that σ in (11.14) can be replaced by the corresponding parabolically induced representation. Proof. Parts 1 and 2.a are proved exactly as in Theorem 11.3, using Theorems 9.7, 10.1, 10.4. As for part 2.b, here, when the sets Ci are pairwise disjoint, we use Theorem 11.3 to construct an irreducible, automorphic, cuspidal, generic representation σ ′ of SO2(n−r+1)−1 (A), which is a generalized endoscopic weak lift of σ1′ ⊗ · · · ⊗ σr′ . We also know that LS (σ ′ , 12 ) = 0, and hence the ψ-theta lift of σ ′ f 2(n−r+1) (A) is nontrivial, cuspidal and ψ-generic. Take such an irreducible to Sp summand σ of this theta lift. Then σ satisfies the requirements of the theorem. 

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3. G = SO2n , Sp2n Let σ be an irreducible, automorphic, cuspidal ψ-generic representation of SO2n (A) (split or quasi-split over F ) or Sp2n (A). Assume that the lift of σ to GL2n (A), or, respectively, to GL2n+1 (A) is not cuspidal. By Theorem 11.2 the lift of σ has the form τ = τ1 × · · · × τr , where the representations τi are pairwise inequivalent, and each τi is an irreducible, automorphic, representation of GLmi (A), cuspidal when mi > 1, such that LS (τi , sym2 , s) has a pole at s = 1. If G = SO2n , then ωτ = χG . First, let us consider the case, where there is an index i, such that mi = 1. This means that τi = χi is a character of A∗ /F ∗ , such that χ2i = 1. In this case, σ is roughly in the image of the theta correspondence from Sp2n−2 (A), or SO2n (A), respectively. More precisely, if σ is a representation of SO2n (A), then since LS (σ × χi , s) has a pole at s = 1, LS (σ ⊗ χi , s) has a pole at s = 1, where now we view χi as a character of SO2n (A), by composition with the spinor norm. This implies that the ψ-theta lift of σ ⊗ χi to Sp2n−2 (A) is (nontrivial), cuspidal and ψ-generic. This is proved in [Ginzburg, Rallis and Soudry II (1997)], when SO2n is split. The proof is the same, when it is quasi-split over F . If σ is a representation of Sp2n (A), then, as before, LS (σ × χi , s) has a pole at s = 1. Write χi (t) = (t, αi ), the Hilbert symbol of t with αi ∈ F ∗ /(F ∗ )2 . Then the ψ-theta lift of σ to SO2n,αi (A) is (nontrivial) cuspidal and ψ-generic. This is proved in [Ginzburg, Rallis and Soudry II (1997)], when αi is a square, that is χi is trivial, and SO2n,αi is split. The proof is similar, when αi is not a square. Thus, in both cases, we can carry on by induction, with the cuspidal ψ-generic representation at hand on Sp2n−2 (A), or SO2n,αi (A), respectively. Assume now that mi > 1, for all 1 ≤ i ≤ r. Then we can apply descent to each τi . Let ωτi = χαi , the global Hilbert symbol character, corresponding to the square class of αi in F ∗ . If mi = 2ni + 1, then by Theorem 9.1, and Theorems 10.1, 10.3, the descent of τi ⊗ χαi to Sp2ni (A), with respect to ψ, σψ (τi ⊗ χαi ) := σψ,αi (τi ), provides us with an irreducible, automorphic, cuspidal, ψ-generic representation σi of Sp2ni (A), which lifts to τi ⊗ χαi . If mi = 2ni is even, then by Theorem 9.2 (and its notations), and Theorems 10.1, 10.3, the descent σψ,αi (τi ) provides us with an irreducible, automorphic, cuspidal, ψ-generic representation σi of SO2ni ,αi (A), which lifts to τi . (We do not know that the descent is irreducible; we simply pick any irreducible summand.) We may assume that mi = 2ni + 1 are odd, for i ≤ r′ , and mi = 2ni are even, for r′ + 1 ≤ i ≤ r. Therefore σ is a “generalized endoscopic weak lift” of σ1 ⊗ · · · ⊗ σr on Sp2n1 (A) × · · · × Sp2nr′ (A) × SO2nr′ +1 ,αr′ +1 (A) × · · · × SO2nr ,αr (A).

Assume that σ is a representation of SO2n (A) (split or quasi-split), then r′ must be even, and χG = χα1 · ... · χαr . Denote α = α1 · ... · αr . We can explain this lift in terms of an L-group homomorphism described as follows. Let Ei be the extension of F generated by the square root of αi , for 1 ≤ i ≤ r, and let L be the extension of F generated by E1 , ..., Er . Let us take the L-groups as [SO2n1 +1 (C) × · · · × SO2nr′ +1 (C) × SO2nr′ +1 (C) × · · · × SO2nr (C)] ⋊ ΓL/F , (11.15)

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and SO2n (C) ⋊ ΓL/F .

(11.16)

We will write the elements of SO2ni +1 (C) according to the standard anti-diagonal matrix of size 2ni + 1, i ≤ r′ . Consider C2ni +1 , equipped with the corresponding quadratic form. Similarly for SO2nj (C), r′ + 1 ≤ j ≤ r. We will write the elements of SO2n (C) according to the direct sum of the quadratic forms of C2n1 +1 , ..., C2nr . The Galois group ΓL/F acts as follows. Let ǫ be an element of ΓL/F . Let gi ∈ SO2ni +1 (C), 1 ≤ i ≤ r′ , and gj ∈ SO2nj (C), r′ + 1 ≤ j ≤ r. Assume that r > r′ . Then i

′

−i

′

+1 r +1 gr′ +1 vr′ +1 , ..., vrir gr vr−ir ), ǫ(g1 , ..., gr )ǫ−1 = (g1 , ..., gr′ , vrr′ +1

(11.17)

where, for r′ + 1 ≤ j ≤ r

  01 vj = diag(Inj −1 , , Inj −1 ) ∈ O2nj (C), 10 √ √ and ij = 0, 1 is determined by ǫ( αj ) = (−1)ij αj . For g ∈ SO2n (C), ǫgǫ−1 = v t gv −t ,

√ √ where v = diag(I2(n−nr ) , vr ) ∈ O2n (C); t is determined by ǫ( α) = (−1)t α. Now, define a map f from the L-group (11.15) to the L-group (11.16) by f ((g1 , ..., gr )ǫ) = i

′

i

r−1 +1 ⊕ · · · ⊕ gr−1 vr−1 ((−1)i1 g1 ⊕ ... ⊕ (−1)ir′ gr′ ⊕ gr′ +1 vrr′ +1 ⊕ gr vri1 +···+ir−1 )ǫ. (11.18)

This is an L-homomorphism. Assume that r = r′ . Thus, r is even. Then we take our L-groups as above, with the following action of the Galois group ΓL/F . Let ǫ be an element of ΓL/F . Let gi ∈ SO2ni +1 (C), 1 ≤ i ≤ r. Then (with ir , as before) ǫ(g1 , ..., gr )ǫ−1 = (g1 , ..., gr−1 , ωrir gr ωr−ir )

where, ωr = diag(Inr , −1, Inr ) ∈ O2nr +1 (C). For g ∈ SO2n (C),

ǫgǫ−1 = ω t gω −t ,

where ω = diag(I2(n−nr )−1 , ωr ) ∈ O2n (C), and t is as before. Note that the Lgroup (SO2n1 +1 (C) × ... × SO2nr +1 (C)) · Γ(L/F ) is isomorphic to the direct product (SO2n1 +1 (C) × ... × SO2nr +1 (C)) × Γ(L/F ). Now, we modify f as follows. f ((g1 , ..., gr )ǫ) = ((−1)i1 g1 ⊕ ... ⊕ (−1)ir−1 gr−1 ⊕ gr ωri1 +···+ir−1 )ǫ.

(11.19)

This is an L-homomorphism, and it describes the weak lift from σ1 ⊗ ... ⊗ σr to σ, as above.

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Assume that σ is a representation of Sp2n (A). Then r′ must be odd. In this case, ωτ is trivial, and hence χα1 · ... · χαr = 1. We keep notation as above. We will take the L-group of Sp2n over F to be the direct product SO2n+1 (C) × Γ(L/F ), and the L-group of Sp2n1 × ... × Sp2nr′ × SO2nr′ +1 ,αr′ +1 × ... × SO2nr ,αr as in (11.15) (also in case r = r′ , where the action is trivial). Define now f : [Sp2n1 ×...×Sp2nr′ ×SO2nr′ +1 ,αr′ +1 ×...×SO2nr ,αr ]·Γ(L/F ) 7→ SO2n+1 (C)×Γ(L/F ) by i

′

+1 ⊕ · · · ⊕ gr vrir , ǫ). (11.20) f ((g1 , ..., gr )ǫ) = ((−1)i1 g1 ⊕ ... ⊕ (−1)ir′ gr′ ⊕ gr′ +1 vrr′ +1

Note that here i1 + ... + ir is even, so that this definition is the same as (11.18). This is an L-homomorphism, and it describes the weak lift from σ1 ⊗ ... ⊗ σr to σ, in this case. Conversely, let α1 , ..., αr be elements in F ∗ , and think of them modulo squares in F ∗ . Fix 0 ≤ r′ ≤ r. Assume that α = α1 · ... · αr is a square in F ∗ , when r′ is odd. Let σ1 , ..., σr be irreducible, automorphic, cuspidal, ψ- generic representations of Sp2n1 (A), ..., Sp2nr′ (A), SO2nr′ +1 ,αr′ +1 (A), ..., SO2nr ,αr (A) respectively. Put (2n1 + 1) + · · · + (2nr′ + 1) + 2nr′ +1 + · · · + 2nr = 2n, 2n + 1, according to the parity of r′ . Consider the lifts τ˜i of σi to GL2ni +1 (A), for i ≤ r′ , and put τi = τ˜i ⊗ χαi . Note that ωτ˜i = 1. For r′ + 1 ≤ i ≤ r, consider also the lifts τi of σi to GL2ni (A). We know that each one of the representations τ˜i has the form τ˜i = τ˜i1 × · · · × τ˜iki , i ≤ r′ , as in Theorem 11.2, and similarly for the representations τi . Denote Ci = i i , for r′ + 1 ≤ i ≤ r. If the sets , for i ≤ r′ , and Ci = {τij }kj=1 {τij = τ˜ij ⊗ χαi }kj=1 Ci , i ≤ r′ are pairwise disjoint, and so are also the sets Ci , r′ + 1 ≤ i ≤ r, then i τij lies in the image of the lift from GA = SO2n,α (A), when τ = ×ri=1 τi = ×ri=1 ×kj=1 ′ r is even, or GA = Sp2n (A), when r′ is odd. Thus, by Theorems 9.1, 9.2, 10.1, the descent of τ to GA provides us with an irreducible, automorphic, cuspidal, ψ-generic representation σ of GA , which is a generalized endoscopic weak lift of σ1 ⊗ · · · ⊗ σr , with respect to the L-homomorphism f described above. Summarizing Theorem 11.5. A. Let σ be an irreducible, automorphic, cuspidal, ψ- generic representation of GA , where G = SO2n,α (split, or quasi-split over F ), or G = Sp2n . Assume that the lift of σ to GL2n (A), or, respectively, to GL2n+1 (A) is not cuspidal. Then either there exists a quadratic character (the trivial character included) χβ , such that LS (σ × χβ , s) has a pole at s = 1, or we have the following. There exist 1. integers 0 ≤ r′ ≤ r; r′ odd integers, each one larger than 1, {mi = 2ni + 1}i≤r′ ; r − r′ + 1 positive even integers {mi = 2ni }r′ +1≤i≤r , such that m1 + ... + mr is equal to 2n, or 2n + 1, respectively; 2. r elements α1 , ..., αr in F ∗ , such that, modulo squares in F ∗ , α1 · ... · αr is equal to α, or 1, respectively; 3. irreducible, automorphic, cuspidal, ψ-generic representations σ1 , ..., σr of Sp2n1 (A), ..., Sp2nr′ (A), SO2nr′ +1 ,αr′ +1 (A), ..., SO2nr ,αr (A). The representation σ is a generalized endoscopic lift of σ1 ⊗ ... ⊗ σr , with respect to

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the L-homomorphism f = fα1 ,...,αr described in (11.18) - (11.20). A. Assume that LS (σ × χβ , s) has a pole at s = 1. 1. If σ is on SO2n,α (A), then there is an irreducible, automorphic, cuspidal, ψgeneric representation π of Sp2n−2 (A), whose ψ-theta lift to SO2n,α (A) contains σ ⊗ χβ . In this case, denote the lift of π to GL2n−1 (A) by t. Then the lift of σ to GL2n (A) is (t ⊗ χ αβ ) × χβ .

(11.21)

2. If σ is on Sp2n (A), then there is an irreducible, automorphic, cuspidal, ψ-generic representation π of SO2n,β , whose ψ-theta lift to Sp2n (A) contains π. In this case, denote the lift of π ⊗ χβ to GL2n (A) by t. Then the lift of σ to GL2n+1 (A) is t × χβ .

(11.22)

B. Conversely, start with α1 , ..., αr ∈ F ∗ , and σ1 , σ2 , ..., σr irreducible, automorphic, cuspidal, ψ-generic representations of Sp2n1 (A), ..., Sp2nr′ (A), SO2nr′ +1 ,αr′ +1 (A), ..., SO2nr ,αr (A) respectively, where 0 ≤ r′ ≤ r, α1 , ..., αr , and n are as in the first part. Assume that α = α1 ·...·αr is a square in F ∗ , when r′ is odd. Consider the sets {Ci }ri=1 as above. If these sets are pairwise disjoint, then there is an irreducible, automorphic, cuspidal, generic representation σ of SO2n,α (A), when r′ is even, or of Sp2n (A), when r′ is odd, which is a generalized endoscopic weak lift of σ1 ⊗ · · · ⊗ σr , with respect to the L-homomorphism f = fα1 ,...,αr described in (11.18) - (11.20). Otherwise, cuspidal data for SO2n,α (A), or Sp2n (A), respectively, can be specified, so that σ1 ⊗ · · · ⊗ σr lifts to a constituent of the corresponding parabolically induced representation. 4. G = U2n+1 , U2n Let σ be an irreducible, automorphic, cuspidal ψ-generic representation of Um (A). Assume that the lift (standard base change lift) of σ to GLm (AE ) is not cuspidal. By Theorem 11.2 the lift of σ has the form τ = τ1 × · · · × τr , where the representations τi are pairwise inequivalent, and each τi is an irreducible, automorphic, representation of GLmi (A), cuspidal when mi > 1, such that LS (τi′ ⊗ µm , Asai, s) has a pole at s = 1. Here, µm = 1, when m is odd, and when m is even, µm is a character, γ, which we fix, of A∗E /E ∗ , whose restriction to A∗ is the quadratic character ωE/F . Note that in this case, LS (τi ⊗ γ, Asai, s) is independent of the choice of γ. We also know that the restriction of ωτ to A∗ is trivial. Note, also that LS (τi′ ⊗ µm , Asai, s) = LS (τi ⊗ µm , Asai, s). As before, let us consider first the case, where there is an index i, such that mi = 1. This means that τi = χi is m−1 a character of A∗E /E ∗ , such that its restriction to A∗ is ωE/F . Thus, there is an automorphic character η of U1 (A), such that χi µm = ηE , the (base change) lift of η to A∗E , i.e. ηE (x) = η( xx¯ ). We conclude that σ is roughly in the image of the theta correspondence from Um−1 (A). More precisely, since LS (σ × χ−1 i , s) has a pole at s = 1, the (ψ, γ)-theta lift of σ ⊗ χ−1 to U (A) is (nontrivial), cuspidal m−1 i

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and ψ-generic. This is proved in [Grenie (2004)], when m is odd. The proof is similar for m even. Thus, as before, we can carry on by induction, with the cuspidal ψ-generic representation at hand on Um−1 (A). One can check that if π is an irreducible, automorphic, cuspidal, ψ-generic representation of Um−1 (A), such that the (ψ, γ)-theta lift of π to Um (A) contains σ ⊗ χ−1 i , then the lift of σ to GLm (AE ) −1 is of the form t × ηE µm , where t is the lift to GLm−1 (AE ) of π ⊗ ηµm−1 (det ·). Assume that mi > 1, for 1 ≤ i ≤ r. Then we can apply descent to each τi′ ⊗ µm . Note that the parities of m and mi might be opposite. By Theorems 9.5, 9.7, the complex conjugate of the descent of τi′ ⊗ µm to Umi (A), with respect to (ψ, µmi ), σψ,µmi (τi′ ⊗ µm ), provides us with an irreducible, automorphic, cuspidal, ψ-generic representation σi of Umi (A), such that σi lifts to τi ⊗ µ−1 m µmi . Therefore σ is a “generalized endoscopic weak lift” of σ1 ⊗ · · · ⊗ σr on Um1 (A) × · · · × Umr (A). Assume that mi = 2ni + 1 are odd, for i ≤ r′ , and mi = 2ni are even, for r′ + 1 ≤ i ≤ r. Put mr′ +1 + ... + mr = 2ℓ. We will explain this lift in terms of an L-group homomorphism. Here, we will need to replace the Galois group by the Weil group of F in the formation of the L-group. See [Rogawski (1990)], Sec. 4.6-4.8. Fix an element φ in WE/F which projects to the Galois conjugation in ΓE/F . Then WF = WE ∪ WE φ. We take the L-group of Uk as L

Uk = GLk (C) ⋊ WF ,

where WE acts trivially on GLk (C), and φ acts by φgφ−1 = dk wkt g −1 (dk wk )−1 ,

(11.23)

where wk is the standard k×k anti-diagonal matrix which has 1 on the anti-diagonal and zero elsewhere; dk = Ik , if k is odd, and if k = 2n is even, then dk = diag(In , −In ). Let L

(Um1 × ... × Umr ) = (GLm1 (C) × ... × GLmr (C)) ⋊ WF ,

where the action of WF is coordinate-wise, according to (11.23). Define f :L (Um1 × ... × Umr ) 7→L Um , as follows. Assume that m is even (and so r′ is even). f (g1 , ..., gr ) · wφj ) = (i2 (gr′ +1 ⊕ ... ⊕ gr ) · 1)i1 ((g1 , ..., gr′ ) · wφj ),

(11.24)

where w ∈ WE , j = 0, 1, and the maps i1 , i2 are defined as follows. Write, for r′ + 1 ≤ i ≤ r,   g g gi = i,1 i,2 , gi,j ∈ Mni (C) (1 ≤ j ≤ 4). gi,3 gi,4

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Then

Next,



      i2 (gr′ +1 ⊕ ... ⊕ gr ) =      

gr′ +1,1

gr′ +1,2 ..

. gr,1

gr,2 Im−2ℓ

gr,3

gr,4 ..

.

gr′ +1,3

gr′ +1,4 

 i1 ((g1 , ..., gr′ ) · wφj ) = diag(Iℓ , γ(w)a 



g1 ..

. gr ′

where



      .     

 −1 j j  a α , Iℓ ) · wφ ,

α = diag(−1, 1, −1, 1, ..., −1, 1)wm−2ℓ,

(11.25)

(11.26)

(11.27)

and a ∈ GLm−2ℓ (C) may be chosen (and fixed), such that t

a[wm−2ℓ dm−2ℓ α−1 ]a = diag(wm1 , ..., wmr′ ).

(11.28)

Note that the matrices wm−2ℓ dm−2ℓ α−1 (diagonal with ±1 on the diagonal) and diag(wm1 , ..., wmr′ ) are symmetric. Finally, we view γ, in (11.26), as a character of WE . Assume that m is odd (so that r′ is odd). Then we define f (g1 , ..., gr )·wφj ) = i2 (γ −1 (w)gr′ +1 ⊕ ...⊕ γ −1 (w)gr )i′1 ((g1 , ..., gr′ )β j ·wφj , (11.29) where the embedding i2 is exactly as in (11.25); β = diag(Iℓ , Im−2ℓ , −Iℓ );

 g1  i′1 (g1 , ..., gr′ ) = diag(Iℓ , b  . . .

gr ′

The matrix b ∈ GLm−2ℓ (C) is chosen such that t

(11.30) 

 −1  b , Iℓ ).

bwm−2ℓ b = diag(wm1 , ..., wmr ).

(11.31)

(11.32)

The map f (in each case) is an L-homomorphism, and it describes the “generalized endoscopic weak lift” which takes σ1 ⊗ · · · ⊗ σr to σ from Um1 (A) × · · · × Umr (A) to Um (A). Consider, for example, the case where m is odd, and v is an inert unramified place for all data above. Assume that, for 1 ≤ i ≤ r, σi,v is induced from the Borel subgroup of Umi (Fv ) from the character µi1 ⊗ ... ⊗ µini which takes ¯ diag(t1 , ..., tni , (ǫ), t¯−1 ni , ..., t1

−1

)

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to µi1 (t1 ) · ... · µini (tni ). Here, when mi = 2ni is even, ǫ is omitted, and when mi = 2ni + 1 is odd, ǫ¯ ǫ = 1. Denote ti = diag(µi (p), ..., µini (p)). Then the following element in L (Um1 ×...×Umr ) is a representative of the semisimple class corresponding to σ1 ⊗ · · · ⊗ σr . sσ = (diag(t1 , In1 +1 ); ...; diag(tr′ , Inr′ +1 ); diag(tr′ +1 , Inr′ +1 ); ...; diag(tr , Inr )) · φ. Apply the L-homomorphism f , given in (11.29), to sσ . We get f (sσ ) = diag(tr′ +1 , ..., tr , b · diag(t1 , In1 +1 ; ...; tr′ , Inr′ +1 )b−1 , −Inr′ +1 +...+nr ) · φ. This is conjugate to diag(−tr′ +1 , ..., −tr , b · diag(t1 , In1 +1 ; ...; tr′ , Inr′ +1 )b−1 , Inr′ +1 +...+nr ) · φ.

Note the minus signs multiplying the parameters ti , r′ + 1 ≤ i ≤ r, coming from σr′ +1,v , ..., σr,v . These representations lift to τr′ +1,v ⊗ γv , ..., τr,v ⊗ γv , on GL2nr′ +1 (Ev ), ..., GL2nr (Ev ), respectively. The minus signs in sσ will twist out γv , so that in the lift to GLm (E), the factors at r′ + 1, ..., r will be τr′ +1,v , ..., τr,v , without the twist by γv . Of course, τ1,v , ..., τr′ ,v remain untouched. Conversely, let σ1 , ..., σr be irreducible, automorphic, cuspidal, ψ- generic representations of Um1 (A), ..., Umr (A) respectively. Assume that mi = 2ni + 1 are odd, for 1 ≤ i ≤ r′ , and mi = 2ni are even, for r′ + 1 ≤ i ≤ r. Put m1 + ... + mr = m. Consider the lifts (standard base change lifts) τi of σi to GLmi (AE ), for i ≤ r. Note that the restriction of each ωτi to A∗ is trivial. We know that each one of the representations τi has the form τi = τi1 × · · · × τiki , i ≤ r, as in Theorem 11.2. In particular, for 1 ≤ i ≤ r, LS (τi′j ⊗ µmi , Asai, s) has a pole at s = 1, for all 1 ≤ j ≤ ki . Let ki −1 r τ = ×ri=1 (τi ⊗ µm µ−1 mi ) = ×i=1 ×j=1 (τij ⊗ µm µmi ).

i , for i ≤ r. If the sets Ci , i ≤ r are pairwise disjoint, Denote Ci = {τij ⊗ µmi }kj=1 then τ lies in the image of the lift from Um (A). Indeed, for each ij above,

′ S ′ LS ((τij ⊗ µm µ−1 mi ) ⊗ µm , Asai, s) = L (τij ⊗ µmi , Asai, s)

has a pole at 4s=1. Thus, by Theorems 9.5, 9.7, the descent of τ ′ ⊗ µm to Um (A) provides us with an irreducible, automorphic, cuspidal, ψ-generic representation σ of Um (A), which is a generalized endoscopic weak lift of σ1 ⊗ · · · ⊗ σr , with respect to the L-homomorphism f described above. Summarizing Theorem 11.6. Let σ be an irreducible, automorphic, cuspidal, ψ- generic representation of Um (A). Assume that the lift of σ to GLm (AE ) is not cuspidal. Then either there exists a character χ of A∗E /E ∗ , such that LS (σ × χ−1 , s) has a pole at s = 1, or there exist integers m1 , ..., mr > 1, with m1 , ..., mr′ being odd; mr′ +1 , ..., mr -even; m1 + ... + mr = m, and there exist irreducible, automorphic, cuspidal, ψ-generic representations σ1 , ..., σr of Um1 (A), ..., Umr (A), such that σ is

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a generalized endoscopic lift of σ1 ⊗ ... ⊗ σr , with respect to the L-homomorphism f = fm1 ,...,mr described in (11.24) - (11.32). Assume that LS (σ × χ−1 , s) has a pole at s = 1. Then there is an automorphic character η of U1 (A), such that χ = ηE µ−1 m , and there is an irreducible, automorphic, cuspidal, ψ-generic representation σ ′ of Um−1 (A), such that the (ψ, γ)-theta lift of σ ′ to Um (A) contains σ ⊗ χ−1 . In this case, the lift of σ to GLm (AE ) is of ′ the form t × ηE µ−1 m , where t is the lift to GLm−1 (AE ) of σ ⊗ ηµm−1 (det ·). Conversely, start with σ1 , σ2 , ..., σr irreducible, automorphic, cuspidal, ψ-generic representations of Um1 (A), ..., Umr (A) respectively. Assume that m1 , ..., mr′ are odd, and mr′ +1 , ..., mr are even. Put m1 +...+mr = m. Consider the sets {Ci }ri=1 as above. If these sets are pairwise disjoint, then there is an irreducible, automorphic, cuspidal, ψ-generic representation σ of Um (A), which is a generalized endoscopic weak lift of σ1 ⊗ · · · ⊗ σr , with respect to the L-homomorphism f = fm1 ,...,mr described in (11.24) - (11.32). Otherwise, cuspidal data for Um (A) can be specified, so that σ1 ⊗ · · · ⊗ σr lifts to a constituent of the corresponding parabolically induced representation.

11.4

Base change

Let G be one of the groups SO2n+1 , SO2n,α , Sp2n , Um (over F ). Let K/F be a cyclic extension of prime degree ℓ. For simplicity, we assume that ℓ is odd. We will establish the existence of a weak base change lift from cuspidal generic representations of GA to G(AK ). Choose a nontrivial character η of A∗ /F ∗ NK/F (A∗ ), and a generator ǫ of ΓK/F . In case G = Um , we identify ΓK/F with ΓKE/E . Let σ be an irreducible, automorphic, cuspidal, ψ-generic representation of GA . Lift σ to the automorphic representation τ = τ1 × ... × τr , of GLm (AE ), as in Theorem 11.2. Next, consider the base change lift bc(τ ) = bcKE/E (τ ) = bc(τ1 ) × ... × bc(τr ) to GLm (AKE ) (m is the degree of the standard representation of the L-group.) See [Arthur and Clozel (1989)]. We are going to show that bc(τ ) lies in the image of the lift (restricted to cuspidal generic representations) from G(AK ). Once we show this, we can apply descent to bc(τ ) (followed by taking the complex conjugate) and obtain the base change of σ to G(AK ). Theorem 11.7. The representation bc(τ ) is the lift to GLm (AK ) of an irreducible, automorphic, cuspidal, ψ ◦ trK/F -generic representation π of G(AK ). The representation π can be taken as a constituent of the complex conjugate of the descent of bc(τ ), up to the modification “bc(τ ) 7→ bc(τ )′ ⊗ γ”, which will be specified in the proof. It is a base change lift of σ. Proof. Consider the representation bc(τi ), 1 ≤ i ≤ r. Extend η to an automorphic character ξ of A∗E , trivial on NKE/E (A∗KE ). This is a character of order ℓ. Of course, in case E = F , ξ = η. We need this extension in case of unitary groups. In this

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case, we also fixed an automorphic character γ of A∗E , whose restriction to A∗ is ωE/F . Let γ˜ = γ ◦ NKE/E . Then γ˜ is an automorphic character of A∗KE , whose restriction to A∗K is ωKE/K . There are two cases, according to whether τi = τi ⊗ ξ, or not. See [Arthur and Clozel (1989)]. If τi 6= τi ⊗ξ, then bc(τi ) = θi is cuspidal and ǫ-invariant. Let us denote, for short, by LS (θi , β, s) the L-functions which appear in Table 11.9. More precisely, if G is not unitary then LS (θi , β, s) = LS (θi , α(2) , s); if G = Um , then LS (θi , β, s) = LS (φKE/K (θi ) ⊗ µ ˜m , Asai, s). Here φKE/K denotes the Galois conjugation in the quadratic extension KE/K; for m even, µ ˜ m = γ˜, and for m odd µ ˜ m = 1. We have LS (θi , β, s) =

ℓ−1 Y

k=0

LS (τi , β ⊗ ξ k , s),

(11.33)

where LS (τi , β ⊗ ξ k , s) is defined similarly. When G is symplectic, or even orthogonal, β is the symmetric square representation; when G is odd orthogonal, β is the exterior square, and when G = Um is unitary, LS (τi , β, s) = LS (φE/F (τi ) ⊗ µm , Asai, s), where, as before, φE/F denotes the Galois conjugation in the quadratic extension E/F ; for m even, µm = γ, and for m odd µm = 1. (Previously, we also denoted φE/F (τi ) = τi′ .) Sometimes, in order to make our notation clear we will denote the Asai representation, corresponding to a given quadratic extension L2 /L1 by AsaiL2 /L1 . It is a theorem of Shahidi [Shahidi, F. (1998)] that each factor in the product (11.33) is nonzero at s = 1, when G is symplectic or orthogonal (i.e. β is the symmetric square, or the exterior square representation). Note that, since ℓ is an odd prime, the product in (11.33) is the same as the product ℓ−1 Y

k=0

LS (τi ⊗ ξ k , β, s).

(11.34)

When G is unitary, we have φE/F (ξ) = ξ ◦ φE/F = ξ, so that

LS (τi ⊗ ξ k , β, s) = LS (φE/F (τi ) ⊗ µm ξ k , Asai, s).

Now, we can use Shahidi’s theorem (see [Shahidi, F. (1981)]) on the nonvanishing of LS (τi ⊗ξ k , β, s) at s = 1, which applies to all cuspidal generic representations, which fit into the Langlands-Shahidi set-up (and in particular, for the Asai representation). Since LS (τi , β, s) has a pole at s = 1, we conclude that LS (θi , β, s) has a pole at s = 1. Let us explain (11.34). Since ℓ is an odd prime, the group generated by ξ is the same as the group generated by ξ 2 . When β = ∧2 , sym2 (ξ = η), LS (τi , β ⊗ η 2k , s) = LS (τi ⊗ η k , α(2) , s).

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Now (11.34) is clear. Assume that G = Um is unitary. Denote φE/F (τi ) ⊗ µm = πi . Let v be a place of F , outside of S, which is inert in E. Then s L(πi,v , Asai ⊗ ξvk , s) = L(πi,v , ∧2 ⊗ ξvk , s)L(πi,v ⊗ ξvk , ). 2 Hence ℓ−1 Y

k=0

L(πi,v , Asai ⊗

ξvk , s)

ℓ−1 Y

=

k=0

=

ℓ−1 Y

k=0

2

L(πi,v , ∧ ⊗

ξv2k , s)

ℓ−1 Y

k=0

s L(πi,v ⊗ ξvk , ) 2

L(πi,v ⊗ ξvk , Asai, s).

For a place v of F , outside of S, which splits in E, it is easy to see that L(pii,v , Asai ⊗ ξvk , s) = L(πi,v ⊗ ξvk , Asai, s). This proves (11.34) at all places outside S. Thus, when τi 6= τi ⊗ξ, and bcKE/E (τi ) = θi , we proved that LS (θi , β, s) has a pole at s = 1. (For a unitary group G = Um , this means that LS (φKE/K (θi ) ⊗ µ ˜m , AsaiKE/K , s) has a pole at s = 1.) Assume next that τi = τi ⊗ ξ. Then ℓ divides mi , and bc(τi ) = θi × θiǫ × · · · × θiǫ

ℓ−1

,

where θi is an irreducible, automorphic, cuspidal representation of GLmi /ℓ (AKE ), such that θi 6= θiǫ . Note that φE/F (τi ) = φE/F (τi ) ⊗ ξ. We have, in case G is not unitary, [LS (τi , β, s)]ℓ =

ℓ−1 Y

k=0

=

Y

LS (τi , β ⊗ ξ k , s) = LS (bc(τ ), β, s)

j

0≤j
k

LS (θiǫ × θiǫ , s)

ℓ−1 Y

j

LS (θiǫ , β, s) .

j=0

In case g = Um is unitary, we get [LS (φE/F (τi ) ⊗ µm , AsaiE/F , s)]ℓ = LS (φKE/K (bcKE/E (τi )) ⊗ µ ˜m , AsaiKE/K , s) =

Y

0≤j
j

k

LS (φKE/K (θi )ǫ × θiǫ , s)

ℓ−1 Y

j=0

j

LS (φKE/K (θi )ǫ ⊗ µ ˜m , AsaiKE/K , s).

Note that in the last case, φKE/K (˜ µm ) = µ ˜−1 m . We conclude that the last product (in all cases) has a pole of order ℓ at s = 1. Since τi = φE/F (ˆ τi ) (self-conjugate), ǫk ǫk ) is selfand ℓ is odd, it follows that, for one of the indices k, θ = φ (θc i

conjugate, and hence all

j θiǫ

KE/K

i

j ǫk are self-conjugate. In particular, φKE/K θiǫ 6= θc i , for

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Langlands (weak) functorial lift and descent

0 ≤ j < k ≤ ℓ. We conclude that

ℓ−1 Q

j=0

333

j

LS (θiǫ , β, s) has a pole of order ℓ at s = 1,

and hence LS (θij , β, s) has a pole at s = 1, for 0 ≤ j ≤ ℓ − 1. We showed that in    ǫj , (11.35) bc(τ ) = bc(τ1 ) × · · · × bc(τr ) = ×τi 6=τi ⊗ξ θi × ×τi =τi ⊗ξ ×ℓ−1 j=0 θi

all factors are such that their partial L-function, with respect to β has a pole at s = 1. Finally, we want to check whether all factors in (11.35) are pairwise different. If i 6= j are such that τi 6= τi ⊗ ξ, and τj 6= τj ⊗ ξ, then bc(τi ) = bc(τj ), if and only if there is 0 ≤ k < ℓ, such that τj = τi ⊗ ξ k . In this case, τj = φE/F (ˆ τj ) = φE/F (ˆ τi ) ⊗ φE/F (ξ)−k = τi ⊗ ξ −k = τj ⊗ ξ −2k .

Since ℓ is an odd prime, we must have k = 0, and hence τi = τj , which is a contradiction, and so θi 6= θj . Assume that i 6= j are such that τi = τi ⊗ ξ, and k

k′

τj = τj ⊗ ξ. If there are 0 ≤ k, k ′ < ℓ, such that θiǫ = θjǫ , then k

k

ǫ ℓ−1 {θiǫ }ℓ−1 k=0 = {θj }k=0 ,

and hence

bc(τi ) = θi × θiǫ × ... × θiǫ

ℓ−1

= θj × θjǫ × ... × θjǫ

ℓ−1

= bc(τj ).

But this implies that τi = τj , which is a contradiction. Assume that i, j are such that τi 6= τi ⊗ ξ and τj = τj ⊗ ξ. If there is 0 ≤ k < ℓ, k such that θi = θjǫ , then θjǫ

k+1

k

= θiǫ = θi = θjǫ .

k

This is impossible, since {θiǫ }ℓ−1 k=0 are ℓ different representations. This proves that bc(τ ) lies in the image of the weak lift from cuspidal generic representations of G(AK ), and hence we can apply descent to bc(τ ). This completes the proof.  Remark: We did not include metaplectic groups in the last theorem due to the fact that the condition of nonvanishing of the LS (τi , 21 ), might be destroyed upon tensoring by powers of η, and then bc(τ ) is no longer in the image of the lift from f 2n (AK ), (m = 2n). Sp 11.5

Automorphic induction

We keep the same notations and the same assumptions as in the last section. Let π be an irreducible, automorphic, cuspidal, ψ◦trK/F -generic representation of G(AK ). Lift π to the representation θ = θ1 × ... × θr of GLm (AKE ), as in Theorem 11.2. Again, there are two cases, for each factor θi . Assume that θi , on GLmi (AKE ), is ǫ-invariant. Then, by [Arthur and Clozel (1989)], Sec. 3.6, there is an irreducible, automorphic, cuspidal (when mi > 1) representation τi of GLmi (AE ), such that τi 6= τi ⊗ ξ, and θi = bc(τi ).

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In this case, bc(τi ) = bc(τi ⊗ ξ k ), for all 0 ≤ k < ℓ. Let τ i = τi × (τi ⊗ ξ) × ... × (τi ⊗ ξ ℓ−1 ). This is an irreducible, automorphic representation of GLmi ℓ (AE ). Note that the factors τ ⊗ ξ k are pairwise different. As in (11.33), (11.34), we have ℓ−1 Y LS (θi , β, s) = LS (τi ⊗ ξ k , β, s). k=0

Since LS (θi , β, s) has a pole at s = 1, we may assume that LS (τi , β, s) has a pole at s = 1, and the remaining factors, LS (τi ⊗ ξ k , β, s), 0 < k < ℓ are nonzero and holomorphic at s = 1. We conclude that φE/F (ˆ τi ) = τi (previously, we also denoted φE/F (τi ) = τi′ ). Since φE/F (ξ) = ξ, we get that the remaining factors of τ i are not self-conjugate, but rather, ℓ−1 ℓ−k k . φE/F (τ\ , 1≤k≤ i ⊗ ξ ) = τi ⊗ ξ 2 i Assume now that θ is not ǫ-invariant. Then there is a unique, irreducible, automorphic, cuspidal representation τ i of GLmi ℓ (AE ), such that τ i ⊗ ξ = τ i , and ℓ−1

bc(τ i ) = θi × (θi )ǫ × ... × (θi )ǫ . As in the previous section, we have ℓ−1 Y Y j j k LS ((θi )ǫ , β, s) . [LS (τi , β, s)]ℓ = LS (φKE/K ((θi )ǫ ) × (θi )ǫ , s) 0≤j
j=0

Since LS (θi , β, s) has a pole at s = 1, and the remaining factors do not vanish at s = 1, we conclude that LS (τ i , β, s) has a pole at s = 1. As before, we also get that k LS ((θi )ǫ , α(2) , s) has a pole at s = 1, for all 0 ≤ k < ℓ. Now, it might happen that k there are i 6= j, such that θi is not ǫ-invariant, and θj = (θi )ǫ , for some 0 ≤ k < ℓ (then, of course, θj is also not ǫ-invariant). In such a case, τ i = τ j . Take the representation τ induced by all the τ i collected above. Clearly, if there is i, such that θi is ǫ-invariant, or if there are two factors, θi , θj , which are not ǫ-invariant, k but θj is of the form (θi )ǫ , then τ is not in the image of the lift from cuspidal, generic representations on the appropriate classical group G′ (A). Indeed, not all factors of τ are different, or self-conjugate. Still, we can specify parabolic cuspidal, generic data on G′ (A), which provide automorphic induction of π. In order to get a cuspidal representation, we must assume that all factors of θ are not ǫ-invariant, and that no factor is obtained from another factor by a conjugation of a power of ǫ. In this case, what we explained above shows that τ lies in the image of the lift from cuspidal, ψ-generic representation of G′ (A). We may apply descent to τ and obtain an irreducible, automorphic, cuspidal, ψ-generic representation σ of G′ (A), which is an automorphic induction of π. Note that for G(AK ) = SO2n+1 (AK ), G′ (A) = SO2nℓ+1 (A); for G(AK ) = SO2n (AK ), G′ (A) = SO2nℓ (A) (when we deal with SO2n,α , we take α ∈ F ∗ not a square in F ); for G(AK ) = Sp2n (AK ), G′ (A) = Sp(2n+1)ℓ−1 (A) (ℓ is odd), and for G(AK ) = Um (AK ), G′ (A) = Umℓ (A). Note also f (AK ), that here we could apply this process to metaplectic groups G(AK ) = Sp 2n f (A). G′ (A) = Sp 2nℓ

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Bibliography

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Ginzburg, D. (1990). L-functions for SOn × GLk , J. reine angew. Math.405, pp. 156–180. Ginzburg, D. (2003). A construction of CAP representations for classical groups, IMRN 20, pp. 1123–1140. Ginzburg, D. (2006). Certain conjectures relating unipotent orbits to automorphic representations, Israel J. Math.151, pp. 323–355. Ginzburg, D. (2008). Endoscopic lifting in classical groups and poles of tensor L-functions, Duke Math. J.141, No. 3, pp. 447–503. Ginzburg, D., Jiang, D., Rallis, S. and Soudry, D. (2009). L-functions for symplectic groups using Fourier-Jacobi models, A volume in honor of S. Kudla. Ginzburg, D., Jiang, D., and Soudry, D. (2010). Periods of automorphic forms, poles of L-functions and functorial lifting, Science in China Mathematics 53, No. 9, pp. 2215–2238. Ginzburg, D., Jiang, D., and Soudry, D. (2011). On a correspondence between certain f 2n (A), Preprint. automorphic forms on Sp4n (A) and Sp Ginzburg, D., Piatetski-Shapiro, I. I., and Rallis, S. (1997). L-functions for the orthogonal group, memoirs of the AMS No. 611. Ginzburg, D., Rallis, S. and Soudry, D. (1997). A new construction of inverse Shimura correspondence, IMRN 7, pp. 349–357. Ginzburg, D., Rallis, S. and Soudry, D. (1997). Self dual automorphic modules and construction of a backward lifting from GLn to classical groups, IMRN 14, pp. 687–701. Ginzburg, D., Rallis, S. and Soudry, D. (1997). Periods, poles of L-functions and symplectic-orthogonal theta lifts, J. reine angw. Math.487, pp. 85–114. Ginzburg, D., Rallis, S. and Soudry, D. (1998). L-functions for symplectic groups, Bull. Soc. Math. France 126, pp. 181–244. Ginzburg, D., Rallis, S. and Soudry, D. (1999). On explicit lifts from cusp forms from GLm to classical groups, Annals of Math. 150, pp. 807–866. Ginzburg, D., Rallis, S. and Soudry, D. (1999). On a correspondence between cuspidal f 2n , J of AMS 12,no. 3, pp. 849–907. representations of GL2n to Sp f 2n : the Ginzburg, D., Rallis, S. and Soudry, D. (1999). Lifting cusp forms on GL2n to Sp unramified correspondence, Duke Math. J. 100,no. 2, pp. 243–266. Ginzburg, D., Rallis, S. and Soudry, D. (2001). Generic automorphic forms on SO(2n + 1): functorial lift to GL2n , endoscopy and base change, IMRN 14, pp. 729–764. f . Journal Ginzburg, D., Rallis, S. and Soudry, D. (2002). Endoscopic representations of Sp 2n Inst. Math. Jussieu 1, (1), pp. 77–123. Ginzburg, D., Rallis, S. and Soudry, D. (2005). Construction of CAP representations for symplectic groups using the descent method. In Automorphic Representations, LFunctions and Applications: Progress and Prospects,Ohio-State U. Math. Res. Inst. Publ., 11, de Gruyter, pp. 193–224. Grenie, L. Poles of L-functions, periods and unitary theta. part I: odd to even Israel J. Math. 142, pp. 93–123. Humphreys, J. E.(1995)Conjugacy classes in semisimple algebraic groups Mathematical Surveys and Monographs, Vol. 43 (AMS) Hundley, J. and Sayag, E. (2009). Descent construction for GSpin groups: main results and applications, Electron. Res. Announc. Math. Sci. 16, pp. 30–36. Jacquet, H. Archimedean Rankin-Selberg integrals. Automorphic forms and L-functions II: Local aspects. A volume in honor of S. Gelbart. D. Ginzburg, E. Lapid and D. Soudry, eds. Contemporary Mathematics, Vol. 489, Israel Math. Conference Proc., pp. 57–172. Jacquet, H., Piatetski-Shapiro, I. and Shalika, J. (1983). Rankin-Selberg convolutions.

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Index

exchanging roots, 152

Asai L-function, 21 representation, 21 automorphic induction, 333

functorial lift, 313 Heisenberg group, 7

base change, 330 Jacquet module, 81 Fourier-Jacobi type, 62 Gelfand-Graev, 58

coefficient Fourier-Jacobi, 45 Fourier-Jacobi model, 182 Gelfand-Graev, 42 Gelfand-Graev model, 169 Whittaker, 43

Rankin-Selberg integral, 48, 281 residue representation, 21 Shimura integral, 49, 281

derivative, 87 descent, 53, 54

theta series, 8 tower property, 57, 151, 169, 182

Eisenstein series, 19 endoscopy, 319

Weil representation, 7

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