Automorphic Representations and L-Functions for the General Linear Group: Volume 2 (Cambridge Studies in Advanced Mathematics)

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CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 130 Editorial Board ´ S, W. F U L T O N, A. K A T O K, F. K I R W A N, P. S A R N A K, B. B O L L O B A B. S I M O N, B. T O T A R O

AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR THE GENERAL LINEAR GROUP Volume II This graduate-level textbook provides an elementary exposition of the theory of automorphic representations and L-functions for the general linear group in an adelic setting. Definitions are kept to a minimum and repeated when reintroduced so that the book is accessible from any entry point, and with no prior knowledge of representation theory. The book includes concrete examples of global and local representations of G L(n), and presents their associated L-functions. In Volume I, the theory is developed from first principles for G L(1), then carefully extended to G L(2) with complete detailed proofs of key theorems. Several proofs are presented for the first time, including Jacquet’s simple and elegant proof of the tensor product theorem. In Volume II the higher rank situation of G L(n) is given a detailed treatment. Containing over 250 exercises written by Xander Faber, this book will motivate students to begin working in this fertile field of research. Dorian Goldfeld is a Professor in the Department of Mathematics at Columbia University, New York. Joseph Hundley is an Assistant Professor in the Department of Mathematics at Southern Illinois University, Carbondale.

CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS Editorial Board: B. Bollob´as, W. Fulton, A. Katok, F. Kirwan, P. Sarnak, B. Simon, B. Totaro All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit: http://www.cambridge.org/series/sSeries.asp?code=CSAM Already published 85 J. Carlson, S. M¨uller-Stach & C. Peters Period mappings and period domains 86 J. J. Duistermaat & J. A. C. Kolk Multidimensional real analysis, I 87 J. J. Duistermaat & J. A. C. Kolk Multidimensional real analysis, II 89 M. C. Golumbic & A. N. Trenk Tolerance graphs 90 L. H. Harper Global methods for combinatorial isoperimetric problems 91 I. Moerdijk & J. Mrˇcun Introduction to foliations and Lie groupoids 92 J. Koll´ar, K. E. Smith & A. Corti Rational and nearly rational varieties 93 D. Applebaum L´evy processes and stochastic calculus (1st Edition) 94 B. Conrad Modular forms and the Ramanujan conjecture 95 M. Schechter An introduction to nonlinear analysis 96 R. Carter Lie algebras of finite and affine type 97 H. L. Montgomery & R. C. Vaughan Multiplicative number theory, I 98 I. Chavel Riemannian geometry (2nd Edition) 99 D. Goldfeld Automorphic forms and L-functions for the group GL(n,R) 100 M. B. Marcus & J. Rosen Markov processes, Gaussian processes, and local times 101 P. Gille & T. Szamuely Central simple algebras and Galois cohomology 102 J. Bertoin Random fragmentation and coagulation processes 103 E. Frenkel Langlands correspondence for loop groups 104 A. Ambrosetti & A. Malchiodi Nonlinear analysis and semilinear elliptic problems 105 T. Tao & V. H. Vu Additive combinatorics 106 E. B. Davies Linear operators and their spectra 107 K. Kodaira Complex analysis 108 T. Ceccherini-Silberstein, F. Scarabotti & F. Tolli Harmonic analysis on finite groups 109 H. Geiges An introduction to contact topology 110 J. Faraut Analysis on Lie groups: An introduction 111 E. Park Complex topological K-theory 112 D. W. Stroock Partial differential equations for probabilists 113 A. Kirillov, Jr An introduction to Lie groups and Lie algebras 114 F. Gesztesy et al. Soliton equations and their algebro-geometric solutions, II 115 E. de Faria & W. de Melo Mathematical tools for one-dimensional dynamics 116 D. Applebaum L´evy processes and stochastic calculus (2nd Edition) 117 T. Szamuely Galois groups and fundamental groups 118 G. W. Anderson, A. Guionnet & O. Zeitouni An introduction to random matrices 119 C. Perez-Garcia & W. H. Schikhof Locally convex spaces over non-Archimedean valued fields 120 P. K. Friz & N. B. Victoir Multidimensional stochastic processes as rough paths 121 T. Ceccherini-Silberstein, F. Scarabotti & F. Tolli Representation theory of the symmetric groups 122 S. Kalikow & R. McCutcheon An outline of ergodic theory 123 G. F. Lawler & V. Limic Random walk: A modern introduction 124 K. Lux & H. Pahlings Representations of groups 125 K. S. Kedlaya p-adic differential equations 126 R. Beals & R. Wong Special functions 127 E. de Faria & W. de Melo Mathematical aspects of quantum field theory 128 A. Terras Zeta functions of graphs 129 D. Goldfeld & J. Hundley Automorphic representations and L-functions for the general linear group, I 130 D. Goldfeld & J. Hundley Automorphic representations and L-functions for the general linear group, II 131 D. A. Craven The theory of fusion systems 132 J. V¨aa¨ n¨anen Models and games

Automorphic Representations and L-Functions for the General Linear Group Volume II DORIAN GOLDFELD Columbia University, New York

JOSEPH HUNDLEY Southern Illinois University, Carbondale With exercises by XANDER FABER

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ao Paulo, Delhi, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9781107007994  c D. Goldfeld and J. Hundley 2011 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2011 Printed in the United Kingdom at the University Press, Cambridge A catalog record for this publication is available from the British Library ISBN 978-1-107-00799-4 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

To Ada, Dahlia, and Iris. –D. G. To Melissa, and to my Family. –J. H.

Contents for Volume II

Contents for Volume I Introduction Preface to the Exercises

page ix xv xix

The classical theory of automorphic forms for G L(n, R) 12.1 Iwasawa decomposition for G L(n, R) 12.2 Congruence subgroups of S L(n, Z) 12.3 Automorphic functions of arbitrary weight, level, and character Exercises for Chapter 12

3 13

13

Automorphic forms and representations for G L(n, AQ ) 13.1 Cartan, Bruhat decompositions for G L(n, R) 13.2 Iwasawa, Cartan, Bruhat decompositions for G L(n, Q p ) 13.3 Strong approximation for G L(n) 13.4 Adelic lifts and automorphic forms for G L(n, AQ ) 13.5 The Fourier expansion of adelic automorphic forms 13.6 Adelic automorphic representations for G L(n, AQ ) 13.7 Tensor product theorem for G L(n) 13.8 Newforms for G L(n) Exercises for Chapter 13

16 16 17 22 24 31 36 41 43 47

14

Theory of local representations for G L(n) 14.1 Generalities on representations of G L(n, Q p ) 14.2 Generic representations of G L(n, Q p ) 14.3 Parabolic induction for G L(n, Q p ) 14.4 Supercuspidal representations of G L(n, Q p ) 14.5 The Bernstein-Zelevinsky classification for G L(n, Q p ) 14.6 Classification of smooth irreducible representations of G L(n, Q p ) via the growth of matrix coefficients 14.7 Unitary representations of G L(n, Q p ) 14.8 Generalities on (g, K ∞ )-modules of G L(n, R)

52 52 56 60 66 70

12

1 1 2

75 78 80 vii

viii

Contents Generic representations of G L(n, R) Parabolic induction for G L(n, R) Classification of the unitary and the generic unitary representations of G L(n, Q p ) 14.12 Unramified representations of G L(n, Q p ) and G L(n, R) 14.13 Unitary duals and other duals 14.14 The Ramanujan conjecture for G L(n, AQ ) Exercises for Chapter 14

14.9 14.10 14.11

15

85 88 100 102 105 106 106

The Godement-Jacquet L-function for G L(n, AQ ) 15.1 The Poisson summation formula for G L(n, AQ ) 15.2 The global zeta integral for G L(n, AQ ) 15.3 Factorization of the global zeta integral for G L(n, AQ ) 15.4 The local functional equation for G L(n, Q p ) 15.5 The L-function and local functional equation for the supercuspidal representations of G L(n, Q p ) 15.6 The local functional equation for tensor products 15.7 The local zeta integral for a parabolically induced representation of G L(n, Q p ) 15.8 The local zeta integral for discrete series (square integrable) representations of G L(n, Q p ) 15.9 The local zeta integral for irreducible unitary generic representations of G L(n, R) Exercises for Chapter 15

143 151

Solutions to Selected Exercises References Symbols Index Index

153 169 175 179

114 114 118 124 125 128 128 130 138

Contents for Volume I

Contents for Volume II Introduction Preface to the Exercises

page xiii xv xix

1

Adeles over Q 1.1 Absolute values 1.2 The field Q p of p-adic numbers 1.3 Adeles and ideles over Q 1.4 Action of Q on the adeles and ideles 1.5 p-adic integration 1.6 p-adic Fourier transform 1.7 Adelic Fourier transform 1.8 Fourier expansion of periodic adelic functions 1.9 Adelic Poisson summation formula Exercises for Chapter 1

1 1 2 7 8 12 15 18 23 30 31

2

Automorphic representations and L-functions for GL (1, AQ ) 2.1 Automorphic forms for GL (1, AQ ) 2.2 The L-function of an automorphic form 2.3 The local L-functions and their functional equations 2.4 Classical L-functions and root numbers 2.5 Automorphic representations for GL (1, AQ ) 2.6 Hecke operators for GL (1, AQ ) 2.7 The Rankin-Selberg method 2.8 The p-adic Mellin transform Exercises for Chapter 2

39 39 45 55 60 65 68 69 70 72

3

The classical theory of automorphic forms for GL (2) 3.1 Automorphic forms in general 3.2 Congruence subgroups of the modular group 3.3 Automorphic functions of integral weight k 3.4 Fourier expansion at ∞ of holomorphic modular forms

76 76 77 78 80 ix

x

Contents 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12

Maass forms Whittaker functions Fourier-Whittaker expansions of Maass forms Eisenstein series Maass raising and lowering operators The bottom of the spectrum Hecke operators, oldforms, and newforms Finite dimensionality of the eigenspaces Exercises for Chapter 3

81 84 87 89 90 92 94 97 98

Automorphic forms for GL (2, AQ ) 4.1 Iwasawa and Cartan decompositions for GL (2, R) 4.2 Iwasawa and Cartan decompositions for GL (2, Q p ) 4.3 The adele group GL (2, AQ ) 4.4 The action of GL (2, Q) on GL (2, AQ ) 4.5 The universal enveloping algebra of gl(2, C) 4.6 The center of the universal enveloping algebra of gl(2, C) 4.7 Automorphic forms for GL (2, AQ ) 4.8 Adelic lifts of weight zero, level one, Maass forms 4.9 The Fourier expansion of adelic automorphic forms 4.10 Global Whittaker functions for GL (2, AQ ) 4.11 Strong approximation for congruence subgroups 4.12 Adelic lifts with arbitrary weight, level, and character 4.13 Global Whittaker functions for adelic lifts with arbitrary weight, level, and character Exercises for Chapter 4

103 103 105 107 108 112 117 117 119 126 128 134 136

5

Automorphic representations for GL (2, AQ ) 5.1 Adelic automorphic representations for GL (2, AQ ) 5.2 Explicit realization of actions defining a (g, K ∞ )-module 5.3 Explicit realization of the action of GL (2, Afinite ) 5.4 Examples of cuspidal automorphic representations 5.5 Admissible (g, K ∞ ) × G L(2, Afinite )-modules Exercises for Chapter 5

152 152 161 168 172 173 178

6

Theory of admissible representations of GL (2, Q p ) 6.0 Short roadmap to chapter 6 6.1 Admissible representations of GL (2, Q p ) 6.2 Ramified versus unramified 6.3 Local representation coming from a level 1 Maass form 6.4 Jacquet’s local Whittaker function 6.5 Principal series representations 6.6 Jacquet’s map: Principal series → Whittaker functions

183 183 183 192 193 195 200 205

4

141 147

Contents 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16

7

8

9

The Kirillov model The Kirillov model of the principal series representation Haar measure on GL (2, Q p ) The special representations Jacquet modules Induced representations and parabolic induction The supercuspidal representations of GL (2, Q p ) The uniqueness of the Kirillov model The Kirillov model of a supercuspidal representation The classification of the irreducible and admissible representations of GL (2, Q p ) Exercises for Chapter 6

Theory of admissible (g, K ∞ ) modules for GL (2, R) 7.1 Admissible (g, K ∞ )-modules 7.2 Ramified versus unramified 7.3 Jacquet’s local Whittaker function 7.4 Principal series representations 7.5 Classification of irreducible admissible (g, K ∞ )-modules Exercises for Chapter 7

xi 214 221 228 232 236 238 240 243 252 252 253 259 259 260 260 263 269 275

The contragredient representation for GL (2) 8.1 The contragredient representation for GL (2, Q p ) 8.2 The contragredient representation of a principal series representation of GL (2, Q p ) 8.3 Contragredient of a special representation of GL (2, Q p ) 8.4 Contragredient of a supercuspidal representation 8.5 The contragredient representation for GL (2, R) 8.6 The contragredient representation of a principal series representation of GL (2, R) 8.7 Global contragredients for GL (2, AQ ) 8.8 Integration on GL (2, AQ ) 8.9 The contragredient representation of a cuspidal automorphic representation of GL (2, AQ ) 8.10 Growth of matrix coefficients 8.11 Asymptotics of matrix coefficients of (g, K ∞ )-modules 8.12 Matrix coefficients of GL (2, Q p ) via the Jacquet module Exercises for Chapter 8

277 277

311 316 330 343 353

Unitary representations of GL (2) 9.1 Unitary representations of GL (2, Q p ) 9.2 Unitary principal series representations of GL (2, Q p )

358 358 360

281 283 285 289 294 303 306

xii

Contents 9.3 9.4 9.5

10

11

Unitary and irreducible special or supercuspidal representations of GL (2, Q p ) Unitary (g, K ∞ )-modules Unitary (g, K ∞ ) × G L(2, Afinite )-modules Exercises for Chapter 9

Tensor products of local representations 10.1 Euler products 10.2 Tensor product of (g, K ∞ )-modules and representations 10.3 Infinite tensor products of local representations 10.4 The factorization of unramified irreducible admissible cuspidal automorphic representations 10.5 Decomposition of representations of locally compact groups into finite tensor products 10.6 The spherical Hecke algebra for GL (2, Q p ) 10.7 Initial decomposition of admissible (g, K ∞ ) × G L(2, Afinite )-modules 10.8 The tensor product theorem 10.9 The Ramanujan and Selberg conjectures for GL (2, AQ ) Exercises for Chapter 10 The Godement-Jacquet L-function for GL (2, AQ ) 11.1 Historical remarks 11.2 The Poisson summation formula for GL (2, AQ ) 11.3 Haar measure 11.4 The global zeta integral for GL (2, AQ ) 11.5 Factorization of the global zeta integral 11.6 The local functional equation 11.7 The local L-function for GL (2, Q p ) (unramified case) 11.8 The local L-function for irreducible supercuspidal representations of GL (2, Q p ) 11.9 The local L-function for irreducible principal series representations of GL (2, Q p ) 11.10 Local L-function for unitary special representations of GL (2, Q p ) 11.11 Proof of the local functional equation for principal series representations of GL (2, Q p ) 11.12 The local functional equation for the unitary special representations of GL (2, Q p ) 11.13 Proof of the local functional equation for the supercuspidal representations of GL (2, Q p )

364 365 368 374 378 378 379 381 383 388 396 403 406 413 415 418 418 419 423 425 430 432 434 440 441 444 446 450 452

Contents The local L-function for irreducible principal series representations of GL (2, R) 11.15 Proof of the local functional equation for principal series representations of GL (2, R) 11.16 The local L-function for irreducible discrete series representations of GL (2, R) Exercises for Chapter 11

xiii

11.14

Solutions to Selected Exercises References Symbols Index Index

463 467 471 474 478 531 537 541

Introduction

The theory of L-functions is an old subject with a long history. In the 1940s Hecke and Maass rewrote the classical theory in the setting of automorphic forms, and it seemed as if the theory of L-functions had settled into a fairly final form. This view was effectively overturned with the publication of two major books: [Gelfand-Graev-Piatetski-Shapiro, 1969], [Jacquet-Langlands, 1970], where it was shown that the theory of L-functions could be recast in the language of infinite dimensional complex representations of reductive groups. Another milestone in the recent theory of L-functions was the book by Roger Godement and Herv´e Jacquet, [Godement-Jacquet, 1972], which defined for the first time the standard L-functions attached to automorphic representations of the general linear group, and proved their key properties by generalizing the seminal ideas of [Tate, 1950], [Iwasawa, 1952, 1992]. The proofs in [Godement-Jacquet, 1972] made fundamental use of matrix coefficients associated to automorphic representations. The standard L-functions of the general linear group are often called Godement-Jacquet L-functions. Although several other techniques have since been discovered to obtain the main analytic properties of such L-functions, none is more beautiful and elegant than the method of matrix coefficients, originally devised by Godement and Jacquet, which is a major theme of this book. Modern research in the theory of automorphic representations and L-functions is largely focused in the direction of the Langlands program. Quoting from [Bernstein-Gelbart, 2003]: The Langlands program roughly states that, among other things, any L-function defined number-theoretically is the same as the one which can be defined as the automorphic L-function of some G L(n). In this loose way, every L-function is (conjecturally) viewed as one and the same object. Langlands’ philosophy established the central importance of the general linear group for number theory. A great step forward was obtained recently when Ngo proved the fundamental lemma (see: [Laumon-Ngo, 2004], [Ngo, 2008]). xv

xvi

Introduction

The purpose of this book is to provide an elementary yet extremely rigorous exposition of the theory of cuspidal automorphic representations and L-functions for the general linear group in a textbook form that can be understood by the beginning graduate student with minimal background in representation theory. The theory of Eisenstein series and the L2 decomposition of the space of automorphic forms are omitted for reasons of space. To simplify the presentation, the theory is restricted to the adele group of Q, although in most cases, the proofs can be easily generalized to any number field. Definitions are reintroduced where necessary so that the book is easily accessible from any entry point. Most definitions and key ideas are explained in concert with simple concrete examples. Almost every definition, theorem, and proposition is captioned, so that the flow of ideas is easy to grasp. The book contains over 250 exercises as well as over 50 pages of solutions to exercises. The first chapter introduces the theory of p-adic fields and the adele ring AQ from first principles. A highlight of this chapter is a short, rigorous, and elementary proof of the Fourier expansion of periodic adelic functions which plays such a crucial role in the proof of the analytic continuation and functional equation of L-functions. The second chapter presents the theory of automorphic representations and L-functions for the group G L(1, AQ ). This is essentially Tate’s thesis recast in the language of automorphic representations for G L(1). Chapters 3 through 11 develop the theory of automorphic representations and L-functions for G L(2). Particular care is taken to show the relationship between irreducible cuspidal automorphic representations of G L(2, AQ ) and Hecke-Maass newforms for congruence subgroups of S L(2, Z). Highlights include the classification of the irreducible admissible representations of G L(2, Q p ) given in Chapter 6, the classification of irreducible admissible (g, K ∞ )-modules given in Chapter 7, growth of matrix coefficients given in Chapter 8, Jacquet’s simple and extremely elegant proof of the tensor product theorem in Chapter 10, and the proofs (using matrix coefficients) of the key analytic properties of the Godement-Jacquet L-functions given in Chapter 11. Finally, the entire theory is redone for the more general case of G L(n) in the final Chapters 12 through 15. Chapter 12 presents a classical theory of automorphic forms for G L(n, AQ ), which generalizes the theory presented in [Goldfeld, 2006]. Instead of K -fixed forms, automorphic forms with arbitrary K -type, level, and character are studied. Chapter 14 presents the Bernstein-Zelevinsky classification of the smooth irreducible representations of G L(n, Q p ) as well as Vogan’s classification of the irreducible unitary representations of G L(n, R). The book ends with the theory of the Godement-Jacquet L-function for G L(n). By seeing the theory of automorphic representations and L-functions in three different settings:

Introduction

xvii

• the abelian setting of G L(1); • the rank one setting of G L(2); • the higher rank situation of G L(n); along with many simple concrete examples to investigate, the beginning student can gain deep insight into this beautiful subject. It is hoped that by reading this book students and researchers will be motivated to begin working in this fertile field of research. The authors are deeply indebted to Herv´e Jacquet for walking them through the most difficult steps and showing them new proofs of many results. These proofs substantially simplify the arguments previously available in the literature, and we would like to thank Herv´e Jacquet for allowing us to include them in this exposition. Without his help this book could not have been written. We are especially grateful to Xander Faber for a careful reading of the manuscript, pointing out innumerable errors, and for creating all the exercises and a solutions section, which will be so invaluable for students. The authors would like to specially thank Min Lee for carefully reading the book, correcting many errors, and preparing the index and table of symbols. We would like to thank Gautam Chinta, Ivan Fesenko, Joe Pleso, and Shou-Wu Zhang for many helpful comments. We thank Jacqueline Anderson, Atanas Atanasov, Alberto Baider, Ioan Filip, Timothy Heath, Jeffrey Hoffstein, Thomas Hulse, Eren Mehmet Kiral, Karol Koziol, Chan Ieong Kuan, Li Mei Lim, Matthew Spencer, and Ian Whitehead for patiently reading various chapters and pointing out errors and typos. We thank the NSF and NSA for financial support. Finally we want to thank Roger Astley and Cambridge University Press for encouraging us to write and publish this book. Dorian Goldfeld and Joseph Hundley

Preface to the Exercises

My goal for this project was to remedy some of my ignorance of the theory of automorphic forms. I hope these exercises will aid the reader in doing the same. If an exercise requires some sort of inspiration that isn’t immediately obvious from the text, then I have tried to give at least a hint. I have attempted to write a detailed sketch or a full solution whenever an exercise was particularly difficult (for me). But it will be evident that I have violated both of these guiding principles at times with little rhyme or reason. An exercise marked with a * is particularly tricky (again, for me). My thanks go to Dorian for the opportunity to be a part of this project, and to Joe for patiently answering loads of my questions. It’s been a pleasure working with both of you. A National Science Foundation Postdoctoral Research Fellowship provided my funding during the completion of this project. Finally, I would like to thank my wife, Alana, for her unwavering support of my endeavors, especially those that detract from our time together. — Xander Faber

xix

12 The classical theory of automorphic forms for G L(n, R)

In this chapter we present a classical description of the theory of automorphic forms of arbitrary weight and level for the group G L(n, R). This generalizes the theory of K ∞ -fixed automorphic forms presented in [Goldfeld, 2006]. For the reader who is mainly interested in the theory of automorphic representations, this chapter may be skipped on a first reading.

12.1 Iwasawa decomposition for G L(n, R) Definition 12.1.1 (Generalized upper half plane) Let n ≥ 2. The generalized upper half plane hn associated to G L(n, R) is defined to be the set of all n × n matrices of the form z = x · y where ⎛1 ⎜ ⎜ x =⎜ ⎜ ⎝

x1,2 1

x1,3 x2,3 .. .

··· ···

x1,n x2,n .. . 1

xn−1,n 1

⎞ ⎟ ⎟ ⎟, ⎟ ⎠

⎛

⎞

 yn−1  ⎜ yn−2 ⎜ ⎜ .. y=⎜ . ⎜ ⎝ y1

⎟ ⎟ ⎟ ⎟, ⎟ ⎠ 1

with xi, j ∈ R for 1 ≤ i < j ≤ n and yi > 0 for 1 ≤ i ≤ n − 1. To simplify later formulae and notation in this book, we will always express y in the form: ⎞ ⎛ y y ··· y 1 2 n−1 y1 y2 · · · yn−2 ⎟ ⎜ ⎟ ⎜ .. ⎟, y=⎜ . ⎟ ⎜ ⎠ ⎝ y1 1 with yi > 0 for 1 ≤ i ≤ n − 1. Note that this can always be done since yi =/ 0 for 1 ≤ i ≤ n − 1. 1

2

The classical theory of automorphic forms for GL(n, R)

Proposition 12.1.2 (The Iwasawa decomposition for G L(n, R)) Let n ≥ 2. Every matrix g ∈ G L(n, R) has a factorization of the form g = g · d · k, with  g ∈ hn , uniquely determined, k ∈ O(n, R), and d a non-zero diagonal matrix in the center of G L(n, R). Furthermore, k and d are also uniquely determined up to multiplication by ±In where In is the n × n identity matrix. Proof This is Proposition 1.2.6 of [Goldfeld, 2006].



Definition 12.1.3 (Action of G L(n, R) on hn ) For g ∈ G L(n, R) and z ∈ hn define g . z := g ·z (∀g ∈ G L(n, R), z ∈ hn ). Here, the product on the right hand side is matrix multiplication, and g · z ∈ hn n is the h component of the Iwasawa decomposition of g · z, as in Proposition 12.1.2.

12.2 Congruence subgroups of S L(n, Z) Definition 12.2.1 (Principal congruence subgroup of S L(n, Z)) Let n ≥ 2. Fix an integer N ≥ 1. A principal congruence subgroup of S L(n, Z) of level N is the kernel of the map S L(n, Z) → S L(n, Z/N Z). We denote the principal congruence subgroup of S L(n, Z) of level N by (N ). Definition 12.2.2 (Congruence subgroup of S L(n, Z)) Let n ≥ 2. A subgroup  ⊂ S L(n, Z) is called a congruence subgroup if  contains (N ) for some N ≥ 1. The least such N is called the level of . When n = 2, it had been known for a long time that there exist infinitely many subgroups of finite index in S L(2, Z) which are not congruence subgroups [Magnus, 1974]. It was proved independently in [Bass-Lazard-Serre, 1964], [Mennicke, 1965] that every subgroup of finite index in S L(n, Z) (with n ≥ 3) must be a congruence subgroup. This was further generalized in [BassMilnor-Serre, 1967]. The congruence subgroup problem for other semisimple algebraic groups is an area of active research. See [Ragunathan, 2004] for a survey. Definition 12.2.3 (The congruence subgroup 0 (N )) Let n ≥ 2. For an integer N ≥ 2, the congruence subgroup 0 (N ) is defined to be the multiplicative group of all matrices of determinant 1 which are of the form





A ∈ Mat(n − 1, Z), B ∈ Mat((n − 1) × 1, Z), A B

. ∈ S L(n, Z) C ∈ Mat(1 × (n − 1), N · Z), d ∈ Z C d

12.3 Automorphic functions

3

Here, for an arbitrary ring R, we define Mat(i, R), Mat(i × j, R) (respectively) to denote the set of all i × i, i × j (respectively) matrices with coefficients in R. In addition, we define 0 (1) := S L(n, Z). Definition 12.2.4 (The character  χ of 0 (N )) Let n ≥ 2. Fix an integer × × N ≥ 1, and let 0 (N ) be as in Definition 12.2.3.  Let  χ : (Z/N Z) → C be A B a Dirichlet character (mod N ). Then for γ = C d ∈ 0 (N ) as in Definition 12.2.3, we define  χ (γ ) := χ (d).

12.3 Automorphic functions of arbitrary weight, level, and character Let G L(n, R)+ denote the subgroup of G L(n, R) consisting of those elements of G L(n, R) with positive determinant. The group G L(n, R)+ acts on the generalized upper half-plane hn as in Definition 12.1.3. This action determines a function κ : G L(n, R)+ × hn → S O(n, R) as follows. Definition 12.3.1 (The function κ) Let n ≥ 2. For any γ ∈ G L(n, R)+ and any z = x y ∈ hn , as in Definition 12.1.1, there exists a unique κ(γ , z) ∈ S O(n, R) and a unique γz ∈ hn such that γ z = γz · κ(γ , z) · d where d = δ In with δ > 0. Here In is the n × n identity matrix. Remark The uniqueness of γz and κ(γ , z) follows immediately from the Iwasawa decomposition as given in Proposition 12.1.2 . y x Example 12.3.2 (The function κ for n = 2) Fix n = 2. Let 0 1 ∈ h2 and let   a b ∈ S L(2, R). Then we may explicitly write c d 

a c

b d



y 0

x 1



with z = x + i y and  κ

 =

a c

y x  0 1

b d

y 0



x 1



−cy |cz+d| cx+d |cz+d|

cx+d |cz+d| cy |cz+d|



|cz + d| 0

0 |cz + d|

,

∈ h2 . It follows that



 ,

y 0

x 1



 =

cx+d |cz+d| cy |cz+d|

−cy |cz+d| cx+d |cz+d|

.

Proposition 12.3.3 (The function κ is a one-cocycle) Let n ≥ 2. Recall that if g ∈ G L(n, R), then  g ∈ hn is uniquely determined in Definition 12.1.2.

The classical theory of automorphic forms for GL(n, R)

4

Consider κ : G L(n, R)+ × hn → S O(n, R) as defined in Definition 12.3.1. Then κ satisfies the following one-cocycle relation:    z · κ(γ  , z) κ(γ γ  , z) = κ γ , γ for all γ , γ  ∈ G L(n, R)+ and all z ∈ hn . Proof It follows from Definition 12.3.1 that (γ γ  )z = γ γ  z · κ(γ γ  , z) · d, with unique matrices γ γ  z ∈ hn and d = δ In with δ > 0. On the other hand, we again have from Definition 12.3.1 that    z · κ(γ  , z) · d  , γ (γ  z) = γ · γ

(d  = δ  In with δ  > 0),

  z · κ γ , γ  z · κ(γ  , z) · d  ,  = γ · γ

(d  = δ  In with δ  > 0).

It immediately follows from the uniqueness of the Iwasawa decomposition that z = γ  γ · γ γ z

and

   z · κ(γ  , z).  κ(γ γ  , z) = κ γ , γ

Following Section 3.3 and (3.5.1), one may define automorphic functions of weight k ∈ Z for S L(2, Z) as functions f : h2 → C satisfying  f for all z =

y x 0 1

az + b cz + d

i = x + i y with 

Jk (γ , z) :=

cz + d |cz + d|



 =

y x 0 1

k ,

cz + d |cz + d|

k f (z)

∈ h2 . The function   a ∀γ = c

b d



∈ S L(n, R) ,

is a one cocycle as in Example 3.1.4. One would like to generalize the notion of “weight k” to the higher rank situation of G L(n, R) with n > 2. It will turn out that if n > 2 then the weight k may be realized as a finite dimensional irreducible representation ρ of S O(n, R). We now proceed to develop these concepts. It will be clear that many of the definitions make sense and many of the results hold without the assumption that ρ is irreducible. However, there is no real loss of generality in

12.3 Automorphic functions

5

assuming that ρ is irreducible, as Exercise 12.11(b) shows, and the requirement of irreducibility maintains the analogy with the classical theory for G L(2, R), where an integral weight k corresponds to an irreducible representation of S O(2, R). Definition 12.3.4 (The function Jρ ) Let n ≥ 2 and r ≥ 1 be integers. Let ρ : S O(n, R) → G L(r, C) be an irreducible representation as in Definition 2.5.1. We define a function Jρ : G L(n, R)+ × hn → G L(r, C) as follows. Let γ ∈ G L(n, R)+ and z ∈ hn . Then we define   Jρ (γ , z) := ρ κ(γ , z)−1 , where κ is given by Definition 12.3.1. Proposition 12.3.5 (The function Jρ is a one-cocycle) Let n ≥ 2. Recall that if g ∈ G L(n, R) then  g ∈ hn is uniquely determined in Definition 12.1.2. Consider the function Jρ : G L(n, R)+ × hn → G L(r, C) defined in Definition 12.3.4. Then Jρ satisfies the one-cocycle relation  z) Jρ (γ γ  , z) = Jρ (γ  , z) Jρ (γ , γ for all γ , γ  ∈ G L(n, R)+ and all z ∈ hn . Proof It follows immediately from Definition 12.3.4 and Proposition 12.3.3 that   Jρ (γ γ  , z) = ρ κ(γ γ  , z)−1      z −1 = ρ κ(γ  , z)−1 · κ γ , γ        z −1 = ρ κ(γ  , z)−1 · ρ κ γ , γ  z).  = Jρ (γ  , z) Jρ (γ , γ

Remark The one-cocycle Jρ is the generalization of the classical j-cocycle which appeared in Example 3.1.4. Example 12.3.6 (The function Jρ for n = 2) For every k ∈ Z we may define a one-dimensional representation ρk : S O(2, R) → C× by letting  cos θ sin θ := (cos θ + i sin θ )k , ρk (θ ∈ R). − sin θ cos θ   It then follows from Example 12.3.2 that for every γ = ac db ∈ S L(2, R) and y x all 0 1 ∈ h2 that     cx+d −1  cy y x |cz+d| |cz+d| ρk κ γ , = ρk −cy cx+d 0 1 |cz+d| |cz+d|  cz + d k , (z = x + i y). = |cz + d|

6

The classical theory of automorphic forms for GL(n, R)

With Example 12.3.6 in mind, it should now start to become clear to the reader how to generalize the classical automorphic forms of weight k, studied in Chapter 3, to the higher rank situation of G L(n, R) with n ≥ 3. The precise definition will be given shortly. It will first be necessary, however, to introduce the analogue of the classical slash operator as given in Definition 3.5.6. Fix n ≥ 2, r ≥ 1, and let : hn → Cr be a smooth vector valued function given by ⎞ ⎛ φ1 (z) . (12.3.7) (z ∈ hn ), (z) := ⎝ .. ⎠ , φr (z) where each φi : hn → C, (1 ≤ i ≤ r ) is smooth. Definition 12.3.8 (Slash operator) Fix n ≥ 2, r ≥ 1, and let : hn → Cr be a smooth vector valued function as in (12.3.7). Let ρ : S O(n, R) → G L(r, C) be an irreducible representation. For z ∈ hn and γ ∈ G L(n, R)+ , we define the slash operator |ρ by   ρ γ (z) := Jρ (γ , z)−1 · (γ . z), where Jρ is defined in Definition 12.3.4. We leave

for the reader to show that the slash operator

it as an exercise satisfies ρ γ γ  = ρ γ ρ γ  for all γ , γ  ∈ G L(n, R)+ . Definition 12.3.9 (Vector valued automorphic function of weight ρ, level N , character χ ) Let n ≥ 2, r ≥ 1 be integers. Fix an irreducible representation ρ : S O(n, R) → G L(r, C), an integer N ≥ 1, and a Dirichlet character χ be as in χ (mod N ). Let 0 (N ) be defined as in Definition 12.2.3 and let  Definition 12.2.4. A vector valued automorphic function of weight ρ, level N , and character χ is a smooth function : hn → Cr , as in (12.3.7), which satisfies the automorphy relation   ρ γ (z) =  χ (γ ) (z), for all γ ∈ 0 (N ), z ∈ hn , and if N is the least integer with this property. Remark If the representation ρ is not smooth, then the space of vector valued automorphic functions of weight ρ (and any level and character) is trivial. We do not need to exclude non-smooth representations from consideration. However, the reader may always assume that ρ is smooth in what follows. The representations ρk considered in Example 12.3.6 are not the only irreducible representations of S O(2, R). They are, however, the only smooth irreducible representations of S O(2, R).

12.3 Automorphic functions

7

In order to define automorphic forms (also called Maass forms) on hn it is necessary to develop the G L(n) versions of the conditions (introduced in Section 3.5) which separate automorphic forms from other automorphic functions. The first step is to generalize Definition 3.3.3 (moderate growth). Definition 12.3.10 (Moderate growth) A smooth function hn → C is said to have moderate growth if, for each fixed σ ∈ G L(n, Q), there exist constants c, C and B such that | f (σ · z)|C ≤ C · (y1 · y2 · . . . · yn−1 ) B for all z = x · y ∈ hn , with x, y as in Definition 12.1.1, such that min(y1 , . . . , yn−1 ) ≥ c. Here ||C is the usual absolute value on C. Remarks Suppose β ∈ G L(n, Q) is upper triangular and z = x · y ∈ hn . Write β · z = β · z · κ(β, z) · d. Then each diagonal entry of β · z is equal to the corresponding entry of z, times the absolute value of the corresponding entry of β. It follows that in Definition 12.3.10 we may restrict σ to a set of representatives for G L(n, Q)/Bn (Q), where Bn (Q) denotes the group of invertible upper-triangular n × n matrices with entries in Q. When n = 2, this is the set G L(2, Q)/B2 (Q), in natural one-to-one correspondence with the set Q ∪ {∞} of cusps. The next step is to generalize the action of the weight k Laplacian, defined in Definition 3.5.3. In Chapter 4, it was shown that a classical Maass form f of weight k may be lifted to a function  f : G L(2, R)+ → C satisfying    f g

cos θ  − sin θ 

sin θ  cos θ 



= eikθ  f (g), 

   ∀ θ ∈ R, g ∈ G L(2, R)+ .

(12.3.11) In the same chapter, the universal enveloping algebra U (g) of g = gl(2, C) and its center Z (U (g)) were introduced. It follows from the proof of (5.5.14) that the weight k Laplacian k is induced by the action of certain elements of the universal enveloping algebra. (To be precise, − 12 times the Casimir element, or any other element of Z (U (g)) which is equivalent to this one modulo the f for any f.) This discussion ideal generated by the operator D Z , which kills  motivates the following definition. Definition 12.3.12 (Lift of an automorphic function to G L(n, R)+ ) Let : hn → Cr be a vector valued automorphic function of weight ρ, level N , and character χ as in Definition 12.3.9. Let G L(n, R)+ denote the subgroup of G L(n, R) consisting of all n × n invertible real matrices with positive determinant.

8

The classical theory of automorphic forms for GL(n, R)  : G L(n, R)+ → Cr by Define   (g) := |ρ g (In ),

(g ∈ G L(n, R)+ ),

where In denotes the n × n identity matrix, regarded as an element of hn .  satisfies the following natural It follows from the definitions that generalization of (12.3.11):   (gk) = ρ(k) · (g), ∀g ∈ G L(n, R)+ , k ∈ S O(n, R) . (12.3.13) Next, it is necessary to introduce the universal enveloping algebra U (g) of the Lie algebra g := gl(n, C), as well as its center Z (U (g)). We follow Section 2.3 in [Goldfeld, 2006]. Definition 12.3.14 (Universal enveloping algebra and its center) For each α ∈ gl(n, R) we define a differential operator Dα acting on smooth functions φ : G L(n, R) → C by the formula     1  g ∈ G L(n, R) . Dα φ(g) = lim φ g · exp(tα) − φ(g) , t→0 t We extend this to vector-valued functions by linearity: ⎛ ⎞ ⎛ ⎞ φ1 D α φ1 . . Dα ⎝ .. ⎠ := ⎝ .. ⎠ . φr Dα φr Then the algebra of differential operators with complex coefficients generated by the operators Dα , α ∈ gl(n, R) is a realization of the universal enveloping algebra U (g) of g := gl(n, C). Its center, Z (U (g)) is isomorphic to a polynomial algebra in n generators. One choice of generators is given by the Casimir differential operators: ⎫

⎧

n n  n ⎬ ⎨ 

··· Di1 ,i2 ◦ Di2 ,i3 ◦ · · · ◦ Dim ,i1

1 ≤ m ≤ n , ⎭ ⎩

i 1 =1 i 2 =1 i m =1 where for 1 ≤ i ≤ n, 1 ≤ j ≤ n, we have Di, j := D Ei, j , and E i, j ∈ gl(n, R) is the n × n matrix with a 1 at the i, j entry and zeros everywhere else. In order to have a well-defined action of the center Z (U (g)) of the universal enveloping algebra U (g) on the space of functions G L(n, R)+ → Cr satisfying the ρ-equivariance condition (12.3.13), it is necessary to check that the action of Z (U (g)) on the space of all smooth functions G L(n, R)+ → Cr preserves this space. This follows easily from the next proposition. Proposition 12.3.15 (The elements of Z (U (g)) are invariant differential operators) Let : G L(n, R) → C be a smooth function, and let D be an

12.3 Automorphic functions

9

element of Z (U (g)). For g ∈ G L(n, R) let π (g) denote the action by right translation, given by (π (g) . )(h) := (h · g). Then D . π (g) . = π (g) . D . ,

(∀g ∈ G L(n, R)).

Proof The identity to be proved may be put into the equivalent form π (g) ◦ D ◦ π (g)−1 = D,

(∀g ∈ G L(n, R), D ∈ Z (U (g))),

where ◦ denotes composition of operators. Clearly, it suffices to consider only the generators given in Definition 12.3.14. From the matrix identity g·exp(tα)·g −1 = exp(t·(g·α·g −1 )),

(∀t ∈ R,g ∈ G L(n, R),α ∈ gl(n, R)),

it follows that π (g) ◦ Dα ◦ π (g)−1 = Dg·α·g−1 ,

(∀g ∈ G L(n, R), α ∈ gl(n, R)).

Form an n × n matrix M of differential operators such that the i, j entry is Di, j := D Ei, j where E i, j ∈ gl(n, R) is the matrix with a 1 at the i, j entry and zeros everywhere else. Observe that the degree m generator given in Definition 12.3.14 is simply the trace of the m-fold matrix product M m . It follows easily from the linearity of the map α → Dα that the i, j entry of the matrix g Mg −1 is Dg·Ei, j ·g−1 . The invariance of each of the generators given in Definition 12.3.14 now follows easily from the invariance of the trace of a matrix under action by conjugation.  Example The case n = 2 was considered previously in Section 5.1. The center of the universal enveloping algebra has two generators when n = 2, namely D I2 = D1,1 +D2,2 ,

and

= D1,1 ◦D1,1 +D1,2 ◦D2,1 +D2,1 ◦D1,2 +D2,2 ◦D2,2 .

Here Di, j := D Ei, j , where E i, j ∈ gl(n, R) is the matrix with a 1 at the i, j entry and zeros everywhere else. Invariance of these differential operators was proved by a different method in Section 5.1. To check it via the method of Proposition 12.3.15, we consider the matrix of differential operators  D1,1 D1,2 . M := D2,1 D2,2 Then D1,1 + D2,2 is the trace of M, while is the trace of   D1,1 D1,2 D1,1 D1,2 2 · M = D2,1 D2,2 D2,1 D2,2  D1,1 ◦ D1,1 + D1,2 ◦ D2,1 D1,1 ◦ D1,2 + D1,2 ◦ D2,2 . = D2,1 ◦ D1,1 + D2,2 ◦ D2,1 D2,1 ◦ D1,2 + D2,2 ◦ D2,2

The classical theory of automorphic forms for GL(n, R)

10

Definition 12.3.16 (Action of Z (U (g)) on automorphic functions of weight ρ, level N and character χ ) Let be a vector valued automorphic function  denote the of weight ρ, level N and character χ as in Definition 12.3.9. Let + lift of to G L(n, R) as in Definition 12.3.12. For each D ∈ Z (U (g)), and each z ∈ hn define )(z). (D . )(z) = (D With this in hand, it is finally possible to give a generalization of Definition 3.5.7. Definition 12.3.17 (Vector valued Maass form for G L(n, R)) Let n ≥ 2, r ≥ 1, N ≥ 1 be integers. Fix a Dirichlet character χ (mod N ), and an irreducible representation ρ : S O(n, R) → G L(r, C). A vector valued Maass form of weight ρ and character χ for 0 (N ) is a smooth function : hn → Cr satisfying the following conditions    A B • ρ γ (z) = χ (d) (z), for all γ = ∈ 0 (N ), z ∈ h; C d • the function is an eigenfunction of every element of Z (U (g)), acting as in Definition 12.3.16; • is of moderate growth as in Definition 12.3.10; || (z)||2 d ∗ z < ∞, • 0 (N )\hn

where d ∗ z denotes the G L(n, R)-invariant measure given in [Goldfeld, 2006, 1.5], and || || denotes a positive definite norm on Cr . (It is easy to see that any two such norms give the same condition here.) A Maass form is said to be of level N , if it is a Maass form for 0 (N ), and it is not a Maass form for 0 (M) with M < N . Definition 12.3.18 (Factorization of hn corresponding to an integer m < n) For n ≥ 2, let hn denote the generalized upper half plane associated to G L(n, R) as in Definition 12.1.1. Let m be an integer with 1 ≤ m < n. We define a factorization of hn into two pieces which depend on m. As usual, Ir (for r = 1, 2, 3, . . . ) denotes the r × r identity matrix. Define X mn ⊂ hn to be the set of all real valued matrices x  of the form ⎛ ⎛ ⎞⎞ x1,m+1 . . . x1,n  X Im ⎜ ⎜ . .. ⎟⎟ , .. , x = ⎝where X = ⎝ .. . . ⎠⎠ 0 In−m xm,m+1 . . . xm,n and let Ymn be the set of all real valued matrices y  of the form     zd 0  , where z  ∈ hm , z  ∈ hn−m , d = r Im (with r > 0) . y = 0 z 

12.3 Automorphic functions

11

It is left as an exercise that every element of hn is uniquely expressible as x · y  with x  ∈ X mn and y  ∈ Ymn . Note that X mn is actually a subgroup of G L(n, R). Let X mn (Z) denote the subgroup of elements of X mn with integer entries. Let 0 (N ) be the subgroup of S L(n, Z) defined in Definition 12.2.3. It is of finite index in G L(n, Z). Consequently, for any fixed γ ∈ G L(n, Z), the  n −1 subgroup X m (Z) ∩ γ 0 (N )γ is of finite index in X mn (Z). 

Definition 12.3.19 (Domain associated to m, γ , and N ) Take two integers m, n with 1 ≤ m < n. Let hn denote the generalized upper half plane associated to G L(n, R), and let X mn ⊂ hn be defined as in Definition 12.3.18. For any integer N ≥ 1 and any γ ∈ G L(n, Z), fix a fundamental domain D(m, N , γ ) for X mn (Z) ∩ γ −1 0 (N )γ acting by translation on X mn . 2 Example: that n = 2. Then necessarily   1. The group X 1 is just    Suppose  m is 1 x 1n

x ∈ R , while X 12 (Z) =

n ∈ Z . The action of S L(2, Z) 0 1 0 1 on the set Q ∪ {∞} of cusps extends naturally to an action of G L(2, Z). The stabilizer of ∞ contains X 12 (Z) as a subgroup of finite index.

Given γ ∈ G L(2, Z) let a = γ · ∞. Then it is not difficult to check   that   the group X 12 (Z) ∩ γ −1 0 (N )γ is the cyclic subgroup generated by 10 m1a , where m a is definedas 3.7. A natural fundamental domain for the action of  in  1 x 2 this group on X 1 is

x ∈ [0, m a ) . 0 1 Definition 12.3.20 (Constant terms of a classical automorphic form) Let n ≥ 2, r ≥ 1, N ≥ 1 be integers. Fix a Dirichlet character χ (mod N ), and an irreducible representation ρ : S O(n, R) → G L(r, C), and let : hn → Cr be a vector valued Maass form for G L(n, R) as defined in Definition 12.3.17. For 1 ≤ m < n and  γ ∈ G L(n, Z), let D(m, N , γ ) denote a fundamental domain for X mn (Z) ∩ γ −1 0 (N )γ acting on X mn as in Definition 12.3.19. Identify X mn with Rm(n−m) and let d x denote the product measure. Then we define the constant term 

γ (x y) d x, m,γ (y) := (∀y ∈ Ymn ). D(m,N ,γ )

ρ

Example: Continuing with the example, n = 2, it has already been explained in Example 12.3.6 how, when n = 2, the representation ρ is one-dimensional and determined by an integer k, with the |ρ operator of Definition 12.3.8 reducing to the |k operator of Definition 3.5.6. In this case, the constant term given in Definition 12.3.20 takes the form  ma   y x ϕ1,γ (y) = ϕ |k γ d x, (∀y ∈ (0, ∞)). 0 1 0

12

The classical theory of automorphic forms for GL(n, R)

Let us compare this with the constant term appearing in the Fourier-Whittaker expansion of ϕ, given in Theorem 3.7.4. Recall that the terms  in√this Fourier ma 0 √ −1 , Whittaker expansion depend on a choice of matrix σa = γa 0 ma with γa ∈ S L(2, Z) such that γa . ∞ = a. Then the constant term of Theorem 3.7.4 is equal to

 √ 1

ma 0 y x

ϕ γa · dx √ −1 0 1 ma 0 k 0   √ 1  ma 0 y x = dx ϕ k γa √ −1 0 1 ma 0 0  √ √ 1  max ma y = dx ϕ k γa √ −1 ma 0 0  1  ma y max

= dx ϕ k γa 0 1 0  ma   1 ma y x dx = ϕ k γa 0 1 ma 0 1 = ϕ1,γa (y), (∀y ∈ (0, ∞)). ma By a constant term in Theorem 3.7.4, we mean a term corresponding to n+μa = 0, where μa is the cusp parameter, defined in Definition 3.7.3. Clearly, such a term exists only when μa = 0. Theorem It is also clear that the constant term Aa,0 (y) given in   3.7.4 vana 0 does. The ishes identically if and only if the constant term ϕ1,γa 0 d process of passing from γa to σa does not appear to have a straightforward generalization to arbitrary n and m. Definition 12.3.21 (Vector valued Maass cusp form for G L(n, R)) Let n ≥ 2, N ≥ 1, r ≥ 1 be integers. Fix a Dirichlet character χ (mod N ), and an irreducible representation ρ : S O(n, R) → G L(r, C). Let : hn → Cr be a vector valued Maass form for G L(n, R) as defined in Definition 12.3.17. Then is said to be cuspidal, or to be a cusp form, if m,γ (y) = 0 (see Definition 12.3.20) for all γ ∈ G L(n, Z) and all m with 1 ≤ m < n. Remarks Now let

 A M(Z) = 0

0 D



A ∈ G L(m, Z), D ∈ G L(n − m, Z) ,

and let Pm,n−m (Z) = M(Z)· X mn (Z). Then it is clear that, for any γ ∈ G L(n, Z), γ0 ∈ 0 (N ), ν ∈ X mn (Z), we have m,γ0 γ ν (y) =  χ (γ0 ) m,γ (y),

(∀y ∈ Ymn ).

Exercises for Chapter 12

13

It is not difficult to show that for μ ∈ M(Z) we have μy) , m,γ μ (y) = Jρ−1 (μ, y) · m,γ (

(∀y ∈ Ymn ).

It follows that is cuspidal if and only if m,γ (y) vanishes for all m and γ , where 1 ≤ m < n and for each m, the matrix γ ranges over a set of representatives for 0 (N )\G L(n, Z)/P(m,n−m) (Z). When n = 2, this set is in one-to-one correspondence with the 0 (N ) equivalence classes of cusps.

Exercises for Chapter 12 In Exercises 12.10 and 12.11, you will consider Maass forms of weight ρ, where ρ is no longer assumed to be irreducible. The definition is the same as Definition 12.3.21, without the condition of irreducibility. (One may consider reducible ρ in several other exercises as well.) 12.1 Compute the dimension of hn for n ≥ 2. 12.2 In this exercise we give the proof of the Iwasawa decomposition for G L(n, R) as in Proposition 12.1.2. (a) For g ∈ G L(n, R), show that there exists k ∈ S O(n, R) such that gk is upper triangular. Hint: Think of S O(n, R) as the group of rotations of the (n − 1)-sphere in Rn . (b) If h ∈ G L(n, R) is upper triangular such that all diagonal entries are positive, show that it can be written in the form h˜ · d, where h˜ ∈ hn and d is a diagonal matrix in the center of G L(n, R). (c) Deduce the existence of the Iwasawa decomposition. (d) Prove the uniqueness of the Iwasawa decomposition. 12.3 Show that G L(n, R) acts transitively on hn . 12.4 Prove that every congruence subgroup has finite index in S L(n, Z). 12.5 Prove that [G L(n, Z) : S L(n, Z)] = 2. 12.6 For each n ≥ 2 and N ≥ 1, prove that 0 (N ) is a congruence subgroup of S L(n, Z). (Don’t forget to check that 0 (N ) is a subgroup.) 12.7 Let G be any group, and let 1 , 2 be two subgroups of finite index. Prove that 1 ∩ 2 is also of finite index.

14

The classical theory of automorphic forms for GL(n, R)

12.8 Let r ≥ 1 and n ≥ 2 be integers, let ρ : S O(n, R) → G L(r, C) be a representation, and let : hn → Cr be a vector valued function. Show that the slash operator satisfies



ρ αβ = ρ α ρ β,

(α, β ∈ G L(n, R)+ ).

12.9 By analogy with the G L(2) theory, write Z = In . (a) Verify that D Z lies in the center of the enveloping algebra Z (U (g)) and that for any smooth function φ : G L(n, R) → C, we have D Z φ(g) = r

∂φ (g) ∂r

(g ∈ G L(n, R)),

where g = z · (r In ) · k with z ∈ hn , r ∈ R+ , and k ∈ O(n, R). (b) Let : hn → Cr be a vector valued automorphic function of weight ρ, level N , and character χ as in Definition 12.3.9. Show that D Z . ≡ 0. In particular, every automorphic function is an eigenfunction for D Z . (Compare with Definition 12.3.17.) 12.10 Fix a representation ρ : S O(n, R) → G L(r, C), an integer N ≥ 1, and a character χ (mod N ). If : hn → Cr is a vector valued Maass form of weight ρ, level N , and character χ , then (by definition) it is an eigenfunction of every element of Z (U (g)). Define a function λ : Z (U (g)) → C by the relation D. = λ(D)



 D ∈ Z (U (g)) .

Prove that λ is a C-algebra homomorphism and that it is determined by its values on the Casimir operators as in Definition 12.3.14. We will call λ the eigenvalue of . 12.11 Fix a representation ρ : S O(n, R) → G L(r, C), an integer N ≥ 1, a character χ (mod N ), and a C-algebra homomorphism λ : Z (U (g)) → C. Define Aρ,χ,λ (0 (N )) to be the set of vector valued Maass forms of weight ρ, level N , character χ , and eigenvalue λ (as in the previous exercise). (a) Prove that Aρ,χ,λ (0 (N )) is a complex vector space. (b) Suppose ρ is a reducible representation that decomposes as ρ = ρ1 ⊕ ρ2 . Prove that there exists an isomorphism of complex vector spaces Aρ,χ,λ (0 (N )) ∼ = Aρ1 ,χ,λ (0 (N )) ⊕ Aρ2 ,χ,λ (0 (N )). 12.12 Let m and n be integers with 1 ≤ m < n. (a) Show that the set of matrices X mn as in Definition 12.3.18 is a subgroup of G L(n, R).

Exercises for Chapter 12

15

(b) Show that every element of hn is uniquely expressible as x  · y  with x  ∈ X mn and y  ∈ Ymn . 12.13 Let r, N ≥ 1 and n ≥ 2 be integers with n even, let ρ : S O(n, R) → G L(r, C) be a representation, and let χ (mod N ) be a Dirichlet character. Suppose there exists a vector-valued automorphic function : hn → Cr of weight ρ, level N , and character χ for 0 (N ) whose image in Cr does not lie in a proper linear subspace. Prove that χ (−1)Ir = ρ(−In ). (Compare with Exercise 3.10.) 12.14* Let r, N ≥ 1 and n ≥ 2 be integers, let ρ : S O(n, R) → G L(r, C) be an irreducible representation, and let χ (mod N ) be a Dirichlet character. Suppose there exists a non-zero vector-valued automorphic function : hn → Cr of weight ρ, level N , and character χ for 0 (N ). Is it true that the image of spans Cr ? (Compare with the previous exercise.)

13 Automorphic forms and representations for G L(n, AQ )

13.1 Cartan, Bruhat decompositions for G L(n, R) The Iwasawa decomposition for G L(n, R) has been given in Section 12.1. The goal of this section is to generalize the Cartan decomposition of Section 4.1 from G L(2, R) to G L(n, R), and give one additional decomposition: the Bruhat decomposition. Proposition 13.1.1 (The Cartan decomposition for G L(n, R)) Let n ≥ 2. Every matrix g ∈ G L(n, R) has a factorization g = k1 · a · k2 with k1 , k2 ∈ O(n, R), where a is a diagonal matrix with entries a1 , . . . , an such that ai ≥ ai+1 for 1 ≤ i ≤ n − 1, and an > 0. The values of ai (1 ≤ i ≤ n) are uniquely determined by g. Proof As in the proof of Proposition 4.1.2, let s := t g · g. Then there is an orthonormal basis of Rn consisting of eigenvectors of s. The proof is the same as in the 2 × 2 case: it is impossible for a matrix which has a non-trivial block in its Jordan canonical form to be symmetric, and eigenvectors of a symmetric matrix with distinct eigenvalues are orthogonal with respect to the standard dot product. By definition, s is positive definite, so all of its eigenvalues are positive. Let k be a matrix such that the columns are an orthonormal basis of Rn consisting of eigenvectors of s, ordered so that the eigenvalues are nonincreasing. Because the i, j entry of t k · k is equal to the standard dot product of the i th and j th columns of k, requiring that the columns form an orthonormal basis is the same as requiring that t k · k is the identity matrix, i.e., that k is an element of K ∞ := O(n, R). 16

13.2 Iwasawa, Cartan, Bruhat decompositions for GL(n, Qp )

17

It follows immediately that ⎛ t

k·s·k = ⎝

⎞

λ1 ..

⎠,

. λn

√ where λi is the eigenvalue of s acting on the i th column of k. Let ai = λi , and let a be the diagonal matrix with entries a1 , . . . , an . Then an > 0, and ai ≥ ai+1 for 1 ≤ i ≤ n − 1, because of the way in which the columns of k were ordered. Furthermore t (g · k) · (g · k) = t a · a. Let k  := a · (g · k)−1 . It follows immediately that t k  · k  is the identity matrix, so that k  ∈ K ∞ . To conclude the proof of existence, note that g = k  −1 ak −1 , so we may take k1 = k  −1 and k2 = k −1 . Uniqueness of the real numbers ai (1 ≤ i ≤ n) follows easily from the construction.  The third decomposition which frequently arises in the representation theory of G L(n, R) is the Bruhat decomposition. It is given in terms of two subgroups of G L(n, R), the Weyl group, and the Borel subgroup, which are defined as follows. Definition 13.1.2 (Weyl group of G L(n)) The Weyl group of G L(n), denoted Wn , is the subgroup consisting of all n × n matrices which have exactly one 1 in each row and each column, and zeros elsewhere. Definition 13.1.3 (Borel subgroup of G L(n, R)) The Borel subgroup, denoted Bn (R) ⊂ G L(n, R) is defined to be the group of all invertible upper triangular matrices with real entries. Proposition 13.1.4 (Bruhat decomposition for G L(n, R)) Let n ≥ 2. Every element g ∈ G L(n, R) can be expressed in the form g = b1 · w · b2 , with b1 , b2 ∈ Bn (R), and w ∈ Wn . The element w is uniquely determined by g. Proof This is Proposition 10.3.2 of [Goldfeld, 2006]. See also Exercise 13.1. 

13.2 Iwasawa, Cartan, Bruhat decompositions for G L(n, Q p ) Roughly, the Iwasawa decomposition states simply that every element of the group G L(n, Q p ) is the product of an upper triangular element and an element of G L(n, Z p ). We have preferred decompositions which

18

Automorphic forms and representations for GL(n, AQ )

include uniqueness statements wherever possible, and the following version of the Iwasawa decomposition, with an explicit uniqueness statement, was worked out for us by Min Lee and Xander Faber. We would like to thank them. Proposition 13.2.1 (Iwasawa decomposition for G L(n, Q p )) Let n ≥ 1. Every g ∈ G L(n, Q p ) can be uniquely expressed in the form ⎛1 u 1,2 1 ⎜ ⎜ g=⎜ ⎜ ⎝

u 1,3 u 2,3 .. .

··· ···

u 1,n u 2,n .. . 1

u n−1,n 1

⎞

⎛ ⎟ p e1 ⎟⎜ ⎟⎝ ⎟ ⎠

⎞ ..

⎟ ⎠ k,

. p en

where k ∈ G L(n, Z p ), and ei ∈ Z for 1 ≤ i ≤ n, and for each pair (i, j) with 1 ≤ i < j ≤ n, either u i, j = 0, or ei −e j −1

u i, j =



u i, j (l) pl

l=−Ni, j

with 0 ≤ u i, j (l) ≤ p − 1, and u i, j (−Ni, j ) =/ 0. Remark The condition on u i, j can be stated succinctly by saying that its p-adic expansion as in (1.2.5) terminates at (i.e., has no non-zero terms after) p ei −e j −1 . It is immediate from the p-adic expansion (1.2.5) that for any f ∈ Z, every element of Q p is uniquely expressible as the sum of an element of the fractional ideal p f ·Z p and an element of Q p whose p-adic expansion terminates at p f −1 . Thus, one may regard the elements of Q p whose p-adic expansion terminates at p f −1 as being reduced modulo the fractional ideal p f · Z p . Proof The proof proceeds by induction on n. The case n = 1 is trivial, and the case n = 2 was proved in Proposition 4.2.1. The next key observation is that the following simple operations (called Gaussian elimination) can be performed using right multiplication by elements of G L(n, Z p ): (i) column swaps, (ii) multiplication of a column by an element of Z×p , (iii) addition of a Z p -multiple of one column to another. Now, there is a unique integer e such that the bottom row of ⎞ ⎛ 1 .. ⎟ ⎜ . ⎟·g ⎜ ⎠ ⎝ 1 pe

13.2 Iwasawa, Cartan, Bruhat decompositions for GL(n, Qp )

19

has all of its entries in Z p and at least one entry in Z×p . By performing Gaussian elimination on the columns of this last matrix, we see that there exists a matrix k1 ∈ G L(n, Z p ) such that   In−1 0 A B ·g·k = ,A ∈ G L(n−1, Q p ), B ∈ Mat(n−1×1, Q p ). 1 0 pe 0 1 (13.2.2) Here, In−1 denotes the (n − 1) × (n − 1) identity matrix and Mat(n − 1 × 1, Q p ) denotes the set of all n − 1 × 1 matrices with entries in Q p . By the inductive hypothesis, we may write A = b k  with k  ∈ G L(n − 1, Z p ) and ⎛1 u u 1,3 · · · u 1,n−1 ⎞ ⎛ 1,2 ⎞ 1 u 2,3 · · · u 2,n−1 ⎟ p e1 ⎜ ⎟⎜ ⎜ ⎟ .. .. .. ⎟⎝ b = ⎜ ⎠ . . . ⎟ ⎜ ⎠ ⎝ en−1 p , 1 u n−2,n−1 1 where u i, j ∈ Q p has a p-adic expansion which terminates at ei − e j for 1 ≤ i < j ≤ n − 1. Then     k 0 B b k, where k := · k1−1 . g= 0 p −e 0 1 

Furthermore if

then



A1 0

A2 0

B1 p −e B2 p −e



 k1 =

−1 

A1 0

A2 0 B1 p −e

B2 p −e

k2 ,

= k2 k1−1 ,

and hence A−1 2 A1 ∈ G L(n − 1, Z p ). It follows from the uniqueness part of the inductive hypothesis that the matrix b is uniquely determined by g, i.e., is independent of the choice of A, B and k1 in (13.2.2). Thus, the expression ⎛1 u u 1,3 · · · u 1,n ⎞ ⎛ 1,2 ⎞ 1 u 2,3 · · · u 2,n ⎟ p e1 ⎜ ⎜ ⎟ .. ⎟ .. .. ⎟⎜ g=⎜ ⎠k . . . ⎟⎝ ⎜ ⎠ ⎝ en p 1 u n−1,n 1 follows from simply setting en = −e and ⎞ ⎛ u 1,n ⎜ .. ⎟ e ⎝ . ⎠ = p B. u n−1,n

Automorphic forms and representations for GL(n, AQ )

20

It must be shown that the matrix B in (13.2.2) can be chosen so that the p-adic expansion of u i,n terminates at p ei −e j −1 for 1 ≤ i ≤ n − 1, and, moreover, that if this condition is placed on B, then B is uniquely determined by g. To prove existence, we perform additional column operations on ⎛ p e1 gk

−1

⎜ ⎜ =⎜ ⎜ ⎝

p e2 u 1,2 p e2

p e3 u 1,3 p e3 u 2,3 .. .

··· ···

p en u 1,n p en u 2,n .. . p en−1

p en u n−1,n 1

⎞ ⎟ ⎟ ⎟. ⎟ ⎠

By adding a suitable multiple of the second-to-last column to the second, we may arrange that u n−1,n is reduced modulo p en−1 −en · Z p . Then, by adding a multiple of the third-to-last column, we may arrange that u n−2,n is reduced modulo p en−2 −en and so on. This proves existence. As for uniqueness, if     b B1 b B2 k k2 , = 1 0 p en 0 p en then 

Write

b 0

B1 p en

−1 

b 0

B2 p en



 =

In 0

⎞ p en u 1,n ⎟ ⎜ .. B1 = ⎝ ⎠, . en p u n−1,n ⎛

(b )−1 (B2 − B1 ) 1

∈ G L(n, Z p ).

⎞ p en v1,n ⎟ ⎜ .. B2 = ⎝ ⎠, . en p vn−1,n ⎛

and assume that, for 1 ≤ i ≤ n, both u i,n and vi,n are reduced modulo p ei −en · Z p . The bottom entry of (b )−1 (B2 − B1 ) is p en −en−1 (vn−1,n − u n−1,n ). Since this is in Z p , it follows that vn−1,n − u n−1,n ∈ p en−1 −en Z p . Since both u n−1,n and vn−1,n are reduced modulo p en−1 −en · Z p , it follows that vn−1,n − u n−1,n = 0. Once it is known that the bottom entry of (b )−1 (B2 − B1 ) is zero, it follows that the second entry from the bottom is p en −en−2 (vn−2,n − u n−2,n ), and the argument proceeds as before. The uniqueness of B, and hence of the entire factorization, follows.  Proposition 13.2.3 (Cartan decomposition for G L(n, Q p )) Let n ≥ 2. Every matrix g ∈ G L(n, Q p ) has a factorization ⎞ ⎛ m p 1 ⎟ ⎜ .. g = k1 · ⎝ ⎠ · k2 , . pm n

13.2 Iwasawa, Cartan, Bruhat decompositions for GL(n, Qp )

21

where k1 , k2 ∈ G L(n, Z p ) and m i ∈ Z, (1 ≤ i ≤ n) such that m i ≤ m i+1 for 1 ≤ i ≤ n − 1. The integers m 1 , m 2 , . . . , m n are uniquely determined by g. Proof Let g be an arbitrary element of G L(n, Q p ). We need to prove that the double coset G L(n, Z p ) · g · G L(n, Z p ) contains an element of the form ⎛

⎞

pm 1

⎜ ⎝

..

⎟ ⎠.

. pm n

Arguing by induction, it is enough to prove that G L(n, Z p ) · g · G L(n, Z p ) contains an element of the form  m p 1 , (m 1 ∈ Z, g  ∈ G L(n − 1, Q p )). (13.2.4) g Since G L(n, Z p ) contains the permutation matrices, it follows that G L(n, Z p ) · g · G L(n, Z p ) contains an element such that |g1,1 | p ≥ |g1, j | p for 2 ≤ j ≤ n. and |g1,1 | p ≥ |gi,1 | p for 2 ≤ i ≤ n. Then ⎛

1 − gg1,2 1,1 ⎜ 1 ⎜ ⎝

... ..

− gg1,n 1,1

.

⎞

⎛

⎟ ⎟, ⎠

⎜ − g2,1 ⎜ 1,1 ⎜ . ⎝ .. − ggn,1 1,1

1

⎞

1 g

⎟ ⎟ ⎟ ∈ G L(n, Z p ), ⎠

1 ..

. 1

and ⎛ ⎜ ⎜ ⎝

1

− gg1,2 1,1 1

... ..

− gg1,n 1,1

. 1

⎞

⎛

g ⎜ − g2,1 ⎜ 1,1

⎟ ⎟·g·⎜ ⎠ ⎝

⎞

1 .. .

− ggn,1 1,1

⎟  g1,1 ⎟ ⎟= ⎠

1 ..

.

g

,

1

 g |g | with g  ∈ G L(n − 1, Q p ). Multiplying (on either side) by 1,1 01,1 p (which is in G L(n, Z p )) we obtain a matrix of the form (13.2.4). 

0



In−1

In order to introduce the Bruhat decomposition for G L(n, Q p ) we introduce the Borel subgroup of G L(n, Q p ). (The definition of the Weyl group is the same, since elements of Wn have entries in Z, which is contained in Qv for any v.) Definition 13.2.5 (Borel subgroup of G L(n, Q p )) The Borel subgroup, denoted Bn (Q p ) ⊂ G L(n, Q p ) is defined to be the group of all invertible upper triangular matrices with entries in Q p .

22

Automorphic forms and representations for GL(n, AQ )

Proposition 13.2.6 (Bruhat decomposition for G L(n, Q p )) Let n ≥ 2. Every element g ∈ G L(n, Q p ) can be expressed in the form g = b1 · w · b2 , with b1 , b2 ∈ Bn (Q p ), and w ∈ Wn . The element w is uniquely determined by g. Proof The proof given in the real case in Proposition 10.3.2 of [Goldfeld, 2006] remains valid over any field, because only basic arithmetic operations which make sense in any field are used. See also Exercise 13.1. 

13.3 Strong approximation for G L(n) Formally, the adele group G L(n, AQ ) is the restricted product (relative to the maximal compact subgroups K p = G L(n, Z p )) G L(n, AQ ) = G L(n, R) ×

!

G L(n, Q p )

p

where restricted product (relative to the subgroups K p ) means that all but finitely many of the components in the product are in K p . An element g ∈ G L(n, AQ ) will be denoted in the form g = {g∞ , . . . , g p , . . . } where gv ∈ G L(n, Qv ) for all v ≤ ∞ and g p ∈ K p for all but finitely many primes p. It is understood here that {g∞ , . . . , g p , . . . } represents the infinite vector {gv }v≤∞ = {g∞ , g2 , g3 , g5 , . . . , }. We shall define the diagonal embedding map i diag : G L(n, Q) → G L(n, AQ ) by i diag (γ ) := {γ , γ , γ , . . . },

(∀γ ∈ G L(n, Q)).

When no confusion will result, we may identify elements of G L(n, Q) with their images in G L(n, AQ ) under this embedding. We also define the embedding at ∞, denoted i ∞ : G L(n, R) → G L(n, AQ ), which is defined by i ∞ (g∞ ) = {g∞ , In , In , In ,

··· ,

},

(∀g∞ ∈ G L(n, R)).

Here In is the n × n identity matrix. Then i ∞ (S L(n, R)) denotes the subgroup of all elements of the above form with α ∈ S L(n, R).

13.3 Strong approximation for GL(n)

23

We wish to prove the G L(n) analogue of Lemma 4.11.7. In order to do this, we first need to generalize Lemma 4.4.1. Lemma 13.3.1 S L(n, AQ ).

The group i diag (S L(n, Q)) · i ∞ (S L(n, R)) is dense in

Proof Let H denote the closure of i diag (S L(n, Q))·i ∞ (S L(n, R)) in S L(n, AQ ). It is a subgroup because multiplication is continuous. For each i, j with 1 ≤ i, j ≤ n and i =/ j, we may define a subgroup Ui j ⊂ S L(n, AQ ) consisting of those elements u of S L(n, AQ ) such that every entry of u − In is zero, except possibly the i, j entry. Then it follows from strong approximation for AQ that Ui, j ⊂ H for all i, j. But the subgroups Ui, j generate S L(n, AQ ). (See Exercise 13.3.)  Next we wish to introduce the appropriate generalization of the subgroup K 0 (N ) defined in Definition 4.11.6. Definition 13.3.2 (The compact subgroup K 0 (N )) Fix N ∈ Z with prime "r ei power decomposition N = i=1 pi . Then we define an open compact " subgroup K 0 (N ) ⊂ p G L(n, Z p ) ⊂ G L(n, Afinite ) as follows. For k ∈ " p G L(n, Z p ) write k = {k2 , k3 , . . . , k p , . . . }, and  A p Bp , kp = C p dp ⎧ A p , an (n − 1) × (n − 1) matrix, with entries in Z p , ⎪ ⎪ ⎪ ⎨ B , a column vector, with entries in Z , p p with ⎪ C , a row vector, with entries in Z , p p ⎪ ⎪ ⎩ d p , an element of Z p . " Then K 0 (N ) is the subgroup of p G L(n, Z p ) defined by imposing the conditions (i = 1, 2, . . . , r ). C pi ∈ ( piei Z pi )n−1 , Proposition 13.3.3 (Strong approximation for K 0 (N )) For g ∈ G L(n, AQ ), there exist γ ∈ G L(n, Q), g∞ ∈ G L(n, R) and k ∈ K 0 (N ) such that g = i diag (γ ) · i ∞ (g∞ ) · k. Proof First assume that g ∈ S L(n, AQ ). Take U∞ any open neighborhood of the from Lemma 13.3.1 that the set  identity in S L(n, R). Then it follows   ) for some γ ∈ S L(n, Q) and g∞ ∈ g · U∞ · K 0 (N ) contains i diag (γ )i ∞ (g∞ S L(n, R). But then    ) g · i ∞ (u ∞ ) · k  = i diag (γ )i ∞ (g∞  ∈ S L(n, R). It follows at with γ ∈ S L(n, Q), u ∞ ∈ U∞ , k  ∈ K 0 (N ), and g∞  −1  · u −1 once that g = i diag (γ ) · i ∞ (g∞ ) · k, with k = (k ) and g∞ = g∞ ∞.

Automorphic forms and representations for GL(n, AQ )

24

If det g =/ 1, it follows from strong approximation in AQ that det g may " be written as α · y∞ · yfinite with α ∈ Q× , y∞ ∈ (0, ∞) and yfinite ∈ p Z×p .  ∈ G L(n, R), and k  ∈ K 0 (N ) with Clearly, there exist γ  ∈ G L(n, Q), g∞  −1  −1  −1 . det(γ ) = α , det(g∞ ) = y∞ and det(k ) = yfinite  ) · k  . Then g  ∈ S L(n, AQ ), so g  = i diag (γ  ) · Let g  = i diag (γ  ) · g · i ∞ (g∞     ∈ S L(n, R) and k  ∈ K 0 (N ), i ∞ (g∞ ) · k for some γ ∈ S L(n, Q), g∞ and hence g = i diag (γ ) · i ∞ (g∞ ) · k, where γ = (γ  )−1 γ  ∈ G L(n, Q),   −1 (g∞ ) ∈ G L(n, R) and k = k  (k  )−1 ∈ K 0 (N ).  g∞ = g∞

Corollary 13.3.4 (Unique factorization for G L(n, AQ )) Fix N ∈ Z with N ≥ 1. Define  0 (N ) ⊂ G L(n, Z) as ⎧ ⎪ ⎨

A C ⎪ ⎩

B d



⎫

A is (n − 1) × (n − 1) with entries in Z ⎪ ⎬

B is (n − 1) × 1 with entries in Z . ∈ G L(n, Z)

C is 1 × (n − 1) with entries in N Z ⎪ ⎭

d∈Z

Let D∞ be any fundamental domain for the action of  0 (N ) on hn . Let In denote the n × n identity matrix. Then every element g of G L(n, AQ ) may be expressed uniquely in the form g = i diag (γ ) · i ∞ (z · d) · k with γ ∈ G L(n, Q), z ∈ D∞ , d = r · In (with r > 0) and k ∈ O(n, R)· K 0 (N ). Proof The key observation is the following. For γ ∈ G L(n, Q p ), define i finite (γ ) ∈ G L(n, Afinite ) to be the diagonal embedding at only the finite places. 0 (N ). Then i finite (γ ) ∈ K 0 (N ) ⇐⇒ γ ∈  Now, by Proposition 13.3.3, g = i diag (γ1 ) · i ∞ (g∞ ) · k1 for some γ1 ∈ G L(n, Q), some g∞ ∈ G L(n, R) and some k1 ∈ K 0 (N ). By proposition 12.1.2, g∞ = g˜ · d · k for some g˜ ∈ hn , k ∈ O(n, R), and d a diagonal matrix in the center of G L(n, R). Furthermore, d may be assumed to be of the form r · In with r > 0, in which case g˜ , k and d are all unique. We may choose 0 (N ) so γ2 · g˜ = z · d  · k  with z ∈ D∞ , k  ∈ O(n, R) and d  = r  · In γ2 ∈  with r  > 0. Then g = i diag (γ2 γ1 ) · i ∞ (zdd  ) · (i ∞ (k  k) · k1 ). This completes the proof of existence. Uniqueness is an elementary exercise, making use again of the key fact that 0 (N ).  i finite (γ ) ∈ K 0 (N ) ⇐⇒ γ ∈ 

13.4 Adelic lifts and automorphic forms for G L(n, AQ ) The strong approximation lays the foundation for the definition of the adelic lift of a vector valued automorphic function of weight ρ, level N and character

13.4 Adelic lifts and automorphic forms for GL(n, AQ )

25

χ as defined in Definition 12.3.9. Another necessary ingredient is a G L(n) version of the character  χidelic of Definition 4.12.5. Definition 13.4.1 (The G L(n) idelic lift of a Dirichlet character) Fix N ∈ Z " with prime power decomposition N = ri=1 piei . For each pi |N , let χ pi be a Dirichlet character (mod pi ), and define χ (d) :=

r !

χ pi (d)−1 ,

(∀d ∈ Z, (d, N ) = 1) .

i=1

Assume kfinite ∈ K 0 (N ) , with A p , B p , C p and d p defined, for each finite prime p, as in Definition 13.3.2. Then we define the idelic lift  χidelic of the Dirichlet character χ by r !   χ pi (d pi ).  χidelic kfinite := i=1

Definition 13.4.2 (Adelic lift of a vector valued automorphic function) Let n ≥ 2, r ≥ 1 be integers. Fix an irreducible representation ρ : O(n, R) → G L(r, C), an integer N ≥ 1, and a Dirichlet character χ (mod N ). Let be a vector valued automorphic function of weight ρ and character χ for 0 (N ) as in Definition 12.3.9. By strong approximation (Proposition 13.3.3) every matrix g ∈ G L(n, AQ ) can be represented in the form g = i diag (γ ) i ∞ (g∞ ) kfinite ,

  γ ∈ G L(n, Q), g∞ ∈ G L(n, R), kfinite ∈ K 0 (N ) .

Define the adelic lift adelic : G L(n, AQ ) → Cr by the formula     χidelic (kfinite ) · |ρ g∞ (In ). adelic i diag (γ ) i ∞ (g∞ ) kfinite :=  Here In denotes the n × n identity matrix. The proof that this is well-defined is the same as in the G L(2) case. See Lemma 4.12.10 and the arguments following it. Next, we will define the notion of an adelic automorphic form. This requires that we recall the definitions of “smooth,” “moderate growth,” “K -finite” and “Z (U (g))-finite” which appear in this definition. Each of these terms may be defined for G L(n) simply by replacing 2 by n in the definition from Chapter 4. For the convenience of the reader we repeat the definitions here. Definition 13.4.3 (Smoothness) A function φ : G L(n, AQ ) → C is said to be smooth if for every fixed g0 ∈ G L(n, AQ ), there exists an open set U of U : G L(n, R) → C such G L(n, AQ ), containing g0 , and a smooth function φ∞ U that φ(x) = φ∞ (x∞ ) for all x = {x∞ , x2 , x3 , . . . , x p . . . } ∈ U.

26

Automorphic forms and representations for GL(n, AQ )

Definition 13.4.4 (Moderate growth) For each place v of Q define a norm function || ||v on G L(n, Qv ) by   ||g||v = max {|gi, j |v | 1 ≤ i, j ≤ n} ∪ {| det(g)|v } . " Define a norm function || || on G L(n, AQ ) by ||g|| = v ||gv ||v . Then we say a function φ : G L(n, AQ ) → C is of moderate growth if there exist constants C, B > 0 such that |φ(g)| < C||g|| B for all g ∈ G L(n, AQ ). " Definition 13.4.5 (K -finiteness) Let K = O(n, R) p G L(n, Z p ) be the standard maximal compact subgroup of G L(n, AQ ). A function φ : G L(n, AQ ) → C is said to be right K -finite if the set {φ(gk) | k ∈ K }, of all right translates of φ(g) generates a finite dimensional vector space. Definition 13.4.6 (Z (U (g))-finiteness) Let Z (U (g)) denote the center of the universal enveloping algebra of g = gl(n, C) as in Definition 12.3.14. Then we say a function φ : G L(n, AQ ) → C is Z (U (g))-finite if the set

% $ Dφ(g) D ∈ Z (U (g)) generates a finite dimensional vector space. Definition 13.4.7 (Adelic automorphic form on G L(n, AQ ) with central × character) Fix a unitary Hecke character ω : Q× \A× Q → C as in Definition 2.1.2. An automorphic form for G L(n, AQ ) with central character ω is a smooth function φ : G L(n, AQ ) → C which satisfies the following five properties:   (1) φ(γ g) = φ(g), ∀g ∈ G L(n, AQ ), γ ∈ G L(n,Q) . (2) φ(zg) = ω(z)φ(g), ∀g ∈ G L(n, AQ ), z ∈ A× Q . (3) φ is right K -finite. (4) φ is Z (U (g))-finite. (5) φ is of moderate growth. As remarked after Definition 4.7.7, it is possible to consider automorphic forms without central character, but we will not do so in this book. It is clear that the adelic lift of a vector valued Maass form for G L(n, R) (see Definition 12.3.17), as defined in Definition 13.4.2 is not an adelic automorphic form as defined in Definition 13.4.7. This is because the former takes values in C, while the latter takes values in Cr . However, the following lemma holds. Lemma 13.4.8 Let adelic : G L(n, Q)\G L(n, AQ ) → Cr be the adelic lift, as defined in Definition 13.4.2, of a vector valued Maass form of weight ρ, and character χ for 0 (N ). Express adelic as

13.4 Adelic lifts and automorphic forms for GL(n, AQ ) ⎛ ⎞ φ1 (g) ⎜ ⎟ (g ∈ G L(n, AQ )), adelic (g) := ⎝ ... ⎠ ,

27

φr (g) with φi : G L(n, Q)\G L(n, AQ ) → C. Then each of the functions φi is an adelic automorphic form, as in Definition 13.4.7, with central character χidelic . Proof The proof proceeds in 6 steps and follows closely the proof of Proposition 4.8.4 for the G L(2) case. (1) We must show that φi is smooth. This is the same as in the G L(2) case. (2) We must show that φi (γ g) = φi (g), (∀ γ ∈ G L(n, Q), g ∈ G L(n, AQ )). This follows immediately from Definition 13.4.2.   (3) We must show φi (zg) = φi (gz) = φi (g), (∀ z ∈ Z G L(n, AQ ) ). This is the same as in the G L(2) case. (4) We must show that φi is K -finite. It follows immediately from the definitions, and the irreducibility of ρ, that for each fixed i, the span of {φi (gk) | k ∈ K }, is equal to the r -dimensional space spanned by the functions {φ j | 1 ≤ j ≤ r }. (5) We must show that φi is Z (U (g))-finite. As in the G L(2) case, φi is $an eigenfunction of every% element of

Z (U (g)), so that the span of Dφi (g) D ∈ Z (U (g)) is nothing more than the one-dimensional space spanned by φi itself. In contrast to the G L(2) case, here there is nothing to check, because we have used the action of Z (U (g)) to define the generalization of the weight k Laplacian. (6) We must show that φi has moderate growth. The proof is an easy adaptation of the G L(2) case.



Our next goal is to define adelic automorphic cusp forms. This will require the notion of parabolic subgroup. Definition 13.4.9 (Ordered partition) Let n be a positive integer. An ordered partition of n is a finite sequence κ = (κ1 , κ2 , . . . , κr ) of positive integers such that n = κ1 + · · · + κr . Definition 13.4.10 (Standard parabolic subgroup) Fix an integer n ≥ 1. Let R be a commutative ring containing “1”. Let κ = (κ1 , κ2 , . . . , κr ) be an ordered partition of n. For any positive a, b ∈ Z write Mat(a×b, R) for the set of all a×b matrices with entries in R. The standard parabolic of G L(n, R) associated to κ is

28

Automorphic forms and representations for GL(n, AQ )

⎧⎛ A1 ⎪ ⎪ ⎨⎜ 0 Pκ (R) := ⎜ ⎝ ⎪ ⎪ ⎩ 0 0

⎫ B1,r ⎞

Ai ∈ G L(κi , R), 1 ≤ i ≤ r, ⎪ ⎪ ⎬ B2,r ⎟

⎟ . .. ⎠ Bi, j ∈ Mat(κi × κ j , R), ⎪ .

⎪ 1 ≤ i < j ≤ r ⎭

Ar

... ... .. .

B1,2 A2 0 0

0

Remarks The parabolic corresponding to the ordered partition (n) of n is just G L(n, R) itself. If r = 1, then κ = (n) is the only possibility. Definition 13.4.11 (Standard Levi subgroup, Levi factor) Fix an integer n ≥ 1. Let R be a commutative ring containing “1”. Let κ = (κ1 , κ2 , . . . , κr ) be an ordered partition of n. The standard Levi subgroup of G L(n, R) associated to κ is ⎧⎛ A1 ⎪ ⎪ ⎨⎜ 0 Mκ (R) := ⎜ ⎝ ⎪ ⎪ ⎩ 0 0

⎫ ⎞ 0

⎪

⎪ 0 ⎟ Ai ∈ G L(κi , R), (1 ≤ i ≤ r ) ⎬

⎟ . .. ⎠ ⎪ .

⎪ ⎭

Ar

... ... .. .

0 A2 0 0

0

It is also referred to as the Levi factor of the standard parabolic subgroup Pκ . Definition 13.4.12 (Unipotent radical) Fix an integer n ≥ 1. Let R be a commutative ring containing “1”. Let κ = (κ1 , κ2 , . . . , κr ) be an ordered partition of n. Let Pκ (R) be the standard parabolic of G L(n, R) introduced in Definition 13.4.10. The unipotent radical of Pκ (R) is ⎧⎛ Iκ1 ⎪ ⎪ ⎨⎜ 0 Uκ (R) := ⎜ ⎝ ⎪ ⎪ 0 ⎩ 0

B1,2 Iκ2 0 0

... ... .. . 0

⎫ B1,r ⎞ ⎪

⎪ B2,r ⎟ Bi, j ∈ Mat(κi × κ j , R), ⎬

⎟ , .. ⎠ (1 ≤ i < j ≤ r ) ⎪ .

⎪ ⎭

Iκr

where (for 1 ≤ i ≤ r ), the matrix Iκi denotes the κi × κi identity matrix and Mat(κi × κ j , R) denotes the set of all κi × κ j matrices with entries in R. Definition 13.4.13 (Parabolic subgroup, Levi factor, unipotent radical) Fix an integer n ≥ 1. Let R be a commutative ring containing “1”. A subgroup P of G L(n, R) is said to be parabolic if there exists an ordered partition κ and an element g ∈ G L(n, R) such that P = g Pκ (R)g −1 , where Pκ (R) is the standard parabolic of G L(n, R) associated to κ as in Definition 13.4.10. In this situation, the group g Mκ (R)g −1 is said to be a Levi factor of P, where

13.4 Adelic lifts and automorphic forms for GL(n, AQ )

29

Mκ (R) is the standard Levi subgroup of Pκ (R) as in Definition 13.4.11. Also the group gUκ (R)g −1 is called the unipotent radical of P. If R is a ring, S ⊂ R is a subring, κ is an ordered partition and g ∈ G L(n, S) ⊂ G L(n, R), then we may write P(R) for g Pκ (R)g −1 and P(S) for g Pκ (S)g −1 , and we define U (R), U (S), in the same fashion. Remarks The language “a Levi factor” versus “the unipotent radical” requires some justification. If P is a parabolic subgroup, then the element g ∈ G L(n, R) such that P = g Pκ (R)g −1 is not unique. However, if g1 Pκ (R)g1−1 = g2 Pκ (R)g2−1 , then it immediately follows that g2−1 g1 lies in the normalizer of Pκ (R) in G L(n, R). We leave it as an exercise to the reader that a parabolic subgroup is equal to its own normalizer in G L(n, R) for all of the rings R under consideration here, although it is not true for arbitrary rings R. (See Exercise 13.6.) Consequently, g2−1 g1 ∈ Pκ (R). It is easy to verify that Uκ (R) is normal in Pκ (R), but that Mκ (R) is not (except in some degenerate cases such as κ = (n)). As a consequence, for any fixed parabolic subgroup P, the subgroup U = gUκ (R)g −1 is uniquely determined by P, i.e., is the same for all choices of g. On the other hand, the subgroup P has many different Levi factors, corresponding to different choices of g. Definition 13.4.14 (Constant terms of an adelic automorphic form) Let ϕ : G L(n, AQ ) → C be an adelic automorphic form as in Definition 13.4.7. Let P be a parabolic subgroup of G L(n) with unipotent radical U. The constant term of ϕ along P is the function: ϕ P : G L(n, AQ ) → C, given by ϕ(ug) du.

ϕ P (g) = U (Q)\U (AQ )

Definition 13.4.15 (Adelic cusp form on G L(n, AQ )) An adelic automorphic form ϕ on G L(n, AQ ), as defined in Definition 13.4.7, is said to be a cusp form (or cuspidal) if ϕ P (g) = 0 for all proper parabolic subgroups P(AQ ) of G L(n, AQ ) and for all matrices g ∈ G L(n, AQ ). Remarks on the definition of cuspidal (1) Recall that the group G L(n, AQ ) is a parabolic subgroup of itself. In order to have the right notion, one has to restrict to proper parabolic subgroups. (2) Suppose that P = γ Pκ γ −1 where γ ∈ G L(n, Q). Then ϕ P (g) = ϕ Pκ (γ −1 g). Consequently, it suffices to consider only standard parabolics. (3) Suppose that Uκ ⊂ Uκ  . This is equivalent to Pκ ⊃ Pκ  . Then it follows easily from the definitions that

30

Automorphic forms and representations for GL(n, AQ )     ϕ Pκ (g) = 0, ∀g ∈ G L(n, AQ ) =⇒ ϕ Pκ  (g) = 0, ∀g ∈ G L(n, AQ ) . However, for individual values of g, ϕ Pκ (g) = 0 =⇒  ϕ Pκ  (g) = 0. This is because ϕ Pκ  (g) may be expressed as an integral involving many different values of ϕ Pκ . Consequently, it suffices to consider only maximal standard parabolic subgroups, i.e., those corresponding to ordered partitions κ which are of the form κ = (m, n − m), 1 ≤ m < n. (See Exercise 13.9.) (4) Fix a character ω, and let Aω (G L(n, AQ )) denote the vector space of all adelic automorphic forms on G L(n, AQ ) with central character ω. Let π denote the action of G L(n, AQ ) on functions: G L(n, AQ ) → C by right translation. It follows immediately from the definitions that   π (g) . ϕ P (g  ) = π (g) . ϕ (g  ), P    ∀g, g ∈ G L(n, AQ ), ϕ ∈ Aω (G L(n, AQ )) , and that the space of cusp forms is closed under right translation. Similarly, if D is a differential operator in U (g), then D(ϕ P ) = (Dϕ) P . (5) Let P be a parabolic with unipotent radical U and Levi factor M. The constant term ϕ P satisfies ϕ P (u · m · k) = (π (k) . ϕ) P (m) for all " u ∈ U (AQ ), m ∈ M(AQ ) and k ∈ K := O(n, R) · p G L(n, Z p ). Let V be a space of automorphic forms which is invariant under the actions of U (g), K ∞ and G L(n, Afinite ). Then V is contained in the space of cusp forms if and only if ϕ P (m) = 0 for all ϕ ∈ V and m ∈ M(AQ ). This follows easily from the previous remark and the Iwasawa decomposition. Note: we do need to know that for all k ∈ K and ϕ ∈ V, the function (π (k) . ϕ) is again in V ! (See Exercise 13.11.) (6) The levi factor M(AQ ) is isomorphic to several smaller general linear groups. The definition of automorphic form generalizes in a straightforward way from G L(n, AQ ) to products. The function ϕ P (m), m ∈ M(AQ ), is an automorphic form. Indeed, it is clear that it is smooth, has moderate growth, and is invariant by M(Q). It is easy to see that the space of translates by elements of M(AQ ) ∩ K span a finite dimensional vector space. It is not at all obvious that ϕ P is Z (U (m))-finite, where m is the Lie algebra of M(R), but this can be proved. See [Moeglin-Waldspurger, 1995], I.2.17 for a summary of the argument and references. We remark that ϕ P (m) does not, in general, have a central character.

13.5 The Fourier expansion of adelic automorphic forms

31

13.5 The Fourier expansion of adelic automorphic forms The Fourier expansion of adelic automorphic forms for G L(2, AQ ) was worked out in Section 4.9. It is not immediately obvious that such an expansion also exists for G L(n, AQ ) with n ≥ 3. A real breakthrough was made in [PiatetskiShapiro, 1975] and independently in [Shalika, 1973, 1974], who obtained the Fourier expansions of adelic automorphic forms on G L(n, AQ ) for the first time. These papers led to the development of a theory of Rankin-Selberg L-functions for G L(n, AQ ) (see [Jacquet-Shalika, 1981, 1981], [JacquetPiatetski-Shapiro-Shalika, 1979, 1979, 1983]). A proof of the Fourier expansion, in a classical setting, was given in [Goldfeld, 2006, Chapter 5]. This section will be devoted to presenting the adelic proof. The Fourier expansion of adelic automorphic forms on G L(n, AQ ) requires the clever use of Fourier analysis on the maximal unipotent subgroup Un which is defined below. A major difficulty arises when n > 2 because in this case Un will be a non-abelian group. Getting around this difficulty is not so simple and entails a number of very clever manipulations. Definition 13.5.1 (The maximal unipotent subgroup of G L(n, AQ )) Fix an integer n ≥ 2. The maximal unipotent subgroup of G L(n, AQ ), denoted Un (AQ ), is defined to be the set of all n × n upper triangular matrices in G L(n, AQ ) with ones on the diagonal and arbitrary entries above the diagonal. Definition 13.5.2 (The characters of Un (Q)\Un (AQ )) Let n ≥ 2 and let ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

⎛1

⎜ ⎜ Un (AQ ) = u = ⎜ ⎜ ⎪ ⎪ ⎝ ⎪ ⎪ ⎩

u 1,2 1

⎞ u 2,3 .. .

∗ ..

. 1

u n−1,n 1

⎫ ⎪ ⎪ ⎪ ⎪ ⎬

⎟ ⎟ ⎟ ⊂ G L(n, AQ ) ⎟ ⎪ ⎪ ⎠ ⎪ ⎪ ⎭

be the maximal unipotent subgroup as in Definition 13.5.1. Let e : Q\AQ → C denote the additive character defined in Definition 1.7.1. Fix α := {α1 , α2 , . . . , αn−1 } ∈ Qn−1 . Then the function ψα : Un (AQ ) → C defined by ψα (u) := e(α1 u 1,2 + α2 u 2,3 + · · · αn−1 u n−1,n ),

(u ∈ Un (AQ )),

is a character (continuous homomorphism to C× ) of Un (AQ ), which is trivial on Un (Q). We shall sometimes describe such a character more briefly as a character of Un (Q)\Un (AQ ). The character ψα is said to be generic if αi =/ 0 for i = 1, . . . , n − 1. In order to justify Definition 13.5.2, it is necessary to prove that ψα is actually a character of the group Un (AQ ), i.e., a continuous homomorphism

Automorphic forms and representations for GL(n, AQ )

32

Un (AQ ) → C. It follows easily from the properties of e that ψα is continuous and trivial on Un (Q). This rest is given in the next lemma. Lemma 13.5.3 The function ψα defined in Definition 13.5.2 satisfies ψα (u · u  ) = ψα (u) · ψα (u  ) for all u, u  ∈ Un (AQ ). Proof The lemma follows immediately from the matrix identity ⎛1 ⎜ ⎜ ⎜ ⎜ ⎝

u 1,2 1

u 2,3 .. .

⎞ ⎛1 ⎟ ⎜ ⎟ ⎜ ⎟·⎜ ⎟ ⎜ ⎠ ⎜ ⎝

∗ ..

. 1

u n−1,n 1

⎞

u 1,2 1

u 2,3 .. .

∗ ..

. 1

u n−1,n 1

⎛

1 u 1,2 + u 1,2 ⎜ 1 ⎜ ⎜ =⎜ ⎜ ⎝

⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞

u 2,3 + u 2,3 .. .

∗ ..

. 1

u n−1,n + u n,n−1 1

and the fact that e : Q\AQ → C is an additive character.

⎟ ⎟ ⎟ ⎟, ⎟ ⎠



We are now ready to state the Fourier expansion of an arbitrary adelic cusp form on G L(n, AQ ). Theorem 13.5.4 (Fourier expansion of an adelic cusp form for G L(n, AQ )) Let n ≥ 2 be an integer and let ϕ be an adelic cusp form for G L(n, AQ ) as in Definition 13.4.15. Fix a generic character ψ : Un (AQ ) → C as in Definition 13.5.2. Then ϕ(g) =



 Wϕ

γ ∈ Un−1 (Q)\G L(n−1,Q)

γ 1

g ,

(∀g ∈ G L(n, AQ )),

where ϕ(ug)ψ −1 (u) du,

Wϕ (g) = Wϕ,ψ (g) := Un (Q)\Un (AQ )

for all g ∈ G L(n, AQ ). The series converges absolutely and uniformly on compact subsets of G L(n, AQ ).

13.5 The Fourier expansion of adelic automorphic forms

33

Remarks When n = 2, the Fourier expansion takes the simpler form 

ϕ(g) =

 Wϕ

γ ∈Q γ =/ 0

1

 ϕ

Wϕ (g) =

γ

1 0

x 1

g ,

(∀g ∈ G L(2, AQ )), (13.5.5)

g e(−x) d x,

(∀g ∈ G L(2, AQ )).

Q\AQ

The expansion (13.5.5) appears different from the expansion in Proposition 4.9.5 obtained earlier. The fact that (13.5.5) is identical to Proposition 4.9.5, in the case that ϕ is a cusp form, is left as an exercise for the reader. Definition 13.5.6 (Whittaker coefficient of an adelic cusp form) Let ϕ be an adelic cusp form for G L(n, AQ ) as in Definition 13.4.15, and let ψ be a character of Un (AQ ) as in Definition 13.5.2. The function Wϕ (g) (with g ∈ G L(n, AQ )) occurring in the Fourier expansion in Theorem 13.5.4 above is termed a Whittaker coefficient for the cusp form ϕ, relative to the character ψ. Proof of Theorem 13.5.4 The proof requires the mirabolic subgroup ⎧ ⎪ ⎨ Pn (Q) :=

A ⎪ ⎩ 0

⎛ y

⎜ A ∈ G L(n − 1, Q), y = ⎝ 1

⎫ ⎞ y1 ⎪ ⎬ .. ⎟ ∈ Qn−1 . ⎠ . ⎪ ⎭

yn−1

We also define ⎧ ⎪ ⎨ Yn (Q) :=

⎪ ⎩

In−1 0

⎛

y ⎜ y=⎝ 1

⎫ ⎞ y1 ⎪ ⎬ .. ⎟ ∈ Qn−1 . . ⎠ ⎪ ⎭

yn−1

It is clear that we may express the mirabolic subgroup as a semi-direct product of the form (13.5.7) Pn (Q) ∼ = G L(n − 1, Q)  Yn (Q). The proof of Theorem 13.5.4 will be by induction on n. The first step is to introduce a stronger form of Theorem 13.5.4. The stronger form is in fact easier to prove, because it provides a stronger inductive hypothesis. Claim: The Fourier expansion given in Theorem 13.5.4 holds for any smooth function ϕ : G L(n, AQ ) → C which satisfies ϕ(γ g) = ϕ(g),

  ∀g ∈ G L(n, AQ ), γ ∈ Pn (Q) ,

(13.5.8)

Automorphic forms and representations for GL(n, AQ )

34 and

ϕ(ug) = 0,

(∀g ∈ G L(n, AQ ), m = 1, . . . , n − 1).

U(m,n−m) (Q)\U(m,n−m) (AQ )

(13.5.9)

(Here U(m,n−m) is defined as in Definition 13.4.12.) It is clear that the claim implies Theorem 13.5.4. We now prove the claim by induction. The base case n = 2 is left to the reader. Assuming n > 2, we take ϕ : G L(n, AQ ) → C a smooth function satisfying (13.5.8) and (13.5.9), and, for β = (β1 , . . . , βn−1 ) ∈ Qn−1 define  In−1 y ϕβ (g) = g ... ϕ 0 1 Q\AQ

Q\AQ

· e p (−y1 β1 −· · ·−yn−1 βn−1 ) dy1 . . . dyn−1 . Applying adelic Fourier inversion, Theorem 1.8.10, in each variable y1 , . . . , yn−1 , we find that  ϕ(g) = ϕβ (g), (∀g ∈ G L(n, AQ )). (13.5.10) β∈Qn−1

Absolute convergence of (13.5.10) was shown in Theorem 1.8.10. We leave it to the reader to check that it is also uniform in g, for g restricted to a compact subset of G L(n, AQ ). (See Exercise 13.14.) Next, it is convenient to regard β ∈ Qn−1 as a row vector. In this case we can express y1 β1 +· · ·+ yn−1 βn−1 compactly as β · y (i.e., the product of the row vector β and the column vector y). Now, if γ ∈ G L(n − 1, Q), we compute,  γ 0 ϕβ 0 1   γ 0 In−1 y g e p (−β · y) dy1 . . . dyn−1 = ... ϕ 0 1 0 1 Q\AQ

Q\AQ

...

= Q\AQ

Q\AQ

ϕ

Q\AQ

...

=

  ϕ

In−1 0

γ −1 y 1

In−1 0

y 1

g e p (−β · y) dy1 . . . dyn−1

g e p (−β · γ · y) dy1 . . . dyn−1

Q\AQ

= ϕβ·γ (g). Now, it follows from (13.5.9) that ϕ0 = 0, where 0 = (0, . . . , 0) is the zero vector. For every non-zero element β, and any fixed αn−1 ∈ Q× , there exists

13.5 The Fourier expansion of adelic automorphic forms

35

γ ∈ G L(n − 1, Q) such that β = (0, . . . , 0, αn−1 ) · γ . Furthermore, γ is unique up to an element of the stabilizer of (0, . . . , 0, αn−1 ), which is Pn−1 (Q). Thus, (13.5.10) takes the form  ϕ(0,...,0,αn−1 ) (γ g). (13.5.11) ϕ(g) = γ ∈Pn−1 (Q)\G L(n−1,Q)

Now, fix g ∈ G L(n, AQ ), and define  h 0 ϕg,αn−1 (h) := ϕ(0,...,0,αn−1 ) g 0 1

  ∀h ∈ G L(n − 1, AQ ) .

Then for any fixed character ψ  : Un−1 (Q)\Un−1 (AQ ) → C as in Definition 13.5.2, we may define a character of ψ : Un (Q)\Un (AQ ) → C by   u y ψ = ψ  (u  )e p (−αn+1 u), 0 1 (∀u  ∈ Un−1 (AQ ), y ∈ Mat(n − 1 × 1, AQ )). It is then immediate from the definitions that ϕg,αn−1 (u  h)(ψ  )−1 (u  ) du  Un−1 (Q)\Un−1 (AQ )

  h ϕ u 0

=

0 1

g ψ −1 (u) du.

Un (Q)\Un (AQ )

Clearly, ψ is generic if and only if ψ  is, and every generic character ψ of Un (AQ ) as in Definition 13.5.2 can be obtained in this manner. Now, for each g ∈ G L(n, AQ ) and αn−1 ∈ Q× , the function ϕg,αn−1 : G L(n − 1, AQ ) → C satisfies (13.5.8) and (13.5.9) (with n − 1 substituted for n throughout, of course). Thus, by the inductive hypothesis, ϕg,αn−1 (h) is equal to     γ 0 h (ψ  )−1 (u  ) du  , ϕg,αn−1 u  0 1  γ ∈Un−2 (Q)\G L(n−2,Q) U

n−1 (Q)\Un−1 (AQ )

⎛ ⎛

γ ⎝ ⎝ = ϕ u 0 0 γ  ∈Un−2 (Q)\G L(n−2,Q) U (Q)\U (A ) n n Q 

⎞ ⎞ 0 0  h 0 g⎠ ψ −1 (u  ) du. 1 0⎠ 0 1 0 1 (13.5.12)

After substituting (13.5.12) into (13.5.11), we just need to replace two sums over γ ∈ Pn−1 (Q)\G L(n − 1, Q) and γ  ∈ Un−2 (Q)\G L(n − 2, Q) by a single sum over Un−1 (Q)\G L(n − 1, Q). This is valid, because every element of

Automorphic forms and representations for GL(n, AQ )    Un−1 (Q)\Pn−1 (Q) contains a matrix of the form γ0 01 , where we have γ  ∈ G L(n − 2, Q) unique up to an element of Un−2 (Q) on the left. 

36

13.6 Adelic automorphic representations for G L(n, AQ ) Adelic automorphic representations for G L(1, AQ ) and G L(2, AQ ) were defined in Sections 2.5, 5.1, respectively. We now generalize the construction to G L(n, AQ ) for all n ≥ 1. Definition 13.6.1 (Vector space of adelic automorphic forms) Fix an integer × n ≥ 1 and a unitary Hecke character ω : Q× \A× Q → C as in Definition 2.1.2. Let Aω (G L(n, AQ )) denote the C-vector space of all adelic automorphic forms for G L(n, AQ ) with central character ω, as defined in Definition 13.4.7. Recall that Afinite denotes the finite adeles (as defined in Definition 1.3.3). For an integer n ≥ 1, let G L(n, Afinite ) denote the multiplicative subgroup of all afinite ∈ G L(n, AQ ) of the form afinite = {In , g2 , g3 , g5 ,

... ,

}

where g p ∈ G L(n, Q p ) for all finite primes p and g p ∈ G L(n, Z p ) for all but finitely many primes p. Here In denotes the n × n identity matrix. We shall define an automorphic representation for G L(n, AQ ) as the vector space Aω (G L(n, AQ )) with three natural actions defined as follows. • Action of the finite adeles G L(n, Afinite ) by right translation:   We define an action πfinite : G L(n, Afinite ) → G L Aω (G L(n, AQ )) as follows. For φ ∈ Aω (G L(n, AQ )), let πfinite (afinite ) . φ(g) := φ(g afinite ), for all g ∈ G L(n, AQ ), afinite ∈ G L(n, Afinite ). Here, πfinite (afinite ) . φ denotes the action of afinite on the vector φ. • Action of the group K ∞ = O(n, R) by right translation:   We shall define an action π K∞ : K ∞ → G L Aω (G L(n, AQ )) as follows. The group K ∞ = O(n, R) can be embedded in G L(n, AQ ). If k∞ ∈ K ∞ , then {k∞ , In , In , . . . } is an element of G L(n, AQ ). Now, let φ ∈ Aω (G L(n, AQ )). We define π K∞ (k) . φ(g) := φ(gk),

∀g ∈ G L(n, AQ ),

and all k = {k∞ , In , In , . . . } with k∞ ∈ K ∞ . Here π K∞ (k) . φ denotes the action of k on the vector φ.

13.6 Adelic automorphic representations for GL(n, AQ )

37

• Action of U (g) by differential operators: Let g = gl(n, C) and D ∈ U (g) (universal enveloping algebra) be a differential operator as in Definition 12.3.14. We may define an action πg of U (g) on the vector space Aω (G L(n, AQ )) as follows. For φ ∈ Aω (G L(n, AQ )) let πg (D) . φ(g) := Dφ(g),

g = {g∞ , g2 , g3 , . . . } ∈ G L(n, AQ ),

where πg (D) . φ denotes the action of D on the function φ(g), which is given by the differential operator D acting in the variable g∞ . Remarks The action of the finite adeles by right translation commutes with the action of O(n, R) and the action of the universal enveloping algebra. The action of O(n, R) and the action of the universal enveloping do not commute,  but satisfy the relation πg (Dα ) . π K∞ (k) = π K∞ (k) . πg Dk −1 αk for all α ∈ g and all elements k = {k∞ , In , In , . . . } with k∞ ∈ K ∞ (see (5.1.2)). The action of the finite adeles by right translation defines a group representation of G L(n, Afinite ). The action of K ∞ = O(n, R) by right translation defines a group representation of K ∞ . The action of U (g) does not define a group representation because U (g) is not a group: a differential operator D ∈ U (g) may not have an inverse in U (g). The proof that the space Aω (G L(n, AQ )) is preserved by these three actions is similar to the proof given in Section 5.1. We do not give the details. We now introduce the definitions that characterize the above 3 actions, and that have become customary in the modern theory of automorphic representations. Let g = gl(n, C)). We will define the following two important types of modules which will play a major role in the theory: (g, K ∞ )-module,

(g, K ∞ ) × G L(n, Afinite )-module.

Definition 13.6.2 ((g, K ∞ )-module) Fix an integer n ≥ 1. Let g = gl(n, C), K ∞ = O(n, R), and U (g) denote the universal enveloping algebra as in Section 12.3. We define a (g, K ∞ )-module to be a complex vector space V with actions πg : U (g) → End(V ), π K∞ : K ∞ → G L(V ), such that, for each v ∈ V, the subspace of V spanned by {π K∞ (k) . v | k ∈ K ∞ } is finite dimensional, and the actions πg and π K∞ satisfy the relations   πg (Dα ) . π K∞ (k) = π K∞ (k) . πg Dk −1 αk

38

Automorphic forms and representations for GL(n, AQ )

for all α ∈ g, (Dα as in Definition 12.3.14), and all k ∈ K ∞ . Further, we require that  1 πg (Dα ) . v = lim π K∞ (exp(tα)) . v − v t→0 t for all v ∈ V and α in the Lie algebra k of K ∞ , (see Theorem 4.5.8 and the examples following it) which is contained in g. Note that the limit is defined, without a topology on all of V, because π K∞ (exp(tα)) . v remains within a finite dimensional subspace. We shall denote the pair of actions (πg , π K∞ ) by π and shall also refer to the ordered pair (π, V ) as a (g, K ∞ )-module. Definition 13.6.3 ((g, K ∞ )×G L(n, Afinite )-module) Fix an integer n ≥ 1. Let g = gl(n, C) and K ∞ = O(n, R). We define a (g, K ∞ ) × G L(n, Afinite )-module to be a complex vector space V with actions πg : U (g) → End(V ), π K∞ : K ∞ → G L(V ), πfinite : G L(n, Afinite ) → G L(V ), such that V, πg , and π K∞ form a (g, K ∞ )-module, and in addition the relations πfinite (afinite ) · πg (Dα ) = πg (Dα ) · πfinite (afinite ), πfinite (afinite ) · π K∞ (k) = π K∞ (k) · πfinite (afinite ), are satisfied for all α ∈ g, (Dα as in Definition 12.3.14), k ∈ k∞ , and afinite ∈ G L(n, Afinite ).   We let π = (πg , π K∞ ), πfinite , and refer to the ordered pair (π, V ) as a (g, K ∞ ) × G L(n, Afinite )-module as well. Definition 13.6.4 (Irreducible, smooth, admissible (g, K ∞ ) × G L(n, Afinite )module) Fix an integer n ≥ 1. Let (π, V ) be a (g, K ∞ ) × G L(n, Afinite )module as in Definition 13.6.3. We say that (π, V ) is smooth if every vector v ∈ V is fixed by some open compact subgroup of G L(n, Afinite ) under the action πfinite : G L(n, Afinite ) → G L(V ). We say that (π, V ) is admissible if it is smooth, and, in addition, for any fixed open compact subgroup K  ⊂ G L(n, Afinite ) and any fixed finite-dimensional representation ρ of S O(n, R), the set of vectors in V which are fixed by K  and generate a subrepresentation under the action of S O(n, R) which is isomorphic to ρ spans a finite dimensional space. Finally, the (g, K ∞ ) × G L(n, Afinite )module is said to be irreducible if it is non-zero and has no proper non-zero subspace preserved by the actions πg , π K∞ , πfinite .

13.6 Adelic automorphic representations for GL(n, AQ )

39

Lemma 13.6.5 (The space of all adelic automorphic forms with central character ω is a smooth (g, K ∞ ) × G L(n, Afinite )-module) Fix an integer n ≥ 1 and a unitary Hecke character ω : Q× \AQ → C× , as in Definition 2.1.2. Let Aω (G L(n, AQ )) denote the vector space of all adelic automorphic forms with central character ω as in Definition 13.6.1. Then Aω (G L(n, AQ )) is a smooth (g, K ∞ ) × G L(n, Afinite )-module as defined in Definition 13.6.4. Proof The proof is similar to the proof of Lemma 5.1.7 and is left to the reader as an exercise.  It is also important to define the two notions of isomorphic (g, K ∞ )modules and isomorphic (g, K ∞ ) × G L(n, Afinite )-modules. We shall actually define a more general notion of intertwining which is a type of morphism between these modules. Definition 13.6.6 (Intertwining map of (g, K ∞ )-modules) Let V, V  be two complex vector spaces which define two (g, K ∞ )-modules with associated actions: πg : U (g) → End(V ),

πg : U (g) → End(V  ),

π K∞ : K ∞ → G L(V ),

π K ∞ : K ∞ → G L(V  ).

Let ◦ denote composition of functions. A linear map L : V → V  is said to be intertwining if L ◦ πg (D) = πg (D) ◦ L ,

(∀ D ∈ U (g)),



L ◦ π K∞ (k) = π K∞ (k) ◦ L ,

(∀ k ∈ K ∞ ).

If the linear map L is an isomorphism, then we say the two (g, K ∞ )-modules are isomorphic. Definition 13.6.7 (Intertwining map of (g, K ∞ )×G L(n, Afinite )-modules) Fix an integer n ≥ 1. Let V, V  be two complex vector spaces which define two (g, K ∞ ) × G L(n, Afinite )-modules with associated actions: πg : U (g) → End(V ),

πg : U (g) → End(V  ),

π K ∞ : K ∞ → G L(V ),

π K ∞ : K ∞ → G L(V  ),

πfinite : G L(n, Afinite ) → G L(V ),

πA

finite

: G L(n, Afinite ) → G L(V  ).

A linear map L : V → V  is said to be intertwining if L ◦ πg (D) = πg (D) ◦ L , 

L ◦ π K∞ (k) = π K∞ (k) ◦ L ,

(∀ D ∈ U (g)), (∀ k ∈ K ∞ ),

40

Automorphic forms and representations for GL(n, AQ ) L ◦ πfinite (afinite ) = πA

finite

(afinite ) ◦ L ,

(∀ afinite ∈ G L(n, Afinite )).

If L is an isomorphism, then we say the two (g, K ∞ ) × G L(n, Afinite )-modules are isomorphic. Since linear maps between vector spaces are sometimes called operators, intertwining maps are sometimes called “intertwining operators.” Let V be a (g, K ∞ ) × G L(n, Afinite )-module as in Definition 13.6.3 with actions πg : U (g) → End(V ), π K∞ : K ∞ → G L(V ), πfinite : G L(n, Afinite ) → G L(V ). Let W  ⊂ W ⊂ V be vector subspaces of V . If W, W  are closed under the actions of πg , π K∞ , πfinite , then W/W  (this is a vector space quotient) defines a (g, K ∞ )×G L(n, Afinite )-module as follows. Let w+W  denote a coset in W/W  with w ∈ W. Then for all w ∈ W, D ∈ U (g), k ∈ K ∞ and afinite ∈ G L(n, Afinite ), we may define πg (D) . (w + W  ) := πg (D) . w + W  , π K∞ (k) . (w + W  ) := π K∞ (k) . w + W  , 

(13.6.8) 

πfinite (afinite ) . (w + W ) := πfinite (afinite ) . w + W . It is then easy to show that the three actions given in (13.6.8) will then define a (g, K ∞ ) × G L(n, Afinite )-module which is called a subquotient of V . We are now ready to define the important notion of automorphic representation which is one of the central themes of this book. It will turn out to be a smooth (g, K ∞ ) × G L(n, Afinite )-module as in Definition 13.6.4 which is realized as a subquotient of the complex vector space V of all adelic automorphic forms as in Definition 13.6.1. Definition 13.6.9 (Automorphic representation of G L(n, AQ ) with central character ω) Fix an integer n ≥ 1 and a unitary Hecke character ω : Q× \AQ → C× as in Definition 2.1.2. An automorphic representation with central character ω is defined to be a smooth (g, K ∞ ) × G L(n, Afinite )-module (as in Definition 13.6.4) which is also isomorphic to a subquotient of the complex vector space of adelic automorphic forms Aω (G L(n, AQ )), as defined in Definition 13.6.1. Lemma 13.6.10 (Actions by differential operators and right translation preserve the space of cusp forms) Fix an integer n ≥ 1 and a unitary Hecke character ω : Q× \AQ → C× as in Definition 2.1.2. The action of U (g) by differential operators and the actions of K ∞ and G L(n, Afinite ) by right translation

13.7 Tensor product theorem for GL(n)

41

defined on the space Aω (G L(n, AQ )) preserve the space of adelic cusp forms defined in Definition 13.4.15. Proof For the actions of K ∞ , and G L(n, Afinite ) this is obvious. For the action of g, Definition 13.4.15 tells us that it suffices to show that for any parabolic subgroup P of G L(n) with unipotent radical U, that

U (Q)\U (AQ )

 ∂ 

ϕ u g i ∞ (exp(tα)) du t=0 ∂t ⎛ =

∂ ⎜ ⎝ ∂t

⎞

⎟ ϕ u g i ∞ (exp(tα)) du ⎠



U (Q)\U (AQ )



, t=0

for all g ∈ G L(n, AQ ), all α ∈ gl(n, R), and all ϕ ∈ Aω (G L(n, AQ )). The desired assertion follows by a simple modification of the argument given in the proof of Lemma 5.1.13.  Definition 13.6.11 (Cuspidal automorphic representation with central character ω) Fix an integer n ≥ 1 and a unitary Hecke character ω:Q× \AQ → C× as in Definition 2.1.2. Let g = gl(n, C) and K ∞ = O(n, R). We define a cuspidal automorphic representation with central character ω to be a smooth (g, K ∞ ) × G L(n, Afinite )-module (as in Definition 13.6.4) which is isomorphic to a subquotient of the complex vector space of all adelic cusp forms for G L(n, AQ ) (with central character ω) as in Definition 13.4.15.

13.7 Tensor product theorem for G L(n) In this section we will generalize the tensor product Theorem 10.8.5 to irreducible cuspidal automorphic representations of G L(n, AQ ). Throughout this section we fix an integer n ≥ 1 and define g := gl(n, C) and K ∞ := O(n, R). The first step is to define the sort of product that appears in the tensor product theorem: a restricted tensor product of representations. This is a straightforward generalization of the construction for G L(2) given in Definition 10.3.3. Definition 13.7.1 (Restricted tensor product of representations) Let S be a finite set of primes containing ∞. Let (π∞ , V∞ ) denote a (g, K ∞ )-module. For each prime p ∈ / S, let (π p , V p ) be a representation of G L(n, Q p ) such G L(n,Z p ) that the space V p of G L(n, Z p )-fixed vectors is non-zero with a distinG L(n,Z p ) ◦ guished element ξ p ∈ V p . For each finite prime p ∈ S, let (π p , V p ) be a representation of G L(n, Q p ). $ % $ % & The restricted tensor product v≤∞ Vv of Vv v≤∞ with respect to ξv◦ v∈S / is the space of all finite linear combinations of vectors

Automorphic forms and representations for GL(n, AQ ) ' ξ = ξv

42

v≤∞

where ξv ∈ Vv for all v, and ξv = ξv◦ for all but finitely many v. Further, define actions  πg

: U (g) → End





'



πK∞ : K ∞ → G L

Vv ,

v≤∞





πfinite : G L(n, Afinite ) → G L

'



'

 Vv ,

v≤∞

Vv

v≤∞

as follows. For D ∈ U (g), and πg (D) .

'

&

v ξv

∈

&

v

Vv , we define

    ' ξv = πg (D) . ξ∞ ⊗ ξv .

v

For k ∈ K ∞ , and

&



v ξv

π K∞ (k) .

∈

v<∞

&

v

' v

Vv , we define

    ' ξv = π K∞ (k) . ξ∞ ⊗ ξv . v<∞

Finally, for afinite ∈ G L(n, Afinite ), and 

πfinite (afinite ) .

'

&

v ξv

 ξv = ξ∞ ⊗

v

∈

&

'

v

Vv , we define 

(π p (a p ) . ξ p ) .

p<∞

 Let π  = ((πg , π K ∞ ), πfinite ). Then (π  , V ) is a (g, K ∞ ) × G L(n, Afinite )-module &  & as in Definition 13.6.3. We denote it v≤∞ πv , v≤∞ Vv .

The next step is to give the generalization of Definition 10.8.2, which defines what it means for a (g, K ∞ ) × G L(n, Afinite )-module to be “ramified” or “unramified” at a prime p. Definition 13.7.2 (Ramified or unramified at p) Let (π, V ) be an irreducible, admissible (g, K ∞ ) × G L(n, Afinite )-module, as in Definitions 13.6.3, 13.6.4. For every finite prime p, let K p = i p (G L(n, Z p )). The representation π is termed unramified at p if there exists 0 =/ v ◦ ∈ V satisfying πfinite (k) . v ◦ = v ◦ ,

(∀k ∈ K p ),

and ramified at p if there does not. Alternatively, π is ramified at p if the K p fixed subspace V K p = {0}.

13.8 Newforms for GL(n)

43

Theorem 13.7.3 (Tensor product theorem for G L(n, AQ )) Fix an integer n ≥ 1 and let g := gl(n, C) and K ∞ := O(n, R). Let (π, V ) denote an irreducible admissible (g, K ∞ ) × G L(n, Afinite )-module, as in Definitions 13.6.3, 13.6.4. Let {q1 , . . . , qm } be the finite set of primes where π is ramified, as in Definition 13.7.2, and let S = {∞, q1 , . . . , qm }. Then there exists • an irreducible admissible (g, K ∞ )-module (π∞ , V∞ ) as in Definition 13.6.2, • an irreducible admissible representation (π p , V p ) of G L(n, Q p ) for each finite prime p, • a non-zero G L(n, Z p ) fixed vector v ◦p ∈ V p for each finite prime p ∈ S, such that (π, V ) is isomorphic to the restricted tensor product  & & πv , Vv , defined with respect to {vv◦ }v∈S , as in Definition 13.7.1, and v≤∞

v≤∞

the local representations (πv , Vv ) appearing in this decomposition are unique up to isomorphism. Proof One generalizes the proofs of Theorems 10.7.3, 10.8.5 and 10.8.12.



13.8 Newforms for G L(n) In this section we shall define the concept of a local new vector in an irreducible admissible representation of G L(n, Q p ). Similarly, we define the concept of a global new vector in an irreducible cuspidal automorphic representation of G L(n, AQ ), and sketch a proof that, in the case n = 2, each global new vector is the adelic lift of a newform. In order to define local newforms for G L(n, Q p ), it is necessary to introduce another important family of compact open subgroups of G L(n, Q p ). Definition 13.8.1 Fix a prime p and an integer N . Let K 0,1 (N ) p equal ⎧ ⎪ ⎨

Ap ⎪ ⎩ Cp

Bp dp



⎫ A p ∈ Mat(n, Z p ),

⎪ ⎬

B p ∈ Mat((n − 1) × 1, Z p )

. ∈ G L(n, Z p ) C p ∈ Mat(1 × (n − 1), N · Z p ), ⎪ ⎭ (d p − 1) ∈ N · Z p

In words, K 0,1 (N ) p may be described as consisting of matrices with bottom row congruent to (0, . . . , 0, 1) mod N · Z p . Clearly, this depends only on the power of p that divides N , and if N is prime to p then K 0,1 (N ) p = K 0,1 (1) p = G L(n, Z p ). A local new vector for a smooth representation (π, V ) of G L(n, Q p ) will be defined as a vector which is fixed by K 0,1 ( p m ) p for the smallest possible value of m. However, it is not obvious that, for an arbitrary smooth representation

44

Automorphic forms and representations for GL(n, AQ )

(π, V ) of G L(n, Q p ), there is any value of m such that the K 0,1 ( p m ) p -fixed subspace of (π, V ) is non-trivial. This motivates the next lemma. Lemma 13.8.2 (Existence of a K 0,1 ( p m ) p -fixed vector for some m) Fix a prime p and an integer n ≥ 1. Let (π, V ) be a smooth representation of G L(n, Q p ). Then there exists an integer m ≥ 0 such that

  m

V K 0,1 ( p ) p := v ∈ V π (k) . v = v, (∀ k ∈ K 0,1 ( p m ) p =/ {0}. (13.8.3) Sketch of Proof First assume n = 2 and (π, V ) is irreducible. By Theorem 6.14.2, there is no loss of generality in assuming that (π, V ) is realized as a Kirillov representation as in Definition 6.7.1 consisting of complex-valued functions. According to part (4) of Theorem 6.14.2, the characteristic function of Z×p is an element of V and it is clear from the explicit action given in Definition 6.7.1 that this element is fixed by 

 a b

× a ∈ Z . (13.8.4) , b ∈ Z p p 0 1 By smoothness, it is also fixed by % $  K ( p m ) := k ∈ G L(2, Z p ) | k − I2 ∈ Mat 2, p m · Z p

(13.8.5)

for some m. Since (13.8.5) and (13.8.4) generate K 0,1 ( p m ) p , this completes the proof. This argument generalizes easily to G L(n, Q p ) using theorem E of [Gelfand-Kajdan, 1971]. If (π, V ) is not irreducible, then it has an irreducible quotient (π  , V  ). Then m integration over K 0,1 ( p m ) p defines projection operators: V → V K 0,1 ( p ) p and m m also V  → (V  ) K 0,1 ( p ) p . One easily deduces non-triviality of V K 0,1 ( p ) p from m that of (V  ) K 0,1 ( p ) p .  Definition 13.8.6 (Local new vector) Fix a prime p. Let (π, V ) be an irreducible admissible representation of G L(n, Q p ). Let m ≥ 0 be the smallest integer such that the space of K 0,1 ( p m ) p -fixed vectors in V is non-trivial. A local new vector for the representation (π, V ) is a non-zero K 0,1 ( p m ) p -fixed vector in V. Remarks An equivalent formulation may be given as follows: let K 0 (N ) p equal ⎧ ⎫

A p ∈ Mat(n, Z p ), ⎨ ⎬

A p Bp ∈ G L(n, Z p )

B p ∈ Mat((n − 1) × 1, Z p ), . ⎩ C p dp ⎭ C p ∈ Mat(1 × (n − 1), N · Z p ) Then for (π, V ) a smooth representation of G L(n, Q p ), with central character ω, a vector v ∈ V is K 0,1 (N ) p -fixed if and only if it satisfies

 π

Ap Cp

13.8 Newforms for GL(n)   Bp A p Bp . v = ω(d p ) · v, ∈ K 0 (N ) p . ∀ dp C p dp

45

  The equivalence follows easily because K 0 (N ) p = K 0,1 (N ) p · Z ∩ K 0 (N ) p , where Z is the center of G L(n, Q p ). This formulation makes the analogy with an automorphic form having some character χ more explicit. Note that when (π, V ) is unramified, its non-zero G L(n, Z p )-fixed vectors are local new vectors. Theorem 13.8.7 (Local new vector is unique up to scalar) Fix a prime p and let (π, V ) be an irreducible admissible representation of G L(n, Q p ). Let m ≥ 0 be the smallest integer such that the space of K 0,1 ( p m ) p -fixed vectors in V is non-trivial. Then the space of K 0,1 ( p m ) p -fixed vectors in V is one-dimensional. Consequently, any two local new vectors for (π, V ), as in Definition 13.8.6, are proportional. Proof In the case n = 1, there is nothing to prove, since the whole space V is one-dimensional. In the case n = 2, this result was proved in [Casselman, 1973]. For n ≥ 3, it was proved in [Jacquet-Piatetski-ShapiroShalika, 1981].  Definition 13.8.8 (Global new vector) Fix an integer n ≥ 1. Let (π, V ) be an irreducible cuspidal automorphic representation of G L(n, AQ ), and fix an & isomorphism L : v≤∞ Vv → V of (π, V ) with an infinite restricted tensor  & & 13.7.3. product v≤∞ πv , v≤∞ Vv of local representations as in Theorem  & A global new vector is an element of V which is of the form L v≤∞ ξv , where ξ p is a local new vector for each finite prime p and is the distinguished vector ξ p◦ used to define the restricted tensor product for all but finitely many p, and ξ∞ generates an irreducible representation of S O(n, R) under the action of S O(n, R) by π K ∞ . Theorem 13.8.9 (The global new vector is an adelic lift of a classical newform when n = 2) Let χ be a Dirichlet character, χidelic its idelic lift (as in Definition 2.1.7) and φ : G L(2, AQ ) → C an adelic cusp form with central character χidelic . The following are equivalent: • φ is a global new vector, as in Definition 13.8.8, lying in some irreducible cuspidal automorphic representation, • there is a newform f, as in Section 3.11, such that φ = f adelic . Sketch of Proof For N =

r " i=1

K 0,1 (N ) :=

! p

K 0,1 (N ) p =

piei ∈ N, set r ! i=1

K 0,1 ( piei ) p ·

! pN

G L(2, Z p ) ⊂ G L(2, Afinite ).

Automorphic forms and representations for GL(n, AQ )

46

It is not difficult to check that for N as above and any k ∈ Z,

 

V ∩ f adelic f is a Maass form of weight k for 0 (N ) ⎧ ⎨

⎫

πfinite (k) . φ = φ, (∀k ∈ K 0,1 (N )), ⎬

  = φ ∈ V

, cos θ sin θ ⎩

πK∞ . φ = eikθ · φ, (∀ θ ∈ [0, 2π )) ⎭ − sin θ cos θ

which is just the span of the set  ) (  e

K (p i ) '

ξ pi ∈ V pi 0,1 i pi , (1 ≤ i ≤ r ), ξv , L

ξ p = ξ p◦ , ( p  N ), ξ∞ ∈ V∞,k v≤∞ & Vv → V and where L is an isomorphism

V∞,k := v ∈ V∞

v≤∞



cos θ

π

K∞ − sin θ

sin θ cos θ



 . v = eikθ · v, (∀θ ∈ [0, 2π )) .

The next step is to verify that for any Maass form f, and any prime p not dividing the level of f, 

Tp f

 adelic

1 (g) = √ p

 p 0

G L(2,Z p )

0 1

π (i p (x)) . f adelic (g) d x, G L(2,Z p )

(∀g ∈ G L(2, AQ )). Here T p denotes the classical Hecke operator as in Definition 3.11.2, and i p : G L(2, Q p ) → G L(2, AQ ) denotes inclusion at the place p. From this, it follows easily that if f adelic lies in an irreducible cuspidal automorphic representation   ' ' ∼ πv , Vv , (π, V ) = v≤∞

v≤∞

then f is an eigenfunction of T p with eigenvalue p for all p  N , where 1

 G L(2,Z p )



p 0 G L(2,Z p ) 0 1

 − 12

·ξ p 1

   G L(2,Z p )

p 0 G L(2,Z p ) 0 1

,

is the characteristic function of

  the double coset G L(2, Z p ) 0p 10 G L(2, Z p ), and ξ p is the spherical Hecke character, as in Definition 10.6.14, of the representation π p . Next, we invoke two results from Atkin-Lehner theory. These are given for holomorphic forms in [Atkin-Lehner, 1970], and the same arguments extend to the case of arbitrary Maass forms. The first result states that the

Exercises for Chapter 13

47

space of all Maass forms of weight k for 0 (N ) has a basis consisting of newforms and oldforms which are associated to newforms of strictly lower level (cf. Theorem 5 of [Atkin-Lehner, 1970]). The second states that if two newforms of the same weight and character have all but finitely many of their Hecke eigenvalues in common, then they are proportional (cf. Theorem 4 of [Atkin-Lehner, 1970]). Now, let f be a newform of level N and character χ . Let Aχidelic (G L(2, AQ )) denote the vector space of all adelic automorphic forms of central character χidelic . Consider the subspace V of Aχidelic (G L(2, AQ )) generated by f adelic . By an argument like that used to prove Lemma 6.1.6, this (g, K ∞ ) × G L(2, Afinite )-module has an irreducible quotient. Using the invariant Hermitian form on Aχidelic (G L(2, AQ )) given in Definition 9.5.3 one may deduce that this irreducible quotient is isomorphic to a subrepresentation (cf. Proposition 9.5.8). Project f adelic onto this subspace. Its image is still fixed by K 0,1 (N ) and equivariant by K ∞ , hence it is the adelic lift of some Maass form of weight k and level N . Also, this Maass has the same Hecke eigenvalues as f, at all but finitely many primes. It follows from the results in Atkin-Lehner theory quoted above that this Maass form is actually f, that V is irreducible, and that f adelic is a global new vector.  & & ∼ πv , Vv , is an irreducible cuspiConversely, suppose (π, V ) = v≤∞

v≤∞

dal automorphic representation of G L(2, AQ ) with central character χidelic and φ ∈ V is a global new vector. Then φ is fixed by K 0,1 (N ) for some N and satisfies  cos θ sin θ . φ = eikθ · φ, (∀θ ∈ [0, 2π )), πK∞ − sin θ cos θ for some integer k. (This is because the characters eikθ are the only irreducible continuous representations of S O(2, R).) It follows that φ is the adelic lift of a Maass form f of weight k, and character χ for 0 (N ). Further, this Maass form is an eigenfunction of all but finitely many Hecke operators, and hence is a linear combination of coming from a single newform, f new . One   oldforms may then show that φ = f new adelic . 

Exercises for Chapter 13 13.1 Let k be a field and n ≥ 2 an integer. Prove that every matrix in G L(n, k) has a Bruhat decomposition. That is, any g ∈ G L(n, k) can be written in the form g = b1 · w · b2 with b1 , b2 upper triangular matrices and w a Weyl element as in Definition 13.1.2. Hint: Multiplication on the left and right by upper triangular matrices corresponds to certain types of row and column operations, respectively.

48

Automorphic forms and representations for GL(n, AQ ) 13.2 Let g ∈ G L(2, Q p ) be a matrix that is not upper triangular. Prove   that any Bruhat decomposition of g is of the form g = b1 01 10 b2 with b1 , b2 upper-triangular. If we assume further that g ∈ G L(2, Z p ), does it follow that we can choose b1 , b2 ∈ G L(2, Z p )? 13.3 Let R be a commutative ring with identity and let n ≥ 2 be an integer. For 1 ≤ i, j ≤ n with i = j, write Ui, j (R) for the set of matrices of the form In + A, where A is an n × n matrix with all entries equal to zero except perhaps at the (i, j)-position. (a) Prove that Ui, j (R) is a subgroup of S L(n, R) isomorphic to the additive group of R. (b) Let k be a field. Prove that S L(n, k) is generated by the union of the subgroups Ui, j (k) with 1 ≤ i, j ≤ n and i = j. (c) Deduce that S L(n, AQ ) is generated by the union of the subgroups Ui, j (AQ ) with 1 ≤ i, j ≤ n and i = j. 13.4 Prove the uniqueness part of the factorization statement for G L(n, AQ ) given in Corollary 13.3.4. 13.5 Let R be a commutative ring with 1, let n be a positive integer, and let κ = (κ1 , . . . , κr ) be an ordered partition of n. Recall the definitions of the sets of n × n matrices Pκ (R), Mκ (R), and Uκ (R) given in Definitions 13.4.10–13.4.12. (a) Verify that Pκ (R), Mκ (R), and Uκ (R) are subgroups of G L(n, R). (b) Prove that Uκ (R) is a normal subgroup of Pκ (R). (c) Find an example of an ordered partition κ and a ring R such that Mκ (R) is not a normal subgroup of Pκ (R). (d*) Classify the ordered partitions κ and the rings R for which Mκ (R) is normal in Pκ (R).

13.6* Let R be a commutative ring with 1 and let n be a positive integer. If P is a parabolic subgroup of G L(n, R) as in Definition 13.4.13, the normalizer of P is the largest subgroup of G L(n, R) in which P is normal:

% $ N (P) = g ∈ G L(n, R) g Pg −1 = P . Evidently P ⊂ N (P), and for many rings R, one has N (P) = P for every parabolic subgroup P. (a) If R is a field, prove that N (P) = P for every parabolic subgroup P ⊂ G L(n, R). Deduce the same if R = AQ . (b) If R = Z p for some prime p, prove that N (P) = P for every parabolic subgroup P ⊂ G L(n, R).

Exercises for Chapter 13

49

(c) Let P = P(1,1) (R) ⊂ G L(2, R) be the standard maximal parabolic subgroup. Find an example of a ring R such that P is strictly smaller than its normalizer. 13.7 Let n ≥ 2 be an integer and κ = (κ1 , . . . , κr ) an ordered partition of n. " Let D = [0, 1)· p Z p be the standard fundamental domain for Q\AQ , and write Mat(r × s, D) for the set of r × s matrices with entries in D. Define

⎧⎛ ⎫ Iκ1 B1,2 . . . B1,r ⎞ ⎪ ⎪

⎪ ⎪ ⎨⎜ 0 Iκ2 . . . B2,r ⎟ Bi, j ∈ Mat(κi × κ j , D), ⎬

⎟ . Uκ (D) = ⎜ .. ⎠ .. ⎝ (1 ≤ i < j ≤ r ) ⎪ ⎪ . 0 0 .

⎪ ⎪ ⎩ ⎭

0 0 0 Iκr Prove that Uκ (D) is a fundamental domain for Uκ (Q) \ Uκ (AQ ). Conclude that the quotient space Uκ (Q) \ Uκ (AQ ) is compact. Hint: Try the case n = 3 and κ = (1, 1, 1) first. 13.8 Let n be a positive integer, and suppose κ and κ  are ordered partitions of n. Write κ = (κ1 , . . . , κr ). We say κ  refines κ if κ  = (κ1 , . . . , κs1 , κs1 +1 , . . . , κs2 , . . . , κsr −1 +1 , . . . , κsr ), where κs j−1 +1 + · · · + κs j = κ j for each j = 1, . . . , r . (For notational convenience, we set s0 = 0.) Now let R be a commutative ring with 1. If κ  refines κ, show that Pκ  (R) ⊂ Pκ (R), Mκ  (R) ⊂ Mκ (R), and Uκ (R) ⊂ Uκ  (R). (Note the difference in the last inclusion.) 13.9 Let n ≥ 2 be an integer, let κ  = (κ1 , . . . , κr ) be an ordered partition of n, and let κ = (m, n − m) be another ordered partition of n such that κ  refines κ in the sense of the previous exercise. Suppose that ϕ : G L(n, AQ ) → C is a smooth function such that ϕ(ug) du = 0,



 g ∈ G L(n, AQ ) .

Uκ (Q)\Uκ (AQ )

Prove that ϕ(ug) du = 0,



g ∈ G L(n, AQ )



Uκ  (Q)\Uκ  (AQ )

by unfolding this integral in such a way that an integral of the first type appears as an integrand. Conclude that if ϕ is an automorphic form for

50

Automorphic forms and representations for GL(n, AQ ) G L(n, AQ ), then in order to check that ϕ is cuspidal, it suffices to verify that the constant term of ϕ along P vanishes for all maximal parabolic subgroups P. (See Remark (3) following Definition 13.4.15.)

13.10 Let R be a commutative ring with 1, let n ≥ 2 be an integer, and let κ = (κ1 , . . . , κr ) be an ordered partition of n. Show that Pκ (R) = Uκ (R)  Mκ (R). Deduce the following generalized Iwasawa decomposition: for each g ∈ G L(n, AQ ), there exists u ∈ Uκ (AQ ), " m ∈ Mκ (AQ ), and k ∈ K = O(n, R) · p G L(n, Z p ) such that g = umk. 13.11 Let V be a space of automorphic forms for G L(n, AQ ) that is invariant under the actions of K ∞ and G L(n, Afinite ). Show that V is contained in the space of cusp forms if and only if P (m) = 0 for all ∈ V , all parabolic subgroups P, and all m ∈ M(AQ ), where M is a Levi factor of the parabolic subgroup P. Hint: Use the Iwasawa decomposition in the previous exercise to reduce to evaluating P on elements of a Levi factor. 13.12 Let n ≥ 2 be an integer and let ϕ be an adelic cusp form for G L(n, AQ ) with central character ω as in Definitions 13.4.7 and 13.4.15. Fix a character ψ : Un (AQ ) → C× of the form ψ = ψα for some α = (α1 , . . . , αn−1 ) ∈ (Q× )n−1 as in Definition 13.5.2. Let ϕ(ug)ψ −1 (u) du

Wϕ (g) = Un (Q)\Un (AQ )

be the Whittaker coefficient for the cusp form ϕ. Prove the following statements:   (a) Wϕ (zg) = ω(z) Wϕ (g), g ∈ G L(n, AQ ), z ∈ Z (G L(n, AQ )) . " (b) Wϕ is right K -finite for K = O(n, R) · p G L(n, Z p ). (c) Wϕ is smooth. (d) Wϕ is right Z (U (g))-finite. (e) Wϕ is of moderate growth.   u ∈ Un (AQ ), g ∈ G L(n, AQ ) . (f) Wϕ (ug) = ψ(u)Wϕ (g) 13.13* Let k be a field. A set of matrices in G L(n, k) is called Zariski closed if it is defined by a collection of polynomial equations in the entries. It is a proper Zariski closed set if it is not equal to G L(n, k). For example, the Borel subgroup Bn (k) of upper triangular invertible matrices is a proper Zariski closed subset of G L(n, k) when n ≥ 2. A subset of G L(n, k) is called Zariski open if it is the complement of a Zariski closed set.

Exercises for Chapter 13

51

(a) Let w ∈ G L(n, k) be a Weyl element and write ⎛ w = ⎝

..

.

1

⎞ ⎠.

1 Prove that the double coset Bn (k) · w · Bn (k) is contained in a proper Zariski closed subset of G L(n, k) if and only if w = w . Conversely, show that Bn (k) · w  · Bn (k) is Zariski open. (b) Let v ≤ ∞ be a prime of Q. Show that Bn (Qv ) · w  · Bn (Qv ) is open and dense in (the topology of) G L(n, Qv ). 13.14 Prove that the Fourier expansion of an adelic cusp form, given in Theorem 13.5.4, converges absolutely and uniformly on compact subsets of G L(n, AQ ).

14 Theory of local representations for G L(n)

14.1 Generalities on representations of G L(n, Q p ) Fix an integer n ≥ 1 and a prime p. A representation of G L(n, Q p ) is a pair (π, V ) where V is a complex vector space and π : G L(n, Q p ) → G L(V ) is a homomorphism. Such a representation (π, V ) is smooth if every vector has an open stabilizer (see Definition 6.1.1). For every integer r ≥ 1, define the open subgroup K r := {k ∈ G L(n, Z p ) | k − In ∈ pr · Mat(n, Z p )},

(14.1.1)

where In is the n × n identity matrix and Mat(n, Z p ) is the set of all n × n matrices with coefficients in Z p . Definition 14.1.2 (Admissible representation of G L(n, Q p )) Fix an integer n ≥ 1 and a prime p. Let V be a complex vector space and let (π, V ) be a smooth representation of G L(n, Q p ). We say (π, V ) is admissible if the space

% $ V Kr := v ∈ V π (k) . v = v, for all k ∈ K r , is finite dimensional for each integer r ≥ 0, where K r is given in (14.1.1). Definition 14.1.3 (Finitely generated representation) Fix an integer n ≥ 1 and a prime p. A smooth representation (π, V ) of G L(n, Q p ) is said to be finitely generated if there exists a finite set S ⊂ V such that the only subspace W ⊂ V satisfying S ⊂ W and π (g) . w ∈ W (∀ g ∈ G L(n, Q p ), w ∈ W ) is V itself. The reader should check that if n = 2, this is equivalent to Definition 6.1.15. Definition 14.1.4 (Representation of finite length) A representation (π, V ) of G L(n, Q p ) is said to have finite length if there exists a sequence of subspaces {0} = V0 ⊂ V1 ⊂ · · · Vm = V 52

14.1 Generalities on representations of GL(n, Qp )

53

such that for each i = 1, . . . m, • the subspace Vi is invariant, i.e. π (g) . v ∈ Vi for all v ∈ Vi and for all g ∈ G L(n, Q p ); • the representation of G L(n, Q p ) on the quotient space Vi /Vi−1 is an irreducible representation. Remarks The Definitions 14.1.3, 14.1.4 hold for representations of any group. We previously showed in Theorem 6.1.11 that a smooth and irreducible representation of G L(2, Q p ) is admissible. The proof which was given in chapter 6 does not generalize. However, the result has been generalized to G L(n, Q p ) for any n ≥ 2, and one may show that a smooth representation (π, V ) of G L(n, Q p ) has finite length if and only if it is admissible and is finitely generated. This deep result is Theorem 4.1 in [Bernstein-Zelevinsky, 1976]. See also [Harish-Chandra, 1970], [Jacquet, 1975], [Howe, 1974, 1977] for earlier work on this subject. Definition 14.1.5 (Quotient, subquotient, and subrepresentation of a representation of G L(n, Q p )) Fix an integer n ≥ 1 and a prime p. Let (π, V ) be a smooth representation of G = G L(n, Q p ). We also refer to V as a G-module. A subrepresentation or G-submodule of (π, V ) is a subspace V  ⊂ V such that π (g) . v  ∈ V  for all g ∈ G and v  ∈ V  . For such a subspace V  , one may define an action π  : G → G L(V /V  ), where V /V  = {v + V  | v ∈ V } is the usual quotient space, by the formula π  (g) . (v + V  ) := (π (g) . v) + V  , (∀g ∈ G L(n, Q p ), v ∈ V ).     The representation π , V /V is called a quotient of (π, V ). If V  ⊂ V  ⊂ V are two subrepresentations of (π, V ), then the representation π  : G L(n, Q p ) → G L(V  /V  ) with action defined by π  (g) . (v  + V  ) := (π (g) . v  ) + V  ,

(∀g ∈ G L(n, Q p ), v  ∈ V  ),

is called a subquotient of (π, V ). Proposition 14.1.6 (Finitely generated smooth representations have an irreducible quotient) Fix an integer n ≥ 1 and a prime p. Let (π, V ) be a smooth representation of G = G L(n, Q p ). We also refer to V as a G-module. If V is finitely generated then there exists a proper G-submodule V  ⊂ V such that the quotient representation V /V  , defined as in Definition 14.1.5, is irreducible and smooth. Proof The proof is the same as the proof of Proposition 6.1.16.



Definition 14.1.7 (Ramified/unramified representation of G L(n, Q p )) Fix an integer n ≥ 1 and a prime p. A representation (π, V ) of G L(n, Q p ) is

54

Theory of local representations for GL(n)

termed unramified if there exists a non-zero G L(n, Z p ) fixed vector v ◦ ∈ V . Otherwise it is said to be ramified. Dixmier’s Lemma 6.1.8 generalizes directly to G L(n, Q p ) with the same proof. A consequence of this lemma is the existence of a central character for any irreducible smooth representation of G L(n, Q p ). Proposition 14.1.8 (Central character) Fix an integer n ≥ 1 and a prime p. Let V be a complex vector space and let (π, V ) be an irreducible smooth representation of G L(n, Q p ). Then there exists a unique multiplicative character ωπ : Q×p → C× such that   π (a · In ) . v = ωπ (a) · v, ∀a ∈ Q×p , v ∈ V , where In is the n × n identity matrix. The character ωπ is called the central character associated to the representation (π, V ). Proof The proof is the same as in the case n = 2, given in Proposition 6.1.10.  Proposition 14.1.9 (Characterization of the finite dimensional irreducible representations of G L(n, Q p )) Fix an integer n ≥ 1 and a prime p. Let V be a finite dimensional non-zero complex vector space. Let (π, V ) be a smooth irreducible representation of G L(n, Q p ). Then V ∼ = C and there exists a × multiplicative character ω : Q p → C such that π (g) . v = ω (det g) · v for all g ∈ G L(n, Q p ) and for all v ∈ V. Proof The proof is similar to the proof of Theorem 6.1.7 and is left to Exercise 14.2.  Next we generalize the definition of contragredient representation, given for G L(2, Q p ) in Definition 8.1.4, to G L(n, Q p ). Definition 14.1.10 (Contragredient) Fix an integer n ≥ 1 and a prime p. Let (π, V ) be a smooth representation of G L(n, Q p ). The contragredient representation is defined to be ( π,  V ), where  V is the complex vector space of all linear functionals  : V → C which satisfy (π (k) . v) = (v),

(∀k ∈ K  , v ∈ V ),

π is the action of for some compact open subgroup K  ⊂ G L(n, Q p ), and  G L(n, Q p ) on this space defined by  π (g) . (v) := (π (g −1 ) . v),

(∀ ∈  V , v ∈ V, g ∈ G L(n, Q p )).

14.1 Generalities on representations of GL(n, Qp )

55

As for G L(2, Q p ), the canonical bilinear form V ×  V → C is defined by (∀v ∈ V,  ∈  V ).

v,  := (v),

(14.1.11)

It is invariant, in the sense that π (g) . v,  π (g) .  = v, ,

  ∀v ∈ V,  ∈  V , g ∈ G L(n, Q p ) ,

and, if (π, V ) is irreducible then any other representation (π  , V  ) having a π,  V ). non-zero invariant bilinear form V × V  → C is isomorphic to ( Definition 14.1.12 (Matrix coefficients for G L(n, Q p )) Fix an integer n ≥ 1 π,  V) and a prime p. Let (π, V ) be a smooth representation of G L(n, Q p ). Let ( be the contragredient of (π, V ) as in Definition 14.1.10. Fix v ∈ V and  v∈ V.  Let  ,  : V × V → C be the canonical bilinear form (14.1.11). The function v, βv,˜v (g) := π (g) . v, 

(g ∈ G L(n, Q p )),

is called a matrix coefficient of π. Proposition 14.1.13 (Constructing a model for the contragredient from a pairing) Fix an integer n ≥ 1 and a prime p. Let (π1 , V1 ) and (π2 , V2 ) be two admissible representations of G L(n, Q p ). Suppose that there is a nondegenerate bilinear form  ,  : V1 × V2 → C which is invariant, in the sense that it satisfies π1 (g) . v1 , π2 (g) . v2  = v1 , v2 ,

  ∀ v1 ∈ V1 , v2 ∈ V2 , g ∈ G L(n, Q p ) .

Here nondegenerate means that for any fixed v2 ∈ V2 we have v1 , v2  =/ 0 for some v1 ∈ V1 , and vice versa. Then (π2 , V2 ) is isomorphic to the contragredient representation of (π1 , V1 ). V1 Proof As in the proof of Proposition 8.1.10, we define a map L:V2 →  by L(v2 ) = v2 , (v2 ∈ V2 ), where v2 : V1 → C is given by v2 (v1 ) = v1 , v2 , (∀v1 ∈ V1 ). Then L is an intertwining map. If v2 =/ 0 is in the kernel of L , then v1 , v2  = 0 (∀v1 ∈ V1 ) and  ,  is degenerate. Likewise if the image of L is a proper subspace of  V1 , then there exists v1 ∈ V1 such that v1 , v2  = 0 (∀v2 ∈ V2 ), and  ,  is again degenerate. (Note that the existence of such a vector v1 actually depends on the assumption that V1 and V2 are admissible. See Exercise 14.3.) It follows that the intertwining map L is an isomorphism. 

56

Theory of local representations for GL(n)

14.2 Generic representations of G L(n, Q p ) It is good for the reader to have a concrete example of a representation of G L(n, Q p ) in mind, especially when n > 2, which is a case that we have not considered before. Following the spirit of Section 6.3, we shall now construct an example of a representation of G L(n, Q p ) coming from an adelic automorphic form. Example 14.2.1 (Local representation of G L(n, Q p ) coming from an adelic automorphic form) Fix an integer n ≥ 1 and a prime p. Let ϕ : G L(n, AQ ) → C be an adelic automorphic form as in Definition 13.4.7. We can regard ϕ as a function with domain G L(n, Q p ) by restriction of variables, i.e., consider ϕ(g) where g = {In , In ,

... ,

In ,

g p , In , *+,-

. . . },

(g p ∈ G L(n, Q p )).

pth position

Here In is the n × n identity matrix. By abuse of notation we will write this as ϕ(g p ). Define the complex vector space Vϕ to be the space of all functions: ( Vϕ :=

gp →

m  i=1

)

m ∈ N, ci ∈ C,

ci ϕ(g p h i ) ,

h i ∈ G L(n, Q p ), (i = 1, . . . , m)

with g p ∈ G L(n, Q p ). Let π : G L(n, Q p ) → G L(Vϕ ) be defined by right translation, i.e., π (h p ) . f (g p ) := f (g p h p ),

(∀h p , g p ∈ G L(n, Q p ), f ∈ Vϕ ).

Then (π, Vϕ ) is an example of a representation of G L(n, Q p ). Example 14.2.1 is archetypical. It is the motivating example and was introduced to obtain deep information about automorphic forms, not only adelic automorphic forms, but the much older classical forms studied in Chapter 12. Example 14.2.2 (Whittaker model of a local representation of G L(n, Q p ) coming from an adelic automorphic form) Fix an integer n ≥ 1 and a prime p. Let e : Q\AQ → C denote the additive character defined in Definition 1.7.1, and consider a character ψ of the group Un (AQ ) of the form ⎛⎛ 1 u 1,2 1 ⎜⎜ ⎜⎜ ⎜ ψ⎜ ⎜⎜ ⎝⎝

... u 2,3 .. .

u 1,n ..

. 1

.. . u n−1,n 1

⎞⎞ ⎟⎟ ⎟⎟   ⎟⎟ = e a1 u 1,2 + · · · + an−1 u n−1,n , ⎟⎟ ⎠⎠

14.2 Generic representations of GL(n, Qp )

57

for u i, j ∈ AQ , (1 ≤ i < j ≤ n), with ai ∈ Q× , (1 ≤ i < n). Let ϕ : G L(n, AQ ) → C be an adelic automorphic form as in Definition 13.4.7. Then by Theorem 13.5.4, the function ϕ has a Fourier expansion given by   γ g , (∀g ∈ G L(n, AQ )), Wϕ ϕ(g) = 1 γ ∈ Un−1 (Q)\G L(n−1,Q)

where ϕ(ug)ψ −1 (u) du,

Wϕ (g) = Un (Q)\Un (AQ )

for all g ∈ G L(n, AQ ). The function Wϕ is called a Whittaker function relative to ψ. Abstractly, a Whittaker function relative to ψ is any function W which is smooth, of moderate growth, and satisfies W (ug) = ψ(u)W (g),

(14.2.3)

for all u ∈ Un (AQ ), g ∈ G L(n, AQ ). Clearly, the function Wϕ has these properties. Let Wϕ denote the complex vector space of all functions

( ) m

 m ∈ N, ci ∈ C,

Wϕ := g p → ci Wϕ (g p h i ) ,

h i ∈ G L(n, Q p ), (i = 1, . . . , m) i=1

with g p ∈ G L(n, Q p ). Let π : G L(n, Q p ) → G L(Wϕ ) be defined by right translation, i.e., π (h p ) . w(g p ) := w(g p h p ),

(∀h p , g p ∈ G L(n, Q p ), w ∈ Wϕ ). (14.2.4)

Following Proposition 10.4.5, one may show that the representation (π, Vϕ ) defined in Example 14.2.1 is isomorphic to the Whittaker model (π, Wϕ ). Definition 14.2.5 (Whittaker model of a representation of G L(n, Q p )) Fix an integer n ≥ 1 and a prime p. Let e p : Q p → C be the additive character as in Definition 1.6.3. Fix a character ψ of Un (Q p ) of the form ⎛⎛ 1 u 1,2 1 ⎜⎜ ⎜⎜ ⎜ ψ⎜ ⎜⎜ ⎝⎝

... u 2,3 .. .

u 1,n ..

. 1

.. . u n−1,n 1

⎞⎞ ⎟⎟ ⎟⎟   ⎟⎟ = e p a1 u 1,2 + · · · + an−1 u n−1,n , ⎟⎟ ⎠⎠

(14.2.6) for u i, j ∈ Q p , (1 ≤ i < j ≤ n), with ai ∈ Q×p , (1 ≤ i < n). Let (π, V ) be a complex representation of G L(n, Q p ). A Whittaker model for (π, V ) relative to ψ is a second representation (π  , W) ∼ = (π, V ) where W is a space

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Theory of local representations for GL(n)

of Whittaker functions relative to ψ, i.e., of locally constant functions satisfying (14.2.3), for all u ∈ Un (Q p ), g ∈ G L(n, Q p ), and π  is given by right translation as in (14.2.4). Note that if there exists a character ψ as in (14.2.6) such that (π, V ) has a Whittaker model relative to ψ, then (π, V ) has a Whittaker model relative to ψ for every character of the form (14.2.6). (See Exercise 14.4.) Note also that not every representation of G L(n, Q p ) has a Whittaker model. The simplest example is given by one-dimensional representations which factor through the determinant: these do not have Whittaker models. Nevertheless, it is clear that all representations coming from adelic cusp forms as in Example 14.2.2 do have Whittaker models. This motivates the following definition. Definition 14.2.7 (Generic representation of G L(n, Q p )) Fix an integer n ≥ 1, a prime p, and a character ψ as in (14.2.6). A representation (π, V ) of G L(n, Q p ) is said to be generic relative to ψ if it is non-zero, and has a Whittaker model relative to ψ, as in Definition 14.2.5. It is generally permissible to speak of a representation of G L(n, Q p ) as being simply “generic” or “not generic,” without specifying the character ψ to which this genericity is relative. This is because a given representation will either be generic relative to all characters ψ of the form (14.2.6), or else it will not be generic relative to any of them. Proposition 14.2.8 (Contragredient of a generic representation is generic) Fix an integer n ≥ 1, a prime p, and a character ψ as in (14.2.6). A representation (π, V ) of G L(n, Q p ) is generic relative to ψ if and only if its contragredient ( π,  V ), as in Definition 14.1.10, is generic relative to ψ. Proof We sketch a simple proof which applies only to the case when (π, V ) is unitary, i.e., when there is a positive definite Hermitian form ( , ) : V × V → C such that (π (g) . v, v  ) = (v, π (g −1 ) . v  ) for all v, v  ∈ V and all g ∈ G L(n, Q p ). As in Proposition 8.10.1, (π, V ) may be realized by right translation on a space of its own matrix coefficients. When (π, V ) happens to be unitary, these matrix coefficients may be expressed using the invariant Hermitian form ( , ) : V × V → C, as above. Fix v0 ∈ V, and for each v ∈ V, let βv,v0 (g) = (π (g) . v, v0 ). Then v → βv,v0 is an intertwining map from (π, V ) to a model for (π, V ) on a space of matrix coefficients, with action by right translation. Next, for each v ∈ V, the functional v  → v (v  ) := (v  , v) is an element of  V , and  π (g) . v = π(g) . v . Hence, the functions g → (v  , π (g) . v) are actually matrix coefficients of the contragredient ( π,  V ), and, in fact the map

14.2 Generic representations of GL(n, Qp )

59

π,  V ) to a model for ( π,  V ) on a v → βv,v0 is an intertwining map from ( space of matrix coefficients, with action by right translation. From the previous two paragraphs, it follows that we may assume that V is a space of functions,  V is the space containing the complex conjugates of the functions in V, and π and  π are both right translation. Now let W be a Whittaker model for (π, V ) relative to ψ. Let L : V → W be an intertwining map. Let W be the space containing the complex conjugates of the functions in W. Since W is closed under right translation, it follows that the same is true of W. Define L : V → W by L( f ) = L( f ). It is clear that L is a C-linear isomorphism  V to W. Furthermore, it is an intertwining map for the action by right translation on these two spaces. π,  V ) relative to ψ. Consequently W is a Whittaker model for ( The general case follows from theorem A of [Gelfand-Kazhdan, 1975], which states that  π is isomorphic to the representation . π defined by . π (g) := π (t g −1 ). Indeed, assume that (π, V ) is realized on its Whittaker model relative to ψ,   i.e., that V is a space of Whittaker functions relative to ψ and for all g, h ∈ G L(n, Q p ) and W ∈ V. Then π (g) . W (h) = W (hg) . π (g) . W (h) = W (h · t g −1 ) for all g, h ∈ G L(n, Q p ) and W ∈ V. This realization of the contragredient is not a Whittaker model, even though the space V consists of Whittaker functions, because the action is not by right translation. . (h) := W (J t h −1 ), where However, for each W ∈ V we can define W ⎛ ⎞ −1 1 ⎜ ⎟ ⎜ ⎟ J =⎜ −1 ⎟. ⎝ ⎠ ... n (−1)   . is also a Whittaker function relative to ψ. Let V  = W .

W ∈ V . Then W . is clearly a C-linear isomorphism V → V  . Further The function W → W       . (hg) . π (g) . W (h) = . π (g) . W J t h −1 = W J t h −1 t g −1 = W    . (h), = π (g) . W where π  denotes the action of G L(n, Q p ) on V  by right translation.



Theorem 14.2.9 (Multiplicity one for G L(n, Q p )) Fix an integer n ≥ 1, a prime p, a character ψ as in (14.2.6), and an irreducible admissible representation (π, V ) of G L(n, Q p ). If (π, V ) is a generic representation of G L(n, Q p ) relative to ψ, as in Definition 14.2.7, then its Whittaker model relative to ψ is unique. Equivalently, if W1 and W2 are two spaces of Whittaker

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Theory of local representations for GL(n)

functions relative to the same character ψ, and if (π, W1 ) ∼ = (π, W2 ), where π denotes right translation in both cases, then W1 = W2 . Proof See [Gelfand-Kazhdan, 1975].



14.3 Parabolic induction for G L(n, Q p ) A goal of this chapter is to classify the admissible irreducible representations of G L(n, Q p ) for arbitrary n ≥ 2. This has already been done for G L(2, Q p ) in Theorem 6.13.4 by using parabolic induction for G L(2, Q p ) as discussed in 6.12. We will now generalize parabolic induction to G L(n, Q p ) for all n ≥ 1. Recall Definition 13.4.9, that an ordered partition of an integer n ≥ 1 is a finite sequence κ = (κ1 , κ2 , . . . , κr ) of positive integers such that n = κ1 + · · · + κr . In Definition 13.4.10, we defined the standard parabolic subgroup of G L(n, Q p ), associated to a partition κ = (κ1 , κ2 , . . . , κr ), to be

⎧⎛ ⎫ A1 B1,2 . . . B1,r ⎞ ⎪ ⎪

⎪ ⎪ ⎨⎜ 0 A2 . . . B2,r ⎟ Ai ∈ G L(κi , Q p ), (1 ≤ i ≤ r ), ⎬

⎜ ⎟ . Bi, j ∈ Mat(κi × κ j , Q p ), Pκ (Q p ) := ⎝ .. ⎠ .. ⎪ 0 ⎪ . 0 .

⎪ ⎪ (1 ≤ i < j ≤ r ) ⎩ ⎭

0 0 0 Ar (14.3.1) where Mat(κi × κ j , Q p ) denotes the set of all κi × κ j matrices with coefficients in Q p . The standard Levi subgroup, denoted Mκ (Q p ), which is associated to a standard parabolic subgroup Pκ (Q p ) was defined in Definition 13.4.11 and is ⎫ ⎧⎛ ⎞ A1 0 . . . 0

⎪ ⎪ ⎪ ⎪ ⎬ ⎨ ⎜ 0 A2 . . . 0 ⎟

Ai ∈ G L(κi , Q p ), (1 ≤ i ≤ r )

⎜ ⎟ . Mκ (Q p ) := ⎝ . .. .. ⎠

⎪ ⎪ 0 . 0 ⎪ ⎪ ⎭ ⎩

0 0 0 Ar (14.3.2) Similarly, the unipotent radical, denoted Uκ (Q p ), defined in Definition 13.4.12 is given by

⎧⎛ ⎫ Iκ1 B1,2 . . . B1,r ⎞ ⎪ ⎪

⎪ ⎪ ⎨⎜ 0 Iκ2 . . . B2,r ⎟ Bi, j ∈ Mat(κi × κ j , Q p ), ⎬

⎜ ⎟ , Uκ (Q p ) := ⎝ .. ⎠ .. (1 ≤ i < j ≤ r ) ⎪ ⎪ . 0 .

⎪ ⎪ ⎩ 0 ⎭

0 0 0 Iκr (14.3.3) where Iκi denotes the κi × κi identity matrix, for 1 ≤ i ≤ r. Lemma 14.3.4 (The natural projection: pr : Pκ → Mκ ) Let R be a ring, and let n ≥ 2 be a positive integer. Let κ = (κ1 , κ2 , . . . , κr ) with n = κ1 + · · · + κr

14.3 Parabolic induction for GL(n, Qp )

61

be an ordered partition of n. Let Pκ (R) be the standard parabolic of G L(n, R) introduced in Definition 13.4.10, and Mκ (R) the standard Levi subgroup as in Definition 13.4.11. The function pr : Pκ (R) → Mκ (R) given by ⎛⎛ A B . . . B1,r ⎞⎞ ⎛ A1 0 . . . 0 ⎞ 1 1,2 A2 . . . B2,r ⎟⎟ ⎜ 0 A2 . . . 0 ⎟ ⎜⎜ 0 ⎟ ⎜ ⎜ pr ⎜ .. ⎟ .. ⎟ .. .. ⎠⎠ = ⎝ 0 ⎝⎝ . . 0 . ⎠ 0 0 . 0 0 0 Ar 0 0 0 Ar is a homomorphism, and its kernel is the unipotent radical Uκ (R) of Pκ (R), which was introduced in Definition 13.4.12. The function pr is called the natural projection of Pκ onto Mκ . Proof Follows from a straightforward matrix multiplication.



Clearly, Mκ (Q p ) ∼ = G L(κ1 , Q p ) × . . . , ×G L(κr , Q p ). For each positive integer n, each ordered partition κ = (κ1 , . . . , κr ) of n, and any g1 ∈ G L(κ1 , Q p ), g2 ∈ G L(κ2 , Q p ), . . . , gr ∈ G L(κr , Q p ), we shall identify ⎞ ⎛ g1 . . . 0 ⎟ ⎜ ⎝ 0 . . . 0 ⎠ ∈ Mκ (Q p ) 0

...

gr

with (g1 , . . . , gr ) ∈ G L(κ1 , Q p ) × · · · × G L(κr , Q p ). This identifies representations of G L(κ1 , Q p ) × · · · × G L(κr , Q p ) with representations of Mκ (Q p ). So it will be necessary to consider admissible representations of products of general linear groups. Proposition 14.3.5 (Admissible representations of products) Let r ≥ 1 and fix positive integers κ1 , . . . , κr . Let (π, V ) be an admissible representation of G L(κ1 , Q p ) × · · · × G L(κr , Q p ). Then there exist admissible representations (πi , Vi ) of G L(κi , Q p ) (for i = 1, . . . , r ) such that (π, V ) is isomorphic to r & (πi , Vi ). i=1

Proof Since there are only finitely many factors, this easily follows from the decomposition Theorem 10.5.1.  Definition 14.3.6 (Modular quasicharacter) Let n be an integer and κ an ordered partition of n. The modular quasicharacter of the parabolic subgroup Pκ (Q p ) as in (14.3.1) is defined as ⎛⎛ A B r i−1 / / . . . B ⎞⎞ 1

⎜⎜ 0 ⎜ δ Pκ ⎜ ⎝⎝ 0 0

1,2

A2 0 0

... .. . 0

1,r

r ! B2,r ⎟⎟

det(Ai ) pj=i+1 ⎟ ⎟ = .. ⎠⎠ . i=1

Ar

κj−

κj

j=1

,

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Theory of local representations for GL(n)

for all Ai ∈ G L(κi , Q p ), (1 ≤ i ≤ r ) and for all Bi, j ∈ Mat(κi × κ j , Q p ), with 1 ≤ i < j ≤ r. Next, we define parabolic induction for G L(n, Q p ) for n ≥ 1. This makes use of the notion of “induced representation,” which has already been introduced in Definition 6.12.2. Before doing this, however, there is a minor technical point which, for reasons of clarity, we prefer not to gloss over. Definition 14.3.7 (The trivial extension π TE of a representation π of the Levi subgroup Mκ (Q p )) Fix an integer n ≥ 1, a prime p, and an ordered partition κ = (κ1 , . . . , κr ) of n. Let Mκ (Q p ) ⊂ G L(n, Q p ) denote the standard Levi subgroup as in (14.3.2). Let (π, V ) be a representation of Mκ (Q p ). Let Pκ (Q p ) ⊇ Mκ (Q p ) denote the standard parabolic subgroup of G L(n, Q p ) as in (14.3.1), and let pr : Pκ (Q p ) → Mκ (Q p ) be the natural projection defined in Lemma 14.3.4. Then the formula π TE (ρ) . v := π (pr(ρ)) . v,

(∀ ρ ∈ Pκ (Q p ), v ∈ V ),

defines a homomorphism π TE : Pκ (Q p ) → G L(V ) which defines a representation π TE called the trivial extension of the representation π. Remark The trivial extension (π TE , V ) is smooth if (π, V ) is smooth and it is admissible if (π, V ) is admissible. Definition 14.3.8 (Normalized parabolic induction for G L(n, Q p )) Fix an integer n ≥ 1, a prime p. Let r, κi (i = 1, . . . , r ) be positive integers satisfying κ1 + · · · + κr = n. Set κ = (κ1 , . . . , κr ). For i = 1, . . . r, let (πi , Vi ) be a representation of G L(κi , Q p ). Let (π, V ) be the tensor product representation π :=

r ' i=1

πi ,

V :=

r '

Vi ,

i=1

as in Definition 10.2.1. We regard (π, V ) as a representation of Mκ (Q p ). Define V G L(n,Q p ) to be the space of functions ⎧

⎫

f is locally constant, ⎪ ⎪

⎪ ⎪ 1 ⎨

f (umg) = δ 2 (m) · π (m) . f (g), ⎬

P κ f : G L(n, Q p ) → V , ⎪ ⎪ for all u ∈ Uκ (Q p ),

⎪ ⎪ ⎩

m ∈ M (Q ), g ∈ G L(n, Q ) ⎭ κ p p where δ Pκ is the modular quasicharacter which is defined in Definition 14.3.6. Further, define a homomorphism   π G L(n,Q p ) : G L(n, Q p ) → G L V G L(n,Q p )

14.3 Parabolic induction for GL(n, Qp ) where   π G L(n,Q p ) (g) . f (h) := f (h · g),

63

  ∀g, h ∈ G L(n, Q p ), f ∈ V G L(n,Q p ) .

Then (π G L(n,Q p ) , V G L(n,Q p ) ) is called the representation parabolically induced G L(n,Q ) from (π, V ). We also adopt the notation: π G L(n, Q p ) := Ind Pκ (Q p ) p (π ). Remarks The notation V G L(n,Q p ) for the induced space is essentially the same  as the notation V K for the space of K  -fixed vectors, where K  is a compact open subgroup. Unfortunately, both usages are fairly standard. Since we do not consider G L(n, Q p )-fixed vectors or induced representations of compact open subgroups, the meaning should always be clear from context. In Definition 6.12.2, we gave a formal definition of an induced representation, for any closed subgroup H of G L(2, Q p ). The definition extends naturally to G L(n, Q p ). According to this definition, the representation G L(n,Q ) Ind Pκ (Q p ) p (π ) is an induced representation, but it is induced from the representation (π TE , V ) of Pκ (Q p ), not the representation (π, V ) of Mκ (Q p ). “Parabolic induction” refers to this two-step process of first extending trivially from the Levi to the parabolic, and then inducing. For the student interested in working in representation theory or automorphic forms, it is important to understand the distinction between normalized and non-normalized parabolic induction. The word “normalized” indicates that 1

the factor δ P2 κ is included in the definition of the induced representation. The non-normalized induced representation is defined in the same manner, with this factor omitted. For the remainder of this book, parabolic induction is assumed to be normalized. However, non-normalized induction is also common in the literature. Lemma 14.3.9 (Parabolic induction for G L(n, Q p ) is non-trivial) Fix an integer n ≥ 1 and a prime p. Let r, κi (i = 1, . . . , r ) be positive integers such that κ1 + · · · + κr = n. Set κ = (κ1 , . . . , κr ). For i = 1, . . . r, let (πi , Vi ) be a representation of G L(κi , Q p ). Let (π, V ) be the tensor product representation π :=

r ' i=1

πi ,

V :=

r '

Vi ,

i=1

as in Definition 10.2.1. We regard (π, V ) as a representation of Mκ (Q p ). Then, the space of functions V G L(n,Q p ) , introduced in Definition 14.3.8, is non-zero. Proof Fix non-zero v = v1 ⊗ · · · ⊗ vr ∈ V. Because (πi , Vi ) is smooth for 1 ≤ i ≤ r, there exists an integer j such that πi (k) . vi = vi for all k ∈ G L(κi , Z p ) such that k − Iκi (Z p ) ∈ Mat(κi , p j Z p ), and for all 1 ≤ i ≤ r. Define

64

Theory of local representations for GL(n) ⎧ π (m) . v, ⎪ ⎪ ⎪ ⎨

f (g) =

⎪ ⎪ ⎪ ⎩

if g = umk with u ∈ Uκ (Q p ), m ∈ Mκ (Q p ), and k ∈ G L(n, Z p ), such that k − In ∈ Mat (n, p j Z p ),

0,

if g is not of this form.

This is well-defined because of the manner in which j was chosen, and is easily seen to give a non-zero element of V G L(n,Q p ) .  Example 14.3.10 (Principal series representations of G L(n, Q p ) obtained by parabolic induction) Take n a positive integer, p a prime, and consider the ordered partition n = 1 + · · · + 1, given by κ = (1, 1, . . . , 1). Then Mκ (Q p ) is simply the torus of G L(n, Q p ) consisting of all diagonal matrices, and an irreducible representation is simply a character χ : Mκ (Q p ) → C× given by ⎛⎛ ⎜⎜ ⎜ χ⎜ ⎝⎝

⎞⎞

t1

⎟⎟ ⎟⎟ = χ1 (t1 ) · χ2 (t2 ) · . . . · χn (tn ), ⎠⎠

t2 ..

.

(14.3.11)

tn where χ1 , . . . , χn are characters: Q×p → C× . Following Definition 14.3.8, G L(n,Q )

the parabolically induced representation Ind Pκ (Q p ) p (χ ) has a vector space V G L(n,Q p ) (χ ) consisting of locally constant functions f : G L(n, Q p ) → C satisfying ⎛⎛ ⎜⎜ ⎜ f⎜ ⎝⎝

⎞

t1

∗

t2 ..

⎞

⎟ ⎟ ⎟ · g⎟ = ⎠ ⎠

.

 n !

 n+1 2 −i

|ti | p

· χi (ti ) · f (g) (14.3.12)

i=1

tn for all t1 , . . . , tn ∈ Q×p , g ∈ G L(n, Q p ). Definition 14.3.13 (Principal series representation of G L(n, Q p )) Fix an integer n ≥ 1, a prime p, and characters χ1 , . . . , χn : Q×p → C× . Let χ : M1,1,... ,1 (Q p ) → C be the character defined in (14.3.11). Define V G L(n,Q p ) (χ ) to be the vector space of all locally constant functions satisfying (14.3.12). The  principal series representation of G L(n, Q p ) associated to χ is defined to be π, V G L(n,Q p ) (χ ) where π is the action by right translation. Thus 

 π (h) . f (g) := f (gh),

  ∀g, h ∈ G L(n, Q p ), f ∈ V G L(n,Q p ) (χ ) .

14.3 Parabolic induction for GL(n, Qp )

65

Proposition 14.3.14 (Representations that are parabolically induced from irreducible representations have finite length) Fix an integer n ≥ 1 and a prime p. Let r, κi (i = 1, . . . , r ) be positive integers such that κ1 +· · ·+κr = n. Set κ = (κ1 , . . . , κr ). Let (π, V ) denote an irreducible admissible representation of the standard Levi subgroup Mκ (Q p ) as in (14.3.2). Then G L(n,Q ) the induced representation Ind Pκ (Q p ) p (π ), defined in Definition 14.3.8, is of finite length as in Definition 14.1.4. G L(n,Q )

Idea of Proof To show that Ind Pκ (Q p ) p (π ) is smooth, one exploits the fact that an element f ∈ V G L(n,Q p ) is determined by its restriction to the compact group K := G L(n, Z p ), and mimics the proof of Proposition 6.5.5. To show G L(n,Q ) that Ind Pκ (Q p ) p (π ), one checks that if f ∈ V G L(n,Q p ) is fixed by a compact open subgroup K  ⊂ K , then it is completely determined by its values on any set of representatives for Pκ (Q p )\G L(n, Q p )/K  , which is finite, and that for each representative k, the value f (k) must lie in the finite dimensional space of Mκ (Q p ) ∩ k K  k −1 fixed vectors in V.  In order to work with contragredients of induced representations, a generalization of Proposition 8.2.4 is required. Proposition 14.3.15 (The contragredient of a parabolically induced representation) Fix an integer n ≥ 2 and a prime p. Let r, κi (i = 1, . . . , r ) be positive integers satisfying κ1 + · · · + κr = n. Set κ = (κ1 , . . . , κr ). For i = 1, . . . r, let (πi , Vi ) be an admissible representation of G L(κi , Q p ), Vi ). Define the tensor product representations (π, V ) with contragredient ( πi ,  and ( π,  V ), by π :=

r '

πi ,

V :=

i=1

r '

Vi ,

 π :=

i=1

r '

 πi ,

i=1

 V :=

r '

 Vi ,

i=1

as in Definition 10.2.1. Then the contragredient of the representation (π G L(n,Q p ) , V G L(n,Q p ) ) as in V G L(n,Q p ) ). Definition 14.3.8, is isomorphic to ( π G L(n,Q p ) ,  Vi → C denote the canonical bilinear form. Proof For each i, let  , i : Vi ×   Define a bilinear form  ,  : V × V → C by using the formula 0

r ' i=1

vi ,

r ' i=1

1  vi

:=

r ! vi , vi i

(14.3.16)

i=1

on pure tensors, and then extending to elements which are not pure tensors using bilinearity. It is easily verified that π (m) . v,  π (m) . v = v,  v for all

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Theory of local representations for GL(n)

f ∈  V G L(n,Q p ) , v ∈ V,  v ∈  V , and m ∈ Mκ (Q p ). For f ∈ V G L(n,Q p ) and  define 2 3 f (k),  f (k) d × k.  f,  f G L(n,Q p ) := G L(n,Z p )

A generalization of Proposition 8.2.3 shows that  , G L(n,Q p ) is an invariant pairing, in the sense that 2 3G L(n,Q p )  f,  f G L(n,Q p ) = π G L(n,Q p ) (g) . f,  π G L(n,Q p ) (g) . f , for all f ∈ V G L(n,Q p ) ,  f ∈ V G L(n,Q p ) , and g ∈ G L(n, Q p ). Vi → C (1 ≤ i ≤ r ) are The canonical bilinear forms  , i : Vi ×  nondegenerate by definition. The proof that the bilinear form (14.3.16) is nondegenerate is an exercise in multilinear algebra which we leave to the reader. (See Exercise 14.11.) Now take f ∈ V G L(n,Q p ) . Let v = f (In ) ∈ V, where In denotes the identity. Because V is smooth, there exists an integer j such that π (m) . v = v for all m ∈ Mκ (Q p ) such that (m − In ) ∈ Mat(n, p j Z p ). We define an element  f of  V G L(m,Q p ) by ⎧ π (m) . v, if g = umk, u ∈ Uκ (Q p ), m ∈ Mκ (Q p ), ⎪ ⎨  f (g) := k ∈ G L(n, Z p ), (k − In ) ∈ Mat(n, p j Z p ), ⎪ ⎩ 0, otherwise. Then it is easily verified that  f,  f G L(n,Q p ) =/ 0. To complete the proof that G L(n,Q p ) is nondegenerate, one must prove that for any given  f ∈ V G L(m,Q p ) ,  G L(m,Q p ) G L(n,Q p )  there exists f ∈ V so that  f, f  =/ 0. This is left to the reader. It now follows from Propositions 14.3.14 and 14.1.13 that the contragrediπ G L(n,Q p ) ,  V G L(n,Q p ) ).  ent of (π G L(n,Q p ) , V G L(n,Q p ) ) is isomorphic to (

14.4 Supercuspidal representations of G L(n, Q p ) Parabolic induction gives a means of constructing many representations of the group G L(n, Q p ) from representations of smaller general linear groups. In order to understand the role that this will have in the classification of representations of G L(n, Q p ), it is helpful to review the G L(2) classification from a “generalizable” perspective. In the G L(2) theory, it was shown in Theorem 6.16.1, in a different language, that every irreducible admissible representation of G L(2, Q p ) is isomorphic to one of the following three possibilities:

14.4 Supercuspidal representations of GL(n, Qp )

67

(1) an irreducible representation parabolically induced from the torus T = M(1,1) , (i.e., an irreducible principal series representation); (2) a subquotient of a reducible representation parabolically induced from the torus T = M(1,1) , (i.e., a special representation or a finite dimensional representation); (3) a supercuspidal representation. In order to generalize the above classification, we now introduce the definition of supercuspidal representation for G L(n, Q p ). Definition 14.4.1 (Supercuspidal representation of G L(n, Q p )) Fix a positive integer n and a prime p. Let (π, V ) be an irreducible admissible representation of G L(n, Q p ). The representation (π, V ) is said to be supercuspidal if for every ordered partition κ = (κ1 , . . . , κr ), with n = κ1 + · · · + κr , and every v ∈ V there is an integer m (positive or negative) such that Uκ ( p m Z p )

π (u) . v du = 0.

Here Uκ (Q p ) is the unipotent subgroup defined in (14.3.3) and Uκ ( p m Z p ) denotes the subgroup of Uκ (Q p ) consisting of those elements whose entries all lie in the fractional ideal p m Z p . The integral is an integral over p m Z p in each entry, with respect to the Haar measure on Q p as in Example 1.5.4. Historical Remarks The discovery of supercuspidal representations goes back to [Mautner, 1958, 1964]. The classification of supercuspidal representations of G L(n, Q p ) has a long history, see: [Shalika, 1966], [Moy-Sally, 1984], [Bushnell-Kutzko, 1993], for example. The next theorem gives several equivalent definitions of supercuspidality. Theorem 14.4.2 (Equivalent conditions for supercuspidality) Fix a positive integer n ≥ 1 and a prime p. Let (π, V ) be an irreducible admissible representation of G L(n, Q p ). The following are equivalent: (i) π is supercuspidal, as in Definition 14.4.1; (ii) the matrix coefficients of π, as defined in Definition 14.1.12, are compactly supported modulo the center (this term is defined in the same manner as for G L(2), see Definition 8.4.8); (iii) π is not isomorphic to a subquotient of a representation induced, as in Definition 14.3.8, from a proper parabolic subgroup. The proof of Theorem 14.4.2 makes use of the Jacquet modules as in (6.11.6). A complete proof is beyond the scope of this book. However, we shall introduce Jacquet modules for G L(n, Q p ), and indicate some of the key ideas in the proof of Theorem 14.4.2.

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Definition 14.4.3 (Jacquet module for G L(n, Q p )) Fix an integer n ≥ 1 and a prime p. Let r, κi (i = 1, . . . , r ) be positive integers such that κ1 +· · ·+κr = n. Set κ = (κ1 , . . . , κr ). Let Uκ (Q p ) denote the unipotent radical of the standard parabolic subgroup Pκ (Q p ) as in (14.3.3). Let (π, V ) denote an irreducible admissible representation of G L(n, Q p ). Then the Jacquet module VUκ of V is defined as V /V (Uκ ) where $ % π (u) . v − v | u ∈ Uκ (Q p ), v ∈ V

)

π (u) . v du = 0, ( for some m ∈ Z) . = v∈V

Uκ ( p m Z p )

V (Uκ ) = Span (

It is equipped with an action πUκ : Pκ (Q p ) → G L(VUκ ), given by πUκ (ρ) . (v + V (Uκ )) := (π (ρ) . v) + V (Uκ ),

(∀ρ ∈ Pκ (Q p ), v ∈ V ).

This action may also be restricted to Mκ (Q p ). The equivalence of the two definitions of V (Uκ ) given above is not obvious but is proved in the same manner as for G L(2, Q p ) (see Proposition 6.11.3). Likewise, the well-definedness of the action of Mκ (Q p ) is proved in the same way as for G L(2, Q p ) (6.11.5). The fact that Uκ (Q p ) also fixes V (Uκ ), and, hence, that there is a well-defined action of all of Pκ (Q p ), is obvious. Next, it is desirable to prove that the Jacquet module of an admissible representation is admissible. In order to do this it is necessary to introduce notation for dealing with compact open subgroups of G L(n, Q p ), as well as compact open subgroups of standard Levi subgroups of G L(n, Q p ). For any positive integers n, m, let $ % K m(n) = k ∈ G L(n, Z p ) | k − In ∈ Mat(n, p m Z p ) ,

(14.4.4)

where Mat(n, p m Z p ) denotes the set of n × n matrices with coefficients in p m Z p . Also, for any ordered partition κ = (κ1 , . . . , κr ) with n = κ1 + · · · + κr , let K mκ = K m(n) ∩ Mκ (Q p ) ⎧⎛ A1 0 . . . ⎪ ⎪ ⎨ ⎜ 0 A2 . . . = ⎜ .. ⎝ ⎪ . 0 ⎪ ⎩ 0 0 0 0

0 0 .. . Ar

(14.4.5) ⎫ ⎞

⎪

⎪ ⎟ Ai ∈ K (κi ) ⊂ G L(κi , Z p ), (1 ≤ i ≤ r ) ⎬ m ⎟ . ⎠ ⎪

⎪ ⎭

In addition, it is necessary to introduce the Haar measure for G L(n, Q p ). This was done for the case n = 2 in Section 6.9. The case of general n is the same.

14.4 Supercuspidal representations of GL(n, Qp )

69

Proposition 14.4.6 (Normalized Haar measure on G L(n, Q p )) There is a unique measure μ on the  set of all compact open subsets of G L(n, Q p ) which  satisfies μ G L(n, Z p ) = 1 and     μ g1 · K m(n) = μ K m(n) · g2 = μ(K n ),

(∀ m ∈ N, g1 , g2 ∈ G L(n, Q p )).

The corresponding integral satisfies f (gx) dμ(x) = G L(n,Q p )

f (xg) dμ(x) = G L(n,Q p )

f (x) dμ(x), G L(n,Q p )

(14.4.7) for all g ∈ G L(n, Q p ), and all locally constant functions f : G L(n, Q p ) → C such that any of the integrals in (14.4.7) converge. As in the case n = 2, it is possible to extend the measure in Proposition 14.4.6 to a measure on the ring of all Borel subsets of G L(n, Q p ). The term “Haar measure” properly refers to this extension. Because we only need to integrate locally constant functions we shall not go into measure theory in detail, but we will use the term “Haar measure” for the invariant integral of Proposition 14.4.6. It is clear that a representation is supercuspidal if and only if all of its Jacquet modules vanish. The next proposition shows that if any of the Jacquet modules of a representation (π, V ) is non-zero, then the representation may be embedded into a representation parabolically induced from the same parabolic. Proposition 14.4.8 (The rough classification of the irreducible admissible representations of G L(n, Q p )) Fix an integer n ≥ 1 and a prime p. Let (π, V ) be an irreducible admissible representation of G L(n, Q p ). Then either • (π, V ) is supercuspidal, or • there exists an ordered partition κ =/ (n) of n (see Definition 13.4.9) and an irreducible admissible representation (π  , V  ) of Mκ (Q p ) such that G L(n,Q ) (π, V ) is isomorphic to a subrepresentation of Ind Pκ(Q p ) p (π  ), as defined in Definition 14.3.8. More precisely, for every ordered partition κ such that the Jacquet module VUκ is non-trivial, there exists an irreducible admissible representation (π  , V  ) of Mκ (Q p ) such that (π, V ) is isomorphic to a subrepresentation G L(n,Q ) of Ind Pκ (Q p ) p (π  ). Further, if (π, V ) is isomorphic to a subrepresentation of G L(n,Q )

Ind Pκ (Q p ) p (π  ), then (π, V ) is not supercuspidal.

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Proof The proof is very similar to that of Theorem 6.13.4. For the convenience of the reader we review the key steps. For any compact open subgroup K  ⊂ G L(n, Q p ), the set Pκ (Q p )\G L(n, Q p )/K  is finite. It follows from this fact, together with the irreducibility and smoothness of (π, V ), that V is finitely generated as a Pκ (Q p )-module. From this, one concludes that VUκ is finitely generated as an Mκ (Q p )-module. A straightforward generalization of Proposition 6.1.16 shows that VUκ has an irreducible quotient (π  , V  ), and it can be shown that (π  , V  ) is also admissible. Frobenius reciprocity, Proposition 6.12.3, generalizes directly to G L(n, Q p ) and yields the proof, exactly as in Theorem 6.13.4.  It follows from Proposition 14.4.8 that every irreducible admissible representation of G L(n, Q p ) is either supercuspidal, isomorphic to an irreducible parabolically induced representation, or it is isomorphic to a subrepresentation of a reducible parabolically induced representation.

14.5 The Bernstein-Zelevinsky classification for G L(n, Q p ) Theorem 14.4.2 leaves the following questions open: • When is a parabolically induced representation irreducible? • Under what conditions are distinct parabolically induced representations isomorphic? • When a parabolically induced representation is reducible, what is the length of its Jordan-H¨older series? How are the representations in it described? • Is it possible for a parabolically induced representation to be isomorphic to a proper subrepresentation of another parabolically induced representation? If so, under what conditions does this occur? • Under what conditions do two parabolically induced representations (which are both reducible) have isomorphic subrepresentations? More generally, under what conditions do they have isomorphic subquotients? These problems were resolved in the papers: [Bernstein-Zelevinsky, 1976], [Bernstein-Zelevinsky, 1977], [Zelevinsky, 1980], which are based on earlier work of [Harish-Chandra, 1970], [Jacquet, 1970], [Godement-Jacquet, 1972], [Howe, 1974, 1977], [Gelfand-Kazhdan, 1971], [Bernstein, 1974]. We review some of their results, giving only hints at or sketches of the proofs. A key component in the Bernstein-Zelevinsky classification is the reduction to the case of a supercuspidal representation on the Levi subgroup. This idea goes back to [Harish-Chandra, 1970] and [Jacquet, 1970]. Regarding Harish-Chandra’s work, we quote from [Bernstein-Zelevinsky, 1976]

14.5 The Bernstein-Zelevinsky classification for GL(n, Qp )

71

“By analogy with the real case, he studied those irreducible representations whose matrix elements are finite modulo the centre of the group. We call such representations cuspidal. Harish-Chandra showed that any irreducible unitary representation of a reductive group G can be induced from a cuspidal representation of some subgroup. Thus the study of arbitrary representations reduces, in a certain sense, to that of cuspidal representations.” There is a beautiful analogy between supercuspidal representations of G L(n, Q p ) and rational prime numbers. It was known since the time of Euclid that every integer n > 1 can be uniquely decomposed into a product of prime powers, i.e., given a positive integer n, there exist integers m, d1 , · · · dm ≥ 0 and primes p1 , . . . , pm such that n = p1d1 · · · p2d2 · · · pmdm . Parabolic induction gives us a method to construct representations of larger general linear groups from representations of smaller general linear groups. This may be thought of as analogous to the process of forming larger positive integers from smaller ones by multiplying. The process of analyzing a representation using its Jacquet module is then akin to factoring an integer. Supercuspidals are akin to primes in this analogy, because, according to Proposition 14.4.8 they are the elements which can not be “factored” any further. One would then like to have, for any representation, a “prime factorization,” that is, an expression for how it is built up from objects which can not be broken down any further. In other words, an expression in terms of a representation which is parabolically induced from supercuspidals. This motivates the next proposition. Proposition 14.5.1 (Irreducible admissible representations of G L(n, Q p ) are all parabolically induced from supercuspidal representations) Fix an integer n ≥ 1 and a prime p. Let (π, V ) be an irreducible admissible representation of G L(n, Q p ). Then there exist positive integers r, κi (1 ≤ i ≤ r ) satisfying κ1 + · · · + κr = n, and an irreducible admissible supercuspidal representation π  of Mκ (Q p ), where κ = (κ1 , . . . , κr ), such that G L(n,Q ) π is isomorphic to a subrepresentation of Ind Pκ (Q p ) p (π  ), as defined in Definition 14.3.8. Remarks A representation (π  , V  ) of Mκ (Q p ) which is isomorphic to r & (πi , Vi ) is said to be supercuspidal if (πi , Vi ) is supercuspidal for each i. i=1

In the case when (π, V ) is supercuspidal, κ = (n). Every representation of G L(1, Q p ) is defined to be supercuspidal.

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Idea of Proof The idea of the proof is the following. We have seen that (π, V ) may be embedded into a representation induced parabolically from Mκ if and only if the Jacquet module VUκ does not vanish. As in the case n = 2, described in Section 8.12, there is a functor Jnκ which maps each smooth representation (π, V ) of G L(n, Q p ) to its Jacquet module, and each intertwining map L : V → V  to the induced map L Uκ : VUκ → VU κ as in Proposition 8.12.20. Furthermore, this generalizes in a straightforward way: if λ and κ are two ordered partitions of n such that Mλ (Q p ) ⊂ Mκ (Q p ), then by applying suitable Jacquet functors in each of the components of Mκ , we obtain a functor Jκλ , mapping smooth representations of Mκ (Q p ) to smooth representations of Mλ (Q p ). Finally, Jnλ = Jκλ ◦ Jnκ . Let (π  , V  ) be an irreducible quotient of VUκ , and write (π  , V  ) ∼ = r &     (πi , Vi ), where (πi , Vi ) is an irreducible admissible representation of i=1

G L(κi , Q p ) for i = 1, . . . , r. If any of the representations (πi , Vi ) is not supercuspidal, then there exists λ such that Mλ (Q p )  Mκ (Q p ) and Jnλ ((π, V )) =/ 0. As an immediate consequence, if κ is chosen so that no such λ exists, then (π  , V  ) is supercuspidal.  For representations parabolically induced from supercuspidal representations, the first question posed above: • When is a parabolically induced representation irreducible? has a straightforward answer. In order to state the answer, we recall the notion of a twist, which was introduced for G L(2, Q p ) in Definition 8.10.24. In order to avoid confusion, we use a slightly different notation in the present context. Definition 14.5.2 (Twist of a representation of G L(n, Q p )) Fix an integer n ≥ 1, a prime p, and a representation (π, V ) of G L(n, Q p ). Let χ : Q×p → C× be a character. The twist of (π, V ) by χ , which is denoted by (χ · π ), is defined to be the representation of G L(n, Q p ) on the same space V by the action   χ · π (g) . v := χ (det(g)) · π (g) . v, (∀g ∈ G L(n, Q p ), v ∈ V ). Theorem 14.5.3 (Criterion for reducibility of a representation induced from supercuspidals) Fix an integer n ≥ 1 and a prime p. Let r, κi (i = 1, . . . , r ) be positive integers such that κ1 + · · · + κr = n. Set κ = (κ1 , . . . , κr ). Let πi be an irreducible supercuspidal representation of r & G L(κi , Q p ) for 1 ≤ i ≤ r, and let π = πi be the corresponding superi=1

cuspidal representation of the standard levi Mκ (Q p ) ⊂ G L(n, Q p ) as in (14.3.2).

14.5 The Bernstein-Zelevinsky classification for GL(n, Qp )

73

G L(n,Q )

Then the parabolically induced representation Ind Pκ (Q p ) p (π ) is reducible if and only if there exists integers i, j (with 1 ≤ i, j ≤ n) such that κi = κ j and πi ∼ = (| | p · π j ), where | | p denotes the absolute value character Q×p → C× . Next we wish to tackle the question of when two induced representations are isomorphic or have isomorphic subquotients. Experience with G L(2) suggests that these two matters will be connected. Indeed, if χ1 and χ2 are two characters of Q×p , and B(χ1 , χ2 ) is the principal series representation, then for most values of χ1 and χ2 , the representations B(χ1 , χ2 ) and B(χ2 , χ1 ) are isomorphic and irreducible, while for a few select values, they are both reducible, with isomorphic subquotients. It will turn out that a suitable generalization of this phenomenon holds in the general case as well. In order to formulate it, we introduce a few important concepts. Definition 14.5.4 (Jordan-H¨older series, composition factors) Let (π, V ) be a representation of a group G. A Jordan-H¨older series is a finite ordered set {0} = V0 ⊂ V1 ⊂ . . . ⊂ Vm = V of subspaces of V such that for each i = 1, . . . m, • Vi−1 is an invariant subspace, and • the action of G on Vi /Vi−1 is an irreducible representation. Jordan-H¨older series are also called composition series. The irreducible representations Vi /Vi−1 are called the composition factors of (π, V ). Examples: Clearly, if (π, V ) is irreducible then {0} ⊂ V is a JordanH¨older series. For a non-trivial example, take G =× G L(2, Q p ) and (π, V ) = π, B(χ · | | p , χ ) , for some character χ of Q p . Then it was shown in Chapter 6 that (π, V ) has an irreducible subrepresentation (π, V1 ). This is a special representation. Further, the quotient V /V1 is one-dimensional, hence irreducible. So {0} ⊂ V1 ⊂ V is a composition series in this case. Finally, if G = G L(2, Q p ) and   (π, V ) = π, B(χ , χ · | | p ) , for some character χ of Q×p , then it follows from the results of Chapter 6 that (π, V ) has a composition series {0} ⊂ V1 ⊂ V where V1 is one-dimensional and V /V1 is a special representation.

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Theory of local representations for GL(n)

Not every representation of G L(n, Q p ) has a composition series, and a given representation may have more than one composition series. However, it may be shown that any two composition series for the same representation yield composition factors which are the same up to isomorphism and reordering. In particular, the number of composition factors is independent of the choice of series. The next Theorem 14.5.5 is Theorem 2.9 (part a) in [Bernstein, Zelevinsky, 1977]. Theorem 14.5.5 Fix an integer n ≥ 1 and a prime p. Let κ = (κ1 , . . . , κr ), λ = (λ1 , . . . , λs ), be two ordered partitions of n as in Definition 13.4.9. Let (π, V ) and (π  , V  ) be irreducible admissible supercuspidal representations of Mκ (Q p ) and Mλ (Q p ) respectively. The following are equivalent: (i) r = s and there is a permutation σ : {1, . . . , r } → {1, . . . , r } such that λi = κσ (i) and πi ∼ = πσ (i) for each i; (ii) there is a non-zero intertwining operator Ind Pκ (Q p ) p (π ) −→ Ind Pλ (Q p ) p (π  ); G L(n,Q )

G L(n,Q )

G L(n,Q )

(iii) some subquotient of Ind Pκ (Q p ) p (π ) is isomorphic to a subquotient of G L(n,Q )

Ind Pλ (Q p ) p (π  ); G L(n,Q )

(iv) every irreducible subquotient of Ind Pκ (Q p ) p (π ) is isomorphic to a G L(n,Q )

subquotient of Ind Pλ (Q p ) p (π  ), and vice versa. It follows from Theorem 14.5.5 that two representations of G L(n, Q p ), which are both parabolically induced from the same collection of supercuspidals, but in two different orders, will have the same set of irreducible subquotients. This motivates the following corollary. Corollary 14.5.6 (Supercuspidal support of an irreducible smooth representation of G L(n, Q p )) Fix a positive integer n, and a prime p. Let (π, V ) be an irreducible smooth representation of G L(n, Q p ). Then there exists a unique unordered partition κ = (κ1 , . . . , κr ) of n and an unordered tuple (π1 , . . . , πr ) of supercuspidal representations, unique up to isomorphism, satisfying: • πi is a supercuspidal representation of G L(κi , Q p ) (i = 1, . . . , r ) G L(n,Q ) • π is isomorphic to a subquotient of Ind Pκ (Q p ) p (π1 ⊗ · · · ⊗ πr ). The unordered tuple (π1 , . . . , πr ) is called the supercuspidal support of π.

14.6 Classification of smooth irreducible representations

75

Remark In fact, we are cheating slightly in the above statement, because the definition of Pκ given in (14.3.1) requires that κ be an ordered partition. It follows from Theorem 14.5.5, however, that the set of subquotients of G L(n,Q )

Ind Pκ (Q p ) p (π1 ⊗ · · · ⊗ πr ) is independent of the order. Consequently the supercuspidal support is well-defined. The main Theorem 6.1 in [Zelevinsky, 1980] (see also [Tadi´c, 1986]) gives the classification of the irreducible smooth representations of G L(n, Q p ). Theorem 14.5.7 (Bernstein-Zelevinsky classification) Fix a prime p. (i) Fix two positive integers r, d. Let (π, V ) be an irreducible supercuspidal representation of G L(r, Q p ). Then the parabolically induced representation      G L(r d,Q ) · π Ind P(r,... ,r ) p π ⊗ | | p · π ⊗ · · · ⊗ | |d−1 p * +, d terms

has aunique irreducible subrepresentation, which we denote as π d . Here | |kp · π is the twist of π by the absolute value character | |kp as in Definition 14.5.2. (ii) Fix positive integers m, r1 , . . . , rm and d1 , . . . , dm , and fix irreducible supercuspidal representations πi of  G L(ri , Q p ), (1 ≤ i ≤ m), such that i > j whenever π j ∼ = | |kp · πi for some integer k with 0 < k ≤ di < k + d j . Then the parabolically induced representation G L(n,Q p )

Ind P(r

1 d1 ,... ,rm dm )



 π1d1 ⊗ π2d2 ⊗ · · · ⊗ πmdm ,

(n = r1 d1 + · · · + rm dm ),

m has a unique irreducible subrepresentation determined by (ri , di , πi )i=1 . (iii) Two irreducible subrepresentations, of the type (ii), determined by m m and (ri , di , πi )i=1 are isomorphic if and only if m = m  (ri , di , πi )i=1 and there exists a permutation σ of {1, . . . , m} such that ri = rσ (i) , di = dσ (i) and πi ∼ = πσ (i) for all i = 1, . . . , m. (iv) Every irreducible smooth representation of G L(n, Q p ) is isomorphic to an irreducible subrepresentation of the type (ii).

14.6 Classification of smooth irreducible representations of G L(n, Q p ) via the growth of matrix coefficients In this chapter, two approaches to classifying irreducible smooth representations of G L(n, Q p ) are developed and compared. The first approach was

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Theory of local representations for GL(n)

introduced in the previous section, culminating in Theorem 14.5.7, which effectively solves the classification problem for irreducible smooth representations of G L(n, Q p ), by identifying a “standard representative” for each isomorphism class. In this section, a different sort of classification will be pursued, in which the smooth irreducible representations of G L(n, Q p ) are divided into broad categories based on the growth properties of their matrix coefficients. The process of sorting irreducible smooth representations of G L(n, Q p ) based on growth properties of their matrix coefficients was actually already begun in Theorem 14.4.2 which states that the matrix coefficients of a smooth irreducible representation are compactly supported modulo the center if and only if the representation is supercuspidal. We continue the classification with the following definitions. Definition 14.6.1 (Tempered representation of G L(n, Q p )) Fix an integer n ≥ 1 and a prime p. An irreducible admissible representation (π, V ) of G L(n, Q p ) is tempered if the following two conditions hold: (1) the central character of π (defined in Proposition 14.1.8) is unitary, i.e., it is a continuous homomorphism of absolute value 1; (2) For every  > 0 the matrix coefficients of π (defined in Definition  14.1.12) lie in L2+ Z \G L(n, Q p ) where Z denotes the center of G L(n, Q p ). Remarks Condition (1) ensures that the absolute value of each matrix coefficient β of (π, V ) is well-defined as a function:  Z \G L(n, Qp ) → C. A matrix coefficient β : G L(n, Q p ) → C lies in L2+ Z \G L(n, Q p ) if and only if

Z \G L(n,Q p )

|β(g)|2+ dg < ∞.

The integral here may be defined using the theory of invariant measures on homogeneous spaces as in [Nachbin, 1965], Chapter III, or, equivalently, as an integral over a “fattened up” fundamental domain, analogous to the domain D used in Definition 8.10.18. Definition 14.6.2 (Square integrable representation of G L(n, Q p )) Fix an integer n ≥ 1 and a prime p. An admissible representation (π, V ) of G L(n, Q p ) is square integrable if the following two conditions hold: (1) the central character of π (defined in Proposition 14.1.8) is unitary, i.e., it is a continuous homomorphism of absolute value 1; (2) The matrix coefficients of π , as in Definition 14.1.12, lie in  L2 Z \G L(n, Q p ) , where Z denotes the center of G L(n, Q p ).

14.6 Classification of smooth irreducible representations

77

Definition 14.6.3 (Discrete series representation of G L(n, Q p )) The discrete series representations of G L(n, Q p ) are just the irreducible smooth square integrable representations of G L(n, Q p ) as in Definition 14.6.2. The discrete series of G L(n, Q p ) is the set of all such representations. Clearly, any locally constant function: G L(n, Q p ) → C which is of compact support modulo the center is square integrable modulo the center, and a locally constant function: G L(n, Q p ) → C which is square integrable modulo the center is in L2+ (Z \G L(n, Q p )) for every  > 0. It immediately follows that every supercuspidal representation with a unitary central character is discrete series, and every discrete series representation is tempered. In the case n = 1, condition (2) in Definition 14.6.2 becomes vacuous, because Z \G L(1, Q p ) is a single point. Consequently, the discrete series representations of G L(1, Q p ) are simply the unitary characters of Q×p . Next, the classification given in Theorem 14.5.7 relates to the categories of representations described in Definition 14.6.1 and Definition 14.6.2. Theorems 14.6.4, 14.6.5 are based on material in [Rodier, 1982], [Prasad-Raghuram, 2008], [Wedhorn, 2008]. These classifications are given in terms of irreducible quotients as in Definition 14.1.5. It is customary to call them Langlands classifications by way of analogy with [Langlands, 1989]. Theorem 14.6.4 (Bernstein-Zelevinsky classification of discrete series representations of G L(n, Q p )) Fix a prime p. (i) Fix two positive integers r, d. Let (π, V ) be an irreducible supercuspidal representation of G L(r, Q p ) with unitary central character. Then the parabolically induced representation   1−d   d−1  G L(r d,Q ) Ind P(r,... ,r ) p π ⊗ | | p2 · π ⊗ · · · ⊗ | | p2 · π +, * d terms

has a unique irreducible quotient, as in Definition 14.1.5. (ii) The representation determined by (π, r, d) as in (i) is a discrete series representation (as in Definitions 14.5.2, 14.6.3) of G L(r d, Q p ) for all positive integers r, d and all irreducible supercuspidal representations π with unitary central character. (iii) Fix an integer n ≥ 1. Every discrete series representation of G L(n, Q p ) is isomorphic to a unique representation as in (ii). Theorem 14.6.5 (The Bernstein-Zelevinsky classification of tempered representations of G L(n, Q p )) Fix a prime p. (i) Fix positive integers m, r1 , . . . , rm , and fix irreducible discrete series representations πi of G L(ri , Q p ), (1 ≤ i ≤ m). Then the parabolically induced representation

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Theory of local representations for GL(n)   G L(n,Q ) Ind P(r ,... ,rmp) π1 ⊗ π2 ⊗ · · · ⊗ πm , (n = r1 + · · · + rm ), 1

is irreducible and tempered. (ii) Every irreducible tempered representation is isomorphic a unique representation as in (ii), with π1 , . . . , πr unique up to order and isomorphism.

14.7 Unitary representations of G L(n, Q p ) Recall the definition of unitary representation which was previously introduced for G L(2) in Chapter 9. Definition 14.7.1 (Unitary representation of G L(n, Q p )) Let n ≥ 1 be an integer and let p be a prime. Let (π, V ) be a smooth representation of G L(n, Q p ). Then (π, V ) is said to be unitary if V is equipped with a positive definite Hermitian form ( , ) : V × V → C, and (π (g) . v, π(g) . w) = (v, w),

(∀g ∈ G L(n, Q p ), v, w ∈ V ).

Unitary representations are of crucial importance for the theory of L-functions, because, as we have seen in Corollary 9.5.10, all cuspidal automorphic representations of G L(2, AQ ) are unitarizable. This then implies that all the corresponding local representations (see the tensor product Theorem 10.8.5) are also unitarizable. So, for the study of L-functions it is enough to consider only the unitary representations. It will turn out that this is also the case for G L(n) with n ≥ 3. Lemma 14.7.2 (Contragredient of a unitary representation of G L(n, Q p ) is unitary) Fix an integer n ≥ 1 and a prime p. An irreducible smooth representation (π, V ) of G L(n, Q p ) is unitary if and only if its contragredient, ( π,  V ), as defined in Definition 14.1.10, is unitary. Proof The argument used to prove Proposition 9.1.4 easily generalizes.



Lemma 14.7.3 (Matrix coefficients of unitary representations of G L(n, Q p ) are bounded) Fix an integer n ≥ 1 and a prime p. Let (π, V ) be a smooth irreducible representation of G L(n, Q p ) which is unitary. Then each of the matrix coefficients of π, as defined in Definition 14.1.12, is a bounded function from G L(n, Q p ) to C. Proof This follows from the Cauchy-Schwartz inequality, exactly as in Lemma 9.1.6.  The proof of the following proposition is due to Garrett.

14.7 Unitary representations of GL(n, Qp )

79

Proposition 14.7.4 (Bound on the parameters associated to a unitary principal series representation of G L(n, Q p )) Fix an integer n ≥ 1, a prime p, and characters χ1 , . . . , χn : Q×p → C× . Let π, V G L(n,Q p ) (χ ) be the principal series representation as in Definition 14.3.13. For i = 1, . . . , n, let σi be the unique real number satisfying |χi (t)|C = |t|σpi for all t ∈ Q×p , and define ρi := (n 1)/2.   − 2iG+L(n,Q p) (χ ) is unitary, then σ1 + · · · + σn = 0, and, for If π, V i = 1, . . . , n−1, we have − (ρ1 + · · · + ρi ) ≤ σ1 + · · · +σi ≤ ρ1 + · · · +ρi . Proof The equality σ1 + · · · +σn = 0, follows easily from the requirement that the central character be unitary. Let V G L(n,Q p ) (χ −1 ) denote the vector space of the principal series representation as in Definition 14.3.13, associated to the characters χ1−1 , . . . , χn−1 . By a generalization of Proposition 8.2.3, the pairing f 1 (k) · f 2 (k) d × k,

 f 1 , f 2  := G L(n,Z p )

where f 1 ∈ V G L(n,Q p ) (χ ), f 2 ∈ V G L(n,Q p ) (χ −1 ), is an invariant bilinear form V G L(n,Q p ) (χ ) × V G L(n,Q p ) (χ −1 ) → C. Fix i with 1 ≤ i < n. Define

K i :=

⎧ ⎪ ⎨

A C ⎪ ⎩

B D



⎫ A ∈ G L(i, Z p ),

⎪ ⎬

B ∈ Mat(i × (n − i), Z p ),

∈ G L(n, Z p ) , C ∈ Mat(i × (n − i), p · Z p ), ⎪ ⎭ D ∈ G L(n − i, Z p )

where Mat(i × j, R) denotes the set of all i × j matrices with entries in R. We leave it for Exercise 14.12 to verify that for all t ∈ Z p {0} and all k  ∈ K i , there exist k  ∈ K i and B  ∈ Mat(i × (n − i), Q p ) such that 

t −1 · Ii 0

0 In−i







·k ·

t · Ii 0

0 In−i



 =

Ii 0

B In−i



· k  .

Here, Ii and In−i denote the identity matrices of the indicated sizes. Define a function f 1 ∈ V G L(n,Q p ) (χ ) by

f 1 (b · k) =

χ (b),

if k ∈ K i ,

0,

otherwise,

−1 p) and define f 2 ∈ V G L(n,Q ) similarly.  Consider the matrix coefficient  (χ G L(n,Q p) (χ ) , determined by f 1 and f 2 as in β of our representation π, V Definition 14.1.12. Then we have

80  β

t · Ii 0

Theory of local representations for GL(n)   t · Ii 0 0 = f 2 (k) d × k f1 k · In−i 0 I n−i G L(n,Z p )

=

K i

  t · Ii f1 k · 0

0 In−i



d ×k

= Vol(K i ) · |t|ρp1 +···+ρi χ1 · · · χi (t), for all t ∈ Z p . By Lemma 14.7.3, this is a bounded function. It follows that − (ρ1 + · · · + ρi ) ≤ σ1 + · · · + σi . π,  V ). By Lemma 14.7.2, On the other hand, (π, V G L(n,Q p ) (χ −1 )) ∼ = ( this representation is also unitary. Applying the arguments above to (π, V G L(n,Q p ) (χ −1 )) yields the inequality σ1 + · · · + σi ≤ ρ1 + · · · + ρi .  Proposition 14.7.5 (Tempered representations of G L(n, Q p ) are unitary) Fix an integer n ≥ 1 and a prime p. Every irreducible tempered representation (π, V ) of G L(n, Q p ), as in Definition 14.6.1, is unitary as in Definition 14.7.1. Sketch of Proof An easy generalization of the proof of Proposition 9.3.1 shows that discrete series representations are unitary. A less straightforward generalization of the proof of Proposition 9.2.1 shows that the parabolically induced G L(n,Q ) representation Ind Pκ (Q p ) p (π ), defined in Definition 14.3.8, is unitary whenever the representation π of Mκ (Q p ) is unitary. The result now follows easily from Theorem 14.6.5.  In closing, we make the following important definition of complementary series representation. Definition 14.7.6 (Complementary series representation of G L(n, Q p )) Fix an integer n ≥ 1 and a prime p. A representation (π, V ) of G L(n, Q p ) is said to be complementary series if it is unitary, as defined in Definition 14.7.1, but is not tempered, as defined in Definition 14.6.1. Remarks Complementary series can be constructed by a method of Stein (see [Stein, 1967], [Sahi, 1990]). Analogous results were also attained by Godement. Complementary series representations for G L(2n, Q p ) were also constructed in [Bernstein, 1984].

14.8 Generalities on (g, K ∞ )-modules of G L(n, R) Fix an integer n ≥ 1. The Lie algebra gl(n, R) of G L(n, R) (see [Goldfeld, 2.1]) consists of the additive vector space of all n ×n matrices with coefficients in R, with Lie bracket given by

14.8 Generalities on (g, K∞ )-modules of GL(n, R)

81

[α, β] = α · β − β · α, for all α, β ∈ gl(n, R). The universal enveloping algebra U (gl(n, R)) of gl(n, R) (see [Goldfeld, 2.1, 2.2]) is an associative algebra which contains gl(n, R). The Lie bracket on gl(n, R) and the associative product ◦ on U (gl(n, R)) are compatible, in the sense that [α, β] = α ◦ β − β ◦ α for all α, β ∈ gl(n, R). Furthermore, U (gl(n, R)) is “universal” with respect to this property. The universal enveloping algebra U (gl(n, R)) can be realized as an algebra of differential operators acting on smooth functions F : G L(n, R) → C. Definition 14.8.1 Fix an integer n ≥ 1. Let α ∈ gl(n, R) and F : G L(n, R) → C be a smooth function. Then we define a differential operator Dα acting on F by the rule: Dα F(g) :=





∂  ∂  = F g · exp(tα) F g + t(g · α) . t=0 t=0 ∂t ∂t

Following Definition 4.5.5 we introduce the complexified universal enveloping algebra. Definition 14.8.2 (Complexified universal enveloping algebra) Fix an √ integer n ≥ 1. Set i = −1. Let β ∈ gl(n, R) and F : G L(n, R) → C, a smooth function. Then we define a differential operator Diβ acting on F by the rule: Diβ F(g) := i Dβ F(g). More generally, if α + iβ ∈ g := gl(n, C) with α, β ∈ gl(n, R), then Dα+iβ = Dα + i Dβ . The differential operators Dα+iβ (with α, β ∈ gl(n, R)) generate an algebra of differential operators which is isomorphic to the universal enveloping algebra U (g). The definition of a (g, K ∞ )-module for G L(n, R) is a direct generalization of the definition for G L(2, R) which was previously given in Chapter 7. Definition 14.8.3 ((g, K ∞ )-module of G L(n, R)) Fix an integer n ≥ 1. Let g = gl(n, C), K ∞ = O(n, R), and U (g) denote the universal enveloping algebra as in Definition 14.8.2. We define a (g, K ∞ )-module to be a complex vector space V with actions πg : U (g) → End(V ), π K∞ : K ∞ → G L(V ),

82

Theory of local representations for GL(n)

such that, for each v ∈ V, the subspace of V spanned by {π K∞ (k) . v | k ∈ K ∞ } is finite dimensional, and the actions πg and π K∞ satisfy the relations   πg (Dα ) · π K∞ (k) = π K∞ (k) · πg Dk −1 αk (14.8.4) for all α ∈ g, Dα given by Definition 14.8.2, and k ∈ K ∞ . Further, we require that  1 (14.8.5) πg (Dα ) . v = lim π K∞ (exp(tα)) . v − v t→0 t for all α ∈ gl(n, R), such that exp(α) ∈ O(n, R). We denote this (g, K ∞ )module as (π, V ) where π = (πg , π K∞ ). Next, we give an explicit definition of an admissible (g, K ∞ )-module for G L(n, R) which is a straightforward generalization of Definition 7.1.4. Definition 14.8.6 (Admissible (g, K ∞ )-module of G L(n, R)) Fix an integer n ≥ 1. Let g = gl(n, C), K ∞ = O(n, R), and let (π, V ) be a (g, K ∞ )-module as in % each v ∈ V define a vector space Wv to be the span $ Definition 14.8.3. For of π K∞ (k) . v k ∈ K ∞ , and define a homomorphism ρv : K ∞ → G L(Wv ) given by ρv (k) . w := π K∞ (k) . w,

(∀k ∈ K ∞ , w ∈ Wv ).

This determines a finite dimensional representation (ρv , Wv ) of the group K ∞ . Then (π, V ) is admissible $if, for each finite dimensional representation % (ρ, W ) of K ∞ , the span of v ∈ V (ρv , Wv ) ∼ = (ρ, W ) is finite dimensional. Definition 14.8.7 (Ramified/unramified (g, K ∞ )-module of G L(n, R)) Let n ≥ 1 and let (π, V ) be a (g, K ∞ ) module as in Definition 14.8.3. Then (π, V ) is said to be unramified if there exists a non-zero vector v ◦ ∈ V such that π K∞ (k) . v ◦ = v ◦ ,

(∀k ∈ K ∞ ).

Otherwise, (π, V ) is said to be ramified. Definition 14.8.8 (Contragredient of a (g, K ∞ )-module of G L(n, R)) Fix an integer n ≥ 1 and let g = gl(n, C) and K ∞ = O(n, R). If (π, V ) is a (g, K ∞ )module as in Definition 14.8.3, then the contragredient of (π, V ) is defined as follows: • First, let  V denote the space of$ linear functionals  : V → C %with the prop erty that the set of functions v →  π K∞ (k) . v k ∈ K ∞ (with v ∈ V ) spans a finite dimensional space. V ) is given by • Second, the action  π K∞ : K ∞ → G L(    π K∞ (k) . (v) :=  π K∞ (k −1 ) . v , (∀ k ∈ K ∞ ,  ∈  V , v ∈ V ).

14.8 Generalities on (g, K∞ )-modules of GL(n, R)

83

• Third, the action πg : U (g) → End(V ) is defined by    πg (Dα ) . (v) = − πg (Dα ) . v ,

(∀ α ∈ g,  ∈  V , v ∈ V ).

• Finally, the contragredient of (π, V ) is defined to be ( π,  V ) where we define π K∞ ).  π := ( πg ,  Historically, (g, K ∞ )-modules arose as a tool for the study of unitary group representations of real Lie groups. Their utility is based on a one-to-one correspondence between unitary (g, K ∞ )-modules and unitary representations of the corresponding group. This correspondence will also be very useful in our study of (g, K ∞ )-modules for G L(n, R). We now introduce the required definitions and give the precise statement in the specific case we need. Definition 14.8.9 (Unitary (g, K ∞ )-module of G L(n, R)) Fix an integer n ≥ 1. Let (π, V ), with π = (πg , π K∞ ), be a (g, K ∞ )-module of G L(n, R), as defined in Definition 14.8.3. Then (π, V ) is said to be unitary if there exists a positive definite Hermitian form ( , ) : V × V → C which is invariant, in the sense that     π K∞ (k) . v, v  = v, π K∞ (k −1 ) . v  ,     πg (Dα ) . v, v  = − v, πg (Dα ) . v  , for all v, v  ∈ V, k ∈ K ∞ , α ∈ gl(n, R). Definition 14.8.10 (Unitary representation of G L(n, R)) Fix an integer n ≥ 1. A unitary representation of G L(n, R), consists of a complex vector space V, equipped with a positive definite Hermitian form ( , ) : V × V → C, and a homomorphism π : G L(n, R) → G L(V ), such that the function (g, v) → π (g) . v is a continuous function G L(n, R) × V → V, and     π (g) . v, v  = v, π (g −1 ) . v  ,



 ∀ v, v  ∈ V, g ∈ G L(n, R) .

Further, we assume that V is a Hilbert space with respect to ( , ). Remarks The positive definite Hermitian form ( , ) : V × V → C defines a topology on V. One can then define continuity of (g, v) → π (g) . v. If the last condition is dropped, then V is a pre-Hilbert space, but it may be completed to a Hilbert space. Theorem 14.8.11 (The correspondence between unitary (g, K ∞ )-modules and unitary representations of G L(n, R)) Fix an integer n ≥ 1, and let K ∞ = O(n, R), g = gl(n, C).

84

Theory of local representations for GL(n) (1) If (π, V ) is a unitary representation of G L(n, R), as in Definition 14.8.10, then there is a dense subspace V(g,K∞ ) of V with actions     π K∞ : K ∞ → G L V(g,K∞ ) , πg : g → End V(g,K∞ ) , given by 1 (π (exp(tα)) . v − v) , t    for all v ∈ V(g,K∞ ) , k ∈ K ∞ , α ∈ g. Furthermore, πg , π K∞ , V(g,K∞ ) is a unitary (g, K ∞ )-module, as in Definition 14.8.9, called the underlying (g, K ∞ )-module of (π, V ). π K∞ (k) . v = π (k) . v,

πg (Dα ) . v = lim

t→0

(2) A unitary representation of G L(n, R) is irreducible if and only if its underlying (g, K ∞ )-module is irreducible. (3) If (π, V ) is a unitary (g, K ∞ )-module, as in Definition 14.8.9, then there exists a unitary representation (π  , V  ) of G L(n, R) as in Definition 14.8.10 such that (π, V ) is isomorphic to the underlying (g, K ∞ )-module of (π  , V  ), as in (1). (4) Two irreducible unitary representations of G L(n, R) are isomorphic if and only if their underlying (g, K ∞ )-modules are isomorphic. Remarks The correspondence given in Theorem 14.8.11 is valid because we have required the space of a unitary representation of G L(n, R) to be a Hilbert space. Further, a unitary representation of G L(n, R) is considered to be reducible only if it has a closed invariant subspace. Proof All three parts are essentially proved in [Wallach, 1988]. For (1), see 3.3.5. For (2) and (4), see 3.4.11. For (3), see 6.A.4.2. There are two minor technicalities. First, Wallach’s is for admissible and finitely generated (g, K ∞ )-modules. In applying it to an arbitrary irreducible unitary (g, K ∞ )module, we also use Wallach’s Corollary 3.4.8, which states that irreducible (g, K ∞ )-modules are admissible. Second, Wallach’s Theorem 6.A.4.2 applies to connected groups, while G L(n, R) has two connected components. We sketch a means of filling the gap in Exercise 14.18.  Using Theorem 14.8.11 makes it possible to define matrix coefficients in a more simple manner. Definition 14.8.12 (Matrix coefficient) Fix an integer n ≥ 1 and let (π, V ) be a unitary representation of G L(n, R). For any v, v  ∈ V, the function   (g ∈ G L(n, R)), βv,v (g) := π (g) . v, v  , is called a matrix coefficient of (π, V ). If v and v  are elements of the subspace V(g,K∞ ) as in Theorem 14.8.11, then βv,v is called a matrix coefficient of the underlying (g, K ∞ )-module as well.

14.9 Generic representations of GL(n, R)

85

Remark Since every irreducible unitary (g, K ∞ )-module is isomorphic to the underlying (g, K ∞ )-module of some irreducible unitary representation, it follows that Definition 14.8.12 defines matrix coefficients for every irreducible unitary (g, K ∞ )-module.

14.9 Generic representations of G L(n, R) Let n ≥ 1 be an integer, g = gl(n, C), and K ∞ = O(n, R). Following Section 14.2, we begin with a concrete example of a (g, K ∞ )-module of G L(n, R), as defined in Definition 14.8.3. In this section we shall refer to such a (g, K ∞ )module as a “representation” of G L(n, R). Example 14.9.1 (Local representation of G L(n, R) coming from an adelic automorphic form) Fix an integer n ≥ 1. Let ϕ : G L(n, AQ ) → C be an adelic automorphic form as in Definition 13.4.7. We can regard ϕ as a function with domain G L(n, R) by restriction of variables, i.e., consider ϕ(g) where g = {g∞ , In , In ,

. . . , },

(g∞ ∈ G L(n, R)).

Here In is the n × n identity matrix. By abuse of notation we will write this as ϕ(g∞ ). Define the complex vector space Vϕ to be the space of functions:

( ) m



m ∈ N, ci ∈ C, ki ∈ K ∞ , ci Di ϕ(g∞ ki ) g∞ → ,

Di ∈ U (g) (i = 1, . . . , r ) i=1

with g∞ ∈ G L(n, R). Let π K∞ : O(n, R) → G L(Vϕ ) be defined by right translation, i.e., π K∞ (k∞ ) . f (g∞ ) := f (g∞ k∞ ),

(∀g∞ ∈ G L(n, R), k∞ ∈ K ∞ , f ∈ Vϕ ).

Let πg : U (g) → End(V ) be given by the action by differential operators as in Definition 14.8.2. Then (π, Vϕ ), with π := (πg , π K∞ ) is an example of a (g, K ∞ )-module of G L(n, R). Example 14.9.2 (Whittaker model of a local representation of G L(n, R) coming from an adelic automorphic form) Fix an integer n ≥ 1. Define e∞ (x) := e2πi x , as in Section 1.6, and fix a character ψ of the group Un (AQ ) of the form ⎞⎞ ⎛⎛ 1 u ... u 1,2

⎜⎜ ⎜⎜ ⎜ ψ⎜ ⎜⎜ ⎝⎝

1

1,n

u 2,3 .. .

..

. 1

.. . u n−1,n 1

⎟⎟ ⎟⎟   ⎟⎟ = e∞ a1 u 1,2 + · · · + an−1 u n−1,n , ⎟⎟ ⎠⎠

86

Theory of local representations for GL(n)

for u i, j ∈ AQ , (1 ≤ i < j ≤ n), with ai ∈ Q× , (1 ≤ i < n). Let ϕ : G L(n, AQ ) → C be an adelic automorphic form as in Definition 13.4.7. Then by Theorem 13.5.4, the function ϕ has a Fourier expansion given by 



ϕ(g) =

Wϕ

γ 1

γ ∈ Un−1 (Q)\G L(n−1,Q)

g ,

(∀g ∈ G L(n, AQ )),

where ϕ(ug)ψ −1 (u) du,

Wϕ (g) = Un (Q)\Un (AQ )

for all g ∈ G L(n, AQ ). The function Wϕ is called a Whittaker function relative to ψ. Abstractly, a Whittaker function relative to ψ is any function W which is smooth, of moderate growth, and satisfies Wϕ (ug) = ψ(u)Wϕ (g),

(14.9.3)

for all u ∈ Un (AQ ), g ∈ G L(n, AQ ). Clearly, the function Wϕ has these properties. Let Wϕ denote the complex vector space of functions ( g∞ →

m  i=1

)

m ∈ N, ci ∈ C, ki ∈ K ∞ , ci Di Wϕ (g∞ ki )

Di ∈ U (g), (i = 1, . . . , r )

where g∞ ∈ G L(n, R). Let π K∞ : O(n, R) → G L(Vϕ ) be a homomorphism defined by right translation, i.e., π K∞ (k∞ ) . f (g∞ ) := f (g∞ k∞ ),

(∀g∞ ∈ G L(n, R), k∞ ∈ K ∞ , f ∈ Wϕ ). (14.9.4)

Let πg : U (g) → End(V ) be given by the action by differential operators as in Definition 14.8.2. Finally, let π = (πg , π K∞ ). Following Proposition 10.4.5, one may show that the representation (π, Vϕ ) defined in Example 14.9.1 is isomorphic to the Whittaker model (π, Wϕ ). Definition 14.9.5 (Whittaker model of a representation of G L(n, R)) Fix an integer n ≥ 1, and a character of ψ : Un (R) → C× of the form ⎛⎛ 1 ⎜⎜ ⎜⎜ ⎜ ψ⎜ ⎜⎜ ⎝⎝

u 1,2 1

... u 2,3 .. .

u 1,n ..

. 1

.. . u n−1,n 1

⎞⎞ ⎟⎟ ⎟⎟   ⎟⎟ = e∞ a1 u 1,2 + · · · + an−1 u n−1,n , ⎟⎟ ⎠⎠ (14.9.6)

14.9 Generic representations of GL(n, R)

87

for u i, j ∈ R, (1 ≤ i < j ≤ n), with ai ∈ R× , (1 ≤ i < n). Let (π, V ) be a (g, K ∞ )-module, where g = gl(n, C) and K ∞ = O(n, R). Following the conventions of this section, we refer to (π, V ) as a “representation” of G L(n, R). A Whittaker model for (π, V ) relative to ψ is a second representation (π  , W) ∼ = (π, V ) where W is a space of Whittaker functions relative to ψ, i.e., of smooth functions of moderate growth satisfying (14.9.3), for all u ∈ Un (R), g ∈ G L(n, R). Further π  = (πg , π K ∞ ) with π K ∞ is given by right translation as in (14.9.4), and πg is the action by differential operators as in Definition 14.8.2. Note that if there exists a character ψ as in (14.9.6) such that (π, V ) has a Whittaker model relative to ψ, then (π, V ) has a Whittaker model relative to ψ for every character of the form (14.9.6). Note also that not every representation of G L(n, R) has a Whittaker model. The simplest example is given by onedimensional representations which factor through the determinant: these do not have Whittaker models. Nevertheless, it is clear that all representations coming from adelic cusp forms as in Example 14.9.2 do have Whittaker models. This motivates the following definition. Definition 14.9.7 (Generic representation of G L(n, R)) Fix an integer n ≥ 1, and a character ψ as in (14.9.6). A representation (π, V ) of G L(n, R) is said to be generic relative to ψ if it has a Whittaker model relative to ψ as in Definition 14.9.5. It is generally permissible to speak of a representation of G L(n, R) as being simply “generic” or “not generic,” without specifying the character ψ to which this genericity is relative. This is because a given representation will either be generic relative to all characters ψ of the form (14.9.6), or else it will not be generic relative to any of them. Proposition 14.9.8 (Contragredient of a generic representation is generic) Fix an integer n ≥ 1 and a character ψ as in (14.9.6). A representation (π, V ) of G L(n, R) is generic relative to ψ if and only if its contragredient ( π,  V ), as in Definition 14.8.8 is generic relative to ψ. Proof Each of the two proofs of Proposition 14.2.8 can be adapted to the real case. The real analogue of theorem A of [Gelfand-Kazhdan, 1971] is given in an unnumbered corollary stated at the very end of Section 3 of [Shalika, 1974].  Theorem 14.9.9 (Multiplicity one for G L(n, R)) Fix an integer n ≥ 1, a character ψ as in (14.9.6) and an irreducible admissible representation (π, V ) of G L(n, R). If (π, V ) is generic relative to ψ as in Definition 14.9.7, then its Whittaker model is unique. In other words if W1 and W2 are both spaces

88

Theory of local representations for GL(n)

of Whittaker functions relative to the same character ψ, and if (π, W1 ) ∼ = (π, W2 ), then W1 = W2 . Here, in both cases, π = (πg , π K∞ ) with πg denoting action by differential operators as in Definition 14.8.2 and π K∞ denoting right translation. Proof See [Shalika, 1974].



14.10 Parabolic induction for G L(n, R) Recall Definition 13.4.9, that an ordered partition of an integer n ≥ 1 is a finite sequence κ = (κ1 , κ2 , . . . , κr ) of positive integers such that n = κ1 + · · · + κr . In Definition 13.4.10, we defined the standard parabolic subgroup of G L(n, R), associated to a partition κ = (κ1 , κ2 , . . . , κr ), to be

⎧⎛ ⎫ A1 B1,2 . . . B1,r ⎞ ⎪ ⎪

⎪ ⎪ ⎨⎜ 0 A2 . . . B2,r ⎟ Ai ∈ G L(κi , R), (1 ≤ i ≤ r ), ⎬

⎜ ⎟ , Pκ (R) := ⎝ Bi, j ∈ Mat(κi × κ j , R), .. ⎠ .. ⎪ ⎪ . 0 .

⎪ ⎪ (1 ≤ i < j ≤ r ) ⎩ 0 ⎭

0 0 0 Ar (14.10.1) where Mat(κi × κ j , R) denotes the set of all κi × κ j matrices with coefficients in R. The standard Levi subgroup, denoted Mκ (R), which is associated to a standard parabolic subgroup Pκ (R) was defined in Definition 13.4.11 and is ⎫ ⎧⎛ ⎞ A1 0 . . . 0

⎪ ⎪

⎪ ⎪ ⎬ ⎨ ⎜ 0 A2 . . . 0 ⎟ A ∈ G L(κ , R), (1 ≤ i ≤ r ) i i

⎜ ⎟ . Mκ (R) := ⎝ . .

.. .. ⎠ ⎪ ⎪ 0 ⎪ ⎪ ⎭ ⎩ 0

0 0 0 Ar (14.10.2) Similarly, the unipotent radical, denoted Uκ (R), defined in Definition 13.4.12 is given by

⎧⎛ ⎫ Iκ1 B1,2 . . . B1,r ⎞ ⎪ ⎪

⎪ ⎪ ⎨⎜ 0 Iκ2 . . . B2,r ⎟ Bi, j ∈ Mat(κi × κ j , R), ⎬

⎜ ⎟ , Uκ (R) := ⎝ .. ⎠ .. (1 ≤ i < j ≤ r ) ⎪ 0 ⎪ . 0 .

⎪ ⎪ ⎩ ⎭

0 0 0 Iκr (14.10.3) where Iκi denotes the κi × κi identity matrix for 1 ≤ i ≤ r. Lemma 14.10.4 (The natural projection: pr : Pκ → Mκ ) Fix an integer n. Let κ = (κ1 , κ2 , . . . , κr ) with n = κ1 + · · · + κr be an ordered partition of n. Let Pκ (R) be given by (14.10.1) and Mκ (R) by (14.10.2). Then the function pr : Pκ (R) → Mκ (R) given by

14.10 Parabolic induction for GL(n, R) ⎛⎛ A

1

⎜⎜ 0 ⎜ pr ⎜ ⎝⎝ 0 0

... ... .. .

B1,2 A2 0 0

0

B1,r ⎞⎞ ⎛ A1 B2,r ⎟⎟ ⎜ 0 ⎟ ⎜ .. ⎟ ⎠⎠ = ⎝ 0 . 0 Ar

0 A2 0 0

89 ⎞ 0 0 ⎟ .. ⎟ . ⎠

... ... .. . 0

Ar

is a homomorphism and its kernel is the unipotent radical Uκ (R) of Pκ (R), as in (14.10.3). The function pr is called the natural projection of Pκ (R) onto Mκ (R). Proof Follows from a straightforward matrix multiplication.



Clearly, Mκ (R) ∼ = G L(κ1 , R) × . . . , ×G L(κr , R). For any integer n ≥ 1, each ordered partition κ = (κ1 , . . . , κr ) of n, and gi ∈ G L(κi , R) (i = 1, . . . , r ), we shall identify ⎛

g1

⎜ ⎝0 0

... .. . ...

0

⎞

⎟ 0 ⎠ ∈ Mκ (R) gr

with (g1 , . . . , gr ) ∈ G L(κ1 , R) × · · · × G L(κr , R). Parabolic induction is a method of constructing representations and (g, K ∞ )-modules for G L(n, R) from representations or (g, K ∞ )-modules, respectively, for G L(κi , R), (i = 1, . . . , r ). By Theorem 14.8.11, unitary representations determine unitary (g, K ∞ )-modules and vice-versa, so it suffices to give a construction using unitary representations. The first step is to build a representation of Mκ (R) as a product of unitary representations of G L(κi , R), (i = 1, . . . , r ). This motivates the following definition. Definition 14.10.5 (Tensor product of unitary representations) Fix two integers n, n  ≥ 1, and let (π, V ), (π  , V  ) be unitary representations of G L(n, R), G L(n  , R) respectively, as in Definition 14.8.10. Let V ⊗ V  denote the ordinary tensor product of V and V  , as first introduced in Section 2.5. Define an action of G L(n, R) × G L(n  , R) on pure tensors by     (π ⊗ π  )(g, g  ) . v ⊗ v  = π (g) . v ⊗ π  (g  ) . v  . Extend this to an action on V ⊗ V  by linearity. Let ( , ) and ( , ) denote the invariant positive definite Hermitian forms on V and V  respectively. Define a positive definite Hermitian form ( , ) : (V ⊗ V  ) × (V ⊗ V  ) → C by the formula (v1 ⊗ v1 , v2 ⊗ v2 ) = (v1 , v2 ) · (v1 , v2 ),

(∀ v, v2 ∈ V, v1 , v2 ∈ V  ),

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on pure tensors, and by linearity in the first argument and conjugate linearity in the second argument for arbitrary elements of V ⊗ V  × V ⊗ V  . Clearly, ( , ) .V  is invariant under the action of G L(n, R) × G L(n  , R) by π ⊗ π  . Let V ⊗  denote the Hilbert space tensor product of V and V , i.e., the completion of V ⊗ V  with respect to ( , ) . For each g ∈ G L(n, R), g  ∈ G L(n  , R), the operator (π ⊗ π  )(g, g  ) : V ⊗ V  → V ⊗ V  . π  )(g, g  ) : V ⊗ .V → V ⊗ . V be the extension of is bounded. Let (π ⊗   . . π  is an action (π ⊗ π )(g, g ) to an operator on V ⊗V by continuity. Then π ⊗ . V  ). . π  : G L(n, R) × G L(n  , R) → G L(V ⊗ π⊗ . V  ) is called the tensor product of (π, V ) and . π , V ⊗ The representation (π ⊗   (π , V ). Definition 14.10.5 extends in a natural way to several factors. If (π1 , V1 ),

... ,

(πr , Vr ),

are unitary representations of G L(κ1 , R), . . . , G L(κr , R), respectively, then r 4 & the tensor product πi is a unitary representation of G L(κ1 , R) × · · · × i=1

G L(κr , R), or of the standard Levi Mκ (R), as in (14.10.2). It turns out that the construction in Definition 14.10.5 is not quite general enough, and requires the following minor extension. Definition 14.10.6 (Twist of a tensor product representation of Mκ (R)) Let n ≥ 1 be an integer and κ = (κ1 , . . . , κr ) an ordered partition of n. For i = 1, . . . , r, let si ∈ C, and (πi , Vi ) be a unitary representation of G L(κi , R). Let (π, V ) be the tensor product of the representations (π1 , V1 ), . . . (πr , Vr ), defined using Definition 14.10.5. We define the twist of (π, V ) by s = (s1 , . . . , sr ) to be the representation of G L(κ1 , R), . . . , G L(κr , R) on the same vector space V with an action | |s∞ · π defined by r !   s | det gi |s∞i · π (g1 , . . . , gr ) . v. | |∞ · π (g1 , . . . , gr ) . v := i=1

The representation | |s∞ · π defined in Definition 14.10.6 is not, in general, unitary. It is, however, continuous, in the sense that the function (g, v) → π (g) . v is a continuous function with respect to the natural topology on Mκ (R) and the Hilbert space topology on V. The next step is to construct representations of G L(n, R) from representations of its standard Levis.

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Definition 14.10.7 (Modular quasicharacter) Let n be an integer and κ an ordered partition of n. The modular quasicharacter of the parabolic subgroup Pκ (R) as in (14.10.1) is defined as ⎛⎛ A B r i−1 . . . B1,r ⎞⎞ 1 1,2 / / r A2 . . . B2,r ⎟⎟ ! ⎜⎜ 0

κj− κj

j=1 ⎜ ⎟

det(Ai ) j=i+1 δ Pκ ⎜ , .. ⎟ .. ⎝⎝ ⎠⎠ = ∞ . 0 0 . i=1

0

0

0

Ar

for all Ai ∈ G L(κi , R), (1 ≤ i ≤ r ) and all Bi, j ∈ Mat(κi × κ j , R), (1 ≤ i < j ≤ r ). Next, we define parabolic induction for G L(n, R) for n ≥ 1. This is modeled on the notion of “induced representation,” which has already been introduced in Definition 6.12.2 for the p-adic case. In the real case, there are some technical points. Our treatment follows [Knapp-Vogan, 1995]. As a first step, we introduce an important space of functions. Definition 14.10.8 (Parabolically induced pre-Hilbert space) Fix an integer n ≥ 1. Let κ = κ1 , . . . , κr with n = κ1 + · · · + κr be an ordered partition of n. For i = 1, . . . r, let si ∈ C,and let (πi , Vi ) be a unitary representation of r r 4 4 & & G L(κi , R). Let (π, V ) = | |s∞ · πi , Vi be the twist of the tensor prodi=1

i=1

uct of the representations (πi , Vi ), by the tuple (s1 , . . . , sr ) (see Definitions 14.10.5, 14.10.6). We regard (π, V ) as a representation of Mκ (R). Define the parabolically induced pre-Hilbert space V G L(n,R) to be the space of continuous functions ⎧

⎫

f (umg) = δ 12 (m) · π (m) . f (g), ⎬ ⎨

Pκ f : G L(n, R) → V

V G L(n,R) := ∀u ∈ Uκ (R), m ∈ Mκ (R), ⎭ . ⎩

g ∈ G L(n, R) Remarks According to Theorem 14.8.5, we can choose between working with (g, K ∞ )-modules and working with representations of G L(n, R). It is clear that one can not define a space such as V G L(n,R) without an action of Mκ (R). However, working with representations of G L(n, R) is not without its difficulties. Indeed, Theorem 14.8.5 applies to unitary representations of G L(n, R) on Hilbert spaces. The space V G L(n,R) is not a Hilbert space. In order to give a definition of parabolic induction using representations, it is necessary to complete the space in Definition 14.10.8 to a Hilbert space. Definition 14.10.9 (Parabolically induced positive definite Hermitian form) Fix an integer n ≥ 1. Let κ = κ1 , . . . , κr with n = κ1 + · · · + κr be an ordered partition of n.

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For i = 1, . . . r, let si ∈ C, and (πi , Vi ) be unitary representations of G L(κi , R). Set 5  r r 5 ' ' s (π, V ) = | |∞ · πi , Vi i=1

i=1

to be the twist of the tensor product of the representations (πi , Vi ), by the tuple (s1 , . . . , sr ) (see Definitions 14.10.5, 14.10.6). We regard (π, V ) as a representation of Mκ (R) with a positive definite Hermitian form ( , ) : V × V → C (see Definition 14.10.5). Let V G L(n,R) denote the parabolically induced pre-Hilbert space introduced in Definition 14.10.8. We define the parabolically induced positive definite Hermitian form to be the Hermitian form: ( , )G L(n,R) : V G L(n,R) ×V G L(n,R) → C, defined by   ∀ f 1 , f 2 ∈ V G L(n,R) .

( f 1 (k), f 2 (k)) d × k,

( f 1 , f 2 )G L(n,R) := O(n,R)

Definition 14.10.10 (Parabolically induced Hilbert space) Fix an integer n ≥ 1. Let κ = κ1 , . . . , κr with n = κ1 + · · · + κr be an ordered partition of n. For i = 1, . . . r, let si ∈ C, and (πi , Vi ) be a unitary representation of G L(κi , R). Let 5  r r 5 ' ' s (π, V ) = | |∞ · πi , Vi i=1

i=1

be the twist of the tensor product of the representations (πi , Vi ), by the tuple (s1 , . . . , sr ) (see Definitions 14.10.5, 14.10.6). We regard (π, V ) as a representation of Mκ (R). Let V G L(n,R) denote the parabolically induced pre-Hilbert space introduced in Definition 14.10.8, and let ( , )G L(n,R) be the positive definite Hermitian form given in Definition 14.10.9. Define . V G L(n,R) to be the G L(n,R) with respect to the norm induced by ( , )G L(n,R) . completion of V Now, that a Hilbert space has been defined, the next step is to define an action on that Hilbert space. The pre-Hilbert space V G L(n,R) is equipped with an action of G L(n, R) by right translation. In order to define an extension of this action to . V G L(n,R) , the following proposition is required. Proposition 14.10.11 (Right translation is bounded on a parabolically induced pre-Hilbert space) Fix an integer n ≥ 1. Let κ = κ1 , . . . , κr with n = κ1 + · · · + κr be an ordered partition of n. For i = 1, . . . r, let si ∈ C, and (πi , Vi ) be a unitary representation of G L(κi , R). Let

14.10 Parabolic induction for GL(n, R) (π, V ) = |

|s∞

93

5  r r 5 ' ' · πi , Vi i=1

i=1

be the twist of the tensor product of the representations (πi , Vi ), by the tuple (s1 , . . . , sr ) (see Definitions 14.10.5, 14.10.6). We regard (π, V ) as a representation of Mκ (R). Let V G L(n,R) denote the parabolically induced pre-Hilbert space introduced in Definition 14.10.8, and let ( , )G L(n,R) be the positive definite Hermitian form given in Definition 14.10.9. For g ∈ G L(n, R), let π  (g) : V G L(n,R) → V G L(n,R) be the operator corresponding to right translation by g, i.e.,   π (g) . f (h) = f (hg), ∀g, h ∈ G L(n, R), f ∈ V G L(n,R) . Then (i) the operator π (g) is bounded for all g ∈ G L(n, R), (ii) if each of the complex numbers s1 , . . . , sr is pure imaginary, then π (g) is unitary for all g ∈ G L(n, R). Proof See [Knapp-Vogan, 1995], Proposition 11.41 and Corollary 11.39.



Definition 14.10.12 (Normalized parabolic induction for G L(n, R)) Fix an integer n ≥ 1. Let κ = κ1 , . . . , κr with n = κ1 + · · · + κr be an ordered partition of n. For i = 1, . . . r, let si ∈ C, and (πi , Vi ) be a unitary representation of G L(κi , R). Let 5  r r 5 ' ' (π, V ) = | |s∞ · πi , Vi i=1

i=1

be the twist of the tensor product of the representations (πi , Vi ), by the tuple (s1 , . . . , sr ) (see Definitions 14.10.5, 14.10.6). We regard (π, V ) as a representation of Mκ (R). Let . V G L(n,R) be the Hilbert space given in Definition 14.10.10. For g ∈ V G L(n,R) → . V G L(n,R) denote the unique bounded G L(n, R), let π G L(n,R) (g) : . operator satisfying   π G L(n,R) (g) . f (h) = f (gh), ∀g, h ∈ G L(n, R), f ∈ V G L(n,R) . (The existence of such an operator is assured by Proposition 14.10.11.) Then (π G L(n,R) , V G L(n,R) ) is called the representation parabolically induced from (π, V ). We also adopt the notation: π G L(n, R) := IndGPκL(n,R) (R) (π ). Remarks For the student interested in working in representation theory or automorphic forms, it is important to understand the distinction between normalized and non-normalized parabolic induction. The word “normalized”

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indicates that the factor δ P2 κ is included in the definition of the induced representation. The non-normalized induced representation is defined in the same manner, with this factor omitted. Here, parabolic induction is assumed to be normalized. However, non-normalized induction is also common in the literature. Example 14.10.13 (Principal series representations of G L(n, R) obtained by parabolic induction) Take n a positive integer, and consider the ordered partition of n given by κ = (1, 1, . . . , 1). Then Mκ (R) is simply the torus of G L(n, R) consisting of all diagonal matrices. It is isomorphic to G L(1, R)n . Now, G L(1, R) is abelian, and an irreducible representation of an abelian group is necessarily one-dimensional. (See the proof of Lemma 6.1.7.) Consequently, if (π1 , V1 ), . . . , (πn , Vn ) are irreducible unitary representations of G L(1, R), then, for each i, the space Vi is one-dimensional and there is a unitary character of ωi : R× → C× such that πi (x) . v = ωi (x) · v, for all x ∈ G L(1, R) ∼ = R× and all v ∈ Vi , (i = 1, . . . n). The tensor product n & Vi is again one-dimensional, and hence must be equal to its completion i=1

n 4 &

Vi . There is no loss of generality in simply identifying

i=1

n 4 &

Vi with C. Fix i=1 si |t|∞ · ωi (t), (∀t ∈ t).

(s1 , . . . , sn ) ∈ C , and, for i = 1, . . . , n define  χi (t) := r r 4 4 & & Then the representation (π, V ) = | |s∞ · πi , Vi n

i=1

acts on C by the

i=1

character χ : Mκ (R) → C× which satisfies ⎛⎛ ⎜⎜ ⎜ χ⎜ ⎝⎝

⎞⎞

t1

⎟⎟ ⎟⎟ = χ1 (t1 ) · χ2 (t2 ) · . . . · χn (tn ). ⎠⎠

t2 ..

.

(14.10.14)

tn Following Definition 14.10.12, the parabolically induced representation G L(n,R) (χ ) consisting of continuous IndGPκL(n,R) (R) (χ ) has a pre-Hilbert space V functions f : G L(n, R) → C satisfying ⎛⎛ ⎜⎜ ⎜ f⎜ ⎝⎝

⎞

t1

 n  ! n+1 −i ⎟ ⎟ 2 ⎟ · g⎟ = |ti |∞ · χi (ti ) · f (g) (14.10.15) ⎠ ⎠

∗

t2 ..

⎞

.

i=1

tn for all t1 , . . . , tn ∈ R× , g ∈ G L(n, R). Such a function is clearly determined by its restriction to O(n, R), and the positive definite Hermitian form given in Definition 14.10.9 corresponds to the L2 inner product on O(n, R). Elements of the Hilbert space . V G L(n,R) (χ ) may be thought of as

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functions: G L(n, R) → C satisfying (14.10.15), with the property that the restriction to O(n, R) is an L2 function (considered up to L2 -equivalence). Definition 14.10.16 (Principal series representation of G L(n, R)) Fix an integer n ≥ 1, and characters χ1 , . . . , χn : R× → C× . Let χ : M1,1,... ,1 (R) → C be the character defined in (14.10.14). Define . V G L(n,R) (χ ) to be the vector space of all functions satisfying (14.10.15), such that the restriction to O(n, R) series repreis an L2 function (considered up to L2 -equivalence). The  principal sentation of G L(n, R) associated to χ is defined to be π, . V G L(n,R) (χ ) where π is the action by right translation. Thus   π (h) . f (g) := f (gh),

  ∀g, h ∈ G L(n, R), f ∈ V G L(n,R) (χ ) .

The preceding theory may be put into an even more explicit context by specializing to the case when n = 2. A function f : G L(2, R) → C which satisfies 

a 1 a b

2 f · g = χ1 (a)χ2 (d) 0 d d ∞ for some characters χ1 , χ2  : R× → Cis determined by its restriction to cos θ sin θ . The function f is an element of S O(2, R). Write F(θ ) = f − sin θ cos θ the (g, K ∞ )-module B∞ (χ1 , χ2 ) introduced in Definition 8.6.7 if and only if F is of the form  cn einθ , (cn ∈ C, S ⊂ Z, a finite set). F(θ ) = n∈S

On the other hand, if F is any L2 function, we may express F as a convergent Fourier series ∞  cn einθ . F(θ ) = n=∞

The space of all such L functions corresponds to . B∞ (χ1 , χ2 ). Each such function can be expressed as a limit of partial sums, and these partial sums are, of course, in the underlying (g, K ∞ )-module B∞ (χ1 , χ2 ). Note that the space smooth (χ1 , χ2 ) given in Definition 8.11.1 is equal to the set of smooth elements B∞ of . B∞ (χ1 , χ2 ). This subspace is invariant under the action of G L(2, R). Nevertheless, (π, . B∞ (χ1 , χ2 )) is considered an irreducible unitary representation of G L(2, R), because this invariant subspace is not closed. This also explains why one does not get a good correspondence in Theorem 14.8.11 without restricting to Hilbert space representations of G L(n, R). 2

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To finish our treatment of representations of G L(n, R) and (g, K ∞ )modules, we content ourselves with stating, without proof, the classification of the irreducible unitary representations of G L(n, R), and specifying which of the irreducible unitary representations appearing in this classification are generic. The classification of irreducible unitary representations of G L(n, R) was first obtained in [Vogan, 1986]. A different method, closer to that of Bernstein-Zelevinsky theory as discussed in Section 14.5, as well as the classification of unitary representations of G L(n, Q p ) [Tadi´c, 1986], was employed in [Tadi´c, 1993]. This approach relied on a hypothesis (U0) (see [Tadi´c, 1993], Section 8,9) which was not proved until later, by [Baruch, 2003]. The problem of translating back and forth between Vogan’s classification and Tadi´c’s was addressed briefly in Section 7.4 of [Clozel, 1987] (in French). A more detailed discussion, in English, appears in Section 12 of [Badulescu-Renard, 2010]. Our treatment follows [Tadi´c, 1993]. We wish to state a theorem which explicitly describes the classification of the irreducible unitary representations of G L(n, R) (and hence also of the irreducible unitary (g, K ∞ )-modules of G L(n, R)). First of all, it is convenient to introduce a collection of specific representations, in terms of which this classification can be cleanly formulated. An important class of representations which appears in this classification is the discrete series representations of G L(2, R). A unitary representation of G L(2, R) is said to be discrete series if its underlying (g, K ∞ )-module is one of the (g, K ∞ )-modules introduced in Definition 7.4.10. Proposition 14.10.17 (Four basic types of unitary representations of real general linear groups) (i) Fix a unitary character ω of R× , and an integer d ≥ 1. Let u(ω, d) denote the representation of G L(d, R) on C with action given by u(ω, d)(g) . v = ω(det g) · v for all g ∈ G L(d, R), v ∈ C. Then u(ω, d) is irreducible and unitary. (ii) Fix a discrete series representation π of G L(2, R) and an integer d ≥ 1. Let  := π · · ⊗ π* ⊗ ·+, d times

denote the tensor product of π with itself d times, regarded as a representation Levi M(2,... ,2) (R) ⊂ G L(2d, R). Let   d−1 d−3 of the1−dstandard ρ = 2 , 2 , . . . , 2 , and let π  = | |ρ∞ · denote the twist of  by ρ as in Definition 14.10.6. Then the parabolically induced representaL(2d,R)  tion IndGP(2,...,2) (R) (π ), as in Definition 14.10.16, has a unique irreducible quotient. This quotient is unitary, and is denoted u(π, d).

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(iii) Fix a unitary character ω of R× , a real number α ∈ (0, 12 ), and an integer d. Let u(ω, d) be defined as in (i). Then the parabolically induced representation   L(2d,R) | |(α,−α) · (u(ω, d) ⊗ u(ω, d)) IndGP(d,d) (R) is irreducible and unitary. (iv) Fix a discrete series representation π of G L(2, R), a real number α ∈ (0, 12 ), and an integer d. Let u(π, d) be defined as in (i). Then the parabolically induced representation   L(4d,R) (α,−α) | | · d) ⊗ u(π, d)) IndGP(2d,2d) (u(π, (R) is irreducible and unitary. Proof Part (i) is obvious. Part (ii) is due to [Speh, 1983]. Parts (iii) and (iv) were proved independently in [Tadi´c, 1985] and [Vogan, 1986].  Remarks (1) The representation u(ω, d) of (i) can also be described in a similar manner to the way u(π, d) of (ii) was described. This goes as follows: replacing the representation π in (ii) by a one-dimensional representation corresponding to a character transforms the induced representaL(2d,R)  G L(d,R) (χ )) as tion IndGP(2,...,2) (R) (π ) into a principal series representation (π, V d+1

−i

in Definition 14.10.16 where χ = (χ1 , . . . , χd ) with χi (t) = ω(t) · |t|∞2 (1 ≤ i ≤ d). To show that the one-dimensional representation corresponding to ω ◦ det is a quotient of (π, V G L(d,R) (χ )), it suffices to show that the onedimensional representation corresponding to ω−1 ◦ det is a subrepresentation of the contragredient of (π, V G L(d,R) (χ )). We lead the reader through the details of this in Exercise 14.19. The fact that (π, V G L(d,R) (χ )) has a unique irreducible quotient is a special case of the main result of [Langlands, 1989]. (2) Speh’s proof that the representations given in (ii) are unitary is a globalto-local argument similar to the proof of Corollary 9.5.11. She constructs a realization of u(π, d) (or rather, its underlying (g, K ∞ )-module) on a space of L2 automorphic forms. Theorem 14.10.18 (Classification of irreducible unitary representations of G L(n, R)) Fix an integer n ≥ 1. (i) Let κ = (κ1 , . . . , κr ) be an ordered partition of G L(n, R). For each i = 1, . . . , r, let πi denote a representation of G L(κi , R), which is of one of the four basic types introduced in Proposition 14.10.17. Let π = π1 ⊗ · · · ⊗ πr . Then the induced representation IndGPκL(n,R) (R) (π ) is irreducible and unitary. It is denoted π1 × · · · × πr .

98

Theory of local representations for GL(n) (ii) Two representations π1 × · · · × πr and π1 × · · · × πr  , as in (i) are isomorphic if and only if r = r  and there is a permutation σ : {1, . . . r } → {1, . . . r } such that πi ∼ = πσ (i) for all 1 ≤ i ≤ r. (iii) Every irreducible unitary representation of G L(n, R) is isomorphic to a representation as in (i).

Proof See [Tadi´c, 1993].



Definition 14.10.19 (Four basic types of generic irreducible unitary representations of G L(n, R)) The following four types of representations will be called the four basic types of generic irreducible unitary representations of G L(n, R): • unitary characters of G L(1, R), • discrete series representations of G L(2, R), that is, the irreducible unitary representations of G L(2, R) which correspond, via Theorem 14.8.11, to the (g, K ∞ )-modules introduced in Definition 7.4.10, • complementary series representations of G L(2, R), that is, the irreducible unitary representations of G L(2, R) which correspond, via Theorem 14.8.11, to the (g, K ∞ )-modules introduced in Definition 9.4.6,   L(4,R) (α,−α) | | • the representations IndGP(2,2) · ⊗ π , where π is a discrete (π ) (R) series representation of G L(2, R). Remarks (1) Four basic types of irreducible unitary representations of G L(n, R) were described in Proposition 14.10.17. The description of each of these types involves an integer d. Each of the four basic types of generic irreducible unitary representations of G L(n, R) can be obtained by taking one of the four basic types of irreducible unitary representations of G L(n, R) and requiring that the integer d equal 1. (2) Characters of G L(1, R) are a degenerate sort of generic representation: since the maximal unipotent subgroup U1 (R) ⊂ G L(1, R) is trivial, the condition (14.9.3) becomes vacuous in the case n = 1. Hence every smooth function of moderate growth is a Whittaker function, and every representation is its own Whittaker model. These technicalities may not present an entirely persuasive case for calling characters “generic.” The next theorem will show that characters truly must be included among the essential building blocks of generic representations. Theorem 14.10.20 (The classification of the generic irreducible unitary representations of G L(n, R)) Fix an integer n ≥ 1. Let κ = (κ1 , . . . , κr ) be an ordered partition of G L(n, R). For i = 1, . . . , r, let πi denote a representation of G L(κi , R), which is of one of the four basic types introduced in Proposition 14.10.17. The representation π1 ×· · ·×πr , as in Theorem 14.10.18 (i) is generic if and only if each of the representations πi is actually one of the

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four basic types of generic irreducible unitary representations of G L(n, R), as in Definition 14.10.19. Proof This may be deduced from the “Proof that (a) =⇒ (f),” appearing on p. 93 of [Vogan, 1978]. See also the very last remark of [Vogan, 1978], on p. 98.  Next we define discrete series representations of G L(n, R). This requires that we perform integration over the quotient space Z \G L(n, R), where Z is the center of G L(n, R). This integral is defined in much the same way as the adelic analogue given, for the case n = 2, in Definition 8.8.3. (See also the definition of “square integrable modulo the center” for G L(2, Q p ), given in Definition 8.10.18.) The extension to R and to general n is left to the reader. Definition 14.10.21 (Discrete series representation of G L(n, R)) Let n ≥ 1 be an integer. A unitary representation (π, V ) of G L(n, R) is said to be square integrable if and only if

βv,v (g) 2 dg < ∞, C Z \G L(n,R)

for each of the matrix coefficients βv,v of (π, V ) as in Definition 14.8.12. The discrete series of G L(n, R) is the set of all irreducible square integrable representations. The representations in the discrete series are also called discrete series representations. The discrete series representations of G L(1, R) are just the unitary characters. Theorem 14.10.22 (G L(n, R) has no discrete series representations unless n = 1, 2) If (π, V ) is a discrete series representation of G L(2, R) then its underlying (g, K ∞ )-module is isomorphic to one of the (g, K ∞ )-modules given in Definition 7.4.10. If n > 2, then G L(n, R) has no discrete series representations. Proof The fact that matrix coefficients of the (g, K ∞ )-modules given in Definition 7.4.10 are indeed square integrable follows from Theorem 8.11.10 and the formula for the Haar measure in terms of the Cartan decomposition given in the proof of Theorem 11.16.1. It follows from Theorem 13 on p. 90 of [Harish-Chandra, 1966] that S L(n, R) does not have discrete series representations for n > 2. To pass from S L(n, R) to G L(n, R), we note that the center of G L(n, R) acts by scalars on any irreducible representation, and hence that an irreducible representation remains irreducible when restricted to the subgroup S L ± (n, R) of matrices with determinant ±1. Restricting further to S L(n, R), an irreducible representation may become reducible, but it will have at most two components, because the index of S L(n, R) in S L ± (n, R) is 2. In this fashion, any discrete series

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representation of G L(n, R) would give rise to a discrete series representation of S L(n, R). And so Harish-Chandra’s result implies that there are no discrete series representations of G L(n, R) for n ≥ 3.  Definition 14.10.23 (Tempered representation of G L(n, R)) Let n ≥ 1 be an integer. A unitary representation (π, V ) of G L(n, R) is said to be tempered if

βv,v (g) 2+ dg < ∞, C

Z \G L(n,R)

for every  > 0, and for each of the matrix coefficients βv,v of (π, V ) as in Definition 14.8.12.

14.11 Classification of the unitary and the generic unitary representations of G L(n, Q p ) In this short section we present the p-adic analogues of Theorems 14.10.18 and 14.10.20. We continue to follow [Tadi´c, 1993]. Proposition 14.11.1 (Two basic types of p-adic irreducible unitary representations of general linear groups) (i) Fix positive integers m, d and a discrete series representation π of := · · ⊗ π-, G L(m, Q p ), as in Definition 14.6.3. Let  *π ⊗ ·+, d times

which is regarded as a representation of the standard Levi subgroup M(m,... ,m) (Q p ) ⊂ G L(md, Q p ).  d−1 ρ , . . . , 1−d , and let π  = | |∞ ·  denote the twist Let ρ = 2 , d−3 2 2 of  by ρ as in Definition 14.10.6. Then the parabolically induced representation G L(md,Q ) Ind P(m,...,m) (Qpp ) (π  ) as in Definition 14.10.16 has a unique irreducible quotient. This quotient is unitary, and is denoted u(π, d). (ii) Fix integers m, d ≥ 1, a discrete series representation π of G L(m, Q p ), as in Definition 14.6.3 and a real number α ∈ (0, 12 ). Let u(π, d) be defined as in (i). Then the parabolically induced representation   G L(2md,Q ) Ind P(md,md) (Q pp) | |(α,−α) · (u(π, d) ⊗ u(π, d)) is irreducible and unitary. Proof Uniqueness of the quotient is a general phenomenon, already noted in Theorem 14.6.5. The rest of (i) is proved in [Tadi´c,1986], Theorem A.8, p. 375, while (ii) is proved in [Tadi´c,1986], Proposition 4.2, p. 357. 

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101

As noted after Definition 14.6.3, a discrete series representation of G L(1, Q p ) is just a unitary character of Q×p . Further, Theorem 14.10.22 tells us that G L(n, R) has no discrete series representations unless n = 1 or 2. Thus, Proposition 14.11.1 is an exact p-adic analogue of Proposition 14.10.17. Next, we give the analogues of Theorems 14.10.18 and 14.10.20. Theorem 14.11.2 (The classification of the smooth irreducible unitary representations of G L(n, Q p )) Fix a positive integer n and a prime p. (i) Let κ = (κ1 , . . . , κr ) be an ordered partition of G L(n, Q p ). For each i = 1, . . . r, let πi denote a representation of G L(κi , Q p ), which is of one of the two basic types introduced in Proposition 14.11.1. Let G L(n,Q ) π = π1 , ⊗ · · · ⊗ πr . Then the induced representation Ind Pκ (Q p ) p (π ) is irreducible and unitary. It is denoted π1 × · · · × πr . (ii) Two representations π1 × · · · × πr and π1 × · · · × πr  , as in (i) are isomorphic if and only if r = r  and there is a permutation σ of {1, . . . r } such that πi ∼ = πσ (i) for all 1 ≤ i ≤ r. (iii) Every irreducible unitary representation of G L(n, Q p ) is isomorphic to a representation as in (i). Proof See [Tadi´c, 1993].



Definition 14.11.3 (The two basic types of generic irreducible unitary representations of G L(n, Q p )) Fix a positive integer n and a prime p. The following two types of representations will be called the two basic types of generic irreducible unitary representations of G L(n, Q p ): • discrete series representations of G L(n, Q p ),

 G L(2n,Q ) • the representations Ind P(n,n) (Q p )p | |(α,−α) · (π ⊗ π ) , where π is a discrete series representation of G L(n, Q p ). Theorem 14.11.4 (The classification of the generic irreducible unitary representations of G L(n, Q p )) Fix an integer n ≥ 1 and a prime p. Let κ = (κ1 , . . . , κr ) be an ordered partition of G L(n, Q p ). For each i = 1, . . . , r, let πi denote a representation of G L(κi , Q p ), which is of one of the two basic types introduced in Proposition 14.10.17. The representation π1 × · · · × πr , as in Theorem 14.10.18 (i) is generic if and only if each representation πi is actually of one of the two basic types of generic irreducible unitary representations of G L(n, Q p ), as in Definition 14.10.19. Proof This follows from Theorem 8.1 on p. 195 of [Zelevinsky, 1980]. The interested reader will also need to consult p. 452 of [Bernstein-Zelevinsky, 1977] for the usage of the term “highest derivative.” 

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14.12 Unramified representations of G L(n, Q p ) and G L(n, R) This section will describe the classification of the irreducible, generic, unramified, unitary representations of both G L(n, Q p ) and G L(n, R). Recall from Definition 14.1.7 that a representation (π, V ) is said to be unramified if V contains a non-zero vector fixed by the maximal compact G L(n, Z p ). In a similar manner, a unitary representation or (g, K ∞ )-module of G L(n, R) is said to be unramified if it contains a non-zero K ∞ -fixed vector. It turns out that any K ∞ -fixed vector in an irreducible unitary representation of G L(n, R) must lie in the underlying (g, K ∞ )-module, so unitary representations of G L(n, R) and (g, K ∞ )-modules can be treated jointly. Unramified representations and (g, K ∞ )-modules are sometimes called spherical. That is, the terms “spherical” and “unramified” are interchangeable in this context. Theorem 14.12.1 (Classification of the irreducible, generic, unramified, unitary representations of G L(n, Q p ) and G L(n, R)) Fix an integer n ≥ 1 and a prime v ≤ ∞. Suppose that (π, V ) is an irreducible, generic, unramified, unitary representation of G L(n, Qv ). Then there exist: • an integer 0 ≤ r < n2 , × • unramified unitary characters ω1 , . . . , ωn−r : Q× v →C , 1 • real numbers α1 , . . . , αr ∈ (0, 2 ), (   if v < ∞, π G L(n,Qv ) , VG L(n,Qv ) (χ ) , such that (π, V ) ∼ =  G L(n,Q ) G L(n,Q )  v v . π ,V (χ ) , if v = ∞. where   αr −αr 1 χ = ω1 · | |αv 1 , ω1 · | |−α v , . . . , ωr · | |v , ωr · | |v , ωr +1 , . . . , ωn−r . Remarks Recall that a unitary character of Q× v may be regarded as a onedimensional representation of G L(1, Qv ). It is unramified if it is trivial on the maximal compact subgroup G L(1, Zv ) = Z× v in the case v < ∞ or O(1, R) = {±1} in the case v = ∞. An equivalent condition is the following: a unitary character of Q× v is unramified if and only if it is of the form × (∀t ∈ Q ) for some r ∈ R. t → |t|ir v v Furthermore, a unitary character is a discrete series representation of G L(1, Qv ). The classification of the generic irreducible unitary representations of G L(n, Q p ), given in Theorem 14.11.4, states that each such representation is induced from a collection of representations, each of which is of one of the two basic types given in Definition 14.11.3. The classification of generic irreducible unitary representations of G L(n, R) given in Theorem 14.10.20 is stated similarly, and in fact, according to Theorem 14.11.2, can be stated in exactly the same way. Now, Theorem 14.12.1 may be reformulated as a refinement of Theorems 14.10.20 and 14.11.4 as follows: a generic irreducible

14.12 Unramified representations of GL(n, Qp ) and GL(n, R)

103

unitary representation of G L(n, Qv ) (v ≤ ∞) is unramified if and only if each of the discrete series representations which appear in the “basic” representations πi , (1 ≤ i ≤ r ) used to construct it, is actually an unramified character of Q× v. Sketch of Proof One first checks that for any integer n ≥ 1, any ordered partition κ = (κ1 , . . . , κr ) of n, and any irreducible unitary representations πi of G L(κi ) (1 ≤ i ≤ r ), the induced representation v) IndGPκL(n,Q (Qv ) (π1 ⊗ · · · ⊗ πr )

is unramified if and only if each of the representations πi is unramified. The case v = ∞ then follows from Theorems 14.11.2 and 14.11.4. The case v = p < ∞ can be proved by the same steps given in the case n = 2. We briefly review these steps. Step 1: Recall that the spherical Hecke algebra, denoted K H K , is the set of all locally constant, compactly supported functions f : G L(n, Q p ) → C satisfying   ∀k1 , k2 ∈ G L(n, Z p ) . f (k1 gk2 ) = f (g), The spherical Hecke algebra acts on (π, V ) by π( f ) . v =

f (g)π (g) . v dg,

(∀ f ∈

K

H K , v ∈ V ).

G L(n,Q p )

Then π ( f 1 )◦π ( f 2 ) = π ( f 1 ∗ f 2 ),

where

f 1 (gh −1 ) f 2 (h) dh,

f 1 ∗ f 2 (g) = G L(n,Q p )

for all g ∈ G L(n, Q p ) and all f 1 , f 2

K

H . K

Step 2: Show that the spherical Hecke algebra is abelian. The spherical Hecke algebra consists of finite linear combinations of characteristic functions of individual G L(n, Z p ) double-cosets. It follows easily from the Cartan decomposition given in Proposition 13.2.3 that each of these is fixed by the involution ι f (g) := f ( t g). On the other hand, a simple change of variable shows that ι ( f 1 ∗ f 2 ) = ι f 2 ∗ ι f 1 . Thus, f 1 ∗ f 2 (g) = ι ( f 1 ∗ f 2 ) = ι f 2 ∗ ι f 1 = f 2 ∗ f 1 . Step 3: Deduce that the space of G L(n, Z p )-fixed vectors in any irreducible spherical representation of G L(n, Q p ) is one-dimensional, with each element f of the spherical Hecke algebra acting by a scalar ξ ( f ) ∈ C on this onedimensional space. Furthermore the function f → ξ ( f ) is a C-linear ring homomorphism, called the spherical Hecke character. Step 4: Show that all values of a distinguished matrix coefficient of (π, V ) can be computed from the spherical Hecke character. If (π, V ) is unramified then

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the contragredient ( π,  V ) is also unramified (by a straightforward generalizav◦ ∈  V such that tion of (8.1.7)). Fix G L(n, Z p )-fixed vectors v ◦ ∈ V and  ◦ ◦  v  = 1. Here,  ,  : V × V → C is the canonical pairing as in (14.1.11). v , Let β(g) be the associated matrix coefficient v ◦ . β(g) = π (g) . v ◦ ,  v ◦ that Then it follows from the invariance of v ◦ and  π (g) . π (k1 ) . v ◦ ,  π (k2 ) . v ◦  d × k1 d × k2

β(g) = G L(n,Z p ) G L(n,Z p )

π (k2−1 gk1 ) . v ◦ ,  v ◦  d × k1 d × k2

= G L(n,Z p ) G L(n,Z p )

= π ( f g ) . v ◦ ,  v ◦  = ξ ( f g ), where f g is equal to the characteristic function of G L(n, Z p ) · g · G L(n, Z p ), normalized by the measure of this set. Step 5: Deduce that any two unramified representations with the same spherical Hecke character have a matrix coefficient in common. But, exactly as in Proposition 8.10.1, every irreducible representation of G L(n, Q p ) can be realized as the representation given by right translation acting on its own matrix coefficients. Consequently, any two irreducible representations which have a matrix coefficient in common are isomorphic. Step 6: Show that every possible spherical Hecke character is the spherical Hecke character of some principal series representation induced from unramified characters. This may be accomplished by finding a set of generators for the spherical Hecke algebra, as was done in the case n = 2, in the proof of Proposition 11.6.3. The generalization was obtained in [Tamagawa, 1963] and states that the spherical Hecke algebra is generated by the characteristic functions 1

 G L(n,Z p )·



p·Ir In−r

·G L(n,Z p )

(1 ≤ r ≤ n)

and

1 p−1 ·G L(n,Z p ) . (14.12.2)

The only relation among these generators is that the generator 1G L(n,Z p )· p·In ·G L(n,Z p )

(14.12.3)

corresponding to the choice r = n is equal to the inverse of 1 p−1 ·G L(n,Z p ) . It follows that spherical Hecke characters are in one-to-one correspondence with n-tuples of complex numbers, corresponding to the eigenvalues of the first

14.13 Unitary duals and other duals

105

n generators in (14.12.2). Further, the eigenvalue of the generator (14.2.3) must be non-zero. The fact that any n-tuple of eigenvalues with the last one being non-zero can be obtained from a principle series representation is also proved (in another language) in [Tamagawa, 1963], where it is credited to Satake. Finally, it follows from Theorem 14.10.20 that the principal series representation acquired at Step 6 is irreducible, which completes the proof.  Remark In order to classify all unramified representations of G L(n, Q p ), one has to consider the reducible principal series representations induced from unramified characters. Each has a unique unramified subquotient, and every unramified representation is isomorphic to some such representation.

14.13 Unitary duals and other duals The purpose of this short section is to familiarize the reader with the notion of a “dual.” The motivation comes from the theory of locally compact abelian groups. Theorem 14.13.1 (Pontryagin dual of a locally compact abelian group) Let G be a locally compact abelian group. Then the set G  of all continuous homomorphisms χ : G → C× with |χ (g)|C = 1 (∀ g ∈ G) is itself a locally compact abelian group, called the Pontryagin dual of G, with the group law being (χ1 · χ2 )(g) = χ1 (g) · χ2 (g),

(∀g ∈ G, χ1 , χ2 ∈ G  ),

and the topology being the compact-open topology. Furthermore (G  ) ∼ = G.

(14.13.2)

Remark Identity (14.13.2) gives a clear rationale for the use of the term “dual” in this case. Proof See [Rudin, 1962].



It is clear that Theorem 14.13.1 can not generalize in a direct way to nonabelian groups. For example, any homomorphism χ : G L(n, Q p ) → C× will be trivial on the commutator subgroup S L(n, Q p ), and consequently will be of the form χ (g) = χ  (det g) for some homomorphism χ  : Q×p → C× . Thus G L(n, Q p ) ∼ = G L(1, Q p ) for all values of n. It follows that no analogue of (14.13.2) is possible. In order to see a workable generalization to the nonabelian case, note that the elements of G  may be regarded as one-dimensional irreducible unitary

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representations of G. Thus one might hope that a satisfactory dual object can be obtained by taking irreducible unitary representations of arbitrary dimension. Definition 14.13.3 (Unitary dual of a locally compact group) Let G be a locally compact group. The unitary dual of G is the set of all isomorphism classes of irreducible unitary Hilbert space representations of G. (See, e.g., [Vogan, 1986], [Tadi´c, 1993].) Note that in the nonabelian case, the unitary dual does not have the structure of a group. It can still be equipped with a topology but we will not go into this here. One also encounters allusions to the “admissible dual,” the “spherical dual,” etc., in the literature. The admissible dual is defined as the set of isomorphism classes of irreducible admissible representations, the spherical dual spherical representations, etc. The Langlands dual, on the other hand, is a different sort of dual group, based on the classification of linear algebraic groups. It is also called the L-group. See [Borel, 1979] for a precise definition. For the relevant classification results, see [Springer, 1998].

14.14 The Ramanujan conjecture for G L(n, AQ ) We conclude this chapter with a statement of the Ramanujan conjecture for G L(n, AQ ). See Section 10.9 for a history of the Ramanujan conjecture and a discussion of the meaning of the conjecture in the case of G L(2, AQ ). Conjecture 14.14.1 (Ramanujan conjecture for G L(n, AQ )) Fix an integer n ≥ 1. Let π be an automorphic cuspidal representation of G L(n, AQ ), as defined in Definition 13.6.11, where ' πv , π ∼ = v≤∞

as in the tensor product Theorem 13.7.3. Then for each v ≤ ∞ the representation πv is tempered, as in Definitions 14.6.1, 14.10.23.

Exercises for Chapter 14 14.1 Let R be a commutative ring with 1. Prove that the commutator subgroup of G L(n, R) is precisely S L(n, R). (See Exercise 6.7 for the definition of commutator.) 14.2 In this exercise we lead the reader through a proof of Proposition 14.1.9. Let n ≥ 1 be an integer and let p be a prime. Suppose (π, V ) is a finite dimensional continuous representation of G L(n, Q p ). (See Exercise 6.11.) (a) Prove that π is smooth.

Exercises for Chapter 14

107

(b) Prove that the kernel of π contains S L(n, Q p ). (c) Now suppose (π, V ) is irreducible. Prove that V ∼ = C and that there × × exists a character ω : Q p → C such that π (g).v = ω(det(g)) · v for all g ∈ G L(n, Q p ) and v ∈ V . 14.3 Let (π, V ) be a smooth representation of G L(n, Q p ). (a) Prove that the canonical bilinear pairing  ,  : V ×  V → C is nondegenerate. Hint: The proof of Proposition 8.1.5 may be useful. (b) Prove that the map L : V →  V defined by L(v) = v, · is an injective intertwining map.  K   V (c) Prove that the map L in the previous part satisfies L(V K ) ⊂  for each compact open subgroup K  ⊂ G L(n, Q p ), and that equal ity holds if and only if V K is finite dimensional. Deduce that (π, V ) is isomorphic to its double contragredient if and only if it is admissible. (Compare with Corollary 8.1.13.) (d) Prove that Proposition 14.1.13 is not valid for smooth representations which are not admissible. 14.4 Let a, a  ∈ (Q×p )n−1 , and let ψ and ψ  be the associated characters of Un (Q p ) as in Definition 14.2.5. (a) Show that there exists a diagonal matrix γ ∈ G L(n, Q p ) such that ψ  (u) = ψ(γ uγ −1 ) for all u ∈ Un (Q p ). (b) Let Wψ : G L(n, Q p ) → C be a local Whittaker function with associated character ψ, as in Definition 14.2.5. Prove that the function Wψ  (g) := Wψ (γ g)

(g ∈ G L(n, Q p ))

is a local Whittaker function with associated character ψ  . Conclude that a representation (π, V ) of G L(n, Q p ) has a Whittaker model with respect to ψ if and only if it has a Whittaker model with respect to ψ  . (c) Not all linear algebraic groups behave as in (a). For example, it fails for S L(2, Q p ). Fix a, a  ∈ Q×p , and define characters ψ, ψ  on U2 (Q p ) by   1 x 1 x = e p (a  · x). ψ = e p (a · x) ψ 0 1 0 1 Prove there is γ ∈ S L(2, Q p ) as in part (a) if and only if a/a  is a square. Exercises 14.5–14.10 will develop a number of explicit forms for the Haar measure on G L(n, Q p ). Its existence was asserted in Proposition 14.4.6. We

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fix an integer n ≥ 2 and a prime p for the duration of these exercises. A function f : G L(n, Q p ) → C will always be assumed locally constant and compactly supported. 14.5 For a matrix g = (gi, j ) ∈ Mat(n, Q p ), write dgi, j for the additive Haar measure on Q p . Show that the integral " 1≤i, j≤n

f (g)

dgi, j

| det g|np

G L(n,Q p )

satisfies the invariance properties of the Haar measure for G L(n, Q p ), and hence is equal to the integral with respect to the Haar measure up to constant multiple. Finally, compute the constant. , the dimension of the maximal unipotent subgroup 14.6 Set N = n(n−1) 2 Un (Q p ). Prove that the integral against the normalized Haar measure on G L(n, Q p ) may be expressed by

(Q p ) N (Q×p )n G L(n,Z p )

⎛⎛ 1 u 1,2 1 ⎜⎜ ⎜ · f⎜ ⎝⎝

... ... .. .

⎞ ⎛

u 1,n u 2,n .. .

⎟ ⎜ ⎟·⎜ ⎠ ⎝

⎞

t1

⎟ ⎟ ⎟ · k⎟ ⎠ ⎠

t2 ..

⎞

. tn

1 · δ B (t)−1 d × k

n ! i=1

d × ti

!

du i, j ,

1≤i< j≤n

where d × k denotes the Haar measure on G L(n, Z p ), d × ti denotes the multiplicative Haar measure on Q×p for 1 ≤ i ≤ n, and du i, j denotes the additive Haar measure on Q p for 1 ≤ i < j ≤ n. Also, the factor δ B is given by n ! δ B (t) = |ti |n+1−2i . p i=1

14.7 Now let κ = (κ1 , . . . , κr ) be an ordered partition of n. Prove that the integral against the normalized Haar measure on G L(n, Q p ) may be expressed as f (u · m · k) δ P (m)−1 d × k d × m du, Uκ (Q p ) Mκ (Q×p ) G L(n,Z p )

Exercises for Chapter 14

109

where d × k is the normalized Haar measure on G L(n, Z p ), d × m is the product of the normalized Haar measures on the components of Mκ (Q p ) ∼ = G L(κ1 , Q p ) × · · · × G L(κr , Q p ), and the measure du on Uκ (Q p ) is simply the product measure corresponding to the additive Haar measure on Q p in each variable. Finally, ⎛⎛ ⎞⎞ /r /i−1 m1 r ! κ − κ j=i+1 j j=1 j . ⎝ ⎝ ⎠ ⎠ . δP | det m i | p . = . i=1 mr 14.8 Fix 1 ≤ r ≤ n. Show that the function G L(r, Q p ) × G L(n − r, Q p ) × Mat((n − r ) × r, Q p ) × Mat(r × (n − r ), Q p ) → G L(n, Q p )     In−r X 0 In−r Ir m1 0 (m 1 , m 2 , X, Y ) → Ir 0 0 Ir 0 m2 0

Y



In−r

is injective and has an open dense image in G L(n, Q p ). Let d × m 1 = normalized Haar measure on G L(r, Q p ), d × m 2 = normalized Haar measure on G L(n − r, Q p ), d X = product measure on Mat((n − r ) × r, Q p ) (additive Haar measure on Q p in each coordinate), dY = product measure on Mat(r × (n − r ), Q p ) (additive Haar measure on Q p in each coordinate). Show that the integral against the normalized Haar measure on G L(n, Q p ) can be expressed as a scalar multiple of

G L(r,Q p ) G L(n−r,Q p ) Mat(r ×(n−r ),Q p ) Mat((n−r )×r,Q p )



· f

In−r 0



X Ir ·

0 Ir

In−r 0

| det m 1 |n−r p

·



m1 0

| det m 2 |−r p

Then compute the scalar. 14.9 Define the n × n Weyl element ⎛ w=⎝

.. 1

0 m2

.

1

⎞ ⎠.



Ir 0

Y



In−r ×

dY d X d m 2 d × m 1 .

110

Theory of local representations for GL(n) Let Un (Q p ) denote the maximal unipotent subgroup of G L(n, Q p ), and let T (Q p ) denote the group of all diagonal matrices in G L(n, Q p ). Write t ∈ T (Q p ) and u ∈ Un (Q p ) as ⎛ t =⎝

⎛1 u 1,2 1 ⎜ u=⎜ ⎝

⎞

t1 ..

⎠,

. tn

... ... .. .

u 1,n u 2,n .. .

⎞ ⎟ ⎟. ⎠

1

Let du denote the measure on Un (Q p ) corresponding to the product of the additive Haar measures in each variable u i, j ∈ Q p for 1 ≤ i < j ≤ n. Let d × t denote the measure on T (Q p ) corresponding to the product of the multiplicative Haar measures on each variable ti ∈ Q×p for 1 ≤ i ≤ n. Let δ B : T (Q p )n → R be as in Exercise 14.6. Prove that f (uwu  t)δ B (t) d × t du du  Un (Q p ) Un (Q p ) T (Q p )

satisfies the invariance property of a Haar measure, and deduce that it is a scalar multiple of the normalized Haar measure. Then compute the scalar. (Compare with Exercise 13.14.) 14.10 Generalize Exercises 14.8 and 14.9 to obtain expressions for the normalized Haar measure involving Pκ (Q p ) for an arbitrary ordered partition κ of n. 14.11 In this exercise, we complete a multilinear algebra detail in the proof of Proposition 14.3.15. (a) Let V and V  be complex vector spaces equipped with a nondegenerate bilinear pairing  ,  : V × V  → C. Suppose v1 , . . . , vn ∈ V are linearly independent vectors. Show that there exist vectors v1 , . . . , vn ∈ V  such that vi , v j  =



1 if i = j 0 if i =/ j.

(b) Let r ≥ 1 be an integer and let Vi , Vi be complex vector spaces for i = 1, . . . , r equipped with bilinear pairings  , i : Vi × Vi → C. Define the tensor product pairing ,  :

 r ' i=1

 Vi

×

 r ' i=1

 Vi

→C

Exercises for Chapter 14

111

by the formula v1 ⊗ · · · ⊗

vr , v1

⊗ ··· ⊗

vr 

=

r !

vi , vi i

i=1

on pure tensors and extending using bilinearity. Prove that  ,  is nondegenerate if and only if each  , i is nondegenerate for i = 1, . . . , r . 14.12 Let 1 ≤ i < n be integers and let p be a prime. Let K i be the group of matrices as in the proof of Proposition 14.7.4. Prove that for each t ∈ Z p {0} and each k  ∈ K i , there exist k  ∈ K i and B  ∈ Mat(i × (n − i), Q p ) such that 

t −1 Ii 0

0 In−i



· k ·



t · Ii 0

0 In−i



 =

Ii 0

B In−i



· k  .

14.13 The goal of this exercise is to give a partial generalization of Proposition 8.3.1: “The contragredient of a special representation is the other special representation.” Let n ≥ 2 be an integer, let (π, V ) be an admisπ,  V ) be its contragredient. sible representation of G L(n, Q p ), and let ( Suppose that (π, V ) is reducible, and let {0} = W0  W1  · · ·  W = V be a composition series for it. Prove that ( π,  V ) has a composition series {0} = W0  W1  · · ·  W =  V  ∼  such that W i /Wi−1 = W−i+1 /W−i for 1 ≤ i ≤ .

14.14 Generalize Exercises 14.5, 14.8, 14.9, and 14.10 to give various explicit forms for the invariant measure on G L(n, R). (The usual method of normalizing the Haar measure is to generalize Section 14.6, but this generalization is less straightforward. See [Knapp, 1986], V.6 and [Bump, 1996], Proposition 2.1.5.) 14.15 Let (π, V ) be a (g, K ∞ )-module of G L(n, R). For each v ∈ V , define

Wv ⊂ V to be the span of the set of K ∞ -translates of v: {π (k).v k ∈ K ∞ }. Define ρv : K ∞ → G L(Wv ) by ρv (k).w = π (k).w

(k ∈ K ∞ , w ∈ Wv ).

112

Theory of local representations for GL(n) Then (ρv , Wv ) is a finite dimensional representation of K ∞ . If (ρ, V ) is any irreducible finite dimensional representation of K ∞ , consider the set Vρ = {v ∈ V | (ρv , Wv ) ∼ = (ρ, V )} ∪ {0}. Intuitively, it is the subspace of V on which π acts in the same way as ρ. (a) Prove that Vρ is a subspace of V and that it is closed under the action of K ∞ . 6 (b) Prove that V decomposes as a direct sum V = ρ Vρ over all irreducible finite dimensional representations of K ∞ . (Compare with Exercise 7.1.) (c) If (π, V ) is unitary, show that the subspaces Vρ are pairwise orthogonal.

14.16 Let (π, V ) be an irreducible admissible (g, K ∞ )-module of G L(n, R), and let ( π,  V ) be its contragredient as defined in Section 14.8. (a) Show that the contragredient ( π,  V ) is also irreducible and admissible. (b) Show that the canonical bilinear form V ×  V → C is invariant and nondegenerate. (c) If (π  , V  ) is another irreducible admissible (g, K ∞ )-module such that there is an invariant nondegenerate bilinear form V × V  → C, prove that (π  , V  ) is isomorphic to the contragredient of (π, V ). 14.17 Generalize Exercise 14.13 to G L(n, R). 14.18 Fix an integer n ≥ 2 and let g = gl(n, C). In this exercise we relate (g, K ∞ )-modules for G L(n, R) and (g, K ∞ )-modules for the connected group G L(n, R)+ . Since the two groups have different maximal compacts, we shall avoid the notation “K ∞ ,” instead writing out “O(n, R)” or “S O(n, R)” as appropriate. (See the remarks in the proof of Theorem 14.8.11.) (a) Let (π, V ) be an irreducible (g, O(n, R))-module, where π = (πg , π O(n,R) ). Let π  = (πg , πS O(n,R) ), where πS O(n,R) : S O(n, R) → G L(V ) is simply the restriction of π O(n,R) to S O(n, R). Show that (π  , V ) is either irreducible, or else it is the direct sum of two irreducible (g, S O(n, R))-modules. In the case that n is odd, show that it is always irreducible. (b) Let (π, V ) be an irreducible (g, O(n, R))-module, where π = (πg , π O(n,R) ). Suppose that (π, V ) is a direct sum of two irreducible (g, S O(n, R))-modules (π  , V  ) and (π  , V  ). Fix δ1 ∈ O(n, R) 1 ), where with det δ1 = −1, and define π δ1 = (πgδ1 , πSδO(n,R) πgδ1 : U (g) → End(V  ),

1 πSδO(n,R) : S O(n, R) → G L(V  )

Exercises for Chapter 14

113

are defined by πgδ1 (Dα ) = πg (Dδ1 ·α·δ−1 ), 1

  1 πSδO(n,R) (k) = πS O(n,R) δ1 · k · δ1−1 .

Show that (π  , V  ) ∼ = (π δ1 , V  ). (c) Keep the setup in (b), but let us further assume that δ12 = In . .) be an irreducible unitary representation of G L(n, R)+ Let ( π .,V such that the underlying (g, K ∞ )-module is isomorphic to (π  , V  ). Define V) . π δ1 : G L(n, R)+ → G L(. by

π (δ1 · g · δ1−1 ). . π δ1 (g) = .

V ) can be made into Show that the direct sum of (. π, . V ) and (. π δ1 , . a unitary representation of G L(n, R) by decreeing that δ1 acts on . V ⊕. V by swapping the two components. It can be shown that the underlying (g, O(n, R))-module of this representation of G L(n, R) is isomorphic to our original (g, O(n, R))-module, namely (π, V ). 14.19 This exercise addresses the details of Remark (1) following Proposition 14.10.17. Fix a unitary character ω of R× , and an integer d. Set δω−1 (g) = ω−1 (det(g)). For i = 1, . . . , n, define characters n+1

χi (t) = ω(t) · |t|∞2

−i

i− n+1 2

χi (t) = ω(t)−1 · |t|∞

.

Let χ = (χ1 , . . . , χd ) and χ  = (χ1 , . . . , χd ). We have shown that the principal series representations (π, V G L(n,R) (χ )) and (π, V G L(n,R) (χ  )) are isomorphic to the contragredients of one another. (a) Prove that δω−1 is an element of (π, V G L(n,R) (χ  )). (b) Prove that the one-dimensional subspace of V G L(n,R) (χ  ) spanned by δω−1 is invariant, and that G L(n, R) acts on this subspace by the character ω−1 ◦ det . (c) Deduce that (π, V G L(n,R) (χ )) has a quotient that is one-dimensional with G L(n, R) acting by the character ω ◦ det.

15 The Godement-Jacquet L-function for G L(n, AQ )

15.1 The Poisson summation formula for G L(n, AQ ) The Poisson summation formulae for G L(1, AQ ) and G L(2, AQ ) were presented in (1.9.1) and Theorem 11.2.3, respectively. We now generalize the Poisson summation formula to G L(n, AQ ) for all n ≥ 1. This will require several preliminary definitions. Definition 15.1.1 (The ring Mat(n, R)) Let n ≥ 1 be an integer. For any ring R, define Mat(n, R) to be the ring of all n ×n matrices with entries in R, under the usual addition and multiplication of matrices. (i, j)

Let n ≥ 1 be an integer. For each v ≤ ∞ and each 1 ≤ i, j ≤ n, let av ∈ Qv and let (i, j) (i, j) j) , a2 , . . . , a (i, . . . } ∈ AQ . a (i, j) = {a∞ p  (i, j)   (i, j)  We adopt the notation that a = a denotes the n × n matrix 1≤i, j≤n (i, j) th at the (i, j) place. Then the matrix with coefficient a  (i, j)   (i, j)   (i, j)  = a∞ , a2 a

 j)  . . . , a (i, , p

...



∈ Mat(n, AQ )

is an element of the restricted product of the groups {Mat(n, Qv ) | v ≤ ∞} with respect to the compact subgroups {Mat(n, Z p ) | p a finite prime }. The ring Mat(n, Q) will also be identified with its image under the diagonal embedding into Mat(n, AQ ). That is:

% $ Mat(n, Q) := i diag (γ ) γ ∈ Mat(n, Q) , where i diag (γ ) := {γ , γ , γ , 114

. . . }.

15.1 The Poisson summation formula for GL(n, AQ )

115

Definition 15.1.2 (Bruhat-Schwartz function on Mat(n, AQ )) Let n ≥ 1 be an integer. A complex valued function : Mat(n, AQ ) → C is said to be Bruhat-Schwartz if it is a finite sum of functions of the form () (m) =

! v

() v (m v ),

∀m = {m ∞ , m 2 , . . . , m p , . . . } ∈ Mat(n, AQ ),

where, for each  = 1, 2, 3, . . . , () at ±∞ • () ∞ : Mat(n, R) → C is smooth and (g∞ ) has rapid decay   in (i, j) (i, j) ∈ each of the variables a∞ (with 1 ≤ i, j ≤ n) where g∞ = a∞ Mat(n, R); • () p : Mat(n, Q p ) → C is locally constant and compactly supported for each finite prime p; • () p is the characteristic function of Mat(n, Z p ) for almost all primes p.

Theorem 15.1.3 (The Poisson summation formula for Mat(n, AQ )) Let n ≥ 1 be an integer and let be a Bruhat-Schwartz function on Mat(n, AQ ) as in Definition 15.1.2. Let e : AQ → C be given as in Definition 1.7.1, and let Mat(n, Q) be as in Definition 15.1.1. Then we have the identity 



(ξ ) =

ξ ∈Mat(n,Q)

.(ξ )

ξ ∈Mat(n,Q)

. is given by where the Fourier transform   (a) e − Tr(aξ ) da,

. (ξ ) :=

    a = a (i, j ∈ Mat(n, AQ ) ,

Mat(n,AQ )

 for all ξ ∈ Mat(n, AQ ). In the above, Mat(n,AQ ) indicates integration in each of the n 2 entries a (i, j) with respect to the adelic integral as in Definition 1.7.5 and Tr(aξ ) denotes the trace of the matrix aξ. Proof Let ξ (i, j) ∈ AQ for 1 ≤ i, j ≤ n. Define ⎛⎛

ξ (1,1) . 1 ⎝⎝ ..

ξ (1,2) .. .

···

ξ (n,1)

ξ (n,2)

···

⎞⎞ ξ (1,n) .. ⎠⎠ := . ξ (n,n)

AQ

⎛⎛

a (1,1) . ⎝⎝ ..

ξ (1,2) .. .

···

⎞⎞ ξ (1,n) .. ⎠⎠ .

ξ (n,2) · · · ξ (n,n)  (1,1) (1,1)  · e −ξ da (1,1) . a

ξ (n,1)

  Write ξ ∈ Mat(n, Q) as ξ = ξ (i, j) . If we fix all the variables ξ (i, j) except ξ (1,1) then we may consider (ξ ) as an adelic Bruhat-Schwartz function (as defined in Definition 1.7.4) in the variable ξ (1,1) . Its Fourier transform, defined as in

The Godement-Jacquet L-function for GL(n, AQ )

116

Definition 1.7.7, will be 1 (ξ ) . By the Poisson summation formula (1.9.1), it follows that for fixed ξ (i, j) ∈ AQ with 1 ≤ i, j ≤ n and (i, j) =/ (1, 1) we have 

⎛⎛

ξ (1,1) . ⎝⎝ ..

ξ (1,2) .. .

···

ξ (n,1)

ξ (n,2)

···

ξ (1,1) ∈ Q



=

⎞⎞ ξ (1,n) .. ⎠⎠ . ξ (n,n) ⎛⎛

ξ (1,1) . 1 ⎝⎝ ..

ξ (1,2) .. .

···

ξ (n,1)

ξ (n,2)

···

ξ (1,1) ∈ Q

⎞⎞ ξ (1,n) .. ⎠⎠ . . ξ (n,n)

Summing over all ξ (i, j) with (i, j) =/ (1, 1), yields   (ξ ) = 1 (ξ ). ξ ∈ Mat(n,Q)

ξ ∈ Mat(n,Q)

Now, the formula which defines the function 1 may be described as taking the Fourier transform of “in the variable ξ (1,1) .” Repeating these steps and taking the Fourier transform in the remaining variables ξ (1,2) , ξ (1,3) , . . . , ξ (n,n) it may be deduced that 



(ξ ) =

ξ ∈Mat(n,Q)



1 (ξ ) =

ξ ∈Mat(n,Q)



2 (ξ ) = · · · =

ξ ∈Mat(n,Q)

n 2 (ξ ),

ξ ∈Mat(n,Q)

where ⎛⎛

2 (ξ ) := AQ AQ

a (1,1) ⎜⎜ ξ (2,1) ⎜⎜ (3,1) ⎜ξ ⎜ ⎜⎜ . ⎝⎝ ..

a (1,2) ξ (2,2) ξ (3,2) .. .

ξ (n,1)

ξ (n,2)

⎛⎛

3 (ξ ) := AQ AQ AQ



ξ (1,3) ξ (2,3) ξ (3,3) .. .

··· ··· ··· .. .

⎞⎞ ξ (1,n) ξ (2,n) ⎟⎟ ⎟⎟ ξ (2,n) ⎟⎟ ⎟ .. ⎟ . ⎠⎠

ξ (n,3) · · · ξ (n,n)  (1,1) (1,1)  · e −ξ a − ξ (1,2) a (1,2) da (1,1) da (1,2)

a (1,1) ⎜⎜ ξ (2,1) ⎜⎜ (3,1) ⎜ξ ⎜ ⎜⎜ . ⎝⎝ ..

a (1,2) ξ (2,2) ξ (3,2) .. .

a (1,3) ξ (2,3) ξ (3,3) .. .

··· ··· ··· .. .

⎞⎞ ξ (1,n) (2,n) ξ ⎟⎟ ⎟⎟ ξ (2,n) ⎟⎟ ⎟ .. ⎟ . ⎠⎠

ξ (n,1)

ξ (n,2)

ξ (n,3)

···

ξ (n,n)

· e −ξ (1,1) a (1,1) − ξ (1,2) a (1,2) − ξ (1,3) a .. .

 (1,3)

da (1,1) da (1,2) da (1,3)

15.1 The Poisson summation formula for GL(n, AQ ) ⎛⎛

n 2 (ξ ) :=

··· AQ AQ

AQ

a (1,1) ⎜⎜ a (2,1) ⎜⎜ (3,1) ⎜a ⎜ ⎜⎜ . ⎝⎝ ..

a (1,2) a (2,2) a (3,2) .. .

a (1,3) a (2,3) a (3,3) .. .

117

⎞⎞ a (1,n) (2,n) ⎟⎟ a ⎟⎟ a (2,n) ⎟⎟ ⎟ .. ⎟ . ⎠⎠

··· ··· ··· .. .

a (n,1) a (n,2) a (n,3) · · · a (n,n)  (1,1) (1,1)  · e −ξ a − ξ (1,2) a (1,2) − · · · − ξ (n,n) a (n,n) da (1,1) da (1,2) · · · da (n,n) . .(ξ ) is simply n 2 (t ξ ). This completes the proof.  Furthermore, For our later purposes it is convenient to break the sums occurring on both sides of the Poisson summation formula (Theorem 15.1.3) into n + 1 pieces according as the rank of the matrix ξ ∈ Mat(n, Q) is 0, 1, . . . , n. Recall that the rank of a matrix ξ ∈ Mat(n, Q) is the dimension of {ξ · x | x ∈ Qn }. The rank of ξ will be n if and only if ξ is an invertible matrix, i.e., ξ ∈ G L(n, Q). Only the zero matrix (0) will have rank 0. It follows that we may rewrite the Poisson summation formula of Theorem 15.1.3 in the form   (ξ ) + (ξ ) + ((0)) ξ ∈G L(n,Q)

ξ ∈Mat(n,Q) 0
=



.(ξ ) +

ξ ∈G L(n,Q)



.(ξ ) + . ((0)) .

ξ ∈Mat(n,Q) 0
(15.1.4)

Actually, we shall need a variant of (15.1.4) for our applications which we present in the form of a proposition. Proposition 15.1.5 (The general Poisson summation formula for G L(n, AQ )) Let n ≥ 1 be an integer and let be a Bruhat-Schwartz function on Mat(n, AQ ) as in Definition 15.1.2. Let e : AQ → C be given as in Definition 1.7.1. Let G 1 = {g ∈ G L(n, A) | |det(g)| =1}, andMat(n, Q) be as t

in Definition 15.1.1. Then for h 1 , h 2 ∈ G 1 and τ =

with t > 0, we

. t

have 

..

  + h −1 2 ξ τ h1

ξ ∈G L(n,Q)

  + ((0)) h −1 2 ξ τ h1

ξ ∈Mat(n,Q) 0
⎡ 2 ⎢ = t −n ⎢ ⎣



 ξ ∈G L(n,Q)

  −1 . h −1 h2 + 1 ξτ

⎤  ξ ∈Mat(n,Q) 0
⎥ −1 .(h −1 . ((0))⎥. h2) + 1 ξτ ⎦

The Godement-Jacquet L-function for GL(n, AQ )

118

Proof Fix h 1 , h 2 ∈ G 1 , t > 0, and a Bruhat-Schwartz function as in Definition 15.1.2, and define a new function 1 : Mat(n, AQ ) → C by   1 (m) = h −1 2 mτ h 1 . Then 1 is a Bruhat-Schwartz function. Furthermore, in order to prove the present proposition, it suffices to prove that −1 .1 (m) = t −n 2 .(h −1 h 2 ), 1 mτ

(∀m ∈ Mat(n, A)).

n " n   " d x (i, j) . It is left as an Let x = x (i, j) ∈ Mat(n, AQ ) and define d x := i=1 j=1

exercise that (g · x · h) d x = | det g|−n · | det h|−n ·

··· AQ

AQ

··· AQ

(x) d x,

AQ

for all g, h ∈ G L(n, AQ ), and any function  : Mat(n, AQ ) → C such that these integrals are defined (including Bruhat-Schwartz functions as defined in Definition 15.1.2). In other words, the “Jacobian” of the change of variables x → g −1 · x · h −1 is | det g|−n · | det h|−n . Now, for m ∈ Mat(n, Q), .1 (m) =

    h −1 2 xτ h 1 e − Tr(x · m) d x.

··· AQ

AQ

We make the simple change of variables x → h 2 xτ −1 h −1 1 , and exploit the fact −1 −1 m) = Tr(x · h mτ h ) to obtain that Tr(h 2 xτ −1 h −1 2 1 1 .1 (m) = t −n 2

··· AQ

  −1 (x) e − Tr(x · h −1 h 2 ) d x. 1 mτ

AQ

The general Poisson summation formula follows immediately from (15.1.4), but with the function 1 instead of . 

15.2 The global zeta integral for G L(n, AQ ) Following the ideas that appeared in Tate’s thesis [Tate, 1950], Godement and Jacquet introduced a global zeta integral associated to a cuspidal automorphic representation of G L(n, AQ ) by representing it as an integral transform of a matrix coefficient.

15.2 The global zeta integral for GL(n, AQ )

119

Let (π, V ) be a cuspidal automorphic representation of G L(n, AQ ) with central character χπ as defined in Definition 13.6.9. We want to explicitly construct global matrix coefficients associated to π in the same way we did this for the G L(2)-case in Definition 9.5.12. It is enough to define an invariant pairing on the space of adelic cusp forms with central character χπ . To define an invariant pairing by an integral we need to know that such adelic cusp forms are L2 modulo the center. This motivates the following proposition. Proposition 15.2.1 (Adelic cusp forms are of rapid decay) Fix an integer n ≥ 2. Let φ be an adelic cusp form for G L(n, AQ ) as in Definition 13.4.15. Fix g ∈ G L(n, AQ ), and M = (M1 , . . . , Mn−1 ) ∈ Nn−1 . Then there is a constant cg,M > 0 such that −M

|φ (i ∞ (y) · g)|C ≤ cg,M y1−M1 y2−M2 · · · yn−1n−1 , for all ⎛ ⎜ ⎜ y=⎜ ⎜ ⎝

⎞

y1 y2 · · · yn−1 ..

⎟ ⎟ ⎟ ∈ G L(n, R). ⎟ ⎠

. y1 y2 y1 1

Furthermore, the constant cg,M can be made uniform over g in a compact set. Proof The proof is similar to that of Proposition 8.9.2. We omit the details.



Definition 15.2.2 (The matrix coefficient of a cuspidal automorphic representation for G L(n, AQ )) Fix an integer n ≥ 1. Let (π, V ) be a cuspidal automorphic representation of G L(n, AQ ) as defined in Definition 13.6.9. By a straight-forward generalization of Proposition 9.5.8, we may assume that V is realized as a space of adelic cusp forms. Fix φ1 , φ2 ∈ V. Then the map φ1 (hg) φ2 (h) d × h,

β(g) :=

(g ∈ G L(n, A)),

Z (A)G L(n,Q)\G L(n,A)

is called a matrix coefficient of (π, V ). Remarks The convergence of the integral in Definition 15.2.2 is assured by Proposition 15.2.1. This generalizes the matrix coefficients introduced in Definition 9.5.12. Definition 15.2.3 (Global zeta integral for G L(n, AQ )) Fix an integer n ≥ 1. Let (π, V ) be a cuspidal automorphic representation of G L(n, AQ ) as in Definition 13.6.11. Fix f 1 , f 2 ∈ V , and let

The Godement-Jacquet L-function for GL(n, AQ )

120

f 1 (hg) f 2 (h) d × h,

β(g) :=

(g ∈ G L(n, A)),

Z (AQ )G L(n,Q)\G L(n,AQ )

be a matrix coefficient as in Definition 15.2.2. Let be a Bruhat-Schwartz function as in Definition 15.1.2. For s ∈ C with (s) sufficiently large, we define the global zeta integral Z (s, , β) :=

(g) β(g) |det(g)|s+

n−1 2

d × g,

G L(n,AQ )

where, for g = {gv }v≤∞ ∈ G L(n, AQ ), we define d × g :=

" v≤∞

d × gv . Here

d × gv is the multiplicative Haar measure on G L(n, Qv ), such that the measure of G L(n, Zv ) is one for all non-archimedean v. We leave it as an exercise to the reader to check that the integral defining Z (s, , β) converges absolutely for (s) sufficiently large. Following [Godement-Jacquet, 1972], [Jacquet, 1979], it will be shown that Z (s, , β) has a holomorphic continuation to all s ∈ C and satisfies a functional equation. Furthermore, if the representation (π, V ) is irreducible, then we choose the matrix coefficient β so that it factors as a product of local matrix coefficients, and then show that the global zeta integral can be represented by an Euler product. Thus, the global zeta integral has the same general properties as the Riemann zeta function. Theorem 15.2.4 (Functional equation of the global zeta integral) Fix an integer n ≥ 1. Let (π, V ) be a cuspidal automorphic representation of G L(n, AQ ) as in Definition 13.6.11. Let Z (s, , β) denote the global zeta integral defined in Definition 15.2.3. Then Z (s, , β) has a holomorphic continuation to all s ∈ C and satisfies the functional equation ∨

., β ) Z (s, , β) = Z (1 − s, ∨

. is given in Theorem 15.1.3, and β (g) = β(g −1 ). where ∨

Remark The function β is actually a matrix coefficient of the contragredient representation ( π,  V ). Proof Recall that G 1 = {g ∈ G L(n, AQ ) | |det(g)| = 1}. As in the proof of Theorem 11.4.2, we compute ⎞ ⎛

s+ n−1 ⎟ ⎜ (g) ⎝ f 1 (h 2 g) f 2 (h 2 ) d × h 2 ⎠ det(g) 2 d × g Z (s, , β) = G L(n,AQ )

G L(n,Q)\G 1

15.2 The global zeta integral for GL(n, AQ )

121

s+ n−1 ×

  ×

2 d g

h −1 2 g f 1 (g) f 2 (h 2 ) d h 2 det(g)

= G L(n,AQ ) G L(n,Q)\G 1 +∞

=

⎡

⎤ 

⎢ ⎣

0

  × × ⎥ h −1 2 ξ τ h 1 f 1 (h 1 ) f 2 (h 2 ) d h 1 d h 2⎦

ξ ∈G L(n,Q)

G L(n,Q)\G 1 G L(n,Q)\G 1

t In the above we changed g → ξ τ h 1 , with τ =

· t ns+

 ..

n(n−1) 2

d × t.

(t > 0), ξ ∈

. t ×

G L(n, Q), and h 1 ∈ G L(n, Q)\G 1 . Then d × g = d × h 1 d t with d × t = See Lemma 11.3.7.

dt . t

As in Riemann’s proof of the functional equation of ζ (s), we break the 1  +∞ integral over R+ above into the sum of two integrals 0 and 1 . It follows that Z (s, , β) is equal to ⎡ ⎤ 1    ⎢ × × ⎥ h −1 ⎣ 2 ξ τ h 1 f 1 (h 1 ) f 2 (h 2 ) d h 1 d h 2⎦ 0

G L(n,Q)\G 1 G L(n,Q)\G 1

+∞

+ 1

ξ ∈G L(n,Q)

· t ns+

⎡ 

⎢ ⎣ G L(n,Q)\G 1 G L(n,Q)\G 1

n(n−1) 2

d ×t ⎤

  × × ⎥ h −1 2 ξ τ h 1 f 1 (h 1 ) f 2 (h 2 ) d h 1 d h 2⎦

ξ ∈G L(n,Q)

· t ns+

n(n−1) 2

d × t.

(15.2.5)

The second integral above converges for all s ∈ C. In the first integral above, we make the transformation t → t −1 and then use the general Poisson summation formula of Proposition 15.1.5. It follows that ⎡ ⎤ 1    ⎢ × × ⎥ h −1 ⎣ 2 ξ τ h 1 f 1 (h 1 ) f 2 (h 2 ) d h 1 d h 2 ⎦ 0

G L(n,Q)\G 1 G L(n,Q)\G 1

ξ ∈G L(n,Q)

· t ns+ = I1 + I2 + I3 − I4 − I5 ,

n(n−1) 2

d ×t

(15.2.6)

where the integrals I1 , I2 , I3 , I4 , I5 are defined as ⎡ ⎤ +∞    ⎢ × × ⎥ . h −1 I1 = ⎣ 1 ξ τ h 2 f 1 (h 1 ) f 2 (h 2 ) d h 1 d h 2⎦ 1

G L(n,Q)\G 1 G L(n,Q)\G 1

ξ ∈G L(n,Q)

· tn

2

− n(n−1) −ns × 2

d t,

The Godement-Jacquet L-function for GL(n, AQ )

122 ⎡ +∞



⎢ ⎢ ⎣

I2 = 1

G L(n,Q)\G 1 G L(n,Q)\G 1

ξ ∈Mat(n,Q) 1≤rank(ξ )
  . h −1 1 ξ τ h2 ⎤

⎥ n 2 − n(n−1) −ns × 2 · f 1 (h 1 ) f 2 (h 2 ) d × h 1 d × h 2 ⎥ d t, ⎦t

+∞

I3 =

⎡

⎤

⎢ ⎣

. ((0)) f 1 (h 1 ) f 2 (h 2 ) d × h 1 d × h 2 ⎥ ⎦

1

G L(n,Q)\G 1 G L(n,Q)\G 1

· tn

⎡ +∞

I4 = 1



⎢ ⎢ ⎣ G L(n,Q)\G 1 G L(n,Q)\G 1

2

− n(n−1) −ns × 2

d t,

  −1 h −1 h1 2 ξτ

ξ ∈Mat(n,Q) 1≤rank(ξ )
⎤

⎥ −ns− n(n−1) × 2 · f 1 (h 1 ) f 2 (h 2 ) d × h 1 d × h 2 ⎥ d t, ⎦t +∞

I5 = 1

⎡

⎤

⎢ ⎣

n(n−1) ⎥ ((0)) f 1 (h 1 ) f 2 (h 2 ) d × h 1 d × h 2⎦ t −ns− 2 d × t.

G L(n,Q)\G 1 G L(n,Q)\G 1

We now show that the integrals I2 , I3 , I4 , I5 above are all equal to zero. In . ((0)), vanish because the fact, the integrals I3 , I5 , containing ((0)) and function f 1 is orthogonal to the constant function on G 1 . To deal with the remaining two integrals, fix τ, h 2 , and an integer 1 ≤ r < n, and consider, for example,    × . h −1 (15.2.7) 1 ξ τ h 2 f 1 (h 1 ) d h 1 . G L(n,Q)\G 1

ξ ∈Mat(n,Q) rank(ξ )=r

We can split {ξ ∈ Mat(n, Q) | rank(ξ ) = r } up into orbits for the action of G L(n, Q) by matrix multiplication γ · ξ (γ ∈ G L(n, Q p ), ξ ∈ Mat(n, Q), rank(ξ ) = r ). By an analogue of (11.2.4), each orbit contains a representative of the form  Ir 0 γ2 , (γ2 ∈ G L(n, Q)). (15.2.8) 0 0

15.2 The global zeta integral for GL(n, AQ )

123

Fix such a representative for each orbit. Define the groups  P=

Ir 0

∗ ∗



 ,

U=



∗

Ir 0

In−r

.

Then the stabilizer of each of the representatives (15.2.8) is precisely P(Q). Writing  I 0 γ2 , ξ = γ1−1 r 0 0 we can replace the sum over ξ in (11.4.5) by a sum over γ1 ∈ P(Q)\G L(n, Q) and a sum over the orbit representatives (15.2.8). For each fixed representative, we obtain    Ir 0 −1 −1 . h 1 γ1 γ2 τ h 2 f 1 (h 1 ) d × h 1 0 0 G L(n,Q)\G 1

γ1 ∈P(Q)\G L(n,Q)

=

  1 −1 . h1 0

0 0



γ2 τ h 2

f 1 (h 1 ) d × h 1 . (15.2.9)

P(Q)\G 1

    Ir 0 . h −1 γ does not change if we replace h 1 τ h Note that the function 2 2 1 0 0 by p · h 1 with p ∈ P(AQ ). Since    =  P(Q)\G 1 = U (Q)\U (AQ ) · U (AQ ) · P(Q) G 1 , it follows from (15.2.9) that  G L(n,Q)\G 1

γ1 ∈P(Q)\G L(n,Q)

  Ir −1 −1 . h 1 γ1 0

  Ir −1 . h1 0

= (U (AQ )·P(Q))\G 1

0 0

0 0



γ2 τ h 2

f 1 (h 1 ) d × h 1



γ2 τ h 2

f 1 (uh 1 ) du d × h 1

U (Q)\U (AQ )

= 0. The integral

 U (Q)\U (AQ )

f 1 (uh 1 ) du is the constant term of f along the para-

bolic P(r,n−r ) , defined as in Definition 13.4.14. Because f 1 is a cusp form, this integral is zero. This concludes the proof that the last four integrals in (15.2.6) vanish. It follows that (15.2.6) reduces to the following simpler identity:

The Godement-Jacquet L-function for GL(n, AQ )

124 Z (s, , β) = ⎡ 1

= 0

1



⎢ ⎣ G L(n,Q)\G 1 G L(n,Q)\G 1

+∞

=

⎤   × × ⎥ h −1 2 ξ τ h 1 f 1 (h 1 ) f 2 (h 2 ) d h 1 d h 2 ⎦

ξ ∈G L(n,Q)

· t ns+

⎡ 

⎢ ⎣ G L(n,Q)\G 1 G L(n,Q)\G 1

n(n−1) 2

d ×t ⎤

  × × ⎥ . h −1 1 ξ τ h 2 f 1 (h 1 ) f 2 (h 2 ) d h 1 d h 2 ⎦

ξ ∈G L(n,Q)

· tn

2

− n(n−1) −ns × 2

d t.

The above expression is absolutely convergent for all s ∈ C and is invariant under the transformation s → 1 − s,

∨

. →

This completes the proof of Theorem 15.2.4.

β → β. 

15.3 Factorization of the global zeta integral for G L(n, AQ ) Let β be a matrix coefficient, as in Definition 15.2.2, of an irreducible cuspidal automorphic representation π, and let be a Bruhat-Schwartz function as in Definition 15.1.2. Then for s ∈ C, with (s) sufficiently large, we may attach a global zeta integral Z (s, , β) associated to π (as in Definition 15.2.3), where (g) β(g) |det(g)|s+

Z (s, , β) =

n−1 2

d × g.

(15.3.1)

G L(n,AQ )

As in the G L(2) case, given in Proposition 11.5.2, it will turn out that irreducible smooth representations of G L(n, Q p ) possess factorizable matrix coefficients. Given such a matrix coefficient β, if the Bruhat-Schwartz function is also factorizable, then the global zeta integral Z (s, , β) can be factored as a product of local zeta integrals, i.e., the global zeta integral (15.3.1) is Eulerian. We present this as a formal proposition. Proposition 15.3.2 (Factorization of the global zeta integral) Fix an integer n ≥ 1. Let (π, V ) be an irreducible cuspidal automorphic representation of G L(n, AQ ) as in Definition 13.6.11, and let β be a matrix coefficient of π as in Definition 15.2.2, defined using two cusp forms φ1 , φ2 ∈ V, which are mapped & πv given in Theorem 13.7.3. to pure tensors under the isomorphism π ∼ = v≤∞ " Then there exist matrix coefficients βv of πv (v ≤ ∞) so that β = βv . v≤∞

15.4 The local functional equation for GL(n, Qp )

125

Further, if : Mat(n, AQ ) → C is a factorizable Bruhat-Schwartz function, i.e., can be represented in the form (m) =

!

v (m v ),

(m = (m v )v≤∞ ∈ Mat(n, AQ )),

v≤∞

then for s ∈ C with (s) sufficiently large, we have the factorization Z (s, , β) =

!

Z v (s, v , βv )

v≤∞

where s+ n−1 2

Z v (s, v , βv ) :=

v (g) βv (g) |det(g)|v

d × g.

G L(n,Qv )

Proof The proof is similar to that of Proposition 11.5.2.



15.4 The local functional equation for G L(n, Q p ) Fix an integer n ≥ 1 and a rational prime p and let (π, V ) be an admissible irreducible representation of G L(n, Q p ) as in Definition 14.1.2. We denote by  π the representation contragredient to π and  V the space on which it acts as in Definition 14.1.10. Let  ,  denote the canonical invariant bilinear form on V ×  V and for a pair of fixed vectors v ∈ V,  v ∈  V , consider a matrix coefficient β defined by β(g) := π (g) . v,  v,



 g ∈ G L(n, Q p ) .

(15.4.1)

The matrix coefficient (15.4.1) will be the first ingredient in a local zeta integral, analogous to the global zeta integral of (15.3.1). The second ingredient is a local analogue of the Bruhat-Schwartz function for Mat(n, AQ ), defined as in Definition 15.1.2. Definition 15.4.2 (The Bruhat-Schwartz space associated to Mat(n, Q p )) Fix an integer n ≥ 1. Let Mat(n, Q p ) denote the ring of all n × n matrices with coefficients in Q p as in Definition 15.1.1. A function : Mat(n, Q p ) → C is said to be Bruhat-Schwartz if it is locally constant and compactly supported. The vector space of all such Bruhat-Schwartz functions : Mat(n, Q p ) → C is denoted S(Mat(n, Q p )). Let : Mat(n, Q p ) → C be a Bruhat-Schwartz function, i.e., it is locally constant and compactly supported.

126

The Godement-Jacquet L-function for GL(n, AQ )

For s ∈ C with (s) sufficiently large, the local zeta integral s+ n−1 2

Z p (s, , β) =

(g) β(g) |det(g)| p

d×g

(15.4.3)

G L(n,Q p )

was introduced in Proposition 15.3.2. We would like to show that Z p (s, , β) has a meromorphic continuation to all complex s and also satisfies a functional equation. The following theorem, which generalizes part of Tate’s thesis (see Theorem 2.3.4), is the basis of the theory of local L-functions for G L(n, Q p ). Theorem 15.4.4 (Local functional equation for G L(n, Q p )) Fix an integer n ≥ 1 and a prime p and let (π, V ) be an admissible irreducible representation of G L(n, Q p ) as in Definition 14.1.2. Let β : G L(n, Q p ) → C be a matrix coefficient as in (15.4.1) and let : Mat(n, Q p ) → C be a locally constant and compactly supported (Bruhat-Schwartz function). Consider, Z p (s, , β), the local zeta integral defined in (15.4.3). Then we have the following. (1) There exists s0 ∈ R such that the integral (15.4.3) converges absolutely for (s) > s0 . (2) For each Bruhat-Schwartz function , and each matrix coefficient β, the integral (15.4.3) represents a rational function of p −s . The set of such rational functions obtained admits a common divisor, L p (s, π ), which is characterized (up to scalar multiple) by the property that the ratio Z p (s, , β) L p (s, π ) is an entire function of s which is identically 1 for suitable choice of , β. Further L p (s, π ) = Q( p −s )−1 for some polynomial Q satisfying Q(0) = 1. (3) There exists a rational function of p −s (which does not depend on or β), denoted γ (s, π ), such that the local zeta integral (15.4.3) satisfies the functional equation ∨

., β ) = γ (s, π ) Z p (s, , β). Z p (1 − s, ∨

. is the Fourier transform as defined in Theorem 15.1.3, and β Here is defined by ∨

β (g) := β(g −1 ) for all g ∈ G L(n, Q p ). Further, there exists a local root number p (s, π ) (which is a rational function of p −s ) such that we have the functional equation

15.4 The local functional equation for GL(n, Qp )

127

∨

., β ) Z p (1 − s, Z p (s, , β) = p (s, π ) . L p (1 − s,  π) L p (s, π ) Proof The theorem will be proved over the course of the next four sections for irreducible generic unitary representations, using an inductive argument. In the next section, we prove Theorem 15.4.4 for irreducible supercuspidal representations. The next step is to consider parabolically induced representations. This is done in two stages: first we will show how to pass from two representations of smaller general linear groups to their tensor product, and then we will show how to pass from the tensor product to the parabolically induced representation. After that, we will prove Theorem 15.4.4 for discrete series representations. It will then immediately follow that Theorem 15.4.4 holds for representations which are parabolically induced from discrete series representations, which, by Theorem 14.11.4, includes all irreducible generic unitary representations of G L(n, Q p ).  Remark It is worth noting that Theorem 15.4.4 holds, and will be proved, for certain representations which are not irreducible. Specifically, if a representation is parabolically induced from irreducibles, then Theorem 15.4.4 will be true, even if the representation is reducible. Proposition 15.4.5 (The zeta integral of a subquotient of a reducible representation of G L(n, Q p )) Fix an integer n ≥ 1 and a prime p. Let (π, V ) be a reducible smooth representation of G L(n, Q p ), such that Theorem 15.4.4 holds for (π, V ). If (π  , V  ) is any subquotient of (π, V ), as in Definition 14.1.5, then Theorem 15.4.4(1) and Theorem 15.4.4(3) hold for (π  , V  ). Further, (1) the domain of absolute convergence as in Theorem 15.4.4(1) for π  contains the domain of absolute convergence for π. (2) For each Bruhat-Schwartz function , and each matrix coefficient β, the integral (15.4.3) represents a rational function of p −s , and the function L p (s, π )−1 · Z p (s, , β) is entire. (3) γ (s, π  ) = γ (s, π ), where γ (s, ∗) is defined as in Theorem 15.4.4(3), Proof This follows immediately from the fact that every matrix coefficient of π  is a matrix coefficient of π.  Proposition 15.4.6 (Behavior of zeta integrals and L-functions under unramified twists) Fix an integer n ≥ 1 and a prime p. Let (π, V ) be a smooth representation of G L(n, Q p ), such that Theorem 15.4.4 holds for (π, V ). Then for ν ∈ C, the twist | |νp · π of π by the unramified character t → |t|νp as defined in Definition 14.5.2, also satisfies Theorem 15.4.4.

128

The Godement-Jacquet L-function for GL(n, AQ )

Further, the domain of absolute convergence for | |νp · π is equal to that of π shifted by (ν), and we have     L p s, | |νp · π = L p (s + ν, π ), γ s, | |νp · π = γ (s + ν, π ),    p s, | |νp · π =  p (s + ν, π ). Proof This follows easily from the fact that the matrix coefficients of | |νp · π are precisely the functions g → | det g|νp β(g), where β is a matrix coefficient of π. 

15.5 The L-function and local functional equation for the supercuspidal representations of G L(n, Q p ) Theorem 15.5.1 (The L-function of a supercuspidal representation of G L(n, Q p )) Fix an integer n ≥ 2 and a prime p. Let (π, V ) denote an irreducible supercuspidal representation of G L(n, Q p ). Then (i) The local zeta integral Z p (s, , β) is entire for all matrix coefficients β of π and all Bruhat-Schwartz functions : Mat(n, Q p ) → C. (ii) There exist a matrix coefficient β of π and a Bruhat-Schwartz function : Mat(n, Q p ) → C such that the local zeta integral Z p (s, , β) is identically equal to 1. (iii) The local L function L p (s, π ) = 1. (iv) The local functional equation given in Theorem 15.4.4 holds. Sketch of Proof The proof of part (i) is essentially the same as that of Proposition 11.13.2, while part (ii) can be proved in a similar fashion as Theorem 11.8.3. Together, (i) and (ii) imply (iii) according to the definition of L p (s, π ) in Theorem 15.4.4. The proof of (iv) follows from generalizing Proposition 11.13.5 and its proof.  Remark When giving the classification results for representations of G L(n, Q p ), in Chapter 14, we observed the convention that characters of Q×p = G L(1, Q p ) are “supercuspidal.” It is important to note that Theorem 15.5.1 does not apply to characters of G L(1, Q p ).

15.6 The local functional equation for tensor products Proposition 15.6.1 (Local functional equation for tensor products) Fix an integer n ≥ 2 and a prime p. Let 1 ≤ m < n be an integer. Take two smooth irreducible representations (π  , V  ) of G L(m, Q p ) and (π  , V  ) of G L(n − m, Q p ). Below we take A1 ∈ G L(m, Q p ), A2 ∈ G L(n − m, Q p ). v∈ V ⊗  V  , define the associated matrix coefficients For v ∈ V  ⊗ V  ,  v, β(A1 , A2 ) := π  ⊗ π  (A1 , A2 ) . v, 

  ∨ −1 . β (A1 , A2 ) := β A−1 1 , A2

15.6 The local functional equation for tensor products

129

For  : Mat(m, Q p )×Mat(n−m, Q p ) → C, a Bruhat-Schwartz (i.e., locally constant and compactly supported) function, define the Fourier transform   .(A1 , A2 ) = (A, D)e p − Tr(A A1 )− Tr(D A2 ) d D d A.  Mat(m,Q p ) Mat(n−m,Q p )

In addition, define the local zeta integral s+ m−1 2

Z p (s, , β) :=

(A1 , A2 )β(A1 , A2 )| det A1 | p G L(m,Q p ) G L(n−m,Q p ) s+ n−m−1 2

· | det A2 | p

d × A2 d × A1 .

Then the integral defining Z p (s, , β) converges absolutely for (s) sufficiently large, represents a rational function of p −s , and satisfies the functional equation  ∨ ., β , Z p (s, , β) = γ (s, π  )γ (s, π  )Z p 1 − s, 

(15.6.2)

where γ (s, π  ) and γ (s, π  ) are defined as in Theorem 15.4.4(3). v∈ V ⊗  V  as finite sums of pure tensors, Proof Write v ∈ V  ⊗ V  and  

v=

r 



vi ⊗ vi ,

 v=

i=1

r 

 v j ⊗  v j ,

j=1

and also, write  as a finite sum of functions (A1 , A2 ) =

r 

 (A1 ) ·  (A2 ),

=1

where  : Mat(m, Q p ) → C,

 : Mat(n − m, Q p ) → C,

are Bruhat-Schwartz functions. It is possible to express  in this manner because every compact open subset of Mat(m, Q p ) × Mat(n − m, Q p ) can be written as a finite disjoint union of products of open compact sets of the form U × V with U ∈ Mat(m, Q p ) and V ∈ Mat(n − m, Q p ). Clearly, r  . (A1 ) · . (A2 ). .(A1 , A2 ) =  

=1 

For i = 1, . . . , r and j = 1, . . . , r , define   βi, j (A1 ) := π  (A1 )vi ,  v j , A1 ∈ G L(m, Q p ) ,   βi, j (A2 ) := π  (A2 )vi ,  v j , A2 ∈ G L(n − m, Q p ) .

130

The Godement-Jacquet L-function for GL(n, AQ )

Then



β(A1 , A2 ) =



r r  

βi, j (A1 ) · βi, j (A2 ),

i=1 j=1

and



∨

β (A1 , A2 ) =



r r  

∨

∨

β i, j (A1 ) · β i, j (A2 ),

i=1 j=1

because of the fact that v  ⊗v  ,  v  ⊗ v   = v  ,  v  ·v  ,  v  ,

(∀ v  ∈ V  , v  ∈ V  ,  v ∈  V ,  v  ∈  V  ).

Consequently, by Theorem 15.4.4(3) 

Z p (s, , β) =



r  r r  

Z p (s,  , βi, j ) · Z p (s,  , βi, j )

i=1 j=1 =1 

=



r  r r  

. , β i, j ) · Z p (1 − s, . , β i, j ) Z p (1 − s, ∨

∨

i=1 j=1 =1 ∨

., β ).  = Z p (1 − s, 

15.7 The local zeta integral for a parabolically induced representation of G L(n, Q p ) According to the Bernstein-Zelevinsky classification, given in Theorem 14.5.7, each smooth irreducible representation of G L(n, Q p ) is isomorphic to a subrepresentation of a representation which is parabolically induced, as in Definition 14.3.8, from supercuspidal representations of smaller general linear groups. In this section, we will study the local zeta integral and L-function of a smooth representation (π, V ) of G L(n, Q p ) which can be obtained by parabolic induction from two representations (π  , V  ) of G L(m, Q p ) and (π  , V  ) of G L(n − m, Q p ). We express local zeta integrals for (π, V ) in terms of local zeta integrals for (π  , V  ) and (π  , V  ), and deduce that the local L-function L p (s, π ) is given by the product L p (s, π  ) · L p (s, π  ). Theorem 15.7.1 (The L-function of a representation of G L(n, Q p ) which is parabolically induced from two smaller general linear groups) Fix an integer n ≥ 2 and a prime p. Let 1 ≤ m < n be an integer. Take admissible representations (π  , V  ) of G L(m, Q p ) and (π  , V  ) of G L(n − m, Q p ), and let G L(n,Q p ) (π  ⊗ π  ). π = Ind P(m,n−m) (Q p) Here P(m,n−m) (Q p ) is the standard parabolic subgroup



A1 0

15.7 The local zeta integral for induced representations 131



∗ ∈ G L(n, Q p )

A1 ∈ G L(m, Q p ), A2 ∈ G L(n − m, Q p ) , A2 G L(n,Q )

p and Ind P(m,n−m) (Q denotes normalized parabolic induction as in Definition p) 14.3.8. Assume that Theorem 15.4.4 holds for π  and π  . Then Theorem 15.4.4 also holds for π with

L p (s, π ) = L p (s, π  ) · L p (s, π  ),

 p (s, π ) =  p (s, π  ) ·  p (s, π  ),

where L p (s, ∗) and  p (s, ∗) are defined in Theorem 15.4.4. Proof of Theorem 15.7.1 The proof we will present is a generalization of the proof of the local functional equation for principal series representations, given in Section 11.11. The proof will be given in 6 steps. Step 1, Initial computation of the zeta integral: According to Proposition 14.3.14, a matrix coefficient for (π, V ) has the form >

β(g) =

? f (kg),  f (k) d × k,

G L(n,Z p )

where f : G L(n, Q p ) → V  ⊗ V  satisfies  n−m −m A1 X · g = |det A1 | p 2 |det A2 | p2 (π  ⊗ π  )(A1 , A2 ) . f (g), f 0 A2 (15.7.2) for all A1 ∈ G L(m, Q p ), A2 ∈ G L(n − m, Q p ), g ∈ G L(n, Q p ), and all matrices X ∈ Mat(m × (n − m), Q p ), where Mat(m × (n − m), Q p ) denotes the set of all m × (n − m) matrices with coefficients in Q p . Set K = G L(n, Z p ), and consider the integral   A1 X k k −1 0 A2 G L(m,Q p ) G L(n−m,Q p ) Mat(m×(n−m), Q p ) K K

3 2 s+ m−1 s+ n−m−1 f (k) |det A1 | p 2 |det A2 | p 2 · (π  ⊗ π  )(A1 , A2 ) . f (k  ),  · d × A1 d × A2 d X d × k  d × k.

(15.7.3)

For now we shall assume that (15.7.3) converges absolutely and uniformly for (s) sufficiently large, and show that it is equal to Z p (s, , β). In order to do so, it is necessary to replace the integral over G L(n, Q p ) which appears in the definition of Z p (s, , β) by a multiple integral of the type that was introduced in Proposition 6.9.6. Accordingly, we present the following proposition.

132

The Godement-Jacquet L-function for GL(n, AQ )

Proposition 15.7.4 (Factorization of Haar measure on G L(n, Q p )) Fix a prime p and integers n ≥ 2 and 1 ≤ m < n. Let F : G L(n, Q p ) → C be an integrable function. Then F(g) d × g = G L(n,Z p ) G L(m,Q p ) G L(n−m,Q p ) Mat(m×(n−m), Q p )

G L(n,Q p )



· |det

A1 |m−n p

F

A1 0

X A2

k d X d × A2 d × A1 d × k,

where d X is the additive Haar measure on Mat(m × (n − m), Q p ) and d × A1 , d × A2 , d × k are multiplicative Haar measures. Proof The generalization of the proof of Proposition 6.9.6 is given in Exercise 14.7.  It follows from Proposition 15.7.4 that the integral (15.7.3) is equal to 3

s+ n−1 2 f (k) d × g d × k. (k −1 g) det k −1 g p 2 f (g), 

(15.7.5)

G L(n,Z p ) G L(n,Q p )

To show that (15.7.5) is equal to Z p (s, , β) one has only to make the change of variables g → kg and then reverse the order of integration. This is justified, under the assumption of absolute convergence. Step 2, Expressing the zeta integral as a sum of products: In this step we will show that (15.7.3) is actually a sum of products of zeta integrals associated to the representations π  , π  of G L(m, Q p ) and G L(n − m, Q p ), respectively. The absolute and uniform convergence of (15.7.3) for (s) sufficiently large will then follow, because we have assumed that Theorem 15.4.4 holds for π  and π  , and this will also complete the proof that (15.7.3) is equal to Z p (s, , β) in the domain of convergence. More explicitly, we will show that there exists an integer  ≥ 1 and Bruhat-Schwartz functions i : Mat(m, Q p ) → C,

i : Mat(n − m, Q p ) → C,

(1 ≤ i ≤ ),

and matrix coefficients βi : G L(m, Q p ) → C,

βi : G L(n − m, Q p ) → C,

(1 ≤ i ≤ ),

of π  , π  , respectively such that Z p (s, , β) =

  i=1

Z p (s, i , βi ) · Z p (s, i , βi ).

(15.7.6)

15.7 The local zeta integral for induced representations

133

For each fixed k, k  ∈ G L(n, Z p ) we have f (k  ) ∈ V  ⊗ V  ,

 f (k) ∈  V ⊗  V  .

It follows from smoothness, that as k, k  vary over the compact set G L(n, Z p ) f (k) take only finitely many values. Each is a finite sum of pure that f (k  ),  tensors. For each fixed k, k  ∈ G L(n, Z p ), the Bruhat-Schwartz function    A1 X A1 X k k −1 := k k 0 A2 0 A2 is a sum of characteristic functions of sets of the form

  A1 X

A ∈ C , A ∈ C , X ∈ C 1 2 2 3 , 0 A2 1 where C1 , C2 and C3 are compact subsets of Mat(m, Q p ), Mat(n − m, Q p ), and Mat(m × (n − m), Q p ), respectively. Equation (15.7.6) follows from these remarks and (15.7.3). Step 3, Finding a common divisor of all the zeta integrals: Following Theorem 15.4.4(2), we must show that the functions Z p (s, , β) (as and β vary) admit a greatest common divisor which will define L p (s,  π−s). More precisely, we need to show that there exists a polynomial Q π p , with constant term 1, such that   Q π p −s · Z p (s, , β) is entire for all , β and equal to one for suitable , β. In this case  −1 L p (s, π ) := Q π p −s .   The proof of the existence of a greatest common divisor Q π p −s is in some sense inductive. We have already proved this for G L(1, Q p ) and G L(2, Q p ) in Section 2.3 and in Theorem 11.6.5(2), respectively. We have also proved it for supercuspidal representations of G L(n, Q p ) (for all n = 1, 2, . . . ) in Section 15.5. In the present context, we have assumed that Theorem 15.4.4 is valid for the representations π and π  . Thus, we know that there exists a polynomial Q π  ( p −s ) such that Q π  ( p −s ) · Z p (s,  , β  ) is entire for all Bruhat-Schwartz functions  : Mat(m, Q p ) → C and all matrix coefficients β  of π  , and equal to one for suitable,  , β  . Likewise, we know that there exists a polynomial Q π  ( p −s ) such that Q π  ( p −s ) · Z p (s,  , β  )

134

The Godement-Jacquet L-function for GL(n, AQ )

is entire for all Bruhat-Schwartz functions  : Mat(n − m, Q p ) → C and all matrix coefficients β  of π  , and equal to one for suitable,  , β  . It follows immediately from (15.7.6) that Q π  ( p −s )Q π  ( p −s )Z p (s, , β) is entire for all Bruhat-Schwartz functions : Mat(n, Q p ) → C and all matrix coefficients β of π. To complete the proof of Theorem 15.4.4(2), we need to show that Q π  ( p −s )Q π  ( p −s )Z p (s, , β) is equal to one for suitable , β. Step 4, The common divisor from Step 2 is the greatest common divisor: In this step, we show that a Bruhat-Schwartz function and a matrix coefficient β of π can be chosen, so that Q π  ( p −s )Q π  ( p −s )Z p (s, , β) ≡ 1.

(15.7.7)

Because Theorem 15.4.4 holds for π  and π, there exist Bruhat-Schwartz functions  : Mat(m, Q p ) → C and  : Mat(n − m, Q p ) → C, and matrix coefficients β of π  and β  of π  such that Q π  ( p −s )Z p (s,  , β  ) ≡ Q π  ( p −s )Z p (s,  , β  ) ≡ 1. Hence, to prove (15.7.7), it suffices to construct and β such that Z p (s, , β) = Z p (s,  , β  ) · Z p (s,  , β  ).

(15.7.8)

The construction of and β satisfying (15.7.8) is modeled after the proof of Lemma 14.3.9. According to the definition of matrix coefficient, given in Definition v ∈  V  , v  ∈ V  , and  v  ∈  V  such 14.1.12, there exist vectors v  ∈ V  ,  that β  (g) = π  (g) . v  ,  v  , 







v , β (g) = π (g) . v , 

(g ∈ G L(m, Q p )), (g ∈ G L(n − m, Q p )).

Because π  and π  are both smooth, there is an integer j such that π  (k) . v  = v  ,

 π  (k) . v =  v,

for all k ∈ G L(m, Q p ) with k − Im ∈ Mat(m, p j · Z p )), and π  (k) . v  = v  ,

 π  (k) . v  =  v  ,

15.7 The local zeta integral for induced representations

135

for all k ∈ G L(n − m, Q p ) with k − In−m ∈ Mat(n − m, p j · Z p ). Enlarging j if necessary, we can assume that  is constant on each additive coset x + Mat(m, p j · Z p ), and  is constant on each additive coset x + Mat(n − m, p j · Z p ). These conditions ensure that  (k −1 hk  ) = (h) for all k, k  ∈ G L(m, Z p ) such that k − Im , k  − Im ∈ Mat(m, p j · Z p ), and  (k −1 hk  ) = (h) for all k, k  ∈ G L(n − m, Z p ) such that k − In−m , k  − In−m ∈ Mat(m, p j · Z p ). For each A1 ∈ G L(m, Q p ), each A2 ∈ G L(n − m, Q p ), and each X ∈ f ∈ V as in Lemma 14.3.9, by Mat(m × (n − m), Q p ), define f ∈ V and  ⎧ n−m −m ⎪ | det A1 | p 2 | det A2 | p 2 ⎪ ⎪ ⎪ ⎨ · π  (A1 ) . v  ⊗ π  (A2 ) . v  , f (g) = ⎪ ⎪ ⎪ ⎪ ⎩ 0, ⎧ n−m −m ⎪ | det A1 | p 2 | det A2 | p 2 ⎪ ⎪ ⎪ ⎨ · π  (A1 ) . v ⊗  π  (A2 ) . v  ,  f (g) = ⎪ ⎪ ⎪ ⎪ ⎩ 0,

if g =

 A1

X 0 A2



k with k ∈ G L(n, Z p ),

such that k − In ∈ Mat (n, p j · Z p ), if g is not of this form.

if g =

 A1

X 0 A2



k with k ∈ G L(n, Z p ),

such that k − In ∈ Mat (n, p j · Z p ), if g is not of this form.

For f and  f defined in this manner, (15.7.3) takes the form   A1 k −1 0

Z p (s, , β) = G L(m,Q p ) G L(n−m,Q p ) Mat(m×(n−m), Q p ) K j K j

X A2

k

3 2 3 2 s+ m−1 s+ n−m−1 v  π  (A2 ) . v  ,  v  |det A1 | p 2 |det A2 | p 2 · π  (A1 ) . v  ,  · d × A1 d × A2 d X d × k  d × k, where

$ % K j := k ∈ G L(n, Z p ) | k − In ∈ Mat(n, p j · Z p ) .

Next, choose Bruhat-Schwartz functions 3 : Mat (m × (n − m), Q p ) → C,

(15.7.9)

136

The Godement-Jacquet L-function for GL(n, AQ )

constant on each coset of Mat (m × (n − m), p j · Z p ), and 4 : Mat (n − m × m, Q p ) → C, constant on each coset of Mat ((n − m) × m, p j · Z p ). Then we may define a Bruhat-Schwartz function : Mat (n × n Q p ) → C by the formula  A1 X :=  (A1 )  (A2 ) 3 (X ) 4 (Y ). Y A2 It follows that satisfies (k −1 hk  ) = (h) for all h ∈ G L(n, Q p ) and all k, k  ∈ K j . It further follows that (15.7.9) is equal to Vol(K j ) · Vol(K j ) ·

3 (X ) d X

Mat(m×(n−m), Q p )

3 2 s+ m−1  (A1 ) π  (A1 ) . v  ,  v  |det A1 | p 2 d × A1

· G L(m,Q p )

· G L(n−m,Q p )

3 2 s+ n−m−1  (A2 ) π  (A2 ) . v  ,  v  |det A2 | p 2 d × A2 . (15.7.10)

The second integral above is the zeta integral Z p (s,  , β  ) while the third integral above is Z p (s,  , β  ). It immediately follows that (15.7.10) is a constant times L p (s, π  ) · L p (s, π  ). The constant can be made precisely equal to one by choosing 3 in a suitable manner. Step 5, Proof of two lemmas: Lemma 15.7.11 Take k, k  ∈ G L(n, Z p ), : Mat(n, Q p ) → C a Bruhat Schwartz function. Define k k (X ) := (k −1 X k). Then  k5 k

=

k  .k

.

Proof We compute:  k5 k (X )

(k −1 Y k  ) · e p (Tr (X · Y )) d X

= Mat(n,Q p )

   (Y ) · e p Tr X kY (k  )−1 d X

= Mat(n,Q p )

   (Y ) · e p Tr (k  )−1 X kY d X

= Mat(n,Q p )

  . (k  )−1 X k = =

k  .k

(X ). 

15.7 The local zeta integral for induced representations

137

Lemma 15.7.12 Let  : Mat(n, Q p ) → C be a Bruhat-Schwartz function. For A1 ∈ Mat(m, Q p ) and A2 ∈ Mat(n − m, Q p ), define  A1 X T (A1 , A2 ) := d X.  0 A2 Mat(m×(n−m),Q p )

Then . T.  (A1 , A2 ) = T (A1 , A2 ) T (A, D)e p ( Tr(A A1 ) + Tr(D A2 )) d D d A.

:= Mat(m,Q p ) Mat(n−m,Q p )

Proof Writing a matrix in Mat(n, Q p ) as  A B A ∈ Mat(m, Q p ), B ∈ Mat(m × (n − m), Q p ), , C D C ∈ Mat((n − m) × m, Q p ), D ∈ Mat(n − m, Q p ), we compute T.  (A1 , A2 ) = Mat(m×(n−m),Q p ) Mat(m,Q p ) Mat(m×(n−m),Q p ) Mat((n−m)×m,Q p )



·



A C

B D



  A e p Tr C

B D



A1 0

X A2



Mat(n−m,Q p )

· d D dC d B d A d X. Note that  A Tr C

B D



A1 0

X A2

= Tr (A A1 ) + Tr (C X ) + Tr (D A2 ) .

Fix all variables except X, C. Then the integration in C takes the Fourier transform in that variable. By Fourier inversion, the integral in X will then return the value of the original function at 0. Lemma 15.7.12 now follows readily from the definitions.  Step 6, Proof of the local functional equation for parabolically induced representations: In Step 4, we showed that the local zeta integrals Z p (s, , β) (as , β vary) admit a greatest common divisor L p (s, π ) = L p (s, π  )L p (s, π  ). This immediately establishes a functional equation for the induced representation G L(n,Q p ) π = Ind P(m,n−m) (Q (π  ⊗ π  ), p)

The Godement-Jacquet L-function for GL(n, AQ )

138

.,  → . ,  → . , β → β , but we still need to show that the maps → ∨

∨

∨

β  → β  , and β  → β  , distribute properly to make theorem 15.4.4 (3) hold. This may be deduced from the two lemmas which were proved in Step 5, together with the local functional equation for tensor products, given in Proposition 15.6.1. Translating the zeta integral (15.7.3) into the notation introduced in Step 5, Z p (s, , β) =

Tk k  (A1 , A2 ) βk,k  (A1 , A2 ) K K G L(m,Q p ) G L(n−m,Q p ) s+ m−1 2

· |det A1 | p

s+ n−m−1 2

|det A2 | p

· d × A1 d × A2 d X d × k  d × k, (15.7.13)

for (s) sufficiently large, where, for each fixed k, k  ∈ K := G L(n, Z p ), 2 3 βk,k  (A1 , A2 ) := (π  ⊗ π  )(A1 , A2 ) . f (k  ),  f (k) . The function βk,k  is a matrix coefficient of π  ⊗ π  , as in Proposition 15.6.1. ∨ ., β ), we find that for (1 − s) Repeating the same steps for Z p (1 − s, sufficiently large, ∨

∨

., β ) = Z p (1 − s,

k (A 1 , A 2 ) β k,k  (A 1 , A 2 ) Tk .

K K G L(m,Q p ) G L(n−m,Q p ) s+ m−1 2

· |det A1 | p

s+ n−m−1 2

|det A2 | p

· d × A1 d × A2 d X d × k  d × k, (15.7.14)



where for each fixed k, k ∈ K := G L(n, Z p ), 2 3 ∨ −1   β k,k  (A1 , A2 ) := (π  ⊗ π  )(A−1 1 , A2 ) . f (k ), f (k) . It follows immediately from the local functional equation for π  ⊗ π  , as in Proposition 15.6.1, that (15.7.13) has a meromorphic continuation to C, as does (15.7.14). Further, by the two lemmas in the previous section, 5 k = T Tk . k k  , and so, by Proposition 15.6.1, the meromorphic continuations of (15.7.13) and (15.7.14) coincide. This completes the proof of Theorem 15.7.1. 

15.8 The local zeta integral for discrete series (square integrable) representations of G L(n, Q p ) The Bernstein-Zelevinsky classification of the irreducible discrete series representations of G L(n, Q p ) was given in Theorem 14.6.4. In this section, we describe how to compute the L and  factors of such a representation.

15.8 The local zeta integral for discrete series representations

139

As a first step, we show that the local zeta integrals given in (15.4.3), converge absolutely for (s) > 0, for any discrete series representation. Proposition 15.8.1 (Region of absolute convergence of the local zeta integral for a discrete series representation of G L(n, Q p )) Fix an integer n ≥ 1 and a prime p. Let (π, V ) be an irreducible discrete series representation of G L(n, Q p ), as in Definition 14.6.3. Take : Mat(n, Q p ) → C a Bruhat-Schwartz function and β : G L(n, Q p ) → C a matrix coefficient of π, as in Definition 14.1.12. Then the local zeta integral, given by s+ n−1 2

Z p (s, , β) =

(g) β(g) |det(g)| p

d × g,

G L(n,Q p )

converges absolutely for (s) > 0. Proof This will follow directly from the definition of discrete series, given in Definition 14.6.3, which asserts that the matrix coefficients of the special representations are square integrable modulo the center of G L(n, Q p ). Consider first the integral over the center

%   $ Z G L(n, Q p ) = z(r ) := r · In r ∈ Q×p ∼ = Q×p , where In is the n × n identity matrix. For (s) sufficiently large, define   s+ 1 z(r )g |det(g)| p 2 |r |2s+1 ωπ (r ) d ×r,

Fs (g) := Q×p

where ωπ is the central character of π. It follows that β(z(r )g) = ωπ (r )β(g) for all r ∈ Q×p , g ∈ G L(n, Q p ). Consequently Z p (s, , β) =

Fs (g) β(g) dg,

(15.8.2)

Q×p \G L(n,Q p )

where |β(g)|2 dg < ∞, Q×p \G L(n,Q p )

so by (15.8.2) the proof of Proposition 15.8.1 will follow from the CauchySchwartz inequality if we can show that |Fs (g)|2C dg Q×p \G L(n,Q p )

converges absolutely for (s) > 0. This is established as follows.

140

The Godement-Jacquet L-function for GL(n, AQ )

Let K = G L(n, Z p ). By the Cartan decomposition (Proposition 13.2.3), we have ⎞ ⎞ ⎛ ⎛ m1 p ..  ⎟ ⎟ ⎜ ⎜ . ⎟ K⎟ ⎜ |Fs (g)|2C dg = Vol ⎜ K ⎠ ⎠ ⎝ ⎝ m n−1 p m 1 ≥m 2 ≥···≥m n−1 ≥0 Q×p \G L(n,Q p ) 1

⎛ ⎛ m1 ⎞ ⎞ 2 p



..

⎜ ⎜ ⎟ ⎟ .

Fs ⎜k1 ⎜ ⎟ k2 ⎟ dk1 dk2 . ·

⎝ ⎝ ⎠ ⎠ K K p m n−1

(15.8.3)

1 C Fix integers m 1 ≥ m 2 ≥ . . . ≥ m n = 0, and set ⎛ m1 p .. ⎜ . d := ⎜ ⎝ p m n−1

⎞ ⎟ ⎟. ⎠ 1

Then

 n−1 /

Fs (k1 · d · k2 ) = p

−

mi

(s+ n−1 2 )

i=1

ns+ n(n−1) 2

(k1 · d · (r In ) · k2 ) |r | p

ωπ (r ) d ×r,

Q×p

and for (s) > 0, the integral ns+ n(n−1) 2

(k1 · d · (r In ) · k2 ) |r | p

ωπ (r ) d ×r

Q×p

is bounded by a constant which is independent of d. Consequently, n / − m i ((s)+ n−1 2 ) |Fs (k1 · d · k2 )|C  p i=1 . To complete the proof, we must estimate Vol(K · d · K ). First, note that this volume is equal to the number of distinct left K -cosets which are contained in K ·d · K , because each such coset has volume 1. The number of distinct left K cosets is, in turn, equal to the index of K ∩d K d −1 in K , since k1 ·d·K = k2 ·d·K if and only if k1−1 k2 ∈ K ∩ d K d −1 . To estimate [K : K ∩ d K d −1 ], note that an element of K lies in d K d −1 if and only if ki j ∈ p m j −m i · Z p whenever i > j. In particular, K ∩ d K d −1 . Since K ∩ d K d −1 contains K ( p m 1 ) p := {k ∈ K | k − In ∈ Mat(n, p m 1 · Z p )},

15.8 The local zeta integral for discrete series representations

141

which is the kernel of the reduction map G L(n, Z p ) = G L(n, Z p / p m 1 · Z p ) ∼ = G L(n, Z/ p m 1 Z), we have [K : K ( p m 1 ) p ] [K ∩ d K d −1 : K ( p m 1 ) p ] #G L(n, Z/ p m 1 Z) . = #(image of K ∩ d K d −1 in G L(n, Z/ p m 1 Z))

[K : K ∩ d K d −1 ] =

Here, we have written #X for “the number of elements of the set X.” It is easy to check that #G L(n, Z/ p m 1 Z) = p (m 1 −1)·n · 2

n−1 !



 pn − pi .

(15.8.4)

i=0

First, assume for simplicity that i < j =⇒ m i > m j . Then every entry of k which is below the diagonal lies in p · Z p . Thus, when viewed modulo p · Z p , an element of K ∩ d K d −1 corresponds to an element of the minimal parabolic subgroup consisting of upper triangular matrices. It follows that the entries on the diagonal lie in Z×p , and one easily checks that an element x of Mat(n, Z/ p m 1 Z) lies in the image of K ∩ d K d −1 if and only if ⎧ m1 × m 1 −1 · ( p − 1) possibilities), if i = j, ⎪ ⎨ (Z/ p Z) , ( p m j −m i m1 m 1 −m j +m i p xi j ∈ · (Z/ p Z) ( p possibilities), if i > j, ⎪ ⎩ (Z/ p m 1 Z) ( p m 1 possibilities), i > j. So that the number of elements of the image of K ∩ d K d −1 in G L(n, Z/ p m 1 Z) is given by ! 2 p m i −m j . p n m 1 · (1 − p −1 )n · 1≤ j
If the inequalities m 1 ≥ m 2 · · · ≥ m n−1 ≥ 0 are not all strict, then the situation is similar, except when we view an element of K ∩ d K d −1 modulo p · Z p , only block upper triangular. That is, it will lie in a parabolic subgroup of G L(n, Z/ p m 1 Z) which is no longer minimal, and (15.8.4) must be used on the blocks of the Levi in order to obtain a precise formula. What is important, however, is that one can bound Vol(K d K ) by a constant times ! p m j −m i . 1≤ j
Consequently, the convergence of (15.8.3) will follow from that of n / −2 m i ((s)+ n−1  ! 2 ) i=1 · p p m j −m i . m 1 ≥m 2 ≥···≥m n−1 ≥m n :=0

1≤ j
The Godement-Jacquet L-function for GL(n, AQ )

142

Substituting νi = m i − m i+1 for i = 1, . . . , n − 1 yields  n−1 n−1 / / ∞ ∞ −2 (n−i)·ν i·(n−i)·νi ((s)+ n−1 i   2 ) i=1 · p i=1 ··· p , ν1 =0

νn−1 =0

which is easily seen to converge in (s) > 0.



In order to describe the local L-function and root number for discrete series representations, we review the classification of such representations, given in Theorem 14.6.4. Briefly, every such representation is isomorphic to the unique irreducible quotient of  1−d  3−d   d−1  G L(n,Q ) (15.8.5) Ind P(r,... ,r ) p | | p2 · π ⊗ | | p2 · π ⊗ · · · ⊗ | | p2 · π , +, * d terms

for some integers r, d ≥ 1 with r d = n, and some irreducible unitary supercuspidal representation π of G L(r, Q p ). It will turn out that the answer is slightly different in the case when r = 1 and π is a character than it is in the general case. Theorem 15.8.6 (The L-function of a discrete series representation of G L(n, Q p )) Fix an integer n ≥ 1 and a prime p. Choose a divisor d of n. Let (π, V ) be an irreducible unitary supercuspidal representation of G L(r, Q p ) with r := n/d, and (π  , V  ) the irreducible discrete series quotient of (15.8.5). Then Theorem 15.4.4 holds for (π  , V  ). Further, ( 1−d L p (s, | | p2 · π ), r = 1,  L p (s, π ) = 1, r > 1. Proof Theorem 15.4.4(1) was proved in Proposition 15.8.1. First assume r > 1. Theorems 15.5.1, 15.7.1, and Proposition 15.4.5 imply Theorem 15.4.4(3) and most of 15.4.4(2). To complete the proof of Theorem 15.4.4(2), one has only to show that Z p (s, , β) = 1 for some Bruhat-Schwartz function and matrix coefficient β of π  . This can be accomplished simply by choosing β arbitrary, and then taking to be supported on a compact set where both β and | det | are constant. In the case when r = 1, Proposition 15.4.5 implies that n   ! n+1 1 − π ( p) p −s− 2 +i Z p (s, , β) i=1

=

n+1  !

1 − π ( p)−1 p s+

n+1 2 −i



 ∨ ., β . Z p 1 − s,

i=2

(15.8.7)

15.9 The local zeta integral for GL(n, R)

143

(Keep in mind that if r = 1 then π is a character of Q×p , and π ( p) is a complex number.) Since,     n+1 n+1 n+1 1 − π ( p)−1 p s+ 2 −i = π ( p)−1 p s+ 2 −i 1 − π ( p) p −s− 2 +i , we may cancel some terms in (15.8.7), yielding   n−1 1 − π ( p) p −s− 2 Z p (s, , β)    ∨ n(n+1) n+1 ., β . = π ( p)−n+1 p s(n−1)+ 2 −1 1 − π ( p)−1 p s− 2 Z p 1 − s, (15.8.8) Now, it follows from Proposition 15.8.1 that the left hand side of (15.8.8) is holomorphic in (s) > 0, and that the right hand side is holomorphic in (s) < 1. Consequently, both sides are entire. It immediately follows that   1−d n−1 −1 , or 1. L p (s, π  ) is equal to either L p (s, | | p2 · π ) = 1 − π ( p) p −s− 2 To prove that it is not 1, we have to prove that there exist and β such that Z p (s, , β) is not entire. This may be accomplished by taking β randomly, and ∨

. to have such small support that Z p (1 − s, ., β ) is constant. then choosing , and it immeThen the right hand side of (15.8.8) will not vanish at s = − n−1 2 diately follows that the meromorphic continuation of Z p (s, , β) must have a pole at this point. 

15.9 The local zeta integral for irreducible unitary generic representations of G L(n, R) In this section we state the analogue of Theorem 15.4.4 for the group G L(n, R), and give an indication of the proof. Theorem 15.9.1 (The local L-function for G L(n, R)) Fix an integer n ≥ 1 and let (π, V ) be an irreducible generic unitary representation of G L(n, R) as in Definition 14.8.10. Let β : G L(n, R) → C be a matrix coefficient as in Definition 14.8.12, defined using vectors v, v  ∈ V which are both K ∞ -finite. Let : Mat(n, R) → C be a Schwartz function of the form −π

(g) = e

n / i, j=1

gi2j

P(g),

(15.9.2)

with P being a polynomial in the entries gi j of g. Consider the local zeta integral s+ n−1 2

Z ∞ (s, , β) =

(g)β(g)| det g|∞ G L(n,R)

d × g.

(15.9.3)

The Godement-Jacquet L-function for GL(n, AQ )

144 Then we have.

(1) There exists s0 ∈ R such that the integral (15.9.3) converges absolutely for (s) > s0 . (2) There exist complex numbers ν1 , . . . , νn such that integral (15.9.3) is equal to the product of n s + ν  ! s+νi i L ∞ (s, π ) := (15.9.4) π− 2  2 i=1 and a polynomial Q(s, , β), or each Schwartz function , of the form (15.9.2) and each matrix coefficient β, formed using K ∞ -finite vectors, and, further, such that there exist a Schwartz function and a matrix coefficient β, formed using K ∞ -finite vectors, such that Q(s, , β) = 1. (3) There exists a meromorphic function (not depending on ), denoted γ (s, π ), such that the local zeta integral satisfies the functional equation  ∨ ., β = γ (s, π ) Z ∞ (s, , β). Z ∞ 1 − s, . is the Fourier transform Here   (y) e − Tr(x · y) dy,

. (x) := Mat(n,R)

∨ ∨ " with x ∈ Mat(n, R), dy = i,n j=1 dyi j , and β is defined by β (g) := β(g −1 ) for all g ∈ G L(2, R). Further, there exists a local root number ∞ (s, π ) such that we have the functional equation

 ∨ ., β Z ∞ 1 − s, Z ∞ (s, , β) = ∞ (s, π ) . L ∞ (1 − s,  π) L ∞ (s, π ) (4) The precise values of L ∞ (s, π ), γ (s, π ), and ∞ (s, π ) can be given explicitly in terms of the classification of irreducible unitary generic representations as in Theorem 14.10.20. Indeed, if π = π1 × · · · × πr as in Theorem 14.10.20, then r r ! ! L ∞ (s, π ) = L ∞ (s, πi ), γ (s, π ) = γ (s, πi ), i=1

∞ (s, π ) =

i=1 r ! i=1

∞ (s, πi ).

15.9 The local zeta integral for GL(n, R)

145

Sketch of proof of Theorem 15.9.1 According to Theorem 14.10.20, every irreducible unitary generic representation of G L(n, R) is built up, using parabolic induction, from characters and representations of G L(2, R). Since the theorem is already proved for these “building blocks,” we just need to generalize Proposition 15.4.6, and the results of Sections 15.6 and 15.7 to the real case. In order to handle complementary series representations, it is necessary to show that Theorem 15.9.1 also holds for a nonunitary representation π, if it is the twist of a unitary representation by an unramified character. Let (π, V ) be a unitary representation of G L(n, R) and χ : R× → C an unramified character. That is χ (x) = |x|ν∞ for some complex number ν. If χ is unitary then χ · π is a unitary representation and the matrix coefficients of χ · π are simply the functions of the form β(g) = χ (det g) · β  (g),

(β  a matrix coefficient of π ).

(15.9.5)

If χ is nonunitary, then χ · π is nonunitary and we have not defined its matrix coefficients yet. We define them to be all functions of the form (15.9.5). The generalizations of Proposition 15.4.6 and Section 15.6 are left as exercises. This includes, of course, defining local zeta integrals for products π  ⊗ π  of representations of G L(n, R). We also leave convergence to the reader. In order to generalize Section 15.7, we take two representations (π  , V  ) and (π  , V  ) of G L(m, R) and G L(n − m, R), respectively, and assume that Theorem 15.9.1 holds for π  and π  . We further assume that each of these representations is a twist, as in Definition 14.10.6, of a unitary representation, say   π  = | |ν∞ ·πu and π  = | |ν∞ ·πu , with πu and πu unitary and ν  , ν  ∈ R. Recall from Definitions 14.10.5 and 14.10.6 that, technically, one should first use uni.πu of G L(m, R) × G L(n − m, R), on tarity to extend πu ⊗ πu to an action πu ⊗   ,ν )   the completion of V ⊗ V , and then twist that action by | |(ν . But it is rea∞ sonable to think of the representation thus obtained as the product of π  and π  ,   ,ν ) .πu ), · (πu ⊗ so, we shall abuse notation and write π  ⊗ π  for the twist | |(ν ∞ defined as in Definition 14.10.6. L(n,R)  We take β a K ∞ -finite matrix coefficient of π := IndGP(m,n−m) (R) π ⊗ π and as in (15.9.2), and consider the local zeta integral Z ∞ (s, , β) as in (15.9.3). Step 1: Express Z ∞ (s, , β) as a finite linear combination of zeta integrals for π  ⊗ π  . First write   A1 X k k −1 Z ∞ (s, , β) = 0 A2 G L(m,R) G L(n−m,R) Mat(m×(n−m), R) K ∞ K ∞

2 3 s+ m−1 s+ n−m−1 · (π  ⊗ π  )(A1 , A2 ) . f (k  ),  f (k) |det A1 | p 2 |det A2 | p 2 · d × A1 d × A2 d X d × k  d × k.

(15.9.6)

146

The Godement-Jacquet L-function for GL(n, AQ )

(Cf. (15.7.3) above.) Because we have required f and  f to be K ∞ -finite, and to be of the form (15.9.2), each of the following spaces of functions is finite dimensional:   k k | k, k  ∈ K ∞ , W := Span 

where k k (g) := (k −1 gk  ), % $ U := Span π (k  ) . f | k  ∈ K ∞ , % $  := Span π (k) .  U f | k ∈ K∞ .

(g ∈ G L(n, R)),

Now, take g ∈ G L(n, R) and k ∈ K ∞ and consider the integral (k −1 gk  ) f (gk  ) dk  .

I (g) := K∞

The integral which defines I (g) is certainly convergent, since the integrand is continuous and the domain is compact. And clearly, it will converge to some element of Span ({ · h |  ∈ W, h ∈ U }) . Thus I (g) is of the form I (g) =

N 

ci · (g)h(g).

i=1

Using the same reasoning to treat the integral over k in (15.9.6), we find that (15.9.6) is equal to a finite sum of integrals of the form  A1 X  Z ∞ (s, , β) = 0 A2  K K G L(m,R) G L(n−m,R) 2  A1 · h 0

Mat m×(n−m), R

X A2



∞

∞

3 s+ m−1 s+ n−m−1 ,  h(In ) |det A1 | p 2 |det A2 | p 2 · d × A1 d × A2 d X,

 . From this, it follows easily that Z ∞ (s, , β) with  ∈ W, h ∈ U, and  h∈U as a finite linear combination of zeta integrals for π  ⊗ π  . Consequently, L ∞ (s, π  )−1 · L ∞ (s, π  )−1 · Z ∞ (s, , β) is a polynomial for each , β. Step 2: Prove the functional equation. Define k k (g) := (k −1 gk  ), T (A1 , A2 ) :=



Mat m×(n−m),Q p

(g ∈ G L(n, R))  A1 X d X,  0 A2 

15.9 The local zeta integral for GL(n, R)

147

for A1 ∈ G L(m, R), A2 ∈ G L(n − m, R). Also, define 2 3 f (k) βk,k  (A1 , A2 ) := (π  ⊗ π  )(A1 , A2 ) . f (k  ),  (which we regard as a matrix coefficient of π  ⊗ π  ). We find that (15.9.6) is equal to Tk k  (A1 , A2 ) βk,k  (A1 , A2 ) K ∞ K ∞ G L(m,R) G L(n−m,R) s+ m−1 2

· |det A1 | p

s+ n−m−1 2

|det A2 | p

d × A2 d × A1 d × k  d × k.

This may be expressed as an integral of local zeta integrals for π  ⊗ π  : Z ∞ (s, Tk k  , βk,k  ) dk dk  . K∞ K∞

By similar steps, one finds that in the domain of convergence for the function ∨

., β ), one has Z ∞ (1 − s, ∨

∨

., β ) = Z ∞ (1 − s,

 Z ∞ (1 − s, Tk . k , β k  ,k ) dk dk . K∞ K∞

Lemmas 15.7.11 and 15.7.12 are valid in the real case as well, and the local functional equation for π  ⊗ π  implies that for each fixed k, k  ∈ K ∞ we have ∨

Z ∞ (s, Tk k  , βk,k  ) = γ (s, π  )γ (s, π  )Z ∞ (1 − s, Tk . k , β k  ,k ).

(15.9.7)

Finally, it may be verified that both sides of (15.9.7) depend continuously on k and k  . From this we can deduce the local functional equation, part (3), as well as the precise values for γ and ∞ given in (4). Step 3: Statement of the key technical result. To complete the proof of Theorem 15.9.1, we must explain how to choose and β so that Z ∞ (s, , β) = L ∞ (s, π  ) · L ∞ (s, π  ). This will require some preparation. We will first state the main technical result, Proposition 15.9.8 which enables us to choose and β. Then we explain how to choose and β given Proposition 15.9.8. Finally, we give an indication of the proof of Proposition 15.9.8. Proposition 15.9.8 (Main technical result) Let π and π  be representations of G L(m, R) and G L(n − m, R) which satisfy Theorem 15.9.1. Take four vectors v ∈  V  , v  ∈ V  ,  v  ∈  V  , as well as Schwartz functions v ∈ V , 

148

The Godement-Jacquet L-function for GL(n, AQ ) 

−π



(x ) = e  (x  ) = e

−π

m /

(xi j )2

i, j=0 m /

(xij )2

i, j=0

P  (x  ),

P  (x  ),

x  ∈ Mat(m, R), x  ∈ Mat(n − m, R),

   such that Z (s,  , βv , v  ) = L ∞ (s, π ), and Z (s, , βv  , v  ) = L ∞ (s, π ).        v  for g ∈ G L(m, R), and βv , (Here βv , v  (g ) = π (g ) . v , v  (g ) =     v  for g ∈ G L(n − m, R).) Define : G L(n, R) → C by π (g ), m /

−π

(x) = e

i, j=0

xi2j

P(x),

(x ∈ Mat(n, R)),



A B P := P  (A)P  (D), (15.9.9) C D     for A ∈ Mat(m, R), B ∈ Mat m × (n − m), R , C ∈ Mat (n − m) × m, R , and D ∈ Mat(n − m, R). Then there exist O(n, R)-finite functions H, H  : O(n, R) → C satisfying where

K∞

H (k) (k −1 g) dk =  −1 κ1 H 0

O(m,R) O(n−m,R)

 H O(m,R) O(n−m,R)

κ1−1 0

H  (k  ) (gk  ) dk  = (g), K∞ 0 ·k κ2−1 · (π  ⊗ π  )(A1 κ1 , A2 κ2 ) · v  ⊗ v  dκ2 dκ1 =/ 0, 0 ·k κ2−1 · ( π ⊗  π  )(A1 κ1 , A2 κ2 ) ·  v ⊗  v  dκ2 dκ1 =/ 0. (15.9.10)

Step 4: Show that Z ∞ (s, , β) = L ∞ (s, π  )L ∞ (s, π  ) for suitable and β. The Schwartz function which is “suitable” for these purposes is precisely the one already defined by (15.9.9). Further, the second and third lines L(n,R)  ⊗ π  ) and of (15.9.10) define two elements f ∈ IndGP(m,n−m) (R) (π G L(n,R)  ( π  ⊗  π  ), respectively. We take f ∈ Ind P(m,n−m) (R)

 f (kg),  f (k) d × k.

β(g) = K∞

Then by steps which are by now familiar we can express Z ∞ (s, , β) in the form (15.9.6). Then, plugging in the definitions of f and  f we obtain

15.9 The local zeta integral for GL(n, R)

149

Z ∞ (s, , β) = G L(m,R) G L(n−m,R) Mat(m×(n−m), R) K ∞ K ∞ O(m,R) O(n−m,R) O(m,R) O(n−m,R)   −1  −1  κ3 0 κ1 0 A1 X −1   k H · k H · k 0 A2 0 κ2−1 0 κ4−1



·k

2

· (π  ⊗ π  )(A1 κ1 , A2 κ2 ) . (v  ⊗ v  ), ( π ⊗  π  )(κ3 , κ4 ) . ( v ⊗  v  ) s+ m−1 2

· |det A1 | p

s+ n−m−1 2

|det A2 | p

3

d × κ4 d × κ3 d × κ2 d × κ1

· d × A1 d × A2 d X d × k  d × k. After placing the integrals  4 on the outside we make the  in κi , i = 1, 2, 3, κ 0 κ 0 changes of variable k → 03 κ4 · k, k  → 01 κ2 · k  . Then, after using the invariance property 2

3 (π  ⊗ π  )(A1 κ1 , A2 κ2 ) . (v  ⊗ v  ), ( π ⊗  π  )(κ3 , κ4 ) . ( v ⊗  v  ) 3 2 v ⊗  v  ) , = (π  ⊗ π  )(κ3−1 A1 κ1 , κ4−1 A2 κ2 ) . (v  ⊗ v  ), (

we make the changes of variable A1 → κ3 A1 κ1−1 , κ2 → κ4 A2 κ2−1 . The integrand becomes independent of κi , i = 1, 2, 3, 4, and since the Haar measure on each compact group is normalized so that the total volume is one, the integration in these variables may be erased. It then follows from the definitions, along with the first identity of (15.9.10) that the remaining integral is exactly Z ∞ (s,  , βv ,˜v ) · Z ∞ (s,  , βv ,˜v ). Step 5: Prove the main technical result. The crucial fact is known as Schur orthogonality. Before stating it, let us mention a few additional points for convenience. First, it can be shown that any continuous representation π : K → G L(V ), of a compact group K on a topological vector space V such that (k, v) → π (k) . v is continuous is unitary. Further, it can be shown that any irreducible unitary representation of a compact group is finite dimensional, and that every unitary representation of a compact group on a Hilbert space is a direct sum of irreducible invariant subspaces. We refer the reader to [Bump, 1996], 2.4, or [Knapp, 1986], Chapter I, 5 for these facts. We do not, strictly speaking, require them for our purposes, but it will allow us to state some results in a more general and less technical setting, without adding unnecessary hypotheses. Lemma 15.9.11 (Schur orthogonality) Let (ρ, W ) and (ρ  , W  ) be two irreducible unitary representations of O(n, R), and let ( , )W , ( , )W 

The Godement-Jacquet L-function for GL(n, AQ )

150

denote the corresponding invariant positive definite Hermitian forms. Then for w1 , w2 ∈ W and w1 , w2 ∈ W  ,

O(n,R)

(ρ(k) . w1 , w2 )(ρ  (k) . w1 , w2 ) d × k

(15.9.12)

is zero, unless there is an isomorphism L : W → W  , in which case it is a scalar multiple of (w1 , L(w1 )) · (L(w2 ), w2 ). Proof The proof of Lemma 15.9.11 is similar to that of (11.13.21), and is left to the reader.  Remarks When (ρ, W ) ∼ = (ρ  , W  ), the isomorphism L is unique up to scalar, and the best way to normalize it is to require that (L(w1 ), L(w2 ))W = (w1 , w2 )W  for all w1 , w2 ∈ W  . Then (15.9.12) is equal to (dim W )−1 · (w1 , L(w1 )) · (L(w2 ), w2 ). See [Knapp, 1986], Corollary 1.10(b). Proposition 15.9.13 (Projection operator associated to an irreducible representation of O(n, R) Let n ≥ 1 be an integer and let (ρ, W ) be an irreducible unitary representation of O(n, R). Fix an orthonormal basis w1 , . . . , wd of W and define Hρ (k) :=

d 

(ρ(k) . wi , wi ),

(k ∈ O(n, R)).

i=1

If (π, V ) is any continuous representation of O(n, R), define the operator Pρ : V → V,

Hρ (k) · π (k) . v dk,

Pρ (v) = dim W

(∀v ∈ V ).

O(n,R)

Then Pρ projects V onto the maximal subspace of V which is isomorphic to the direct sum of some number of copies of (ρ, W ). Proof Exercise.



Idea of proof of Proposition 15.9.8: Consider the action of O(n, R) on the space of Schwartz functions Mat(n, R) → C by ρ(k) . (x) = (xk),

(∀ x ∈ Mat(n, R), k ∈ O(n, R)).

The function generates a finite dimensional space under this action which we denote (ρ, R( )). It decomposes as a finite direct sum of irreducible unitary representations, and we choose for H  the sum of the functions Hρi , as in Proposition 15.9.13, with ρi ranging over a maximal set of nonisomorphic representations of O(n, R) appearing in the decomposition of (ρ, R( )).

Exercises for Chapter 15

151

We choose H in a similar fashion using instead the representation (λ, L( )) generated by under the action of O(n, R) by λ(k) . (x) = (k −1 x),

(∀ x ∈ Mat(n, R), k ∈ O(n, R)).

It follows from an analogue of Proposition 8.10.1 that each of these functions is K ∞ -finite. The first identity of (15.9.10) is immediate from Proposition 15.9.13.  −1  κ1 0 is To prove the second identity of (15.9.10), we note that H  −1 0 κ 2

equal to the sum of the functions Hρ  (κ1 )Hρ  (κ2 ), such that ρ  is an irreducible unitary representation of O(m, R), ρ  is an irreducible unitary representation of O(n − m, R), and the representation which is dual to ρ  ⊗ ρ  appears in the decomposition of one of the representations ρi used to define H  , when the action is restricted to O(m, R) × O(n − m, R). The nonvanishing given in the second line of (15.9.10) is then tantamount to the assertion that one such representation ρ  ⊗ ρ  appears in the representation of O(m, R) × O(n − m, R) generated by v  ⊗ v  . To prove this, we suppose that it is false and show, using Lemma 15.9.11 that Z (s,  , βv , v ) · ) vanishes identically. The proof of the third part of (15.9.10) is Z (s,  , βv ,  v similar. 

Exercises for Chapter 15 

n " n  " 15.1 Fix n ≥ 2, let x = x (i, j) ∈ Mat(n, AQ ), and define d x = d x (i, j) to i=1 j=1

be the additive Haar measure on Mat(n, AQ ). Prove that ··· AQ

AQ

(g · x · h) d x = | det g|−n · | det h|−n ·

··· AQ

(x) d x,

AQ

for all g, h ∈ G L(n, AQ ), and any function  : Mat(n, AQ ) → C such that these integrals are defined. × 15.2 Fix n ≥ 2, a prime p, and a unitary Hecke character ω : Q× \A× Q →C . Let (π, V ) be a cuspidal automorphic representation of G L(n, AQ ) with central character ω as in Definition 13.6.11. (a) Prove that (π, V ) is unitarizable. (b) Prove that the matrix coefficients of (π, V ) are bounded functions.

15.3 Prove that the global zeta integral in Definition 15.2.3 converges absolutely for (s) sufficiently large (depending on and β). Hint: Use the previous exercise.

152

The Godement-Jacquet L-function for GL(n, AQ )

15.4 Set n = 2 and m = 1 in the factorization of the Haar measure given by Proposition 15.7.4. Does it contradict Proposition 6.9.6? 15.5 Prove that the local integral (15.9.3) converges absolutely for (s) sufficiently large (depending on and β). 15.6 Generalize Proposition 15.4.6 to G L(n, R). 15.7 Generalize Proposition 15.6.1 to G L(n, R).

Solutions to Selected Exercises

12.1

1 (n 2

+ 2)(n − 1)

12.2 (d) Suppose z · d · k = z  · d  · k  with z, z  ∈ hn , d, d  in the center of G L(n, R), and k, k  ∈ O(n, R). Then z −1 z  d  d −1 = k(k  )−1 ∈ O(n, R). The matrix on the left is upper triangular, so its inverse is upper triangular. But it is also orthogonal, and hence its inverse equals its transpose, which is lower triangular. Thus z −1 z  d  d −1 is diagonal, and since it’s orthogonal, it can only have ±1 on the diagonal. Write d = δ In and d  = δ  In . The diagonal entries of z and z  are all positive, so the diagonal entries of z −1 z  d  d −1 all agree in sign with δδ  . That is, z −1 z  d  d −1 = ±In , or z  = ±δ(δ  )−1 z. The bottom right entries of z and z  are both 1, so we see ±δ(δ  )−1 = 1 and z  = z. It follows also that d = ±d  and k = ±k  . 12.3 Let z, z  ∈ hn , viewed as matrices in G L(n, R). Then the matrix g = z  z −1 has the property that g.z = z  . 12.4 See the solution to Exercise 3.6, and note that (N ) has finite index in S L(n, Z) because it is the kernel of the reduction homomorphism S L(n, Z) → S L(n, Z/N Z). 12.5 Note that S L(n, Z) is the kernel of the surjective homomorphism det : G L(n, Z) → {±1}. 12.6 Use Cramer’s rule to check that 0 (N ) is closed under inversion. Also note that 0 (N ) ⊃ (N ). 153

154

Solutions to Selected Exercises

12.7 Partition G into left cosets of 1 as G=

@

gi 1 .

i

Without loss of generality, we may assume that gi ∈ 2 whenever / ∅. Now observe that if (gi 1 )∩2 = / ∅, then (gi 1 )∩2 = (gi 1 )∩2 = gi (1 ∩2 ). Consequently, if we intersect the above partition of G with the subgroup 2 , we get @

2 = G ∩ 2 =

gi (1 ∩ 2 ).

i (gi 1 )∩2 =/ ∅

Now partition G into left cosets of 2 as G=

@ j

h j 2 =

@

@

j

i (gi 1 )∩2 =/ ∅

h j gi (1 ∩ 2 ).

Since 1 and 2 both have finite index in G, both of the above unions are over finitely many indices, and hence 1 ∩ 2 has finite index in G as well. 12.9 (a) See the solution to Exercise 5.10. (b) Write g = z · (r In ) · k with z ∈ hn , r ∈ R, and k ∈ S O(n, R). By definition, we see that   (g) = g (In ) = ρ(κ(g, In )) (g.In ) = ρ(k) (z). ρ The right side is independent of the parameter r , and hence  ≡ 0, which shows D Z . ≡ 0. DZ 12.11 (a) The first, third, and fourth conditions of the definition of a Maass form of weight ρ, level N , and character χ (Definition 12.3.17) are clearly preserved under linear combinations. By fixing the eigenvalue λ, we see also that the second condition is preserved. 12.12 (a) That X mn is closed under multiplication follows immediately from I X  the way that block matrices multiply. If x  = 0m In−m with X an  I −X  m × (n − m) matrix with real entries, then (x  )−1 = 0m In−m . (b) Observe that hn is precisely the set of upper triangular matrices in G L(n, R) with positive entries on the diagonal and   a 1 in the lower right corner. If z ∈ hn , we can write A0 DB , where A is m × m upper triangular with positive entries on the diagonal and

Solutions to Selected Exercises

155

entry a > 0 in its lower right corner, B is an m × (n − m) matrix, and D is (n −m)×(n −m) upper triangular with positive entries on the diagonal and a 1 in the lower right corner. That is, if z  = a −1 A    and z  = D, then z = az0 zB with z  ∈ hm and z  ∈ hn−m . Set X = B(z  )−1 to find    X az 0 Im . z= 0 In−m 0 z  12.13 Since we have assumed n is even, we find −In ∈ S O(n, R). Set γ = −In in the definition of automorphic function to get

χ (γ ) (z) = χ (−1) (z). ( ρ γ )(z) = ρ(−In ) (z) =  Since the image of does not lie in a proper subspace, we are finished. 13.2 For the first statement, see the solution to Exercise 8.14. For the second, if      1 0 a b 0 1 a b , = 0 c p 1 0 c 1 0 then a  c = p, which implies that one of a  or c does not lie in Z×p . 13.3 (b) Let g ∈ G L(n, k). Left multiplication by elements of Ui j (k) amounts to adding scalar multiples of the jth row of g to the ith row of g. Right multiplication is equivalent to adding scalar multiples of the ith column of g to the jth column of g. By adding some multiple of a row to the first row, we may assume that it has at least two non-zero entries in it. By adding some multiple of a column to the first column of g, we may assume that the (1, 1)entry is equal to 1. Now perform row additions to eliminate all of the other (i, 1)-entries for i > 1, and then perform column operations to eliminate all of the (1, j)-entries for j > 1. (Note that in order to do these steps we need to divide, which is where we use the fact that k is a field.) The resulting matrix has a 1 at the (1, 1)position and zeros below and to the right of it. This procedure may be continued inductively to find matrices x and y that are products of elements from Ui j (k) for various choices of i, j such that ⎛ ⎜ x·g·y =⎜ ⎝

⎞

1 1

..

⎟ ⎟. ⎠

. det(g)

156

Solutions to Selected Exercises If we now specialize to the case where g ∈ S L(n, k), then we see g = x −1 y −1 is generated by matrices of the desired form. (c) The procedure described in (b) runs in a finite number of steps depending only on n. So we may run it simultaneously for each prime on a matrix g ∈ S L(n, AQ ), which amounts to multiply" ing on the left and right by matrices in Ui, j ( v G L(n, Qv )). The final detail to check is that if g p ∈ S L(n, Z p ) (which it is for all but finitely many primes p), then the procedure above can be performed with matrices in Ui, j (Z p ). Hence g can be generated by matrices in Ui, j (AQ ).

13.5 (a) The fact that these sets are closed under multiplication is immediate from the way one multiplies block matrices. We will compute the inverse of a matrix in Pκ (R). The other cases are similar. Suppose we have a matrix B ∈ Pκ (R) given by ⎛B B ... B ⎞ 1,1

⎜ 0 B=⎜ ⎝ 0 0

1,2

B2,2 0 0

... .. . 0

1,r

B2,r ⎟ .. ⎟ ⎠. . Br,r

We want to solve for a matrix B  ∈ Pκ (R) such that ⎛B B1,2 . . . B1,r ⎞ 1,1 B2,2 . . . B2,r ⎟ ⎜ 0 B B = ⎜ .. ⎟ .. ⎠ ⎝ . 0 0 . 0  B1,1 ⎜ 0 ⎜ ·⎜ ⎝ 0 0 ⎛

0  B1,2  B2,2 0 0

0 Br,r  ⎞ . . . B1,r  ⎟ . . . B2,r ⎟ = In . .. ⎟ .. . . ⎠  0 Br,r

Multiplying this out shows that we want to solve the equations j  k=i

 Bi,k Bk, j =



1 if i = j 0 if i < j

.

We fix j and perform a descending induction on i < j. The base case is B j, j = B −1 j, j . Note that B j, j is invertible by definition of Pκ (R). Now assume i < j and that we have been able to solve   for Bi+1, j , . . . , B j, j . To satisfy the above equation, we set  j   −1   Bi,k Bk, j . Bi, j = −Bi,i k=i+1

Solutions to Selected Exercises

157

Continuing in this way, we are able to construct the desired inverse B  . (b) If B ∈ Pκ (R), B  = B −1 as in the last part, and C ∈ Uκ (R), then one verifies immediately that BC B  ∈ Uκ (R). Indeed, these are all upper triangular block matrices, and the jth block on the diagonal is given by B j, j Iκ j B −1 j, j = Iκ j . (c) Let R = R for simplicity, although this example works for much more general rings. We also assume that r ≥ 2, or equivalently that κ = / (n). Suppose A ∈ Mκ (R) is given by ⎛A

0 A2,2

1,1

⎜ 0 A=⎜ ⎝ 0 0

0 0

⎞

... ... .. .

0 0 .. .

0

Ar,r

⎟ ⎟. ⎠

Suppose further that B ∈ Pκ (R) has the special form ⎛I

0 Iκ2

κ1

⎜ 0 B=⎜ ⎝ 0 0

0 0

... ... .. .

X 0 .. .

0

Iκr

⎞ ⎟ ⎟, ⎠

where X ∈ Mat(κ1 × κr , R). Then B −1 is given by ⎛I

0 Iκ2

κ1

⎜ 0 B −1 = ⎜ ⎝ 0 0

0 0

... ... .. . 0

−X ⎞ 0 ⎟ .. ⎟ ⎠, . Iκr

and we have ⎛A

1,1

⎜ 0 B AB −1 = ⎜ ⎝ 0 0

0 A2,2 0 0

... ... .. . 0

X Ar,r − A1,1 X ⎞ 0 ⎟ ⎟. .. ⎠ . Ar,r

For this product to lie in Mκ (R), it must be true that X Ar,r = A1,1 X . So in order for Mκ (R) to be normal in Pκ (R), it is necessary that g X = X h for all matrices g ∈ G L(κ1 , R), h ∈ G L(κr , R), and X ∈ Mat(κ1 × κr , R). Take X to be any nonzero matrix, g = Iκ1 , and h = 2Iκr to see clearly that this cannot hold. Hence Mκ (R) is not normal in Pκ (R). (d*) Observe first that when κ = (n), we have Mκ (R) = G L(n, R) = Pκ (R), so normality holds trivially in this case. Next note that

158

Solutions to Selected Exercises if κ = (1, . . . , 1), and if R is any commutative ring with trivial unit group — i.e., R × = {1} — then Mκ (R) = {In }, which is clearly a normal subgroup of Pκ (R). For example, we could take R = F2 or the polynomial ring F2 [x], where F2 is the field with 2 elements. We claim that the examples in the previous paragraph are the only cases for which Mκ (R) is normal in Pκ (R). To see it, suppose first that Mκ (R) is normal for some ring R and some ordered partition κ with r ≥ 2 and some κ j ≥ 2. We first treat the case that j = 1. We can proceed just as in part (c) to conclude that g X = X h for all g ∈ G L(κ1 , R), h ∈ G L(κr , R), and X ∈ Mat(κ1 × κr , R). Take h = Iκr and g = Iκ1 + g  , where g  has a 1 in the upper right entry and zeros elsewhere. Then for any κ1 × κr matrix X , we have 0 = g X − X h = g  X. Considering the form of g  , this implies the bottom row of X is identically zero, which is a contradiction since X was arbitrary. If κ j ≥ 2 for some j > 1, then a similar argument to the above one will succeed upon replacing the conjugating matrix B by one that has an undetermined block matrix X in the (1, j)position. Finally, suppose that κ = (1, . . . , 1). Then one can use the above argument again to deduce that g X = X h for all X ∈ R and g, h ∈ R × . Taking X = g −1 shows 1 = g −1 h, or h = g. But h was arbitrary, so this can only hold if R × consists of a single element, namely R × = {1}, which is what we claimed.

13.6* (a,b) Suppose R is a field or R = Z p . After conjugation, the general case follows from the case of a standard parabolic Pκ (R). Let M ∈ N (P). Choose an integer 1 ≤ m < n such that Pκ(R) ⊂  P(m,n−m) (R). (See also Exercise 13.8.) We write M = CA DB , where A ∈ Mat(m, R), B ∈ Mat(m × (n − m), R), C ∈ Mat((n − m) × m, R), D ∈ Mat(n − m, R). Our first goal is to show that M ∈ P(m,n−m) (R). For any X ∈ Mat(m ×(n−m), R), there exist matrices E, F, G of appropriate sizes such that     E F A B X A B Im = . 0 In−m 0 G C D C D Comparing lower right corners, we find that C X + D = G D.

(†)

Solutions to Selected Exercises

159

The R-submodule of R n−m generated by the rows of D agrees with that of G D, since G is invertible. Now suppose that C has at least one invertible entry. We have two cases to consider. If the (n − m) rows of D generate R n−m , then they form an R-basis. We may choose the matrix X so that C X + D has a zero row, which means that the rows of C X + D cannot generate R n−m , contradicting (†). If, on the other hand, the rows of D do not generate R n−m , then we can choose X so that C X + D has a row lying outside of the span of the rows of D, which again contradicts (†). We therefore conclude that C contains no invertible entry. If R is a field, then C = 0 and the normalizing matrix M is block upper triangular. If R = Z p , then every entry of C is divisible by p. Hence



det A

C

B

= | det A| p · | det D| p = / 0, D p

so that A and D must be invertible. Let A B . Then C A + DC  = 0, and C D 

A C

B D



Im 0

X In−m



A C

B D

 A

B C  D





 =

be the inverse of

∗ C XC 

∗ ∗

.

  The condition that CA DB is a normalizing matrix means C XC  = 0. Now D is invertible, so the equality C A + DC  = 0 implies C  = −D −1 C A . Hence C XC  = 0 =⇒ C X D −1 C A = 0 =⇒ C X  C = 0, where we have defined X  = X D −1 . As this must hold for all m × (n − m) matrices X  with coefficients in Z p , we conclude C = 0.   So far we have shown that the normalizing matrix M = CA DB must have C = 0. Now repeat this argument on the m ×m matrix A and the (n − m) × (n − m) matrix D. By induction, one learns that a normalizing matrix must lie in Pκ (R). The result for R = AQ follows immediately from the case for fields by considering the shape of a normalizing matrix at each prime v ≤ ∞. (c) Let R = F2 [ε]/(ε2 ), where F2 is the field of two elements. The units in this ring are R × = {1, 1 + ε}, so that P consists of the

160

Solutions to Selected Exercises a x  matrices of the form 01 a2 for x ∈ R and a1 , a2 ∈ {1, 1 + ε}. Now one can directly verify that 

1 0 ε 1

∈ N (P) P.

13.7 Here we explain the case κ = (1, . . . , 1). The general case requires only notational generalization. Let 1 ≤  ≤ n − 1 be an integer, and let u ∈ Uκ (AQ ). An entry u i, j of u with i < j is said to be at level  if j − i = . The proof is by induction on the level. For level 1, by strong approximation for AQ we may choose α1,2 , . . . , αn−1,n ∈ Q such that u i, j − αi, j ∈ D. Set ⎛ 1 −α 1,2 1 ⎜ ⎜ a1 = ⎜ ⎜ ⎝

0 −α2,3 .. .

··· ··· .. .

0 0 .. .

1

−αn−1,n 1

⎞ ⎟ ⎟ ⎟ ∈ Uκ (Q). ⎟ ⎠

Then all entries of ua1 at level 1 lie in the fundamental domain D. Now suppose that we have constructed a matrix a ∈ Uκ (Q) such that the product ua has all of its entries at level at most  − 1 lying in D. To prove the induction step, the key observation is that if v, b ∈ Uκ (AQ ), where b has the property that its only non-zero entries above the diagonal lie at level , then the product vb agrees with v at all levels 1, . . . ,  − 1, and the entries at level  are given by adding the entries of v and the corresponding entries of b at level . So if v = ua, and if b is chosen with entries at level  that are good rational approximations to the corresponding entries of ua, then the product uab will have all entries at level at most  lying in D. Proceeding by induction allows one to see that any coset of Uκ (Q)\Uκ (AQ ) can be represented by a matrix in Uκ (D). Uniqueness of this representation can also be proved by induction on the level. 13.9 By the previous exercise, we know that Uκ (AQ ) ⊂ Uκ  (AQ ). Write m =  + · · · + κr , and write μ = (κ1 , . . . , κs ) and κ1 + · · · + κs and n − m = κs+1  , . . . , κr ) for the corresponding ordered partitions of m and ν = (κs+1   n − m, respectively. Then any u ∈ Uκ  (AQ ) can be written u = A0 CB with A ∈ Uμ (AQ ) ⊂ Um (AQ ), C ∈ Uν (AQ ) ⊂ Un−m (AQ ).

B ∈ Mat(m × (n − m), AQ ),

Solutions to Selected Exercises Now observe that u =

 Im

BC −1 In−m

161

 A  , so that C

ϕ(ug) du Uκ  (Q)\Uκ  (AQ )

 ϕ

= A∈Uμ (Q)\Uμ (AQ ) ∗ C∈Uν (Q)\Uν (AQ )

Im BC −1 In−m





A C

· g dB dA dC

The inner integral is over all B ∈ Mat(m × (n − m), Q)\Mat(m × (n − m), AQ ). Make the change of variables B → BC to see that the inner integral is the constant term of ϕ along the maximal parabolic subgroup Pκ (AQ ), which we know vanishes by hypothesis. 13.10 Let G be a group and let N , H be subgroups of G with N normal. Then G = N  H if and only if G = N · H and N ∩ H = {1}. (There are other equivalent definitions.) By Exercise 13.5, Uκ (R) is a normal subgroup of Pκ (R). If g ∈ Pκ (R), and if m ∈ Mκ (R) is defined by taking the invertible matrices on the “diagonal” of g, then gm −1 lies in Uκ (R). Hence Pκ (R) = Uκ (R) · Mκ (R). Evidently Uκ (R) ∩ Mκ (R) = {In }. Thus Pκ (R) = Uκ (R)  Mκ (R). For the second part of the exercise, let g ∈ G L(n, AQ ). The usual Iwasawa decomposition gives an upper triangular matrix p and an element k ∈ K such that g = pk. Then p ∈ Pκ (AQ ), and the result follows from the first part of the exercise. 13.13* (a) For an n × n matrix g, write fr (g) for the lower left r × r minor of g. If we view fr as a function of the entries of g, then fr is a homogeneous polynomial of degree r . It suffices to show that @ Bn (k) · w · Bn (k) w =/ w 

= {g ∈ G L(n, k) fr (g) = 0 for some r = 1, . . . , n − 1}.

To prove this claim, consider the (r − 1) × (r − 1) matrix ⎛ ⎞ 1 . ⎠. Jr −1 = ⎝ .. 1 If w = / w  , then we may write w uniquely in the form  w0 w= Jr −1

162

Solutions to Selected Exercises for some Weyl element w0 ∈ G L(n −r +1, k) with lower left entry zero. Let A1 , C3 ∈ Bn−r +1 (k), A3 , C1 ∈ Br −1 (k), A2 ∈ Mat((n − r +1)×(r −1), k), and C2 ∈ Mat((r −1)×(n −r ), k). An arbitrary element of Bn (k) · w · Bn (k) is of the form    w0 C1 C2 A1 A2 . A3 Jr −1 C3 A direct (albeit messy) calculation shows that the lower r ×r minor of this matrix vanishes. (b) A Zariski open subset is always open and dense in the topology of G L(n, Qv ).

14.1 It is clear that every commutator element aba −1 b−1 lies in S L(n, R). For the opposite inclusion, Exercise 13.3 shows that S L(n, R) is gen/ j. So it suffices to erated by the union of the subgroups Ui, j (R) for i = show that any element I + A ∈ Ui, j (R) is a commutator, where A is a matrix whose only non-zero entry ai j ∈ R lies at the (i, j)-position. To that end, let X be the diagonal matrix with all 1’s on the diagonal aside from a −1 at the ith diagonal position. Let A be the matrix with all entries equal to zero except for −ai j /2 at the (i, j)-position. Then one verifies easily that (A )2 = 0, that X A = −A , and that A X −1 = A . Therefore, X (In + A )X −1 (In + A )−1 = X (In + A )X −1 (In − A ) = In −2A = In + A. 14.2 (a) Follow the solution to Exercise 6.11. (b) It suffices to show that ker(π ) contains the subgroup Ui, j (Q p ) for each i = / j by Exercise 13.3. The technique in the proof of Theorem 6.1.7 carries over immediately. (c) Any irreducible representation of an abelian group is onedimensional (Exercise 2.9). Observe that G L(n, Q p )/S L(n, Q p ) ∼ = × Q p , where the isomorphism is given by the determinant. Hence π factors through this isomorphism, and so can be factored as π = det ◦ω, where ω is a character of Q×p . 14.3 (a) If  ∈  V is such that v,  = 0 for all v ∈ V , then  = 0 by definition of linear functional. Hence the pairing is nondegenerate on the right. For the left, suppose v ∈ V {0}. We will construct an element  ∈  V such that v,  = / 0. Observe that v is fixed by some compact open subgroup K  ⊂ G L(n, Q p ). The argument in Proposition 8.1.5 shows that there is a linear projection Proj K  :  V → V K such that Proj K  (π (k).w) = Proj K  (w)

(w ∈ V, k ∈ K  ).

Solutions to Selected Exercises

163



/ 0. (This Choose any linear map  : V K → C such that  (v) = is possible by linear algebra since we are not requiring  to be  V K , and (v) =  ◦ Proj K  (v) = smooth.) Then  :=  ◦ Proj K  ∈  / 0.  (v) = (b) Follow the proof of Proposition 14.1.13. (c) The first claim is a consequence of the definition of intertwining map. The proof of Proposition 8.1.5 shows that    VK ∼ = Hom(V K , C) for any compact open subgroup K  . Here Hom(A, B) is the space of complex linear maps A → B (with  no smoothness requirement). If V K is finite-dimensional, then  K  applying this observation twice shows  V has the same (finite) 

dimension as V K . Since we know any finite-dimensional vector space is isomorphic to its double dual via the map determined by  the canonical pairing, the map L restricted to V K must also be an isomorphism.  Conversely, if some V K is infinite-dimensional, then   Hom(V K , C) has dimension strictly greater than V K (in the sense   of cardinality of a basis). Thus  V K has larger dimension than V K ,  K  while  V must have even larger dimension still. So L restricted 

to V K cannot be an isomorphism. (d) Take V1 to be any smooth representation which is not contragredient, and V2 to be its contragredient. By part (a) the canonical invariant pairing on these two spaces is nondegenerate, but by part (c), V1 is not isomorphic to the contragredient of V2 . 14.5 It suffices to verify the invariance property for elementary matrices since any h ∈ G L(n, Q p ) is a product of them. Each type of elementary matrix corresponds to making a different sort of change of variables in the integral. We will consider changes of the form g → hg in the integral; multiplication on the right by h is similar. If h permutes rows r and s of the matrix g, then the change to the integral amounts to swapping the orders of integration for the variables gr, j and gs, j for j = 1, . . . n. Such an h has determinant 1. If instead h corresponds to adding a multiple of the r th row of g to its sth row, then the integral transforms by the change of variables gs, j → gs, j + cgr, j for j = 1, . . . , n. The additive Haar measure is invariant under such transformations, and det(h) = 1 again. Finally, if h corresponds to multiplying row r by the scalar c, then the integral transforms by gr, j → cgr, j for j = 1, . . . , n. The additive Haar measure transforms by dgr, j → |c| dgr, j for j = 1, . . . , n, while | det h|−n = |c|−n .

164

Solutions to Selected Exercises Hence the integral is invariant under multiplication by all three types of elementary matrix. Consider the compact open group of matrices

K 1 = {g ∈ G L(n, Z p ) g − In ∈ p · Mat(n, Z p )}. The additive measure of pZ p or 1 + pZ p is p −1 . It follows that the measure of K 1 with respect to the measure given in the exercise is 2 p −n . (Note that all elements of K 1 have | det g| p = 1.) Its normalized Haar measure is the reciprocal of its index in G L(n, Z p ), which is equal to the cardinality of G L(n, Z/ pZ), which is ( p n − 1)( p n − p) . . . ( p n − p n−1 ). (Proof: There are p n − 1 non-zero vectors that can serve as the first column. If we have chosen the first i columns, then there are pi vectors in their span. The (i + 1)st cannot lie in their span, so there are p n − pi choices for it. Continue by induction.) It follows that " ×

n2

d g = p ( p − 1)( p − p) . . . ( p − p n

n

n

n−1

)

1≤i, j≤n

dgi, j

| det g|n

.

14.7 Plug the result of Exercise 14.6 into each block of Mκ , then collect the u’s and the t’s, and absorb the integration over Mκ (Z p ) into the integral over G L(n, Z p ). 14.8 Multiply the product out: 

  In−r X m1 0 In−r Ir 0 0 Ir 0  X m1 m2 + X m1Y . = m1Y m1

0 m2



Ir 0

Y



In−r

Injectivity is trivial now, and one can check that the image is the set ⎧ ⎨

A ⎩ C

B D

⎫

A ∈ Mat((n − r ) × r, Q p ), B ∈ Mat(n − r, Q p ), ⎬

.

C ∈ Mat(r, Q p ), D ∈ Mat(r × (n − r ), Q p ),

⎭ det(C) = / 0, det(B − AC −1 D) = /0

The image is a nonempty Zariski open subset of G L(n, Q p ) (see Exercise 13.14), and so it is open and dense in the p-adic topology. Express d × m 1 and d × m 2 using Exercise 14.5, and then make the −1 changes of variables X → X m −1 1 and Y → m 1 Y .

Solutions to Selected Exercises

165

14.11 (a) Let W ⊂ V be the span of the vectors v1 , . . . , vn . Construct an ndimensional subspace W  ⊂ V  such that the pairing restricted to W × W  is nondegenerate. In the finite-dimensional setting, such a pairing induces an isomorphism W  ∼ = W ∗ , where W ∗  is the dual of W . Now choose vectors v1 , . . . , vn ∈ W  that correspond under this isomorphism to the basis of W ∗ dual to {v1 , . . . , vn }. (b) Suppose first that all of the component pairings are nondegenerate. / For non-zero v ∈ V we may write v = j v1 j ⊗ · · · ⊗ vr j , where vi j ∈ Vi for each i, j. By multilinearity properties of the tensor product, we may assume that the set of vectors {vi1 , vi2 , . . . } is linearly independent in Vi for each i = 1, . . . , r . For each index   , vi2 , . . . ∈ Vi such that i, apply part (a) to choose vectors vi1  vi j , vik i = 1 or 0, depending on whether j = k or not. Now /    set v  = j v1 j ⊗ · · · ⊗ vr j . Then v, v  is equal to the number of terms in the sum defining v, which is non-zero. Hence  ,  is nondegenerate on the left. By symmetry, it is also nondegenerate on the right. Conversely, suppose one of the pairings  , i is degenerate. By symmetry, we may assume that i = 1 and that there exists a nonzero v1 ∈ V1 such that v1 , w  1 = 0 for all w  ∈ V1 . Choose non-zero vectors vi ∈ Vi for i = 2, . . . , r . If v = v1 ⊗ · · · ⊗ vr , &  Vi . then clearly v, v   = 0 for all v  ∈   14.12 Write k  = CA DB as in the definition of the group K i . Take B  = t −1 B D −1 . Then   −1   Ii −B  t Ii A − B D −1 C 0 0 0 t Ii  ·k · = . 0 In−i 0 In−i 0 In−i tC D Now A − B D −1 C ∈ Mat(i, Z p ). By hypothesis, C ≡ 0 (mod p). Hence tC ≡ 0 (mod p) and A − B D −1 C ≡ A (mod p), which shows that | det(A − B D −1 C)| p = | det(A)| p = 1. Thus A − B D −1 C ∈ K i as desired. 14.13 Define Wi to be the space of smooth linear functionals V → C that vanish identically on W−i for each i. Evidently this gives a chain of G L(n, Q p )-invariant subspaces V. {0} = W0  W1  · · ·  W =  The space Wi is canonically identified with the contragredient of W /W−i . Restriction of functions gives a surjection

166

Solutions to Selected Exercises   (W  /W−i ) → (W−i+1 /W−i ). Two elements of Wi map to the same  /W−i ) if and only if they differ by an element in element of (W−i+1   /W−i ). W , which is to say that W  /W  ∼ = (W−i+1 i

i−1

i−1

14.15 (a) By definition, (ρv , Wv ) = (ρπ(k).v , Wπ(k).v )

(v ∈ V, k ∈ K ∞ ).

Hence Vρ is closed under the K ∞ -action. Evidently 0 ∈ Vρ , and if v ∈ Vρ and λ ∈ C {0}, then (ρv , Wv ) = (ρλ·v , Wλ·v ), so that Vρ is closed under scalar multiplication. Now suppose v, v  ∈ Vρ are linearly independent vectors, and let’s show v + v  ∈ Vρ . Set W = Wv ∩ Wv ⊂ V . Then W is K ∞ -invariant, so by irreducibility we know W = Wv = Wv or W = 0. In the former case, v+v  ∈ Wv , and again by irreducibility we see (ρv+v , Wv+v ) = (ρv , Wv ). If instead W = 0, then the linear algebraic direct sum Wv ⊕ Wv is in fact a K ∞ -equivariant direct sum. As Wv+v ⊂ Wv ⊕ Wv , we get a non-trivial intertwining projection map Wv+v → Wv , which induces an isomorphism (ρv+v , Wv+v ) ∼ = (ρv , Wv ) by irreducibility. (b) Given any v ∈ V , let (ρv , Wv ) be the representation of K ∞ generated by it. By the representation theory of compact groups, it decomposes as a direct sum of irreducible representations of K ∞ as (ρv , Wv ) ∼ = (ρ1 , W1 ) ⊕ · · · ⊕ (ρr , Wr ). For a given irreducible representation (ρ, W ) of K ∞ , the sum of the terms in the above decomposition that are isomorphic to (ρ, W ) injects into Vρ . The projection of Wv onto this subspace determines the component of v inside Vρ . 14.18 (a) Fix a non-zero vector v ∈ V. Every element of V is a linear combination of elements, each of which is a word in the operators π O(n,R) (k) for k ∈ O(n, R) and πg (Dα ) for α ∈ gl(n, R), applied to v. We know how to commute elements of the first type with those of the second, so we can collect all the elements of O(n, R) at the beginning, and just take a product. If the result is in S O(n, R), we have an element of the (g, S O(n, R))-module generated by v. If not, for any fixed δ1 ∈ O(n, R) with det δ1 = −1 we have an element of the (g, S O(n, R))-module generated by π O(n,R) (δ1 ) . v.

Solutions to Selected Exercises

167

If π  is irreducible, we’re done. Otherwise, the previous paragraph shows that v and π O(n,R) (δ1 ) . v generate linearly disjoint (g, S O(n, R))-modules that are swapped by the action of π O(n,R) (δ1 ), and that (π  , V ) is the direct sum of these submodules. When n is odd, −In ∈ O(n, R) − S O(n, R). It commutes with everything and therefore it has to act by a scalar on an irreducible; i.e., π O(n,R) (δ1 ) . v = ±v. So (π  , V ) is generated by v. (b) The map π O(n,R) (δ1 ) is a linear isomorphism V  → V  . All we’re doing is pulling back the actions on V  through this map. 15.1 See Exercise 14.5. 15.2 (a) Follow the strategy of Corollary 9.5.10. You will use (15.3.1) and Proposition 15.3.2 to define an invariant Hermitian form on the space of all cuspidal automorphic forms with central character ω. Then prove that every irreducible cuspidal representation is isomorphic to a subrepresentation of the space of all cuspidal automorphic forms. Now we may simply restrict the invariant Hermitian form to the image of (π, V ) inside the space of all cuspidal automorphic forms. (b) To show the matrix coefficients are bounded, follow the strategy of Lemma 9.1.6. 15.4 Of course not! Note that the upper triangular element is factored in Proposition 6.9.6, while it is expanded in Proposition 15.7.4. Using the notation from Proposition 6.9.6, this gives  Q×p

Q×p

f Qp

G L(2,Z p )

a1 0

· d k d x d × a1 d × a2

0 a2

 1 · 0

x 1



·k

×

=

Q×p

Q×p

 f

Qp

G L(2,Z p )

a1 0

a1 x a2



·k

· d × k d x d × a1 d × a2 . Now make the change of variables x → a1−1 x and the form given by Proposition 15.7.4 appears. 15.5 First show that β is bounded (because π is unitary). 15.6 Twists were defined for the real case in Definition 14.10.6 (for tensor products, but one may specialize to the case when there is only one term). Matrix coefficients were defined for unitary representations in

168

Solutions to Selected Exercises Definition 14.8.12 and twists of unitary representations in (15.9.5). With (15.9.5) in hand, the proof is the same as in the p-adic case.

15.7 To generalize the statement, replace Q p by R, and take (π  , V  ) and (π  , V  ) to be twists of unitary representations (including the “trivial twists”— the representations themselves). Instead of arbitrary Schwartz functions, consider those of the form (15.9.2), or the natural extension of this form to G L(m, R) × G L(n − m, R). Further, use matrix coefficients defined using K ∞ -finite vectors. Then the proof of Proposition 15.6.1 goes through word-for-word.

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Symbols Index

Page numbers for volume 2 are indicated by (2). p finite prime (prime integer); v prime (finite or infinite) 2 Q p 3, 6; Z p 4, 6; Z× p 8 AQ adeles; A× ideles 7 Q Afinite , finite adeles 8 F p , F p [T ], F p [[T ]], F p (T ), F p ((Q)), AF p (T ) 33–34 {a}, fractional part of a, for a ∈ Q p 15–16 R, commutative ring with “1” 181 absolute values: | |, | |v 1, 2 | |trivial trivial absolute value 1 | |∞ archimedean absolute value 1 | | p non-archimedean absolute value 2 | |C , 40 | |A adelic absolute value 31 additive character: " e∞ , e p , ev , 15; e(x) = e (x ) for v≤∞ v v

x = {xv }v≤∞ ∈ AQ 18–19 ψ 15, 18, 126, 127, 130, 144, 420, 449, (2) 32, (2) 57, (2) 86 adelic element, local components $ % g = g∞ , g2 , . . . , g p , . . . 7, 107, (2) 22 automorphic form, ϕ 40 ϕ P (g), constant term of ϕ along a parabolic subgroup P (2) 29 Wϕ (g), Whittaker coefficient for ϕ (2) 32, (2) 33 Wϕ,ψ (g), Whittaker function for ϕ, relative to the character ψ (2) 32  , : bilinear form 279; Petersson inner product 103  , Bruhat , Bruhat pairing 322

Borel subgroup Bn (R), Borel subgroup of G L(n, R) (2) 17 Bn (Q p ), Borel subgroup of G L(n, Q p ) (2) 21 Bruhat-Schwartz S 188 S X (Q× p ) 215 S(Q× p ) 215





S Mat(n, Q p ) , Bruhat-Schwartz space associated to Mat(n, Q p ) 433, (2) 125 category of smooth representation of G : SR(G) 350 character : χ , ω local characters : χ∞ , χv , χ p 43–44 χidelic , Dirichlet character, idelic lift 41–42 ωπ , central character 187 sign, sign character on R× 64, 263 characteristic function on A : 1 A 17 classical automorphic form associated to a given adelic automorphic form φ: φclassical 175 cocycle j(γ , z), j-cocycle 76, (2) 5 Jρ (γ , z) (2) 5 Jk (γ , z) (2) 4 κ(γ , z) (2) 3 (maximal) compact subgroup S O(n, R), O(n, R) (2) 2, (2) 3, (2) 16, (2) 36 K ∞ 103 (2) 16; K p 105, (2) 22; K 111, (2) 26 K finite 158 (compact) open subgroups K r 184, (2) 52 K 0 (N ), compact subgroup 135, 170 K (N ), K 1 (N ) 170, 171 K 0 (N ) p , Iwahori subgroup 134 K 0,1 (N ), K 0,1 (N ) p (2) 43, (2) 45 (n) K m , K mκ (2) 68 congruence subgroup  77 (N ), congruence subgroup 77 0 (N ), principal congruence subgroup 77 cusp cusps(), set of cusps 78 a, cusp 87 μa , cusp parameter 87 m a 87 σa , γa 87

175

176 a , stability subgroup 87 ga , generator 87

Symbols Index Hecke algebra K H K , space of bi-K -invariant functions for K = G L(2, Z p ) 396 K S H K S , S-spherical Hecke algebra for a finite set S of places of Q 408 π ( f ), action of the Hecke algebra 399 T pk (g), R p (g) 435 Hecke operator : Tn , Tn∗ 95, 96 Hermitian form : ( , ) 358, 359, 366, 369

differential operator

, Casimir operator 164

k , weight k Laplacian 82. Relationship with Casimir 177 E i, j , standard basis element of g 114, (2) 8 Dα differential operator 112, 115, 161, (2) 8 Di, j , D Ei, j , differential operator 164, (2) 8 imaginary part of s : Im(s) ; real part of s : (s) R, L raising/lowering operator, 162. induced representation of π Relationship with Maass operators 162–163 IndGH (π ) 239 D Z , D X , DY , D H 161 G L(n,Q p ) distinguished nonzero element : ξv◦ 381 (π ) parabolically induced Ind Pκ (Q p ) dual of G : G ∗ (2) 105 representation (2) 63   G L(n,Q p ) G L(n,Q p ) π ,V Eisenstein series : E a 89 embeddings: i diag , i ∞ 108; i finite 142; i p 144 G L(n,R) , parabolically induced Hermitian (, ) even part : even ; odd part : odd 337 form (2) 91–92 intertwining map : L 57 factorization of hn invariant form : ( , )W (2) 149–150 n , Y n (2) 10 Xm invariant linear functional : L s,ω ( f ) 235 m finite dimensional representation of S O(n, R) : involution : ι 398 ρ, (ρ, W ) (2) 4 fixed vector Jacquet v ◦ 192 VN , V /V (N ), VUκ , V /V (Uκ ), Jacquet module ◦ ξ 382 of V 238, (2) 68 Flat section J, Jnκ , Jacquet functor 350 f (∗; s1 , ss ) flat section 206 Js1 ,s2 , Js,ω , Jacquet’s integral 205, 206 Flat p (ω) space of all flat sections, 206 Bn (ω) basis for space of K n -fixed flat sections, K-Bessel function : K ν (y) 85 207–208 Kirillov n n f 1,d (g; s1 , s2 ), f pc,1 (g; s1 , s2 ) flat sections, K, Kirillov space 214, 225, 244, 252, 253 elements of Bn (ω), 207–208 K p (s, ω), Kirillov space of the principal series function: , , ϕ, φ, f , g 221, 225  representation  . , Fourier transform of 15, 17, 22, 420, π  , K , Kirillov representation 214 (2) 115  , Mellin transform of 53, 70–71, 356 L fundamental domain L2 , square integrable 83, 364 D 78 L2+ 363 D N 81 L-function D∗ 78 L(s, φ) 45 D∞ 109, 309, 310 γ (s, ω) 56 D(m, N , γ ) (2) 11 γ (s, π ) 434, (2) 126 F∞ 310 v (s, ω), (s, ω) 62 L p (s, π ) local L-function, 433, 434, 439, 440, G S (outside of S), G S (including S), for a finite 444, 452, (2) 126, (2) 128, (2) 131, (2) 142 set S of places of Q 408 L ∞ (s, π ) local L-function, 466, (2) 144 Gamma  p (s, π ) local root number, 434, 452, (2) 126, Gamma function : (s) 47 (2) 127 Gamma factor : G(s) 468–469 ∞ (s, π ) local root number, 468, 471, (2) 144 156 (g,K ∞ )-module  L ∗ (s, ω) 62 πg , π K ∞ 153, 156, 259 L v (s, χ ), L(s, χ ), Dirichlet series 54, 378 (π∞ , V∞ ) 380 Lie algebra group of$matrices G, gl(n, C) (2) 8; gl(2, C), 115 % G 1 = g ∈ G L(2, AQ ) : |detg| = 1 426 gl(n, R) (2) 80; gl(2, R), 112

Symbols Index g 115, (2) 8; k 156, (2) 38 [ , ], Lie bracket 112, (2) 81 U (g), universal enveloping algebra 112–115, 164, 165 Z (U (g)), center of the universal enveloping algebra 117 lift χ, ˜ lift of χ χ˜ idelic , idelic lift of the character χ 137, 140  f , lift from upper half plane to G L(n, R)+ 136 f adelic , adelic lift of an automorphic form f for G L(2) 120, 126, 138, 140 adelic , adelic lift of a vector valued automorphic function (2) 25

177

μ, measure 12 μHaar , dμHaar , Haar measure 13 d x, d × x"14 d × g = v≤∞ d × gv 47, 306 Mellin transform M(s; ) 336 Mextended (s, ), extended Mellin transform 336, 337 modular quasicharacter : δ Pκ (g) (2) 61, (2) 91 newform, oldform f new , newform 97, (2) 47 f old , oldform 97 norm : || || (2) 10, (2) 26

partition : κ = (κ1 , . . . , κr ) (2) 60 permutation: σ (2) 74 principal series s = (s1 , s2 ), pair of complex numbers 201 ω = (ω1 , ω2 ), pair of characters 201, 270 (s, ω) 201, 270 B p (χ1 , χ2 ) 281 V p (s, ω) 201 ∨ (π, B p (χ1 , χ2 )) 282 β (g) = β(g −1 ), matrix coefficient of the B∞ (χ1 , χ2 ) 296 contragredient representation 426 V∞ (s, ω) 263 β(A1 , A2 ), matrix coefficient of tensor product (π, B∞ (χ1 , χ2 )) 296 (2)128 (π, V∞ (s, ω)) 264 matrix, entries of a matrix product In , n × n identity matrix (2) 2 ◦, associative product 112 kfinite " element of K 0 (N ) 135, (2) 25, element ∗, convolution product 396; associative product G L(n, Z ) of p p 112 κθ 161 projection : Projn 278 (i, j) (i, j) (i, j) a (i, j) , a∞ , a p , av (2) 114 gi j (2) 143 quadratic torus : Tτ 188 t g, transpose of the matrix g 104 quotient space W/W  159, V /V  (2) 40 matrix, group Mat(a × b, R), set of all a × b matrices with representation entries in R (2) 3 π 65, 66 Mat(a, R), set of all a × a matrices with ∨ entries in R 148, (2) 3 π 455 Pκ (R), standard parabolic (2) 28 π ∗ , dual 277, 289 Uκ (R), unipotent radical (2) 28  π , contragredient 277, 290 Mκ (R), standard Levi subgroup of Pκ (R) π TE , trivial extension of a representation π (2) 28 of the Levi subgroup Mκ (Q p ) (2) 62 Un (R), Un maximal unipotent subgroup of | | p · π , unramified twists (2) 127 G L(n, R) 31 π H , restricted representation 239 Pn , mirabolic subgroup (2) 33 πg , π K ∞ , πfinite 153, 156, 157  P(R), parabolic subgroup of G L(n, R) (2) 28 πg , π K ∞ , πfinite 157 N (P), normalizer (2) 48 G L(n,Q p ) , parabolically induced π M1,1,... ,1 , minimal standard Levi subgroup representation (2) 63 (2) 64 π ⊗ χ , twist by a character 330, χ · π , twist [G, G], commutator subgroup 181 by a character (2) 72 B, upper triangular matrices 238 (π, V ) 65, 66 B(Q p ), Borel subgroup 319 (π H , VH ), restricted 239 Z (R) center of the ring R, 117, (2)8, center ResGH (π ), restricted representation of Z (G L(2, R)) 103, 307 239 measure Maass operator L k , Maass lowering operator 90, 91, 162 Rk , Maass raising operator 90, 91, 162 matrix coefficient: βv,˜v (g) = π (g).v, v˜  317 β f, f˜ 299, β f,˜v 302, β(g) 327, 433 βv (gv ), local 430

178

Symbols Index

G G (π  , V ), induced 239  π, V , contragredient 277, 290

 TE  π , V (2) 62 &r

(πi , Vi ) (2) 61, (2) 62 i=1 u(ω, d), u(π, d) (2) 96, (2) 100 Riemann zeta function : ζ (s) 46, 50 slash operator: |k , 83; |ρ (2) 6 supercuspidal support : (π1 , . . . , πr ) (2) 74 Tate series : T (s, x, ) 69 tensor & product &, tensor product 67 , restricted tensor product 67, 381, v≤∞ 382 V ⊗ V  , tensor product ; (π ⊗ π  , V ⊗ V  ) 379, (2) 89 ⊗v ξv , pure tensor 67 upper half plane : h 78 h∗ , extended upper half plane 78 hn , generalized upper half plane (2) 1 vector space G L(V ) 65 End(V ) 114 HomG (V, W ) 239 V f 167, 172, 384 V K , fixed space 173 VN , V (N ) 236–237 vector space of  Aω G L(n, AQ ) , vector space of adelic automorphic forms (2) 30, (2) 36 Ak,χ () 79 A∗k,χ (0 (N )) 82





Aω G L(2, AQ ) , vector space of adelic automorphic forms 152 Aρ,χ ,λ (0 (N )) (2) 14 L2 (0 (N )\h, k, χ ) 83 L2ωπ Q× p \G L(2, Q p ) 364

Sλ (N , k, χ ), space of Maass forms Scusp,ω , adelic cusp forms 279, 369, 370 vector⎛ valued⎞ function : (z) := φ1 (z) ⎝ .. ⎠ , (z ∈ hn ) . φr (z) Weyl group of G L(n): Wn (2) 17 Whittaker Wϕ , W : Whittaker function space (2) 57 W  f , 384:  Whittaker function space π, Wϕ (2) 57 W  tensor 387  π, W p (s, ω) , Whittaker model of a principal series representation 223 (π, W), Whittaker model (2) 57 W p (∗; f ), local Whittaker function 194 W p ( f ), space of Whittaker functions 194 W p (∗; s1 , s2 ), Jacquet’s local Whittaker function 196, 199 W∞ (∗; s1 , s2 ), Jacquet’s local Whittaker function 261, 262 W (∗; f ), (global) Whittaker function 130 Wα,ν (y), Whittaker function 84 zeta integral: Z ∞ (s, , β), at ∞ 464–471, (2) 143–150 Z (s, , ω), global 61 Z (s, , β), global 426, (2) 120 Z v (s, v , ωv ), local 55 Z v (s, v , βv ), local 433, 438–471, (2) 125, "(2) 130–150 Z (s, v , βV ) 431, (2) 125 v≤∞ v det, determinant 105 inf, infimum 424 ker, kernel 72, 222 rank, rank (of a matrix) 421 red, reduction map 253 restr, restriction function 221 sgn, sign (of a real number) 167 Span, (linear) span (of vector or vectors) 222, 267 sup, supremum 424 Tr, trace (of a matrix) 150, 373

Index

Page numbers for volume 2 are indicated by (2). absolute value 1–2 archimedean absolute value 1 non-archimedean absolute value 1 trivial absolute value 1 action by differential operator (2) 37, (2) 40, 153, 160, 172 by right translation (2) 36, (2) 40, 153, 160, 172 of Maass operators on Whittaker functions 92 of Q on the adeles and ideles 8 of the Hecke algebra 399 on the Kirillov space 224 additive character 15 adelic 18 adele 7 finite adeles 8 group 107–108 adelic automorphic form 118, 119, (2) 26 automorphic representation 152–161 automorphic representation for G L(n, AQ ) (2) 36–41 Bruhat-Schwartz function 20 Bruhat-Schwartz space 46 Cartan decomposition 111 cusp form 119, 311, (2) 29 Whittaker coefficient of (2) 33 space of 160, 311, (2) 40 Fourier transform 18–22 integral 20 Iwasawa decomposition 110 lift 119–126, 136–141 of a classical automorphic form 136–141 of a vector valued automorphic function (2) 25 of Maass form 119–126, 166 matrix 419 Poisson summation formula 30–31 admissible (g, K ∞ ) × G L(2, Afinite )-module 173–178, 403–406

(g, K ∞ ) × G L(n, Afinite )-module (2) 38 (g, K ∞ )-module 259–260, (2) 82 dual (2) 106 admissible representations of G L(2, Afinite ) 404 of G L(2, Q p ) 183–192, 252–253, 404 of G L(n, Q p ) 2(52) algebra of differential operators 114 simple 418 algebraic direct sum of vector spaces 265 archimedean 1 associative algebra 112 asymptotic behavior 84 of matrix coefficient 323, 330–343 Atkin-Lehner theory 97, (2) 46 automorphic cuspidal representation 160, 172–173, 177, 372, (2) 41 form 40, 76–77, 118–119, (2) 26 adelic 40, 118, 119, (2) 26 classical (2) 11, see also Maass form classical associated to an adelic automorphic form 175 for G L(n, AQ ) (2) 26 K ∞ -fixed (2) 1 L2 - 83–84, 92–94, 119, 126, 369, 374, (2) 97, (2) 119 non-holomorphic 414, 418 function (2) 3–13 for 0 (N ) 79 of integral weight 78–80 vector space of 82 vector valued (2) 6 with multiplier ψ 76 relation 82, 115, 117, (2) 6 representation 65–68, 152–178, (2) 36–41 adelic 152–161 with central character 160, (2) 40 automorphy relation 82, 115, 117, (2) 6

179

180 basic types of generic irreducible unitary representations of G L(n, Q p ) (2) 101 generic irreducible unitary representation of G L(n, R) (2) 98 p-adic irreducible unitary representations of general linear groups (2) 100 unitary representations of real general linear groups (2) 96 basis 207, 265 Bernstein-Zelevinsky classification (2) 75 for G L(n, Q p ) (2) 70–75 of discrete series representations of G L(n, Q p ) (2) 77 of tempered representations of G L(n, Q p ) (2) 77 Bessel function 85, 261 bi K -invariant function 396 bilinear form canonical 280, 294 invariant 279, 280, 293, 294, 315, 331 nondegenerate (2) 55 on S(Q× p ) 285 on the vector space of adelic cusp forms 311 Borel set 424 subgroup (2) 17, (2) 21 of G L(n, R) (2) 17 of G L(n, Q p ) (2) 21 subset 309 bottom of the spectrum 92–94 bound on parameters, unitary representation 361, (2) 79 boundedness of matrix coefficients, unitary representation 375, (2) 78 Bruhat decomposition for G L(2, Q p ) 319 for G L(2, R) 299 for G L(n, Q p ) (2) 22 for G L(n, R) (2) 17 Bruhat-Schwartz function/space for AQ 20, 46 for G L(2, Q p ) 188 for Mat(2, AQ ) 419, 430 for Mat(n, AQ ) (2) 125 for Mat(2, Q p ) 433 for Mat(n, Q p ) (2) 125 for Qxp 215 for R 15 Cartan decomposition for G L(n, Q p ) (2) 20 for G L(2, R) 104 for G L(n, R) (2) 16 p-adic 107 Casimir 164 Cauchy-Riemann condition 82

Index Cauchy-Schwartz inequality 360 cell big 319 little 319 center of the universal enveloping algebra 117, 269, (2) 8 central character 118, 119, 187, (2) 54 of an adelic lift 140 character additive 15 central 118, 119, 187, (2) 54 Dirichlet 41 Hecke 40 normalized unitary character 71, 201, 263 ramified 44 sign 263 spherical Hecke 402, (2) 103 unramified 43 characterization of irreducible tempered (g, K ∞ )-modules 368 of irreducible unitary principal series representations 363 of representations of G L(2, Q p ) via the growth of matrix co-efficients 326 of the finite dimensional irreducible representation of G L(n, Q p ) (2) 54 of the irreducible tempered representations of G L(2, Q p ) 363 of the Kirillov space 225 of unitary special representations of G L(2, Q p ) 365 Chinese Remainder Theorem 9 class field theory 51 classical automorphic form 175 classical L-function 60–65 classification Bernstein-Zelevinsky (2) 70–75 Langlands (2) 77 of irreducible admissible (g, K ∞ )-modules 269–274 of irreducible unitary representations of G L(n, R) (2) 97 of linear algebraic groups (2) 106 of smooth irreducible representations of G L(n, Q p ) via the growth of matrix coefficients (2) 75–78 of supercuspidal representations (2) 67 of the generic irreducible unitary representations of G L(n, Q p ) (2) 101 of the generic irreducible unitary representations of G L(n, R) (2) 98 of the irreducible, generic, unramified unitary representations of G L(n, Qv ) (2) 102–105 of the smooth irreducible representations (2) 75–78 of the smooth irreducible unitary representations of G L(n, Q p ) (2) 101

Index Rough classification of the irreducible admissible representations of G L(n, Q p ) (2) 69 cocycle j-cocycle 76 classical (2) 5 one-cocycle (2) 3, (2) 5 relation (2) 4, (2) 5 commutative ring containing “1” (2) 27, (2) 28 commutator subgroup 254 compact-open topology (2) 105 compactly supported function 70 modulo the center 287 complementary series representation 367, (2) 98 of G L(2, Q p ) 363 of G L(2, R) 367, (2) 98 of G L(n, Q p ) (2) 80 complex parameter 361, (2) 79 complexified universal enveloping algebra 115, (2) 81 composition factor (2) 73 conductor 44 of a locally constant compactly supported function 70 congruence subgroup 77, (2) 2–3 non-congruence subgroup 79 principal congruence subgroup 77 conjecture Ramanujan conjecture 413–414 Ramanujan-Petersson conjecture 414 Selberg conjecture 413–414 constant term of a classical automorphic form (2) 11 of an adelic automorphic form (2) 29 continuation analytic 48, 54, 199, 206, 419 meromorphic 47, 60 continuous homomorphism 40 continuous representation 254 contragredient of a (g, K ∞ )-module 290, (2) 82 of a (g, K ∞ ) × G L(2, Afinite )-module 303–306, 369 of a generic representation (2) 58, (2) 87 of a parabolically induced representation (2) 65 of a principal series representation 283 of a smooth representation of G L(n, Q p ) (2) 54 of a special representation 283–285 of a supercuspidal representation 285–288 of a unitary (g, K ∞ )-module 366 of a unitary representation 358, (2) 78 of an irreducible admissible cuspidal automorphic representation 315 of irreducible (g, K ∞ )-modules 298 contragredient representation 277–281, 289–294

181 of a cuspidal automorphic representation 311–316 of a principal series representation 281–283, 294–303 convolution algebra 396 coset representative 80 cusp 78 form 370 holomorphic 414 space of cusp forms 160, (2) 40 inequivalent cusps 80 parameter 87 decomposition Bruhat 299, 319, (2) 17, (2) 22 Cartan 104, 107, 111, (2) 16, (2) 20 Iwasawa 103, 105, 110, (2) 2, (2) 18 decomposition of representations 388–396 decomposition theorem 389 degree 37 dense 108, (2) 23 diagonal embedding 8 diagonal matrix 110 differential form 77 differential operator 112, 161, 162 Dirichlet character 41 Dirichlet L-function 45 discrete series 268, (2) 77 essentially 268 limit of 268 of G L(n, Q p ) (2) 77 representation 372 (2) 77 discrete series representation 372 of G L(2, R) 471–474, (2) 99 of G L(n, Q p ) (2) 77 of G L(n, R) (2) 99 division ring 418 divisor 37 Dixmier’s lemma 186 double coset 203 dual (2) 105–106 admissible (2) 106 Langlands (2) 106 of a (g, K ∞ )-module 289 of a (g, K ∞ ) × G L(2, Afinite )-module 303 Pontryagin (2) 105 representation 277 spherical (2) 106 unitary (2) 105–106 -factor 451, see also root number eigenfunction 370 eigenvalue 174 Eisenstein series 89–90 elliptic curve 52 embedding 168 diagonal 8, 108 equivalence of absolute values 1 equivalent conditions for supercuspidality (2) 67

182 Euler product 378–379 factorization 378 exact 350 explicit computation of γ (s, ω) 57 factor of automorphy 76 factorizable function 20, 39, 430, (2) 125 factorizability for representations 67, 68, 378–413, (2) 41–43 factorization Euler 378 initial factorization of a (g, K ∞ ) × G L(2, Afinite )-module 404 Iwahori 343 of adelic integral 21 of hn (2) 10 of Haar measure 230, 425, (2) 132 of integral 295 of the global Whittaker function 133 of the global zeta integral 430–432, (2) 124–125 of unramified irreducible admissible cuspidal automorphic representation 383–388 unique factorization of adeles 111 field of formal Laurent series 34 finite, finiteness finite adeles 8 finiteness property of a (g , k∞ )-module 156 K ∞ (2) 26 right K 118 Z (U (g)) 118, (2) 17 finite dimensional representation continuous of G L(n, Q p ) 106 of K ∞ (2) 4, (2) 82 finite length (2) 52, (2) 65 finitely generated representation 191, (2) 52 smooth 191, (2) 53 fixed vector G L(2, Z p ) fixed/k-fixed vector 192, 400 K 0,1 ( p m ) p (2) 44, (2) 45 K ∞ 260 flat section 206 Fourier coefficient 96 Fourier expansion 23–30, 80–81 of an adelic automorphic form 126–128, (2) 31–36 of an adelic cusp form (2) 32 Fourier inversion 453 on Q p 17 on R 15 on the adeles 22 Fourier transform adelic 18–22 on R 15–18 p-adic 15–18 Fourier-Whittaker expansion 87–89 Frobenius reciprocity 239 function adelic 19

Index automorphic (2) 3–13 Bruhat-Schwartz 20, see Bruhat-Schwartz function/space Bessel 85, 261 characteristic 17 compactly supported 12, 70 entire 49 Euler’s 71, 172 factorizable 20 holomorphic 80–81 K-Bessel 85, 261 K -invariant 396 locally constant 12 locally integrable 307 periodic adelic 23 periodized Bruhat-Schwartz 23 radially symmetric 398 restriction 221 Schwartz 340 smooth vector valued (2) 6 spherical 398 test 47, 438, 440, 441, 463 vector valued (2) 8 vector valued automorphic (2) 6 functional 13 functional equation Dirichlet L-function 45 for the induced representation (2) 137 global 62 local 56, 432–434, (2) 125–128 of Jacquet’s Whittaker function 262 of the global zeta integral 62, 426, (2) 120 fundamental domain 9–11, 78, 109,111 fundamental identity for local integrals of unitary supercuspidal representations of G L(2, Q p ) 454 (g, K ∞ )-module 156, 259, 289, (2) 37 (g, K ∞ )-module of G L(n, R) (2) 81 (g, K ∞ ) × G L(n, Afinite )-module (2) 38 (g, K ∞ ) × G L(2, Afinite )-module 157, 368–374 function field 33 G-module 191, (2) 53 G-submodule 192, (2) 53 Gauss sum 58 generic (2) 31, (2) 58 generic representation (2) 56–60 irreducible unitary (2) 98, (2) 101 of G L(n, Q p ) (2) 56–60 of G L(n, R) (2) 85–88 global conductor 54 functional equation 62 matrix coefficients 374, (2) 119 new vector (2) 45 Whittaker function 128–134, 141–147 zeta integral 425–430 factorization of 430–432, (2) 124–125

Index for G L(2, AQ ) 425–430 for G L(n, AQ ) (2) 118 Tate’s 61 Godement-Jacquet L-function 418–419 Godement-Jacquet method 419 growth condition 76 moderate 40, 79, 118, (2) 7, (2) 26 of matrix coefficients 316–330, (2) 75–78 polynomial 81 property 311, 316–330 Haar measure 423–425 additive 13, 455 existence and uniqueness of 229, 424 factorization of 230, 425 left and right 424, 425 multiplicative 14, 426 on G L(2, Q p ) 228–232, 318, 319 on G L(n, Q p ) (2) 69 on G L(2, R) 299 on Q p 13 harmonic analysis 52 Hecke algebra 396–403, 408, (2) 103 character 66 newform 172 operator 68, 95, 378, 396 Hecke algebra spherical 396–403, (2) 103 S-spherical 408 Hecke character 66 unitary 40 Hermitian form 358, (2) 91 invariant 369 on adelic cusp form 369 Hilbert space (2) 92 idele, ideles 7 idelic absolute value 47 differential 47 lift 41, (2) 25 G L(n) idelic lift of a Dirichlet character (2) 25 of a Dirichlet character 41 index 77 induced map 349 non-normalized representation (2) 94 representation 238–240, 257 space (2) 63 inner product 372 integrable locally integrable function 307 Integration Integral on Z (AQ )\G L(2, AQ ) 307

183 Integral on Z (AQ )G L(2, Q)\G L(2, AQ ) 309 on G L(2, AQ ) 306–311 on G L(2, Q p ) 228–232 on G L(2, R) 299 intertwining map 66 injective (2) 107 of (g, K ∞ )-modules 158, (2) 39 of (g, K ∞ ) × G L(n, Afinite )-modules 159, (2) 39 invariance of the integral 310 invariant bilinear form 315, 331 on principal series of G L(2, Q p ) 282 on principal series of G L(2, R) 296, 300 differential operator 155, (2) 8 Hermitian form on cusp forms 369 linear functional 234 pairing 280, 294, 306 inverse limit 6 involution 398 irreducible 188, 279 (g, K ∞ ) × G L(2, Afinite )-module 157 (g, K ∞ ) × G L(n, Afinite )-module (2) 38 quotient, (2) 53 representation 66, 184 admissible 252–253 automorphic 174 cuspidal automorphic 177 discrete series 471–474 special 364–365 spherical 435 square integrable 329 supercuspidal 364–365, 440 tempered 363, 368 unitary Hilbert space (2) 106 subquotient (2) 74 Iwahori factorization of K n 343 subgroup 134 Iwasawa decomposition 103, 105–107 adelic 110 for G L(n, Q p ) (2) 26 generalized (2) 50 p-adic 105 Jacquet functor 350 Jacquet integral 205 analytic continuation of 206 Jacquet module 236–238 for G L(n, Q p ) (2) 68 of a principal series representation 347 of a special representation 351 Jacquet’s Whittaker function 86 local 195–200, 260–263 local K ∞ -invariant 261 Jordan-H¨older series (2) 73

184

Index

K n -symmetrization 344 K -invariance 436 Kirillov model 214–221, 243–252 of a supercuspidal representation 252 of the principal series representation 221–228 representation 214 space 221, 225 uniqueness of 243–252 L2 automorphic form 83–84, 92–94, 119, 126, 369, 374, (2) 97, (2)119 equivalence (2) 95 modulo the center (2) 119 space 83, 84, 374 Langlands parameter 439 Langlands program 51 Laplace operator (Laplacian), weight K 82 relation with casimir 176–177 left and right translation invariant measure 424 level 84, (2)6 Levi factor (2) 28 subgroup (2) 28 L-functions 45–55, 60–65 analytic theory of 418–419 associated to automorphic representation 62 classical Dirichlet 48 Dirichlet 45 Godement-Jacquet 418–419 Hasse-Weil 52 local 55–60, 64, 434–440, (2) 143 of a discrete series representation (2) 142 of a supercuspidal representation (2) 128 of an automorphic form on G L(2) 378 L-group (2) 106 Lie algebra 116, (2) 80–85 bracket 112, (2) 80–85 group 77 lift adelic 119–126, (2)24–25 central character of 140 from the upper half-plane h to G L(2, R)+ ) 136 idelic 41, (2) 25 of an automorphic function (2) 7 limits of discrete series 268 linear fractional transformation 103 local component at p 444 functional equation 56 for G L(1) 56 for parabolically induced representation (2) 137 for principal series representations 446–450, 467–471

for tensor products (2) 128–130 for the supercuspidal representations 452–462, (2) 128 for unitary special representations 450–452 L-function 55–60, 64, 434–440, (2) 126–144 as a common divisor 64 associated to an irreducible discrete series representation 471–474 for the unitary special representation 444–446 of an irreducible principal series representation 441–444, 463–466 of an irreducible supercuspidal representation 440 of an unramified irreducible admissible representation 439 matrix coefficient 426 new vector (2) 44 representation 193–195, 223, 226, (2) 85 from a Maass form 193–195, 223, 226 root number 62 unitary character 55 Whittaker function 195, 260 zeta integral 55 associated to a supercuspidal representation 440, 452 evaluation of 438, 440, 441, 463 for a parabolically induced representation (2) 130–138 for discrete series (2) 138–143 for irreducible unitary generic representation of G L(n, R) (2) 143–151 meromorphic continuation of 60 operator valued 455 locally compact group 388–396 abelian (2) 105 totally disconnected 389 locally constant 12 Maass cusp form 89 non-holomorphic 396 vector valued (2) 12 Maass form 81–84, 98 cusp 89 even 119 odd 126 Maass raising and lowering operators 90–92, 162 map induced 349 intertwining 66 linear 277, 399 matrix coefficient 281, 301, (2) 84 factorizable 374, 430, (2) 124 for (g, K ∞ )-module 299, 302 for G L(2, Q p ) 281, 343–353 for G L(n, Q p ) (2) 55 global (2) 119 growth of 316–353, (2) 75–78

Index of a cuspidal automorphic representation 374, (2) 119 of a principal series representation 318 of a special representation 316, 328 of a supercuspidal representation 288, 326, 440, 461, 462 of a unitary representation 359, (2) 78 of an automorphic representation 374, 425–426, 430–431, (2) 119–120, (2) 124–125 of the contragredient representation 426 maximal compact subgroup 105 maximal standard parabolic subgroup (2) 30 maximal unipotent subgroup (2) 31 measure Haar 423–425 on Q p 12 product 425 Mellin inversion 336 p-adic 71, 70 Mellin transform 336 extended 336 p-adic 70–72 minimal parabolic (2) 141 mirabolic subgroup (2) 33 model 213 consisting of matrix coefficients 317 Kirillov 214–221, 243–252 of a representation 213 Whittaker 213, 223, (2) 56, (2) 57, (2) 85, (2) 86 moderate growth 40, 79, 118, (2) 7, (2) 26 modular form Fourier expansion of a holomorphic 80–81 holomorphic 77 modular group 77 modular quasicharacter (2) 61, (2) 91 modular relation 418 module (g, K ∞ )- 156, 259, 289 (g, K ∞ ) × G L(2, Afinite ) 157, 368–374 (g, K ∞ ) × G L(n, Afinite ) (2) 38 Jacquet 236–238 multilinear algebra (2) 66 multiplicative Haar measure on Q× p 14 on G L(2, Qv ) 426 multiplicativity of the Fourier coefficients 96 multiplicity one for G L(n, Q p ) (2) 59 for G L(n, R) (2) 87 natural projection (2) 60, (2) 88 newform 97, 169 classical 97, (2) 45 for G L(n) (2) 43–47 Hecke 172 holomorphic Hecke 172 local new vector (2) 44

185 non-archimedean 1 non-normalized induction (2) 63 nondegenerate (2) 55 nonunitary representation (2) 145 norm-square 316, 329 normalized Haar measure 230 Haar measure on G L(n, Q p ) (2) 69 parabolic induction for G L(n, Q p ) (2) 62 parabolic induction for G L(n, R) (2) 93 unitary character 71 of Q× p 201 of R× 263 normalizer (2) 29 odd 126, 337 oldform 97, 169 one-dimensional 400 operator Casimir 164, 176–177, (2) 7–8 classical Hecke 95 differential 90–94, 112–117, 153–155, 160–166, 174, 176–177, 469, 497, (2) 8–10, (2) 37, (2) 40, (2) 81–82 Hecke 68, 95, 396, 378, 435–437 intertwining 66, 159, (2) 40 Laplace 82 lowering 162 Maass lowering 90, 162 Maass raising 90, 162 projection 390 raising 162 slash (2) 6, 83 ordered partition (2) 27 orthogonality relation 72, 177, 220, 461, (2) 149 p-adic 2 Cartan decomposition 107 field 3 Fourier transform 15–18 integer 6 integration 12–15 Iwasawa decomposition 105 Mellin inversion 71, 70 Mellin transform 70–72 number 2 representation 414 pairing, see also bilinear form Bruhat 322 canonical bilinear (2) 107, 280, 294 invariant 280 nondegenerate bilinear (2) 110, 306 tensor product (2) 110 uniqueness of invariant 294, 280 parabolic induction 238–240 for G L(n, Q p ) (2) 60–66 for G L(n, R) (2) 88–100

186 normalized (2) 62, (2) 93 subgroup (2) 27, (2) 28 parabolically induced Hilbert space (2) 92 positive definite Hermitian form (2) 91 pre-Hilbert space (2) 91 representation (2) 65 parity of an irreducible admissible (g, K ∞ )-module 260 partition (2) 27 periodic adelic function 23 periodized Bruhat-Schwartz function 23, 24 Petersson inner product 83 Poisson summation formula 30–31 for G L(2, AQ ) 419–423 for G L(n, AQ ) (2) 114–118 Pontryagin dual of a locally compact abelian group (2) 105 positive definite Hermitian form 358 parabolically induced (2) 91 positive definite norm on Cr (2) 10 pre-Hilbert space (2) 91 prime 2 infinite 2 principal series representation 200–205, 226, 263–268 Kirillov model of the 221–228 of G L(n, R) 296, (2) 95 the vector space V∞ (s, ω) 263 the vector space V p (s, ω) 201 product associative 112 convolution 396 direct 425 infinite 62 product formula 2 products of general linear groups (2) 61 projection operator 191, 278, 291, 344, 390, (2) 62, (2) 150 pure tensor 67 quasicharacter (2) 61, (2) 91 quotient (2) 53 of a representation (2) 56–60 space (2) 53 unique irreducible (2) 142 Ramanujan conjecture 439, see also Selberg conjecture at ∞ 466 for G L(2, AQ ) 413–414 for G L(n, AQ ) (2) 106 Petersson 414 ramification 54 ramified 44 (g, K ∞ )-module for G L(n, R) 260, (2) 82 at p 406, (2) 42 character 44

Index representation 192 representation of G L(n, Q p ) (2) 53 rank of a matrix 421, (2) 117 Rankin-Selberg method 69 rapid decay 311, (2) 119 rational function field 33 reducibility of a representation induced from supercuspidal (2) 72 of principal series 267 reducible (2) 70 parabolically induced representation (2) 127 representation (2) 127 region of absolute convergence 445, (2) 127, (2) 128, (2) 139 representation automorphic 65–68, 152–178 automorphic cuspidal 172–173, 372 complementary series 363, 367, (2) 80 essentially square integrable 330 group 65 induced 238–240, 257 induced from supercuspidals (2) 72 irreducible 66, 184, 252–253 isomorphic 66 Kirillov 214 nonunitary (2) 145 of finite length (2) 52 p-adic 414 parabolically induced (2) 65 parabolically induced from supercuspidal representations (2) 71 principal series 200–205, 226, 263–268 quotient of a (2) 56–60 ramified 192 restricted 239 smooth 183, 184, 404 special 232–236, 364–365 spherical 402 square integrable (2) 76, 329 supercuspidal 240–243, 364–365 tempered 363, 368, (2) 76, (2) 80 unitarizable 358 unitary group (2) 83 unramified 192 residues of Eisenstein series 83 restricted direct product 7, 67, 107, (2) 22 representation 239 tensor product of local representations 382, (2) 41 tensor product of vector spaces 381 Riemann hypothesis for curves 414 Riemann-Roch formula 38 Riemann zeta function 46, 50 ring of Borel sets 424 root number 54–55, 62–65, 434, 450–452, 468, (2) 126–128, (2) 131, (2) 144 local 62

Index scaling matrix 89 Schur orthoganality (2) 149 for unitary supercuspidal representations 461 Schur’s lemma for irreducible admissible (g, K ∞ )-module 292 for irreducible and admissible (g, K ∞ ) × G L(2, Afinite )-modules 304 for irreducible smooth representations 186 Schwartz function 340 Selberg conjecture 413–414 Selberg spectral decomposition 83 self-adjoint 370 extension 93 slash operator 83, (2) 6 smooth, smoothness 109, 188, (2) 25 adelic function 24 (g, K ∞ ) × G L(2, Afinite )-module 157 (g, K ∞ ) × G L(n, Afinite )-module (2) 38 H -module 236 linear map 277 representation 183, 184, 404 vector valued function (2) 6 special representation 232–236, 364–365 spectrum bottom of the 92–94 continuous 89 spherical, see also unramified dual (2) 106 function 398 Hecke algebra 396–403, (2) 103 Hecke character 402, (2) 103 representation 402 square integrable, see also direct series essentially 330 modulo the center 326 representation 329, (2) 76 stability subgroup 84 stabilizer open (2) 52 standard Levi subgroup (2) 28 maximal parabolic subgroup (2) 49 parabolic subgroup (2) 27 representative (2) 76 strong approximation 109 for adeles 10 for congruence subgroups 134–135 for G L(n) (2) 22–24 for ideles 11 for K 0 (N ) (2) 23 for prime power level subgroups 134 subquotient 159, (2) 40, (2) 53 subrepresentation (2) 53 supercuspidal equivalent conditions for supercuspidality (2) 67 representation 240–243, 364–365

187 representation of G L(n, Q p ) (2) 66–70, (2) 71 support (2) 74 Tate series 69 Tate’s thesis 51 tempered representation 363, 368, (2) 78, (2) 80, (2) 100 tensor product infinite 381–383 of a (g, K ∞ )-module and a representation of G L(2, Afinite ) 404 of (g, K ∞ )-modules and representations 379–380 of local representations 380, 381–383 of 2 group representations 379 of unitary representations (2) 89 pairing (2) 110 theorem 406–413 theorem for G L(n) (2) 41–43 topological group 8, 389 topological space 7, 389 topological vector space 66 torus 238 quadratic 188 totally disconnected group 388, 389 translation additive 13 right 153, 65 trivial extension (2) 62 twist of a G L(2, Q p ) representation by a character 330 of a representation of G L(n, Q p ) (2) 72 of a representation with unitary central character 330 of a tensor product representation of Mκ (R) (2) 90 type K ∞ -type 270, 272 uniformity of behavior of matrix coefficients 317 unimodular group 425 unipotent radical (2) 28 uniqueness of factors (tensor product theorem) 412, (2) 43 unitarizable representation 358 unitary central character 330 dual (2) 105–106 (g, K ∞ )-module 365–368, (2) 83 Hecke character 40 principal series representation 360–364 representation 358–365, (2) 78–83 universal enveloping algebra 112–117, (2) 8 complexified 115, (2) 81 unordered partition (2) 74

188

Index

unramified 260 at p 406, (2) 42 character 43 local character 43 representation 192, 202, (2) 53 twists (2) 127 unramified representation 192, 202, (2) 53 admissible irreducible 400 of G L(n, Q p ) and G L(n, R) (2) 97 upper half plane 78 extended 78 generalized (2) 1

Whittaker function classical 84–87, 92 global 128–134 Jacquet’s 86, 141–147, 195–200, 260–263 local 195, 260 relative to ψ (2) 57, (2) 86 Whittaker model of a local representation of G L(n, R) (2) 85–86 of a local representation of G L(n, Q p ) 56–57 of a principal series representation 213, 223 Whittaker’s differential equation 84, 88

vector valued automorphic function (2) 6

Z (U (g))-finite 118 Zariski closed (2) 50 open (2) 50 zeta function 418 Riemann 46, 50 zeta integral, see also local zeta integral global 419–423, (2) 118 Zorn’s lemma 192

weak approximation 10 weight k Laplacian 82, 176, (2) 7 Weyl element (2) 47, (2) 109 group of G L(n) (2) 17 Whittaker coefficient of an adelic cusp form (2) 33