Automorphic Representations and L-Functions for the General Linear Group: Volume 1 (Cambridge Studies in Advanced Mathematics)

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CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 129 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 I 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 I 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/9780521474238  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 978-0-521-47423-8 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 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 vii

viii

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

ix 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

x

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

xi

11.14

Solutions to Selected Exercises References Symbols Index Index

463 467 471 474 478 531 537 541

Contents for Volume II

Contents for Volume I Introduction Preface to the Exercises

page vii 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 xiii

xiv

Contents 14.9 14.10 14.11 14.12 14.13 14.14

15

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 ) Unramified representations of G L(n, Q p ) and G L(n, R) Unitary duals and other duals The Ramanujan conjecture for G L(n, AQ ) Exercises for Chapter 14

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

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

1 Adeles over Q

1.1 Absolute values Definition 1.1.1 (Absolute value) An absolute value on a field F is a nonnegative real valued function | | on F which satisfies the conditions: (i) |x| = 0 if and only if x = 0, (ii) |x y| = |x| · |y|, (iii) |x + y| ≤ |x| + |y|, (triangle inequality) for all x, y ∈ F. If an absolute value | | on a field F satisfies the stronger condition   |x + y| ≤ max |x|, |y| ,

(1.1.2)

then it is called a non-archimedean absolute value. If condition (1.1.2) fails for some x, y ∈ F, then | | is called an archimedean absolute value. It is always possible to define a trivial absolute value | |trivial on any field F where  1, if x =/ 0, |x|trivial = 0, otherwise. Since | |trivial is not very interesting, we shall usually exclude it in our discussions. Definition 1.1.3 (Equivalence of absolute values) Two absolute values | |1 and | |2 , defined on the same field F, are termed equivalent if there exists c > 0 such that |x|1 = |x|c2 for all x ∈ F. Example 1.1.4 The field Q of rational numbers has the classical (and very ancient) archimedean absolute value which we denote by | |∞ which is defined by  x, if x ≥ 0, (1.1.5) |x|∞ = −x, if x < 0, 1

2

Adeles over Q

for all x ∈ Q. For each prime p one may define the non-archimedean absolute value | | p as follows. Given x ∈ Q with x = p k · mn with p  | mn, and k ∈ Z, we define  m   (1.1.6) |x| p =  p k ·  = p −k . n The definition of | | p has the effect that the non-archimedean absolute values of numbers divisible by high powers of p become small. Theorem 1.1.7 (Ostrowski) The only non-trivial absolute values on Q are those equivalent to the | | p or the ordinary absolute value | |∞ . 

Proof See [Cassels, 1986], [Murty, 2002].

Theorem 1.1.8 (Product formula) Let α ∈ Q with α =/ 0. The absolute values | |v , given by (1.1.5), (1.1.6), satisfy the product formula 

|α|v = 1

v

where the product is taken over all v ∈ {∞, 2, 3, 5, 7, 11, 13, . . . }, i.e., v = ∞ or v is a prime. Proof The proof is elementary and left to the reader.



Definition 1.1.9 (Finite and infinite primes) Following the modern tradition we shall call v = 2, 3, 5, 7, 11, 13 . . . the finite primes and v = ∞ the “infinite “or” archimedean prime.” Henceforth, we shall adhere to the convention that v refers to an arbitrary prime v (with v finite or infinite), while p refers specifically to a finite prime.

1.2 The field Q p of p -adic numbers An absolute value | | on a field F allows us to define the notion of distance between two elements x, y ∈ F as |x − y|. We may also introduce a topology on F where the basis of open sets consists of the open balls Br (a) with center a ∈ F and radius r > 0:    Br (a) = x  |x − a| < r . A sequence of elements x1 , x2 , x3 , . . . ∈ F is termed Cauchy provided |xm − xn | −→ 0,

(m, n → ∞).

(1.2.1)

A field F with a non-trivial absolute value | | is said to be complete if all Cauchy sequences of elements x1 , x2 , x3 , . . . ∈ F have the property that there

1.2 The field Qp of p-adic numbers

3

exists an element x ∗ ∈ F such that |xn − x ∗ | → 0 as n → ∞, i.e., all Cauchy sequences converge. If a field F is not complete, it is possible to complete it by standard methods of analysis. In brief, one adjoins to the incomplete field F all the elements arising from equivalence classes of Cauchy sequences, where two Cauchy sequences {x1 , x2 , . . . }, {y1 , y2 , . . . } are equivalent if limi→∞ |xi − yi | = 0. The original elements α ∈ F are then realized as the equivalence class of the constant Cauchy sequence {α, α, α, . . . }. Addition, subtraction, and multiplication of the representatives {xi } = {x1 , x2 , . . . }, {yi } = {y1 , y2 , . . . } of two equivalence classes of Cauchy sequences are defined by {xi } ± {yi } = {xi ± yi },

{xi } · {yi } = {xi · yi }.

The definition of division is the same, except one has to be careful to not divide by zero because in a Cauchy sequence {x1 , x2 , x3 , . . . }, some of the xi may be 0. Happily, this is not a problem, because every Cauchy sequence is equivalent to a Cauchy sequence without any zero terms and we always choose such a representative for performing division. The sequence of quotients will be Cauchy, provided the Cauchy sequence by which we divide does not converge to zero. Definition 1.2.2 (pp -adic fields) Let p be a prime number. The completion of Q with respect to the p-adic absolute value | | p , defined by (1.1.6), is denoted as Q p and called the p-adic field. We now present two explicit constructions of Q p . Analytic construction of Q p : The first construction we present is based on the notion of Cauchy sequences. Let k < n be any two integers (positive or negative) and for each i satisfying k ≤ i ≤ n let 0 ≤ ai < p also be an integer. If we assume ak =/ 0, then it easily follows from (1.1.6) that  n   ai pi   i=k

    = p −k . 

(1.2.3)

p

Fix k ∈ Z. An infinite sequence {ak , ak+1 , ak+2 , . . . }, where ai ∈ {0, 1, . . . , p − 1} for each i ≥ k, and ak =/ 0, determines an infinite sequence x 1 = ak p k x2 = ak p k + ak+1 p k+1 x3 = ak p k + ak+1 p k+1 + ak+2 p k+2 .. .

Adeles over Q

4

of elements in Q. By (1.2.3) it is easy to see that the sequence x1 , x2 , x3 . . . is a Cauchy sequence. Formally, we may define lim xi =

i→∞

∞

ai pi ,

(with |xi | p = p −k for all i = 1, 2, . . . ).

i=k

Let Z p denote the set of all elements x of the completed field Q p which satisfy |x| p ≤ 1. By (1.1.2) it easily follows that Z p must be a ring with maximal ideal  

 π = x ∈ Z p  |x| p < 1 . It is easy to check that π = p · Z p . Every x ∈ Z p can be uniquely realized as the equivalence class of a Cauchy sequence of the form

 a0 , a0 + a1 p, a0 + a1 p + a2 p 2 , a0 + a1 p + a2 p 2 + a3 p 3 , . . . where 0 ≤ ai < p for i = 0, 1, 2, . . .. One may check this by first showing that every element of Z p contains a sequence consisting entirely of integers. Every integer may be expressed as a finite sum a0 + · · · + a N p N . One then shows that for the sequence to be Cauchy, the “digit” ai must be eventually constant for each i. The ring Z p can thus be realized as the set of all sums of the type: ∞

ai pi

(1.2.4)

i=0

where 0 ≤ ai < p for each i ≥ 0. Suppose x ∈ Q p does not satisfy |x| p ≤ 1. Then we can multiply x by a suitable power p n with n > 0 so that | p n x| p ≤ 1. It immediately follows that the field Q p , can thus be realized as the set of all sums of the type: ∞

ai pi

(1.2.5)

i=k

where 0 ≤ ai < p for each i ≥ k and k ∈ Z arbitrary. The actual mechanics of performing addition, subtraction, multiplication, and division in the field Q p is very similar to what we do in the field R where every element is of the form ak 10k + ak−1 10k−1 + · · ·

(1.2.6)

with 0 ≤ ai ≤ 9 for all i ≥ k. The main difference in Q p is that the expansion ak p k + ak+1 p k+1 + ak+2 p k+2 · · · goes up instead of down as in (1.2.6).

1.2 The field Qp of p-adic numbers

5

Here is an example of multiplication in Q5 . Note that the multiplication and carrying procedures mimic the case of multiplication in R except that we move from left to right instead of right to left. 2 · 5−1 + 4 · 50 + 3 · 51 + 2 · 52 + · · · × 1 · 5−2 + 3 · 5−1 + 2 · 50 + 1 · 51 + · · · 2 · 5−3 + 4 · 5−2 + 3 · 5−1 + 2 · 50 + · · · + 1 · 5−2 + 3 · 5−1 + 1 · 50 + 3 · 51 + · · · + 4 · 5−1 + 3 · 50 + 2 · 51 + · · · + 2 · 50 + 4 · 51 + · · · 2 · 5−3 + 0 · 5−2 + 1 · 5−1 + 0 · 50 + 1 · 51 + · · ·. We give one more example of the type of infinite expansion that occurs in Q p which is analogous to the expansion 13 = 0.33333 . . . that occurs in R. Example 1.2.7 Let a be an integer coprime to the prime p. Let f ≥ 1 be a ¯ a1 , a2 , . . . such that fixed integer. Then there exist integers a, 1 = a¯ + a f p f + a f +1 p f +1 + a f +2 p f +2 + · · · ∈ Q p a where a · a ≡ 1 (mod p f ) with 0 < a¯ < p f and 0 ≤ ai < p for i = f, f + 1, f + 2, . . .. Since |a −1 | p = 1 it follows that a −1 must be in Z p and, thus, have an expansion of type (1.2.4). We require   a · a¯ + a f p f + a f +1 p f +1 + · · · = 1 from which it easily follows that a a¯ ≡ 1 (mod p f ). Note that p-adic expansions of p-adic numbers are always unique. This is not the case for decimal expansions of real numbers. For example: 1.000 . . . = 0.999 . . .. Algebraic construction of Q p : Let A1 , A2 , A3 , . . . be an infinite set of groups, rings, or fields. We assume that for every pair of positive integers i, j with i > j there exists a homomorphism f i, j : Ai → A j .

(1.2.8)

Adeles over Q

6

Assume also that whenever i, j, k are positive integers satisfying i > j > k, that (1.2.9) f i,k = f j,k ◦ f i, j . Definition 1.2.10 (Inverse limit) Let A1 , A2 , A3 , . . . be an infinite set of groups, rings, or fields. Assume that for all positive integers i > j that homomorphisms f i, j exist satisfying (1.2.8), (1.2.9). Then the inverse limit of the Ai , denoted lim Ai ←−

is defined to be the set of all infinite sequences (a1 , a2 , a3 , . . . ) where ai ∈ Ai for all i ≥ 1 and f i, j (ai ) = a j for all i > j ≥ 1. The inverse limit inherits the algebraic structure of the sets Ai . It will be either a group, ring or field. In the algebraic approach to the construction of Q p we first construct (using the inverse limit) the ring of p-adic integers, denoted Z p . The field Q p is then constructed as the field of quotients of Z p , consisting of all elements of the form a/b with a, b ∈ Z p and b =/ 0. Note that Z p is an integral domain. Let p be a prime and let i be a positive integer. Then the set 

  (1.2.11) Ai := a0 + a1 p + · · · ai−1 pi−1  0 ≤ a < p for all 0 ≤  < i determines a finite ring with pi elements which is canonically identified with the quotient ring (Z/ pi Z). The algebraic operations are addition and multiplication modulo pi . For every i > j, we have the canonical homomorphism f i, j : Ai → A j defined by   f i, j a0 + a1 p + · · · ai−1 pi−1 = a0 + a1 p + · · · a j−1 p j−1 , which simply drops off the tail end terms in the sum. It easily follows from Definition 1.2.10 that an element of the inverse limit is a sequence of the form (a0 , a0 + a1 p, a0 + a1 p + a2 p 2 , a0 + a1 p + a2 p 2 + a3 p 3 ,

. . . ).

∞ ai pi to be the sequence above. Then Formally, we define the infinite sum i=0   ∞    i  i  lim Z p Z = ai p  0 ≤ ai < p for all i ≥ 0 . (1.2.12) ←−  i=0

Definition 1.2.13 (Ring of p -adic integers Z p ) Let p be a prime number. The ring of p-adic integers Z p is defined to be the inverse limit of finite rings given by (1.2.11).

1.3 Adeles and ideles over Q

7

1.3 Adeles and ideles over Q The completion of Q with respect to the archimedean absolute value | |∞ is just R which we also denote as Q∞ . Formally, the ring of adeles over Q, denoted AQ , is a ring determined by the restricted product (relative to the subgroups Z p )  AQ = R × Qp, p

where restricted product (relative to the subgroups Z p ) means that all but finitely many of the components in the product are in Z p . Definition 1.3.1 (Adeles) The ring of adeles over Q, denoted AQ , is defined by 

 AQ := {x∞ , x2 , x3 , . . . }  xv ∈ Qv (∀ v ≤ ∞), x p ∈ Z p ,  (∀ but finitely many p) . Given two adeles x = {x∞ , x2 , x3 ,

x = {x∞ , x2 , x3 ,

. . . },

. . . },

we define addition and multiplication (the ring operations) as follows

, x2 + x2 , x3 + x3 , x + x := {x∞ + x∞

x · x := {x∞ ·

x∞ ,

x2 ·

x2 ,

x3 ·

x3 ,

...} . . . }.

Recall that a topological space X is called locally compact if every point of X has a compact neighborhood. For example, Q p is locally compact and Z p is compact. Furthermore, AQ can be made into a locally compact topological ring by taking as a basis for the topology all sets of the form  U× Zp p∈ S

where S is any finite set of primes containing ∞, and U is any open sub set in the product topology on the finite product v∈S Qv . (This follows the Tychonoff theorem, see [Munkres, 1975].) The ideles of Q are defined to be the multiplicative subgroup of AQ , denoted A× Q. Definition 1.3.2 (Ideles) The multiplicative group of ideles over Q, denoted A× Q , is defined by 

 × × A× := {x , x , . . . } ∈ A ∞ 2 Q  x v ∈ Qv (∀v), x p ∈ Z p , Q  (∀ but finitely many p) .

Adeles over Q

8

Here Z×p denotes the multiplicative group of units of Z p . Clearly, u ∈ Z×p if and only if |u| p = 1. The ideles over Q also form a locally compact topological group with the basis of the topology consisting of the open sets  Z×p U× 

p∈ S

where U is an open set in v∈S Q× any finite set of primes containing v and S is  ∞. Here, the topology on the finite product v∈S Q× v is the product topology. Warning: The topology of the ideles is not the topology induced from the adeles. It is quite different. Definition 1.3.3 (Finite adeles) The ring of finite adeles over Q, denoted Afinite , is defined by 

 Afinite := {x2 , x3 , . . . }  x p ∈ Q p (∀ p < ∞), x p ∈ Z p ,  (∀ but finitely many p) . There is a natural embedding of Afinite into AQ given by {x2 , x3 , . . . } → {0, x2 , x3 , . . . }. Definition 1.3.4 (Finite ideles) The group of finite ideles over Q, denoted , is defined by A× finite 

 := {x , x , . . . } A×  x p ∈ Q×p (∀ p < ∞), x p ∈ Z×p , 2 3 finite  (∀ but finitely many p) . There is a natural embedding of A× into A× Q given by finite {x2 , x3 , . . . } → {1, x2 , x3 , . . . }.

1.4 Action of Q on the adeles and ideles The ring Q can be embedded in the adeles as follows. It is clear that for any fixed q ∈ Q that |q|v > 1 for only finitely many v ≤ ∞. Thus q lies in Z p for all but finitely many p < ∞. Let q ∈ Q. Then {q, q, q, . . . } ∈ AQ . This is usually referred to as a diagonal embedding. It follows that Q may be considered as a subring of AQ . Viewing AQ and Q as additive groups, it is

1.4 Action of Q on the adeles and ideles

9

then natural to take the quotient Q\AQ . Another way to view this quotient is to define an additive action (denoted +) of Q on AQ by the formula q + x := {q + x∞ , q + x2 , q + x3 , . . . } for all x = {x∞ , x2 , x3 , . . . } ∈ AQ and all q ∈ Q. Here q + xv denotes addition in Qv . This is a continuous action and Q is a discrete subgroup of AQ in the sense that for each q ∈ Q, there is a subset U ⊂ AQ , which is open in the topology on AQ , such that U ∩ Q = {q}. We now introduce the notion of a fundamental domain for the action of an arbitrary group on an arbitrary set X . Definition 1.4.1 (Fundamental domain) Let a group G act on a set X (on the left). A fundamental domain for this action is a subset D ⊂ X which satisfies the following two properties: (1) For each x ∈ X , there exists d ∈ D and g ∈ G such that gx = d. (2) The choice of d in (1) is unique. Remarks A fundamental domain is precisely a choice of one point from each orbit of G. If G\X is the quotient space with the quotient topology and π : X → G\X is the quotient map, then the fundamental domain is the image of a section σ : G\X → X . (This is a set theoretic section, it need not be continuous.) The construction of an explicit fundamental domain for the action of the additive group Q on the adele group AQ is equivalent to a generalization of the ancient Chinese remainder theorem. Theorem 1.4.2 (Chinese Remainder Theorem) Let p1 , p2 , . . . pn be distinct primes. Let e1 , e2 , . . . , en be positive integers and c1 , c2 , . . . , cn be arbitrary integers. Then the system of linear congruences x ≡ c1 (mod p1e1 ) x ≡ c2 (mod p2e2 ) .. . x ≡ cn (mod pnen ) has a unique solution x (mod p1e1 p2e2 · · · pnen ). Proof A simple proof can be obtained by explicitly constructing a solution to the system of linear congruences. Set N = p1e1 p2e2 · · · pnen . For each 1 ≤ i ≤ n define an integer u i by the condition N · ui ≡ 1 piei

(mod piei ).

Adeles over Q

10

Then one easily checks that the element x ≡ c1

N N e1 · u 1 + c2 e2 · u 2 + p1 p2

···

+ cn

N · un pnen

satisfies x ≡ ci (mod piei ) for all 1 ≤ i ≤ n. We leave the proof of uniqueness to the reader.  Example 1.4.3 Consider the system of linear congruences x ≡ 2 (mod 32 ) x ≡ 1 (mod 53 ) x ≡ 3 (mod 7). Then u 1 is defined by the congruence 53 · 7 · u 1 ≡ 1 (mod 32 ), and u 1 = 5. Similarly, 32 · 7 · u 2 ≡ 1 (mod 53 ) and u 2 = 2, while 32 · 53 · u 3 ≡ 1 (mod 7) and u 3 = 3. It follows that x ≡ 2 · 53 · 7 · 5 + 32 · 7 · 2 + 3 · 32 · 53 · 3 ≡ 3251

(mod 32 · 53 · 7).

A modern version of the Chinese Remainder Theorem (Theorem 1.4.2) can be given in terms of p-adic absolute values. Theorem 1.4.4 (Weak approximation) Let p1 , p2 , . . . , pn be distinct primes. Let ci ∈ Q pi for each i = 1, 2, . . . , n. Then for every  > 0, there exists an α ∈ Q such that |α − ci | pi <  for all 1 ≤ i ≤ n. Furthermore, α may be chosen so that the denominator, when written in lowest terms, is not divisible by any primes other than p1 , . . . , pn . Proof The general case follows easily from the case when ci ∈ Z pi for all i. As Z is dense in Z p , we may then replace ci by ci ∈ Z. At this point the statement reduces to the classical form, given in Theorem 1.4.2.  Proposition 1.4.5 (Strong approximation for adeles) A fundamental domain D for Q\AQ is given by 

  D = {x∞ , x2 , x3 , . . . }  0 ≤ x∞ < 1, x p ∈ Z p for all finite primes p  Zp. = [0, 1) · p

That is, we have AQ =

 β∈Q

{β + D},

(disjoint union).

1.4 Action of Q on the adeles and ideles

11

Proof Following Definition 1.4.1, it is enough to show that every element in AQ can be uniquely expressed as d + q for d ∈ D and q ∈ Q. Fix x = {x∞ , x2 , x3 , . . . , } ∈ AQ . Apply Theorem 1.4.4 with p1 , . . . , pn being the finite set of primes such that x pi ∈ / Z pi , ci = x pi and  = 1. We obtain α ∈ Q such that |x p − α| p ≤ 1 for all p. For t ∈ R, let [t] denote the greatest integer not exceeding t. Since α ∈ Z p for all p =/ pi , it follows that x p − α − [x∞ −α] ∈ Z p for all finite primes p and x∞ − α − [x∞ −α] ∈ [0, 1). We have thus found q = −α − [x∞ − α] ∈ Q and d ∈ D such that x + {q, q, q, . . . } = d. Next, we consider uniqueness. Suppose there exists q ∈ Q and d ∈ D such that x + {q , q , q , . . . } = d . This implies {q, q, q, . . . } − {q , q , q , . . . } = d − d . But then q − q is an integer at all finite places and at ∞ we must have −1 < q − q < 1. It immediately follows that q = q . Finally, the proof that the union of all rational translates of the fundamental domain D gives AQ follows immediately from Definition 1.4.1 of a fundamental domain.  Next, we consider the multiplicative action of Q× on the ideles A× Q which we denote by · which is defined by q · x = {q · x∞ , q · x2 , q · x3 , . . . } × for all x = {x∞ , x2 , x3 , . . . } ∈ A× Q and q ∈ Q . Here q · x v denotes multiplication in Qv .

Proposition 1.4.6 (Strong approximation for ideles) A fundamental domain D for Q× \A× Q is given by  

 D = {x∞ , x2 , x3 , . . . }  0 < x∞ < ∞, x p ∈ Z×p for all finite primes p  Z×p . = (0, ∞) · p

That is, we have A× Q =



α · D,

(disjoint union).

α∈Q×

Proof Following Definition 1.4.1, it is enough to show that every element in × A× Q can be uniquely expressed as d · q for some d ∈ D and some q ∈ Q . The proof is very similar to the proof of Proposition 1.4.5 and is left to the reader. 

Adeles over Q

12

1.5 p -adic integration Let us consider complex valued continuous functions f defined on Q p . We  would like to define the notion of the integral of f , denoted A f (x) d x, taken over a subset A ⊂ Q p . Definition 1.5.1 (Measure on Q p ) By a measure on Q p , we shall mean a nonzero map from the set of compact open subsets of Q p to the non-negative real numbers which satisfies ∞ ∞   Un = μ(Ui ), μ n=1

n=1

for all compact open subsets U1 , U2 , . . . , of Q p which are pairwise disjoint, ∞  such that Un is still compact. n=1

It is easy to see that a measure μ on Q p must satisfy     μ a + bp n + p n+1 Z p μ a + pn Z p = p−1

b=0

for all 0 ≤ a ≤ p − 1 and n ∈ Z, because the compact set on the left side is the disjoint union of the ones on the right. Definition 1.5.2 (Locally constant function) A function f : Q p → C is said to be locally constant on a subset V ⊂ Q p if for every x ∈ V there exists an open set U ⊂ V containing x such that f (x) = f (u) for all u ∈ U. The function f is said to be locally constant if it is locally constant on all of Q p . Note that any locally constant function f : Q p → C can be expressed as a linear combination of characteristic functions of the form f (x) =

∞

ci · 1Ui (x),

i=1

where ci ∈ C and 1Ui is the characteristic function of Ui . Here Ui are open subsets of Q p for i = 1, 2, . . . The locally constant functions are the analogue of step functions in the classical integration theory on R. Definition 1.5.3 (Integration of locally constant, compactly supported functions on Q p ) Let f : Q p → C be a locally constant function. Let μ be a measure on Q p as in Definition 1.5.1 and assume A = U1 ∪ U2 ∪ · · · ∪ Un

1.5 p-adic integration

13

is a disjoint union of compact open sets Ui ⊂ Q p such that f is the constant function ci ∈ C on each Ui for i = 1, 2, . . . , n. Then we define  f (x) dμ (x) = c1 μ (U1 ) + c2 μ (U2 ) + · · · + cn μ (Un ) . A

Remarks (1) We shall refer to dμ (x) as the “differential induced from” the measure μ. This is perhaps best thought of as a common and extremely useful abuse of notation. What has been defined rigorously is a linear functional on the space of locally constant, compactly supported functions f. We shall be considering other, similar, functionals, coming from closely-related measures. The descriptions are easiest to grasp when presented as relationships among the “differentials.” The interested reader should have no difficulty recovering the definition in terms of measures. (2) It is possible, in a very straightforward manner, to extend the definition of the integral given in Definition 1.5.3 to an infinite disjoint union ∞ ∞  Ui provided the sum ci μ(Ui ) converges absolutely. Furthermore, A = i=1

i=1

one may extend a measure as above to a measure on a σ -algebra of sets, define associated classes of measurable and integrable functions, etc. See [Halmos, 1950], [Hewitt-Ross, 1979], [Bourbaki, 2004]. We shall not need to do this, however, since the construction of the standard automorphic L-functions, which is the main theme of this book, only requires the integration of locally constant functions. (3) Because the compact open sets a + p n Z p form a basis of open sets for Q p , we are able to integrate any locally constant, compactly supported function using only the values of μ on these sets. Example 1.5.4 (Haar measure on Q p ) Let a ∈ Q p , n ∈ Z. We define   μHaar a + p n Z p = p −n . We also set dμHaar (x) = d x. Haar measure is (obviously) invariant under additive translations. Note that μHaar can be arbitrarily large on the compact sets a + p n Z p when n → −∞. We now give an example of a simple p-adic integral. Let s be a complex number with (s) > −1. Then the function |x|sp is a locally constant function on Q p − {0}. We compute the integral of |x|sp over Z p − {0}, the non-zero p-adic integers. Let       Z×p = 1 + pZ p ∪ 2 + pZ p ∪ · · · ∪ p − 1 + pZ p

Adeles over Q

14

denote the units (invertible elements) in Z p , which are characterized by the ∞  fact that u ∈ Z×p if and only if |u| p = 1. Clearly Z p − {0} = p n Z×p is a n=0   disjoint union and μHaar Z×p = p−1 . p Example 1.5.5 Let d x be the differential induced from the Haar measure as in Example 1.5.4. We have  |x|sp d x = Z p −{0}

 ∞ n=0

∞

|x|sp d x =

pn Z×p

p − 1 −n −ns p−1 p ·p = . p n=0 p(1 − p −1−s )

In Example 1.5.5 we have reduced the integral over Z×p − {0} to an infinite sum of integrals over compact sets which can be computed as in Definition 1.5.3. The condition (s) > −1 ensures that the  above infinite sum converges |x|sp d x = |x|sp d x since the integral over absolutely. We also note that Z p −{0}

the point {0} is 0.

Zp

As noted above, the Haar measure given in Example 1.5.4 is invariant by additive changes of variable. When we make a multiplicative change of variables, we get   f (ax) d x = |a|−1 f (x) d x. (1.5.6) p Qp

Qp

Definition 1.5.7 (Multiplicative Haar measure on Q×p ) Let d x be as in Example 1.5.4. For x ∈ Q×p = Q p − {0}, we define d×x =

−1 d x  dx p = 1 − p −1 . p − 1 |x| p |x| p

The differential d × x satisfies the following two important properties. First of all, it is invariant under transformations x → yx for any fixed y ∈ Q×p . That is, for any locally constant function f such that the integral  f (x)d × x Q×p

converges, and for any y ∈ Q×p , we have   × f (x y)d x = Q×p

Q×p

f (x)d × x,

for any y ∈ Q×p . (The general case reduces to the special case when the function f is the characteristic function of an open ball, which is a straightforward

1.6 p-adic Fourier transform

15

exercise.) Thus, d × x is invariant under multiplication, which is why it is called a multiplicative Haar measure. Secondly, it satisfies d × x = 1. ×

Zp

Example 1.5.8 Let d × x be as in Definition 1.5.7. Then for (s) > 0, we have  Z p −{0}

 ∞ ∞ −1  dx 1 |x|sp d × x = 1 − p −1 |x|sp = p −ns = . |x| p 1 − p −s n=0 n=0 pn Z×p

1.6 p -adic Fourier transform In order to motivate the p-adic Fourier transform we begin by reviewing the Fourier transform on R. Let f : R → C be a function. We shall say that f has rapid decay at ∞ if for each m > 0 there exists a fixed constant C > 0 such that |x|m ∞ | f (x)|C < C for |x|∞ sufficiently large. Here |x|∞ is the ordinary absolute value on R as in (1.1.5), and | |C is the ordinary absolute value on C. A function f : R → C is said to be Schwartz if it is smooth (infinitely differentiable) and all of its derivatives have rapid decay at infinity. The Fourier transform of f , denoted  f , is defined by  f (x) =

 f (y)e∞ (−x y) dy,

(1.6.1)

R

where e∞ (x) = e2πi x for all x ∈ R. Theorem 1.6.2 (Fourier inversion on R) Let f : R → C be a Schwartz function. Let  f be defined by (1.6.1). Then  f is again a Schwartz function and  f (x) = f (−x). Proof See [Lang, 1983]. We want to generalize (1.6.1) and Theorem 1.6.2 to p-adic fields. The first step required to do this is to obtain an analogue of the additive character e∞ (x) which satisfies e∞ (x + y) = e∞ (x)e∞ (y) for all x, y ∈ R. Accordingly, we define the function e p : Q p → C. Definition 1.6.3 (Additive character on Q p ) Let e p : Q p → C be defined by e p (x) = e−2πi{x}

Adeles over Q

16 where

{x} =

⎧ −1 ∞ ⎪ ⎪ ⎨ ai pi , if x = ai pi ∈ Q p with k > 0, 0 ≤ ai ≤ p − 1, ⎪ ⎪ ⎩

i=−k

i=−k

0,

otherwise.

Remarks We think of {x} as the fractional part of x ∈ Q p . Clearly e p (x + y) = e p (x) · e p (y) for all x, y ∈ Q p . Note the minus sign in the definition of e p . The minus sign plays an important role in the adelic Fourier theory. Let us mention that Tate and some other authors include a minus sign in e∞ , rather than e p . Lemma 1.6.4 Let n ∈ Z. Then  −n  p , e p (x) d x = 0,

if n ≥ 0, otherwise.

pn Z p

Proof If n ≥ 0 then the integrand is identically equal to 1, and hence the integral is equal to the measure of the domain of integration, which is p −n . If n < 0, then there exists y ∈ p n Z p such that e p (y) = 1. Making the change of variables x → x + y in the integration, we obtain 

 e p (x) d x = e p (y) pn Z

e p (x) d x. pn Z

p

p

Since e p (y) =/ 1, the integral must be 0. Proposition 1.6.5 Let n ∈ Z. Then  pn Z×p

⎧ −n   1 − p −1 , ⎪ ⎨ p e p (x) d x = −1, ⎪ ⎩ 0,

if n ≥ 0, if n = −1, if n < −1.

Proof Since Z×p = Z p − pZ p it follows, after multiplying by p n , that we may write p n Z×p = p n Z p − p n+1 Z p . Consequently 



pn Z×p

 e p (x) d x −

e p (x) d x = pn Z p

e p (x) d x. pn+1 Z p

(1.6.6)

1.6 p-adic Fourier transform

17

If n ≥ 0 then e p (x) ≡ 1 in both of the integrals on the right side of (1.6.6 ) so the value of the integral is given by μ H aar ( p n Z p ) − μ H aar ( p n+1 Z p ) =

  1 1 − n+1 = p −n 1 − p −1 . pn p

If n = −1, then e p is non-trivial on p n Z p and trivial on p n+1 Z p , so the value of the integral in (1.6.6 ) is just 0 − μ H aar (Z p ) = −1. If n < −1, then since e p is non-trivial in both of the integrals on the right side of (1.6.6 ) the integral is just 0 in this case.  Proposition 1.6.7 Let  1 A (x) =

1,

if x ∈ A,

0,

otherwise,

denote the characteristic function of a subset A ⊂ Q p . Let n ∈ Z. Then we have  −n  p , if y ∈ p −n Z p , 1 pn Z p (x) · e p (−x y) d x = 0, otherwise. Qp

Proof If y ∈ p −n Z p then x y ∈ Z p for all x ∈ p n Z p , so the integral is  p −n Z p then e p is non-trivial and the integral just pn Z p d x = p −n . If y ∈ vanishes.  Theorem 1.6.8 (Fourier inversion on Q p ) Let f : Q p → C be a locally constant compactly supported function as in Definition 1.5.2. Let  f be defined by   f (y)e p (−x y) dy. f (x) = Qp

Then  f is again a locally constant compactly supported function and  f (x) = f (−x). Proof We first show that  f is a locally constant compactly supported function. Every locally constant compactly supported function on Q p can be expressed as a finite linear combination of characteristic functions of compact open sets of the form a + p n Z p with a ∈ Q p . and n ∈ Z. Since integration is a linear function it suffices to show that the Fourier transform of the characteristic function 1a+ pn Z p is again a locally constant compactly supported function. We compute

Adeles over Q

18  1a+ pn Z p (y) =



 1a+ pn Z p (x) · e p (−x y) d x =

Qp

e p (−x y) d x a+ pn Z



=

p

  e p − (a + x)y d x

pn Z p



= e p (−ay)

e p (−x y) d x pn Z

= e p (−ay) p

p

−n

· 1 p−n Z (y), p

(1.6.9) where the last step in the above calculation follows from Proposition 1.6.7. To show that  f (x) = f (−x) it suffices to check it for the case that f = 1a+ pn Z p is a characteristic function of the compact open set a + p n Z p . It follows from (1.6.9) that for any y ∈ Q p ,   1a+ pn Z p (y) =  1a+ pn Z p (x) · e p (−x y) d x Qp

= p −n

 1 p−n Z (x) · e p (−(a + y)x) d x p

Qp

= 1 pn Z p (a + y), where the last step follows from Proposition 1.6.7 (with n replaced by −n). But 1 pn Z p (a + y) = 1a+ pn Z p (−y) because − p n Z p = p n Z p . 

1.7 Adelic Fourier transform Recall Definition 1.3.1 which states that the adele ring AQ is defined by 

 AQ := {x∞ , x2 , x3 , . . . }  xv ∈ Qv (∀ v ≤ ∞), xv ∈ Zv ,

 (∀ but finitely many v) .

In order to define a Fourier transform on the global ring AQ it is first necessary to construct an appropriate additive character as in the local Definition 1.6.3. Definition 1.7.1 (Additive adelic character) We shall define an additive adelic character e : AQ → C as follows. For x = {x∞ , x2 , x3 , . . . } ∈ AQ let e(x) =

 v≤∞

ev (xv )

1.7 Adelic Fourier transform

19

with e p (x p ) = e−2πi{x p } as in (1.6.3) if p < ∞ and e∞ (x∞ ) = e2πi x∞ if v = ∞. Note that only finitely many of the terms in the product are not equal to 1. Proposition 1.7.2 The function e(x) which is defined in Definition 1.7.1 satisfies the following two properties: (1) additivity: e(x + y) = e(x) · e(y) for all x, y ∈ AQ ; (2) periodicity: e(x + α) = e(x) for all x ∈ AQ and α ∈ Q. Proof (1) The additivity follows from the additivity of each of the local exponentials e∞ , e2 , e3 , . . . as explained in the remark after Definition 1.6.3. (2) Next we show the periodicity. Let a α = f1 f2 f p1 p2 · · · p  f

f

f

where a ∈ Z and p1 , p2 , . . . , p are primes and p1 1 p2 2 · · · p  is the prime factorization of the denominator of α. Now there exist integers b1 , b2 , . . . b such that b1 b2 b 1 = f1 + f2 + · · · + f . f1 f2 f p1 p2 · · · p p1 p2 p Note that this partial fraction decomposition easily follows from the theorem of Euclid which says that for any two non-zero integers r, s there exist integers x, y such that r x +sy = (r, s), where (r, s) denotes the greatest common divisor of r and s. It follows that e(x + α) = e(x)e(α) and e(α) = e2πiα

 

e pi (α) = e2πiα

i=1

 

fi

e−2πiabi / pi = 1.

i=1

Here we have crucially used the minus sign in Definition 1.6.3 and also used the fact that e p (x + z) = e p (x) for all x ∈ Q and any p-adic integer z ∈ Z p . For example, in Q p1 , the element b2f2 + · · · + bf is a p-adic integer (see Example 1.2.7). 

p2

p

We now consider adelic functions f : AQ → C. We would like to extend the definition of Fourier transform to adelic functions and, in addition, we want to define a suitable space S of adelic functions so that the Fourier transform  f of f ∈ S is given by an absolutely convergent integral and  f is again in S. Such a space S was introduced by Laurent Schwartz and Franc¸ois Bruhat in the more general context of locally compact abelian groups [Osborne, 1975]. The above goals can be achieved by making the following definitions.

Adeles over Q

20

Definition 1.7.3 (Factorizable function) An adelic function f : AQ → C is factorizable if there exist local functions: f v : Qv → C (∀v ≤ ∞) where f p ≡ 1 on Z p for all but finitely many p < ∞, and where 

f (x) = f ∞ (x∞ ) f 2 (x2 ) f 3 (x3 ) · · · =

f v (xv ).

v

for all x = {x∞ , x2 , x3 , . . . } ∈ AQ . Definition 1.7.4 (Adelic Bruhat-Schwartz function) An adelic function is said to be Bruhat-Schwartz if it can be expressed as a finite linear combina tion (with complex coefficients) of factorizable functions f = v≤∞ f v , as in Definition 1.7.3, where the f v satisfy the following conditions: (1) f ∞ is Schwartz (as defined in the beginning of Section 1.6); (2) each f p is a locally constant compactly supported function at all p < ∞; (3) f p is the characteristic function of Z p for all but finitely many p < ∞. Next, we wish to define an integral on a suitable space of adelic functions.  Definition 1.7.5 (Adelic integral) Suppose that f = v f v is a factorizable function as in Definition 1.7.3, that f ∞ is an integrable function on R, that for each p, the function f p is the characteristic function of a compact set C p , and that C p = Z p for all p outside of some finite set S. Then we define the adelic integral 

 f (x) d x = AQ

R

f ∞ (x∞ ) d x∞ ·

 p∈S

Qp

f p (x p ) d x p .

We further define the adelic integral of finite or countably infinite linear combinations of factorizable functions of the same type, with disjoint supports by linearity, provided (in the infinite case) that the corresponding sum is absolutely convergent. Finally, we define 

 f (x) d x =

AQ

U

f (x)1U (x) d x,

for any subset U of AQ such that f · 1U is integrable as defined above. Remarks (1) Note that for a function f as in Definition 1.7.5, the p-adic integral  Qp

f p (x p ) d x p

1.7 Adelic Fourier transform

21

is simply the p-adic Haar measure of the set C p . In particular, it is equal to one for all p not in S. This means that we replace S by a larger set S , the product is the same. (2) Suppose that f takes values in the positive reals, and that K i , i = 1, 2, 3, . . . is an increasing family of compact subsets of AQ , such that the union is all of AQ . Then it is easily verified that 

 f (x) d x = sup

AQ

f (x) d x.

i

Ki

To extend to complex valued f, we write f = (u + − u − ) + i(v + − v − ) where u + , u − , v + , v − are positive-real-valued.

 Lemma 1.7.6 (Factorization of adelic integral) Suppose that f = v f v is a factorizable function as in Definition 1.7.3, f ∞ is an integrable function on R, that for each p, the function f p is a locally constant function as in Definition 1.5.2 with a convergent p-adic integral as in Definition 1.5.3, and that for almost all p, the function f p is identically equal to 1 on Z p . Then 

 f (x) d x =

AQ

R

f ∞ (x∞ ) d x∞ · lim

N →∞

provided the limit lim

N →∞

 p
Qp

 p
Qp

f p (x p ) d x p ,

| f p (x p )| d x p ,

is convergent. Proof For f with positive real values, this follows from remark (2) above, together with the observation that any compact subset of AQ is contained in R·



Qp ·

p


Zp

p≥N

for some N . The supremum over all compact sets corresponding to one fixed N , is   f ∞ (x∞ ) d x∞ · f p (x p ) d x p . R

p
Qp

Then taking a supremum over all N yields the limit. More generally, this follows from Lebesgue’s dominated convergence theorem [Lang, 1969]. 

Adeles over Q

22

  Remark At first glance, v Qv f v (xv ) d xv looks like it ought to be the integral of f over the full infinite cartesian product of the fields Qv – a space which is properly larger than AQ . However, one may see that this is indeed the correct definition of an integral over AQ by reasoning as follows. Suppose that f is a positive function. Then its integral over all of AQ will be the supremum of the integrals over all compact subsets of AQ . If we restrict x to any fixed compact subset of AQ , then x p is restricted to Z p for all p outside some finite set S. However, this set S depends on the compact set. Thus taking a supremum over all compact sets of AQ is equivalent to taking a supremum over compact sets of Qv for each v and taking a supremum over the finite set S, and does indeed yield the full infinite product of local integrals. Definition 1.7.7 (Adelic Fourier transform) Let f : AQ → C be a factorizable adelic Bruhat-Schwartz function as in Definition 1.7.4. Let e : AQ → C be given as in Definition 1.7.1. Then we define the Fourier transform  f by the formula   f v (yv )ev (−xv yv ) dyv . f (x) = Qv

v

This definition may be extended to arbitrary adelic Bruhat-Schwartz functions by linearity. Note that by Proposition 1.6.7 and Definition 1.7.4 (3), the integral  Qv

f v (yv )ev (−xv yv ) dyv

has the value 1 for all but finitely many v so the infinite product above is well  defined. If we let dy = v dyv , then we may think of dy as a differential on the adeles and we may succinctly write  f (x) =

 f (y)e(−x y) dy. AQ

Theorem 1.7.8. (Fourier inversion on the adeles) Let f : AQ → C be a Bruhat-Schwartz function as in Definition 1.7.4. Let  f denote the Fourier transform as in Definition 1.7.7. Then  f is again a Bruhat-Schwartz function and  f (x) = f (−x). Proof This follows immediately from Definitions 1.7.4 and 1.7.7, and Theorems 1.6.2 and 1.6.8. 

1.8 Fourier expansion of periodic adelic functions

23

1.8 Fourier expansion of periodic adelic functions A function f : R → C is said to be periodic if f (x + n) = f (x) for all integers n. We want to generalize this notion to the adele group and develop a Fourier theory on the adele group. Definition 1.8.1 (Periodic adelic function) Let f : AQ → C be a complex valued adelic function. The function f is said to be periodic if f (x + α) = f (x) for all x ∈ AQ and all α ∈ Q. We have shown in Proposition 1.7.2 that the additive adelic character e : AQ → C given in Definition 1.7.1 is periodic. The Fourier theory of locally compact groups tells us that any periodic adelic function (satisfying certain smoothness hypotheses) can be represented as an infinite linear combination of the form bα e(αx) (1.8.2) α∈Q

with bα ∈ C. We shall present here a short simple proof first shown to the first author by Jacquet [Anshel-Goldfeld, 1996], (see also [Garrett, 1990]). A natural way to construct periodic adelic functions is to take all translates by elements in Q of a given adelic Bruhat-Schwartz function. Proposition 1.8.3 (Periodized Bruhat-Schwartz function) Let h : AQ → C be an adelic Bruhat-Schwartz function as in Definition 1.7.4. Then the sum

h(x + α),

(x ∈ AQ ),

α∈Q

converges absolutely and uniformly on compact subsets of AQ to a periodic adelic function f which is termed a periodized Bruhat-Schwartz function. Proof It is enough to prove the theorem for Bruhat-Schwartz functions h which are factorizable as in Definition 1.7.3. Following Definition 1.7.4, we may represent  h v (xv ) h(x) = v≤∞

where x = {x∞ , x2 , x3 , . . . } and h p is the characteristic function of Z p for all but finitely many p < ∞. Fix x = {x∞ , x2 , x3 , . . . } ∈ AQ . Let S = {∞, p1 , p2 , . . . , p } denote the finite set of primes such that x p ∈ Z p and h p is the characteristic function of Z p for p ∈ S. If α ∈ Q, it follows that h p (x p + α) = 0 for p ∈ S unless α is an integer in Z p . Since h v is a locally

Adeles over Q

24

constant compactly supported function for v ∈ S this implies that there exists a rational integer M so that h(x + α) = 0 unless α = Mn with n ∈ Z. Therefore (at least formally)  n . h(x + α) = h x+ M α∈Q n∈Z

  Finally, for fixed x ∈ AQ we must have that h x + Mn has rapid decay in n as n → ±∞. This is because h ∞ is a classical Schwartz function and for the finitely many primes v ∈ S the function h v is absolutely bounded. In all other cases h v is either 1 or 0. The stated uniformity can be obtained because M can be chosen independent of x, for x in a compact set.  Definition 1.8.4 (Smooth adelic function) An adelic function f : AQ → C is said to be smooth if for any point x0 ∈ AQ , there exists an open set U U U : R → C such that f (x) = f ∞ (x∞ ) (containing x0 ) and a smooth function f ∞ for all adeles x = {x∞ , x2 , . . . } ∈ U. Remark Using the fact that AQ is the union of a countable increasing family of compact sets, it is not difficult to show that any smooth adelic function is a countable linear combination of functions of the type considered in Definition 1.7.5, with disjoint supports. In particular, the adelic integral of a smooth adelic function is defined, provided the relevant infinite sum is convergent. We now show that every smooth periodic adelic function can, in fact, be realized as a periodized Bruhat-Schwartz function. Proposition 1.8.5 (Smooth + periodic =⇒ periodized Bruhat-Schwartz) Let f : AQ → C be smooth as in Definition 1.8.4. Assume that f (x + α) = f (x),

∀ α ∈ Q, x ∈ AQ .

Then there exists an adelic Bruhat-Schwartz function h : AQ → C, (as in Definition 1.7.4) such that f (x) =



h(x + α).

α∈Q

Proof Assume there exists an adelic Bruhat-Schwartz function h 0 : AQ → C such that   h 0 (x + α) = 1, ∀ x ∈ AQ . (1.8.6) α∈Q

Then, if f : AQ → C is any smooth periodic adelic function, we see that we may define h(x) := h 0 (x) f (x).

1.8 Fourier expansion of periodic adelic functions

25

Consequently, Proposition 1.8.5 immediately follows if we can construct a Bruhat-Schwartz function h 0 so that (1.8.6) holds, and, in addition, we could show that the function h(x) = h 0 (x) f (x) was an adelic Bruhat-Schwartz function as in Definition 1.7.4. To get h 0 , we first need a Schwartz function h 0,∞ : R → R satisfying ∞

h 0,∞ (x + n) = 1,

(∀ x ∈ R).

(1.8.7)

n=−∞

Such a function may be constructed, for example, by letting g(x) be any smooth function such that supp (g) = [− 14 , ∞) and g(x) = 1 for all x ≥ 0. Then put h 0,∞ (x) = g(x) − g(x − 1).  Now, for x = {x∞ , . . . , x p , . . . } ∈ AQ define h 0 (x) = h 0,∞ (x∞ )· p 1Z p (x p ) where 1Z p denotes the characteristic function of Z p . We need to show that h 0  satisfies (1.8.6), for any x ∈ R × p Z p . Any other x may be written as β + y  for y ∈ R × p Z p and β ∈ Q, so one has only to make a change of variables in the summation. But this reduces to (1.8.7) and we are done provided we can show that h(x) = h 0 (x) f (x) is an adelic Bruhat-Schwartz function as in Definition 1.7.4. The proof that h(x) is an adelic Bruhat-Schwartz function will follow directly from Lemma 1.8.8 below.  Lemma 1.8.8 Every compactly supported smooth adelic function as in Definition 1.8.4 is an adelic Bruhat-Schwartz function as in Definition 1.7.4. Proof Let h : AQ → C be smooth and compactly supported. We first prove a refinement of smoothness. Recall the Definition 1.3.3 of the finite adeles  

 Afinite = {x2 , x3 , . . . }  {0, x2 , x3 . . . } ∈ AQ . Claim: For all xfinite ∈ Afinite there exists an open set Ufinite of Afinite containing xfinite and a smooth function h ∞ : R → C such that h(y) = h ∞ (y∞ ) for every y = {y∞ , yfinite } in R × Ufinite . Furthermore, Ufinite may be assumed to be of the  form p U p where U p = Z p for almost all p, and is of the form a + p n Z p for some a ∈ Q p and n ∈ Z at the remaining p. Proof of Claim First, fix xfinite ∈ Afinite . It is fixed for the entirety of this proof. Consider all points of the form x = {x∞ , xfinite } which are in the support of h. Each element of this set is contained in a set U with the following properties: (1) U is open, and there is a smooth function h U∞ such that h(x) = h U∞ (x∞ ) for all x ∈ U. (From smoothness.)  (2) U = U∞ · p U p where U∞ is open, U p = Z p for almost all p, and is of the form a + p n Z p for some a ∈ Q p and n ∈ Z at the remaining p. (Every open set contains one of this form.)

Adeles over Q

26

The set of all points of the form x = {x∞ , xfinite } which are in the support of h is a compact set. Using compactness, we get a finite subcover consisting of sets with properties (1) and (2). Let us say that they are U (1) , . . . U (N ) and that  (i) N × p U p(i) . Let U p = ∩i=1 U p(i) . Suppose U p(i) = ai + p ni Z p for each U (i) = U∞ p, i. (So for almost all p, i the value of n i is 0 and ai is an integer, so that this is Z p !) Then U p = a + p n Z p where n = maxi n i and a = ai0 where i 0 is any of the values of i such that n i = n. The sets U p(i) can not be disjoint since all of the  (i) sets Ufinite := p U p(i) contain our fixed xfinite . If two sets of the form a + p n Z p (different as same n) intersect, then they are the same.  Now, we simply define Ufinite = p U p , and we also define (i)

h ∞ (x∞ ) = h U∞ (x∞ ),

  ∀x∞ such that {x∞ , xfinite } ∈ U (i) .

One must check that this gives a well defined function, because the sets U (i) (i) overlap. From the definition of h U∞ as in property (1) above, this function may also be described as   N  (i) ∀ x∞ ∈ U∞ . h ∞ (x∞ ) = h({x∞ , xfinite }), i=1

From the second description, it is clear that the function is well defined. From (i) . It may be extended the first, it is clear that it is smooth at every point in U∞ to a function on all of R by setting it equal to zero everywhere else, and one then has (∀ x∞ ∈ R). h ∞ (x∞ ) = h({x∞ , xfinite }), This completes the proof of the claim. Now we turn to the proof of the main Lemma. The support of h is contained  in a set of the form v K v where K v is compact for all v and K p = Z p for  almost all p < ∞. Let K finite = p K p ⊂ Afinite . It is a compact set and is covered by the sets Ufinite from the refined form of smoothness, so there is a finite subcover. Recall that if a1 + p n 1 Z p and a2 + p n 2 Z p intersect, then one of them contains the other. If n 1 = n 2 they coincide. Otherwise, suppose n 1 > n 2 . Then the ball a2 + p n 2 Z p is also a coset of the ideal p n 2 Z p and a finite disjoint union of cosets α + p n 1 Z p with a1 + p n 1 Z p . Using these remarks, it is clear that we may subdivide the elements of our finite subcover to obtain a cover with sets which are of the same form as in the claim, and pairwise disjoint. (1) (M) , . . . , Ufinite , say, and the corresponding funcLet us number the sets Ufinite (1) (M) tions h ∞ , . . . , h ∞ . Then   (i) . ∀ x∞ ∈ R, xfinite ∈ Ufinite h({x∞ , xfinite }) = h (i) ∞ (x ∞ ),

1.8 Fourier expansion of periodic adelic functions

27

But then because the sets are pairwise disjoint this is the same as h(x) =

M

h (i) ∞ (x ∞ ) · 1U (i)



finite

xfinite



i=1

=

M

h (i) ∞ (x ∞ ) ·

i=1



  1U p(i) x p .

p

(Recall that = Z p for almost all p for each i.) In this final form, h is seen to be Bruhat-Schwartz, as defined in Definition 1.7.4.  U p(i)

We shall now present a simple proof of the Fourier expansion (1.8.2) which holds for smooth periodic adelic functions as in Proposition 1.8.5. Recall Proposition 1.4.5 which states that a fundamental domain for Q\AQ is given   by [0, 1) · p Z p . This allows us to define an integral Q\AQ as an integral over this fundamental domain. If f is factorizable, this integral will factor as the 1   infinite product of local integrals 0 · p Z p . If D is any other fundamental  domain for Q\AQ , and f is periodic, then it may be shown that Q\AQ is also equal to the integral over D . Lemma 1.8.9 Let f : AQ → C be a smooth adelic function, as in Definition 1.8.4, such that the adelic integral  f (x)d x AQ  given in Definition 1.7.5 is convergent. Let D = [0, 1) · Z p denote the fundamental domain for Q\AQ given in 1.4.5. Then • for any α ∈ Q, the function f · 1α+D is integrable, • the infinite sum  f (x) d x, α∈Q α+D

converges absolutely to

 f (x) d x, AQ

independently of the order in which the sum over Q is performed. Proof The first statement is clear, since | f · 1U | ≤ | f | for any U. We have only to observe that for any α ∈ Q, the set α + Z p is compact for all p and equal to Z p for almost all p, while the set α + [0, 1) is Lebesgue-measurable. The second statement follows easily from remark (2) after Definition 1.7.5.  Theorem 1.8.10 (The Fourier expansion of smooth periodic adelic functions) Let f : AQ → C be a smooth periodic adelic function as in Definition 1.8.4. Then  f (x) = f α · e(αx) α∈Q

Adeles over Q

28

where the above sum converges absolutely for all x ∈ AQ and 

 fα =

f (x)e(−αx) d x =  h(α),

( for all α ∈ Q).

Q\AQ

Here h is any adelic Bruhat-Schwartz function such that f (x) = β∈Q h(x +β) as in Proposition 1.8.5, and  h is the adelic Fourier transform of h as in Definition 1.7.7. Proof The proof is presented in 6 steps. By Proposition 1.8.5, we may assume that h(x + β) (x ∈ AQ ) f (x) = β∈Q

for some adelic Bruhat-Schwartz function h : AQ → C. It is enough to give the proof for the case when h is factorizable as in Definition 1.7.3. h(α). Step 1: We prove  f α = This follows from the computation shown below. We use the fact that e(αβ) = 1 for αβ ∈ Q (Proposition 1.7.2) and the fact that the union of all rational translates of the fundamental domain for Q\AQ is just AQ (Proposition 1.4.5).   h(x + β)e(−αx) d x fα = Q\AQ β∈Q

=





h(x + β)e(−αx) d x

β∈Q Q\A Q

=





h(x)e(−αx)e(αβ) d x

β∈Q −β+ Q\A Q



h(x)e(−αx) d x

= AQ

= h(α). Step 2: We show there exists fixed N ∈ Z such that  f α = 0 unless α = Nn with n ∈ Z. We may think of the minimal positive N satisfying this condition as the conductor of f . If p is a prime and h p is the characteristic function of Z p , then it follows from Proposition 1.6.7 that   h p (x p )e p (−αx p ) d x p = e p (−αx p ) d x p = 0 Qp

Zp

1.8 Fourier expansion of periodic adelic functions

29

unless |α| p ≤ 1. Since h p = 1Z p is the characteristic function of Z p for all but finitely many primes p, it follows that  h(α) = 0 unless α = Nn (with n ∈   ai Z) where N = pi and p1 , p2 , . . . , p are the finitely many primes where i=1

h pi =/ 1Z pi for i = 1, 2, . . . , . The exponents ai ∈ Z are determined by the fact that each h pi is a locally constant compactly supported function for i = 1, 2, . . . , . Step 3: Next we show that there exists a fixed constant C > 0 (depending at most on f ) such that      f Nn  < Cn −2 where | | denotes the ordinary absolute value on C. This will establish the  f α e(αx). absolute convergence of the Fourier series α∈Q

This follows immediately from the fact that the Fourier transform of an adelic Bruhat-Schwartz function is again a Bruhat-Schwartz function which has rapid decay properties at ∞. From the properties of a Bruhat-Schwartz function, one may actually obtain the stronger bound Cn −B for any fixed constant B > 0. Step 4: It is enough to prove that f (0) =



 f α.

(1.8.11)

α∈Q

To see this fix x0 ∈ AQ and define a new function g(x) = f (x + x0 ) for x ∈ AQ . Then g is again a periodized Bruhat-Schwartz function, so that by (1.8.11), we have  gα . f (x0 ) = g(0) = α∈Q

But



 gα =

 f (x + x0 )e(−αx) d x = e(αx0 )

Q\AQ

f (x)e(−αx) d x = e(αx0 )  f α,

Q\AQ

from which it follows that f (x0 ) =



 f α e(αx0 ).

α∈Q

Step 5: It is enough to prove (1.8.11) for functions f which satisfy the condition f (0) = 0. If this is not the case, consider the new function f (x) − f (0) which vanishes at 0. Here, we use the fact that (1.8.11) is easy to prove for constant functions.

Adeles over Q

30

Step 6: We are reduced to proving that

 f Nn = 0

(1.8.12)

n∈Z

where f satisfies f (0) = 0. Here N is the conductor of f as in Step 2. Define a new function g(x) =

f (x) . 1 − e (x/N )

By definition, g is again a smooth periodic adelic function. We compute   n   f Nn = f (x)e − x d x N Q\AQ



= Q\AQ



= Q\AQ

 n    x  g(x)e − x d x 1−e N N   n  g(x)e − x d x − N

Q\AQ

  n−1 x dx g(x)e − N

= g Nn − g n−1 . N

It follows that

    g Nn −  g n−1 = 0, f Nn = N

n∈Z

n∈Z

since the latter is a telescoping sum where all the terms cancel.



1.9 Adelic Poisson summation formula Let h be an adelic Bruhat-Schwartz function as in Definition 1.7.4. The adelic Poisson summation formula states that α∈Q

h(α) =



 h(α)

(1.9.1)

α∈Q

where  h is the adelic Fourier transform of h as defined in Definition 1.7.7. For applications, we require the following generalization of (1.9.1):   1  α h(αy) = h |y|A α∈Q y α∈Q



(1.9.2)

Exercises for Chapter 1 which holds for any idele y = {y∞ , y2 , y3 , . . . } and where |y|A = the adelic absolute value.

31  v

|yv |v is

Proof of (1.9.1) Define the periodized Bruhat-Schwartz function h(x + α) f (x) = α∈Q

as in Proposition 1.8.3. The Fourier expansion in Theorem 1.8.10 can be applied to f and we obtain  h(α) · e(αx). (1.9.3) f (x) = α∈Q

Letting x = 0 in (1.9.3) immediately establishes (1.9.1).  Proof of (1.9.2) For a fixed idele y, the function g(x) = h(x y) is again an adelic Bruhat-Schwartz function. The result follows on using the relation   1  x .  h  g (x) = |y|A y

Exercises for Chapter 1 1.1 Show that any absolute value on a finite field is trivial. 1.2 Let F be a field and write n for the element of F given by adding 1 to itself n times. Prove that an absolute value | · | is non-archimedean if and only if |n| ≤ 1 for all n ∈ Z. Hint: For the sufficiency statement, compare |x + y|n and max(|x|, |y|)n .  1.3 For a ∈ Q p and r > 0, define B(a, r ) = {x ∈ Q p  |x − a| p ≤ r } to be the closed ball of radius r centered at a. For example, B(a, p −m ) = a + pm Z p . (a) For any b ∈ B(a, r ), show that B(a, r ) = B(b, r ). That is, every point of a closed ball in Q p can act as the center. (b) Show that any two closed balls in Q p are either disjoint, or else one contains the other. (c) Show that closed balls are also open in the p-adic topology. (d) Find a set that is closed but not open, and one that is open but not closed. 1.4 Is the rational number 1−1 p an element of Z p ? What is its p-adic power series expansion? √ 1.5 Does −1 exist in Q3 ? In Q5 ? In Q2 ? Hint: Said another way, can one solve the equation x 2 = −1 in these fields?

Adeles over Q

32

1.6 This exercise characterizes all of the locally constant and compactly supported functions on Qv for v ≤ ∞. (a) For p a prime, show that any compact open subset of Q p is just a finite union of neighborhoods of the form a + p m Z p , where a ∈ Q p and m ∈ Z. (b) Suppose h : Q p → C is locally constant and compactly supported. Prove that h is a finite linear combination of characteristic functions of the form 1a+ pm Z p . (c) If h : R → C is locally constant and compactly supported, prove that h is identically zero. 1.7 This exercise proves that the topology on A× Q is strictly finer than the subspace topology induced by AQ . (a) Define an injective map i: A× Q → AQ × AQ via the rule i(x) = (x, x −1 ). Show that the topology on A× Q coincides with the subspace topology of i(A× ) ⊂ A × A . Q Q Q (b) Show that if V ⊂ AQ is an open subset, then V ∩ A× Q is open in the topology of A× . Q (c) Show that the sets in the basis for the topology specified after Definition 1.3.2 are not open in the subspace topology of A× Q ⊂ AQ .  1.8 Consider the adelic function f : AQ → C defined by f = v f v , where  2 exp(−π x∞ ), if v = ∞, f v (xv ) = 1Z p (x p ) + 1 p−1 + p2 Z p (x p ), if v = p is prime. Show that the Fourier transform  f is a well-defined function, but that  f is not Bruhat-Schwartz. Why does this not contradict Theorem 1.7.8?  Hint: It may be useful to know that an infinite product (1 + an ) converges if the series an converges absolutely. 1.9 Show that a smooth adelic function f : AQ → C is continuous (for the adelic topology). 1.10 Let α ∈ Q, and let e : AQ → C be the additive adelic character as in Definition 1.7.1. Prove that   1, if α = 0, e(αu) du = 0, if α = 0. Q\AQ

1.11 The goal of this exercise is to illustrate a technical detail from step  6 of the proof of Theorem 1.8.10. Suppose that h(x) = v h v (x) is a factorizable adelic Bruhat-Schwartz function, and suppose

Exercises for Chapter 1

33

f (x) = α∈Q h(x + α) has conductor N (as in step 2 of the proof of Theorem 1.8.10). We will show there is a smooth periodic adelic function g : AQ → C such that g(x) =

f (x) − f (0) , 1 − e(x/N )

(†)

whenever this expression makes sense. As each h v is a finite linear combination of characteristic functions, one can observe that it suffices in the proof of Theorem 1.8.10 to assume that each finite factor of h is of the form h p = 1a+ pm Z p for some a ∈ Q p and some integer m (that depend on the prime p). (a) Prove that the function f (N x) is smooth and periodic with conductor 1. Deduce that it suffices to prove (†) when N = 1. (b) Deduce from part (a) that it suffices to prove (†) when N = 1 and  1a( p)+Z p (x p ), h(x) = h ∞ (x∞ ) p

for some a( p) ∈ Q p such that a( p) = 0 for all but finitely many p. By replacing h(x) with h(x + β) for a clever choice of rational number β, show that we may even assume a( p) = 0 for all primes p.  (c*) Assume now that h = h ∞ p 1Z p and N = 1. When e(x) = 1, define g(x) by the formula (†); when e(x) = 1, set g(x) =

−1 h (n). 2πi n∈Z ∞

Show that g(x) is smooth and periodic. 1.12* In this exercise, we develop a few of the interesting analogies between the rational numbers Q and the field of rational functions F p (T ). It will require a little more in the way of abstract algebra than some of the other exercises. Let p be a prime number, and F p the field of p elements. The rational function field with coefficients in F p , denoted F p (T ), is the set of all quotients of the form f (T ) =

P1 (T ) , P2 (T )

where P1 , P2 are polynomials with coefficients in F p , and P2 is not the zero polynomial. Equip this set with the usual addition and multiplication operations of rational functions. If this concept is new to you, verify that F p (T ) is a field.

Adeles over Q

34

(a) Prove that any non-trivial absolute value on F p (T ) is equivalent (in the sense of Definition 1.1.3) to one of | · | Q or | · |∞ , defined as follows (i) Fix a monic irreducible polynomial Q ∈ F p [T ]. Since F p [T ] is a unique factorization domain, any non-zero rational function f ∈ F p (T ) can be written uniquely as f = Q r g for some integer r and some rational function g such that Q does not divide the numerator or denominator of g. Define an absolute value on F p (T ) by | f | Q = p −r deg(Q) and |0| Q = 0. (ii) Let f = P1 /P2 be a non-zero rational function. Define | f |∞ = p deg(P1 )−deg(P2 ) , where deg denotes the degree of a polynomial. Set |0|∞ = 0. (b) Let Q be a monic irreducible polynomial with coefficients in F p , let d = deg(Q), and let q = p d . Define Fq ((Q)) to be the field of formal Laurent series in the variable Q with coefficients in Fq : ∞    i  Fq ((Q)) = ai Q  N ∈ Z, ai ∈ Fq . i=N

This field has an absolute value defined by ∞      ai Q i  = q −N , (a N = 0).    i=N

Q

Prove that the completion of F p (T ) with respect to the absolute value |·| Q can be identified with Fq ((Q)). Hint: The field Fq arises naturally in this context as Fq = F p [T ]/(Q(T )). (c) Let Fq ((1/T )) be the ring of formal Laurent series in the variable 1/T . This field has an absolute value defined by ∞    1 i   ai (a N = 0).  = p −N ,   T  i=N

∞

Prove that the completion of F p (T ) with respect to the absolute value | · |∞ can be identified with F p ((1/T )). (d) Let the symbol v denote either ∞ or a monic irreducible polynomial Q ∈ F p [T ]. Show that the following product formula holds for any non-zero rational function f ∈ F p (T ):  | f |v = 1. v

Exercises for Chapter 1

35

(e) For each monic irreducible polynomial Q ∈ F p [T ], we define a subring of the completion by 

  O Q = f ∈ Fq ((Q))  | f | Q ≤ 1 . Similarly, we have O∞ =



   f ∈ F p ((1/T ))  | f |∞ ≤ 1 .

Define the rational function field adeles to be the restricted product (relative to the subgroups O Q ) of AF p (T ) = F p ((1/T )) ×



Fq ((Q)).

Q∈F p [T ] monic irreducible

Show that a fundamental domain for the additive action of F p (T ) on AF p (T ) is given by 

 1 O∞ × T



OQ .

Q∈F p [T ] monic irreducible

1.13* In this exercise we indicate all of the necessary adjustments in order to make sense of Fourier transforms, the Fourier inversion theorem, and the Poisson summation formula for the function field adeles AF p (T ) . Parts (d-g) are essentially verifying that the proofs in the case of AQ carry over to the function field setting. We continue with the notation from Exercise 1.12. ∞ ai (1/T )i for an element of F p ((1/T )). Define a (a) Write x = i=N function ψ∞ : F p ((1/T )) → C× by ψ∞ (x) = e−2πia1 / p . The map ψ∞ is well-defined if we interpret a1 as an integer modulo character on the additive group of p. Show that ψ∞ is a unitary   2 F p ((1/T )) and that ψ∞ (1/T ) O∞ = 1. (b) Let Q be a monic irreducible polynomial in F p [T ]. Write ∞ ai Q i for an element of Fq ((Q)). Via the identification x = i=N Fq = F p [T ]/(Q(T )), we can express the coefficient a−1 ∈ Fq as a−1 ≡ c0 + c1 T + · · · + cdeg(Q)−1 T deg(Q)−1

(mod Q(T )),

36

Adeles over Q where each c j ∈ F p . Define a function ψ Q : Fq ((Q)) → C× by ψ Q (x) = e2πic0 / p . Show that ψ Q is a unitary  character on the additive group of Fq ((Q)) and that ψ Q O Q = 1. (c) Now write x = {xv } ∈ AF p (T ) . Define an adelic unitary character ψ : AF p (T ) → C× by the formula ψ(x) =



ψv (xv ),

v

where the local characters ψv were defined in parts (a) and (b) above. Show that ψ is non-trivial and that ψ( f ) = 1 for every f ∈ F p (T ). Hint: Follow the strategy of Theorem 1.7.2. (d) Normalize the Haar measure on F p ((1/T )) so that μHaar (O∞ ) = p, and let d x = dμHaar (x). A Bruhat-Schwartz function on F p ((1/T )) is a locally constant compactly supported function. Define the Fourier transform of a Bruhat-Schwartz function f : F p ((1/T )) → C by 

 f (x) =

f (y)ψ∞ (−x y)dy. F p ((1/T ))

Show that  f is Bruhat-Schwartz and that the Fourier inversion formula holds:  f (x) = f (−x),

(x ∈ F p ((1/T ))).

(e) Let Q be a monic irreducible polynomial in F p [T ]. Normalize the Haar measure on Fq ((Q)) so that μHaar (O Q ) = 1 and d x = dμHaar (x). Define the Fourier transform of a Bruhat-Schwartz function f : Fq ((Q)) → C by  f (x) =

 f (y)ψ Q (−x y)dy. Fq ((Q))

Show that  f is Bruhat-Schwartz and that the Fourier inversion formula holds as in part (d). (f) Conclude that we may define a Fourier transform on adelic BruhatSchwartz functions (as in Definition 1.7.7) and that Fourier inversion holds as in Theorem 1.7.8.

Exercises for Chapter 1

37

(g) Define a smooth adelic function on AF p (T ) to be the same thing as an adelic Bruhat-Schwartz function. Modify the statements and proofs in Sections 1.8 and 1.9 to conclude that the Poisson summation formula holds for function field adeles. Hint: Replace Q by F p (T ) and replace Z by F p [T ]. This is especially important in Step 2 of the proof of Theorem 1.8.10. 1.14* The goal of this exercise is to sketch the proof of a special case of the statement “The Poisson summation formula for function field adeles yields the Riemann-Roch theorem for curves.” We continue with the notation from Exercises 1.12 and 1.13. Recall that the symbol v is allowed to denote either ∞ or a non-trivial monic irreducible polynomial Q ∈ F p [T ]. The degree of v, denoted deg(v), is given by  deg(v) =

if v = ∞,

1,

deg(Q), if v = Q is a monic irreducible polynomial.

Define a divisor D to be a finite formal linear combination of all possible symbols v with coefficients in the integers: D=



n v .v

(n v ∈ Z).

v

The degree of the divisor D is the integer deg(D) = We let

L(D) =

v

n v deg(v).

   f ∈ F p (T )  | f |v ≤ p n v deg(v) for all v .

A rational function f ∈ L(D) is bounded v-adically in terms of the divisor D. We will see that L(D) is an F p -vector space, and the essence of the Riemann-Roch theorem is that we can calculate its dimension. (a) Verify that L(D) is a finite-dimensional F p -vector space. (b) Write x = {xv } for an element of AF p (T ) . Define an adelic BruhatSchwartz function h D : AF p (T ) → C by the rule  h D (x) = Deduce that

1,

if |xv |v ≤ p n v deg(v) for all v,

0,

otherwise.

f ∈F p (T )

h D ( f ) = p dim L(D) .

Adeles over Q

38

(c) Show that the Fourier transform  h D is given by  h D (x) =

⎧ deg(D)+1 , ⎪ ⎨ p ⎪ ⎩

if |x∞ |∞ ≤ p −n ∞ −2 and |xv |v ≤ p −n v deg(v) for all v = ∞, otherwise.

0,

Hint: As h D is factorizable, one can compute the Fourier transforms for each v separately. Now follow Proposition 1.6.7. (d) Define a divisor by K = −2.∞. Show that

 h D ( f ) = p dim L(K −D)+deg(D)+1 .

f ∈F p (T )

(e) By the previous exercise, we know that the Poisson summation formula holds in this context. Deduce that the Riemann-Roch formula holds for the function field F p (T ): dim L(D) − dim L(K − D) = deg(D) + 1.

2 Automorphic representations and L-functions for GL(1, AQ )

2.1 Automorphic forms for GL(1, AQ ) Let AQ denote the adele ring over Q as in Definition 1.3.1. The key point is that G L(1, AQ ) is just A× Q , the multiplicative subgroup of ideles of Q. In conformity with modern notation we shall use the notation g = {g∞ , g2 , g3 , . . . } to denote an element of the group G L(1, AQ ). Here gv ∈ Q× v for all v and g p ∈ Z×p for all but finitely many finite primes p. × The multiplicative group Q× is diagonally embedded in A× Q and acts on AQ by left multiplication. Proposition 1.4.6 tells us that a fundamental domain for this action is given by Q× \A× Q = (0, ∞) ·



Z×p ,

p

where the product is over all finite primes p. An idelic function f : A× Q → C is said to be factorizable if it is determined × by local functions f v : Q× v → C (∀v ≤ ∞), where f p ≡ 1 on Z p for all but finitely many finite primes p, and where f (g) =

 v≤∞

f v (gv ),



 ∀ g = {g∞ , g2 , g3 , . . . } ∈ A× Q .

(2.1.1)

Note that this definition is slightly different than the notion of “factorizability of adelic functions” which was given in Definition 1.7.3. 39

40

Automorphic representations and L-functions for GL(1, AQ )

× Definition 2.1.2 (Unitary Hecke character of A× Q ) A Hecke character of AQ is defined to be a continuous homomorphism × ω : Q× \A× Q →C .

A Hecke character is said to be unitary if all its values have absolute value 1. A unitary Hecke character of A× Q is characterized by the following four properties:   ∀ g, g ∈ A× (i) ω(gg ) = ω(g)ω(g ), Q ;  (ii) ω(γ g) = ω(g), ∀γ ∈ Q× , ∀g ∈ A× Q ; (iii) ω is continuous at {1, 1, 1, . . . }. (iv) |ω|C = 1. Definition 2.1.3 (Moderate growth) We say that a function φ : A× Q → C is of moderate growth if, for each g = {g∞ , g2 , g3 , . . . } ∈ A× , there exist positive Q constants C and M such that    φ {tg∞ , g2 , g3 , . . . } 

C

< C (1 + |t|∞ ) M

for all t ∈ R. Definition 2.1.4 (Automorphic form) Fix a unitary Hecke character ω as in Definition 2.1.2. An automorphic form for G L(1, AQ ) with character ω is a function φ : G L(1, AQ ) → C which satisfies the following conditions:   (1) φ(γ g) = φ(g), ∀g ∈ A× , ∀γ ∈ Q× ; Q   × (2) φ(zg) = ω(z)φ(g), ∀g ∈ A× Q , ∀z ∈ AQ ; (3) φ is of moderate growth as in Definition 2.1.3. Let Sω denote the set of all automorphic forms for G L(1, AQ ) with character ω as in Definition 2.1.4. If c1 , c2 ∈ C, are arbitrary complex constants, and φ1 , φ2 ∈ Sω , then it is easy to see that c1 φ1 + c2 φ2 is again automorphic with character ω. The space Sω is, therefore, a vector space over C. Setting g = {1, 1, 1, . . . } it immediately follows from Definition 2.1.4 (2) that φ(z) = cω(z), with c = φ({1, 1, 1, . . . }). Thus Sω is a one-dimensional space. The reader may ask why we simply do not define an automorphic form as a Hecke character as in Definition 2.1.2? The reason is that we want to give a uniform definition of automorphic form for G L(n, AQ ) that holds for all n = 1, 2, 3, . . .. In the case of n = 1, Definition 2.1.4 (2) becomes superfluous since z, g

2.1 Automorphic forms for GL(1, AQ )

41

both lie in the same space. This is not the case for n > 1 as we shall see later. Definition 2.1.4 may seem rather imposing at first sight but it turns out that the automorphic forms for G L(1, AQ ) are just classical Dirichlet characters in disguise. For a fixed integer q > 1, a Dirichlet character χ (mod q) is a homomorphism (2.1.5) χ : (Z/qZ)× → C× . That is, χ (ab) = χ (a)χ (b),

  ∀a, b ∈ (Z/qZ)× .

(2.1.6)

Because a ϕ(q) = 1 for all a ∈ (Z/qZ)× , such a function must take values in the ϕ(q)th roots of unity (where ϕ is Euler’s ϕ function). In particular, |χ (a)|C = 1,

  ∀a ∈ (Z/qZ)× .

It is standard practice in analytic number theory (see [Davenport, 2000]) to lift a Dirichlet character χ to Z by defining a new function χ1 : Z → C which satisfies χ1 (a) = 0,

(∀a ∈ Z with (a, q) = 1) ;

χ1 (a + mq) = χ (a), χ1 (ab) = χ1 (a)χ1 (b),

(∀a, m ∈ Z with (a, q) = 1) ; (∀a, b ∈ Z) .

We shall follow the standard practice of denoting the character χ1 by the sym× bol χ . Remarkably, it is also possible to lift χ to the idele group A× Q . Since AQ is a purely multiplicative group, the value of the lifted character can never be 0. The result is an automorphic form as in Definition 2.1.4. We now explicitly describe and prove the existence of this lifting which was found by Tate and appeared in his thesis [Tate, 1950]. Definition 2.1.7 (Idelic lift of a Dirichlet character) Let χ be a Dirichlet character (mod p f ) as in (2.1.5), (2.1.6) where p f is a fixed prime power. We define the idelic lift of χ to be the unitary Hecke character χidelic : Q× \A× Q → × C defined as χidelic (g) = χ∞ (g∞ ) · χ2 (g2 ) · χ3 (g3 ) · · · , where χ∞ (g∞ ) =

⎧ ⎪ ⎨ ⎪ ⎩



 g = {g∞ , g2 , g3 , . . . } ∈ A× Q ,

1, 1,

χ (−1) = 1, χ (−1) = −1, g∞ > 0,

−1,

χ (−1) = −1, g∞ < 0,

42

Automorphic representations and L-functions for GL(1, AQ )

and where

⎧ m m × and v =/ p, ⎪ ⎨ χ (v) , if gv ∈ v Z v  −1 χv (gv ) = χ ( j) , if gv ∈ p k j + p f Z p with j, k ∈ Z, ( j, p) = 1, ⎪ ⎩ and v = p.

To see that Definition 2.1.7 actually defines a unitary Hecke character satisfying Definition 2.1.2 we make the following observations. First of all, for every prime v ≤ ∞, it is clear from the definition that χv (gv gv ) = χv (gv ) · χv (gv ) for all gv , gv ∈ Q× v . Consequently, χidelic must satisfy Definition 2.1.2 (i). Secondly, if  =/ p is any finite prime, then χ () = χ (). Also χ p () = χ ()−1 , χv ( p) = 1 for any finite prime v and χv () = 1 for any finite prime v =  and v = p. It follows that χidelic () = 1     for all primes . Also χidelic {−1, −1, −1, . . . , } = χidelic {1, 1, 1, . . . , } = 1. When combined with Definition 2.1.2 (i), this establishes Definition 2.1.2 (ii). Thirdly, we can see directly that the kernel of χidelic is an open neighborhood of {1, 1, 1, . . . }. It is also obvious that |χidelic | = 1 since χ has this property on (Z/ p n Z)× . The above observations establish that χidelic is indeed a Hecke character satisfying the four conditions of Definition 2.1.2. r fi More generally, every Dirichlet character χ (mod q), with q = i=1 pi , where p1 , p2 , . . ., pr are distinct primes and f 1 , f 2 , . . ., fr ≥ 1 can be factored as r  χ = χ (i) , i=1 f

where χ is a Dirichlet character of conductor pi i . It follows that χ may be lifted to a Hecke character χidelic on A× Q where (i)

χidelic =

r 

(i) χidelic .

(2.1.8)

i=1

Theorem 2.1.9 Every automorphic form φ on G L(1, AQ ), as in Definition 2.1.4, can be uniquely expressed in the form   ∀g ∈ A× φ(g) = c · χidelic (g) · |g|itA , Q , where c ∈ C, t ∈ R, are fixed constants, and χidelic is an idelic lift of a fixed Dirichlet character χ as in Definition 2.1.7 and (2.1.8). Here,  |gv |v is the idelic absolute value. if g = {g∞ , g2 , g3 , . . . }, then |g|A = v≤∞

2.1 Automorphic forms for GL(1, AQ )

43

Proof It follows from Definition 2.1.4 that we may take φ(g) = c · ω(g),

(∀g ∈ A× Q)

  with c = φ {1, 1, 1, . . . } , and where ω is a unitary Hecke character satisfying Definition 2.1.2. For each prime v ≤ ∞, consider the embedding i v (gv ) = {1, . . . , 1,



gv , 1, . . . }, !"#

 gv ∈ Q× v .

v th position

Then, if we define ωv (gv ) := ω(i v (gv )),

(∀gv ∈ Q× v ),

then ωv is a character of Q× v for every prime v ≤ ∞. Furthermore,  ωv (gv ) ω(g) =

(2.1.10)

v≤∞

where ω p (g p ) ≡ 1, (∀g p ∈ Z×p ) for all but finitely many primes p. This can be obtained from the continuity of ω at {1, 1, . . . }. The Hecke characters ω can then be determined if we can classify the local characters ωv for all primes v. First of all, every unitary continuous multiplicative character ω∞ : R× → C× is of the form ω∞ (g∞ ) = |g∞ |it∞ ,

or

ω∞ (g∞ ) = |g∞ |it∞ · sign(g∞ ), 

for some fixed constant t ∈ R, where sign(g∞ ) =

+1,

(2.1.11)

if g∞ > 0,

−1, if g∞ < 0. To continue the proof and classify the characters ωv for v < ∞, we introduce some definitions. Definition 2.1.12 (Unramified local character) Fix a finite prime p. A local character ω p , occurring in the decomposition (2.1.10), is said to be unramified if ω p (u) = 1 for all u ∈ Z×p . At ∞, the local character |g∞ |it∞ (for some fixed t ∈ R) is said to be unramified. Fix a finite prime p. The unramified local characters of Q×p are easy to describe. Every g p ∈ Q×p is an element of p m Z×p for some integer m. Let g p = p m · u with u ∈ Z×p . Then if ω p is unramified, it follows that ω p (g p ) = ω p ( p m · u) = ω p ( p m ) = ω p ( p)m .

(2.1.13)

So once we know the value of ω p at the one point p, we know its value everywhere.

44

Automorphic representations and L-functions for GL(1, AQ )

Definition 2.1.14 (Ramified local character and its conductor) Fix a finite prime p. We say a local character ω p , occurring in the decomposition (2.1.10), is ramified if ω p (u) =/ 1 for some u ∈ Z×p . The conductor of ω p is defined to be p k where k is the smallest positive integer such that 1+ p k Z p is contained in the kernel of ω p . We say the local character ω∞ is ramified if ω∞ (u) = −ω∞ (−u) for all u ∈ R× . Fix a finite prime p. The ramified local characters of Q×p can be described as follows. Let g p = p m · u with u ∈ Z×p . Then ω p (g p ) = ω p ( p m · u) = ω p ( p)m ω p (u). Since ω p (u) =/ 1 for some u ∈ Z×p it follows, by continuity, that the kernel of ω p contains an open subgroup of the form 1 + p k Z p . We shall assume k is minimal and term p k the conductor of ω p .    × Claim: Z×p / 1 + p k Z p ∼ = Z/ p k Z . To see this note that the multiplicative   cosets in Z×p / 1 + p k Z p are all of the form   j 1 + p k Z p = j + p k Z p , (1 ≤ j < p k , ( j, p) = 1). Furthermore       j 1 + pk Z p · j 1 + pk Z p = ( j · j ) 1 + pk Z p ,  × from which it follows that we may assume j, j , ( j · j ) ∈ Z/ p k Z . The above claim implies that the ramified local character ω p must satisfy: ω p (g p ) = ω p ( p)m χ ( p ) ( j)−1 , (2.1.15)    × × ( pk ) k k of Z/ p Z , for all j ∈ Z/ p Z , for some fixed Dirichlet character χ   m k and whenever g p ∈ p · j + p Z p . The relations (2.1.11), (2.1.13) and (2.1.15) specify all the possible local characters that may occur in the factorization (2.1.10) of ω. Now the local characters ωv must be chosen so that  ωv (γ gv ) = ω(g) (2.1.16) ω(γ g) = k

v≤∞ × for all g ∈ A× Q , and for all γ ∈ Q . The relation ω(γ g) = ω(γ )ω(g) = ω(g) implies that ω(γ ) = 1 for all γ ∈ Q× . Let S = { p1 , p2 , . . . , pr } denote the finite set of primes where ω pi is ramified with conductor piki (for i = 1, 2, . . . , r ). Define

χ=

r  i=1

ki

χ ( pi ) ,

2.2 The L-function of an automorphic form

45

ki

where each χ ( pi ) is determined by (2.1.15) for i = 1, 2, . . . , r. Let  > 0 be a prime where  ∈ S. Then combining (2.1.16) with (2.1.11), (2.1.13), (2.1.15), it follows that ω() = 1 = it · ω () ·

r 

ki

χ ( pi ) ()−1 .

i=1

This forces ω () = −it

r 

ki

χ ( pi ) () = −it · χ ().

(2.1.17)

i=1

On the other hand, if  ∈ S, we have ω() = 1 = it · ω () ·

r 

ki

χ ( pi ) ()−1 .

i=1 pi =/ 

So in this case, ω () = −it

r 

ki

χ ( pi ) ().

(2.1.18)

i=1 pi =/ 

Finally we consider the case where γ = −1 in (2.1.16). It follows that ω(−1) = 1 = ω∞ (−1)

r 

ki

χ ( pi ) (−1)−1 = ω∞ (−1)χ (−1).

(2.1.19)

i=1

Equations (2.1.17), (2.1.18), (2.1.19) together with Definition 2.1.7 imply Theorem 2.1.9. 

2.2 The L-function of an automorphic form We have shown in Theorem 2.1.9 that every automorphic form φ on G L(1, AQ ) is associated to a uniquely defined Dirichlet character χ . To each such automorphic form φ we shall attach an L-function:   χ ( p) −1 1− s , L(s, φ) = p p

(2.2.1)

where the above product converges absolutely for (s) > 1. The L-function (2.2.1) is just the classical Dirichlet L-function associated to the Dirichlet character χ . What we have done here is to take some classical objects: Dirichlet character, Dirichlet L-function,

46

Automorphic representations and L-functions for GL(1, AQ )

and dress them in very fancy clothes so that they become almost unrecognizable extremely elegant objects with a new personality. Nevertheless, we are undaunted because we know them for what they truly are. However, there is much to be gained by such an approach and the later rewards will be very significant. Our next goal is to obtain the analytic continuation and functional equation of L(s, φ) by the adelic method of [Tate, 1950], [Iwasawa, 1952, 1992]. In general, the L-function (2.2.1) can be constructed by integrating (over the idele group) the automorphic form φ against a suitable test function. Let us begin by discussing the simplest case when the automorphic form is the trivial function, i.e., the constant one. In this case, the L-function is just the Riemann zeta function  ∞  1 −1 1− s n −s = . ζ (s) = p p n=1 In order to construct the Riemann zeta function as an idelic integral, we must introduce a set of test functions and appropriate measures to do idelic integration. Here are the precise definitions we need. Definition 2.2.2 (Adelic Bruhat-Schwartz space) Let S denote the set of all adelic Bruhat-Schwartz functions as defined in Definition 1.7.4. Recall that these are just linear combinations of functions  : AQ → C where  =  v (with v the characteristic function of Zv for all but finitely many

v≤∞

v ≤ ∞) where ∞ is Schwartz on R and v is Bruhat-Schwartz on Qv , i.e., it is a locally constant compactly supported function at all finite primes v.

Definition 2.2.3 (Idelic integral) As in Definition 1.7.5, we define the idelic  integral for factorizable idelic functions  = v v such that  p is the × characteristic function 1Z p for almost all primes p by  A× Q

(x)d × x =

 v∈S

Q× v

v (xv )d × xv ,

where S is a finite set containing ∞ and all the primes such that f p =/ 1Z×p . Also,  d x∞ , if v = ∞, |x∞ |∞ × d xv = dxp 1 , if v = p is a finite prime. 1− p−1 |x p | p Thus, d × x p is normalized so that



d × x p = 1. In the usual manner, we  extend the definition of adelic integration to define E (x)d × x for functions ×  : A× Q → C, and subsets E of AQ such that  · 1 E is a linear combination of Z×p

2.2 The L-function of an automorphic form

47

factorizable functions of the type considered above. As in the additive theory, we have    v (xv ) d × x = v (xv ) d × xv . A× Q v≤∞

× v≤∞ Qv

We summarize this information as defining an idelic differential: d×x =



d × xv .

v≤∞

Definition 2.2.4 (Idelic absolute value) For x = {x∞ , x2 , . . . } ∈ A× Q , we recall the definition  |xv |v . |x|A = v≤∞

Definition 2.2.5 (Special choice of test function) For x = {x∞ , x2 , . . . } ∈ A× Q , we define the test function h(x) = e−π x∞ 2



1Zv (xv ) ∈ S

v<∞

where 1Zv is the characteristic function of Zv at all the finite primes v < ∞. The function h has the nice property that h =  h, i.e., it is its own Fourier transform. Let s ∈ C with (s) > 1. Following Example 1.5.8, the Riemann zeta function, ζ (s), then appears naturally in the following computation:  A× Q

h(x)|x|sA d × x =



2

R×

=π

e−πt |t|s∞

− 2s



= π−2  s

s  2

s  2

 dt · |t|∞ p



|x p |sp d × x p

Z p −{0}

 −1 · 1 − p −s

(2.2.6)

p

ζ (s).

The meromorphic continuation and functional equation of ζ (s) can be obtained by use of the adelic Poisson summation formula (1.9.2). Since h =  h, and more specifically,  h(0) = h(0) = 1, we may rewrite the adelic Poisson summation formula (1.9.2) in the form: 1+

α∈Q×

h(αx) =

1 1 α  . + h |x|A |x|A x × α∈Q

(2.2.7)

Automorphic representations and L-functions for GL(1, AQ )

48

Recall Proposition 1.4.6 which says that 

A× Q =

  α · Q× \A× Q .

α∈Q×

To obtain the analytic continuation and functional equation of ζ (s) we proceed as follows:  s

A× Q

×

h(x)|x|A d x = 



= Q× \A× Q

α∈Q×



x∈Q× \A× Q

α∈Q×



 ×

h(x) |x|sA d × x

α· Q× \AQ

h(αx) |αx|sA d × x



=





α∈Q×



h(αx) |x|sA d × x +



x∈Q× \A× Q

|x|A ≤ 1

α∈Q×

h(αx) |x|sA d × x.

(2.2.8)

|x|A ≥ 1

In the above we have used the product formula (Theorem 1.1.8) which says that |α|A = 1 for α ∈ Q× . We apply the Poisson summation formula (2.2.7) to the first term in the last line of (2.2.8) to obtain 



x∈Q× \A× Q

α∈Q×

|x|A ≤ 1

 =

h(αx) |x|sA d × x =

x∈Q× \A× Q



=

α∈Q

|x|A ≤ 1

  s |x|s−1 d×x + − |x| A A

x∈Q× \A× Q |x|A ≤ 1

⎞ α  1 1 ⎝ − 1⎠ + h |x|A |x|A x × ⎛



 x∈Q× \A× Q

  s |x|s−1 d×x + − |x| A A

|x|A ≤ 1



x∈Q× \A× Q

x∈Q× \A× Q

|x|A ≤ 1

|x|A ≥ 1

α∈Q×

α∈Q×

h

α  x

· |x|sA d × x |x|s−1 d×x A

h(αx)|x|1−s d × x. A

(2.2.9)

2.2 The L-function of an automorphic form

49

By Proposition 1.4.5 (strong approximation for ideles) we have Q× \A× Q = (0, ∞) ·



Z×p .

p

It follows that for (s) > 0, 

1

×

|x|A d x = s

x∈Q× \A× Q

ys

1 dy = . y s

0

|x|A ≤ 1

Combining the above computations with (2.2.6), (2.2.8), (2.2.9) yields π−2  s

s  2





ζ (s) = x∈Q× \A× Q

=

α∈Q×

h(αx) |x|sA d × x

1 1 − + s−1 s





x∈Q× \A× Q

α∈Q×

  d × x. h(αx) |x|sA + |x|1−s A

|x|A ≥ 1

(2.2.10) Note that the integral on the right side of (2.2.10) converges for all s ∈ C and defines an entire function. Consider x on the right side of (2.2.10). Since x is restricted to be in the fundamental domain for Q× \A× Q , it follows that x p ∈ Z×p for all finite primes p. It further follows from the definition of h that αx p ∈ Z p for all finite primes p. This implies that α ∈ Z. After some elementary manipulations it is easy to show that equation (2.2.10) is really the same as the classical identity

π

− 2s



s  2

ζ (s) =

∞ 0

e−πn

2 2

y

|y|s∞

n∈Z n =/ 0

1 1 = − + s−1 s

dy y

∞ 1

e−πn

n∈Z n =/ 0

2 2

y

  dy . |y|s∞ + |y|1−s ∞ y (2.2.11)

One immediately obtains the following theorem.

50

Automorphic representations and L-functions for GL(1, AQ )

Theorem 2.2.12 The Riemann zeta function, ζ (s) =

∞

n −s , has a meromor-

n=1

phic continuation to all s ∈ C with a simple pole at s = 1. It satisfies the functional equation   s  1−s − 2s − 1−s 2 ζ (s) = π ζ (1 − s). π   2 2 Tate, in his thesis [Tate, 1950] realized that an identity of type (2.2.10) could be obtained for any test function  in the adelic Bruhat-Schwartz space S defined in Definition 2.2.2. If we rewrite the adelic Poisson summation formula (1.9.2) in the form (0)  1 α    − (0), (αx) = + |x|A |x|A x × × α∈Q

α∈Q

and we replicate the steps in the proof of (2.2.10), using a more general function  instead of h, it immediately follows that  (αx) |x|sA d × x (2.2.13) x∈Q× \A× Q

α∈Q×

(0)  (0) = − + s−1 s



  (αx)|x|1−s d × x. (αx)|x|sA +  A

x∈Q× \A× Q

α∈Q×

|x|A ≥ 1

Note that the right side of (2.2.13) does not change if we make the simultaneous transformations: . s → 1 − s, → Now, let us assume that  ∈ S is factorizable, as in Definition 1.7.3. Then  p is the characteristic function of Z p for all but finitely many primes p. Let S = { p1 , p2 , . . . , p } denote the finite set of primes pi where  pi is not equal to the characteristic function of Z pi (i = 1, 2, . . . , ). It follows that     d x∞ (x)|x|sA d × x = ∞ (x∞ ) |x∞ |s∞ ·  p (x p ) |x p |sp d × x p × × |x | ∞ ∞ AQ R p Q×p

⎞   −1 ⎟  (∞ (s) · ⎜ 1 − p −s =  p (x p ) |x p |sp d × x p ⎠ · ⎝ ⎛

p∈S

⎛ (∞ (s) · ⎝ =

 p∈S

p∈ S

Q×p



Q×p

 p (x p ) |x p |sp d × x p (1 − p −s )−1

⎞ ⎠ ζ (s), (2.2.14)

2.2 The L-function of an automorphic form where ((s) = 

 R×

(y) y s

51

dy . y

We immediately conclude that the right hand side of (2.2.14) is invariant under the simultaneous transformations s → 1 − s,

. →

INTERLUDE (Three remarks of Ivan Fesenko): Remark (1) Equations (2.2.13) and (2.2.14) are further generalized in Tate’s thesis [Tate, 1950]. The method is a very powerful and amazingly simple way to deduce the key properties of zeta and L-functions. Exactly the same reasoning as above gives the functional equation and meromorphic continuation of the completed zeta function of an algebraic number field K . The analogue of the right hand side of (2.2.10) is  c

1 1 − s−1 s



 + x∈K × \A× K |x|A ≥1



 ×  d x h(αx) |x|sA + |x|1−s A

α∈K ×

where c is the volume of K × \A K , and A K = {x ∈ A× K | |x|A = 1}. As first observed in [Iwasawa, 1952], the above formula implies the finiteness of c which in turn implies the compactness of K × \A K and the finiteness of the class group of K . It also easily implies the Dirichlet theorem of units in K . So, the most fundamental theorems of classical algebraic number theory follow as easy and fast corollaries of the adelic computation of the zeta function. Remark (2) Ideles were first introduced by Chevalley in the 1930s to simplify and make more conceptual the exposition of class field theory. The adelic method of Tate and Iwasawa works with functions on adelic objects but does not use any result of class field theory. An extension of the adelic method from G L(1) to G L(n) was found by Godement and Jacquet (see [GodementJacquet, 1972]) and is a major theme of this book. The extension gives a noncommutative theory over global fields, which is related to noncommutative class field theory and the Langlands program. The latter is about 40 years old and is still actively involved in the discovery of new fundamental concepts. Remark (3) Instead of going from G L(1) to G L(n), it is also possible to extend the work of Tate and Iwasawa in an entirely different “aspect.” From the point of view of modern algebraic geometry, the field Q may be thought of as the

Automorphic representations and L-functions for GL(1, AQ )

52

field of rational functions on the one-dimensional arithmetic scheme Spec(Z). A two-dimensional adelic analysis seeks to develop and use harmonic analysis on adelic spaces associated to an arithmetic surface (i.e., a two-dimensional object). For example, to an elliptic curve E over a global field K , given by equation Y 2 = X 3 + a X + b, one can associate an arithmetic surface E. Two-dimensional (abelian) class field theory [Parshin, 1976], [Kato-Saito, 1986], [Fesenko, 2000] (and others) describes abelian extensions of the field of rational functions on E via certain adelic objects and their quotients. The adelic objects are restricted products of certain two-dimensional local fields, e.g., Q p ((t)), R((t)), etc., which are not locally compact groups. See [Fesenko, 2003, 2008] for a development of a two-dimensional adelic theory and a generalization of (2.2.10) in this context. The two-dimensional adelic theory studies a zeta integral which is closely related to the arithmetic zeta function of E which itself is essentially ζ K (s) times ζ K (s − 1) divided by the Hasse-Weil L-function of E. Note that, as in the classical, one-dimensional case, the study of zeta functions uses adelic objects which naturally show up in class field theory, but does not depend on results from class field theory. We shall now show how the adelic method of Tate and Iwasawa can be used to obtain the meromorphic continuation and functional equation of all G L(1, AQ ) L-functions. Let φ be a non-zero automorphic form for G L(1, AQ ) as in Definition 2.1.4. By Theorem 2.1.9, the automorphic form φ is determined by a constant c = 0, a real number t, and the idelic lift χidelic of a Dirichlet character χ . Let f ∈ S be a factorizable adelic Bruhat-Schwartz function as in Definition 2.2.2. For s ∈ C with (s) > 1, the Dirichlet L-function (2.2.1) associated to φ will appear naturally in the computation of the adelic integral   f (x) χidelic (x) |x|sA d × x = c−1 f (x) φ(x) |x|s−it d × x. (2.2.15) A A× Q

A× Q

Let S = { p1 , p2 , . . . , p } be the finite set of primes where χ p is ramified (see Definition 2.1.14). Then χ p is unramified for p ∈ S. We shall now also make the assumption that f p is equal to the characteristic function of Z p for p ∈ S. The computation of (2.2.14) easily generalizes to the situation of (2.2.15). We have  f (x) χidelic (x) |x|sA d × x A× Q



= R×

f ∞ (x∞ ) χ∞ (x∞ ) |x∞ |s∞

  d x∞ · |x∞ |∞ p

f p (x p ) χ p (x p ) |x p |sp d × x p

Q×p

(2.2.16)

2.2 The L-function of an automorphic form 53 ⎞ ⎛   −1 ⎟  ⎜ =  f ∞ χ∞ (s) · ⎝ 1 − χ ( p) p −s f p (x p ) χ p (x p ) |x p |sp d × x p ⎠ · p∈S

p∈ S

Q×p



⎛ ⎜ =  f ∞ χ∞ (s) · ⎜ ⎝

f p (x p ) χ p (x p ) |x p |sp d × x p

Q×p

(1 −

p∈S

χ ( p) p −s )−1

⎟ ⎟ L(s, χ ) ⎠

⎞

⎛

 ⎜ =  f ∞ χ∞ (s) · ⎝ p∈S

⎞

⎟ f p (x p ) χ p (x p ) |x p |sp d × x p ⎠ L(s, χ )

Q×p

where



d x∞ |x∞ |∞ R× denotes the Mellin transform of f ∞ χ∞ . Note, also, that if χ p is ramified then χ ( p) = 0. This fact is used to obtain the last line in (2.2.16). Furthermore, equation (2.2.13) also easily generalizes to this situation and we may obtain  f (x) χidelic (x) |x|sA d × x  f ∞ χ∞ (s) =

A× Q





= x∈Q× \A× Q



α∈Q×

f ∞ (x∞ ) χ∞ (x∞ ) |y|s∞

f (αx)χidelic (αx) |x|sA d × x

   f (0) |x|s−1 − f (0) |x|sA χidelic (x) d × x A

= x∈Q× \A× Q |x|A ≤ 1



+ x∈Q× \A× Q

 α∈Q×

 f (αx) χidelic (x)|x|sA +  d × x. f (αx) χidelic (x)|x|1−s A

|x|A ≥ 1

(2.2.17) In the last line above we have used the fact that χidelic (1/x) = χidelic (x), which holds because χidelic is unitary, i.e., has absolute value = 1. Note that the last integral in (2.2.17) converges absolutely for all s ∈ C and defines an entire function. It is clear that the rightmost integral on the right side of equation (2.2.17) is invariant under the simultaneous transformations: s → 1 − s,

f → f,

χidelic → χidelic .

(2.2.18)

Automorphic representations and L-functions for GL(1, AQ )

54

One may also show that     f (0)χidelic (x) |x|s−1 − f (0)χidelic (x) |x|sA d × x = 0 A

(2.2.19)

x∈Q× \A× Q |x|A ≤ 1

if χidelic has ramification (as in Definition 2.1.14) at some finite prime. The fact that the left hand side of (2.2.19) vanishes is essentially due to the fact that a non-trivial Dirichlet character χ (mod q) satisfies q

χ ( j) = 0.

j=1 ( j,q)=1

It follows from (2.2.17) that if χidelic has ramification, then the L-function L(s, χ ) has a holomorphic continuation to all s ∈ C and satisfies a functional equation: s →1−s f → f, χidelic → χidelic . Note that the holomorphy follows since there is a particular choice of f in (2.2.17) for which  f ∞ is everywhere nonvanishing. If χidelic is everywhere unramified, then we are in the situation of the Riemann zeta function and the formulas take the form of (2.2.9). Let χ be a primitive Dirichlet character (mod q). It is known (see [Davenport, 2000]) that the Dirichlet L-function L(s, χ ) has holomorphic continuation to the entire complex plane and satisfies the explicit functional equation ξ ∗ (s, χ ) :=

 − 12 (s+a)  τ (χ ) π s + a L(s, χ ) = a √ ξ ∗ (1 − s, χ )  q 2 i q

where τ (χ ) =

q

χ (a)e

(2.2.20)

2πia q

a=1 (a,q)=1

is the Gauss sum and  a=

0,

if χ (−1) = 1,

1,

if χ (−1) = −1.

Definition 2.2.21 (Global conductor and root number) The integer q in √ is called (2.2.20) is called the conductor of L(s, χ ) while the constant iτa(χ) q

2.3 The local L-functions and their functional equations

55

the root number. Tate defined a different root number (which depends on s and  s− 12 √ . · iτa(χ) is not of absolute value one). Tate’s root number is defined to be πq q Tate realized that the identities (2.2.16) and (2.2.17) which are invariant under the transformations (2.2.18) must encode the classical functional equation (2.2.20) of the Dirichlet L-function. The fact that (2.2.16) and (2.2.17) hold for any adelic Bruhat-Schwartz function f ∈ S suggests that there must be further symmetries hidden in these identities. This led him to discover the so called local functional equations and the fact that the analytic conductor and the root number can be determined by local conditions. In the next two sections we shall introduce the local L-functions and their functional equations and explain how the global functional equation (2.2.20) follows from them.

2.3 The local L-functions and their functional equations In this section let v ≤ ∞ determine a local field Qv . Definition 2.3.1 (Local zeta integral) Fix s ∈ C with (s) > 0. Let  : Qv → C be a locally constant compactly supported function as in Definition × 1.5.2 if v < ∞, and a Schwartz function if v = ∞. Let ω : Q× v → C be a local unitary character, i.e., a continuous homomorphism of absolute value 1. We define  (x) ω(x) |x|sv d × x, (2.3.2) Z v (s, , ω) = Q× v

to be the local zeta integral associated to ω and . Here d × x denotes the differential associated to the multiplicative Haar measure as in Definition 1.5.7 if v < ∞ and d × x = d x/|x|∞ if v = ∞. Remark 2.3.3 The local zeta integral (2.3.2) is absolutely convergent if (s) > 0. To see this we write   s × (x) ω(x) |x|v d x + (x) ω(x) |x|sv d × x. Z v (s, , ω) = x∈Q× v |x|v >1

x∈Q× v |x|v ≤1

The first integral on the right above is clearly convergent. Since  is bounded in the region |x|v ≤ 1, the second integral is bounded by  |x|σv d × x, x∈Q× v |x|v ≤1

an integral which converges absolutely for (s) = σ > 0 for any fixed v ≤ ∞. We now state the main theorem in the local theory.

Automorphic representations and L-functions for GL(1, AQ )

56

Theorem 2.3.4 (Local functional equation) Let s ∈ C with 0 < (s) < 1. The local zeta integral Z v (s, , ω), as defined in Definition 2.3.1, satisfies the functional equation , ω) Z v (s, , ω) = γ (s, ω) · Z v (1 − s,   denotes the v-adic Fourier transform of  (see (1.6.1), Theorem where  1.6.8), ω = ω−1 is the complex conjugate of ω, and γ (s, ω) is a meromorphic function which is independent of the choice of . Proof To show that γ (s, ω) is independent of  it is enough to show that Z v (s, , ω) Z v (s, , ω) = , ω) Z v (1 − s,  , ω) Z v (1 − s,  for any function  : Qv → C with the same properties as . We will prove that , ω) = Z v (1 − s,  , ω) Z v (s, , ω) Z v (s, , ω) Z v (1 − s,  by showing that

, ω) Z v (s, , ω) Z v (1 − s, 

(2.3.5)

is symmetric in  and . By Remark 2.3.3, if 0 < (s) < 1, we can write (2.3.5) as an absolutely convergent double integral     (y)ω x y −1 |x|sv |y|1−s (x) d × x d × y. (2.3.6) v Q× v

Q× v

The measure d × y is invariant under the transformation y → x y. It follows that (2.3.6) can be rewritten in the form   (x y) ω (y) |x|v |y|1−s (x) d × x d × y. v Q× v

Q× v

But (x y) = 

 Qv

(z)ev (−x yz)dz.

Consequently   (x y) ω (y) |x|v |y|1−s (x) d×x d× y v Q× v

=

Q× v



Q× v

 Q× v

 Qv

(x)(z)ev (−x yz) ω (y) |y|1−s v

which is clearly symmetric in  and .



×

d x d y dz ·



v , v−1

1,

if v =/ ∞, if v = ∞,

2.3 The local L-functions and their functional equations

57

Proposition 2.3.7 Let s ∈ C with 0 < (s) < 1. Let γ (s, ω) be defined as in Theorem 2.3.4. Then γ (s, ω) · γ (1 − s, ω) = ω(−1). , ω) = ω(−1) · Proof A simple change of variables shows that Z v (s,  Z v (s, , ω). By definition , ω), Z v (s, , ω) = γ (s, ω) · Z v (1 − s,  , ω) = γ (1 − s, ω) · ω(−1) · Z v (s, , ω). Z v (1 − s,  Consequently Z v (s, , ω) = ω(−1) γ (s, ω) γ (1 − s, ω) · Z v (s, , ω). One may choose  as in the proof of the next theorem so that Z v (s, , ω) = 0, which implies that γ (s, ω) · γ (1 − s, ω) = ω(−1).  Theorem 2.3.8 (Explicit computation of γ (s, ω)) Let s ∈ C, 0 < (s) < 1. The function γ (s, ω) can be explicitly given as follows. Case 1: v = ∞

γ (s, ω) =

⎧ ⎪ ⎪ ⎪ ⎨

− 1−s π 2

⎪ ⎪ ⎪ ⎩

1 i

π

·

−s 2

( 2s ) ( 1−s 2 ) π

− s+1 2

(1−s)+1 − 2 π

,

if ω(x) = 1 for x ∈ R,

( s+1 2 ) 

 (1−s)+1  , if ω(x) = sign(x) for x ∈ R. 2

Case 2: v = p < ∞ +  ⎧ ω( p) ω( p) ⎪ 1 − 1 − , if ω is unramified, ⎪ s 1−s p p ⎪ ⎪ ⎪ ⎨ γ (s, ω) = pr pr (s−1) ⎪ −2πi j p−r ⎪ , ω is ramified and has conductor pr . ⎪ ⎪ ω( p)r j=1 ω( j)e ⎪ ⎩ ( j, p)=1

Remarks The local zeta integral Z v (s, , ω) can easily be = 0 if the function  is chosen in a stupid way. In the proof below we have made smart choices  = ◦ in each situation. The factor 1/i that appears in γ (s, ω) when

58

Automorphic representations and L-functions for GL(1, AQ )

ω(x) = sign(x), occurs in the global root number, of (2.2.20) for the case when χ (−1) = −1. When ω is a character of Q×p which is ramified and has conductor pr then γ (s, ω) contains the Gauss sum r

p

j

ω( j)e−2πi pr ,

j=1 ( j, p)=1

and when these Gauss sums are multiplied together over all ramified primes we obtain the Gauss sum occurring in the root number (2.2.20). Proof of Theorem 2.3.8 We first consider the case v = ∞, ◦ (x) = e−π x , and ω ≡ 1. Then the local zeta integral is 2

∞

◦

Z ∞ (s,  , ω) =

e−π x |x|s∞ 2

−∞

s  dx s = π−2  |x|∞ 2

(2.3.9)

◦

 and ω = ω, it follows that Since ◦ = 

  s π − 2  2s Z ∞ (s, ◦ , ω) = γ (s, ω) =  1−s  . Z ∞ (1 − s, ◦ , ω) π − 1−s 2  2

Next, consider the case v = ∞, ◦ (x) = xe−π x , and ω(x) = sign(x) = Consequently 2

∞

◦

Z ∞ (s,  , ω) =

e−π x |x|s+1 ∞ 2

−∞

  dx s+1 s+1 . = π− 2  |x|∞ 2

◦

 (x) = A simple computation shows that  ◦

 , ω) = i Z ∞ (1 − s, 

∞ −∞

e−π x |x|(1−s)+1 ∞ 2

∞ −∞

|x|∞ . x

(2.3.10)

◦ (y)e2πi x y dy = i◦ (x). Hence

  (1−s)+1 dx (1 − s) + 1 . = iπ − 2  |x|∞ 2

It follows that

  s+1 π − 2  s+1 1 Z ∞ (s, ◦ , ω) .  2 = γ (s, ω) = ◦ , ω) i π − (1−s)+1 Z ∞ (1 − s,  2  (1−s)+1 2

We now consider the non-archimedean situation where v = p is a finite prime and ω is unramified. Choose  1, if x ∈ Z p , ◦  (x) = 0, otherwise.

2.3 The local L-functions and their functional equations

59

Then Z p (s, ◦ , ω) =

p p−1



◦p (x)ω(x) |x|sp

Q×p

dx p = |x| p p−1

 ω(x) |x|sp Z p −{0}

ω( p) ω( p ) ω( p ) + + + ··· ps p 2s p 3s   ω( p) −1 = 1− s . p 2

dx |x| p

3

=1+

(2.3.11)

◦

 = ◦ , we also have Since  

ω( p)  , ω) = 1 − Z p (1 − s,  p 1−s ◦

Consequently

 1− γ (s, ω) =  1−

ω( p) p1−s ω( p) ps

−1 .

 .

Finally, we consider v = p and ω is ramified with conductor pr as in Definition 2.1.14. In this case, we choose ◦



 (x) =

e−2πi{x} , if x ∈ p −r Z p , 0,

otherwise.

where {x} denotes the fractional part of x ∈ Q×p as in Definition 1.6.3. With these choices we obtain  dx p ◦ Z p (s,  , ω) = e−2πi{x} ω(x) |x|sp p−1 |x| p p−r Z p −{0} r

p r p = p − 1 =1 j=1 ( j, p)=1



e−2πi{x} ω(x) |x|sp

p− ( j+ pr Z p )

dx |x| p

r

= p

−r

p r −2πi j p s − · p ω( p) e p ω( j) p − 1 =1 j=1 ( j, p)=1 pr

=

− 2πi j p −r +1 · pr s ω( p)−r · e pr ω( j). p−1 j=1 ( j, p)=1

(2.3.12)

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Automorphic representations and L-functions for GL(1, AQ )

In the above computation, we used the fact that r

p

e

−2πi j p

ω( j) = 0

j=1 ( j, p)=1

if  < r. ◦ , ω) we need to compute the Fourier In order to compute Z p (1 − s,  ◦  . We have transform    ◦ ◦ 2πi{x y}   (x) =  (y)e dy = e−2πi{y(1−x)} dy p−r Z p

Qp



=

p , 0, r

if x ∈ 1 + p Z p , otherwise. r

It follows that ◦ , ω) = pr Z p (1 − s,  = pr

p p−1 p p−1

 ω(x) |x|1−s p 1+ pr Z p



1+ pr Z

dx |x| p

dx p . = |x| p p−1



p

Theorem 2.3.8 can be used to show that the local L-functions, as defined in Definition 2.3.1, have meromorphic continuation to the whole complex plane. Theorem 2.3.13 (Meromorphic continuation of the local zeta integrals) Fix s ∈ C with (s) > 0. Let Z v (s, , ω) denote the local zeta integral as in Definition 2.3.1. Then Z v (s, , ω) has meromorphic continuation to all s ∈ C. Proof The explicit computation of γ (s, ω) given in Theorem 2.3.8 shows that in all cases, γ (s, ω) has meromorphic continuation to all s ∈ C. When this fact is combined with the functional equation of Theorem 2.3.4 we immediately obtain the meromorphic continuation of the local L-function to the whole complex plane. 

2.4 Classical L-functions and root numbers The founders of the theory of L-functions over Q, Riemann, Dirichlet, Hecke, etc., defined L-functions so that their Euler products were as simple as possible and so that their functional equations were as symmetric as possible. The

2.4 Classical L-functions and root numbers

61

classical L-functions over Q are just the Dirichlet L-functions as in (2.2.1) or the Riemann zeta function. If we are given a Bruhat-Schwartz function  : AQ → C as in Definition 2.2.2 and an adelic automorphic form ω : G L(1, AQ ) → C as in Definition 2.1.4, then we have intensively studied Tate’s global zeta integral which is defined by  (x) ω(x) |x|sA d × x Z (s, , ω) := G L(1,AQ )

for (s) > 1. It is natural to ask which choices of  and ω lead to the classical L-functions.  Let s ∈ C with (s) > 1. If we assume that  = v v is factorizable and  ω = v ωv then    Z (s, , ω) = v (xv ) ωv (xv ) |xv |sv d × xv = Z v (s, v , ωv ), (2.4.1) v≤∞

Q× v

v≤∞

which reduces Tate’s global zeta integral to a product of local zeta integrals. × Recall that a local character ωv : Q× v → C may be ramified or unramified as in Definition 2.1.14. Accordingly, we say the prime v is ramified or unramified, respectively. In order to construct the classical L-functions it is necessary to choose v (for every v ≤ ∞) so that the local zeta integral (2.4.1) is as simple as possible and the global functional equation of Z (s, , ω) is as symmetric as possible. It is easy to do this if v is unramified. In this case, we choose v v = v , i.e., v is its own Fourier transform. This (for all v ≤ ∞) so that  amounts to the choices:  2 e−π x∞ , if v = ∞ is unramified, v (xv ) = 1Z p (x p ), if v = p < ∞ is unramified. Define the local L-function, denoted L v (s, ωv ), by setting L v (s, ωv ) = Z v (s, v , ωv ) for the above choice of v . It follows from (2.4.1) and the proof of Theorem 2.3.8 that the local L-functions take the form ⎧ −s s  if v = ∞ is unramified, ⎨ π 2 2 , −1 L v (s, ωv ) =  (2.4.2) ⎩ 1 − ω p (sp) , if v = p < ∞ is unramified. p It is not as clear, however, how to make the choices when v is ramified. Again, using Theorem 2.3.8 as our guide, we will choose v so that L v (s, ωv ) := Z v (s, v , ωv ) and  − s+1  s+1  π 2  2 , if v = ∞ is ramified, (2.4.3) L v (s, ωv ) = 1, if v = p < ∞ is ramified.

62

Automorphic representations and L-functions for GL(1, AQ )

We may then define the classical completed L-function  L v (s, ωv ) L ∗ (s, ω) =

(2.4.4)

v≤∞

where L v (s, ωv ) is defined in (2.4.2), (2.4.3). Then ⎧  −1    s ω ( p) ⎪ π − 2  2s 1 − pps , if ∞ is unramified, ⎪ ⎪ ⎨ p unramified ∗ L (s, ω) =  −1 ⎪    s+1 ⎪ ω p ( p) ⎪ 1 − , if ∞ is ramified. ⎩ π − 2  s+1 s 2 p p unramified

Definition 2.4.5 (Local root number) Fix v ≤ ∞. Let L v (s, ω) (where ω = ωv ) be given by (2.4.2), (2.4.3), and for a Bruhat-Schwartz function  : Qv → C, let Z v (s, , ω) be a non-identically vanishing local zeta integral as in (2.3.2). We define the local root number to be the complex valued function v (s, ω) which satisfies , ω) Z v (1 − s,  Z v (s, , ω) = v (s, ω) . L v (1 − s, ω) L v (s, ω)

(2.4.6)

Note that by Theorem 2.3.4, the local root number v (s, ωv ) is independent of the choice of . Furthermore, by Theorem 2.3.8, the local root number v (s, ωv ) takes the value 1 at unramified primes p. It follows that the infinite product  (s, ω) := v (s, ωv ) (2.4.7) v≤∞

is really just a finite product which can be explicitly computed using Theorem 2.3.8. Proposition 2.4.8 (Global functional equation) For (s) > 1, let L ∗ (s, ω) be defined as in (2.4.4). Then L ∗ (s, ω) has a meromorphic continuation to all s ∈ C with at most simple poles at s = 0, 1, and satisfies the functional equation L ∗ (s, ω) = (s, ω)L ∗ (1 − s, ω) where (s, ω) is given by (2.4.7). If ω is not the trivial character then L ∗ (s, ω) is an entire function of s. Proof For each prime v ≤ ∞, define a local zeta integral Z v (s, ◦v , ωv ) as in (2.3.9), (2.3.10), (2.3.11), (2.3.12). For (s) > 1 let  Z v (s, ◦v , ωv ). Z (s, ◦ , ω) = v≤∞

2.4 Classical L-functions and root numbers

63

Then by (2.2.17), we know that Z (s, ◦ , ω) has a meromorphic continuation to all s ∈ C, with at most simple poles at s = 0, 1, and satisfies the global functional equation ◦ , ω). Z (s, ◦ , ω) = Z (1 − s, 

(2.4.9)

Now Z (s, ◦ , ω) is almost the same as L ∗ (s, ω). When these functions are defined for (s) > 1 as products, then they differ only at the primes p where ω p is ramified. In fact, for (s) > 1, we may write Z (s, ◦ , ω) = L ∗ (s, ω)



Z p (s, ◦p , ω p )

p ramified

L p (s, ω p )

=



Z p (s, ◦p , ω p ).

(2.4.10)

p ramified

It follows from (2.3.12) that if ω p is ramified then Z p (s, ◦p , ω p ) is just a constant times a power of p s . Thus the right side of (2.4.10) is holomorphic everywhere and never zero. This shows that L ∗ (s, ω) has the same properties as Z (s, ◦ , ω). In particular it has meromorphic continuation to all complex s with at most simple poles at s = 0, 1. The simple poles can only occur when ω is trivial and we are in the situation of the Riemann zeta function. Further, by the definition of the local root number (2.4.6), (2.4.7), we obtain from (2.4.10) that ◦p , ω p )  Z p (1 − s,  Z (s, ◦ , ω) −1 = δω · (s, ω) , L ∗ (s, ω) L p (1 − s, ω p ) p ramified

(2.4.11)

where δω = i if ω is ramified at infinity and is equal to one otherwise. The function on the right side of (2.4.11) has meromorphic continuation to (s) < 0, and in that region is equal to (s, ω)−1 ·

◦

 , ω) Z (1 − s,  , ∗ L (1 − s, ω)

from which we deduce that ◦

 , ω) Z (1 − s,  Z (s, ◦ , ω) = (s, ω) . L ∗ (1 − s, ω) L ∗ (s, ω) When (2.4.12) is combined with (2.4.9), Proposition 2.4.8 follows.

(2.4.12) 

Remarks on roots numbers The global root number (s, ω) is precisely Tate’s root number as defined in Definition 2.2.21. Before closing this section, we record a couple of additional observations which will be useful in the generalization to G L(n).

64

Automorphic representations and L-functions for GL(1, AQ )

Theorem 2.4.13 (The local L-function as a “common divisor” in the p-adic case) Fix a rational prime p and let  : Q p → C be a locally constant compactly supported function as in Definition 1.5.2. Let ω : Q×p → C× be a unitary character, i.e., a continuous homomorphism of absolute value 1. Then the local zeta integral: Z p (s, , ω), defined in Definition 2.3.1, is a rational function in p −s . Furthermore, Z p (s, , ω) · L p (s, ω)−1 is a polynomial in p s and p −s . Remark Theorem 2.4.13 may be viewed as saying that L p (s, ω p ) is a “common divisor” for the rational functions Z p (s, , ω) corresponding to various choices of locally constant compactly supported functions . Indeed, Z p (s, , ω) · L p (s, ω p )−1 is an entire function of s, and by the results of Section 2.4, it is possible to choose  so that Z p (s, , ω) · L p (s, ω)−1 ≡ 1. Proof It is easy to see that Z p (s, , ω) is a polynomial in p s and p −s whenever  is a locally constant compactly supported function: Q p → C which vanishes at 0, for in this case |x|sp takes only finitely many values on the domain of . On the other hand, every locally constant compactly supported function may be expressed as a multiple of 1Z p plus a function which vanishes at 0. In the ramified case, it is easily verified that Z p (s, 1Z p , ω) vanishes identically. It follows that Z p (s, , ω) is always simply a polynomial in p s and p −s . In the unramified case, Z p (s, , ω) is equal to a polynomial plus a multiple of L p (s, ω) = Z p (s, 1Z p , ω).  It is clear that Theorem 2.4.13 does not extend directly to the real case. However, we have the following analogue. Theorem 2.4.14 (The local L function as a “common divisor” in the real 2 case) Let  : R → C be given by (x) = P(x)e−π x , where P(x) is a polynomial. Let ω(t) equal 1 or sign(t) for all t ∈ R× . Then Z ∞ (s, , ω)· L ∞ (s, ω)−1 is a polynomial in s. Proof Exercise.



For purposes of the G L(1) theory, it suffices to consider only unitary characters of Q×p and only two characters of R× (namely, the trivial character and the sign character. This is because an arbitrary character of Qv , for v ≤ ∞ is of the form χ (t) = ω(t)|t|λv (∀t ∈ Qv ) for some λ ∈ C (which is unique in the real case and unique up to 2πi/ log p in the p-adic case) and some character ω of the type that has been considered above. It is clear that a local zeta integral Z v (s, , χ ), may be defined exactly as in Definition 2.3.1, and that it will satisfy Z v (s, , χ ) = Z v (s + λ, , ω), whenever χ (t) = ω(t)|t|λv . However, for purposes of applying the G L(1) theory inductively in higher rank, it will be convenient to have the following extension to arbitrary characters. The proof is left to the reader.

2.5 Automorphic representations for GL (1, AQ )

65

Theorem 2.4.15 (Extension to arbitrary characters) Fix v ≤ ∞ and let χ : Qv → C be a character, that is, a continuous homomorphism. Write χ in the form (x ∈ Qv ), χ (x) = ω(x) · |x|λv , with λ ∈ C, and ω a unitary character, such that ω is either trivial or the sign character in the real case. Define L v (s, χ ) := L v (s + λ, ω) and v (s, χ ) := v (s + λ, ω) with L v (s, ω), v (s, ω) given as in Definition 2.4.5. Let  : Qv → C be a function which is locally constant when v < ∞ and Schwartz when v = ∞. Then the local zeta integral  (x)χ (x)|x|sv d × x Z v (s, , χ ) := Q× v

converges for (s) sufficiently large, has meromorphic continuation to all s, and satisfies a functional equation , χ −1 ) Z v (1 − s,  Z v (s, , χ ) = v (s, χ ) . L v (s, χ −1 ) L v (s, χ )

2.5 Automorphic representations for G L(1, AQ ) We begin with the abstract definition of a representation of a group on a vector space. Definition 2.5.1 (Representation of a group on a vector space) Let G be a group and let V be a vector space. A representation of G on V is a homomorphism π : G → G L(V ) = {group of all invertible linear maps: V → V }. Here π (g) . v denotes the action of π (g) on v and π (g g

) = π (g ) . π (g

) for all g , g

∈ G. We call V the space of π and refer to the ordered pair (π, V ) as a representation. Remarks Very often we shall consider representations (π, V ) where V is a space of functions f : G → C and the action is given by right translation, i.e., π (g ) . f (g) = f (gg ),

(∀ g, g ∈ G).   In this situation we see that π (g g

) . f (g) = π (g ) . π (g

) . f (g) . Note that a representation defined by a left action on a space of functions would satisfy the reverse identity π (g g

) = π (g

) . π (g ) for g , g

∈ G.

66

Automorphic representations and L-functions for GL(1, AQ )

If the group G and the vector space V are equipped with topologies, then we shall also require the map G × V → V, given by (g, v) → π (g) . v, to be continuous. Definition 2.5.2 (Irreducible representation) A representation (π, V ) as in Definition 2.5.1 is said to be irreducible if V =/ 0, and V has no closed π invariant subspace other than 0 and V . Definition 2.5.3 (Intertwining maps and isomorphic representations) Let π1 : G → G L(V1 ),

π2 : G → G L(V2 ),

be two representations as in Definition 2.5.1. An intertwining map, or intertwining operator, is a linear map L : V1 → V2 such that   L . π1 (g) . v = π2 (g) . (L . v) for all g ∈ G, v ∈ V1 . Here L . v denotes the action of L on v ∈ V1 . If there is an intertwining operator L : V1 → V2 which an isomorphism of vector spaces, then the two representations are said to be isomorphic. Roughly speaking, an automorphic representation of G L(n) (for n = 1, 2, . . . ) is an irreducible representation of G L(n, AQ ) on a topological vector space V which consists of complex valued functions on G L(n, AQ ) satisfying certain growth, smoothness, and invariance properties. We shall now make the definition precise for the case of G L(1, AQ ). Definition 2.5.4 (Automorphic representation for G L(1, AQ )) Fix a unitary Hecke character ω of A× Q as in Definition 2.1.2. Define Vω to be the onedimensional vector space (over C) of all automorphic forms with character ω as defined in Definition 2.1.4. We may define a representation π : G L(1, AQ ) → G L(Vω ) by requiring that π (g) . φ(x) := φ(x · g) = ω(g)φ(x) for all φ ∈ Vω and all g, x ∈ G L(1, AQ ). Remarks It is clear that the Hecke character ω is factorizable, as a function, in the sense described at the beginning of this chapter. But in fact, the representation (π, Vω ) also factors as a representation. Once everything is viewed in the right manner, this fact is very close to being a tautology. We use this as an opportunity to introduce the notion of “factorizable” which is appropriate to representations. We first introduce the appropriate product, which is the

2.5 Automorphic representations for GL (1, AQ )

67

restricted tensor product. In a sense, this product is to tensor products what the restricted direct product used to form the adeles is to Cartesian products. We first recall the ordinary tensor product of two vector spaces (see [AtiyahMacdonald, 1969]). Suppose V and W are vector spaces over a field k, with (possibly infinite) bases B and C. Then the tensor product of V and W is a vector space, denoted V ⊗ W, generated by a basis consisting of the symbols v ⊗ w, where v ∈ B and w ∈ C. It may also be realized as the vector space spanned by the set of all symbols v ⊗ w with v ∈ V and w ∈ W, such that the only relations are those coming from relations in V, relations in W, or relations of the form α · (v ⊗ w) = (α · v) ⊗ w = v ⊗ (α · w),

(for all α ∈ k, v ∈ V, w ∈ W ),

(v1 + v2 ) ⊗ w = v1 ⊗ w + v2 ⊗ w,

(for all v1 , v2 ∈ V, w ∈ W ),

v ⊗ (w1 + w2 ) = v ⊗ w1 + v ⊗ w2 ,

(for all v ∈ V, w1 , w2 ∈ W ).

An element of the form v⊗w is called a “pure tensor.” Because the pure tensors span, it is often enough to prove theorems only for pure tensors. This construction generalizes in a straightforward way to threefold and n-fold tensor products, and even to infinite tensor products. The tensor products of interest to us will be infinite and indexed by the set of primes. For each prime v let Vv be a vector space. For these, we write ⊗v Vv for the tensor product and ⊗v ξv for a pure tensor, where ξv ∈ Vv for each v. To define a restricted tensor product, one must fix a non-zero vector ξv◦ in Vv for all but finitely many v. Then the restricted tensor product of {Vv } relative to {ξv◦ } is the subspace of ⊗v Vv spanned by the set of pure tensors which are of the form ξv = ξv◦ for almost all v. ξ = ⊗v ξv This vector space is denoted ⊗ v Vv . Definition 2.5.5 (Factorizability for representations of A× Q ) Let V be a vector space. A representation (π, V ) of A× is factorizable if there exists: Q (1) For each v, a representation πv of Q× v , on a vector space Vv . (2) For almost all v, a distinguished vector ξv◦ ∈ Vv . (3) An isomorphism of vector spaces  : ⊗ v Vv → V, such that

    ⊗v πv (gv ) . ξv = π (g) . (⊗v ξv ),

(2.5.6)

◦ for all g = {g∞ , g2 , g3 , . . . } ∈ A× Q and all ⊗v ξv such that ξv = ξv for almost all v.

Remarks The linear map  in Definition 2.5.5 gives an isomorphism of ⊗ v Vv and V . Note that (2.5.6) is precisely what is required (see Definition 2.5.3)

68

Automorphic representations and L-functions for GL(1, AQ )

  to make the representation ⊗v πv , ⊗ v Vv isomorphic to the representation (π, V ). Proposition 2.5.7 (All automorphic representations for G L(1, AQ ) are factorizable) The representation (π, Vω ) as defined in Definition 2.5.4 is factorizable. Proof For each v, we take an abstract one-dimensional complex vector space, spanned by a vector, which we denote ξv◦ . Thus Vv is simply {cξv◦ | c ∈ C} for each v. We define an action of Q× v on Vv by   πv (g) . cξv◦ = c ωv (g) ξv◦ ,

(g ∈ Q× v ).

Now, we need to define a linear map  from ⊗ v Vv to Vω . By linearity, it is enough to define it on the element ⊗v ξv◦ , which spans ⊗ v Vv . Recall that Vω is a one-dimensional vector space spanned by the function ω. The definition of  is simply    ⊗v ξv◦ = ω. One then easily verifies that       ⊗v πv (gv ) . ξv = π (g) .  ⊗v ξv . (Both sides are equal to ω(g)ω, i.e, the function in Vω which takes the value ω(g)ω(x) at the point x.) 

2.6 Hecke operators for G L(1, AQ ) The theory of Hecke operators for automorphic forms on G L(2, R) is well known (see [Goldfeld, 2006]). Very roughly, Hecke operators act on automorphic forms and transform them to other automorphic forms. If an automorphic form is an eigenfunction of all the Hecke operators then it is called a Hecke eigenform. Hecke proved the remarkable theorem that the L-function associated to a Hecke eigenform has an Euler product. We shall now show that a similar result can be established for automorphic forms for G L(1, AQ ). Definition 2.6.1 (Hecke operator for G L(1, AQ )) Let φ be an automorphic form for G L(1, AQ ) with unitary Hecke character ω as in Definition 2.1.4. For every idele n = {n ∞ , n 2 , . . . } ∈ A× Q we define the Hecke operator Tn which acts on φ as follows: Tn φ(g) = φ(ng), for all ideles g ∈ G L(1, AQ ).

2.7 The Rankin-Selberg method

69

By Definition 2.1.4 (2), we see that Tn φ(g) = ω(n)φ(g) so that every automorphic form is a Hecke eigenform. Actually, Definition 2.6.1 is not very interesting because the space of automorphic forms on G L(1, AQ ) with fixed character ω is just a one-dimensional space. Hecke’s theorem is trivial in this situation because we have already shown that the only possible L-functions we can obtain for G L(1, AQ ) are Dirichlet L-functions which do have Euler products. We state Hecke’s theorem for completeness. Theorem 2.6.2 Let φ be an automorphic form for G L(1, AQ ) with unitary Hecke character ω as in Definition 2.1.4. If φ is an eigenfunction of all the Hecke operators Tn as defined in Definition 2.6.1 then the L-function associated to φ has an Euler product.

2.7 The Rankin-Selberg method The purpose of Section 2.7 is to show that the classical Rankin-Selberg method (see [Goldfeld, 2006]) has an analogue on G L(1, AQ ). This very brief section is for cognoscenti who already have familiarity with the classical Rankin-Selberg method. Let s ∈ C with (s) > 0. Let  : AQ → C be an adelic Bruhat-Schwartz function as in Definition 1.7.4. For x ∈ A× Q , the Tate series associated to  is defined to be T (x, s, ) :=

α∈Q×

(αx) · |αx|sA =

α∈Q×

(αx) · |x|sA

Because we are averaging over the multiplicative group Q× it is easy to see that T (γ x, s, ) = T (x, s, ) for all γ ∈ Q× and x ∈ A× Q . If φ1 , φ2 are automorphic forms for G L(1, AQ ) with characters ω1 , ω2 , respectively, as in Definition 2.1.4, then it is clear that φ1 · φ2 is again an automorphic form with character ω1 · ω2 . The classical Rankin-Selberg unfolding computation takes the following form on G L(1, AQ ) :  Q× \A× Q

φ1 (x)φ2 (x)T (x, s, ) d × x =



φ1 (x)φ2 (x)(x)|x|sA d × x

(2.7.1)

A× Q

where the right side of (2.7.1) is the completed L-function (product of local L-functions) associated to the automorphic form φ1 · φ2 .

Automorphic representations and L-functions for GL(1, AQ )

70

2.8 The p-adic Mellin transform Definition 2.8.1 ( p-adic Mellin transform) Let f : Q×p → C be a locally constant compactly supported function. Assume that f (y) = h(|y| p ) for some other function h : { p  |  ∈ Z} → C. For s ∈ C, we define the p-adic Mellin transform  ( f (u) |u|sp d × u. f (s) := Q×p

Proposition 2.8.2 ( p-adic Mellin inversion) Let f : Q×p → C be a locally constant compactly supported function. Assume that f (y) = h(|y| p ) for some other function h : { p  |  ∈ Z} → C. Let f( be the Mellin transform as in Definition 2.8.1. Then 2π

log p

log p f (y) = 2π

f((it) |y|−it p dt.

0

Proof It follows from Definition 2.8.1 that for |y| p = p  , we have 2π

log p 2π

log p

-

log p , 2π

log p f((it) |y|−it p dt = 2π

0

Q×p

0

   = h pm m∈Z



 

=h p .

p−m Z×p

f (u) |u|itp ⎡ ⎢ log p ⎣ 2π

×

d u 2π

log p

p −it dt ⎤

⎥ pi(m−)t dt ⎦ d × u

0



The above Mellin transform and its inverse can be generalized to compactly supported locally constant functions f : Q×p → C where f (y) is not necessarily a function of |y| p . It is convenient to make the following definition. Definition 2.8.3 (Conductor of a locally constant compactly supported function) Let f : Q×p → C be a locally constant compactly supported function. We define the conductor of f to be the smallest integer N such that f is constant on y · (1 + p N Z p ) for each fixed y ∈ Q×p . (To see that such an integer exists, note that f must be identically equal to zero on p m Z×p for all but a finite number of integers m. For each m such that f does not vanish identically on p m Z×p , there exists an integer Nm such that the function f ( p m y) is constant on y0 · (1 + p Nm Z p ) for each fixed y0 ∈ Z×p . Choose N to be the largest of all such Nm .)

2.8 The p-adic Mellin transform

71

Definition 2.8.4 (Normalized unitary character of Q×p ) A continuous function ψ : Q×p → C× which satisfies   • ψ(yy ) = ψ(y)ψ(y ), ∀y, y ∈ Q×p ;   • |ψ(y)|C = 1, ∀y ∈ Q×p , • ψ( p) = 1, is called a normalized unitary multiplicative character of Q×p . Let N be the smallest integer k ≥ 0 such that 1 + p k Z p is contained in the kernel of ψ. Then ψ is said to have conductor p N . We also call ψ a character (mod p N ). The number of ψ (mod p N ) is ϕ( p N ), where ϕ is Euler’s function. This follows from the claim before (2.1.15), along with the corresponding fact about classical Dirichlet characters. With these preliminaries in place, we may now present the more general Mellin transform. Definition 2.8.5 (General p-adic Mellin transform) Let f : Q×p → C be a locally constant compactly supported function. For s ∈ C, and ψ : Q×p → C× , a normalized unitary character as in Definition 2.8.4, we define the Mellin transform  f (u) ψ(u) |u|sp d × u. f((s, ψ) := Q×p

Proposition 2.8.6 (General p-adic Mellin inversion) Let f : Q×p → C be a locally constant compactly supported function of conductor p N as in Definition 2.8.3. Let ( f be the Mellin transform as in Definition 2.8.5. Then we have 2π



f (y) =

ψ (mod p N )

log p 2π

log p

f((it, ψ) ψ(y)−1 |y|−it p dt.

0

Proof It follows from Definition 2.8.5 that if |y| p = p  , then 2π

ψ (mod p N )

log p 2π

log p

f((it, ψ) ψ(y)−1 |y|−it p dt

0 2π

=

ψ (mod p N )

=

ψ (mod p N )

log p 2π

log p 0

 Q×p

⎡ ⎢ ⎣

 Q×p

⎤ ⎥ f (u) ψ(u) |u|itp d × u ⎦ ψ(y)−1 |y|−it p dt ⎡

⎢ log p f (u) ψ(u) ψ(y)−1 ⎣ 2π

2π

log p 0

⎤ ⎥ |u/y|itp dt ⎦ d × u

72

Automorphic representations and L-functions for GL(1, AQ )  = f (u) ψ(u) ψ(y)−1 d × u ψ (mod p N )

p− Z×p



= ϕ( p N )

f (u) d × u =

f (y).

y·(1+ p N Z p )

The last step in the above argument follows from the orthogonality relation 1 ϕ( p N )

ψ (mod p N )

ψ(u)ψ(y)

−1

 =

1,

if uy −1 ∈ 1 + p N Z p ,

0,

otherwise,

the fact that f is constant on the set y · (1 + p N Z p ), and the fact that the volume of this set is ϕ( p N )−1 . 

Exercises for Chapter 2 2.1 Let t be a non-zero real number, and consider the automorphic form φ : G L(1, AQ ) → C defined by φ(g) = |g|itA . Prove that φ is continuous, but show that its kernel is not an open subgroup. Deduce that φ cannot be the idelic lift of a Dirichlet character. 2.2 From Theorem 2.19, we see that any automorphic form for G L(1, AQ ) can be written as φ(g) = c χidelic (g) |g|itA for some Dirichlet character χ and some constants c and t. Find an explicit formula for t in terms of “nice” values of φ. 2.3 Let G be a topological group (e.g., R, Q p , or AQ ) and let ψ : G → C× be a homomorphism (not necessarily continuous). (a) Show that ψ is continuous if and only if it is continuous at the identity of G. (b) If the image of ψ is a finite set, show that ψ is continuous if and only if ker(ψ) is an open subgroup. (For example, G = AQ and ψ = χidelic for some Dirichlet character χ .) 2.4 Suppose ω : Qv → C× is a continuous unitary character such that ω(−1) = −1, and let  be an even Bruhat-Schwartz function (i.e., (−x) = (x) for all x ∈ Qv ). Prove that the local zeta integral satisfies Z v (s, , ω) = 0.

Exercises for Chapter 2

73

× 2.5 Let ωv : Q× v → C be a ramified continuous unitary character. Recall from (2.4.3) that we defined the local factor of the classical completed L-function to be

 L v (s, ωv ) =

π−

s+1 2



 s+1  2

, if v = ∞, if v = p < ∞.

1,

Find a Bruhat-Schwartz function ◦v : Q× v → C such that Z v (s, ◦v , ωv ) = L v (s, ωv ). 2.6 For 0 < (s) < 1 and ωv : Qv → C× any continuous unitary character, show that L v (s, ωv ) γ (s, ωv )−1 . v (s, ωv ) = L v (1 − s, ωv ) Deduce that v (s, ωv ) ≡ 1 if v is unramified. If v = ∞ is ramified, show that v (s, ωv ) ≡ i. If v = p < ∞ and ω = ω p is ramified with conductor p f , then ⎛

⎞−1 pf

⎜ f ⎟  p (s, ω p ) = p − f (s−1) ω p ( p) f ⎜ ω p (a)e−2πia/ p ⎟ ⎝ ⎠

.

a=1 (a, p f )=1

× 2.7 For any continuous unitary character ω : Q× v → C and any complex number s with 0 < (s) < 1, show that

γ (s, ω) = ω(−1) γ (s, ω). Conclude that |γ (1/2, ω)| = 1. When v = p < ∞ and ω is ramified with conductor p f , deduce the following classical fact about Gauss sums:     f   p  4  −2πia/ p f  f  ω(a)e = p .    a=1  (a, p)=1 2.8 Let χ (mod p f ) be a non-trivial Dirichlet character for some prime p and some integer f ≥ 1.

74

Automorphic representations and L-functions for GL(1, AQ ) p f

(a) Show that

χ ( j) = 0.

j=1 ( j, p)=1

Hint: Make the change of variable j → a j for some a ∈ Z with χ (a) = 1. (b) Show that

 Z×p

p f

χidelic (x p ) d × x p =

χ ( j) = 0.

j=1 ( j, p)=1

(c) Deduce equation (2.2.19). 2.9 Let G be an abelian group, V a finite-dimensional complex vector space, and π : G → G L(V ) an irreducible representation. Prove that dim(V ) = 1. Hint: For a fixed g0 ∈ G, consider how G acts on the eigenspaces of π (g0 ). 2.10 In this exercise we will show that any continuous unitary character ω : R× → C× is of the form ω(g) = |g|it∞

or

ω(g) = |g|it∞ · sign(g),

for some fixed constant t ∈ R as in (2.1.11). In order to accomplish this, we will assume the following fact from the theory of covering spaces in algebraic topology: for any continuous map f : R+ → C× with values in the unit circle, there exists a continuous map λ : R+ → R such that λ(1) = 0 and f (g) = eiλ(g) . Here R+ is the multiplicative group of positive real numbers. The existence of such a map λ is reasonably obvious, but the non-trivial part is that we may take λ to be continuous. For the remainder of the problem, let ω be any continuous unitary character as above, and let f = ω|R+ be the restriction of ω to the subgroup of positive reals. (a) Define a map  : R+ × R+ → R by the formula (g, h) = λ(gh) − λ(g) − λ(h). Show that ei(g,h) = 1 for all g, h ∈ R+ . Conclude that λ is a group homomorphism. (b) Show that there exists t ∈ R such that λ(er ) = r t for all real numbers r . Deduce that f (g) = g it for all g ∈ R+ . (c) Show that  ω(g) =

|g|it∞ , |g|it∞

if ω(−1) = 1,

· sign(g), if ω(−1) = −1.

Exercises for Chapter 2

75

2.11 Let p be a prime number. Let ω be a continuous character of Q×p with image in the positive real numbers. Show that ω is unramified. 2.12 Fix a prime number p. Prove that an unramified continuous unitary character ω : Q×p → C× can be written as ω(x) = |x|itp for some real number t. Show also that t is uniquely determined up to addition of an integer multiple of 2π/ log p. 2.13 In this exercise, we give two examples of representations of the topological group R, the additive group of real numbers endowed with its usual topology. One example is a continuous representation, while the other is not. (a) Let R2 be the standard 2-dimensional vector space over the field R endowed with its usual topology.   Show that the map π : R → G L(2, R) defined by π (u) = 10 u1 determines a continuous representation of R on R2 . That is, verify that π (uu ).v = π (u).(π (u ).v) for all u, u ∈ R and that the associated map R × R2 → R2 given by (u, v) → π (u).v is continuous. (b) Prove that there exists a homomorphism f : R → R that is not continuous. Conclude that the map π : R → G L(1, R) given by π (a).v = e f (a) v is a group representation, but that the associated map R × R → R given by (a, v) → e f (a) v is not continuous.

3 The classical theory of automorphic forms for GL(2)

3.1 Automorphic forms in general A very general notion of automorphic form was given by [Borel, 1966]. Let G be a group which acts on a topological space X . Let us also fix a function ψ : G × X → C. An automorphic function with multiplier ψ is a function f : X → C which satisfies f (gx) = ψ(g, x) f (x),

(∀g ∈ G, x ∈ X ).

(3.1.1)

Often, it is assumed that f satisfies certain growth conditions. It immediately follows from (3.1.1) that if g1 , g2 ∈ G and x ∈ X , then we must have f (g1 g2 x) = ψ(g1 g2 , x) f (x) = ψ(g1 , g2 x) f (g2 x) = ψ(g1 , g2 x)ψ(g2 , x) f (x). (3.1.2) If we further assume that f (x) =/ 0, then we see from (3.1.2) that ψ satisfies the cocycle relation ψ(g1 g2 , x) = ψ(g2 , x) · ψ(g1 , g2 x)

(3.1.3)

for all g1 , g2 ∈ G. If ψ(g, x) does not depend on x, then ψ is just a character of G. In [Borel, 1966], the cocycle ψ is called a factor of automorphy. The cocycle ψ occurs naturally in the theory of differential forms. Example 3.1.4 ( j-cocycle)   Let h = {x + i y | x ∈ R, y > 0} be the upper half plane. Then for γ = ac db ∈ S L(2, R), z ∈ h, define j(γ , z) = cz + d. It is easy to verify with a brute force computation that j(γ1 γ2 , z) = j(γ2 , z) · j(γ1 , γ2 z), 76

(∀ γ1 , γ2 ∈ S L(2, R)).

3.2 Congruence subgroups of the modular group

77

The j-cocycle appears in the definition of holomorphic modular forms. For example, a function f : h → C is a holomorphic form of weight   az+b modular 2k if it is holomorphic and satisfies f (γ z) = f cz+d = j(γ , z)2k f (z) for all   z ∈ h, γ = ac db ∈ S L(2, Z). In this case, the differential form f (z)(dz)k is an S L(2, Z)-invariant differential form. We may formulate the following very general notion of automorphic form for the situation where X is Lie group (such as S L(2, R)) and  is a discrete subgroup (such as S L(2, Z)). • An automorphic form is an automorphic function f : X → C, satisfying (3.1.1), which is an eigenfunction of certain invariant differential operators and satisfies some growth conditions at infinity. We have already encountered automorphic forms in Definition 2.1.4. In the case of Definition 2.1.4, we have G = G L(1, Q) with automorphy factor ψ = 1. Then G acts on X = G L(1, AQ ). The theory of automorphic forms has a long history. We shall now briefly review the classical G L(2) theory.

3.2 Congruence subgroups of the modular group Fix an integer N ≥ 1. The group  (N ) :=

a c

b d



 5  ∈ S L(2, Z)  a ≡ d ≡ 1 (mod N ), b ≡ c ≡ 0 (mod N )

is called the principal congruence subgroup of level N . A subgroup  ⊂ S L(2, Z) is called a congruence subgroup if  contains (N ) for some N ≥ 1. The least such N is called the level of . The index of the principle congruence subgroup (N ) in S L(2, Z) is given by the formula [Shimura, 1971] [S L(2, Z) : (N )] = N 3

  1 − p −2 . p|N

An important congruence subgroup of level N is the group  0 (N ) :=

a c

b d



 5  ∈ S L(2, Z)  c ≡ 0 (mod N ) .

For this subgroup the index is given by [S L(2, Z) : 0 (N )] = N

  1 + p −1 . p|N

78

The classical theory of automorphic forms for GL(2)

3.3 Automorphic functions of integral weight k The group S L(2, R) acts on the upper half plane h := {x + i y | x ∈ R, y > 0}. If γ =



a b c d



∈ S L(2, R) and z ∈ h, then the action is explicitly given by γz =

az + b . cz + d

It is convenient to extend the upper half-plane by adding the point at infinity. We think of the point at ∞ as lim x + i y, which is independent of x ∈ R. y→∞   It follows that for γ = ac db ∈ S L(2, R) we have a(x + i y) + b γ ∞ = lim = y→∞ c(x + i y) + d



a/c, ∞,

c =/ 0, c = 0.

Thus, ∞ is equivalent to a/c under the action of γ . If γ ∈ S L(2, Z), then a/c is always a rational number equivalent to ∞. Definition 3.3.1 (Extended upper half plane) We define the extended upper half plane to be h∗ = h ∪ Q ∪ ∞. Definition 3.3.2 (Cusps) Let  be a subgroup of finite index in S L(2, Z). A cusp for  is defined to be ∞ or any rational number a/c. Clearly a/c is equivalent to ∞ under the action of S L(2, Z). Let  be a subgroup of index n in S L(2, Z) and let \S L(2, Z) = {γ1 , γ2 , . . . , γn } denote a set of coset representatives. Let D denote a fundamental domain for \h and let D∗ = D ∪ cusps() denote a fundamental domain for \h∗ . Here cusps() denotes a set of cusps which are inequivalent with respect to the action of . Define  

 D = z ∈ h  |(z)| ≤ 1/2, |z| ≥ 1 . A fundamental domain for S L(2, Z)\h, (denoted D) is just the domain D with only half the boundary included, i.e., it has congruent boundary points with respect to the imaginary axis. Let D∗ = D ∪ ∞ be a fundamental domain for n γi D∗ . Each translated domain γi D∗ will contain S L(2, Z)\h∗ . Then D∗ = i=1 exactly one cusp.

3.3 Automorphic functions of integral weight k

79

Definition 3.3.3 (Moderate growth) A smooth function f : h → C is said to have moderate growth at a cusp a ∈ Q ∪ ∞ if f (σa (x + i y)) is bounded by a power of y (as y → ∞) for any fixed σa ∈ S L(2, R) satisfying σa ∞ = a. The function f is said to have moderate growth if it has moderate growth at every cusp. Definition 3.3.4 (Automorphic function of integral weight k) Let  be a subgroup of finite index in S L(2, Z). Let k ∈ Z, k ≥ 0. An automorphic function of weight k and character ψ :  → C is a smooth function f : h → C, of moderate growth, which satisfies f (γ z) = ψ(γ )(cz + d)k f (z)

(3.3.5)

for all γ ∈  and z ∈ h. We let Ak,ψ () denote the C-vector space of all automorphic functions of weight k and character ψ satisfying (3.3.5). Example 3.3.6 (Automorphic functions for 0 (N )) Fix N ∈ Z with N ≥ 1. Let χ denote a Dirichlet character of conductor N as in (2.1.5). We may lift χ to a character ( χ of 0 (N ) by defining  ( χ

a c

b d

 := χ (d)

  for ac db ∈ 0 (N ). It is easy to see that ( χ (γ1 γ2 ) = ( χ (γ1 )( χ (γ2 ) for γ1 , γ2 ∈ 0 (N ) because 

a1 c1

b1 d1



a2 c2

b2 d2



 =

∗ ∗

∗ b2 c1 + d1 d2

 .

Let k ∈ Z, k ≥ 0. An automorphic function of weight k and character χ (mod N ) for 0 (N ) will satisfy (3.3.5) if ψ = ( χ . By abuse of notation, we shall denote the space of these automorphic functions by Ak,χ (0 (N )) instead of Ak,( χ (0 (N )). If  is a congruence subgroup as in Section 3.2, then the automorphic functions which satisfy (3.3.5) which are either holomorphic or anti-holomorphic functions are the classical modular forms. We shall show that all these forms can be lifted to adelic modular forms which can then be used to construct automorphic representations for G L(2, AQ ) in a manner which is analogous to the construction of automorphic representations for G L(1, AQ ) given in Section 2.5. The modular forms associated to non-congruence subgroups of S L(2, Z) are of an entirely different species. They will not appear in the adelic representation theory.

80

The classical theory of automorphic forms for GL(2)

Let Ak,ψ () denote the space of automorphic functions of weight k and character ψ (for a subgroup  of finite index in S L(2, Z)) as in Definition 3.3.4. Henceforth, we shall restrict our attention only to the subgroups 0 (N ). This restriction is not too onerous. We shall see later that, with translates and linear combinations, it captures all automorphic forms which arise from the adelic theory. Proposition 3.3.7 (Coset representatives for 0 (N )) Let N ≥ 1 with N ∈ Z. A set of coset representatives for 0 (N )\S L(2, Z) is given by  5    a b ∈ S L(2, Z)  d|N , 0 < c ≤ N /d , c d where for each c, d we choose a, b so that ad − bc = 1. Proof See [Shimura, 1971]. Proposition 3.3.8 (Inequivalent cusps for 0 (N )) A set of inequivalent cusps for 0 (N ) is given by

u      w|N , (u, w) = 1, u mod (w, N /w) . w Here (u, w) denotes the greatest common divisor of u and w, and the integer 0 may not be chosen to represent the trivial residue class (mod (w, N /w)). Proof See [Shimura, 1971].

3.4 Fourier expansion at ∞ of holomorphic modular forms Fix integers k ≥ 0, N ≥ 1. Let f be an automorphic function of weight k and character χ (mod N ) for 0 (N ) as in Example 3.3.6. We say f is a holomorphic modular for z ∈  if f (z) is a holomorphic function of z   h.   form 11 11 z = Now the matrix 0 1 lies in 0 (N ). It follows that f (z) = f 01 f (z +1) so that f (z) is periodic in x. We may now obtain the Fourier expansion of f . Theorem 3.4.1 (Fourier expansion of a holomorphic modular form) Let N , k ∈ Z with k ≥ 0, N ≥ 1. Fix a character χ (mod N ). Let f be a holomorphic modular form of weight k and character χ for the congruence subgroup 0 (N ) as in Example 3.3.6. Then f (z) has a Fourier expansion of the form f (z) =

∞

an e2πinz ,



 z ∈ h, an ∈ C, for n = 1, 2, 3, . . . .

n=0

Furthermore, an has a polynomial growth in n as n → ∞.

3.5 Maass forms

81

Proof We have already pointed out that f (z) is periodic in x with period one. Consequently, it has a Fourier expansion of the form f (z) = n∈Z an (y)e2πinx with an (y) ∈ C. Since f is holomorphic, we must have an (y) = an e−2πny for constants an ∈ C. Further, an = 0 for n < 0 because f is of moderate growth. To show that an has polynomial growth in n as n → ∞, we compute 

1

an = e ·

  f x + i/2π n e−2πinx d x.

(3.4.2)

0

Let D N be a connected fundamental domain for 0 (N )\h. Then the translated fundamental domains γ D N with γ ∈ 0 (N ) cover h. Now, the line {x + i/2π n | 0 ≤ x < 1} will intersect only finitely many such translated fundamental domains. Each such translated fundamental domain contains finitely many cusps and may be considered as the union of a compact set and a finite number of neighborhoods containing the cusps. For a rational cusp a ∈ Q the neighborhood is chosen as {z ∈ h | |z − a| < } with sufficiently small . It is called an -neighborhood of the cusp a. Let 0 ≤ x < 1. Either x + i/2π n is in an -neighborhood of a cusp a or  not. If not, then x + i/2π n lies in a compact set and f x + i/2π n is bounded. Otherwise, there exists σa ∈ S L(2, R) with σa ∞ = a, such that   f (x + i/2π n) = f σa · σa−1 (x + i/2π n) will have at most polynomial growth in n. This follows from Definition 3.3.3 and the fact that σa is chosen from a finite set of matrices associated to the inequivalent cusps for 0 (N ) given in Proposition 3.3.8. Plugging these bounds into (3.4.2) gives the required estimate for an . 

3.5 Maass forms Let f be an automorphic function of weight k and character χ (mod N ) for 0 (N ) as in Example 3.3.6. We shall now consider the situation where f (z) is not necessarily a holomorphic function of z for z ∈ h. Such functions arose in the work of [Maass, 1946, 1947, 1983] which we shall now briefly describe. Fix integers k ≥ 0, N ≥ 1, and a character χ (mod N ). Let Ak,χ (0 (N )) denote the C-vector space of automorphic functions introduced in Example 3.3.6. Then f ∈ Ak,χ (0 (N )) satisfies  f

az + b cz + d

 = χ (d) (cz + d)k f (z)

82

The classical theory of automorphic forms for GL(2)   for all ac db ∈ 0 (N ). We may fix a specific branch of the log function by letting log z = log |z| + i arg(z), with −π < arg(z) ≤ π . Then setting k z w = exp(w log z), it is easy to see that the function f ∗ (z) := y 2 f (z) satisfies

f

∗



az + b cz + d



 = χ (d)

cz + d |cz + d|

k



∗

f (z) = χ (d)

cz + d c¯z + d

 k2

f ∗ (z)

= χ (d)eikarg(cz+d) f ∗ (z)  for all

a b c d



(3.5.1)

∈ 0 (N ).

Definition 3.5.2 (Vector space of automorphic functions) Let N , k ∈ Z with N ≥ 1. Fix a character χ (mod N ). We define A∗k,χ (0 (N )) to be the C-vector space of all smooth functions f ∗ : h → C of moderate growth (as in Definition 3.3.3) which satisfy the automorphic relations (3.5.1). Definition 3.5.3 (Laplace operator) For an integer k, we define the weight k Laplace operator  k := −y

2

∂2 ∂2 + ∂ x 2 ∂ y2

 + iky

∂ . ∂x

Lemma 3.5.4 Let N , k ∈ Z with N ≥ 1. Fix a character χ (mod N ). The weight k Laplace operator k given in Definition 3.5.3 has the property that  k f

az + b cz + d





cz + d = χ (d) |cz + d|

k k f (z),

(z ∈ h),

for all f ∈ A∗k,χ (0 (N )) as in Definition 3.5.2. Thus k maps A∗k,χ (0 (N )) to itself. Proof For z = x + i y ∈ h, define d 1 = dz 2



∂ ∂ −i ∂x ∂y

 ,

d 1 = d z¯ 2



∂ ∂ +i ∂x ∂y

 .

d Then dz z = 1, ddz¯ z = 0, so a function f is holomorphic if it satisfies the Cauchy Riemann condition ddz¯ f = 0. It is easy to see that we may write   d d 2 d d + ik Im(z) + . k = −4 Im(z) dz d z¯ dz d z¯

3.5 Maass forms

83

It follows that  k f

az + b cz + d



  cz + d k = k χ (d) f (z) |cz + d|   k cz + d 2 = χ (d) · k f (z) . c¯z + d 



The result follows after a routine computation.



In a groundbreaking paper, [Selberg, 1956, 1989], the spectral decomposition of the Laplace operator  2  ∂ ∂ ∂2 + iky + k = −y 2 2 2 ∂x ∂y ∂x on the space, L2 (0 (N )\h, k, χ ) was obtained. Here L2 (0 (N )\h, k, χ ) is the completion of the space of all functions f ∈ A∗k,χ (0 (N )) satisfying the L2 condition  d xd y | f (z)|2 < ∞, y2 0 (N )\h

with respect to the Petersson inner product, which is defined on the space of such functions as follows. Definition 3.5.5 (Petersson inner product) Let N , k ∈ Z with N ≥ 1. Fix a character χ (mod N ). For f, g ∈ A∗k,χ (0 (N )), the Petersson inner product of f and g (denoted  f, g) is defined to be  d xd y f (z) g(z) .  f, g = y2 0 (N )\h

Remark Note that f · g¯ is well defined on 0 (N )\h even though neither f nor g is. The differential operator k has the property that it maps automorphic forms of weight k into automorphic forms of weight k. The Selberg spectral decomposition (see [Goldfeld, 2006]) states that the Hilbert space L2 (0 (N )\h, k, χ ) decomposes into Maass cusp forms, Eisenstein series, and residues of Eisenstein series. We shall now define Maass forms. In order to simplify notation it is very convenient to introduce the slash operator. Definition 3.5.6 (Slash operator)   Fix an integer k. Let f : h → C be a function. For z ∈ h and α = ac db ∈ G L(2, R) having positive determinant, we define the slash operator |k by

84

The classical theory of automorphic forms for GL(2)

 cz + d −k f (αz). |cz + d|    The slash operator satisfies f k αβ = f k α k β for all α, β ∈ G L(2, R) having positive determinant.    f k α (z) =



Definition 3.5.7 (Maass form) Let N , k ∈ Z with N ≥ 1. Let ν ∈ C. Fix a character χ (mod N ). A Maass form of type ν of weight k and character χ for 0 (N ) is a smooth function f : h → C satisfying the following conditions:      a b  • f k γ (z) = χ (d) f (z), for all γ = ∈ 0 (N ), z ∈ h; c d • k f = ν(1 − ν) f, where k is the Laplace operator given in Definition 3.5.3; • f is of moderate growth as in Definition 3.3.3;  y | f (z)|2 d xd < ∞. • y2 0 (N )\h

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 .

3.6 Whittaker functions Let α ∈ R and ν ∈ C. The Whittaker functions Wα,ν (y) (see [WhittakerWatson, 1935], [Gradshteyn-Ryzhik, 2007]) are solutions of Whittaker’s differential equation

(y) = y 2 Wα,ν



 1 2 1 y − αy + ν 2 − Wα,ν (y) 4 4

(3.6.1)

which have the asymptotic behavior  α −y/2

Wα,ν (y) ∼ y e

 2   ∞ k 1  2 1 1+ ν − α+ − , k!y k =1 2 k=1 (as y → ∞).

The differential equation (3.6.1) has two linearly independent solutions (over C) given by Wα,ν (y) and W−α,ν (−y). The behavior of the Whittaker functions as y → 0 is given by  Wα,ν ∼

y 2 −|(ν)| , 1 1 2

y | log y|,

if ν =/ 0, if ν = 0.

(3.6.2)

3.6 Whittaker functions

85

An integral representation for the Whittaker function is y

y ν+ 2 e− 2    ν − α + 12 1

Wα,ν (y) =



∞

e−yt t ν−α− 2 (1 + t)ν+α− 2 dt. 1

1

(3.6.3)

0

The integral representation (3.6.3) holds provided (ν −α)>− 12 , | arg(y)|< π. We also require the following Mellin transform formula: 

∞

e

     s + 12 + ν  s + 12 − ν dy −s = 2 , y Wα,ν (2y) y  (s − α + 1)

−y s

0

(3.6.4)

  which holds for  s + 12 ± ν > 0. When α = 0, the Whittaker function is just the K- Bessel function: W0,ν (y) =

 y  12 π

Kν

y 2

1 K ν (y) = 2

,

∞

du 1 −1 . e− 2 y (u+u ) u ν u

(3.6.5)

0

The Whittaker function simplifies considerably when example, we have y

1 2

+ α ± ν ∈ N. For y

Wα, ±(α− 1 ) (y) = y α e− 2 ,

W1−α, ±(α− 1 ) (y) = y 1−α e− 2 .

2

2

(3.6.6)

The Whittaker functions satisfy the following recurrence relations (see [Whittaker-Watson, 1935]):   2    1 1

yWα,ν (y) = α − y Wα,ν (y) − ν 2 − α − Wα−1,ν (y), 2 2   1

yWα,ν (y) = − α − y Wα,ν (y) − Wα+1,ν (y), 2   1 √ − α + ν Wα−1,ν (y), Wα,ν (y) = y Wα− 1 ,ν− 1 (y) + 2 2 2   1 √ − α − ν Wα−1,ν (y). (3.6.7) Wα,ν (y) = y Wα− 1 ,ν+ 1 (y) + 2 2 2 It was shown in [Maass, 1946, 1947, 1983] that the Whittaker functions Wα,ν appear in the Fourier expansion of Maass forms of type ν as defined in Definition 3.5.7. Let f : h → C be a Maass form as in Definition 3.5.7. For each n ∈ Z, let us define  Wn (z) = 0

1

f (z + u)e−2πinu du.

(3.6.8)

86

The classical theory of automorphic forms for GL(2)

It immediately follows from (3.6.8) and Definition 3.5.7 that Wn satisfies the following properties: • Wn (z + u) = e2πinu Wn (z),

(for u ∈ R),

• k Wn (z) = ν(1 − ν)Wn (z), • Wn (z) has moderate growth.

(3.6.9)

Definition 3.6.10 (Jacquet’s Whittaker function) Let n ∈ Z, ν ∈ C. Set ψn (x) = e2πinx , an additive character of R. A smooth function Wn : h → C which satisfies (3.6.9) is termed a Jacquet Whittaker function of type ν and character ψn . If we let Wn (z) = Wn (y)e2πinx then it is easy to see that the differential equation k Wn = ν(1 − ν)Wn implies that 6   7 −y 2 −4π 2 n 2 Wn (y)+Wn

(y) − 2π kynWn (y) e2πinx = ν(1−ν)Wn (y)e2πinx . Thus Wn satisfies Whittaker’s differential equation and, in particular, Wn (y) = cW sign(n) k , ν− 1 (4π |n| · y), 2

(for c =/ 0)

2

if n =/ 0. Following [Goldfeld, 2006] we now show how to explicitly construct Jacquet’s Whittaker function as a simple integral. The beauty of this construction is that it extends easily to higher rank situations. The key point to the construction is that if we define J (γ , z) =

cz + d , |cz + d|

 for γ =

a c

b d

 ∈ 0 (N ), z ∈ h,

then for any γ ∈ S L(2, R), s ∈ C, we have   k J (γ , z)−k Im(γ z)s = s(1 − s)J (γ , z)−k Im(γ z)s . Thus J (γ , z)−k Im(γ z)s is an eigenfunction of k with eigenvalue s(1 − s). To create a Jacquet Whittaker function we also require the first condition of (3.6.9) that the function is multiplied by a character after translation. Formally, the integral ∞ Wn (z) := −∞

 s J (γ , z + u)−k Im γ (z + u) e−2πinu du

(3.6.11)

3.7 Fourier-Whittaker expansions of Maass forms

87

satisfies the requirement that Wn (z + u) = e2πinu Wn (z), provided  integral  the 0 −1 (3.6.11) converges. This will be the case if we choose γ = 1 0 . In this case, the integral (3.6.11) takes the explicit form ∞  Wn (z) = −∞

=e

z+u |z + u| ∞ 

2πinx −∞

−k 

y |z + u|2

iy + u |i y + u|

−k 

s

e−2πinu du.

y y2 + u2

s

e−2πinu du. (3.6.12)

3.7 Fourier-Whittaker expansions of Maass forms Fix an integer N ≥ 1.  Let a ∈ Q ∪ ∞ be a cusp for 0 (N ) as in Proposition 10 3.3.8. Let I2 = 0 1 . Consider the stability subgroup a = {γ ∈ 0 (N ) | γ a = a} which is the product of the multiplicative group of two  elements a b {±I2 } times an infinite cyclic group with generator ga = caa daa , say. We may choose a matrix σa ∈ S L(2, R) satisfying   1 1 σa−1 ga σa = . (3.7.1) σa ∞ = a, 0 1 We may first choose γa ∈ S L(2, Z) such that γa ∞ = a. Then γa−1 a γa contains −I2 and is a subgroup  of finite index in ∞ . It follows that it is generated by −I2 and 10 m1a for some positive integer m a . We take σa = √  ma 0 √ −1 . It follows that for a Maass form f , we have γa 0 m a

   1  f  σa 0 k

1 1



      (z) = f k ga σa (z) = χ (da ) f k σa (z).

(3.7.2)

Definition 3.7.3 (Cusp parameter) We define the cusp parameter μa ∈ [0, 1) by the identity e2πiμa = χ (da ). It immediately    follows from (3.7.2) and Definition 3.7.3 that the function e−2πiμa x f k σa (z) is periodic in x with period one. The following expansion was found by Maass (see [Maass, 1983], [Roelcke, 1966]). Theorem 3.7.4 (Fourier-Whittaker expansion of a Maass form) Let N , k ∈ Z with N ≥ 1. Fix a character χ (mod N ). Let f be a Maass form of type ν, weight k, and character χ as in Definition 3.5.7. Let a ∈ Q ∪ ∞ be a cusp for 0 (N ), let σa be defined by (3.7.1), and let μa be the cusp parameter as in Definition 3.7.3. Then, for z = x + i y ∈ h, we have

88

The classical theory of automorphic forms for GL(2)      f k σa (z) = Aa,0 (y) + aa,n W sgn(n)k , ν− 1 4π |n + μa | · y e2πi(n+μa )x , 2

n+μa =/ 0

2

where aa,n are complex constants, Aa,0 (y) = 0 if μa =/ 0, and 

aa,0 y ν + aa,0 y 1−ν , if ν =/ 12 , μa = 0, Aa,0 (y) = 1 1

y 2 log y, if ν = 12 , μa = 0, aa,0 y 2 + aa,0

∈ C. Furthermore, aa,n has a polynomial growth in n as with aa,0 , aa,o |n| → ∞.    −2πiμa x f k σa (z) is periodic in Proof It follows from equation (3.7.2) that e     x of period one. Consequently, f k σa (z), has a Fourier expansion of the form

   Ba,n (y)e2πi(n+μa )x . f k σa (z) =

(3.7.5)

n∈Z

with Ba,n (y) ∈ C for all integers n. Each term Ba,n (y)e2πi(n+μa )x in the expansion (3.7.5) must satisfy the differential equation k Ba,n (y)e2πi(n+μa )x = ν(1 − ν)Ba,n (y)e2πi(n+μa )x , with

 k = −y 2

∂2 ∂2 + ∂ x 2 ∂ y2

 + iky

∂ . ∂x

Consequently, 6   7

− y 2 − 4π 2 (n + μa )2 Ba,n (y) + Ba,n (y) − 2π ky(n + μa )Ba,n (y) · e2πi(n+μa )x = ν(1 − ν)Ba,n (y)e2πi(n+μa )x . This implies that Ba,n (y) satisfies the following variation of Whittaker’s differential equation (3.6.1):  

(y) = 4π 2 (n + μa )2 y 2 − 2π k(n + μa )y + ν 2 − ν Ba,n (y). (3.7.6) y 2 Ba,n The space of solutions to the differential equation (3.7.6) is a two-dimensional C- vector space. However, the subspace of solutions which do not increase exponentially in y as y → ∞ is one dimensional and spanned by W sgn(n)k . 2 Since Definition 3.5.7 of a Maass form requires moderate growth, it follows that the solution of (3.7.6) must take the form   Ba,n (y) = aa,n W sgn(n)k , ν− 1 4π |n + μa | · y 2

2

3.8 Eisenstein series

89

for some constant aa,n ∈ C provided n =/ 0. The case when n = 0 is left to the reader as an exercise. The argument to show that aa,n has polynomial growth in n as |n| → ∞ is the same as in the proof of Theorem 3.4.1. We leave it to the reader to fill in the details.  Definition 3.7.7 (Maass cusp form) Let N , k ∈ Z with N ≥ 1. Fix a character χ (mod N ). Let f be a Maass form of type ν, weight k, and character χ as in Definition 3.5.7. Let Aa,0 (y) be the constant term at the cusp a in the Fourier-Whittaker expansion of Theorem 3.7.4. If Aa,0 (y) = 0 for all cusps a then f is called a cusp form.    Remark If f is a Maass cusp form then f k σa (z) decreases exponentially to 0 as y → ∞.

3.8 Eisenstein series Fix integers N ≥ 1, k ≥ 0, and let χ (mod N ) be a Dirichlet character. Recall that the scaling matrix of a cusp a, denoted σa is uniquely characterized by (3.7.1). The cusp a is said to be singular if     1 1 −1 σa = 1 or (−1)k . (3.8.1) χ σa 0 1 The Eisenstein series are indexed by the singular cusps and are defined as follows. Definition 3.8.2 (Eisenstein series at a cusp) Let N , k ∈ Z with N ≥ 1. Fix a character χ (mod N ). Let a be a singular cusp of 0 (N ) as in (3.8.1) and let σa be defined by (3.7.1). For (s) > 1 and z ∈ h, the Eisenstein series at the cusp a, of weight k, and character χ , for the congruence subgroup 0 (N ) is defined by the absolutely convergent series   −k  s χ (γ ) J σa−1 γ , z Im σa−1 γ z E a (z, s, χ ) = γ ∈a \0 (N )



where J

a c

b d



 , z

=

cz + d . |cz + d|

The Eisenstein series satisfy the differential equation k E a (z, s, χ ) = s(1 − s)E a (z, s, χ ) where k is given in Definition 3.5.3, and so are eigenfunctions of k with eigenvalue s(1 − s). The continuous spectrum of k is spanned by the Eisenstein series E a . It was shown in [Selberg, 1956] that the Eisenstein series have

90

The classical theory of automorphic forms for GL(2)

meromorphic continuation in s to the whole complex plane. If we form a vector of Eisenstein series, indexed by the cusps, then the vector valued automorphic form will have a functional equation s → 1 − s.

3.9 Maass raising and lowering operators The Maass raising and lowering operators are differential operators found by [Maass, 1953] which have the property that when they are applied to an automorphic function of weight k as in Definition 3.5.2 then they produce a new automorphic function whose weight is either raised or lowered by 2. Without further ado, let’s define these differential operators. Definition 3.9.1 (Maass raising operator) Let k ∈ Z. We define the Maass raising operator Rk to be the differential operator Rk := i y

∂ k ∂ k ∂ +y + = (z − z¯ ) + . ∂x ∂y 2 ∂z 2

Definition 3.9.2 (Maass lowering operator) Let k ∈ Z. We define the Maass lowering operator L k to be the differential operator L k := −i y

∂ k ∂ k ∂ +y − = −(z − z¯ ) − . ∂x ∂y 2 ∂ z¯ 2

The following identities may be easily verified. L k = R−k , Rk = L −k ,     k k k k 1+ = −Rk−2 L k + 1− k = −L k+2 Rk − 2 2 2 2 k+2 Rk = Rk k ,

k−2 L k = L k k .

(3.9.3) (3.9.4) (3.9.5)

Furthermore, the raising and lowering operators Rk , −L k+2 are adjoint operators with respect to the Petersson inner product (see Definition 3.5.5) and satisfy (see [Roelcke, 1966], [Bump, 1997])       d xd y d xd y Rk f (z) · g(z) = f (z) · − L k+2 g (z) (3.9.6) 2 y y2 0 (N )\h

0 (N )\h

which can be succinctly written in the form 8 9 8 9 Rk f, g = f, (−L k+2 g) , where f ∈ A∗k,χ (0 (N )) and g ∈ A∗k+2,χ (0 (N )).

3.9 Maass raising and lowering operators

91

Proposition 3.9.7 (Rk raises weights by 2, L k lowers weights by 2) Fix k, N ∈ Z (with N ≥ 1) and fix a character χ (mod N ). Let A∗k,χ (0 (N )) be the C-vector space of automorphic functions of weight k and character χ for 0 (N ) as in Definition 3.5.2. If f ∈ A∗k,χ (0 (N )) then Rk f ∈ A∗k+2,χ (0 (N )) ,

L k f ∈ A∗k−2,χ (0 (N )) .

Furthermore, if k f = λ f for some eigenvalue λ ∈ C, then         k−2 L k f = λ L k f . k+2 Rk f = λ Rk f ,

(3.9.8)

(3.9.9)

Proof First, note that (3.9.9) follows from (3.9.5). Next, we will prove that        Rk f k+2 α (z) = Rk f k α (z) ,        L k f k−2 α (z) = L k f k α (z)

(3.9.10)

for any α ∈ 0 (N ) and any smooth function f : h → C. It is easy to see that (3.9.10) implies (3.9.8). For example,   since we assume that f ∈ A∗k,χ (0 (N )), one obtains immediately that f k α (z) = χ (d) f (z)      for α = a b ∈ 0 (N ). Consequently Rk f  α (z) = χ (d)(Rk f )(z). c d

k+2

We shall now prove (3.9.10) for the Maass raising operator Rk . The proof is very similarfor the  lowering operator and we leave the details to the reader. Let α = ac db . Using the identity c(z − z¯ ) = (cz + d) − (c¯z + d), and the fact that

∂ z¯ ∂z

= 0, we compute

 , k       k c¯z + d 2 ∂ az + b  Rk f k α (z) = (z − z¯ ) + f ∂z 2 cz + d cz + d   k2 :  ; c¯z + d 1 k z − z¯ k

= + f (αz) + · − c(z − z¯ ) · f (αz) cz + d 2 cz + d 2 (cz + d)2    k+2 k c¯z + d 2 k c¯z + d 2 z − z¯ = f (αz) + f (αz). (3.9.11) 2 cz + d (cz + d)2 cz + d In a similar manner we have  k+2     c¯z + d 2  (Rk f )  α (z) = k+2 cz + d  k+2  c¯z + d 2 = cz + d

    k ∂ + f (w)  · (w − w) ¯ ∂w 2 w= az+b cz+d     k f (w) + (w − w) ¯ f (w)  · . 2 w= az+b cz+d

(3.9.12)

92

The classical theory of automorphic forms for GL(2)

One immediately observes that (3.9.11) and (3.9.12) are the same because   (w − w) ¯ 

w= az+b cz+d

=

az + b a z¯ + b z − z¯ .  − = cz + d cz + d (cz + d)2

Proposition 3.9.13 (Action of Maass operators on Whittaker functions) Let k ∈ Z and let Rk , L k , be the Maass raising and lowering operators, respectively, as in Definitions 3.9.1, 3.9.2. Let r ∈ R with r > 0. Then the action of the Maass operators Rk , L k on the Fourier-Whittaker expansion (Theorem 3.7.4) is given by   Rk W k2 ,ν (4πr y) · e2πir x = −W k+2 ,ν (4πr y) · e2πir x , 2       k−1 2 2πir x 2 =− ν − L k W k2 ,ν (4πr y) · e W k−2 ,ν (4πr y) · e2πir x 2 2 If r < 0, the action is given by 

Rk W− k2 ,ν (4π |r |y) · e

2πir x









=− ν − 2



k+1 2

2  W− k+2 ,ν (4π |r |y) · e2πir x , 2

L k W− k2 ,ν (4π |r |y)e2πir x = −W− k−2 ,ν (4π |r |y) · e2πir x . 2

Proof The proof follows from the Definitions 3.9.1, 3.9.2, and the recurrence relations (3.6.7) after a routine calculation. 

3.10 The bottom of the spectrum Fix integers k, N with N ≥ 1, and let χ be a Dirichlet character (mod N ). To recapitulate, we have been studying the Hilbert space of smooth functions f : h → C which transform by  f  for any

a b c d



az + b cz + d





cz + d = χ (d) |cz + d|

k f (z)

(3.10.1)

∈ 0 (N ) and all z ∈ h. We have defined L2 (0 (N )\h, k, χ )

to be the space of all smooth functions satisfying (3.10.1) and the L2 condition  d xd y | f (z)|2 < ∞. y2 0 (N )\h

3.10 The bottom of the spectrum

93

A much simpler space than L2 (0 (N )\h, k, χ ) is the space L2 (Z\R) consisting of all smooth functions satisfying f (x + 1) = f (x), (∀x ∈ R) together 1 with the L2 condition 0 | f (x)|2 < ∞. We showed in Chapter 1 that every function in L2 (Z\R) has a Fourier expansion f (x) = n∈Z an e2πinx , so that a basis for the space is given by the exponential functions e2πinx with n ∈ Z. The 2 exponential function is an eigenfunction of the Laplacian − ddx 2 with eigenvalue 4π 2 n 2 , i.e., d2 − 2 e2πinx = 4π 2 n 2 e2πinx . dx The eigenvalues comprise the spectrum. The bottom of the spectrum is the 2 smallest eigenvalue. In the case of − ddx 2 acting on L2 (Z\R), the bottom of the spectrum is 0 and this corresponds to the constant eigenfunction. 2 Similarly, [Selberg, 1956] decomposed  2  the space L (0 (N )\h, k, χ ) into 2 ∂ ∂ ∂ eigenfunctions of k = −y 2 ∂ x 2 + ∂ y 2 + iky ∂ x . Such an eigenfunction f satisfies the second order partial differential equation k f = λ f, where λ = λ(ν) = ν(1 − ν). This conforms with Definition 3.5.7. Proposition 3.10.2 (Bottom of the spectrum) Fix integers k and N ≥ 1. Let χ be a Dirichlet character (mod N ). The operator k acting on the Hilbert space L2 (0 (N )\h, k, χ ) has a self-adjoint extension and is bounded below by     |k| |k| |k| := 1− . λ 2 2 2   If there exist elements of L2 (0 (N )\h, k, χ ) which have eigenvalue λ |k| , 2 k | | then they are given by y 2 f (z) where f is a holomorphic modular form of weight k and character χ satisfying (3.3.5) if k > 0, or the complex conjugate of such a function if k < 0. Proof For a proof of the standard fact that the Laplace operator k has a selfadjoint extension see [Iwaniec, 2002]. Now, consider a non-zero function f ∈ L2 (0 (N )\h, k, χ ) satisfying k f = μf for some eigenvalue μ ∈ C. Since k is a self-adjoint operator with respect to the Petersson inner product (see Definition 3.5.5) we have 9 8 9 8 9 8 9 8 μ f, f = k f, f = f, k f = μ f, f . 8 9 Because f =/ 0, f, f > 0 it follows that μ = μ ∈ R. To show that the classical holomorphic modular forms and their conjugates lie at the bottom of the spectrum, we require the Maass raising and lowering operators Rk , L k defined in Definitions 3.9.1 and 3.9.2. There is a natural connection between L k and holomorphic modular forms. Indeed, it follows easily from the expression for

94

The classical theory of automorphic forms for GL(2)

L k in terms of

∂ ∂z

that a Maass form of weight k is killed by L k if and only if it k | | is of the form y 2 f 0 (z) where f 0 (z) is a holomorphic modular form of weight k. Similarly, a Maass form of weight k is killed by Rk if and only if it is of the k form y | 2 | f 0 (z). Using the Petersson inner product (Definition 3.5.5) together with (3.9.4) and (3.9.6) we obtain for any f, g ∈ L2 (0 (N )\h, k, χ ) that 8 9 8 9 f, k g = f, (−Rk−2 L k + λ (k/2)) g 9 8 9 8 = L k f, L k g + λ(k/2) f, g . Let f = g =/ 0 and assume further that k f = μf . Then 8 9 8 9 f, k f = μ f, f . It immediately follows from the above that 9 8 L k f, L k f 9 ≥ 0. μ − λ(k/2) = 8 f, f 9 8 Since  f, f  > 0 8and L k f, 9L k f ≥ 0, we see that μ ≥ λ(k/2) always. If λ(k/2) = μ then L k f, L k f = 0 and L k f = 0. This implies that f (z) = k y | 2 | f 0 (z) where f 0 (z) is a holomorphic modular form of weight k. A similar argument using the other identity in (3.9.4) together with (3.9.6) k shows that μ ≥ λ(−k/2), with equality if and only if f (z) = y | 2 | f 0 (z). We deduce that the bottom of the spectrum is in all cases greater than or equal to max(λ(k/2), λ(−k/2)) = λ(|k|/2). We also deduce that holomorphic forms must have non-negative weight. Finally, we deduce that the bottom of the spectrum is exactly at λ(|k|/2) if and only if there are holomorphic cusp forms of weight |k|.  Remark There are many choices of integers k, N with k ≥ 0, N ≥ 1, and Dirichlet characters χ , such that there are no holomorphic forms of weight k, level N , and character χ . For example, if N = 1 then χ must be trivial and there are no holomorphic cusp forms for k < 12. There exist holomorphic forms f which are not cusp forms, but such elements do not yield elements of L2 (0 (N )\h, k, χ ) . Determining the bottom of the spectrum for such cases is much more difficult.

3.11 Hecke operators, oldforms, and newforms Fix integers k, N with N ≥ 1, and let χ be a Dirichlet character (mod N ). Let f ∈ A∗k,χ (0 (N )) be an automorphic function as in Definition 3.5.2. Then     cz + d k az + b = χ (d) f (z) (3.11.1) f cz + d |cz + d|

3.11 Hecke operators, oldforms, and newforms

95

  for any ac db ∈ 0 (N ) and all z ∈ h. In order that the space of non-zero functions f which satisfy (3.11.1) is not empty, we also require the consistency property that χ (−1) = (−1)k . The Hecke operators act on automorphic functions and are defined as follows. Definition 3.11.2 (Hecke operators) Let n ∈ Z, n ≥ 1. Let f ∈ A∗k,χ (0 (N )) be an automorphic function as in Definition 3.5.2 and (3.11.1). The Hecke operator Tn is defined by   d 1 az + b , Tn f (z) = √ χ (a) f d n ad=n b=1

(for z ∈ h).

By a slight extension of the proofs in [Goldfeld, 2006] one may show that the Hecke operators satisfy the following multiplicative relations: Tm Tn = χ (d) T mn2 (3.11.3) d

d|(m,n)

for all m, n ≥ 1. In particular, the Hecke operators commute with each other. The Hecke operators also commute with k . Let q be a prime with q|N . Because of the presence of the character χ (mod N ) in the Hecke operator Tq we see that   q 1 z+b . (3.11.4) f Tq f (z) = √ q b=1 q If f is a Maass form of type ν, weight k, and character χ (mod N ) as in Definition 3.5.7 with Fourier-Whittaker expansion   f (z) = a(n) W sgn(n)k , ν− 1 4π |n| · y e2πinx n =/ 0

2

2

as in Theorem 3.7.4, then a simple computation shows that   √ Tq f (z) = q a(nq) W sgn(n)k , ν− 1 4π |n| · y e2πinx . n =/ 0

2

(3.11.5)

2

The Hecke operators Tq with q|N require special consideration. Following [Goldfeld, 2006] it can be shown that the Hecke operators Tn with (n, N ) = 1 are endomorphisms of the space A∗k,χ (0 (N )). Furthermore, if we assume (n, N ) = 1, then Tn will be a normal operator because it may be shown for all f, g ∈ L2 (0 (N )\h, k, χ ) that   d xd y d xd y Tn f (z) · g(z) = χ (n) f (z) · Tn g(z) . (3.11.6) 2 y y2 0 (N )\h

0 (N )\h

96

The classical theory of automorphic forms for GL(2)

In particular, the adjoint operator Tn∗ is defined by Tn∗ = χ (n)Tn .

(3.11.7)

The Hecke operator Tq with q|N will, in general, not be a normal operator. Because of this, we will restrict our attention, for the moment, to the Hecke operators Tn with (n, N ) = 1. The Hecke operators Tn with (n, N ) = 1 are normal operators which commute with each other and with k . We may, therefore, simultaneously diagonalize the space L2 (0 (N )\h, k, χ ) with respect to all these operators. This procedure will give a basis for the C-vector space of all Maass cusp forms as defined in Definition 3.7.7 where each basis element is an eigenfunction of all the Hecke operators Tn with (n, N ) = 1. Such a Maass eigenform is characterized by the fact that its Fourier coefficients satisfy certain multiplicativity relations. This is the main theorem of Hecke theory which we now state. The proof is a slight generalization of the proof in [Goldfeld, 2006] where only Maass forms of weight 0, level N = 1, and trivial character were considered. Theorem 3.11.8 (Muliplicativity of the Fourier coefficients) Fix integers N , k with N ≥ 1. For z = x + i y ∈ h, let   a(n) W sgn(n)k , ν− 1 4π |n| · y e2πinx f (z) = √ 2 2 |n| n =/ 0 be the Fourier-Whittaker expansion at ∞ of a Maass form of type ν, weight k, and character χ (mod N ) for 0 (N ) as in Definition 3.5.7 and Theorem 3.7.4 √ (note that we have renormalized by a factor of |n| to make later formulae simpler). Assume that f is an eigenfunction of all the Hecke operators Tn with (n, N ) = 1. If a(1) = 0, then a(n) = 0 for all n ≥ 1 with (n, N ) = 1. Assume a(1) =/ 0 and f is normalized so that a(1) = 1. Then Tn f = a(n) · f,

(∀n = 1, 2, . . . , with (n, N ) = 1) .

Furthermore, for integers m, n ≥ 1 with (m, N ) = (n, N ) = 1, we have the following multiplicativity relations: a(m)a(n) = a(mn), if (m, n) = 1,  mn  a(m)a(n) = . χ (d) a d2 d|(m,n) Remark The Hecke operators also act on the space of Eisenstein series introduced in Section 3.8. If χ is a primitive character, then the vector of Eisenstein series E a (z, s, χ ) indexed by the cusps a will be an eigenvector of all the Hecke

3.12 Finite dimensionality of the eigenspaces

97

operators. This breaks down, however, if χ is not primitive. In a series of papers (see [Rankin, 1990, 1992, 1993, 1994]) the problem of diagonalizing the space of Eisenstein series when χ is not primitive was investigated. We now return to the more delicate issue of the Hecke operators Tn in the case when (n, N ) =/ 1. A complete theory was developed in [Atkin-Lehner, 1970]. To illustrate the difficulties that can arise, consider a Maass form   a(n) W sgn(n)k , ν− 1 4π |n| · y e2πinx , f (z) = √ 2 2 |n| n =/ 0 of type ν, weight k, and level 1. Then f (N z) is a Maass form of type ν, weight     −1    a bN = c/N d k, and level N . This is due to the fact that N0 01 · ac db · N0 10 so that       az + b N 0 a b = f z = f (γ N z) = f (N z) f N· 0 1 c d cz + d     a bN provided γ = c/N d ∈ S L(2, Z) which is equivalent to ac db ∈ 0 (N ). By a simple computation, we observe that if N > 1, then the first Fourier coefficient of f (N z) is zero. This is the troublesome case of Theorem 3.11.8. Atkin and Lehner called f (N z) an oldform. We now give a formal definition of oldforms and newforms. Definition 3.11.9 (Oldforms) Fix integers N , k with N ≥ 1. Let χ (mod N ) be a Dirichlet character. Let f ∈ A∗k,χ (0 (N )) be an automorphic form of weight k, level N , and character χ as in Definition 3.5.2. We say f is an oldform if there exist positive integers M, d with M < N and Md|N , and a character χ M (mod M) and a function f old ∈ Ak,χ M (0 (M)) so that f (z) = f old (dz) for all z ∈ h. The smallest such integer M is termed the conductor of f . The orthogonal complement (with respect to the Petersson inner product as defined in Definition 3.5.5) of the space of oldforms is called the space of newforms. The main result of Atkin-Lehner theory is that the space of newforms has a basis where each basis element is an eigenfunction of all the Hecke operators Tn , n = 1, 2, 3, . . . For the newforms, there is no restriction that (n, N ) = 1. For proofs of these results see [Atkin-Lehner, 1970], [Miyake, 1971], [Miyake, 2006].

3.12 Finite dimensionality of the eigenspaces Fix integers N , k with N ≥ 1, and let χ (mod N ) be a Dirichlet character. Let Sλ (N , k, χ ) denote subspace of all f ∈ L2 (0 (N )\h, k, χ ) which are Maass forms (as in Definition 3.5.7) of type ν with λ = ν(1 − ν).

98

The classical theory of automorphic forms for GL(2)

Theorem 3.12.1 (Maass) The space Sλ (N , k, χ ) is finite dimensional for any fixed λ ≥ 0. Proof By Proposition 3.7.4, every f ∈ Sλ (N , k, χ ) has a Fourier-Whittaker expansion (at a cusp a) of type      f k σa (z) = Aa,0 (y) + aa,n W sgn(n)k , ν− 1 4π |n + μa | · y e2πi(n+μa )x 2

n+μa =/ 0

2

(3.12.2) with λ = ν(1 − ν). If the dimension of Sλ (N , k, χ ) were infinite, it would be possible, for every integer n 0 > 1, to construct a non-zero finite linear combination of Maass forms of type ν which have Fourier-Whittaker expansions of the form given in (3.12.2) where aa,n = 0 for all |n| < n 0 and all cusps a. To complete the proof of Theorem 3.12.1 it is enough to prove the following lemma. Lemma 3.12.3. Assume that the function f : h → C satisfies      aa,n W sgn(n)k , ν− 1 4π |n + μa | · y e2πi(n+μa )x , f k σa (z) = n+μa =/ 0 n>n 0

2

2

for every cusp a and that f ∈ Sλ (N , k, χ ). Then, if n 0 is sufficiently large, it follows that f (z) = 0 for all z ∈ h. Proof See Lemma 3.8.3 in [Goldfeld, 2006] for a proof in the case of level N = 1 and k = 0. For the more general situation, see [Maass, 1983]. 

Exercises for Chapter 3 3.1 Let X be a set and G a group acting on X . Let H ⊂ G be a subgroup. Notice that H also acts on X . Fix a collection of right coset representatives {gi | i ∈ I }. If D H , DG ⊂ X are fundamental domains for the actions of H and of G on X , respectively, prove that D H can be written as a union of translates of DG :  DH = gi DG . i∈I

Show also that gi DG ∩ g j DG =/ ∅ if and only if gi−1 g j fixes some element of DG . 3.2 Let X be a topological space and let G be a topological group that acts continuously on X . Fix a continuous function ψ : G × X → C. (a) Prove that the set of continuous automorphic functions f : X → C with multiplier ψ forms a complex vector space.

Exercises for Chapter 3

99

(b) Let f : X → C be a continuous automorphic function with multiplier ψ. Suppose there exists a point x ∈ X such that f (x) = 0. Let G x = {g ∈ G | gx = x} be the stabilizer subgroup of x. Prove that ψ(g, x) = 1,

(g ∈ G x ).

(c) Suppose there is a point x ∈ X that has dense G-orbit; that is, the set {gx | g ∈ G} intersects every nonempty open neighborhood in X . Suppose further that f : X → C is a continuous automorphic function with multiplier ψ, and that f (y) = 0 for some point y ∈ X . Prove that f is identically zero. (Note that y need not be in the orbit of X , although it must be in the closure of the orbit.) (d) In part (b), it was crucial that we assume there exists a point x at which f does not vanish. Find an example of a topological space X , a group G that acts continuously on X , and a multiplier function ψ such that the zero function is the only automorphic function with multiplier ψ. 3.3 Let σ : C → C be defined by  z  z/ω+ 1 (z/ω)2 2 e 1− , σ (z) = z ω ω =/ 0 where the product is over all non-zero elements ω = r + si for r, s ∈ Z. (This is an example of an object that arises in the theory of elliptic functions. See, for example, Chapter 7 of [Ahlfors 1979]). (a) Verify that the above product converges absolutely on any compact subset of the complex plane. Conclude that σ is holomorphic on C. Observe that σ vanishes exactly at the points ω = r + si for r, s ∈ Z, and that σ is an odd function. (b) Prove that 1 z  σ (z) 1  1 = + + + 2 . σ (z) z ω =/ 0 z − ω ω ω (c) Show that there exist constants η1 , η2 ∈ C such that σ (z + r + si) σ (z) = + r η1 + sη2 , σ (z + r + si) σ (z)

(r, s ∈ Z).

Hint: Beware of convergence issues! (d) Observe that the group Z + Zi acts on C by translation. Define a multiplier function by  5  r + si . ψ(r + si, z) = − exp (r η1 + sη2 ) z + 2 Show that σ is an automorphic function with multiplier ψ.

100

The classical theory of automorphic forms for GL(2)

3.4 Fix N ≥ 2. In this exercise we compute the index of the congruence subgroups (N ) and 0 (N ) in S L(2, Z). (a) Showthat the group S L(2, Z/N Z) is generated by matrices of the   1 0 form 10 x1 and y 1 . (b) Consider the map π : S L(2, Z) → S L(2, Z/N Z) defined by reducing the coefficients modulo N . Prove that π is a surjective homomorphism with kernel (N ). Deduce that (N ) is a normal subgroup. (c) Give a formula for the number of elements in the group S L(2, Z/N Z). Using part (b), deduce a formula for the index of (N ) in S L(2, Z). (d) Let B be the subgroup of S L(2, Z/N Z) consisting of upper triangular matrices. Compute the index of B in S L(2, Z/N Z). (e) Using part (d), deduce a formula for the index of 0 (N ) in S L(2, Z). (f) Is 0 (N ) normal inside S L(2, Z)? 3.5 Prove that (2)\S L(2, Z) ∼ = S3 , where S3 is the permutation group on 3 symbols. Hint: The group S L(2, Z) is generated by the matrices 10 11   and −10 10 . 3.6 Recall that a subgroup H of a group G is said to have finite index if the collection of cosets H \G is finite (or equivalently, that G/H is finite). Prove that every congruence subgroup has finite index in S L(2, Z).   3.7 If z ∈ C and γ = ac db ∈ S L(2, R), prove that Im(γ z) =

Im(z) . |cz + d|2

3.8 Let f : h → C be given by f (x + i y) = e−y . Show that f has moderate growth at ∞ in the sense of Definition 3.3.3. 3.9 Let f (z) be an automorphic function of weight 2 and trivial character for the group S L(2, Z). Prove that the differential form f (z)dz is invariant under the transformations z → γ z for γ ∈ S L(2, Z). 3.10 Let N ≥ 1 and k ≥ 0 be integers, and suppose χ (mod N ) is a Dirichlet character such that χ (−1) = (−1)k . Prove that the space A∗k,χ (0 (N )) is trivial. 3.11 Fix an integer k and a function f : h → C. Show that the slash operator satisfies    (α, β ∈ S L(2, R)). f  αβ = f  α  β, k

k

k

Exercises for Chapter 3

101

3.12 Fix N , k ∈ Z with N ≥ 1, and fix a Dirichlet character χ (mod N ). Let f : h → C be an automorphic function of weight k and character χ for 0 (N ) as in Definition 3.3.4. Set f ∗ (x + i y) = y k/2 f (x + i y). Prove that  ( f ∗ k γ )(z) = χ (d) f ∗ (z),



 z ∈ h, γ =

a c

b d



 ∈ 0 (N ) .

3.13 Fix N , k ∈ Z with N ≥ 1, and fix a Dirichlet character χ (mod N ). If f ∈ Ak,χ (0 (N )) is holomorphic on h, show that f ∗ (z) = y k/2 f (z) is a Maass form of type k2 , weight k, and character χ (mod N ) for 0 (N ). 3.14 Fix integers k, N with N ≥ 1 and a Dirichlet character χ (mod N ). Prove that the complex vector spaces Ak,χ (0 (N )) from Example 3.3.6 and A∗k,χ (0 (N )) from Definition 3.5.2 are isomorphic. 3.15 Verify that the measure the action of S L(2, Z).

d xd y y2

on the upper half-plane is invariant under

3.16 The integral representation (3.6.3) of the Whittaker function Wα,ν is valid when (ν − α) > −1/2. Use it to prove two of the formulas (3.6.6). (The other two formulas lie outside the scope of the integral representation.) 3.17 Fix N ≥ 1 and let a ∈ Q ∪ {∞} be a cusp for 0 (N ). (a) Recall that the stability subgroup of a is a = {γ ∈ 0 (N ) | γ a = a}. Prove that a is isomorphic to Z × {±1}. (b) Fix a Dirichlet character χ (mod N ), and suppose that a and b are equivalent cusps; that is, there is γ ∈ 0 (N ) such that γ a = b. Prove that their associated cusp parameters agree. (See Definition 3.7.3.) (c) Let N = p be prime, and fix a Dirichlet character χ (mod p). For each cusp a for 0 ( p), prove that the cusp parameter satisfies μa = 0. (d) Find an example of an integer N ≥ 1, a Dirichlet character χ (mod N ), and a cusp a for 0 (N ) such that μa = 0. 3.18 Show ker(L k ) = {y k/2 f (z) | f is holomorphic}. Compute the kernel of Rk .

102

The classical theory of automorphic forms for GL(2)

3.19 Fix N , k ∈ Z with N ≥ 1, and fix a Dirichlet character χ (mod N ). Let a be a cusp for 0 (N ). Prove that the Eisenstein series E a (z, s, χ ) (as in Definition 3.8.2) converges absolutely for (s) > 1 and that it is a Maass form of type s(1 − s), weight k, level N and character χ . 3.20 Fix , M, N , k ∈ Z with , M, N ≥ 1, and fix a Dirichlet character χ (mod M). If f ∈ A∗k,χ (0 (M)) and if M | N , prove that g(z) = f (z) lies in A∗k,χ N (0 (N )). Here χ N is the canonical lift of the character χ to the group (Z/N Z)× .

4 Automorphic forms for GL(2, AQ )

4.1 Iwasawa and Cartan decompositions for G L(2, R) The group S L(2, R) acts transitively on the upper half plane h = {x + i y | x ∈ R, y > 0} by linear fractional transformations gz :=

az + b , cz + d

 g=

a c

b d

 ∈ S L(2, R).

It follows that h can be realized as the orbit of any point z ∈ h, i.e., h = {gz | g ∈ S L(2, R)}. The point z = i is stable under the action of the special orthogonal group K ∞ = S O(2, R) = {k ∈ S L(2, R) | ki = i}     cos θ sin θ  =  0 ≤ θ < 2π . − sin θ cos θ  It follows that the upper half plane h may be identified with S L(2, R)/K ∞ or with G L(2, R)/O(2, R) · Z (R), where Z (R) denotes the center of G L(2, R). Proposition 4.1.1 (Iwasawa decomposition) Every matrix g ∈ G L(2, R) has a unique factorization       1 x y 0 cos θ sin θ ±1 0 r 0 g= , 0 1 0 1 − sin θ cos θ 0 1 0 r with x ∈ R, y > 0, 0 ≤ θ < 2π, r > 0. Proof To see this, it is enough to show that every g = unique factorization of the form



a b c d



∈ S L(2, R) has a

103

104 

a c

Automorphic forms for GL(2, AQ )     1 cos θ b 1 x 0 y2 = 1 − sin θ d 0 1 0 y− 2

sin θ cos θ

 .

This can be shown directly by taking y = (c2 +d 2 )−1 , then choosing θ such that 1 1 − sin θ = cy 2 , and cos θ = dy 2 , and finally translating by a suitable x ∈ R. The uniqueness can be easily verified.  Proposition 4.1.2 (Cartan decomposition) Every matrix g ∈ G L(2, R) has a factorization       y 0 cos θ2 sin θ2 ±1 0 r 0 cos θ1 sin θ1 , g= − sin θ1 cos θ1 0 1 − sin θ2 cos θ2 0 1 0 r with 0 < y ≤ 1, 0 ≤ θ1 < π, 0 ≤ θ2 < 2π, r > 0. The values of y and r are uniquely determined by g. The values of θ1 , θ2 are also uniquely determined, except in the case when y = 1. Proof We consider the symmetric matrix s := t g · g, where t g denotes the transpose of the matrix g. We require two facts about 2×2 symmetric matrices. First, that a symmetric matrix always has two linearly independent eigenvectors, and second that these two eigenvectors may be taken to be orthogonal (with respect to the standard dot product). The proof of each of these facts is elementary, and we include them for completeness. Let A be a 2 × 2 matrix which does not possess two linearly independent eigenvectors. Then it follows from the Jordan decomposition that there exist two linearly independent vectors v, w and a scalar λ such that A · v = λ · v + w and A · w = λ · w. It follows immediately that t v · A · w =/ t w · A · v, and hence that A is not symmetric. Now suppose B is a symmetric matrix with eigenvalues λ1 , λ2 and corresponding eigenvectors v1 , v2 . We may assume λ1 =/ λ2 , for if λ1 = λ2 , then v1 , v2 may be chosen arbitrarily. Then λ1 · t v1 · v2 = t (B · v1 ) · v2 = t v1 · t B · v2 = t v1 · (B · v2 ) = λ2 · t v1 · v2 , whence t v1 · v2 = 0, so that v1 is orthogonal to v2 . Clearly, the matrix s = t g · g is positive definite. We form an orthogonal t t matrix k1 whose columns are unit eigenvectors of s = g · g. Then k√1 · s · k1=  λ1 0 λ1 0 t t √ , with λi > 0, i = 1, 2. So (g·k1 )·(gk1 ) = d·d, where d = . 0 λ2 λ2 0 It follows that g · k1 = k2 · d, for some k2 ∈ O(2, R). Proposition 4.1.2 now follows easily.  Remark Consider O(2, R) acting on G L(2, R) on both the left and the right. It is easily verified that the functions   a b  a 2 + b2 + c2 + d 2 , → g → | det g|∞ , c d

4.2 Iwasawa and Cartan decompositions for GL(2, Qp )

105

are invariant under this action. It follows immediately that the quantities y and r appearing in Proposition 4.1.2 are related to the entries of the matrix  a b g = c d by r 2 (y 2 + 1) = a 2 + b2 + c2 + d 2 , r 2 y = |ad − bc|∞ .

(4.1.3)

4.2 Iwasawa and Cartan decompositions for G L(2, Q p ) Fix a prime p. We would like to generalize the Iwasawa decomposition given in Proposition 4.1.1 to the p-adic group    5 a b  a, b, c, d ∈ Q , |ad − bc| = 0 . G L(2, Q p ) = / p p c d  It is necessary to introduce the maximal compact subgroup K p = G L(2, Z p )    −1 which consists of all matrices ac db where a, b, c, d ∈ Z p and ac db =   d −b 1 is again a matrix with coefficients in Z p . It follows, in ad−bc −c a

particular, that the determinant D = ad − bc ∈ Z×p . Consequently,    5 a b  a, b, c, d ∈ Z Kp = , |ad − bc| = 1 . p p c d 

The following proposition and its proof were shown to us by Min Lee and Xander Faber. We would like to thank them for allowing us to include it here. Proposition 4.2.1 ( p-adic Iwasawa decomposition) Every g ∈ G L(2, Q p ) has a unique factorization    e 0 1 u p1 ·k g= 0 p e2 0 1 e1 −e 2 −1

where e1 , e2 ∈ Z, u =  Proof Let g =

a b c d



=−N

∈ G L(2, Q p ). Then we can find a matrix k 

such that gk

u  p  , with N ∈ Z, 0 ≤ u  < p, and k ∈ K p .

−1

=

t1 0

−1

 := u t2 t2

α γ



β δ 

=

 ∈ Kp

1 u 0 1



t1 0

0 t2

 ,

Automorphic forms for GL(2, AQ )

106

with u , t1 , t2 ∈ Q p . There are units 1 , 2 ∈ Z×p such that t1 = 1 p e1 and   e1      p 0 t 0 1 0  0 with 01 2 ∈ K p . Let t2 = 2 p e2 . Consequently, 01 t2 = 0 pe2 0 2    0 us define the matrix k

:= 01 2 · k ∈ K p . It follows that  e   p1 0 1 u k

. g= (4.2.2) 0 1 0 p e2 Since u ∈ Q p , we may write u =

∞

u  p ,

(0 ≤ u  < p, N ∈ Z).

=−N ∞

Then u = u +

=e1 −e2



1 u 0 1



p e1 0

u  p  where u = 0 p e2



1 u 0 1



⎛

∞

⎝1

∞

p e2 −e1

⎝1

k

=

u  p  . Consequently

=−N

⎞  u  p  ⎠ p e1 = =e1 −e2 0 0 1   e  1 u p1 0 = · k

0 1 0 p e2 

⎛

where

e1 −e 2 −1

=e1 −e2

0

⎞ u  p ⎠

0 p e2



∈ K p.

1

When the above is combined with (4.2.2) the factorization of g is established. It remains to prove the uniqueness. e1 −e2 −1 u  p  , 0 ≤ u  < p, e1 , e2 ∈ Z and Suppose that there are u = =−N

k ∈ K p such that   e 1 u p1 g= 0 0 1 Then



p e1 −e1 0

0 p e2



 k =

p e2 −e1 (u − u )

p e2 −e2



1 u 0 1



p e1 0

0

p e2



k .

= k k −1 ∈ K p .

It follows that p e1 −e1 , p e2 −e2 ∈ Z×p which implies that e1 = e1 and e2 = e2 . Also e −e −1  e1 −e 1 2 2 −1 e2 −e1

e2 −e1 

 p (u − u ) = p u p − u p ∈ Zp. =−N

=−N

Since  + e2 − e1 ≤ −1 this implies that u = u . Finally, k k −1 = k = k . 



10 01

 so that

4.3 The adele group GL(2, AQ )

107 

Proposition 4.2.3 ( p-adic Cartan decomposition) Take g = G L(2, Q p ). Then there exist k1 , k2 ∈ G L(2, Z p ) such that   m 0 p · k2 , g = k1 · 0 pn

a b c d

 in

where m, n ∈ Z (with m ≤ n) are determined by the conditions:   | det g| p = p −m−n , max |a| p , |b| p , |c| p , |d| p = p −m . Proof It is enough to consider the case when max(|a| p , |b| p , |c| p , |d| p ) = 1, because we can replace g by      e a b a b 0 p = , g := c d 0 pe c d where e ∈ Z is chosen so that max(|a | p , |b | p , |c | p , |d | p ) = 1. Thus, we assume that max(|a| p , |b| p, |c| p , |d| p) = 1, and what we need to show is that 1

0

, when we allow G L(2, Z p ) to act on g is in the same orbit as 0 | det g|−1 p G L(2, Q p ) on both the left and the right. Our assumptions imply  that all of the entries of g are in Z p , and at least one

is in Z×p . Acting by 01 10 on one or both sides if needed, we may assume that     1 0 a ∈ Z×p . Then −c/a 1 and 10 −b/a are both in G L(2, Z p ). Acting on the left 1 by the first and on the right by the second, we find that g is in the same orbit  a 0 bc

as 0 d , where d = d − a . Since a and d · | det g| p are clearly in Z×p , the −1  a 0 is in G L(2, Z p ). Acting by this matrix (on either side), matrix 0 d ·| det g| p   1 0 . This completes the proof.  we obtain 0 | det g|−1 p

4.3 The adele group G L(2, AQ ) Formally, the adele group G L(2, AQ ) is the restricted product (relative to the maximal compact subgroups K p = G L(2, Z p ))  G L(2, Q p ) G L(2, AQ ) = G L(2, R) × 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(2, AQ ) will be denoted in the forms g = {gv }v≤∞ = {g∞ , g2 , g3 , g5 ,

... ,

} = {g∞ , . . . , g p , . . . }

108

Automorphic forms for GL(2, AQ )

where gv ∈ G L(2, Qv ) for all v ≤ ∞ and g p ∈ K p for all but finitely many primes p. Given g, g ∈ G L(2, AQ ) with g = {g∞ , g2 , g3 ,

. . . },

g = {g∞ , g2 , g3 ,

. . . },

we define multiplication of these elements as follows:

gg = {g∞ · g∞ , g2 · g2 ,

...

, g p · g p , . . . },

where gv ·gv simply denotes matrix multiplication in G L(2, Qv ) for all v ≤ ∞.

4.4 The action of G L(2, Q) on G L(2, AQ ) The group G L(2, Q) may be diagonally embedded in G L(2, AQ ) as follows. We shall define the diagonal embedding map i diag : G L(2, Q) → G L(2, AQ ) by i diag (γ ) := {γ , γ , γ , . . . },

(∀γ ∈ G L(2, Q)).

We may define i diag (G L(2, Q)) ⊂ G L(2, AQ ) as the subgroup of all elements of the above form with γ ∈ G L(2, Q). By abuse of notation, we will often write i diag (G L(2, Q)) as G L(2, Q) when the meaning is clear from the context. We also define the embedding at ∞, denoted i ∞ : G L(2, R) → G L(2, AQ ), which is defined by        5 1 0 1 0 1 0 i ∞ (α) = α, , , , . . . , , (∀α ∈ G L(2, R)) . 0 1 0 1 0 1 Then i ∞ (S L(2, R)) denotes the subgroup of all elements of the above form with α ∈ S L(2, R). The group i diag (G L(2, Q)) acts on G L(2, AQ ) by matrix multiplication (on the left). It is then natural to take the quotient G L(2, Q)\G L(2, AQ ) and consider a fundamental domain for this action as in Definition 1.4.1. In order to obtain an explicit fundamental domain, we shall require the following lemma. Lemma 4.4.1 The group i diag (S L(2, Q))·i ∞ (S L(2, R)) is dense in S L(2, AQ ). Proof Let H denote the closure of i diag (S L(2, Q))·i ∞ (S L(2, R)) in S L(2, AQ ). It is a subgroup because multiplication is continuous. It contains all elements of the form



1 0

4.4 The action of GL(2, Q) on GL(2, AQ )    x 1 0 or , (x, y ∈ AQ ) 1 y 1

109

by the weak approximation theorem for AQ (Theorem 1.4.4). In particular, it contains those of this form with xv = 0 for all but a single prime v. Over any field, matrices of these two forms generate S L(2). Indeed,        1 0 1 a−1 1 0 1 a −1 − 1 a 0 = , −a −1 1 0 1 1 1 0 1 0 a −1       0 −1 1 0 1 −1 1 0 = , 1 0 1 1 0 1 1 1       a b 1 0 a 0 1 b/a = , (a =/ 0), c d c/a 1 0 a −1 0 1        a b 1 a/c 0 −1 c 0 1 d/c = , (c =/ 0). c d 0 1 1 0 0 c−1 0 1 It follows that H contains all elements g = {gv }v of S L(2, AQ ) satisfying gv = I2 for all but finitely many v. But such elements are dense in S L(2, AQ ) and H is closed. Thus H = S L(2, AQ ), as desired.  Remark If one replaces S L(2) by G L(2) in the above lemma, the assertion is false. Proposition 4.4.2 (Strong approximation) Let D∞ be a fundamental domain for G L(2, Z)\G L(2, R). A fundamental domain D for G L(2, Q)\G L(2, AQ ) is given by  

 D = {g∞ , g2 , g3 , . . . }  g∞ ∈ D∞ , g p ∈ G L(2, Z p ) for all finite primes p  G L(2, Z p ). = D∞ · p

Proof We must show two things. (1) Every g ∈ G L(2, AQ ) can be expressed in the form g = γ d for some d ∈ D and some γ ∈ i diag (G L(2, Q)). γ d2 for (2) If d1 =   γ ∈ i diag (G L(2, Q)), (d1 , d2 ∈ D), then we have γ = i diag

10 01

.

To show (1), we first apply Proposition 1.4.6 (strong approximation for  ideles) to the determinant of g, obtaining α ∈ Q× and x ∈ (0, ∞) · p Z×p such that det g = α · x. Write x = {x∞ , x2 , x3 , . . . }, and let  −1   −1  α 0 x 0 ·g· . g = i diag 0 1 0 1 Then g ∈ S L(2, AQ ). Let U be any open subset of S L(2, R) containing the identity matrix. The set

110 

g · U·

 p

 = g · 

Automorphic forms for GL(2, AQ )  S L(2, Z p )  5    = {∞ , 2 , . . . , }, ∞ ∈ U,  p ∈ S L(2, Z p ) ∀ p 

is an open neighborhood of g in S L(2, AQ ) and, therefore, contains (by Lemma 4.4.1) an element of the form i diag (γ )i ∞ (g∞ ) where γ ∈ S L(2, Q) and g∞ ∈ S L(2, R). Thus i diag (γ )i ∞ (g∞ ) = g · {∞ , 2 , . . . }, where ∞ ∈ U and  p ∈ S L(2, Z p ) for all p. At this point, we have      α 0 x 0 −1

−1 −1 g = i diag · γ · {g∞ ∞ , 2 , . . . ,  p . . . } · . 0 1 0 1   −1 x∞ 0 = γ

d. Then we Choose γ

∈ G L(2, Z) and d ∈ D∞ such that g∞ ∞ 0 1 obtain      α 0

· γ · γ · d, (γ

)−1 2 , . . . , (γ

)−1  p , . . . , g = i diag 0 1   xp 0 for all primes p. The above expression for g is of the where  p =  −1 p · 0 1 desired form. The proof of (2) is left as an exercise.  The Iwasawa decompositions given in Propositions 4.1.1, 4.2.1 can be combined to give an adelic Iwasawa decomposition. For x = {x∞ , . . . , x p , . . . } ∈ AQ , we shall adopt the convention that       5 1 x 1 x∞ 1 xp = , ... , , ... . 0 1 0 1 0 1 We will also adopt a similar convention for diagonal matrices so that       5 y 0 y∞ 0 yp 0 = , ... , , ... , 0 1 0 1 0 1       5 r 0 r∞ 0 r 0 = , ... , p , ... , 0 r 0 r∞ 0 rp for r, y ∈ A× Q. Proposition 4.4.3 (Adelic Iwasawa decomposition) Every g ∈ G L(2, AQ ) can be uniquely written in the form     1 x y 0 r 0 g= ·k 0 1 0 1 0 r where x = {x∞ , . . . , x p , . . . } ∈ AQ , y = {y∞ , . . . , y p , . . . } ∈ A× Q, r = {r∞ , . . . , r p , . . . } ∈ A× , and k = {k , . . . , k , . . . } ∈ K ∞ p Q

4.4 The action of GL(2, Q) on GL(2, AQ ) with K = O(2, R) ·

 p

111

G L(2, Z p ). Furthermore r∞ , y∞ > 0, r p = p e2 ( p) ,

y p = p e1 ( p)−e2 ( p) , and

e1 ( p)−e2 ( p)−1

xp =

u  ( p) p 

=−N

with e1 ( p), e2 ( p), N ∈ Z, 0 ≤ u  ( p) < p. Strong approximation (Proposition 4.4.2) can be combined with the above adelic Iwasawa decomposition to give unique factorization of adeles. We now state and prove a unique factorization theorem for adeles which will play a major role in the subsequent theory. Theorem 4.4.4 (Unique factorization of adeles) Every g ∈ G L(2, AQ ) can be uniquely written in the form  g=γ ·

y∞ 0

x∞ 1



r∞ 0

0 r∞



5 , I2 ,

...

, I2 ,

...

·k

where γ ∈ G L(2, Q) (diagonally embedded in G L(2, AQ )), − 12 ≤ x∞ ≤ 0,   2 2 y∞ > 0, x∞ + y∞ ≥ 1, r∞ > 0, I2 = 10 01 , and k ∈ K where we have  K = O(2, R) · p G L(2, Z p ). Proof This follows from strong approximation of Proposition 4.4.2 and the Iwasawa decomposition at ∞ in Proposition 4.1.1. Let h = G L(2, R)/O(2, R) · R× . Then G L(2, R), and in particular G L(2, Z), acts naturally on h by left matrix y x  multiplication. The conditions for 0∞ 1∞ to be in a fundamental domain for 2 2 S L(2, Z)\h are − 12 ≤ x∞ < 12 , y∞ > 0, x∞ + y∞ ≥ 1 (see Example 1.1.9 in [Goldfeld, 2006]). In view of the identity       −1 0 y∞ x∞ −1 0 y∞ −x∞ = , 0 1 0 1 0 1 0 1   y x  one sees that the action of −10 01 on 0∞ 1∞ ∈ h is to simply map   x∞ → − x∞ . Since G L(2, Z) is generated by S L(2, Z) and −10 01 , it follows that a fundamental domain for G L(2, Z)\h is one half of the fundamental domain for S L(2, Z)\h.  Proposition 4.4.5 (Adelic Cartan decomposition) Every g ∈ G L(2, AQ ) may be written in the form g = k1 · a · k2 and ki ∈ K where we have  K = O(2, R) · p G L(2, Z p ), i = 1, 2, and

112  a=

y∞r∞ 0

Automorphic forms for GL(2, AQ )   m   m 2 2 0 p p 0 , , . . . , n2 r∞ 0 2 0

0 pn p



5 , ...

with y∞ , r∞ > 0, and for each p, we have m p ≤ n p . Proof This follows immediately from Propositions 4.1.2 and 4.2.3.



4.5 The universal enveloping algebra of gl(2, C) The Lie algebra gl(2, R) of G L(2, R) (see 2.1 in [Goldfeld, 2006]) consists of the additive vector space of all 2 × 2 matrices with coefficients in R with Lie bracket [α, β] = α · β − β · α, for all α, β ∈ gl(2, R). The universal enveloping algebra U (gl(2, R)) of gl(2, R) (see Sections 2.1, 2.2 in [Goldfeld, 2006]) is an associative algebra which contains gl(2, R). The Lie bracket on gl(2, R) and the associative product ◦ on U (gl(2, R)) are compatible, in the sense that [α, β] = α ◦ β − β ◦ α, for all α, β ∈ gl(2, R). Furthermore, U (gl(2, R)) is “universal” with respect to this property. This means the following. Let A be an R-algebra, with associative product ∗, and ϕ : gl(2, R) → A a linear map, such that ϕ([α, β]) = ϕ(α) ∗ ϕ(β) − ϕ(β) ∗ ϕ(α). Then there is a unique extension of ϕ to an R-linear ring homomorphism from U (gl(2, R)) to A. One of the most important applications of this universal property is to give an isomorphism between U (g) and an algebra of differential operators acting on smooth functions F : G L(2, R) → C. We now explain this. Definition 4.5.1 Let α ∈ gl(2, R) and F : G L(2, R) → C, 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 ∞ k Remark Recall that exp(tα) = I + (tα) , where I denotes the identity matrix k! k=1

on gl(2, R). Since we are differentiating with respect to t and then setting t = 0, only the first two terms in the Taylor series for exp(tα) matter. The differential operator Dα satisfies the usual properties of a derivation: Dα (c · F(g) + G(g)) = c · Dα F(g) + Dα G(g), Dα (F(g) · G(g)) = Dα F(g) · G(g) + F(g) · Dα G(g),

(addition rule), (product rule),

for all smooth F, G : G L(2, R) → C, c ∈ C, and g ∈ G L(2, R).

4.5 The universal enveloping algebra of gl(2, C)

113

The differential operators Dα with α ∈ gl(2, R) generate an associative algebra D2R , defined over R. Every element of D2R is a linear combination (with coefficients in R) of differential operators Dα1 ◦ Dα2 ◦ · · · ◦ Dαk with α1 , α2 , . . . , αk ∈ gl(2, R), where ◦ denotes multiplication in D2R , which is explicitly given by composition (successive application) of differential operators. Proposition 4.5.2 Let Dα , Dβ ∈ D2R with α, β ∈ gl(2, R). Then Dr α+β = r · Dα + Dβ , Dα ◦ Dβ − Dβ ◦ Dα = D[α,β] , where r ∈ R, [α, β] = α · β − β · α, denotes the Lie bracket in gl(2, R), i.e., · denotes matrix multiplication. Proof Let

 g=

g1,1 g2,1

g1,2 g2,2



  = gi, j

1≤i≤2,1≤ j≤2

∈ G L(2, R).

A smooth complex-valued function F(g), defined on G L(2, R), can be thought of as a function of 4 real variables gi, j with 1 ≤ i ≤ 2, 1 ≤ j ≤ 2. It immediately follows from the chain rule for functions of several real variables and Definition 4.5.1 that 2 

Dα F(g) =

g·α

 i, j

·

i, j=1

∂ F(g). ∂gi, j

(4.5.3)

  Here, g · α i, j denotes the i, j entry of the matrix g · α. It immediately follows from (4.5.3) that Dr α+β = r · Dα + Dβ . If we now apply Dβ to the above expression (4.5.3), we see that Dβ ◦ Dα F(g) : ;  2  ∂  ∂  (g + t(g · β)) · α = · F(g + t(g · β))   i, j ∂gi, j ∂t i, j=1 =

2 

g·β ·α

 i, j

i, j=1

·

t=0

n 

   ∂ F(g) ∂ 2 F(g) g · β i , j g · α i, j · + . ∂gi, j ∂gi, j ∂gi , j i , j =1

Consequently, 

Dβ ◦ Dα − Dα ◦ Dβ



F(g) =

2 

g·(β ·α−α·β)

i, j=1

which completes the proof of the proposition.



 i, j

·

∂F = D[β,α] F(g), ∂gi, j

114

Automorphic forms for GL(2, AQ )

Proposition 4.5.2 tells us that the map α → Dα is a linear map from   gl(2, R) → End(V ) = set of all linear maps V → V , where V is the vector space C ∞ (G L(2, R)) of all smooth functions G L(2, R) → C. Furthermore, Dα ◦ Dβ − Dβ ◦ Dα = D[α,β] . It follows immediately from the universal property of U (gl(2, R)) that there is an algebra homomorphism δ : U (gl(2, R)) → End(C ∞ (G L(2, R))) such that δ(α) = Dα for each α ∈ gl(2, R). This algebra homomorphism constitutes an action of U (gl(2, R)) on C ∞ (G L(2, R)). Clearly, the algebra D2R is the image of δ. In the next lemma we show that the kernel of δ is trivial. Lemma 4.5.4 (The map δ from the universal enveloping algebra to the algebra of differential operators is an isomorphism) Let U (gl(2, R)) denote the universal enveloping algebra of gl(2, R). Let δ denote the algebra homomorphism from U (gl(2, R)) into the set of all endomorphisms of C ∞ (G L(2, R)) such that δ(α) = Dα for all α ∈ gl(2, R). Then the kernel of δ is trivial. Consequently, δ maps U (gl(2, R)) isomorphically onto its image, the algebra of differential operators D2R . Proof Note that every element of U (gl(2, R)) may be expressed as a finite linear combination of words in the elements of gl(2, R). (This expression is not unique.) Using linearity, it is enough to consider words in the elements of the standard basis {E i, j | 1 ≤ i, j ≤ 2} of gl(2, R). Let U (gl(2, R))d denote the subspace of U (gl(2, R)) consisting of the span of the set of words of length at most d. Write E i,n j for the nth iterate of E i, j . We claim that for each d, the elements 

 d3 d1 d2 d4  E 1,1 ◦ E 1,2 ◦ E 2,1 ◦ E 2,2  d1 + d2 + d3 + d4 ≤ d form a basis for U (gl(2, R))d . In the case d = 1 this is obvious. For general d, it is clear that they are linearly independent. Furthermore, we know that the set of all words in the elements E i, j is a spanning set. Finally, it follows from the relation α ◦ β = β ◦ α + [α, β] that any word in the E i, j is equivalent to a word in the designated order modulo the subspace U (gl(2, R))d−1 ⊂ U (gl(2, R))d . Our claim now follows by induction on d. It follows that every element of U (gl(2, R)) may be expressed uniquely as d3 d1 d2 d4 ◦ E 1,2 ◦ E 2,1 ◦ E 2,2 . We need to linear combination of words of the form E 1,1

4.5 The universal enveloping algebra of gl(2, C)

115

prove that, for an arbitrary linear combination of elements of this form, there is a smooth function G L(2, R) → C which is not killed by the corresponding differential operator. Now, the differential operator δ(E i, j ) is simply ∂g∂i, j where gi, j denotes the (i, j) entry of g ∈ G L(2, R). It is easily seen that the polynomial function b3 b1 b2 b4 P(g) = g1,1 · g1,2 · g2,1 · g2,2 d3 d1 d2 d4 is killed by δ(E 1,1 ◦ E 1,2 ◦ E 2,1 ◦ E 2,2 ) for all d1 , d2 , d3 , d4 with d1 + d2 + d3 + d4 ≥ b1 + b2 + b3 + b4 except for d1 = b2 , d2 = b2 , d3 = b3 , d4 = b4 . Using this observation, we easily construct, for any linear combination of the d3 d1 d2 d4 ◦ E 1,2 ◦ E 2,1 ◦ E 2,2 , a monomial in the entries gi, j which is not words E 1,1 killed. 

Because the map δ : U (gl(2, R)) → D2R is an isomorphism, we say that the algebra of differential operators is a realization of the algebra U (gl(2, R)). From now on, we shall identify the (abstract) universal enveloping algebra U (gl(2, R)) with its (concrete) realization as differential operators. We may extend the action of gl(2, R) on smooth functions F : G L(2, R) → C as explained above to an action of the complexification gl(2, R) ⊗R C = gl(2, C) with the following definition √ Definition 4.5.5 (Complexified universal enveloping algebra) Set i = −1. Let β ∈ gl(2, R) and F : G L(2, 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(2, C) with α, β ∈ gl(2, R), then Dα+iβ = Dα + i Dβ . The differential operators Dα+iβ (with α, β ∈ gl(2, R)) generate an algebra of differential operators which is isomorphic to the universal enveloping algebra U (g). As with U (gl(2, R)), we shall henceforth identify each element of U (g) with the corresponding differential operator. An important reason for introducing the algebra U (g) is that the differential operators in U (g) are well defined on smooth functions f : G L(2, R) → C which satisfy the automorphic relations f (γ gz) = f (g),

(4.5.6)

Automorphic forms for GL(2, AQ )

     r 0 × r ∈ R for all γ ∈ G L(2, Z), z ∈ Z (G L(2, R)) := , and g ∈  0r G L(2, R).

116

Proposition 4.5.7 Let f : G L(2, R) → C be a smooth function which is invariant on the left by G L(2, Z), and invariant by the center Z (G L(2, R)). Then for all D ∈ U (g), the function D f is also invariant on the left by G L(2, Z) and invariant by Z (G L(2, R)). Proof It is enough to consider the case when D = Dα1 ◦ Dα2 ◦ · · · ◦ Dαm with m ≥ 1, and αi ∈ gl(2, R) for i = 1, 2, . . . , m. Since f is left invariant by G L(2, Z), i.e., f (γ · g) = f (g) for all g ∈ G L(2, R), γ ∈ G L(2, Z), it immediately follows from Definition 4.5.1 that   ∂  ∂ ∂ ··· f γ get1 α1 · · · etm αm  D f (γ g) = t1 =0,... ,tm =0 ∂t1 ∂t2 ∂tm =

  ∂ ∂ ∂  t1 α1 ··· f ge · · · etm αm  t1 =0,... ,tm =0 ∂t1 ∂t2 ∂tm

= D f (g). Similarly, let z ∈ Z (G L(2, R)). Then zg = gz for all g ∈ G L(2, R). It follows, as above, that   ∂  t1 α1 ∂ ∂ ··· f gze · · · etm αm  D f (gz) = t1 =0,... ,tm =0 ∂t1 ∂t2 ∂tm =

  ∂ ∂ ∂  t1 α1 ··· f ge · · · etm αm z  t1 =0,... ,tm =0 ∂t1 ∂t2 ∂tm

=

  ∂ ∂ ∂  t1 α1 ··· f ge · · · etm αm  t1 =0,... ,tm =0 ∂t1 ∂t2 ∂tm

= D f (g).  We conclude this section with one final theorem. Theorem 4.5.8 (Lie algebra of a subgroup of G L(2, R)) Let G be a closed subgroup of G L(2, R). The set g of all matrices α ∈ gl(2, R) such that exp(α) ∈ G is a subalgebra, i.e., it is a linear subspace and is closed under the Lie bracket. This subalgebra is called the Lie algebra of G . Proof See [Fulton-Harris, 1991, 8.1]

4.7 Automorphic forms for GL(2, AQ )

117

Examples: If G is the center Z (G L(2, R)), then g is the span of theidentity  0 1 . matrix. If G is the group S O(2, R), then g is the span of the matrix −1 0

4.6 The center of the universal enveloping algebra of gl(2, C) Let g = gl(2, C). The center of the universal enveloping algebra of g is denoted Z (U (g)). Then D ∈ Z (U (g)) is characterized by the property that D ◦ D = D ◦ D for all D ∈ U (g), and where ◦ denotes composition of differential operators. A key feature of Z (U (g)) is that it is well defined on smooth functions f : G L(2, R) → C which satisy the automorphic relations f (γ gzk) = f (g),



 g ∈ G L(2, R) ,

(4.6.1)

  for all γ ∈ G L(2, Z), all z ∈ Z G L(2, R) , and all k ∈ O(2, R). The functions satisfying (4.6.1) are invariant on the right by the maximal compact subgroup O(2, R). These are the functions that are well defined on the classical upper half plane h  G L(2, R)/O(2, R) · R× as explained in Section 4.1. The following proposition is proved in Section 2.3 in [Goldfeld, 2006]. Proposition 4.6.2 Let D ∈ Z (U (g)). ThenD is well defined on the space of  smooth functions f : G L(2, Z)\G L(2, R)/ O(2, R) · R× −→ C, i.e., (D f )(γ · g · z · k) = D f (g),   for all g ∈ G L(2, R), γ ∈ G L(2, Z), z ∈ Z G L(2, R) , and k ∈ O(2, R). Remarks It may be shown (see Chapter 2, [Goldfeld, 2006]) that Z (U (g)) is a polynomial algebra in two generators over C. One of the generators is a differential operator of degree one. This generatoris uniquely determined up  10 to scalar, and may be chosen to be D Z where Z = 0 1 . The other generator must be of degree two, and may be chosen so that it acts on smooth  2 functions    2 G L(2, R)/ O(2, R) · R× → C by the Laplacian  := −y 2 ∂∂x 2 + ∂∂y 2 . It should be kept in mind, however, that the degree two generator is not uniquely determined, even up to scalar. The space of degree two elements of Z (U (g)) has dimension two. Any element which is not in the span of D Z ◦ D Z may be used.

4.7 Automorphic forms for G L(2, AQ ) We want to give a definition of automorphic forms for G L(2, AQ ) which generalizes the definition for G L(1, AQ ) given in Definition 2.1.4. The most useful definition will allow us to extend the methods introduced by [Tate, 1950] to the

Automorphic forms for GL(2, AQ )

118

more complicated group G L(2, AQ ). Very roughly, automorphic forms are functions (4.7.1) φ : G L(2, Q)\G L(2, AQ ) → C which satisfy certain additional conditions. The additional conditions are: “smoothness,”

“moderate growth,” “right K -finite,”

“Z (U (g))-finite.”

We shall now define these terms one by one. Definition 4.7.2 (Smoothness) A function φ : G L(2, AQ ) → C is said to be smooth if for every fixed g0 ∈ G L(2, AQ ), there exists an open set U of U : G L(2, R) → C such G L(2, AQ ), containing g0 , and a smooth function φ∞ U that φ(g) = φ∞ (g∞ ) for all g ∈ U.   Definition 4.7.3 (Moderate growth) Let g = ac db ∈ G L(2, AQ ) where we have a = {a∞ , a2 , . . . }, b = {b∞ , b2 , . . . }, c = {c∞ , c2 , . . . }, and d = {d∞ , d2 , . . . } ∈ AQ . Define a norm function by 

 . max |av |v , |bv |v , |cv |v , |dv |v , |av dv − bv cv |−1 ||g|| := v v≤∞

We say a function φ : G L(2, Q)\G L(2, AQ ) → C is of moderate growth if there exist constants C, B > 0 such that |φ(g)|C < C||g|| B for all g ∈ G L(2, AQ ).  Definition 4.7.4 (K -finiteness) Let K = O(2, R) p G L(2, Z p ) be the maximal compact subgroup of G L(2, AQ ). A function φ : G L(2, Q)\G L(2, 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 4.7.5 (Z (U (g))-finiteness) Let Z (U (g)) denote the center of the universal enveloping algebra of g = gl(2, C) as in Section 4.6. A smooth function φ : G L(2, Q)\G L(2, AQ ) → C    is said to be Z (U (g))-finite if the set Dφ(g)  D ∈ Z (U (g)) generates a finite dimensional vector space. Without further ado, we will now give the precise definition of an automorphic form for the adele group. Definition 4.7.6 (Adelic automorphic form for G L(2, 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(2, AQ ) with central character ω

4.8 Adelic lifts of weight zero, level one, Maass forms

119

is a smooth function φ : G L(2, AQ ) → C which satisfies the following 5 properties:   (1) φ(γ g) = φ(g), ∀g ∈ G L(2, AQ ), γ ∈ G L(2, Q) . (2) φ(zg) = ω(z)φ(g), ∀g ∈ G L(2, AQ ), z ∈ A× Q . (3) φ is right K -finite. (4) φ is Z (U (g))-finite. (5) φ is of moderate growth. Definition 4.7.7 (Adelic cusp form for G L(2, AQ ) with central character) An adelic automorphic form φ, with central character, as in Definition 4.7.6 is called a cusp form if     1 u φ g du = 0 0 1 Q\AQ        1 u∞ 1 up for all g ∈ G L(2, AQ ). Here 10 u1 = , . . . , , . . . , 0 1 0 1 for u = {u ∞ , . . . , u p , . . . } ∈ AQ , and du is the Haar measure on AQ as defined in Definition 1.7.5, and, as described just before Lemma 1.8.9, we define an integral over Q\AQ as an integral over the fundamental domain given in Proposition 1.4.5. Remark In this book, we shall only consider automorphic forms with central character. Note, however, that it is possible to consider automorphic forms which do not have central characters. These are defined exactly as above, but with condition (2) of Definition 4.7.6 omitted. A slight weakening of (2) follows from (3) and (4). See the exercises.

4.8 Adelic lifts of weight zero, level one, Maass forms Let h  G L(2, R)/O(2, R) · R× denote the upper half-plane. By the Iwasawa decomposition of Proposition   4.1.1, every g ∈ h has a unique representay 0 with x ∈ R and y > 0. We recall tion in the form g = 10 x1 0 1 Definition 3.3.1 (see [Goldfeld, 2006]) of an even weight zero Maass form for S L(2, Z). Definition 4.8.1 (Even weight zero Maass form for S L(2, Z)) Let ν ∈ C. An even weight zero Maass form of type ν for S L(2, Z) is a non-zero smooth function f ∈ L2 (S L(2, Z)\h) which satisfies:   y x • f (γ g) = f (g), for all γ ∈ S L(2, Z), g = ∈ h; 0 1  2  2 •  f = ν(1 − ν) f,  = −y 2 ∂∂x 2 + ∂∂y 2 ;

Automorphic forms for GL(2, AQ )

120 •

1 0

• f

 f 

y 0

  x g d x = 0, for all g ∈ G L(2, R); 1      −x y x y x = f , for all ∈ h. 1 0 1 0 1

1 0

Remarks (1) Identifying h with G L(2, R)/O(2, R) · R× , we may regard a Maass form f (as in Definition 4.8.1) as a function from G L(2, R) to C which is left invariant by S L(2, Z), right invariant by K ∞ = O(2, R), and invariant by the center of G L(2, R) which is isomorphic to R× . We express these conditions as: f (γ gkz) = f (g)   for all γ ∈ S L(2, Z), g ∈ G L(2, R), k ∈ K ∞ , and z = r0 r0 with r ∈ R× . (2) Note that this is a slightly different usage from Definition 3.5.7. In fact, it is a special case of Definition 3.7.7. From now on all Maass forms are explicitly assumed to be cuspidal. We will now show how starting from such a Maass form f we may construct an adelic cusp form f adelic . The construction uses unique factorization of adeles coming from strong approximation as in Theorem 4.4.4. We now define f adelic . Definition 4.8.2 (Adelic lift of an even weight zero Maass form) Let f be an even weight zero Maass form of type ν for S L(2, Z) as in Definition 4.8.1. For g ∈ G L(2, AQ ), let    5 r∞ 0 y∞ x∞ , I2 , . . . , I2 , . . . g=γ · · k, 0 1 0 r∞ as in Theorem 4.4.4, where γ ∈ G L(2, Q) (diagonally embedded  in 2 2 G L(2, AQ )), − 12 ≤ x∞ < 0, y∞ > 0, x∞ + y∞ ≥ 1, r∞ > 0, I2 = 10 01 , and k ∈ K . Define the lifted function f adelic : G L(2, AQ ) → C by setting   y∞ x∞ . (4.8.3) f adelic (g) := f 0 1 Proposition 4.8.4 Let f be an even weight zero Maass form of type ν for S L(2, Z) as in Definition 4.8.1. Then the adelic lift f adelic as defined in Definition 4.8.2 is an adelic cusp form with trivial central character as in Definition 4.7.7. Remarks A crucial point in the definition of f adelic is that in order to evaluate f adelic (g) for g ∈ G L(2, AQ ) it is necessary to first put g in the standard form of Theorem 4.4.4. It should be remarked, however, that the function f adelic obtained does not depend on a particular choice of fundamental domain.

4.8 Adelic lifts of weight zero, level one, Maass forms

121

The adelic function f adelic will not be factorizable so the methods initiated by [Tate, 1950] in his thesis for G L(1) will not directly generalize to adelic automorphic forms of the type f adelic for G L(2). This is one of the reasons why automorphic representations (which will turn out to be pure tensors) were introduced into the theory. We shall soon define the notion of automorphic representation and show that it is a much more natural construction than automorphic form from the point of view of Euler products. Proof of Proposition 4.8.4 We must check that f adelic is smooth, that the 5 conditions in Definition 4.7.6 are satisfied and that, in addition, the cuspidality condition of Definition 4.7.7 is also satisfied. We check these conditions one by one. (1) We must show that f adelic is smooth. For each fixed γ ∈ G L(2, Q), f adelic is given by (4.8.3) uniformly on  the whole open set i diag (γ ) · G L(2, R) · p G L(2, Z p ). Since f is smooth in the ordinary sense, it follows that f adelic is smooth as defined in Definition 4.7.2. (2) We must show that f adelic (γ g) = f adelic (g), ∀ γ ∈ G L(2, Q), g ∈ GL(2, AQ ). This follows immediately from Definition 4.8.2.   (3) We must show f adelic (zg) = f adelic (gz) = f adelic (g), ∀ z ∈ Z G L(2, AQ ) . This doesn’t immediately follow from Definition 4.8.2 because if you multiply by an element of the center, you don’t end up in the standard to define the lift. Note, however, that if z ∈   form necessary Z G L(2, AQ ) then there exists k ∈ K , α ∈ Q× , and r∞ ∈ R× , such that     r∞ 0 · k, z = i diag α0 α0 i ∞ 0 r∞  where i ∞

r∞ 0 0 r∞



 :=

we have f adelic (gz) = f adelic (g).



 , I2 , I2 , . . . . By Definition 4.8.2       r∞ 0 k = f adelic g · i diag α0 α0 i ∞ 0 r∞ r∞ 0 0 r∞

(4) We must show that f adelic is K -finite. Actually, Definition 4.8.2 tells us that f adelic (gk) = f adelic (g) for all k ∈ K , i.e., that f adelic is actually K -invariant on the right. (5) We must show that f adelic is Z (U (g))-finite. Since,  f = ν(1 − ν) f by Definition 4.8.1, we see that f adelic is also an eigenfunction of . It follows   easily from the definitions that D Z f adelic = 0, where Z = 10 01 . Thus, the translates of f adelic by

Automorphic forms for GL(2, AQ )

122

elements of Z (U (g)) generate a one-dimensional space. (See the description of the structure of Z (U (g)) at the end of Section 4.6.) (6) We must show that f adelic has moderate growth. The crux of the matter is to relate the classical notion of moderate growth given in Definition 3.3.3 to the adelic one given in Definition 4.7.3. Take g as in Theorem 4.4.4. That is,   5  r∞ 0 y∞ x∞ , I2 , . . . , I2 , . . . ·k g=γ · 0 1 0 r∞ where γ ∈ G L(2, Q) (diagonally embedded in G L(2, AQ )), and  1 2 2 − 2 ≤ x∞ ≤ 0, y∞ > 0, x∞ + y∞ ≥ 1, r∞ > 0, I2 = 10 01 ,  and k ∈ K = O(2, R) · p G L(2, Z p ). Then   y∞ x∞ , f adelic (g) := f 0 1 and this is bounded by a power of y∞ for all y∞ sufficiently large. It B for suitable constants C and B for follows that it is bounded by C · y∞ all g, because f is smooth and the fundamental domain for G L(2, Z) on G L(2, R) bounds y∞ away from zero. Thus we need to relate y∞ and ||g||. We first concentrate on ∞. Write   ||h ∞ ||∞ := max |a∞ |∞ , |b∞ |∞ , |c∞ |∞ , |d∞ |∞ , |a∞ d∞ − b∞ c∞ |−1 ∞  for any h ∞ =

a∞ b∞ c∞ d∞



∈ G L(2, R). Then

     r∞ 0  γ · y∞ x∞  0 1 0 r ∞  ∞   −1 −2 ≥ max |γ1,1 |∞ y∞r∞ , | det(γ )|−1 ∞ y∞ r ∞ . Here γ1,1 denotes the 1, 1 entry of the matrix γ . Right multiplying an element of G L(2, R) by an element of O(2, R) preserves the L 2 norm of each of√the rows, as well as the determinant. It follows that ||h ∞ · k∞ ||∞ ≥ 22 · ||h ∞ ||∞ for all h ∞ ∈ G L(2, R) and k∞ ∈ O(2, R). Similarly, write 

||h p || p := max |a p | p , |b p | p , |c p | p , |d p | p , |a p d p − b p c p |−1 p   a b for any h p = c pp d pp ∈ G L(2, Q p ). Right multiplying an element of G L(2, Q p ) by an element of G L(2, Z p ) preserves the L ∞ norm (maximum of the absolute values of the entries) of each of its rows,

4.8 Adelic lifts of weight zero, level one, Maass forms

123

and preserves the determinant. It follows that ||γ · k p || p = ||γ || p ≥ max(|γ1,1 | p , | det(γ )|−1 p ), for all γ ∈ G L(2, Q) and k p ∈ G L(2, Z p ) Multiplying, we have  ||gv ||v ||g|| = v

√

≥

  2 −1 −2 max |γ1,1 |∞ y∞r∞ , | det(γ )|−1 y r ∞ ∞ ∞ 2    max |γ1,1 | p , | det(γ )|−1 · p p

√   2 2 13 −1 −2 ≥ max y∞r∞ , y∞ r∞ ≥ y∞ . 2 2 Hence, the bound in terms of y given by Definition 3.3.3 implies the bound in terms of ||g|| required by Definition 4.7.3. √

(7) The final point that needs to be checked is the cuspidality condition given in Definition 4.7.7. Let g ∈ G L(2, AQ ). We must show that     1 u g du = 0. f adelic 0 1 





Q\AQ

   1 u∞ 1 up Here 10 u1 = , . . . , , 0 1 0 1 and u p ∈ Z p for all finite primes p.

...



(4.8.5)

with 0 ≤ u ∞ < 1,

Unfortunately, it is not possible to directly use the definition of the lift f adelic given in Definition 4.8.2 to instantly evaluate the integral on the left side of (4.8.5). The problem is that even if we write g in canonical  as in Theorem  form 1u 4.4.4, we lose the canonical form when we multiply g by 0 1 on the left. This is not a really serious problem, however, and we get around it by a simple trick involving matrix multiplication. Let g = {g∞ , g2 , g3 , . . . } ∈ G L(2, AQ ). By the adelic Iwasawa decomposition of Proposition 4.4.3, we may uniquely express g in the form     1 x y 0 r 0 g= ·k (4.8.6) 0 1 0 1 0 r with k ∈ K and x ∈ AQ , y, r ∈ A× Q satisfying certain conditions. We may assume that y p = p e p with e p ∈ Z and e p = 0 for all but finitely many primes p. It now follows from (4.8.6) after making the change of variables u → u −x, and using the fact that f adelic is left invariant by G L(2, Q) and right invariant by K and Z (G L(2, AQ )), that

124 



1 0

f adelic Q\AQ

Automorphic forms for GL(2, AQ )   u g du 1  

 =

f adelic Q\AQ

 ...

,

1 up 0 1

 

 f adelic

=

1 u∞ 0 1





1 u∞ 0 1

pe p 0



Q\AQ

 ...

y∞ 0

,

0 1 0 1





y∞ 0

0 1

rp 0

0 r∞ 0 rp



 ,



1 up 0 1

r∞ 0

pe p 0

 k∞ , 5 . . . du

 k p,

1 u2 0 1 0 1



2e2 0

0 1

 ,

5 . . . du.

 ,

It follows that 

 f adelic Q\AQ

1 0

  g du

u 1

 

 =

f adelic Q\AQ

 ...  

 = 

f adelic

p −e p 0





Q\AQ

p −e p 0

0 1

1 u2 0 1  

 f adelic

= 



1 u∞ 0 1

Q\AQ

1 u 2 p −e p 0 1



2e2 0

,

0 1

pe p 0



0 1



2e2 0

0 1

1 u ∞ p −e p 0 1

0 1

y∞ 0

p −e p 0

0 1





1 u∞ 0 1

 ,

1 0

u p p −e p 1





y∞ 0

 , ... ,



0 1

p −e p 0



0 1 

, ... ,

1 u2 0 1

0 1



0 1

 ,

5

 ,

...

du

 ,

1 u p p −e p 0 1 

2e2 0

y∞ 0

0 1

1 u p p −e p 0 1



5 , . . . du

 , 5

 , ...

du.

4.8 Adelic lifts of weight zero, level one, Maass forms

125

In the above integral, we make the transformation u → pe p u

du →

=⇒

du = du, | p e p |A

because | p e p |A = 1 since p e p ∈ Q× . This change of variable also replaces the standard fundamental domain for Q\AQ which was fixed in Definition 1.4.6  with a different fundamental domain: [0, p e p ) · p e p Z p · v =/ ∞, p Zv . However the value of the integral of a periodic function over Q\AQ this new fundamental domain is the same as the integral over the standard fundamental domain. It follows that     1 u g du f adelic 0 1 Q\AQ

 



=

f adelic Q\AQ



1 u2 0 1

1 0



u∞ 1

2e2 0



0 1





p −e p 0

0 1

p −e p 0

0 1

y∞ 0

0 1





 , ... ,

, 1 up 0 1

 ,

5 . . . du.

 −e p  Note that p 0 10 ∈ K v for all v =/ p, ∞. The above procedure can be repeated at each prime p where e p =/ 0. If we define  t= pe p , p e p =/ 0

  it then follows from Definition 4.8.2 and the fact that 10 u1p ∈ K p for u p ∈ Z p that          1 u∞ 1 u t 0 y∞ 0 g du = , f adelic f adelic 0 1 0 1 0 1 0 1 Q\AQ

 



1

f

= 0

= 0.

1 0

Q\AQ

1 0

 u2 , 1 

u∞ 1

 ... t y∞ 0

, 0 1

1 0

up 1

 du ∞ ·

5

 ,

 p

...

du

du p

Zp



Next, we discuss the adelic lift of an odd weight zero Maass form for S L(2, Z). We recall the definition of an odd weight zero Maass form.

126

Automorphic forms for GL(2, AQ )

Definition 4.8.7 (Odd weight zero Maass form for S L(2, Z)) Let ν ∈ C. An odd weight zero Maass form of type ν for S L(2, Z) is a non-zero smooth function f ∈ L2 (S L(2, Z)\h) which satisfies:   y x • f (γ g) = f (g), for all γ ∈ S L(2, Z), g = ∈ h, 0 1  2  2 •  f = ν(1 − ν) f,  = −y 2 ∂∂x 2 + ∂∂y 2 ,    1 1 x • f g d x = 0, for all g ∈ G L(2, R), 0 1 0       y −x y x y x • f =−f , for all ∈ h. 0 1 0 1 0 1 Definition 4.8.8 (Adelic lift of an odd weight zero Maass form) Let f be an odd weight zero Maass form of type ν for S L(2, Z) as in Definition 4.8.7. For g ∈ G L(2, AQ ), let    5 r∞ 0 y∞ x∞ , I2 , . . . , I2 , . . . g=γ · · k, 0 1 0 r∞ as in Theorem 4.4.4, where γ ∈ G L(2, Q) (diagonally embedded  in 1 2 2 G L(2, AQ )), − 2 ≤ x∞ < 0, y∞ > 0, x∞ + y∞ ≥ 1, r∞ > 0, I2 = 10 01 , and k ∈ K . Define the lifted function f adelic : G L(2, AQ ) → C by setting   y∞ x∞ · det(k∞ ). f adelic (g) := f (4.8.9) 0 1 Proposition 4.8.10 Let f be an odd weight zero Maass form of type ν for S L(2, Z) as in Definition 4.8.7. Then the adelic lift f adelic as defined in Definition 4.8.8 is an adelic cusp form with trivial central character as in Definition 4.7.7. Proof The proof is almost the same as the proof of Proposition 4.8.4. All one uses is the fact that       y −x −1 0 y x −1 0 = 0 1 0 1 0 1 0 1   = −1.  and det −10 10

4.9 The Fourier expansion of adelic automorphic forms Recall that we have defined additive characters ev : Qv → C for v ≤ ∞ as: e∞ (x∞ ) = e2πi x∞ ,   e p x p = e−2πi{x p } ,

(for x∞ ∈ R), (for x p ∈ Q p ),

4.9 The Fourier expansion of adelic automorphic forms

127

where {x p } denotes the fractional part of x p as in Definition 1.6.3. These additive characters can be multiplied together to give a global additive character e : AQ → C where  e(x) = ev (xv ), (for all x = {x∞ , x2 , x3 , . . . } ∈ AQ ). v≤∞

It was shown in Proposition 1.7.4 that the additive character e(x) is trivial on Q, i.e., it satisfies   e(α) := e {α, α, α, . . . } = 1 for all α ∈ Q. The functions of the form e(αx) with α ∈ Q fixed are periodic in x ∈ AQ , i.e., they satisfy   e α(x + β) = e(αx) for all β ∈ Q. Theorem 1.8.10 tells us that the above functions form a basis for the space of all nice adelic functions which are periodic. Here “nice” means smooth, for example. If f : AQ → C is a periodic adelic function satisfying f (x + α) = f (x), for all α ∈ Q, and f satisfies suitable smoothness properties then by Theorem 1.8.10, we have the following Fourier expansion:  f (x) = (4.9.1) f α · e(αx) α∈Q

where  fα =

 f (x)e(−αx)d x. Q\AQ

Now, let φ : G L(2, AQ ) → C be an adelic automorphic form with central character ω : Q× \AQ → C as in Definition 4.7.6. Then φ satisfies the relation    1 α φ g = φ(g) (4.9.2) 0 1 for all α ∈ Q and all g ∈ G L(2, AQ ). By the adelic Iwasawa decomposition of Proposition 4.4.3 we have the decomposition     1 x y 0 r 0 g= ·k (4.9.3) 0 1 0 1 0 r where x ∈ AQ , r, y ∈ A× Q , and k ∈ K . If we combine (4.9.2) and (4.9.3), it immediately follows that      1 x +α y 0 r 0 φ ·k 0 1 0 1 0 r      1 x y 0 r 0 =φ ·k . 0 1 0 1 0 r

(4.9.4)

Automorphic forms for GL(2, AQ )

128

If we fix y, r, k and consider φ(g) as a function of x ∈ AQ , we see from (4.9.4) that φ is a periodic function of x and has a Fourier expansion of the type (4.9.1). We, therefore, obtain the following proposition. Proposition 4.9.5 (Fourier expansion of adelic automorphic forms) Let φ : G L(2, AQ ) → C be an adelic automorphic form with central character ω as in Definition 4.7.6. Then      1 u α (g) · e(αu), φ ∀g ∈ G L(2, AQ ), ∀u ∈ AQ , φ g = 0 1 α∈Q

and where α (g) = φ



 φ

1 u 0 1

  g e(−αu) du.

Q\AQ

In particular, φ(g) =

α (g) φ α∈Q

for all g ∈ G L(2, AQ ).

4.10 Global Whittaker functions for G L(2, AQ ) Let φ : G L(2, AQ ) → C be an adelic automorphic form with central character ω as in Definition 4.7.6. By Proposition 4.9.5 the function φ has a Fourier expansion α (g), φ (∀g ∈ G L(2, AQ )), (4.10.1) φ(g) = α∈Q

where α (g) = φ



 φ

1 u 0 1

  g e(−αu) du.

(4.10.2)

Q\AQ

α (g) inherits certain properties which come from The Fourier coefficient φ the definition of an adelic automorphic form given in Definition 4.7.6. We list these properties one by one and explain why they must hold. α (zg) = ω(z) φ α (g), φ

   ∀g ∈ G L(2, AQ ), z ∈ Z G L(2, AQ ) .

(4.10.3)

Note that (4.10.3) follows immediately from Definition 4.7.6 (2). α (g) is right K -finite. φ

(4.10.4)

This is a consequence of Definition 4.7.6 (3). Here K -finiteness means that the α (gk) with k ∈ K is finite dimensional. C-vector space generated by the φ

4.10 Global Whittaker functions for GL(2, AQ ) α (g) is right Z (U (g))-finite. φ

129 (4.10.5)

This follows immediately from Definition 4.7.6 (4). Here, the notion of Z (U (g))-finiteness means that the C-vector space generated by the derivaα (g) with D ∈ Z (U (g)) is finite tives of the Fourier coefficients D φ dimensional. α (g) is of moderate growth. φ (4.10.6) This is a consequence of Definition 4.7.6 (5) and Definition 4.7.3. α φ



1 v 0 1

  α (g), g = e(αv) · φ

  ∀v ∈ AQ , ∀g ∈ G L(2, AQ ) . (4.10.7)

This last condition (4.10.7) is a defining characteristic which is easily obtained from the integral (4.10.2). In fact,        1 v 1 v+u α g = φ φ g e(−αu) du. 0 1 0 1 Q\AQ

Then (4.10.7) follows on making the transformation u → u − v in the above integral. α (g) = φ 1 φ



α 0

0 1

  g ,

(for α = / 0 and ∀g ∈ G L(2, AQ )). (4.10.8)

To prove (4.10.8) make the change of variables u → α −1 u in (4.10.2) to obtain     1 α −1 · u  g e(−u) du. φ α (g) = φ 0 1 Q\AQ

Since φ(γ g) = φ(g) for any γ ∈ G L(2, Q), it immediately follows that  −1   α 0  φα g 0 1   −1      α 0 α 0 1 α −1 · u g e(−u) du = φ 0 1 0 1 0 1 Q\AQ





φ

= Q\AQ

1 0

u 1

  1 (g). g e(−u) du = φ

Automorphic forms for GL(2, AQ )

130

Definition 4.10.9 (Global Whittaker function for G L(2, AQ )) A function on G L(2, AQ ) with values in C which satisfies the same properties 1 is termed a global Whittaker function for (4.10.3)–(4.10.8) as φ G L(2, AQ ). α is determined for all α ∈ Q× Note that the identity (4.10.8) tells us that φ 1 . The main reason for the terminology: “Whittaker function,” is that by φ the differential operators which occur in (4.10.5) were originally studied by Whittaker who explicitly solved certain second order differential equations coming from these operators. The solutions of these second order differential equations are also called Whittaker functions (by classical analysts) and are very similar to Bessel functions. To get a feel for what a global Whittaker function is, we shall now explicitly compute the global Whittaker function for an adelic lift f adelic of a Maass form f of type ν as in Definition 4.8.2. We let W (∗; f ) : G L(2, AQ ) → C denote this Whittaker function. Theorem 4.10.10 (Global Whittaker function for Maass forms of S L(2, Z)) Let f be an even Maass form of weight 0 for S L(2, Z) of type ν as in Definition 4.8.1 with classical Whittaker expansion    √ y∞ 0 1 x∞ = A(n) y∞ · K ν− 1 (2π |n|y∞ )e2πinx∞ f 2 0 1 0 1 n =/ 0



   1 x y 0 r 0 · k be the adelic 0 1 0 1 0 r Iwasawa decomposition of g ∈ G L(2, AQ ) as in Proposition 4.4.3. Define  |y p |−1 t= p ,

for all x∞ ∈ R, y∞ > 0. Let g =

p

with the product of absolute values ranging over the finite primes p. Then the lifted adelic Maass form f adelic as in Definition 4.8.2 has a global Whittaker function     1 u g e(−u) du W (g; f ) = f adelic 0 1 Q\AQ

 =

4 A(t) t −1 y∞ K ν− 1 (2π y∞ ) · e(x),

if t ∈ Z,

0,

otherwise,

2

where e(x) is the global additive character defined in Definition 1.7.3. Proof The computation we need to do is very similar to the computation we already did of the constant term in the Fourier expansion given in (4.8.5). By definition:

4.10 Global Whittaker functions for GL(2, AQ )     1 u g e(−u) du. f adelic W (g; f ) = 0 1

131 (4.10.11)

Q\A

By the adelic Iwasawa decomposition of Proposition 4.4.3, every g ∈ G L(2, AQ ) may be uniquely expressed in the form     1 x y 0 r 0 g= ·k 0 1 0 1 0 r where we have x = {x∞ , . . . , x p , . . . } ∈ AQ , y = {y∞ , . . . , y p , . . . } ∈ × A× Q , r = {r ∞ , . . . , r p , . . . } ∈ AQ , and k = {k∞ , . . . , k p , . . . } ∈ K , and where xv , yv , rv , kv satisfy the conditions of Proposition 4.4.3 for every v ≤ ∞. After making the change of variables u → u − x in the integral (4.10.11), and noting that f adelic has trivial central character and is right invariant by K , it immediately follows that       1 u y 0 r 0 W (g; f ) = e(x) · f adelic · k e(−u) du 0 1 0 1 0 r Q\A





= e(x) ·

f adelic

1 0

u 1



y 0

0 1

 e(−u) du. (4.10.12)

Q\A

In (4.10.12), we may assume that y = {y∞ , y2 , . . . } with yq = q eq with eq ∈ Z and eq = 0 for all but finitely many primes q. Now fix a prime p where e p = 0. Since   e    e  1 up pp 0 pp 0 1 u p · p −e p , = 0 1 0 1 0 1 0 1 it follows from (4.10.12) that     y∞ 1 u∞ W (g; f ) = e(x) f adelic 0 1 0 Q\AQ

 ... ,

pe p 0

0 1

0 1



 ,

1 u2 0 1



y2 0

0 1

 ,

  1 u p · p −e p , . . . , e(−u) du. 0 1 !" #



pth place

In the above integral make the change of variables u → p e p u which implies = du, where | p e p |A = 1, since p e p ∈ Q× . Clearly, the intethat du → | pdu ep |A gral is also independent of the fundamental domain Q\AQ so that the change of variables u → p e p · u does not change the domain of integration. Since f adelic is left invariant by G L(2, Q), we see, after making the above change of variables, that

Automorphic forms for GL(2, AQ )     1 u ∞ · pe p y∞ 0 , f adelic 0 1 0 1

132  W (g; f ) = e(x) Q\AQ



pe p 0

... ,  

 f adelic

= e(x) Q\AQ



... ,

p −e p 0

0 1

 

 f adelic

= e(x)

p −e p 0

1 u∞ 0 1

Q\AQ

 ... ,



0 1



0 1

pe p 0



1 up 0 1





y∞ 0

0 1

1 0

up 1



 , ... ,

5   e − p e p u du

, ... ,

1 u ∞ · pe p 0 1 0 1



1 up 0 1



y∞ 0

0 1



 ,

5 , ... ,

p −e p 0

0 1

  e − p e p u du

 ,

5   e − p e p u du.

 −e p  Note that p 0 10 ∈ K v unless v = p, ∞. The above procedure can be repeated at each prime p where e p =/ 0. Observe that  pe p . t= p e p =/ 0

It  follows from Proposition 1.4.5, Definition 4.8.2 and the fact that  then 1 up ∈ K p for u p ∈ Z p that 0 1  

 W (g; f ) = e(x)

f adelic

Q\AQ







1

= e(x)

f 0

1 u2 0 1

1 0

  1 , 0

1 u∞ 0 1



u∞ 1 u3 1



t −1 0

0 1



y∞ 0

=

t −1 y∞ 0

0 1

,

 e∞ (−tu ∞ ) du ∞  · e p (−tu p ) du p

4 A(t) t −1 y∞ K ν− 1 (2π y∞ ) e(x),

t ∈ Z,

0,

t ∈ Z.

2



   5 1 u5 , , . . . , e(−tu) du 0 1

p



0 1

Zp



4.10 Global Whittaker functions for GL(2, AQ )

133

We will now show that the global Whittaker function which was computed in Theorem 4.10.10 has a factorization into local Whittaker functions. Theorem 4.10.13 (Factorization of the global Whittaker function) Let f be a Maass form for S L(2, Z) of type ν as in Definition 4.8.1 with classical Whittaker expansion    √ y∞ 0 1 x∞ A(n) y∞ · K ν− 1 (2π |n|y∞ )e2πinx∞ = f 2 0 1 0 1 n =/ 0

for all x∞ ∈ R, y∞ > 0, normalized so that A(1) = 1. Assume f is an eigenfunction of all the Hecke operators. Let W (g; f ) be the global Whittaker function for f as in Theorem 4.10.10. Then  W (g; f ) = Wv (gv ; f ) (4.10.14) v≤∞

provided we make the definitions      1 x∞ y∞ 0 r∞ 0 · k∞ ; f W∞ 0 1 0 1 0 r∞ √ = y∞ K ν− 1 (2π y∞ )e∞ (x∞ ), 2

and

 Wp

1 0

xp 1



yp 0

0 1 =

 

rp 0

0 rp



(4.10.15)

 · k p; f

1   |y p | p2 · A |y p |−1 e p (x p ), if y p ∈ Z p , p

0,

otherwise, (4.10.16)

at all the finite primes p. Proof Since f is an eigenfunction of all the Hecke operators, the arithmetic Fourier coefficients A(n) (with n ∈ Z) satisfy the multiplicativity relations (see Theorem 3.12.8, [Goldfeld, 2006]) A(mn) = A(m) · A(n),

(∀(m, n) = 1).

The result now follows immediately from Theorem 4.10.10.  Note that W p (g p ; f ) = 1 for all but finitely many primes p, so that the product (4.10.14) is really a finite product and is well defined. The Whittaker function at ∞, defined in (4.10.15), is precisely the Whittaker function for G L(2, R) defined in Section 5, [Goldfeld, 2006]. The functions defined in (4.10.16) may be considered as p-adic generalizations of (4.10.15). It is clear from the definition that

134  Wp

1 0

Automorphic forms for GL(2, AQ )     up g p ; f = e p (u p )W p (g p ; f ), ∀u p ∈ Q p , g p ∈ G L(2, Q p ) , 1 (4.10.17)

which is the key defining property of an abstract Whittaker function. It is also worthwhile to point out that W∞ , given in (4.10.15), depends only on the Laplace eigenvalue ν(1 − ν) of f , while W p , given in (4.10.16), depends only on the Hecke eigenvalue A( p).

4.11 Strong approximation for congruence subgroups Our objective in this section is to generalize the strong approximation result given in Proposition 4.4.2. This will be needed to generalize the adelic lift of a Maass form of weight zero and level 1 given in Section 4.8. In Section 4.12 we will consider adelic lifts of classical automorphic forms of arbitrary weight, level, and character. To this end, we now introduce the Iwahori subgroups and the group G L(2, R)+ . Definition 4.11.1 We define G L(2, R)+ to be the group of all matrices in G L(2, R) with positive determinant. Definition 4.11.2 (Iwahori subgroup) Let p be a prime and let N be an integer. The Iwahori subgroup of G L(2, Z p ), denoted K 0 (N ) p , is defined to be      a b  ∈ G L(2, Z p )  c ∈ Z p . K 0 (N ) p := N ·c d  Remarks Note that K 0 (N ) p only depends on the highest power of p that divides N and that if p  N then K 0 (N ) p = G L(2, Z p ). Theorem 4.11.3 (Strong approximation for prime power level subgroups) Fix a prime power p e with e ≥ 1. Every g ∈ G L(2, AQ ) can be written in the form g = i diag (γ ) · i ∞ (g∞ ) · kfinite  where γ ∈ G L(2, Q), g∞ ∈ G L(2, R)+ , and kfinite ∈ K 0 ( p e ) p G L(2, Zv ). v =/ p,∞

Remark There is no uniqueness in this theorem. Proof The proof is a straightforward adaptation of that of Proposition 4.4.2 (strong approximation). The only differences are: (1) we write the final answer in a slightly different form, and (2) one will need to use a neighborhood of the form ⎛ ⎞    S L(2, Z p )⎠ , g · ⎝U · K 0 p e p · p =/ p

4.11 Strong approximation for congruence subgroups in place of



g · U·



135

 S L(2, Z p ) . 

p

Observe that 0 ( p e ) is precisely the set of matrices γ in G L(2, Q) having the property that    G L(2, Zv ). i diag (γ ) ∈ G L(2, R)+ · K 0 p e p · v =/ p,∞

Lemma 4.11.4 Define K 0 ( p e ) = K 0 ( p e ) p ·

 v =/ p,∞

G L(2, Zv ). Suppose that

i diag (γ ) · i ∞ (g∞ ) · kfinite = i diag (γ ) · i ∞ (g∞ ) · kfinite ,

with γ , γ ∈ G L(2, Q), g∞ , g∞ ∈ G L(2, R)+ , and kfinite , kfinite ∈ K 0 ( p e ). Then γ −1 γ ∈ 0 ( p e ).

Proof Immediate from above observation.



Theorem 4.11.3 easily generalizes as follows. Theorem 4.11.5 (Strong approximation for congruence subgroups) Fix  an integer N with prime power decomposition N = ri=1 piei . Every g ∈ G L(2, AQ ) can be written in the form g = i diag (γ ) · i ∞ (g∞ ) · kfinite where γ ∈ G L(2, Q), g∞ ∈ G L(2, R)+ , and kfinite ∈

 p

K 0 (N ) p .

Definition 4.11.6 (The compact subgroup K 0 (N )) Fix N ∈ Z. Then we define  K 0 (N ) p , K 0 (N ) := p

where K 0 (N ) p is the Iwahori subgroup defined in Definition 4.11.2. Similarly, Lemma 4.11.4 generalizes to the following lemma. Lemma 4.11.7 Let N =

r  i=1

piei . Suppose that

i diag (γ ) · i ∞ (g∞ ) · kfinite = i diag (γ ) · i ∞ (g∞ ) · kfinite ,

with γ , γ ∈ G L(2, Q), g∞ , g∞ ∈ G L(2, R)+ , and kfinite , kfinite ∈ K 0 (N ). Then γ −1 γ ∈ 0 (N ).

Automorphic forms for GL(2, AQ )

136

4.12 Adelic lifts with arbitrary weight, level, and character Fix integers k ≥ 0, N ≥ 1, and let χ be a Dirichlet character (mod N ). The main purpose of this section is to generalize the adelic lift of a Maass form of weight zero and level one (given in Section 4.8) to the more general situation of an automorphic form for 0 (N ) with weight k and character χ . Let h = {x +i y | x ∈ R, y > 0} denote the upper-half plane. For f : h → C recall the definition of the weight k slash operator given in Definition 3.5.6:    f k g (z) =



 az + b , f cz + d   a b for z ∈ h, g = ∈ G L(2, R)+ . c d

cz + d |cz + d|

−k



The first step in the adelic lift of an automorphic form on the upper half plane h will be an intermediate lift from h to G L(2, R)+ , the group of matrices in G L(2, R) with positive determinant. Definition 4.12.1 (Lift from the upper half-plane h to G L(2, R)+ ) Given f : h → C, we define a function ( f : G L(2, R)+ → C by    ( f (g) := f k g (i). By the Iwasawa decomposition, every g ∈ G L(2, R)+ can be uniquely written in the form      1 x y 0 r 0 cos θ sin θ g= , 0 1 0 1 0 r − sin θ cos θ with x, y, r, θ ∈ R, y, r > 0, 0 ≤ θ < 2π. Then ( f (g) = (cos θ + i sin θ )k f (x + i y) = eikθ f (x + i y). It follows that   cos θ ( f g − sin θ

sin θ cos θ

 =e

ikθ (

f (g),

 

+ ∀ θ ∈ R, g ∈ G L(2, R) . (4.12.2)

Lemma 4.12.3 Fix a prime p, an integer e ≥ 0, and another integer k. Let χ (mod p e ) be a Dirichlet character. Let f : h → C be given. Assume that        a b e  f k γ = χ (d) f, ∀γ = ∈ 0 ( p ) . c d + Let ( f be the lift of f from the upper half plane to G L(2,  R) defined in a b Definition 4.12.1. Then ( f (γ g) = χ (d)( f (g) for all γ = c d ∈ 0 ( p e ).

4.12 Adelic lifts with arbitrary weight, level, and character

137

Proof

          ( f (g).  f (γ g) = f k γ g (i) = f k γ k g (i) = χ (d) f k g (i) = χ (d)(

Fix integers k ≥ 0, N ≥ 1, and let χ be a Dirichlet character (mod N ). Let f be an automorphic form for 0 (N ) with weight k and character χ . The final step in the adelic lift of f is to lift ( f to an automorphic function f adelic on the adele group G L(2, AQ ). We first treat the case where N = p e is a prime power with e ≥ 1. Recalling Definition 4.11.6, let    G L(2, Zv ) ⊂ G L(2, Afinite ). K 0 ( pe ) = K 0 pe p · v =/ p,∞

χidelic of K 0 ( p e ). Let χidelic = We want to associate to χ (mod p e ) a character (  v χv be the idelic lift of χ defined as in Definition 2.1.7. It follows from Definition 2.1.7 that for gcd(d, p) = 1 we have  χv (d) = χ p (d)−1 . (4.12.4) χ (d) = v =/ p

Definition 4.12.5 Fix a prime power p e with e ≥ 1, and a character χ (mod p e ). Let kfinite ∈ K 0 ( p e ) have the form        5  a3 b3 a5 b5 ap bp a2 b2 , , ,..., ,... . kfinite = c2 d2 c3 d3 c5 d5 pe c p d p Then we define

  ( χidelic kfinite := χ p (d p ).

Lemma 4.12.6 (Lift of χ to a character of K 0 ( p e )) The function ( χidelic , defined in Definition 4.12.5, is a character of K 0 ( p e ). Proof Take  a2 kfinite = c2

b2 d2

  a3 , c3

b3 d3

  a3 , c3

b3 d3

  a5 , c5

b5 d5



 ,...,

ap pe c p

bp dp



5 ,... ,

     5 ap b p a b5 , 5 , . . . , , . . . , p e c p d p c5 d5     two elements of K 0 ( p e ). So ( = χ (d p )−1 . χidelic kfinite = χ (d p )−1 , and ( χidelic kfinite  

To compute ( χidelic kfinite · kfinite we only need to compute the lower right entry of     ap b p ap bp · , p e c p d p pe c p d p  

which is d p d p + c p b p p e . So ( χidelic kfinite · kfinite = χ (d p d p + c p b p p e )−1 . It follows

−1 from (4.12.4) that this equals χ (d p d p ) , which is equal to χ (d p )−1 χ (d p )−1 because χ is a multiplicative character. 

kfinite =



a2 c2

b2 d2

138

Automorphic forms for GL(2, AQ )

Now take f to be an automorphic form of weight k, level p e , and character χ . We define ( f : G L(2, R)+ → C as in Definition 4.12.1. Then by Lemma ( 4.12.3, f has the property that     a b ( (4.12.7) f (γ g) = χ (d)( f (g), ∀γ = ∈ 0 ( p e ) . c d Definition 4.12.8 (Adelic lifts for prime power level) Fix a prime p, an integer e ≥ 1, and another integer k. Let χ (mod p e ) be a Dirichlet character. Assume that      a b ∀γ = ∈ 0 ( p e ). f k γ = χ (d) f, c d Let ( f be the lift of f from the upper half plane to G L(2, R)+ as in Definition 4.12.1. Let ( χidelic be as in Definition 4.12.5. Define the lifted function f adelic : G L(2, AQ ) → C by setting     f (g∞ ) · ( f adelic i diag (γ ) · i ∞ (g∞ ) · kfinite = ( χidelic kfinite , where γ ∈ G L(2, Q), g∞ ∈ G L(2, R)+ , and kfinite ∈ K 0 ( p e ), as in Definition 4.11.6. Remarks The strong approximation Theorem 4.11.3 for congruence subgroups tells us that every g ∈ G L(2, AQ ) can be expressed in the form g = i diag (γ ) · i ∞ (g∞ ) · kfinite .

(4.12.9)

The representation (4.12.9) is not unique, so it is necessary to check that f adelic is well defined. This is the goal of the next lemma. Lemma 4.12.10 Assume

i diag (γ ) · i ∞ (g∞ ) · kfinite = i diag (γ ) · i ∞ (g∞ ) · kfinite ,

with γ , γ ∈ G L(2, Q), g∞ , g∞ ∈ G L(2, R)+ , and        a3 b3 a5 b5 ap a2 b2 kfinite = , , ,..., c2 d2 c3 d3 c5 d5 pe c p        ap a3 b3 a5 b5 a2 b2

, , , . . . , kfinite = p e c p c2 d2 c3 d3 c5 d5

5 ,... ,  5 b p , . . . , d p bp dp



in K 0 ( p e ). By Lemma 4.11.4, we have (γ )−1 γ ∈ 0 ( p e ). Define a, b, c, d ∈ Z by requiring that   a b

−1 . (γ ) γ = cp e d

4.12 Adelic lifts with arbitrary weight, level, and character Then

139

    = χ (d)−1 ( χidelic kfinite . ( χidelic kfinite

Proof It follows from the definitions of a, b, c, d, that for every prime p, we have      b p ap ap a b bp = . c p p e d p c p pe d p cp e d Consequently   = χ (d p )−1 ( χidelic kfinite = χ (d)−1 χ (d p )−1   = χ (d)−1( χidelic kfinite .  Proof that f adelic is well defined in Definition 4.12.8: We must show that    

( =( f (g∞ )( )( χidelic kfinite χidelic kfinite , f (g∞ provided

i diag (γ ) · i ∞ (g∞ ) · kfinite = i diag (γ ) · i ∞ (g∞ ) · kfinite ,

Take γ , g∞ , kfinite , and γ , g∞ , kfinite as above. Then (γ )−1 γ ∈ 0 ( p e ) and we define a, b, c, d by   a b

−1 . (γ ) γ = pe c d     Then, by Lemma 4.12.10, it follows that ( χidelic kfinite = χ (d)−1( χidelic kfinite . But

g∞ = (γ )−1 γ g∞ , so

( ) = χ (d) ( f (g∞ ), f (g∞

by (4.12.7).

 r

Let N = i=1 piei be the prime power decomposition of an integer N , and let χ (mod N ) be a Dirichlet character. Let k be an integer. We now want to generalize the adelic lift given in Definition 4.12.8 to the case of an automorphic form of arbitrary level N , character χ , and weight k. The extension of (4.12.4) will be χ (d) =

 v| N

χv (d) =

r 

χ pi (d)−1 ,

(∀d ∈ Z, (d, N ) = 1) .

i=1

Similarly, Definition 4.12.5 can be generalized as follows.

(4.12.11)

Automorphic forms for GL(2, AQ )

140

 Definition 4.12.12 Let N = ri=1 piei , and let χ (d) be given by (4.12.11). Assume kfinite ∈ K 0 (N ) has the form  kfinite =

a2 c2

b2 d2

Then we define

  a3 , c3

b3 d3

  a5 , c5

b5 d5



5 ,

...,

.

r    ( χidelic kfinite = χ pi (d pi ). i=1

The adelic lift can now be given in general. The analogue of Definition 4.12.8 will be as follows. Definition 4.12.13 (Adelic lifts for arbitrary weight, level, and  character) Fix an integer N with prime power factorization N = ri=1 piei . Let χ (mod N ) be a Dirichlet character, and let k be an integer. Assume that        a b ∀γ = ∈ 0 (N ) . f k γ = χ (d) f, c d Let ( f be the lift of f from the upper half plane to G L(2, R)+ as in Definition 4.12.1. Let ( χidelic be as in Definition 4.12.12. Define the lifted function f adelic : G L(2, AQ ) → C by setting     f (g∞ ) · ( χidelic kfinite , f adelic i diag (γ ) · i ∞ (g∞ ) · kfinite = ( where γ ∈ G L(2, Q), g∞ ∈ G L(2, R)+ , and kfinite ∈ K 0 (N ), as defined in Definition 4.11.6. Proposition 4.12.14 (Central character of an adelic lift) Let f be an automorphic form of weight k, level N , and character χ (mod N ), as in Example 3.3.6. Then f adelic is an adelic automorphic form with central character χidelic where χidelic is the idelic lift of χ as defined in Definition 2.1.7. Proof To prove the proposition, we have to verify that      r 0 g = χidelic (r ) f adelic (g), . f adelic ∀ g ∈ G L(2, AQ ), r ∈ A× Q 0 r × For r ∈ A× Q , write r = i diag (α)i ∞ (r ∞ )r f , with α ∈ Q , r ∞ ∈ R, r ∞ > 0, × and r f = {1, r2 , r3 , . . . , r p , . . . , } with r p ∈ Z p for all p. Using the strong approximation Theorem 4.11.5 for K 0 (N ) we may express any g ∈ G L(2, AQ ) in the form

g = i diag (γ )i ∞ (g∞ )kfinite ,

  γ ∈ G L(2, Q), g∞ ∈ G L(2, R)+ , kfinite ∈ K 0 (N ) .

4.13 Global Whittaker functions for adelic lifts   Then the corresponding expression for r0 r0 g is        α 0 r∞ 0 rf 0 γ i∞ g∞ kfinite . i diag 0 α 0 r∞ 0 rf

141

By Definition 4.12.13, then,

     f adelic (g) = f k g∞ (i) · ( χidelic kfinite

and f adelic



r 0 0 r

      r∞ g = f 0 k

Since r∞ > 0,



0 r∞

   r∞ f 0 k

so what we have to show is that  rf ( χidelic 0

0 r∞ 0 rf





g∞ (i) · ( χidelic



rf 0

0 rf



 kfinite .

 = f,

 = χidelic (r ).

Now χidelic (i diag (α)) is trivial because χidelic is a Hecke character. Furthermore, reviewing the precise definition of χidelic , given in Definition 2.1.7, we see that χ∞ (r∞ ) = 1 because r∞ > 0 and χ p (r p ) = 1 for p | N because r p ∈ Z×p . Thus    rf 0 . χidelic (r ) = p|N χ p (r p ). But this is precisely the definition of ( χidelic 0 rf We have, of course, only verified one of the conditions for being an adelic automorphic form as in Definition 4.7.6. The other conditions can be established in essentially the same manner as the proof of Proposition 4.8.4. 

4.13 Global Whittaker functions for adelic lifts with arbitrary weight, level, and character Fix integers k ≥ 0, N ≥ 1, and let χ be a Dirichlet character (mod N ). Let f be a cuspidal automorphic form for 0 (N ) with weight k and character χ as in Definition 3.5.2. We may lift f to an adelic automorphic form f adelic as in Definition 4.12.13. Then by (4.10.1), (4.10.2), (4.10.8), the function f adelic has a Fourier expansion of the form    α 0 f adelic (g) = W g; f , 0 1 α∈Q×     1 u g e(−u) du, (∀g ∈ G L(2, AQ )). W (g; f ) := f adelic 0 1 Q\AQ

(4.13.1)

142

Automorphic forms for GL(2, AQ )

We have already explicitly computed the global Whittaker function W (∗; f ) in Theorem 4.10.10 in the case of level N = 1 and weight k = 0. We shall now redo this computation in the more general case of arbitrary weight k, level N , and character χ (mod N ). In this case, the answer will be given in terms of the Fourier coefficients aa,n of f at all the cusps a of 0 (N ), as given in Theorem 3.7.4. The precise definition depends on a choice of matrix σa ∈ S L(2, R) satisfying   1 1 σa−1 ga σa = . σa ∞ = a, 0 1 It follows from Section 3.7 that we may choose  √ ma 0 σa = γ a · √ −1 , ma 0 for some positive integer m a and γa ∈ S L(2, Z). We assume that σa is of this form for each a. Then, as a ranges over a complete set of inequivalent cusps for for the double cosets 0 (N ), the matrix γa ranges over a set   

 of representatives  1n 0 (N )\S L(2, Z)/∞ , where ∞ = n∈Z . 0 1 We require the following: Lemma 4.13.2 Fix an integer N ≥ 1. Let S be any set of representatives for √  ma 0 √ −1 the equivalence classes of cusps for 0 (N ) and assume σa = γa 0 ma has been fixed, for each a ∈ S. Let K 0 (N ) be the compact subgroup of  G L(2, AQ ) as in Definition 4.11.6. If kfinite = {k2 , k3 , . . . , k p , . . . } ∈ p G L(2, Z p ), then there is a unique cusp a ∈ S, an integer j with 0 ≤ j < m a , and an element k N ∈ K 0 (N ) such that    1 −j γa−1 k N , kfinite = i finite 0 1 where i finite denotes the diagonal embedding of G L(2, Q) into G L(2, Afinite ), the group of finite adeles.  Proof Let N = ri=1 piei be the prime power factorization of N . For i = 1 to r write k pi = γi k pi with γi ∈ S L(2, Z) and k pi ∈ K 0 (N ) pi . Let γ be any solution to the system of congruences γ ≡ γi−1 (mod piei ),

(1 ≤ i ≤ r ).

Then γ k pi ∈ K 0 (N ) pi for all i, and hence  ) · kfinite ∈ K 0 (N ). Now, γ is in   i finite (γ  1n the double coset 0 (N ) · γa ·  n ∈ Z for some unique a, and hence 0 1   is equal to γ · γa · 10 1j for some j ∈ Z and γ ∈ 0 (N ). It follows from the definitions of γa and m a that

4.13 Global Whittaker functions for adelic lifts 143 5 5        −1  1 man  1 n  γa · 0 (N ) · γa ∩ n∈Z , n∈Z = 0 1 0 1 so that if we require  0 ≤j < m a , then j is uniquely determined. Furthermore, 1 −j γa−1 k N , with k N = i finite ((γ )−1 ·γ )·kfinite , which is, indeed, kfinite = i finite 0 1 in K 0 (N ).  Remark Clearly, j and a depend only on k pi for 1 ≤ i ≤ r. That is, they are independent of k p for p  N . We are now ready to state and prove the main theorem of this section which gives the explicit computation of the global Whittaker function for a Maass cusp form of arbitrary weight, level, and character. Warning (Two uses of the symbol k) It has become standard practice in analytic number theory to refer to automorphic forms of weight k. Similarly, it is standard practice in the theory of automorphic representations to let k be an  element of the maximal compact subgroup K = O(2, R) p G L(2, Z p ). We have decided to use both notations simultaneously! We hope the reader will understand the context and not confuse the two uses of k. Theorem 4.13.3 (Global Whittaker function for general adelic lifts) Let N , k ∈ Z with N ≥ 1. Fix a Dirichlet character χ (mod N ). Let f be a Maass cusp form of type ν, weight k, and character χ for 0 (N ), as in Definition 3.5.7, with classical Fourier-Whittaker expansions (at the cusps a)      aa,n W sgn(n)k , ν− 1 4π |n + μa | · y∞ e2πi(n+μa )x∞ , f k σa (z) = 2

n+μa =/ 0

2

 √ ma 0 √ −1 as in lemma 4.13.2. Let for z = x∞ + i y∞ ∈ h and σa = γa · 0 ma         1 x y 0 r 0  0 cos θ sin θ g= ·i diag i∞ ·kfinite 0 1 0 1 0 r 0 1 − sin θ cos θ be the adelic Iwasawa decomposition of g ∈ G L(2, AQ ) as in Proposition  4.4.3. Here  = ±1, 0 ≤ θ < 2π , kfinite ∈ G L(2, Z p ), x ∈ AQ , and r, y = {y∞ , y2 , y3 , . . . } ∈ A× Q. Define  t= |y p |−1 p ,

p

n = m a t − μa .

p

If n ∈ Z and y p ∈ |N | p Z p for all finite primes p, then f adelic , as in Definition 4.8.1, has a global Whittaker function     kθ + t j · ( χidelic (k N ) χidelic (r ). W (g; f ) = aa,n W sgn(n)k , ν− 1 4π y∞ e(x) e∞ 2 2 2π

Automorphic forms for GL(2, AQ )

144

If n ∈ Z or y p ∈ |N | p Z p for some prime p, then W (g; f ) = 0.  Here e = v ev is the global additive character defined in Definition 1.7.3, χidelic is defined in Defthe idelic lift χidelic is defined in Definition 2.1.7, and ( inition 4.12.12. The variables a, k N , j are defined as follows. If we write y = i ∞ (y∞ ) · yfinite with yfinite ∈ Afinite , then by Lemma 4.13.2, there is a unique cusp a (in a fixed equivalence class of cusps) and an element k N ∈ K 0 (N ) such that  −1     t yfinite 0 1 −j γa−1 k N . · kfinite = i finite i finite 0 1 0 1 Remark It can be shown that the integer m a introduced in Lemma 4.13.2 is, in fact, a divisor of N , and that the cusp parameter μa is always a rational number such that mNa · μa is an integer. Proof Our goal is to explicitly compute (4.13.1). We first consider the case when N = p e is a prime power, and  = 1. By the Iwasawa decomposition given in Proposition 4.4.3, every g ∈ G L(2, AQ ) can be written in the form       1 x y 0 r 0 g= · · · k, 0 1 0 1 0 r where x ∈ AQ , r ∈ A× Q, k = {k∞ , k2 ,

... ,

k p,

... ,

, } ∈ O(2, R) ·



G L(2, Z p ),

p

y = {y∞ , 2 f2 ,

... ,

p fp ,

... ,

},

(y∞ > 0).

Because   = 1, the matrix k∞ is in fact in S O(2, R). We write it as  we assume θ sin θ k∞ = −cos , for θ ∈ [0, 2π ). sin θ cos θ Let i p : G L(2, Q p ) → G L(2, AQ ) denote the embedding at the prime p defined by

 , for g p ∈ G L(2, Q p ), i p (g p ) := I2 , . . . , I2 , g p , I2 , . . . , !"# position p

where I2 denotes the two by two identity matrix. Then, by (4.12.2), Proposition 4.12.14, and the elementary transformation u → u − x, it easily follows that    y 0 ikθ · i p (k p ); f . Wα (g; f ) = e(αx)χidelic (r )e Wα (4.13.4) 0 1 Let   f , (the product running over finite primes ). (4.13.5) t= 

  −1   Since f adelic (h) = f adelic i diag t 0 01 h , for any h ∈ G L(2, AQ ), we       −1    y 0 1u t y 0 · i · i obtain that f adelic 10 ut1 (k ) = f (k ) . p p adelic p p 0 1 0 1 0 1 Consequently,

4.13 Global Whittaker functions for adelic lifts    y 0 k; f Wα 0 1      1 u y 0 = f adelic · i p (k p ) e(−αu)du 0 1 0 1 Q\AQ





=

f adelic

1 u 0 1



t −1 y 0

0 1



Q\AQ

145

 · i p (k p ) e(−tαu)du. (4.13.6)

It follows from the definition of t that t −1  f ∈ Z×  , for all finite primes . To evaluate (4.13.6) it will be convenient to use a nonstandard fundamental domain for AQ /Q, namely 

[0, p e ) · p e Z p ·

Zv .

v =/ p,∞

A straightforward modification of the proof of Proposition 1.4.5 shows that this set is indeed a fundamental domain for AQ /Q. It follows that (4.13.6) is equal to  

 = [0, pe )· pe Z p ·

 ,

f adelic



1 0

u∞ 1

0 1

t −1 y∞ 0

0 1

;  1 · kp , 0

0 1

  1 , 0

0 1

 , ...

Zv

v= / p,∞

1 0



 :   −1 1 up t yp , 0 1 0

0 1



5

 ,

... , · e(−tαu)du

 p

f adelic

= pe Z p 0



,

 

e

1 0

0 1

1 u∞ 0 1



t −1 y∞ 0

0 1

  1 , 0

0 1

 ,

...

;   5 1 0 , ... , · kp , 0 1   · e∞ (−αtu ∞ )e p (−αtu p )du ∞ du p ev (−αtu v )du v .

 :   −1 1 up t yp , 0 1 0

0 1



v =/ p,∞ Z v

(4.13.7) This is equal to zero if the denominator of the rational number αt contains a power of the prime  =/ p. Otherwise, the contribution from the primes not equal to p and ∞ is 1, and αt must take the form m/ p j , for some integers m and j. Moreover, since u p ∈ p e Z p , we see that

146  1 0

up 1



 ∈

Automorphic forms for GL(2, AQ )       5  a b 1 0 b ≡ (mod p e ) . ∈ G L(2, Z p )  c d 0 1 d

a c

But this is a normal subgroup of G L(2, Z p ), which is contained in K 0 ( p e ) p , and in the kernel of χidelic . Hence     −1    1 u∞ t y∞ 0 1 0 , , ... f adelic 0 1 0 1 0 1   :   −1 ;   5  1 0 1 up t yp 0 1 0 , , , ... , · kp , 0 1 0 1 0 1 0 1 is actually independent of u p ∈ p e Z p . It follows that (4.13.7) is equal to zero unless αt = m/ p e for some m ∈ Z, while if αt is of this form, then (4.13.7) is equal to 1 pe

p

 

e

f adelic

1 0

0

u∞ 1 

... ,

1 0



   t −1 y∞ 0 1 0 , , 0 1 0 1  : −1 ;    0 t yp 0 1 0 , , · kp , 1 0 1 0 1 · e∞ (−αtu ∞ )du ∞ .

5 ... (4.13.8)

Next, we may choose a, j, and k N ∈ K 0 (N ) (here N = p e ) as in Lemma 4.13.2, such that  −1      t yp 0 1 −j γa−1 k N . · k p = i finite ip 0 1 0 1   Using the fact that 10 1j ∈ G L(2, Z ) for all finite primes , and that      1 j f adelic (h) = f adelic i diag γa · ·h , 0 1 we find that (4.13.8) is equal to e   −1      p 1 t y∞ 0 1 j 1 u∞ ( χidelic (k N ) · e , f adelic γa · · 0 1 0 1 0 1 p 0     5 1 0 1 0 , ... , , ... e∞ (−αtu ∞ )du ∞ . 0 1 0 1 √  ma 0 √ −1 , we have Under the assumption that the matrix σa = γa 0 m a

1 ( χidelic (k N ) · e p

p 0

e

    −1 f k σa m a · [(u ∞ + j) + it −1 y∞ ] e∞ (−αtu ∞ ) du ∞

Exercises for Chapter 4 pe

=( χidelic (k N ) · e∞ (αt j) ·

ma pe

ma

147

    −1 f k σa u ∞ + im −1 y∞ a t

0

· e∞ (−αm a tu ∞ ) du ∞ . The stated result in the case  = 1, and N = p e now follows from Theorem 3.7.4. The modifications for  = −1 are straightforward and we leave r  piei with i > 1, the argument is the same them to the reader. When N = i=1

except that we use the fundamental domain, [0, N ) ·

r 

piei Z pi ·



Zp

p| N

i=1

for AQ /Q. That this is indeed a fundamental domain is once again a straightforward modification of the proof of Proposition 1.4.5, which we leave to the reader. 

Exercises for Chapter 4 

4.1 Find the p-adic Iwasawa decomposition of

p −1 0 1

 .

4.2 Let u ∈ Q p . Compute the p-adic Iwasawa decomposition of Hint: Consider the cases u ∈ Z p and u ∈ Z p separately.



1 0 u 1

 .

4.3 By the p-adic Iwasawa decomposition (Proposition 4.2.1), any g ∈ G L(2, Q p ) can be expressed as  g=

1 0

x 1



y 0

0 1



r 0 0 r

 k,

for some x ∈ Q p , y, r ∈ Q×p and k ∈ G L(2, Z p ). Prove that if g ∈ G L(2, Q), then we may take x ∈ Q, y, r ∈ Q× and k ∈ G L(2, Z p ) ∩ G L(2, Q). 4.4 Using the Iwasawa decomposition (Proposition 4.1.1) show that G L(2, R) is diffeomorphic to the topological space h × S 1 × R× , where h is the usual upper half-plane and S 1 = {z ∈ C : |z| = 1} is the unit circle. In particular, observe that there is a smooth map ι : G L(2, R) → h given by projection on the first factor.

Automorphic forms for GL(2, AQ )

148  4.5 For

a b c d



∈ G L(2, R) and z ∈ h, define 

a c

b d

 .z =

⎧ az + b ⎪ ⎪ ⎨ cz + d , if ad − bc > 0, ⎪ ⎪ ⎩ a z¯ + b , if ad − bc < 0. c¯z + d

a z¯ + b ∈ h if ad − bc < 0. c¯z + d (b) Prove the above definition gives a left of action of G L(2, R) on h. (a) Prove that

4.6 The group G L(2, R) acts on the space G L(2, R) by left multiplication. In Exercise 4.5 we saw that the same group also acts on the upper halfplane h. Show that these two actions are compatible with the smooth map ι : G L(2, R) → h from Exercise 4.4. That is, for each g ∈ G L(2, R), show that g.ι(g ) = ι(g.g ) for any g, g ∈ G L(2, R). We indicate this more geometrically by saying the following diagram commutes: G L(2, R) × G L(2, R)

·

−→

⏐ (id, ι) =

G L(2, R) ⏐ = ι

·

G L(2, R) × h

−→

h

The top horizontal map is matrix multiplication and the bottom one is the action from Exercise 4.5. 4.7 Define Mat(n, AQ ) to be the ring of n × n matrices with coefficients in AQ with the usual notions of matrix addition and multiplication. As an abelian group under addition, it is isomorphic to a product of n 2 copies of AQ , and we give it the topology induced by this isomorphism. Show that sets of the following form constitute a basis for the topology on Mat(n, AQ ):   Uv × Mat(n, Zv ). v∈S

v∈ S

Here S is a finite set of primes containing ∞, Uv is an open subset of Mat(n, Qv ), and Mat(n, Zv ) is the ring of n × n matrices with coefficients in Zv . The topology on Mat(n, Qv ) is given by identifying it (as an 2 abelian group under addition) with Qnv . 4.8 Prove that G L(n, AQ ), as defined in Section 4.3, can be identified with the set of all invertible matrices in Mat(n, AQ ).

Exercises for Chapter 4

149

4.9 Consider the injective map i : G L(n, AQ ) → Mat(n, AQ ) × Mat(n, AQ ) given by the formula i(g) = (g, g −1 ). Then we  define thetopology on G L(n, AQ ) to be the subspace topology on i G L(n, AQ ) induced by the product topology on Mat(n, AQ ) × Mat(n, AQ ). (a) Show that a basis for this topology on G L(n, AQ ) is given by sets   Uv × G L(n, Zv ), v∈S

v∈ S

where S is a finite set of primes containing ∞, Uv is an open subset of G L(n, Qv ), and G L(n, Zv ) is the ring of invertible n × n matrices with coefficients in Zv and determinant in Z× v. The topology on G L(n, Qv ) is the subspace topology inherited from Mat(n, Qv ). (b) Verify that the topology on G L(n, AQ ) that we just defined is strictly finer than the subspace topology on Mat(n, AQ ); i.e., show that every open set of Mat(n, AQ ) is open in G L(n, AQ ), but not conversely. 4.10 In this exercise we investigate some of the subgroups of G L(n, Q p ) for p a prime number. Prove the following statements. (a) Let G be a topological group and H a subgroup. If H is open, then H is also closed. If moreover G is compact, then H is open if and only if H is closed and of finite index (i.e., [G : H ] < ∞). (b) G L(n, Z p ) is a compact open subgroup of G L(n, Q p ). (c) S L(n, Z p ) is a compact subgroup of G L(n, Q p ), but not open. (d) S L(n, Q p ) is a closed subgroup of G L(n, Q p ), but not open. It is not compact if n > 1. 4.11 Show that elements of the form g = {gv } of S L(2, AQ ) satisfying gv = I2 for all but finitely many v are dense in S L(2, AQ ). 4.12 Show that if one replaces S L(2, ·) with G L(2, ·) in Lemma 4.4.1, it is false. Hint: What is the determinant of g ∈ i diag (G L(2, Q)) · i ∞ (G L(2, R))? 4.13 Let α ∈ gl(2, R). (a) Verify that exp(α) is in the center of G L(2, R) if and only if α lies in the linear span of the identity matrix. (b) Prove that exp(α) lies  in S O(2, R) if and only if α is in the linear  0 1 span of H = −1 0 . In particular, we may identify the Lie algebra of S O(2, R) with the subspace of gl(2, R) spanned by H .

150

Automorphic forms for GL(2, AQ )

× 4.14 Suppose ω : A× Q → C is a unitary Hecke character and ω is given by composing ω with complex conjugation. Show that if φ is an adelic automorphic form for G L(2, AQ ) with central character ω, then φ is an adelic automorphic form with central character ω. Show also that if φ is a cusp form, then so is φ.

4.15 Let α ∈ gl(2, R). Prove that det(exp(tα)) = et Tr(α) , where Tr(·) denotes the trace. 4.16 Let det : G L(2, AQ ) → G L(1, AQ ) be the adelic determinant map: det ({g∞ , g2 , g3 , . . . }) = {det(g∞ ), det(g2 ), det(g3 ), . . . } . Suppose ω : A× Q → C is a unitary Hecke character as in Definition 2.1.2, and define φ = ω ◦ det to be the induced function on G L(2, AQ ). (a) Set t = arg ω({e, 1, 1, . . . }) ∈ R. If α ∈ gl(2, C), prove Dα · φ = it Tr(α)φ. (b) Show that φ is an adelic automorphic form on G L(2, AQ ) with central character ω2 . (c) Show that φ is not a cusp form. 4.17 In this exercise we construct examples of “adelic automorphic forms with no central character.” × be distinct unitary Hecke characters. (a) Let ω1 , ω2 : A× Q → C Suppose we are given non-zero adelic automorphic forms φ1 , φ2 : G L(2, AQ ) → C with central characters ω1 and ω2 , respectively. Prove that the function φ = φ1 +φ2 satisfies properties (1) and (3)–(5) of Definition 4.7.6 for an adelic automorphic form, but that there exists no unitary Hecke character ω for which property (2) is satisfied. (b) Let f : R+ → C be a differentiable function such that f (t) and f (1/t) both have moderate growth in t. Prove that the function φ : G L(2, AQ ) → C given by   φ(g) = f | det(g)|A satisfies properties (1) and (3)–(5) of the Definition 4.7.6 for an adelic automorphic form. Give an example of a function f such that φ does not satisfy property (2) for any unitary Hecke character ω. 4.18 Prove that the collection of all global Whittaker functions for G L(2, AQ ) is a complex vector space. (See Definition 4.10.9.) 4.19 Fill in the details of the proof of Theorem 4.11.3.

Exercises for Chapter 4

151

4.20 Prove that the differential operator D Z kills f adelic for any classical Maass form f . 4.21 We have introduced two notions of adelic lift of an even or odd Maass form of weight zero and full level: an ad hoc definition in Section 4.8 and a more systematic one in Section 4.12. Show that they agree.

5 Automorphic representations for GL(2, AQ )

5.1 Adelic automorphic representations for G L(2, AQ ) Recall that Afinite denotes the finite adeles (as defined in Definition 1.3.3), while G L(2, Afinite ) denotes the multiplicative subgroup of all afinite ∈ G L(2, AQ ) of the form      5    a3 b3 a5 b5 1 0 a2 b2 , , , ... , afinite = , c2 d2 c3 d3 c5 d5 0 1     a b a b where c pp d pp ∈ G L(2, Q p ) for all finite primes p and c pp d pp ∈ G L(2, Z p ) for all but finitely many primes p. By abuse of language, we shall also refer to the elements afinite as finite adeles. Definition 5.1.1 (Vector space Aω (G L(2, AQ )) of adelic automorphic forms) For each unitary Hecke character ω as in Definition 2.1.2, we let Aω (G L(2, A)) denote the C-vector space of all adelic automorphic forms for G L(2, AQ ) with central character ω, as defined in Definition 4.7.6. Fix a unitary Hecke character ω. There are basically two natural actions on the vector space Aω (G L(2, AQ )). There is an action given by right translation and there is also an action given by differential operators at ∞. The action by right translation takes two different forms according to whether we are considering the finite adeles or the prime at ∞. Recall that a representation is determined by a vector space and linear actions on that vector space. The following three actions form the foundation for the construction of automorphic representations. We first describe the actions. It is clear that each makes sense as an action on the vector space of all functions on G L(2, AQ ). We then explain why, in fact, they preserve conditions (1) to (5) of Definition 4.7.6, and hence, are well defined as actions on the vector space Aω (G L(2, AQ )). Let us note that there is also a natural action of G L(2, R) on the vector space 152

5.1 Adelic automorphic representations for GL(2, AQ )

153

of all functions on G L(2, R) given by right translation. However, this action does not preserve the space Aω (G L(2, AQ )). This is the reason why it is not considered more often. • Action of the finite adeles G L(2, Afinite ) by right translation:   We define an action πfinite : G L(2, Afinite ) → G L Aω (G L(2, AQ )) as follows. For φ ∈ Aω (G L(2, AQ )), let πfinite (afinite ) . φ(g) := φ(g afinite ), for all g ∈ G L(2, AQ ), afinite ∈ G L(2, Afinite ). Here, πfinite (afinite ) . φ denotes the action of afinite on the vector φ. • Action of the group O(2, R) by right translation:   We define an action π K∞ : K ∞ → G L Aω (G L(2, AQ )) as follows. O(2,R) can in G L(2, AQ ). If k∞ ∈ K ∞ , The group K ∞ =  be embedded 

is an element of G L(2, AQ ). then k∞ , 10 01 , 10 01 , . . . Now, let φ ∈ Aω (G L(2, AQ )). We define π K∞ (k) . φ(g) := φ(gk),

∀g ∈ G L(2, AQ ),

     and all k = k∞ , 10 01 , 10 01 , . . . with k∞ ∈ K ∞ . Here π K∞ (k) . φ denotes the action of k on the vector φ. • Action of U (g) by differential operators Let g = gl(2, C) and D ∈ U (g) (universal enveloping algebra) be a differential operator as in Definition 4.5.4. We may define an action πg of U (g) on the vector space Aω (G L(2, AQ )) as follows. For φ ∈ Aω (G L(2, AQ )) let πg (D) . φ(g) := Dφ(g),

g = {g∞ , g2 , g3 , . . . } ∈ G L(2, AQ ),

where πg (D) . φ denotes the action of D on 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(2, R) and the action of the universal enveloping algebra. The action of O(2, 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 

    with k∞ ∈ K ∞ . Indeed, for all elements k = k∞ , 10 01 , 10 01 , . . . α ∈ gl(2, R), we compute

Automorphic representations for GL(2, AQ )   πg (Dα ) . π K∞ (k) . φ(g) = πg (Dα ) . φ(gk)     5    d 1 0 1 0  , , ... , = φ g · eαt k∞ ,  0 1 0 1  dt t=0       5  d 1 0 1 0  −1 αt = φ gk · k∞ e k∞ , , , ... ,  0 1 0 1  dt t=0     5    −1 d 1 0 1 0  = φ gk · ek∞ αk∞ t , , , ... ,  0 1 0 1  dt t=0     . φ (g). (5.1.2) = π K∞ (k) . πg Dk −1 αk

154

∞

∞

In the above calculation we used the matrix identity ∞  ∞ −1 (αt)m −1 (k∞ αtk∞ )m −1 αt −1 k∞ e k∞ = k∞ = ek∞ αk∞ t . k∞ = m! m! m=0 m=0 The proof of (5.1.2) with α ∈ g = gl(2, C) is immediate from Definition 4.5.4. The action of the finite adeles by right translation defines a group representation of G L(2, Afinite ). The action of K ∞ = O(2, 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). Now, we need to explain why the space Aω (G L(2, AQ )) is preserved by these three actions. It will immediately follow that these three actions are actually well-defined as actions on Aω (G L(2, AQ )). Let φ satisfy (1)–(5) of Definition

  4.7.6, and consider afinite ∈ G L(2, Afinite ) and

k = k∞ , 10 01 , 10 01 , . . . with k∞ ∈ O(2, R). Let α ∈ gl(2, R). We need to show that the three functions πfinite (afinite ) . φ, π K∞ (k) . φ, and πg (Dα ) . φ all satisfy (1)–(5) as well. (It is clear that the extension from gl(2, R) to all of U (g) is immediate.) Now, it is easy to see that φ(g afinite ) satisfies the conditions (1), (2), (4), (5) of Definition 4.7.6. It is only necessary to verify condition (3), which says that φ(g afinite ) is right K -finite.  from the fact that for any finite idele  This follows −1 afinite , the subgroup K ∩ afinite K afinite is of finite index in K , which, in turn, follows from the fact that this subgroup is clearly an open subgroup of K , while K is compact. Next, for k ∈ O(2, R), it is clear that φ(gk) satisfies properties (1), (2), (3), (5) in Definition 4.7.6. We must check, however, that φ(gk) is right Z (U (g)) finite where g = gl(2, C). In fact, the statement is stronger:     πg (D) . π K∞ (k) . φ = π K∞ (k) . πg (D) . φ

5.1 Adelic automorphic representations for GL(2, AQ )

155

for all k ∈ K ∞ , D ∈ Z (U (g)), and all φ ∈ Aω (G L(2, AQ )). (Actually, even the restrictions on k and φ are not necessary.) This amounts to the statement that the elements of Z (U (g)) are invariant differential operators. This can be shown by a direct calculation using the fact that Z (U (g)) is a polynomial algebra in the two generators D I2 ,

and

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

where I2 denotes the 2 × 2 identity matrix, and Di, j = Dα , for α the matrix with a 1 in the i, j position and zeros elsewhere. Indeed, the fact that     πg (D I2 ) . π K∞ (k) . φ = π K∞ (k) . πg (D I2 ) . φ is immediate from (5.1.2). To check , one needs also to use equation (4.5.3). Finally, we come to πg (Dα ) . φ with α ∈ g. That it satisfies (1), (2), and (4) of Definition 4.7.6 is obvious. To see that it satisfies (3), let V denote the finite dimensional spanned  of functions   by the O(2, R) transspace

10 10 with k∞ ∈ O(2, R). lates of φ. Let k = k∞ , 0 1 , 0 1 , . . . By (5.1.2), 

    π K∞ (k) . πg (Dα ) . φ ∈ V := πg (Dβ ) . φ  β ∈ g, φ ∈ V . Clearly, dim V ≤ dim V · dim g < ∞. It is not at all obvious that πg (Dα ) . φ satisfies (5) of Definition 4.7.6. However, it follows easily from the following deep result of Harish-Chandra. Theorem 5.1.3 Let φ : S L(2, R) → C be a smooth function which is both S O(2, R)-finite and Z (U (sl(2, C)))-finite. Then there exists a smooth compactly supported function f : S L(2, R) → C, which satisfies  φ(h) f (h −1 g) dh. φ(g) = SL(2,R)

Here the integral is with respect to the Haar measure on S L(2, R) which is given in Iwasawa coordinates (see Section 8.6) for any compactly supported function F : S L(2, R) → C by  F(h) dh SL(2,R)



∞



∞





2π

F

= −∞

0

0

1 0

x 1



1

y2 0



0 y− 2 1

 cos θ sin θ − sin θ cos θ · dθ d y d x

Automorphic representations for GL(2, AQ )

156

Proof See [Bump, 1997], Section 2.9.



We shall 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(2, 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(2, Afinite )-module. Definition 5.1.4 ((g, K ∞ )-module) Let g = gl(2, C), K ∞ = O(2, R), and U (g) denote the universal enveloping algebra as in Section 4.5. We define a (g, K ∞ )-module to be a complex vector space V with actions   πg : U (g) → End(V ) = set of all linear maps V → V ,   π K∞ : K ∞ → G L(V ) = set of all invertible linear maps : V → 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 for all α ∈ g, Dα (given by Lemma 4.5.4), 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. Remark The requirement that {π K∞ (k) . v | k ∈ K ∞ } spans a finite dimensional subspace for each v ∈ V can be replaced by the following more explicit finiteness property: Finiteness property: For all v ∈ V there exist integers M < N , complex N c v , numbers c , and vectors v ∈ V (with M ≤  ≤ N ) such that v = =M

and

 πK∞

cos θ − sin θ

sin θ cos θ

 . v = eiθ v ,

(∀ M ≤  ≤ N , θ ∈ R).

5.1 Adelic automorphic representations for GL(2, AQ )

157

Definition 5.1.5 ((g, K ∞ ) × G L(2, Afinite )-module) Let g = gl(2, C) and let K ∞ = O(2, R). Also, let G L(2, Afinite ) denote the finite adeles. We define a (g, K ∞ ) × G L(2, 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(2, 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  (g), k ∈ K ∞ , and afinite ∈ G L(2, Afinite ).  α ∈ g, Dα ∈ U We let π = (πg , π K∞ ), πfinite , and refer to the ordered pair (π, V ) as a (g, K ∞ ) × G L(2, Afinite )-module as well. Definition 5.1.6 (Irreducible or smooth (g, K ∞ )×G L(2, Afinite )-module) Let the complex vector space V be a (g, K ∞ ) × G L(2, Afinite )-module as in Definition 5.1.4. We say the (g, K ∞ ) × G L(2, Afinite )-module is smooth if every vector v ∈ V is fixed by some open compact subgroup of G L(2, Afinite ) under the action πfinite : G L(2, Afinite ) → G L(V ). The (g, K ∞ ) × G L(2, 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 . Lemma 5.1.7 (The space of all adelic automorphic forms with central character ω is a smooth (g, K ∞ ) × G L(2, Afinite )-module) Let ω be a unitary Hecke character as defined in Definition 2.1.2. Let Aω (G L(2, AQ )) denote the vector space of all adelic automorphic forms with central character ω as defined in Definition 5.1.1. Then Aω (G L(2, AQ )) is a smooth (g, K ∞ ) × G L(2, Afinite )-module as defined in Definition 5.1.6. Proof We have already seen that Aω (G L(2, AQ )) is a (g, K ∞ )×G L(2, Afinite )module, so what we need to show is that for any adelic automorphic form φ, as in Definition 4.7.6, there is an open compact subgroup K of G L(2, Afinite ) such that φ(gk ) = φ(g) for all g ∈ G L(2, AQ ), and k ∈ K . The proof relies on two properties of φ: it is a smooth function, as defined in Definition 4.7.2, and it is K -finite. Let V denote the complex vector space of functions spanned by all translates of φ by elements of K . This is a

Automorphic representations for GL(2, AQ )

158

finite dimensional vector space. Let d denote its dimension. Choose a basis φ1 , . . . , φd of V. These are linearly independent functions, so we may also choose d points g1 , . . . , gd ∈ G L(2, AQ ) such that, for c1 , . . . , cd ∈ C, d

ci φi (g j ) = 0, (∀ j = 1, 2, . . . , d)

(5.1.8)

i=1

=⇒

d

ci φi (g) = 0, (∀ g ∈ G L(2, AQ )).

i=1

Define K finite = p Kp ⊂ G L(2, Afinite ) (a maximal compact subgroup). Let g1 , . . . , gd be chosen as in (5.1.8). Because φ is a smooth function, it follows that for each j there exists an integer N j ≥ 1 such that φ is constant on the coset g j · K (N j ), where

K (N j ) := k = {I2 , k2 , . . . , k p , . . . } ∈ K finite

   k p ≡ I2 (mod p f p ), (∀ p|N j ) .

because these subgroups form a base for the topology of K finite at the identity. Let N be the least common multiple of the integers N j , ( j = 1, . . . , d). Then for all k ∈ K (N ) and 1 ≤ j ≤ d, we have φ(g j k) = φ(g j ). Writing φ(g) and φ(gk) in terms of the basis φ1 (g), . . . , φd (g), we see by (5.1.8), that φ(g) = φ(gk) for all g ∈ G L(2, AQ ) and all k ∈ K (N ).  It is also important to define the two notions of isomorphic (g, K ∞ )modules and isomorphic (g, K ∞ ) × G L(2, Afinite )-modules. We shall actually define a more general notion of intertwining which is a type of morphism between these modules. Definition 5.1.9 (Intertwining map of (g, K ∞ )-modules) Let V, V be 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.

5.1 Adelic automorphic representations for GL(2, AQ )

159

Definition 5.1.10 (Intertwining map of (g, K ∞ ) × G L(2, Afinite )-modules) Let V, V be complex vector spaces which define two (g, K ∞ )× G L(2, 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 ),

πA

πfinite : G L(2, Afinite ) → G L(V ),

finite

: G L(2, Afinite ) → G L(V ).

A linear map L : V → V is said to be intertwining if L◦πg (D) = πg (D)◦L , (∀ D ∈ U (g)), L ◦ πfinite (afinite ) = πA

finite

L◦π K∞ (k) = π K ∞ (k)◦L , (∀ k ∈ K ∞ ),

(afinite ) ◦ L ,

(∀ afinite ∈ G L(2, Afinite )).

If L is an isomorphism, then we say the two (g, K ∞ ) × G L(2, 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(2, Afinite )-module as in Definition 5.1.6 with actions πg : U (g) → End(V ), π K∞ : K ∞ → G L(V ), πfinite : G L(2, 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(2, 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(2, Afinite ), we may define πg (D) . (w + W ) := πg (D) . w + W , π K∞ (k) . (w + W ) := π K∞ (k) . w + W ,

(5.1.11)

πfinite (afinite ) . (w + W ) := πfinite (afinite ) . w + W . It is then easy to show that the three actions given in (5.1.11) will then define a (g, K ∞ ) × G L(2, 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. Unfortunately, an automorphic representation is not really a representation. It is something much more complex. It will turn out to be a smooth (g, K ∞ ) × G L(2, Afinite )-module

160

Automorphic representations for GL(2, AQ )

as in Definition 5.1.6 which is realized as a subquotient of the complex vector space V of all adelic automorphic forms as in Definition 4.7.6. Definition 5.1.12 (Automorphic representation with central character × ω) Fix a unitary Hecke character ω : Q× \A× Q → C as in Definition 2.1.2. An automorphic representation with central character ω is defined to be a smooth (g, K ∞ ) × G L(2, Afinite )-module (as in Definition 5.1.6) which is also isomorphic to a subquotient of the complex vector space of adelic automorphic forms Aω (G L(2, AQ )), as defined in Definition 5.1.1. Lemma 5.1.13 (Actions by differential operators and right translation preserve the space of cusp forms) Let ω be a unitary Hecke character as in Definition 2.1.2. The action of U (g) by differential operators and the actions of K ∞ and G L(2, Afinite ) by right translation defined on the space Aω (G L(2, AQ )) preserve the space of adelic cusp forms defined in Definition 4.7.7. Proof For the actions of K ∞ , and G L(2, Afinite ) this is obvious. For the action of g, it suffices to show that      ∂ 1 u φ gi ∞ (exp(tα))  du 0 1 ∂t t=0 Q\AQ

⎛ =

∂ ⎜ ⎝ ∂t



 φ Q\AQ

1 0

u 1



 gi ∞ (exp(tα))

⎞   ⎟ du ⎠   

, t=0

for all g ∈ G L(2, AQ ), all α ∈ gl(2, R), and all adelic cusp forms φ. Recall that the integral over Q\A (which appears above) is, by definition,  an integral over the fundamental domain [0, 1) · p Z p as given in Proposition    g 1.4.5. For any fixed g ∈ G L(2, AQ ), the function u finite → φ 10 u finite 1 is a locally constant function of u finite ∈ Afinite . Hence, as u finite ranges over the  compact set p Z p , it will take only finitely many values. Furthermore, if in GL(2, g = (g∞ , gfinite ), with g∞ ∈ G L(2, R) and g finite  Afinite ), then, as u finite  1 u finite ranges over p Z p , the function g∞ → φ 0 1 g ranges over a finite set of smooth functions. The desired assertion, that we may take the derivative under the integral, is now clear.  Definition 5.1.14 (Cuspidal automorphic representation with central character ω) Fix a unitary Hecke character ω : Q× \AQ → C× as in Definition 2.1.2. Let g = gl(2, C) and K ∞ = O(2, R). We define a cuspidal automorphic representation with central character ω to be a smooth (g, K ∞ )×G L(2, Afinite )module (as in Definition 5.1.6) which is isomorphic to a subquotient of the complex vector space of all adelic cusp forms for G L(2, AQ ) (with central character ω) as in Definition 4.7.7.

5.2 Explicit realization of actions defining a (g, K∞ )-module

161

It is important to remember that an automorphic representation is not actually a group representation of G L(2, AQ ), but rather a (g, K ∞ ) × G L(2, Afinite )module. Each of the three actions which are part of the definition a (g, K ∞ ) × G L(2, Afinite )-module structure can be understood in terms of a precursor in the theory of classical automorphic forms. For example, the classical Maass raising and lowering operators correspond to the action of certain elements of U (g), and the classical operator which sends a newform to an oldform corresponds to the action of a certain element of G L(2, Afinite ). We shall explicitly work out these actions in the next sections and then give concrete examples of cuspidal automorphic representations.

5.2 Explicit realization of actions defining a (g, K ∞ )-module Let us fix a convenient basis for gl(2, R) consisting of  Z=

1 0

0 1



 ,

X=

1 0 0 −1



 ,

Y =

0 1

1 0



 ,

H=

 0 1 . −1 0 (5.2.1)

We recall Definition 4.5.1 which we now relabel as Definition 5.2.2. Definition 5.2.2 Let α ∈ gl(2, R) and f : G L(2, R) → C, 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 ∂t t=0 t=0

It follows from Definition 5.2.2 that the basis elements in (5.2.1) define differential operators D Z , D X , DY , and D H . Furthermore, in terms of our usual coordinates on G L(2, R)+ ,  g=

1 0

x 1



y 0

0 1



we have DZ = r ·

r 0 0 r

∂ , ∂r



cos θ − sin θ

DH =

∂ . ∂θ

For 0 ≤ θ < 2π, let  κθ =

cos θ − sin θ

sin θ cos θ

 .

sin θ cos θ

 ,

Automorphic representations for GL(2, AQ )

162 Then 

κθ X κθ−1

=

κθ Y κθ−1

=



cos2 θ − sin2 θ −2 sin θ cos θ 2 sin θ cos θ cos2 θ − sin2 θ

 −2 sin θ cos θ = (cos 2θ ) · X − (sin 2θ ) · Y, − cos2 θ + sin2 θ  cos2 θ − sin2 θ = (cos 2θ ) · Y + (sin 2θ ) · X. −2 sin θ cos θ (5.2.3)

Lemma 5.2.4 Let g = gl(2, C), K ∞ = O(2, R), and consider a (g, K ∞ )module V with actions πg and π K∞ . Then for any v ∈ V , we have     π K∞ (κθ ) . πg (D X + i DY ) . v = e2iθ · πg (D X + i DY ) . π K∞ (κθ ) . v ,     π K∞ (κθ ) . πg (D X − i DY ) . v = e−2iθ · πg (D X − i DY ) . π K∞ (κθ ) . v . Proof It follows from the Definition 5.1.4 of a (g, K ∞ )-module that   πg (Dα ) · π K∞ (k) = π K∞ (k) · πg Dk −1 αk for all α ∈ g, k ∈ K ∞ , D ∈ U (g). The above can also be rewritten in the form π K∞ (k) · πg (Dα ) = πg (Dkαk −1 ) · π K∞ (k), from which it follows that      π K∞ (κθ ) . πg (D X + i DY ) . v = πg Dκθ X κ −1 + i Dκθ Y κ −1 π K∞ (κθ ) . v . θ

θ

But Dκθ X κ −1 + i Dκθ Y κ −1 = Dcos 2θ X −sin 2θ Y + i Dsin 2θ X +cos 2θ Y = e2iθ (D X + i DY ), θ

θ

from (5.2.3) and the linearity of the function α → Dα . The proof for D X −i DY is similar and left to the reader.  Definition 5.2.5 (Raising and lowering operators) We define the raising differential operator R = 12 (D X + i DY ) and the lowering differential operator L = 12 (D X − i DY ). Interlude: The differential operators R, L can be realized as Maass raising and lowering operators We will now explain how the operators R, L, defined in Definition 5.2.5, are related to the classical Maass raising and lowering operators Rk , L k that were defined in Section 3.9. Here Rk raises the weight by 2

5.2 Explicit realization of actions defining a (g, K∞ )-module and L k lowers the weight by 2. This is the reason the differential operators introduced in Definition 5.2.5 were called raising and lowering operators. Fix integers k, N ≥ 1, and let χ be a Dirichlet character (mod N ). Consider the C-vector space of automorphic forms f : h → C which satisfy        ∀ γ = ac db ∈ 0 (N ) , f k γ = χ (d) f, where

   f k g (z) = 

α β γ δ





γz +δ |γ z + δ|

−k

 f

αz + β γz +δ

 ,

∈ G L(2, R)+ , is the slash operator. The first step in the adelic lift of f is the lift, f → ( f , from h to G L(2, R)+ , given in ( Definition 4.12.1. Recall that f is defined by the relation

with g =

   ( f (g) := f k g (i),

  ∀g ∈ G L(2, R)+ .

It follows from (4.12.2) that for any f which is automorphic of weight k, we have    θ sin θ ( = eikθ ( f (g), ∀θ ∈ R, g ∈ G L(2, R)+ . f g · −cos sin θ cos θ When this observation is combined with Lemma 5.2.4 one immediately sees that when R acts on f , it raises the weight by 2 while L lowers the weight by 2. In fact, it can be shown [Bump, Proposition 2.2.5], that when written out in coordinates, R and L are given by   ∂ ∂ 1 ∂ r ∂ 2iθ iy +y + − , R=e ∂x ∂ y 2i ∂θ 2 ∂r   ∂ 1 ∂ r ∂ ∂ L = e−2iθ −i y +y − − . ∂x ∂y 2i ∂θ 2 ∂r From this, together with Definitions 3.9.1 and 3.9.2, and (4.12.2), we see that for any automorphic form of weight k (and any character and level) L ·( f = L> R ·( f = R> k f k f, and hence R · f adelic = (Rk f )adelic .

163

Automorphic representations for GL(2, AQ )

164

Clearly, U (g) is  generated as an algebra over C by D Z , D H , R, L .  −1 0 , then O(2, R) is generated by δ1 and elements Furthermore, if δ1 = 01 of the form κθ as in (5.2.3). Hence, we will be able to understand the structure of a (g, K ∞ ) module much more clearly once we understand the relations among the operators πg (D H ),

πg (D Z ),

πg (R),

πg (L),

π K∞ (δ1 ),

π K∞ (κθ ), θ ∈ [0, 2π ).

Lemma 5.2.4 describes two such relations. In addition, it follows easily from the definitions that π K∞ (δ1 )πg (R)π K∞ (δ1 ) = πg (L),

π K∞ (δ1 )πg (L)π K∞ (δ1 ) = πg (R),

π K∞ (δ1 )πg (D H )π K∞ (δ1 ) = −πg (D H ). Also πg (D H ) commutes with π K∞ (κθ ) for any θ, and πg (D Z ) commutes with all operators currently under consideration. Next, we want to deal with the Casimir operator for gl(2, R) which we denote by . We define it as in [Goldfeld, 2006; p. 50] and we recall the notation Di, j used there. For 1 ≤ i ≤ n, 1 ≤ j ≤ n let E i, j ∈ gl(n, R) denote the matrix with a one at the i, j th component and zeros elsewhere. Define Di, j = D Ei, j with D Ei, j given by Definition 5.2.2. Lemma 5.2.6 We have 1 1  = − DH ◦ DH + R ◦ L + L ◦ R + DZ ◦ DZ . 2 2 Here ◦ denotes composition of differential operators. Proof Recall that  = D1,1 ◦ D1,1 + D1,2 ◦ D2,1 + D2,1 ◦ D1,2 + D2,2 ◦ D2,2 . First of all 7 16 (D X + i DY ) ◦ (D X − i DY ) + (D X − i DY ) ◦ (D X + i DY ) 4 7 16 D X ◦ D X + DY ◦ DY . = 2

R◦L+L◦R=

Now plug in D H = (D1,2 − D2,1 ), DY = (D1,2 + D2,1 ), so that 1 (DY ◦ DY − D H ◦ D H ) = (D1,2 ◦ D2,1 + D2,1 ◦ D1,2 ). 2

5.2 Explicit realization of actions defining a (g, K∞ )-module

165

Furthermore D X = D1,1 − D2,2 , and D Z = D1,1 + D2,2 so 1 (D Z ◦ D Z + D X ◦ D X ) = D1,1 ◦ D1,1 + D2,2 ◦ D2,2 .  2 In the next lemma we use the notation D k for the k th iterate of the operator D. For example, D 3 = D ◦ D ◦ D. We also adopt the convention that D 0 is the identity operator. Lemma 5.2.7 The differential operators R, L defined in Definition 5.2.5 satisfy the following relations   1 2 1 2 1  + DH − DZ − i DH , R◦L = 2 2 2   1 2 1 1 2  + DH − DZ + i DH , L◦R= 2 2 2 D H ◦ R = R ◦ D H + 2i R, D H ◦ L = L ◦ D H − 2i L . Proof A straightforward computation shows (D X + i DY ) ◦ (D X − i DY ) − (D X − i DY ) ◦ (D X + i DY ) = 2i(DY ◦ D X − D X ◦ DY ) = 2i DY ·X −X ·Y , = −4i D H , since Y · X − X · Y = −2H. It follows that R ◦ L − L ◦ R = −i D H .

(5.2.8)

Also H · X − X · H = −2Y, H · Y − Y · H = 2X, from which we may deduce D H ◦ R − R ◦ D H = 2i R, D H ◦ L − L ◦ D H = −2i L .

(5.2.9)

Lemma 5.2.7 now follows from Lemma 5.2.6 and equations (5.2.8), (5.2.9).  Proposition 5.2.10 (Standard form of the differential operators in U (g)) Every word in R, L , D H and D Z , and hence every element of U (g), may be expressed as a finite linear combination of terms of the form D = D ◦ D aH ◦ b ◦ D cZ , where a, b, c are non-negative integers and D is either R k or L k for some k ∈ Z.

166

Automorphic representations for GL(2, AQ )

Proof For a word in D H , D Z , R, L and , we let the number of discrepancies denote the number of pairs of indices (i, j) such that one of the following holds (1) The i th and j th symbols are R and L in some order. (2) For i < j, the i th symbol is D H and the j th symbol is either R or L. We proceed by induction on the number of discrepancies. A word will have no discrepancies if and only if it is equal to one of the form D ◦ D aH ◦ b ◦ D cZ , where a, b, c, are non-negative integers and D is either R k or L k for some integer k. Since  and D Z commute with all differential operators in U (g), it follows that any word W is equal to one of the form W b ◦ D cZ , where W is a word in R, L and D H alone. If a word of this form contains a discrepancy, it contains one in which j = i + 1. First, suppose there is a discrepancy of the first type, with j = i + 1. Then, by Lemma 5.2.7, we plug in either R◦L = or

  1 1 1  + D 2H − D 2Z + i D H , 2 2 2

  1 1 2 1 2 L◦R=  + DH − DZ − i DH , 2 2 2

and pass to consideration of words in which the number of discrepancies of the first type is smaller. Now suppose there are no discrepancies of the first type. Then, by Lemma 5.2.7, we plug in either D H ◦ R = R ◦ D H + 2i R, or D H ◦ L = L ◦ D H − 2i L , and pass to consideration of words in which the total number of discrepancies is smaller.  We end this section with an explicit example of a (g, K ∞ )-module. It is very convenient to adopt the following notation for the raising and lowering operators.   R , if  ≥ 0, (5.2.11) R = L − , if  < 0. Example 5.2.12 ((g, K ∞ )-module associated to adelic lifts of Maass forms) Let N , k ∈ Z with N ≥ 1, and let χ (mod N ) be a Dirichlet character. Fix a Maass form f of type ν, weight k, and character χ for 0 (N ) as

5.2 Explicit realization of actions defining a (g, K∞ )-module

167

in Definition 3.5.7, and consider the adelic lift f adelic as in Definition 4.12.13. Define a vector space   M     m c R f adelic g · k  M ∈ N, m  ∈ Z, c ∈ C . V f :=  =1 

    Here g ∈ G L(2, AQ ), and k = k∞, , 10 01 , 10 01 , . . . with k∞, ∈

K ∞ . The differential operator R  is defined by (5.2.11) and the action of R  on f adelic (g) with g = {g∞ , g2 , g3 , . . . } is given by the differential operator acting on g∞ . Since D Z and  act on f adelic by scalars (see Exercise 5.14), and commute with everything else, they act by the same scalars on the whole vector space V f . The operator D H acts by the scalar ki on f adelic , and by (k + 2)i on R  f adelic . The action on R  f adelic (g · k ) will be by sgn(det(k∞, )) · (k + 2)i. Using Lemma 5.2.7, it is then clear that the vector space V f is closed under the actions by the differential operators R  and  multiplication by ele right 0 sends R  f adelic (g · k ) to ments in K ∞ = O(2, R). The element δ1 = −1 0 1 R − f adelic (g · k · δ1 ). Finally, if  cos θ kθ = − sin θ

sin θ cos θ

 ∈ K∞,

then πg (kθ )R  f adelic (g · k ) = ei(k+2)sgn(det(k ))θ f adelic (g · k ). It now follows from Definition 5.1.4 that the vector space V f and the two actions above form a (g, K ∞ )-module. Remark Because of the equivariance property (4.12.2), we do not even have to allow the elements k∞, to be arbitrary. We could impose the condition that k∞, = 10 01 or δ1 for all , and the space of functions V f we get would be the same space. Proposition 5.2.13 (Explicit action of the element δ1 ) Let N , k ∈ Z with N ≥ 1, and let χ (mod N ) be a Dirichlet character. Fix a Maass form f of type ν, weight k, and character χ for 0 (N ) as in Definition 3.5.7, and consider the adelic lift f adelic as in Definition 4.12.13. Let h : h → C be defined by h(z) = f (−z). Then h is a Maass form of type ν, weight −k and character χ for0 (N), and h adelic (g) = f adelic (g · i ∞ (δ1 )) for all g ∈ G L(2, AQ ), where δ1 =

−1 0 01

.

 y x  Proof First assume g = i ∞ 0 1 . Note that i finite (δ1 ) is in K 0 (N ) for any N , and in the kernel of ( χidelic for any χ modulo N . It follows that        f adelic (g · i ∞ (δ1 )) = f adelic i diag (δ1 )g · i ∞ (δ1 ) = f adelic i ∞ δ1 · 0y x1 · δ1 . y x  y −x  Since δ1 · 0 1 · δ1 = 0 1 , this is equal to h adelic (g).

Automorphic representations for GL(2, AQ )     cos(−θ) sin(−θ) θ sin θ Next, since δ1 · −cos · δ , we find that the = 1 − sin(−θ) cos(−θ) sin θ cos θ function f adelic (g · i ∞ (δ1 )) satisfies     cos θ sin θ · i ∞ (δ1 ) = e−ikθ · f adelic (g · i ∞ (δ1 )) , f adelic g · − sin θ cos θ

168

just as h adelic (g) does. Furthermore, the two functions have the same dependence on the center, and on an element of K 0 (N ) on the right. It follows that they must agree everywhere. 

5.3 Explicit realization of the action of G L(2, Afinite ) The action of G L(2, Afinite ) on the C-vector space of adelic automorphic forms is given by right translation by finite adeles afinite ∈ G L(2, Afinite ). We shall now study this action in the simple case where       −e    1 0 1 0 p 0 1 0 , ... , , , ... , , afinite = 0 1 0 1 0 1 0 1 !" # pth position

i.e., where afinite is the identity matrix at all v ≤ ∞, except when v = p, where  p−e 0 it takes the value 0 1 . We shall assume that e > 0 is an integer. In order to simplify the notation, it is convenient to recall the three embeddings: i diag : G L(2, Q) → G L(2, AQ ), i finite : G L(2, Q) → G L(2, Afinite ), i v : G L(2, Qv ) → G L(2, AQ ). The three embeddings i diag , i finite , i v are defined as follows:  i diag  i finite  iv

a c

b d

a c

b d

a c

b d



 :=



 :=



a c

b d

1 0

0 1

 :=

1 0 0 1



 ,



 ,

a c

b d

a c

b d







 , ... ,



a c

b d

a c

b d





,  ,

1 0 0 1



5 ,

... ,

,

... ,



,

a c

5

1 0

,

 b , d !" #

position v



,

0 1

 ,

... ,

5 .

5.3 Explicit realization of the action of GL(2, A f inite )

169

Proposition 5.3.1 Let φ : G L(2, AQ ) → C be an adelic automorphic form, as defined in Definition 4.7.6, which satisfies φ(gk) = φ(g) for all g ∈ G L(2, AQ ), k ∈ K . Let    y∞ x∞ r∞ 0 ·k (5.3.2) g = i diag (γ ) · i ∞ 0 1 0 r∞ be the factorization of an element g ∈ G L(2, AQ ) as in Theorem 4.11.5. Recall  G L(2, Z p ), (where that in (5.3.2) we must take k ∈ O(2, R)· K 0 ( p e ) p · v =/ p,∞

K 0 ( p e ) p is the Iwahori subgroup defined in Definition 4.11.2). Then we have   −e    e     p 0 p 0 y∞ x∞ r∞ 0 = φ i∞ · . φ g · i finite 0 1 0 1 0 1 0 r∞ Proof We compute   −e  −e    p p 0 0 φ g · i finite = φ g · ip 0 1 0 1      −e   y∞ x∞ r∞ 0 p 0 · k · ip = φ i diag (γ ) · i ∞ 0 1 0 r∞ 0 1      −e   y∞ x∞ r∞ 0 p 0 · ip · k , = φ i∞ 0 1 0 r∞ 0 1 where

k = ip



pe 0

0 1



 · k · ip

p −e 0

0 1

 .

Because k p ∈ K 0 ( p e ) p , it is clear that k p ∈ G L(2, Z p ). Here k p and k p are the entries in p th place in k and k , respectively. Continuing the computation, we get  −e   p 0 φ g · i finite 0 1      −e   y∞ x∞ r∞ 0 p 0 · i finite = φ i∞ 0 1 0 r∞ 0 1    e    −e   p 0 y∞ x∞ r∞ 0 p 0 · i finite · i∞ = φ i diag 0 1 0 1 0 r∞ 0 1         pe 0 y∞ x∞ r∞ 0 · i∞ = φ i∞ .  0 1 0 1 0 r∞ Remark (Right action of G L(2, Afinite ) turns newforms into oldforms) In Proposition 5.3.1, let φ = f adelic be the adelic lift of a classical automorphic

170

Automorphic representations for GL(2, AQ )

form f as in Section 4.12. Because of the conditions we imposed on φ, f must be an even weight 0 Maass form for S L(2, Z). In particular, f is a newform as in Section 3.11. Then Proposition 5.3.1 tells us that   −e   e    p 0 p 0 y∞ x∞ . = f · φ g · i finite 0 1 0 1 0 1  −e  Thus the right action by p0 10 turns the newform f into an oldform (see Definition 3.11.9). This same phenomenon persists for more general right actions by G L(2, Afinite ). For  example, with f as above, the right translate of f adelic by the matrix

1 0 0 p−e



a c

corresponds to a Maass form for the group b d



 5  e  ∈ S L(2, Z)  b ∈ p Z p ,

which may be regarded as an “oldform” in that it is a natural analogue of an oldform for 0 ( p e ). Note that in the adelic setting, the newform is distinguished from its oldform translates because it has the largest stabilizer in K finite . (When f has a non-trivial character, the distinction is not as straightforward.) The next proposition will be a complement to this. It says that in any smooth representation of G L(2, Afinite ), there is a vector with a large stabilizer. It is convenient to introduce the following notation.     Definition 5.3.3 Let ac db , ac db ∈ G L(2, Q p ), and let f ∈ Z. We adopt the notation:      a b a b  ≡ mod p f ,

c d c d   if a − a , b − b , c − c , d − d ∈ p f Z p . We also set I2 = 10 01 . Proposition 5.3.4 Let (π, V ) be a smooth representation of G L(2, Afinite ). Then  there exists a non-zero vector v in V, an integer M = p p f p , and a character λ of  K 0 (M) := k = {I2 , k2 , . . . , k p . . . } ∈ K finite |   ∗ ∗  kp ≡ 0 ∗ mod p f p (∀ p | M) , (5.3.5) which is trivial on  K 1 (M) := k = {I2 , k2 , . . . , k p . . . } ∈ K finite |      k p ≡ 10 ∗1 mod p f p (∀ p | M) , such that π (k) . v = λ(k)v for all k ∈ K 0 (M).

(5.3.6)

5.3 Explicit realization of the action of GL(2, A f inite )

171

Remark The vector v is similar to a newform with character. However, the character λ here is not necessarily of the form ( χidelic as in Definition 4.12.12. Proof Let v1 be any non-zero element of V. Then v1 is fixed by some open compact subgroup K , and this subgroup must contain the subgroup 

  K (N ) := k = {I2 , k2 , . . . , k p , . . . } ∈ K finite  k p ≡ I2 (mod p f p ), (∀ p|N ) (5.3.7)  fp p ∈ N, because these subgroups form a base for the for some N = p

topology of K finite at the identity. Let   N v2 = π i finite 0 Now,



N 0

0 1

−1

 K (N )

N 0

0 1

−1 

0 1

. v1 .

 ⊃ K 1 (N 2 ).

(5.3.8)

It follows that π (k) . v2 = v2 for all k ∈ K 1 (N 2 ). Now, we prove that @2 ? K 0 (M)/K 1 (M) ∼ = (Z/MZ)× . The proof is as follows: for each prime, p and positive integer f there is a natural homomorphism G L(2, Z p ) → G L(2, Z/ p f Z) given by reduction modulo m  f pi i , a p f . If we combine several of these, we obtain, for any integer M = i=1

homomorphism p M : K finite →

m 

f G L(2, Z/ pi i Z) ∼ = G L(2, Z/MZ).

i=1

This map may be described completely explicitly. If k = {I2 , k2 , k3 , . . . , k p , . . . } is in K finite , then p M (k) = (k¯ p1 , . . . , k¯ pm ), where k¯ pi denotes the reducf tion of the matrix k pi modulo pi i . The kernel of p M is precisely K (M), which is contained in K 1 (M). Furthermore, the image of K 0 (M) in G L(2, Z/MZ) is the group of upper triangular matrices, while the image of K 1 (M) is the group of upper triangular matrices with ones on the diagonal. The quotient is isomorphic to [(Z/MZ)× ]2 . For each (a, b) ∈ [(Z/N 2 Z)× ]2 , we fix an element k(a, b) of the preimage of (a, b) in K 0 (N 2 ). Now, for λ a character of K 0 (N 2 ) which is trivial on K 1 (N 2 ), define 1 λ(k(a, b))−1 π (k(a, b)) . v2 . vλ = 2 2 ϕ(N ) 2 × a,b∈Z/N Z

Automorphic representations for GL(2, AQ )

172

Here ϕ is Euler’s function. Observe that the vector vλ obtained is independent of the choice of coset representatives k(a, b). This is because K 1 (N 2 ) fixes v2 and is in the kernel of λ. By finite Fourier inversion, or equivalently, by computing the sum of vλ dir ectly using character relations, we may obtain v2 = λ vλ . Thus, since v2 is nonzero, at least one of the vectors vλ is non-zero and this completes the proof. 

5.4 Examples of cuspidal automorphic representations In this section we shall construct very explicit examples of cuspidal automorphic representations for G L(2, AQ ) coming from Maass forms and holomorphic Hecke newforms on congruence subgroups of S L(2, Z). Eventually, it will turn out that these examples are everything. That is, we will show that f ↔ V f is a one-to-one correspondence between classical newforms as defined in Section 3.11 and irreducible cuspidal automorphic representations of G L(2, AQ ). Let N , k ∈ Z with N ≥ 1, and let χ (mod N ) be a Dirichlet character. Fix a Maass form f of type ν, weight k, and character χ for 0 (N ) as in Definition 3.5.7. Let f adelic be the adelic lift of f as in Definition 4.12.13. Define a vector space   N     m c R f adelic g · h   N ∈ N, m  ∈ Z, c ∈ C, , (5.4.1) V f :=  =1

where g ∈ G L(2, AQ ) and h  ∈ G L(2, Afinite ) · i ∞ (K ∞ ). The differential operator R  is defined by (5.2.11) and the action of R  on f adelic (g) with g = {g∞ , g2 , . . . } is given by the differential operator acting on g∞ . Note that by the results of Sections 5.2, 5.3, a spanning set for V f is given by adelic lifts of oldforms (with the notion suitably extended, as described above) associated to f and adelic lifts of the Maass forms of different weights (constructed by applying Maass raising and lowering operators to f and to the function h(z) = f (−z), and to the generalized oldforms obtained from these functions). Theorem 5.4.2 (Example of a cuspidal automorphic representation) Let N , k ∈ Z with N ≥ 1, and let χ (mod N ) be a Dirichlet character. Fix a Maass form f of type ν, weight k, and character χ for 0 (N ) as in Definition 3.5.7. Assume that f is a newform as defined in Section 3.11. Let f adelic be the adelic lift of f as in Definition 4.12.13. The vector space V f (defined in (5.4.1)) together with the actions (defined in Section 5.1)   πg : U (g) → linear maps V f → V f (action by differential operators) π K∞ : K ∞ → G L(V f ) (action by right translation) πfinite : G L(2, Afinite ) → G L(V f ) (action by right translation) define a cuspidal automorphic representation of G L(2, AQ ).

5.5 Admissible (g, K∞ ) × GL(2, A f inite )-modules

173

Proof It is clear that V f is a subspace of Aχidelic (G L(2, AQ )), which is preserved by the actions of G L(2, Afinite ), O(2, R), and U (g) on Aχidelic (G L(2, AQ )), given in Section 5.1. It was proved in Lemma 5.1.7 that the space Aχidelic (G L(2, AQ )), is a smooth (g, K ∞ )×G L(2, Afinite )-module, and it follows that the same is true of V f . The function f adelic is actually an adelic cusp form. This was proved earlier in Proposition 4.8.4 when f is even and of weight zero, and N = 1. The general case follows from a computation which is simliar to, but easier than, the proof of Theorem 4.13.3. It then follows from Lemma 5.1.13 that V f is contained in the space of cusp forms. This completes the proof. 

5.5 Admissible (g, K ∞ ) × G L(2, Afinite )-modules In Section 5.1 we introduced automorphic representations and described in some detail the sorts of objects that they are: (g, K ∞ ) × G L(2, Afinite )-modules as defined in Definition 5.1.5. In this section we will show that irreducible cuspidal automorphic representations, which were defined in Section 5.1 as (g, K ∞ ) × G L(2, Afinite )-modules, are admissible, under a mild technical hypothesis, which will be removed in Chapter 9. To prepare for the definition of an admissible (g, K ∞ ) × G L(2, Afinite )module, we prove a lemma about smooth (g, K ∞ ) × G L(2, Afinite )-modules, as defined in Definition 5.1.6. Lemma 5.5.1 Let V be a (g, K ∞ ) × G L(2, Afinite )-module with actions πg , π K∞ , and πfinite as in Definition 5.1.5. Assume that (π, V ) is smooth, as in Definition 5.1.6. For n ∈ Z let  

cos θ  Vn := v ∈ V  π K∞ − sin θ

sin θ cos θ

 . v = einθ · v,

 (∀θ ∈ R) . (5.5.2 )

For K a compact open subgroup of G L(2, Afinite ), and W any subspace of V, let 

  

 (5.5.3) W K := w ∈ W  πfinite k . w = w, (∀k ∈ K ) . A K Then every element of V is contained in n∈S Vn for some finite set S ⊂ Z and some compact open subgroup K ⊂ G L(2, Afinite ). Proof This follows easily from combining the explicit finiteness property given after Definition 5.1.4 with the definition of a smooth (g, K ∞ ) × G L(2, Afinite )-module given in Definition 5.1.6.  Definition 5.5.4 (Admissible (g, K ∞ ) × G L(2, Afinite )-module) Let V be a smooth (g, K ∞ ) × G L(2, Afinite )-module as in Definition 5.1.5. Then V

174

Automorphic representations for GL(2, AQ )

is said to be admissible if the space VnK , defined by (5.5.2) and (5.5.3) is finite dimensional for all n ∈ Z and all compact open subgroups K ⊂ G L(2, Afinite ). It is easy to see (or rather, follows easily from a very deep result!) that the vector spaces Aω (G L(2, AQ )), (ω a unitary Hecke character), defined as in Definition 5.1.1, are not admissible (g, K ∞ ) × G L(2, Afinite )-modules. For example, if f is any Maass form of weight 0 and level 1, then f adelic is an  K element of Aωtrivial (G L(2, A))0 finite , where K finite = p G L(2, Z p ) is the usual maximal compact subgroup of G L(2, Afinite ), and ωtrivial is the trivial character (which is identically equal to one). Given that there are infinitely many linearly independent Maass forms of weight zero and level one, it is immediately clear that the (g, K ∞ ) × G L(2, Afinite )-module Aωtrivial (G L(2, AQ )) is not admissible. Our next main goal will be to prove that an irreducible automorphic cuspidal representation is admissible. To prepare for doing this, we prove a few results which are of some independent interest. We first prove the following basic lemma. Lemma 5.5.5 (Irreducible automorphic representations and eigenvalues of the Laplacian) Let V be an irreducible automorphic representation, as defined in Definition 5.1.12. Let  denote the Casimir operator as defined in [Goldfeld, 2006; p. 50] (see also Lemma 5.2.6, above). Then there exists a complex number λ such that πg () . ξ = λ · ξ

(∀ξ ∈ V ).

Proof Take ξ ∈ V. Let W be the subspace of V spanned by the images of ξ under the action of Z (U (g)). By condition (4) of the definition of an adelic automorphic form given in Definition 4.7.6, the dimension of W is finite. Clearly,  maps W to W. Hence W has an eigenvector in W. Let λ be the corresponding eigenvalue and let Vλ be the λ-eigenspace of  in V. Since the action of  commutes with all the other actions, it follows at once that Vλ is a non-trivial invariant subspace. And so, because V is irreducible, Vλ = V.  Next, we give a construction which may be regarded as an inverse to the adelic lift given in Definition 4.12.13. To prepare, we need to analyze characters of K 0 (N )/K 1 (N ). Definition 5.5.6 (Characters of K 0 (N )/K 1 (N )) Let N be a positive integer. (1) (2) χidelic , and ( χidelic Let χ (1) and χ (2) be Dirichlet characters modulo N , and let ( be the characters of K 0 (N ) defined in Definition 4.12.12. Then we define a character of the group K 0 (N ) defined in (5.3.5) which is trivial on the group K 1 (N ) defined in (5.3.6) by setting

5.5 Admissible (g, K∞ ) × GL(2, A f inite )-modules       a b d b a b (1) (2) := ( χidelic ·( χidelic . χ (1) ,χ (2) c d c a c d

175

Lemma 5.5.7 Definition 5.5.6 provides a bijective map from the set of pairs of Dirichlet characters modulo N to the set of characters of K 0 (N ) which are trivial on K 1 (N ). That is, if  : K 0 (N )/K 1 (N ) → C× is a homomorphism, then there exist Dirichlet characters χ (1) , χ (2) modulo N such that  = χ (1) ,χ (2) . Proof A character of K 0 (N ) which is trivial on K 1 (N ) is the same thing as a character of the quotient group, K 0 (N )/K 1 (N ). During the proof of Proposition 5.3.4, it was shown that this group is isomorphic to [(Z/N Z)× ]2 . It follows at once that the set of characters is in one to one correspondence with the set of pairs of Dirichlet characters modulo N . The only issue is to show that the concrete map given in Definition 5.5.6 works. The map (χ (1) , χ (2) ) → χ (1) ,χ (2) . is easily seen to be injective. Since we already know that the two sets have the same number of elements, this completes the proof.  Lemma 5.5.8 Let  be a character of K 0 (N ) which is trivial on K 1 (N ). Let characters modulo N such χ (1) and χ (2) be the uniquely determined   Dirichlet a b (1) (2) that  = χ ,χ . Then, for any γ = c d ∈ 0 (N ), we have i finite (γ ) ∈ K 0 (N ),

and

   i finite (γ ) = χ (d)−1 ,

where χ = (χ (1) )−1 χ (2) . Here i finite is the diagonal embedding of G L(2, Q) into G L(2, Afinite ). Proof That i finite (γ ) ∈ K 0 (N ) isclear from  is  the definitions. In addition  it a b (1) (2) clear from the definitions that  i finite (γ ) = χ (a)χ (d). But for c d ∈ 0 (N ), we know that ad ≡ 1 (mod N ), which allows us to replace χ (1) (a) with (χ (1) )−1 (d).  Next we show how to obtain a classical form from an adelic form of a certain special type. Definition 5.5.9 (The classical automorphic form associated to an adelic automorphic form) Let φ be an adelic automorphic form as in Definition 4.7.6. Define φclassical : h → C by   1 y2 φclassical (x + i y) = φ i ∞ 0

x y− 2 1 y− 2 1

 .

Automorphic representations for GL(2, AQ )

176

Lemma 5.5.10 (Properties of φclassical ) Fix integers N , and k with N ≥ 1. Let  be a character of K 0 (N ) which is trivial on K 1 (N ). Let φ be an adelic automorphic form with central character ω, as in Definition 4.7.6, and assume that φ satisfies φ(gk) = (k) · φ(g), (∀g ∈ G L(2, AQ ), k ∈ K 0 (N )),    cos θ sin θ φ g = eikθ ·φ(g), (∀g ∈ G L(2, AQ ), θ ∈ [0, 2π )). − sin θ cos θ πg () . φ(g) = λ · φ(g), πg (D Z ) . φ(g) = μ · φ(g),

(∀g ∈ G L(2, AQ )), (∀g ∈ G L(2, AQ )),

(5.5.11)

for some λ, μ ∈ C, where    is the Casimir, and D Z is the differential operator 10 corresponding to Z = 0 1 ∈ gl(2, R). Then φclassical satisfies 

φclassical



   γ (z) = χ (d)φ (z), classical k

for all γ =

a c

b d

 ∈ 0 (N ), z ∈ h, (5.5.12)

where χ is the Dirichlet character modulo N , determined by  as in Lemma 5.5.8. In addition φclassical satisfies   λ μ2 · φclassical (z), k . φclassical (z) = − + 2 4

(∀z ∈ h),

(5.5.13)

where k is the weight k Laplacian as in Definition 3.5.3. Furthermore, the map φ → φclassical is injective on φ satisfying (5.5.11), for each fixed . Remark The map φ → φclassical is not injective if  is allowed to vary, or if the central character is not fixed.   Proof Take z = x + i y ∈ h and γ = ac db ∈ S L(2, R). Write  γ·

1

y2 0

x y− 2 1 y− 2 1



 =

1

v2 0

uv − 2 1 v− 2 1

  cos θ · − sin θ

sin θ cos θ

 ,

for some u ∈ R, v > 0, θ ∈ [0, 2π ). A routine calculation reveals that u+iv = az+b and eiθ = |cz+d| . After these observations, identity (5.5.12) follows cz+d cz+d easily from Lemma 5.5.8. To prove (5.5.13), we again compare Definitions 3.9.1 and 3.9.2, and (4.12.2) with the formulae for the operators R and L in coordinates which were written out during the interlude, deducing this time that (πg (R) . φ)classical = Rk . φclassical ,

(πg (L) . φ)classical = L k . φclassical ,

5.5 Admissible (g, K∞ ) × GL(2, A f inite )-modules

177

for any φ satisfying (5.5.11). Using Lemma 5.2.6, and keeping in mind that πg (D H ) acts by the scalar ik on any φ satisfying (5.5.11), we find that (πg () . φ)classical =

  μ2 . φclassical . −2k + 2

(5.5.14)

Identity (5.5.13) follows.  Next, we will show that an irreducible cuspidal representation of G L(2, AQ ) is admissible, under the additional condition that it is assumed to be a sub space of the space of all cuspidal automorphic forms (as opposed to a more general sub quotient), so that its elements are functions, rather than elements of a quotient, i.e., cosets. In fact, this restriction is harmless: in Chapter 9 we will show that every cuspidal automorphic representation with central character ω is isomorphic to a subspace of Aω (G L(2, AQ )). Theorem 5.5.15 (Irreducible cuspidal automorphic representations are admissible) Let V be an irreducible cuspidal automorphic representation, as defined in Definition 5.1.14. Assume that V is a subspace of the space of all cuspidal automorphic forms with central character ω. Then V is an admissible (g, K ∞ ) × G L(2, Afinite )-module, as defined in Definition 5.5.4. Proof We assume that V is a subspace of cuspidal elements, contained in the space Aω (G L(2, AQ )), of all adelic automorphic forms with central character ω, for some unitary Hecke character ω. We proved in Theorem 5.4.1 that the vector space Aω (G L(2, AQ )) is a smooth (g, K ∞ ) × G L(2, Afinite )-module. It follows that the same is true of V. We wish to show that, for any two integers N , k (with N ≥ 1), the space K (N ) definedby (5.5.2) Vk  and (5.5.3) is finite dimensional. First, note that the action of πfinite i finite

N 0 0 1

2

maps VkK (N ) to VkK 1 (N ) . Hence the two spaces

have the same dimension. Next, for each  : K 0 (N 2 ) → C× , trivial on K 1 (N 2 ), let 

 2  Vk := v ∈ VnK 1 (N )  πfinite (k) . v = (k) · v, (∀ k ∈ K 0 (N 2 )) . It follows from Lemma 5.5.7 that the set of such characters  is finite, and it follows from finite Fourier expansion (or orthogonality of characters) on A 2  K 0 (N 2 )/K 1 (N 2 ) that VkK 1 (N ) ⊂  Vk . It follows from Lemma 5.5.5 that there exists a complex number λ such that πg () . φ = λ · φ for all φ ∈ V, and it follows from Definitions 5.2.2 and 4.7.7 that there exists a second complex number μ such that πg (D Z ) . φ = μ · φ for all φ ∈ V. Finally, it follows from Lemma 5.5.10 that the map φ → φclassical sends the space Vk injectively to the space of classical Maass cusp forms of weight k and Laplace eigenvalue

178 Automorphic representations for GL(2, AQ )   2 − λ2 + μ4 for 0 (N 2 ) having a character χ which depends on  as in Lemma 5.5.10. This space is finite dimensional by Theorem 3.12.1. This completes the proof.  Remark The above arguments actually show the following: for any fixed unitary character ω, and any fixed complex number λ, the space of all cusp forms of central character ω which satisfy πg () . φ = λφ is an admissible (g, K ∞ )-module.

Exercises for Chapter 5 5.1 Let ω and θ be unitary Hecke characters, and let φ ∈ Aω (G L(2, AQ )) and ψ ∈ Aθ (G L(2, AQ )) be adelic automorphic forms. (a) Prove that φ · ψ ∈ Aωθ (G L(2, AQ )). Conclude that A1 (G L(2, AQ )) is a commutative ring with identity, where 1 is the trivial Hecke character. α be the Fourier coefficients of (b) As in Proposition 4.9.5, let  φα and ψ φ and ψ, respectively. Prove that the Fourier coefficients of φ · ψ satisfy  γ −α φα · ψ (γ ∈ Q). (φ · ψ)γ = α∈Q −1 ) is a subgroup of finite index in K for 5.2 Verify that K ∩ (afinite K afinite each afinite ∈ G L(2, Afinite ). Conclude that if φ : G L(2, AQ ) → C is a right K -finite function, then so is the map g → φ(gafinite ). Hint: Use Exercise 4.10.

5.3 Let ω be a unitary Hecke character and let φ ∈ Aω (G L(2, AQ )) be an automorphic form. Verify that the actions πg and π K ∞ satisfy  1 πg (Dα ).φ = lim π K ∞ (exp(tα)).φ − φ t→0 t for every α in the Lie algebra k of K ∞ . 5.4 Does the space of global Whittaker functions (Definition 4.10.9) with the natural actions form a (g, K ∞ ) × G L(2, Afinite )-module? 5.5 Show that the (g, K ∞ ) × G L(2, Afinite )-module generated by the function φ(g) = log | det(g)|A is two-dimensional. Is it irreducible? (cf. Exercise 4.16.) 5.6 This exercise asks you to prove one of the elementary facts used in the proof of Lemma 5.1.7. Let X be a nonempty set and V a finitedimensional complex vector space of functions f : X → C. Let n = dim(V ).

Exercises for Chapter 5

179

(a) Show that |X | ≥ n. (b) Prove there exist distinct points x1 , . . . , xn ∈ X with the following property: if f ∈ V satisfies f (x1 ) = · · · = f (xn ) = 0, then f ≡ 0. 5.7 Suppose L : V → V is an intertwining map of (g, K ∞ )-modules. If L is an isomorphism of complex vector spaces, prove that L −1 : V → V is also an intertwining map. 5.8 Let ω be a unitary Hecke character, and let V be an automorphic representation with central character ω. Prove that any subquotient of V is also an automorphic representation with central character ω. Conclude that any subquotient of Aω (G L(2, AQ )) is an automorphic representation with central character ω. 5.9 Let V be a complex vector space endowed with an action π K ∞ of the group K ∞ . Show that the following two properties are equivalent: (1) For each v ∈ V , the subspace spanned by {π K ∞ (k).v | k ∈ K ∞ } is finite-dimensional. (2) For each v ∈ V there exist integers M < N , complex numbers c , N and vectors v ∈ V (with M ≤  ≤ N ) such that v = c v , and k=M

 πK∞

cos θ − sin θ

5.10 Recall that Z = D H = ∂θ∂ .

sin θ cos θ

1 0 01

 . v = eiθ v ,

and H =



0 1 . −1 0

(M ≤  < N , θ ∈ R).

Verify that D Z = r ·

∂ ∂r

and

5.11 Define f : G L(2, R) → R by the formula  f

1 0

x 1



y 0

0 1



r 0 0 r



−1 0 0 1



j κθ

= x yr sin θ,

 θ sin θ  where x, y, r ∈ R, y, r > 0, j = 0 or 1, and κθ = −cos with θ ∈ sin θ cos θ [0, 2π ). Verify directly that f is smooth and that D H f (g) = x yr cos θ and D Z f = f . 5.12 If R and L are the usual differential operators, verify that   ∂ ∂ 1 ∂ r ∂ +y + + , R = e2iθ i y ∂x ∂ y 2i ∂θ 2 ∂r L=e

−2iθ

  ∂ ∂ 1 ∂ r ∂ −i y +y − + . ∂x ∂y 2i ∂θ 2 ∂r

180

Automorphic representations for GL(2, AQ ) Also, show that πg (R). f adelic = (Rk f )adelic for any Maass form of weight k and any character and level.

5.13 Let V be a (g, K ∞ )-module. Prove that the actions πg and π K ∞ satisfy πg (D Z )π K ∞ (k) = π K ∞ (k)πg (D Z ),

(k ∈ O(2, R)),

πg (D H )π K ∞ (k) = π K ∞ (k)πg (D H ),

(k ∈ S O(2, R)),

πg (D H )π K ∞ (δ1 ) = −π K ∞ (δ1 )πg (D H ), πg (D).π K ∞ (k) = π K ∞ (k).πg (D), where δ1 =

(k ∈ O(2, R), D ∈ Z (U (g)),

 −1 0  . 0 1

5.14 Let N , k ∈ Z with N ≥ 1, and let χ (mod N ) be a Dirichlet character. Fix a Maass form f of type ν, weight k, and character χ for 0 (N ) as in Definition 3.5.7, and construct the adelic lift f adelic as in Definition 4.12.13. Prove that πg (D Z ). f adelic = 0 and that πg (). f adelic = λ f adelic for some scalar λ, where  is the Casimir operator (see Lemma 5.2.6). 5.15 Let ω be a unitary Hecke character and let φ = ω ◦ det be the induced adelic automorphic form on G L(2, AQ ) with character ω2 as in Exercise 4.16. Define a 1-dimensional complex vector space V = C.φ. Show that V is an admissible automorphic representation with central character ω2 . 5.16* Give an example of an adelic automorphic form φ such that the function g → φ(g · i ∞ (a)) is not automorphic for some a ∈ G L(2, R). 5.17 In this exercise we compute the center of the enveloping algebra of g = gl(2, C). (a) Prove that D Z commutes with all differential operators Dα . (b) Check that the Casimir operator  = D11 ◦ D11 + D12 ◦ D21 + D21 ◦ D12 + D22 ◦ D22 commutes with all other differential operators. Conclude that C.D Z + C. ⊆ Z (U (g)). (c) Prove that D H ◦ R  = R  ◦ D H + 2iR  for all  ∈ Z. (d) Proposition 5.2.10 asserts that any element of U (g) can be written as a linear combination of terms of the form R  ◦ D aH ◦ b ◦ D cZ with a, b, c non-negative integers and  ∈ Z. This representation is unique provided we collect together all terms with the same

Exercises for Chapter 5

181

ordered set of exponents (, a, b, c); a relatively simple proof is given in Exercise 7.11. Use this fact to conclude that Z (U (g)) is the polynomial algebra C[D Z , ]. 5.18 Prove that√ for any (g, K ∞ )-module, the differential operator D H acts by 0 1 . the scalar −1 · n on the space Vn defined in (5.5.2). Here H = −1 0 5.19 Fix an integer M ≥ 1, and write its prime factorization as M =  fp  p . Recall the subgroups of K finite = p G L(2, Z p ) ⊂ G L(2, AQ ) defined by  K 0 (M) = k = {I2 , k2 , . . . , k p , . . . } ∈ K finite |   ∗ ∗  kp ≡ 0 ∗ mod p f p ∀ p ,  K 1 (M) = k = {I2 , k2 , . . . , k p , . . . } ∈ K finite |      k p ≡ 10 ∗1 mod p f p ∀ p . Prove that K 1 (M) properly contains the commutator subgroup of K 0 (M). (Recall that for a group G, the commutator subgroup [G, G] is generated by the elements of the form aba −1 b−1 for a, b ∈ G.)  5.20 For a natural number N = p f p ≥ 1, recall that K (N ) ⊂ G L(2, Afinite ) is defined by  

 K (N ) = k = {I2 , k2 , . . . , k p , . . . } ∈ K finite  k p ≡ I2 (mod p f p ) ∀ p (a) Prove that K (N ) is a compact open subgroup of G L(2, Afinite ). (b) Prove that every open neighborhood of the identity in G L(2, Afinite ) contains K (N ) for some N , so that the collection {K (N )} N ≥1 forms a base of neighborhoods of the identity for the topology on G L(2, Afinite ). 5.21 Suppose χ is a Dirichlet character (mod N ), s is a real number, and let ω = χidelic | · |is AQ . Let φ = ω ◦ det be the induced adelic automorphic form on G L(2, AQ ) as in Exercise 4.16. (a) Show that φ is a character on K 0 (N ) and that it is trivial on K 1 (N ). (See the previous exercise for the definitions of K 0 (N ) and K 1 (N ).) (b) Show that πg ()φ = −2s 2 φ and πg (D Z )φ = 2isφ. (c) Describe φclassical . 5.22* Let R be a commutative ring with 1 such that G L(2, R) is a topological group. (For example, one could take R = R, Q p , or AQ .) A subgroup K ⊂ G L(2, R) is called a maximal compact subgroup if K is compact and if no other compact subgroup properly contains K . One says that

182

Automorphic representations for GL(2, AQ ) a maximal compact subgroup K is unique up to conjugation if for any maximal compact subgroup K ⊂ G L(2, R), there exists a matrix A ∈ G L(2, R) such that K = AK A−1 . (a) Prove that O(2, R) is a maximal compact subgroup of G L(2, R), and that it is unique up to conjugation. (b) Prove that G L(2, Z p ) is a maximal compact subgroup of G L(2, Q p ), and that it is unique up to conjugation.  (c) Deduce from the previous two parts that K = O(2, R) · p G L(2, Z p ) is a maximal compact subgroup of G L(2, AQ ), and that it is unique up to conjugation.

6 Theory of admissible representations of G L(2, Q p )

6.0 Short roadmap to chapter 6 This is a long chapter. It is imperative to read the basic definitions given in Sections 6.1, 6.2, and then familiarize oneself with principal series which are defined in Definition 6.5.3. If the reader would like to obtain the classification of the smooth irreducible representations of G L(2, Q p ) as quickly as possible, it is recommended to jump directly to Sections 6.11, 6.12, and 6.13. The key to the classification is Theorem 6.13.4.

6.1 Admissible representations of G L(2, Q p ) Fix a finite prime p and a complex vector space V . A representation of G L(2, Q p ) on V is a homomorphism π : G L(2, Q p ) → G L(V ). Here π (g) . v denotes the action of π (g) on the vector v. As in Definition 2.5.1, we call V the space of π and refer to the ordered pair (π, V ) as a representation of G L(2, Q p ). We shall also refer to (π, V ) as a local representation. Definition 6.1.1 (Smooth representation) Let V be a complex vector space and let (π, V ) be a representation of G L(2, Q p ). We say (π, V ) is smooth if for every v ∈ V the function         a b a b a b → π . v, for all ∈ G L(2, Q p ) , c d c d c d is smooth, i.e., is a locally constant function. Remark Recall from Definition 4.7.2 that a smooth function on G L(2, Q p ) is just a locally constant function. This is the motivation for Definition 6.1.1. 183

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Definition 6.1.2 (Alternative definition of smooth representation) Let V be a complex vector space and let (π, V ) be a representation of G L(2, Q p ). We shall say (π, V ) is smooth if for every v ∈ V there exists a compact open subgroup K ⊂ G L(2, Q p ) such that π (k) . v = v for all k ∈ K . Definition 6.1.2 can be made even more explicit. Let I2 denote the 2 × 2 identity matrix and let Mat(2, Z p ) denote the set of all 2 × 2 matrices with coefficients in Z p . Every compact open subset of G L(2, Q p ) is a finite disjoint union of cosets of the subgroup K n = {k ∈ G L(2, Z p ) | k − I2 ∈ p n · Mat (2, Z p )},

(n = 0, 1, 2, . . . ) (6.1.3) for some n. Then (π, V ) is smooth if for every v ∈ V there exists a nonnegative integer n such that π (k) . v = v for all k ∈ K n . Definition 6.1.4 (Admissible representation) Let V be a complex vector space and let (π, V ) be a representation of G L(2, Q p ). We say (π, V ) is admissible if it is smooth (Definition 6.1.1) and, in addition, the space    V K n := v ∈ V  π (k) . v = v, for all k ∈ K n , is finite dimensional for each integer n ≥ 0, where K n is given in (6.1.3). Definition 6.1.5 (Irreducible representation of G L(2, Q p )) Fix a prime p and a complex vector space V . A representation π : G L(2, Q p ) → G L(V ) is irreducible if it is non-zero and has no non-trivial invariant subspace V ⊂ V, V =/ V. It turns out that the finite dimensional irreducible smooth representations of G L(2, Q p ) are not very interesting. Each is a one-dimensional space on which the elements of G L(2, Q p ) act by scalars. This is the content of Theorem 6.1.7. The proof of this theorem requires the following lemma.     1 0 Lemma 6.1.6 The two matrices 10 x1 , y 1 , with x, y ∈ Q p generate S L(2, Q p ).   Proof First of all, every a0 db ∈ S L(2, Q p ) can be written in the form 

a 0

b d



 =

a 0

0 a −1



1 a −1 b 0 1

On the other hand, if c =/ 0 we have      a b 1 c−1 a 0 1 −c = c d 0 1 −1 0 0



0 −c−1

.



1 c−1 d 0 1

 .

6.1 Admissible representations of GL(2, Qp )

185

  The above two identities imply that S L(2, Q p ) is generated by −10 01 and the       y 0 matrices 10 x1 , x1 10 , 0 y −1 , with x, x ∈ Q p and y ∈ Q×p . The proof of the lemma is completed upon noting that       1 0 1 1 1 0 y 0 = y −1 − 1 1 0 1 y−1 1 0 y −1   1 −y −1 , (∀y ∈ Q×p ), · 0 1       0 1 1 0 1 1 1 0 = .  −1 0 −1 1 0 1 −1 1 Theorem 6.1.7 (Characterization of the finite dimensional irreducible smooth representations of G L(2, Q p )) Let V be a finite dimensional nonzero complex vector space. Let (π, V ) be a smooth irreducible representation of G L(2, 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(2, Q p ), v ∈ V. Proof Since π is smooth and V is finite dimensional, it follows that the kernel of π is a non-trivial open subgroup H ⊂ G L(2, Q p ). It is also normal. Now, let x∈ Qp be arbitrary. Choose b ∈ Q p so that |bx| p is sufficiently small so that

1 bx 0 1



is in the kernel of π. Then

1 0

x 1



 =

b 0

0 1

−1 

1 bx 0 1



b 0

0 1



  1 0 must also be in the kernel of π. In a similar manner one shows that y 1 is in the kernel of π for all y ∈ Q p . It immediately follows from Lemma 6.1.6 that the kernel of π contains S L(2, Q p ). Note that g1 g2 g1−1 g2−1 ∈ S L(2, Q p ) for all g1 , g2 ∈ G L(2, Q p ). Thus π (g1 ) π (g2 ) = π (g2 ) π (g1 ) for all g1 , g2 ∈ G L(2, Q p ). Now, let g ∈ G L(2, Q p ) be arbitrary. Since V is finite dimensional, it follows that the linear action of π (g) on V has an eigenvector v with corresponding eigenvalue λg ∈ C such that π (g) . v = λg · v. But π (g) commutes with π (g ) for all g ∈ G L(2, Q p ). Thus     π (g) . π (g ) . v = π (g ) . π (g) . v = λg π (g ) . v, so every π (g ) .v is also an eigenvector. Thus the λg -eigenspace for π (g) is a non-trivial invariant subspace. It follows from the irreducibility of (π, V ) that this eigenspace is all of V. But g was arbitrary, so every element of G L(2, Q p ) acts by a scalar on all of V. Using again the irreducibility of (π, V ), we deduce that the dimension of V is one. We leave it to the reader to show that the

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function which sends g to the eigenvalue λg is a character, i.e., that π (g) . v = ω (detg) · v for all g ∈ G L(2, Q p ), v ∈ V, and some multiplicative character ω : Q×p → C× .  The next lemma is also called Dixmier’s lemma. Lemma 6.1.8 (Schur’s lemma for irreducible smooth representations) Let (π, V ) be an irreducible smooth representation of G L(2, Q p ) as in Definition 6.1.1. Let T : V → V be an intertwining map as in Definition 2.5.3. Then there exists a constant c ∈ C such that T . v = c · v for all v ∈ V. Proof We first show that V must have countable dimension. Choose any nonzero vector v ∈ V. It follows from the fact that V is irreducible that    V = Span π (g) . v  g ∈ G L(2, Q p ) .   Since π is smooth π (gk) . v | k ∈ K = G L(2, Z p ) is a finite set for any fixed element g ∈ G L(2, Q p ). The set G L(2, Q p )/K is countable by Proposition 4.2.1 (the Iwasawa decomposition). It follows that V is spanned by a countable set. It follows from the definition of an intertwining map given in Definition 2.5.3 that T . (π (g) . v) = π (g) . (T . v) for all g ∈ G L(2, Q p ), v ∈ V . Let I denote the identity transformation on V , i.e., I . v = v for all v ∈ V. Then     (T − cI ) . π (g) . v = π (g) . (T − cI ) . v , (∀ g ∈ G L(2, Q p ), v ∈ V ). Now, assume T = / c · I for every c ∈ C. Itfollows that T − cI maps the space V = Span π (g) . v  g ∈ G L(2, Q p ) to itself. By the irreducibility of V , the image of T − c · I is V and the kernel is {0}. Thus T − c · I is a bijective map and the inverse map (T − c · I )−1 is well defined and bijective for every c ∈ C. Fix v ∈ V. Because V has a countable basis and C is uncountable, the vectors (T − c · I )−1 . v (with c ∈ C varying) are not linearly independent. Thus, there exist n ∈ N, c1 , . . . , cn , a1 , . . . , an ∈ C such that c1 , . . . cn are all distinct, a1 , . . . , an are all non-zero and n

ai (T − ci · I )−1 . v = 0.

i=1

Write

n i=1

ai Q(x) . = n  x − ci (x − ci ) i=1

(6.1.9)

6.1 Admissible representations of GL(2, Qp )

187

Observe that the polynomial Q is non-zero. Write it as Q(x) =

d

bi x i ,

  for bi ∈ C, (i = 1, . . . d) with bd =/ 0 .

i=0

Define an operator Q(T ) in the usual way: Q(T ) . v =

d i=0

bi · (T ◦ . . . ◦ T ) . v. !" # i times

(Here ◦ denotes composition of operators.) Applying (T − c1 · I ) ◦ . . . ◦ (T − cn · I ) to both sides of (6.1.9), yields Q(T ) . v = 0. Now factor Q(x) as Q(x) = bd ·

d 

(x − αi )

i=1

for some α1 , . . . , αd ∈ C. Then the operator (T − αi · I ) is noninjective for some αi (with 1 ≤ i ≤ d). This is a contradiction.  An immediate consequence of Dixmier’s lemma is the existence of a central character for every irreducible smooth representation of G L(2, Q p ). Proposition 6.1.10 (Central character) Let V be a complex vector space and let (π, V ) be an irreducible smooth representation of G L(2, Q p ). Then there exists a unique multiplicative character ωπ : Q×p → C× such that     a 0 π . v = ωπ (a) · v, ∀a ∈ Q×p , v ∈ V . 0 a The character ωπ is called the central character associated to (π, V ).   Proof Let z = a0 a0 (with a ∈ Q×p ) be in the center of G L(2, Q p ). Then π (z) . (π (g) . v) = π (g) . (π (z) . v) for all g ∈ G L(2, Q p ), v ∈ V. Thus π (z) : V → V is an intertwining map as in Definition 2.5.3. By Schur’s Lemma 6.1.8 there exists a constant c(a) ∈ C such that π (z) = c(a) · I where  I . v = v (for all v ∈ V ) is the identity transformation. Further, if z = a0 a0 ,   z = a0 a0 are any two elements in the center of G L(2, Q p ), then since π (zz ) . v = π (z) . (π (z ) . v) for all v ∈ V , it follows that c(aa ) = c(a) c(a ) for all a, a ∈ Q×p . We define ωπ (z) := c(a), and leave it to the reader to check continuity.  Schur’s lemma for irreducible smooth representations can be used to prove the following surprising result [Bernstein-Zelevinsky, 1976]. The following

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proof, shown to us by Jacquet, is exemplary of the many extraordinarily clever techniques that have come to pervade this field. We do not give every detail of the proof since we will not need this result. For definitions of Haar measure and integration on the space G L(2, Q p ) see Section 6.9. Theorem 6.1.11 (Smooth + irreducible =⇒ admissible) Let (π, V ) be a smooth irreducible representation of G L(2, Q p ). Then (π, V ) must be admissible. Proof Define the Bruhat-Schwartz space:    S := f : G L(2, Q p ) → C  f is locally constant with compact support . For f ∈ S we define a linear map π ( f ) : V → V given by  π ( f ) . v := π (g) . v f (g) d × g, (∀v ∈ V ).

(6.1.12)

G L(2,Q p )

Claim 1: Fix v ∈ V with v =/ 0. Then for any other vector v ∈ V, there exists f ∈ S such that v = π ( f ) . v. The proof of claim 1 essentially follows from the fact that V is irreducible. Indeed, by a simple change of variables in the integral in (6.1.12), the vector space Vv := {π ( f ) . v | f ∈ S} ⊆ V is invariant under π (h) for any h ∈ G L(2, Q p ). Also the space Vv =/ {0} because, by the smoothness of π , we see that there exists a compact set U such that π (u) . v = v for all u ∈ U. 1 π (g) . v d × g, so Vv is not trivial. By the irreducibility of Hence v = Vol(U ) U V , we must have Vv = V which establishes the claim. Fix an element τ ∈ Q×p which is not the square of another element of √ × Q p . Let E = Q p ( τ ) be the corresponding quadratic extension. Define the quadratic torus    5 a bτ  2 2 Tτ :=  a, b ∈ Q p , a − b τ =/ 0 . b a √ √  Clearly Tτ is a subgroup of G L(2, Q p ). Let t = 1τ − 1 τ . Since     a bτ a + bτ 0√ −1 t = , t b a 0 a−b τ 

    α 0 α ∈ E where α denotes conjugation in E. it follows that t−1 Tτ t =  0 α We may define a multiplicative character χ : Tτ → C by   √ a bτ χ = χ (a + b τ ), b a √ where χ is a character of Q p ( τ ).

6.1 Admissible representations of GL(2, Qp ) 189

     r 0 × r ∈ Q Let Z = denote the center of G L(2, Q p ). We claim that  p 0r Tτ /Z is compact. It suffices to do this under the additional hypothesis that τ is a unit. Observe that if p > 2, then an element of Z×p is a square if and only if its image in (Z p / pZ p ) = (Z/ pZ) is. Now, every t ∈ Tτ can be written in the form     n a bτ 0 p · k, k= , (6.1.13) t= b a 0 pn with a, b ∈ Z p and at least one of a, b ∈ Z×p . If exactly one of a, b is a unit, it follows at once that the determinant is a unit, i.e., that k ∈ G L(2, Z p ). If a and b are both units, then we use the fact that τ is not equivalent to a square in (Z/ pZ) to deduce that a 2 − b2 τ is a unit. This completes the proof that Tτ /Z is compact if p > 2. We leave it to the reader to do the case p = 2. Next, define the spaces: V (χ ) := {v ∈ V | π (t) . v = χ (t) · v, (t ∈ Tτ )} ,   S(χ ) := f ∈ S | f (t1 gt2 ) = χ −1 (t1 t2 ) f (g), (t1 , t2 ∈ Tτ , g ∈ G L(2, Q p )) . Claim 2: Fix v ∈ V (χ ) with v =/ 0. Then for any other vector v ∈ V (χ ), there exists f ∈ S(χ −1 ) such that v = π ( f ) . v. By Claim 1, we know that v = π ( f ) . v for some f ∈ S. We must show that it is possible to find some (possibly other) F ∈ S(χ −1 ) where v = π (F) . v. It is possible to define an invariant measure d × t on the torus Tτ , normalized so that Tτ /Z has measure one. We then find that v, v ∈ V (χ ) implies that  

−1 × π (t) . v χ (t) d t, v= π (t) . v χ −1 (t) d × t. v = Tτ /Z

Tτ /Z

Combining the above with the identity v = π ( f ) . v implies that    π (t1 gt2 ) . v χ −1 (t1 )χ −1 (t2 ) f (g) d × t2 d × g d × t1 v = 

Tτ /Z

G L(2,Q p )

Tτ /Z

:

π (g) . v

= G L(2,Q p )

 Tτ /Z

Tτ /Z

  χ −1 (t1 t2 ) f t1−1 gt2−1 d × t1 d × t2

;

d×g

= π (F) . v, where F(g) =



 Tτ /Z

Tτ /Z

  χ −1 (t1 t2 ) f t1−1 gt2−1 d × t1 d × t2 ∈ S(χ −1 ).

Next, we note that for f 1 , f 2 ∈ S(χ −1 ) we may define the convolution  f 1 (h −1 g) f 2 (h) d × h, (g ∈ G L(2, Q p )), (6.1.14) ( f 1 ∗ f 2 )(g) := G L(2,Q p )

Theory of admissible representations of GL(2, Qp )

190

which clearly satisfies 

f 1 (h −1 t1 gt2 ) f 2 (h) d × h

( f 1 ∗ f 2 ) (t1 gt2 ) = G L(2,Q p )



f 1 (h −1 gt2 ) f 2 (t1 h) d × h = χ (t1 t2 )−1 f 1 ∗ f 2 (g),

= G L(2,Q p )

for all t1 , t2 ∈ Tτ , g ∈ G L(2, Q p ). Hence f 1 ∗ f 2 ∈ S(χ −1 ). Claim 3: We have f 1 ∗ f 2 = f 2 ∗ f 1 for all f 1 , f 2 ∈ S(χ −1 ). Consider the map ι : G L(2, Q p ) → G L(2, Q p ) defined by ι

 g :=

τ 0

0 1



 ·

t

g·

τ 0

0 1

−1

,

t for g ∈ G L(2, Q p ), where of the matrix g. It is   g denotes the  transpose  a bτ a bτ a bτ ι easy to see that b a = b a for all b a ∈ Tτ . It can be shown that the double coset space Tτ \G L(2, Q p )/Tτ is invariant under the action of ι. Consequently ι

f (t1 gt2 ) := f



ι

   (t1 gt2 ) = f t1 · ι g · t2 = f (t1 gt2 ) = χ (t1 t2 )−1 f (g),

for all t1 , t2 ∈ Tτ , g ∈ G L(2, Q p ), and f ∈ S(χ −1 ). A brute force computation shows that ι g1 ι g2 = ι g2 ι g1 for all g1 , g2 ∈ G L(2, Q p ). This immediately implies that ι f 1 ∗ f 2 = ι f 2 ∗ ι f 1 for all f 1 , f 2 ∈ S(χ −1 ). Thus, Claim 3 is proved. It is easy to see that π ( f 1 ∗ f 2 ) = π ( f 1 ) . π ( f 2 ) for all f 1 , f 2 ∈ S(χ −1 ). We may then define the commutative convolution algebra A(χ ) of all operators π ( f ) (as defined in (6.1.12)) with f ∈ S(χ −1 ). The algebra A(χ ) acts on V (χ ) and by Schur’s Lemma 6.1.8, the operators π ( f ) must be scalar. One concludes that dim (V (χ )) = 1. Finally, we want to use the fact that dim (V (χ )) = 1 to prove that (π, V ) is √ admissible. We take τ to be a unit in Q×p so that the extension E = Q p ( τ ) is unramified. We also assume p =/ 2. For p = 2 the proof will require some modifications. It follows from (6.1.13) that G L(2, Z p ) · Z  T with T = Z · (Tτ ∩ G L(2, Z p )). Now, let K be a compact open subgroup of G L(2, Z p ). Con

sider V K . Clearly V K is invariant under the center Z . Further, T normalizes K because G L(2, Z p ) normalizes every K and T = Z · (Tτ ∩ G L(2, Z p )).

Thus V K is invariant under the action of T .

6.1 Admissible representations of GL(2, Qp )

Claim 4: V K =

A χ

191

 V (χ ) ∩ V K .

This will follow from the fact that Tτ /Z is compact in conjunction with the introduction of a certain projection operator. For every character χ of Tτ , we

define a projection operator P(χ ) : V K → V K ∩ V (χ ), given by  π (t) . v χ (t)−1 dt. P(χ ) . v := Tτ /Z

It is easy to see that P(χ ) . P(χ ) = P(χ ) and P(χ ) . P(χ ) = 0 if χ =/ χ . To

prove Claim 4 it is enough to show that every v ∈ V K can be expressed in the form v = χ P(χ ) . v, or equivalently, that λ(v0 ) = 0 for every linear form

λ : V K → C, where v0 = v − χ P(χ ) . v. Each linear form λ : Tτ → C determines a function f λ : Tτ → C defined by f λ (t) := λ(π (t) . v) for t ∈ Tτ . Let ωπ be the central character of π as in Proposition 6.1.10. The function f λ satisfies: f (t z) = ωπ (z) f (t),

(∀t ∈ Tτ , ∀z ∈ Z ),

(∀t ∈ Tτ , ∀t ∈ Tτ ∩ K ).  Since P(χ ) . v0 = 0 for any character χ of Tτ , it follows that Tτ /Z f λ (t)χ (t)−1 dt = 0 for every character χ of T . By standard Fourier theory on the torus Tτ it follows that v0 = 0.

To complete the proof of Theorem 6.1.11, we note that Tτ acts on V K with central character ωπ . Also the action is trivial on T ∩ K . But there are only finitely many characters χ of the torus Tτ for which χ restricted to Z is a central character and χ restricted to T ∩ K is trivial. This is because

(Tτ /Z ) (T ∩K ) is a finite set. Thus, in the decomposition of Claim 4, there are only finitely many characters. Further, we have already shown that each V (χ )

is one-dimensional. Therefore, V K must be finite dimensional and (π, V ) is admissible.  f (tt ) = f (t),

Definition 6.1.15 (Finitely generated representation) Let (π, V ) denote a representation of G L(2, Q p ). We say (π, V ) is finitely generated if there exists a finite set: v1 , v2 , . . . , vr ∈ V such that every v ∈ V can be written in the form v=

ni r

ci j π (gi j ) . vi

i=1 j=1

for ci j ∈ C, gi j ∈ G L(2, Q p ), ( j = 1, 2, . . . n i , i = 1, 2, . . . , r ). Proposition 6.1.16 (Finitely generated smooth representations have an irreducible quotient) Let (π, V ) be a smooth representation of G = G L(2, Q p ) as in Definition 6.1.1. We also refer to V as a G-module. If (π, V )

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Theory of admissible representations of GL(2, Qp )

is finitely generated then there exists a proper G-submodule V ⊂ V such that V /V is irreducible and smooth. Further, V /V is called a quotient of the representation (π, V ). Proof Fix E = {e1 , e2 , . . . , em } to be a finite set of generators of V . The proof that a finitely generated smooth representation has an irreducible quotient follows from Zorn’s lemma. We consider the set of proper G-submodules of V, partially ordered by inclusion. It follows easily from the definitions that V /V is irreducible iff V is a maximal element of this set. Let V1 ⊂ V2 ⊂ V3 ⊂ · · · be an ascending chain of proper G-submodules, i.e., each Vi ⊂ V (i = 1, 2, 3, . . . ) is invariant under the action of π . Then we claim that the union  i Vi is still a proper G-submodule. If not, then each ei , (1 ≤ i ≤ m) must lie in some Vi . Let M be the maximum of the i . Then E ⊂ VM and V = VM , a contradiction.  The proof of Proposition 6.1.16 easily generalizes as follows. Proposition 6.1.17 Let (π, V ) denote a smooth representation of some subgroup H ⊂ G L(2, Q p ). If V is finitely generated as an H -module, then there exists a proper H -submodule V ⊂ V such that V /V is irreducible and smooth.

6.2 Ramified versus unramified Fix a prime p. We would like to classify the admissible representations of G L(2, Q p ) as defined in Definition 6.1.4. We first introduce the important distinction between ramified and unramified representations of G L(2, Q p ). Here are the definitions. Definition 6.2.1 (Unramified representation) Fix a prime p. A representation (π, V ) of G L(2, Q p ), as in Definition 6.1.4, is termed unramified if there exists a G L(2, Z p ) fixed vector v ◦ ∈ V . Remark The G L(2, Z p )-fixed vector v ◦ has the property that π (k) . v ◦ = v ◦ for all k ∈ G L(2, Z p ). Definition 6.2.2 (Ramified representation) Fix a prime p. A representation (π, V ) of G L(2, Q p ), as in Definition 6.1.4, is termed ramified if there does not exist a G L(2, Z p ) fixed vector v ◦ ∈ V . Let us try to motivate the definitions of ramified and unramified representations given above. In Definition 2.1.14 we defined a ramified character on Q×p .

6.3 Local representation coming from a level 1 Maass form

193

The finite set of primes where the character is ramified are called “bad primes” because basic objects, such as the local L-function, which number theorists like to compute, may get complicated and become difficult to compute. We assume the reader is familiar with the fact that associated to a Maass form of level N there are also a finite set of certain “bad primes” (primes dividing N , for example) where the local L-functions are quite different than at all the other places. We thus expect Maass forms of level one (which don’t have bad primes) to lift to automorphic representations which are unramified everywhere, while Maass forms of level N > 1 will lift to automorphic representations which are ramified at the bad primes. If you take an even weight zero Maass form f for S L(2, Z), as in Definition 3.5.7, and lift it to an adelic form f adelic as in Definition 4.8.2, then the lifted  form f adelic will be K -invariant where K = O(2, R) p G L(2, Z p ). Let

 g=

10 01

   , 10 01 ,

... ,



10 01



,



    ∈ G L(2, AQ ), , 10 01 , . . . , !" #

a b c d

pth place

  and consider f adelic (g) as a function of ac db ∈ G L(2, Q p ). By taking all finite linear combinations of all right translates by G L(2, Q p ) of this function we can create a local representation of G L(2, Q p ) which will turn out to be irreducible and smooth (hence admissible). Since f adelic is in this space and f adelic is G L(2, Z p )-invariant, we see that this local representation is unramified according to Definition 6.2.1. On the other hand, if we start with a newform of arbitrary weight, level, and character as in Definition 3.5.7, then the above construction will not give an unramified local representation of G L(2, Q p ) if p divides the level of the newform. It turns out that there will be no G L(2, Z p )-fixed vector in the space.

6.3 Local representation coming from a level 1 Maass form In Chapter 5 it was shown how to construct an automorphic representation from an adelic lift of a classical Maass form for a congruence subgroup of S L(2, Z). The same type of construction can be used to create local representations of G L(2, Q p ). We shall begin with the simplest case of an even Maass form of weight 0 and level 1. Recall that level 1 means it is automorphic for the group 0 (1) = S L(2, Z). Let f be an even Maass form of weight 0 for S L(2, Z) of type ν ∈ C, as in Definition 4.8.1, with classical Whittaker expansion    √ y∞ 0 1 x∞ = A(n) y∞ · K ν− 1 (2π |n|y∞ )e2πinx∞ f 2 0 1 0 1 n=/ 0

194

Theory of admissible representations of GL(2, Qp )

for all x∞ ∈ R, y∞ > 0. Let f adelic denote its adelic lift as defined in Definition 4.8.2. Recall from (4.10.10) that for α ∈ Q, and g ∈ G L(2, AQ ),     1 u Wα (g; f ) = g e(−αu) du f adelic 0 1 Q\A

is a global Whittaker function. We showed in (4.10.12) that the global Whittaker function factors as a product of local Whittaker functions. As in (4.10.14) we define the local Whittaker function (coming from the Maass form f ) by the formula      1 xp yp 0 rp 0 · k p; f Wp 0 1 0 1 0 rp  1   |y p | p2 · A |y p |−1 e p (x p ), if y p ∈ Z p , p = 0, otherwise, (6.3.1) for x p ∈ Q p , y p , r p ∈ Q×p , k p ∈ G L(2, Z p ). To explicitly construct a local representation, we shall create a vector space W p ( f ) by taking linear combinations of right translates of W p (∗; f ) by elements of G L(2, Q p ). Definition 6.3.2 (Vector space of right translates of W p (∗; f )) We define a complex vector space W p ( f ) to consist of all linear combinations, with coefficients in C, of right translates of W p (∗; f ) by elements in G L(2, Q p ). A typical element in W p ( f ) is a function w : G L(2, Q p ) → C of the form w(g) =

r

ci W p (g · h i ; f )

i=1

with ci ∈ C, g, h i ∈ G L(2, Q p ), and r a positive integer. Definition 6.3.3 (Local representation coming from a level 1 Maass form) Let f be a level 1, weight 0 Maass form. Let W p ( f ) be the complex vector space given in Definition 6.3.2. We define π : G L(2, Q p ) → G L W p ( f ) to be the action by right translation, so that   π (h) . w (g) = w(g · h) for w ∈ W p ( f ). Clearly, π (h) . w is another element of the space W p ( f ). Then (π, W p ( f )) is termed the local representation coming from the Maass form f . Proposition 6.3.4 (Local representations (from Maass forms) are smooth) Let (π, W p ( f )) denote the local representation coming from the Maass form f as in Definition 6.3.3. Then (π, W p ( f )) is a smooth representation as in Definition 6.1.1.

6.4 Jacquet’s local Whittaker function

195

Proof We must show that for each w ∈ W p ( f ), as in Definition 6.3.2, there is a non-negative integer n such that w (g · k) = w(g),

(∀g ∈ G L(2, Q p ), k ∈ K n ),

where K n is given by (6.1.3). First, consider the simplest case w = W p (∗; f ). Then it follows from (6.3.1) that this condition is satisfied with n = 0. Next, suppose w(g) = W p (g · h; f ). Then w(gk) = w(g) for all k such that h −1 kh ∈ G L(2, Z p ). The subgroup hG L(2, Z p )h −1 ∩G L(2, Z p ) of G L(2, Z p ) is open, so it contains K n for some n. Here I2 is the 2 × 2 identity matrix and Mat (2, Z p ) denotes the set of all 2 × 2 matrices with coefficients in Z p . This completes the proof in this case. Finally, for a general element w(g) =

r

ci W p (g · h i ; f ),

(ci ∈ C, for i = 1, 2, . . . , r ),

i=1

we obtain, for each i = 1 to r, an integer n i which comes from h i as in the previous case. Then we take n = maxi n i . 

6.4 Jacquet’s local Whittaker function Local and global Whittaker functions arose naturally when computing Fourier coefficients of adelic automorphic forms as in Section 4.10. Here is a formal definition following (4.10.16). Definition 6.4.1 (Local Whittaker function) Fix a prime p. A local Whittaker function for the group G L(2, Q p ) is a smooth function W p : G L(2, Q p ) → C which satisfies    1 u Wp g = e p (u)W p (g), 0 1 for all u ∈ Q p , g ∈ G L(2, Q p ). Here e p is the additive character on Q p defined in Definition 1.6.3. In his thesis [Jacquet, 1967], Jacquet showed how to explicitly construct local Whittaker functions. The construction is given by a simple integral as follows. Definition 6.4.2 (Jacquet’s local G L(2, Z p )-invariant Whittaker function) Fix s1 , s2 ∈ C, with (s1 − s2 ) > 1. Let f s◦1 ,s2 : G L(2, Q p )/G L(2, Z p ) → C be defined by      1 x y 0 r 0 ◦ · k := |y|sp1 · |r |sp1 +s2 f s1 ,s2 0 1 0 1 0 r

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196

for x ∈ Q p , y, r ∈ Q×p , and k ∈ G L(2, Z p ). We define Jacquet’s local Whittaker function  W p (g; s1 , s2 ) :=

f s◦1 ,s2



0 1 1 0



1 u 0 1

  g e p (−u) du,

Qp

for all g ∈ G L(2, Q p ), and where e p is the additive character on Q p defined in Definition 1.6.3. Remarks We must show that this integral is convergent. This is the reason for the restriction on (s1 − s2 ). We will shortly show that the function W p (g; s1 , s2 ) can be holomorphically continued to all s1 , s2 ∈ C, and satisfies a functional equation. These results were also proved in [Jacquet, 1967] for a much more general class of groups. Note, also, that the Whittaker function we have constructed in Definition 6.4.2 is G L(2, Z p )-invariant under right multiplication. In his thesis [Jacquet, 1967], Jacquet constructs more general Whittaker functions which are not necessarily G L(2, Z p )-invariant. We shall return to this theme later. Proposition 6.4.3 (Computation of Jacquet’s local Whittaker function) Fix s1 , s2 ∈ C such that (s1 − s2 ) > 1, and let W p (∗; s1 , s2 ) be Jacquet’s local Whittaker function as defined in Definition 6.4.2. Then  Wp

1 0

x 1 

=



y 0

0 1

1− ps2 −s1 1− ps2 −s1 +1



r 0 0 r



 · k; s1 , s2

  |y|sp2 +1 − |y|sp1 p s2 −s1 +1 · |r |sp1 +s2 · e p (x),

0,

if y ∈ Z p , otherwise,

for x ∈ Q p , y, r ∈ Q×p , and k ∈ G L(2, Z p ). Remarks If p s2 −s1 = 1 then W p (g; s1 , s2 ) = 0 for all g ∈ G L(2, Q p ). Note also that W p (g; s1 , s2 ) is invariant under the center of G L(2, Q p ) if and only if p s1 +s2 = 1.    r 0 Proof Since f s◦1 ,s2 g = |r |sp1 +s2 f s◦1 ,s2 (g) for all g ∈ G L(2, Q p ) and all 0r

r ∈ Q×p , it follows by a simple change of variables that  Wp

1 0

x 1



y 0

0 1



r 0 0 r





· k; s1 , s2  y s1 +s2 = e p (x)|r | p W p 0

0 1



 ; s1 , s2 .

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197

Furthermore,    y 0 ; s1 , s2 Wp 0 1      0 1 1 u y 0 e p (−u) du = f s◦1 ,s2 1 0 0 1 0 1 Qp      0 1 y 0 1 uy −1 ◦ e p (−u) du = f s1 ,s2 0 1 1 0 0 1 Qp      0 1 y 0 1 u e p (−uy) du = |y| p f s◦1 ,s2 1 0 0 1 0 1 Qp      1 0 0 1 1 u e p (−uy) du = |y| p f s◦1 ,s2 0 y 1 0 0 1 Qp     0 1 1 u 1+s2 ◦ e p (−uy) du. = |y| p f s1 ,s2 (6.4.4) 1 0 0 1 Qp Now, if u ∈ Z p , then 

0 1



1 0

01 10





1 0

1u 0 1

u 1



∈ G L(2, Z p ). Also,



 =

u −1 0

1 u



−1 u −1

0 1

 ,

  0 ∈ G L(2, Z p ). This allows us to break the integral and, if u ∈ Z p , then u−1 −1 1 (6.4.4) into two pieces. The first piece is  |y|sp2 +1

Zp

f s◦1 ,s2



1 0 0 1



 e p (−uy) du =

0,

if y ∈ / Zp,

|y|sp2 +1 ,

if y ∈ Z p .

(6.4.5)

This piece is always convergent because the integrand is locally constant and the domain of integration is compact. The second piece of the integral is  |y|sp2 +1

Q p −Z p

f s◦1 ,s2



u −1 0

1 u

 e p (−uy) du.

(6.4.6)

This piece is only convergent because we have assumed (s1 − s2 ) > 1. To see this, let | · |C denote the absolute value on C, and observe for  that, u −1 1 × = all u ∈ Q p −Z p , and all s1 , s2 ∈ C and y ∈ Q p , the identity 0 u  −1   −1  1u u 0 , implies that 0 u 0 1   −1  ◦ u f  s1 ,s2 0

1 u



  (s2 −s1 )        e p (−uy) =  |u|sp2 −s1  =  |u| p  . C C C

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198

The value of |u| p is constant, and equal to p m , on each “annulus” p −m Z×p , and, for each m, this annulus has measure p m · (1 − p −1 ). It follows that the integral in (6.4.6) is convergent if and only if the infinite sum ∞

p m((s2 −s1 )+1)

m=1

is convergent. And this is the case iff (s1 − s2 ) > 1. Let y = p  · u 0 with u 0 ∈ Z× and  ∈ Z. Having established that (6.4.6) is absolutely convergent, we may now perform the following manipulations  Q p −Z p



 u −1 1 e p (−uy) du 0 u  −1   ∞ u 1 ◦ e p (−uy) du f s1 ,s2 = 0 u

f s◦1 ,s2

m=1

=



∞ m=1

=

p−m Z×p

∞

p−m Z×p

p m(s2 −s1 )

m=1

= p

|u|sp2 −s1 e p (−uy) du

∞ m=1

 p−m Z×p

p m(s2 −s1 )

  e p −up  du  e p (−u) du.

(6.4.7)

p−m Z×p

To evaluate the last integral on the right side of (6.4.7) we shall recall Proposition 1.6.5 which states: ⎧ m−   1 − p −1 , if  ≥ m, ⎪  ⎨ p −1, if  = m − 1, (6.4.8) e p (u) du = ⎪ ⎩ × −m 0, if  < m − 1. p Zp It immediately follows from (6.4.8) that the integral in (6.4.7) is zero if  ≤ −1, / Z p . By breaking into cases, we compute, using (6.4.8) or equivalently, if y p ∈ that  −1   u 1 ◦ e p (−uy) du f s1 ,s2 0 u Q p −Z p  0, if y ∈ / Zp,  =  s2 −s1 +1 (+1)(s2 −s1 +1) − p (s2 −s1 )(+1)+ , if y ∈ Z p . (1 − p −1 ) p 1−−pps2 −s1 +1

6.4 Jacquet’s local Whittaker function

199

To complete the computation when y ∈ Z p , we need to multiply this by |y|sp2 +1 , and then add another |y|sp2 +1 , coming from (6.4.5). Thus, we need to simplify (1 − p −1 )( p s2 −s1 +1 − p (+1)(s2 −s1 +1) ) + (1 − p (s2 −s1 )(+1)+ ). 1 − p s2 −s1 +1 Plugging x = p s2 −s1 into the polynomial identity (1 − p −1 )(x p − x (+1) p (+1) ) (1 − x)(1 − (x p)+1 ) + (1 − x +1 p  ) = , (1 − px) 1 − xp 

we obtain the stated formula.

Proposition 6.4.9 (Functional equation of Jacquet’s Whittaker function) Fix s1 , s2 ∈ C, and let W p (∗; s1 , s2 ) be Jacquet’s local Whittaker function as defined in Definiton 6.4.2. Then W p (∗; s1 , s2 ) has a holomorphic continuation to all (s1 , s2 ) ∈ C × C and satisfies the functional equation W p (g; s2 + 1, s1 − 1) =

1 − p s1 −s2 −2 W p (g; s1 , s2 ), 1 − p s2 −s1

for all g ∈ G L(2, Q p ). Proof The holomorphic continuation of W p (∗; s1 , s2 ) follows from the computation in the previous theorem. Note that while W p (∗; s1 , s2 ) is most succinctly written as a rational function, in p s2 and p s2 −s1 +1 , the denominator of this rational function comes from evaluating a finite geometric series. So, in fact, it follows that this “rational function” is a polynomial, albeit one whose degree may be arbitrarily large, depending on |y| p . In particular, W p (g; s1 , s2 ) is defined also when p s2 −s1 +1 = 1, where it takes the value  e p (x) · |r |sp1 +s2 · |y|sp2 +1 · (1 − p −1 ) · (1 − log p |y| p ), if y p ∈ Z p , 0,



To see the functional equation, we may rewrite W p more symmetrical form (1 − p

s2 −s1

s1 +s2 +1 2

) · |y| p

( p|y| p ) p

s1 −s2 −1 2 s1 −s2 −1 2

− ( p|y| p ) −p

otherwise.   y 0 ; s in the , s 1 2 0 1 s2 −s1 +1 2

s2 −s1 +1 2

.

The substitutions s1 → s2 + 1, s2 → s1 − 1 send s1 + s2 to s1 + s2 and s2 − s1 + 1 to s1 − s2 − 1, so they fix e p (x) ·

|r |sp1 +s2

s1 +s2 +1 2

· |y| p

( p|y| p ) p

s1 −s2 −1 2 s1 −s2 −1 2

− ( p|y| p ) −p

s2 −s1 +1 2

s2 −s1 +1 2

and they send 1 − p s2 −s1 to 1 − p s1 −s2 −2 . The result follows. 

,

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Theory of admissible representations of GL(2, Qp )

We will show in the next proposition that Jacquet’s Whittaker function in Definition 6.4.2 (with a suitable choice of s1 , s2 ), agrees, up to a constant, with the local Whittaker function coming from a Maass form as computed in (4.10.14). Let f be a Maass form for S L(2, Z) of type ν as in Definition 4.8.1 with classical Whittaker expansion    √ y∞ 0 1 x∞ = A(n) y∞ · K ν− 1 (2π |n|y∞ )e2πinx∞ f 2 0 1 0 1 (6.4.10) n=/ 0 for all x∞ ∈ R, y∞ > 0, normalized so that A(1) = 1, and assumed to be an eigenfunction of all the Hecke operators. It is well known [Goldfeld, 2006, 3.13] that the L-function associated to f , denoted L f (s) and defined by −s L f (s) = ∞ n=1 A(n)n , has an Euler product representation of the form  −1   −s −1 1 − α p p −s 1 − α −1 . L f (s) = p p It follows that we may write m

A( p ) =

m

α m−2 = p

α m+1 − α −m−1 p p

=0

α p − α −1 p

(6.4.11)

for all m = 0, 1, 2, 3, . . . and some α p ∈ C. The Ramanujan conjecture states that |α p |C = 1, where | |C denotes the absolute value on C. Theorem 6.4.12 Let f be a Maass form for S L(2, Z) as in (6.4.10). Let α p ∈ C be given by (6.4.11). Assume s1 , s2 ∈ C with p s1 −s2 =/ 1, p. Let s1 , s2 be √ defined by p s1 = p · α p , and s2 = −s1 . Then we have the equality W p (g; s1 , s2 ) = (1 − p −2s1 )W p (g; f ),

(∀g ∈ G L(2, Q p )),

where W p (∗; s1 , s2 ) is Jacquet’s Whittaker function as defined in Definition 6.4.2 and W p (∗; f ) is the Whittaker function coming from f as given in (4.10.15). Proof Brute force computation left to the reader.



6.5 Principal series representations One of the reasons representation theory is so powerful is because it is very often possible to show that two representations (π, V ), (π , V ) of a group G are isomorphic even though the vector spaces V, V appear to be totally different. It may be hard to prove something in V but much easier in V . We would like to classify the admissible local representations of G L(2, Q p ). This task will turn out to be much easier by using a vector space that is different from the space of Whittaker functions which came up in

6.5 Principal series representations

201

Section 6.3. In this section we define the principal series representations which are parameterized by a pair of complex numbers and a pair of normalized unitary characters which are defined as follows. Definition 6.5.1 (Normalized unitary character of Q×p ) Fix a prime p. A normalized unitary character of Q×p is a continuous function ω : Q×p → C× which satisfies       ∀y, y ∈ Q×p , • ω yy = ω(y)ω y ,   • |ω(y)|C = 1, ∀y ∈ Q×p , • ω( p) = 1. Let s = (s1 , s2 ) ∈ C2 and let ω = (ω1 , ω2 ) where ω1 , ω2 are normalized unitary characters of Q×p as in Definition 6.5.1. We will show that for a suitable choice of s, ω, the associated principal series representation is isomorphic to the local representation coming from a Maass form. The local representation coming from a Maass form has a vector space of local Whittaker functions. The vector space associated to the principal series representation will be much simpler. It will turn out that the two vector spaces can be related by the integral in Definition 6.4.2 first introduced by Jacquet to explicitly construct Whittaker functions. Definition 6.5.2 (The vector spaces V p (s, ω) and V p (s1 , s2 )) Fix a prime p, a pair of complex numbers s = (s1 , s2 ) ∈ C2 , and a pair of normalized unitary characters ω = (ω1 , ω2 ) of Q×p as defined in Definition 6.5.1. Define V p (s, ω) to be equal to        y1 x s1 s2 f : G L(2, Q p ) → C  f 0 y2 g = |y1 | p ω1 (y1 ) · |y2 | p ω2 (y2 ) · f (g),   ∀y1 , y2 ∈ Q×p , x ∈ Q p , g ∈ G L(2, Q p ), f is locally constant . Furthermore, we also define V p (s1 , s2 ) := V p (s, ωtrivial ), where s = (s1 , s2 ) ∈ C2 and ωtrivial is the pair consisting of two copies of the trivial character which takes the value one for every element in Q×p . Definition 6.5.3 (Principal series representation) Fix a prime p, a pair of complex numbers s = (s1 , s2 ) ∈ C2 , and a pair of normalized unitary characters ω = (ω1 , ω2 ) of Q×p as in Definition 6.5.1. The principal series representation   of G L(2, Q p ), associated to (s, ω), is the representation π, V p (s, ω) where π is the action by right translation. Thus   π (h) . f (g) = f (gh) for all g, h ∈ G L(2, Q p ), and f ∈ V p (s, ω).

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Theory of admissible representations of GL(2, Qp )

Remark It is clear for every f ∈ V p (s, ω) and every fixed h ∈ G L(2, Q p ) that π (h) . f ∈ V p (s, ω).   If ω =/ ωtrivial then π, V p (s, ω) will not have a vector fixed by G L(2, Z p ), and, therefore, will be ramified as in Definition 6.2.2. On the other hand, if ω = ωtrivial then the function f s◦1 ,s2 (Definition 6.4.2) is fixed by the right action of G L(2, Z p ) and is an element of V p (s, ωtrivial ) = V p (s1 , s2 ). Thus, the unramified principal series representations take a somewhat simpler form. Definition 6.5.4 (Unramified principal series representation) Fix a prime representation of p and fix s1 , s2 ∈ C. The unramified principal series   G L(2, Q p ), associated to s1 , s2 , is the representation π, V p (s1 , s2 ) , where the vector space V p (s1 , s2 ) is defined in Definition 6.5.2, and π is the action by right translation. Thus   π (h) f (g) = f (gh) for all g, h ∈ G L(2, Q p ), and f ∈ V p (s1 , s2 ). As we will explain in more detail to follow, Theorem 6.4.12 establishes a relationship between the unramified principal series representation (π, V p (s1 , s2 )) and the local representation (π, W p ( f ))) (coming from a level one Maass form) considered in Definition 6.3.3. Note that, although the vector spaces are different, the action is right translation in both cases, so we use the same notation, π. The advantage of V p (s1 , s2 ) is that one can easily write down many elements of this vector space. This allows us to prove, for example, that the principal  series representation π, V p (s1 , s2 ) is admissible, whereas proving that the representation (π, W p ( f )) of Definition 6.3.3 is admissible is more difficult. Proposition 6.5.5 (Principal series representations are admissible) Fix a prime p, a pair of complex numbers s = (s1 , s2 ) ∈ C2 , and a pair of normalized unitary characters ω = (ω1 , ω2 ) of Q×p as in Definition 6.5.1.   The representation π, V p (s, ω) given in Definition 6.5.3 is admissible as in Definition 6.1.4.   Proof We first prove that π, V p (s, ω) is smooth. Let f be an element of V p (s, ω). Since f is locally constant, it follows that for each g ∈ G L(2, Z p ) there exists n such that f (gk) = f (g) whenever k − I2 ∈ p n · Mat(2, Z p ). (Here I2 denotes the 2 × 2 identity matrix and Mat(2, Z p ) denotes the 2 × 2 matrix with coefficients in Z p .) What we need to show is that n can be made independent of g. Recall from (6.1.3) that 

  K n := k ∈ G L(2, Z p )  k − I2 ∈ p n · Mat(2, Z p )

6.5 Principal series representations

203

is a compact open subgroup for every integer n = 0, 1, 2, . . . . Note that, for each k ∈ G L(2, Z p ), there exists an integer n(k) ≥ 1 such that f (kk ) = f (k),

  ∀k ∈ K n(k) .

The cosets k · K n(k) ,

(k ∈ G L(2, Z p )),

form an open cover of G L(2, Z p ). After choosing a finite subcover, there is a largest value of n(k). This will be our n. Because K n ⊂ K m for n > m, it follows that f (kk ) = f (k),

(∀k ∈ G L(2, Z p ), k ∈ K n ).

It then follows from the Iwasawa decomposition and the definition of V p (s, ω) that f (gk ) = f (g),

(∀g ∈ G L(2, Q p ), k ∈ K n ).

  This completes the proof that π, V p (s, ω) is smooth. Now, fix a positive integer n and consider V pn (s, ω) := { f ∈ V p (s, ω) | f (gk) = f (g), ∀g ∈ G L(2, Z p ), k ∈ K n }. We must prove that this space is finite dimensional for each n. First, assume for simplicity that ω = ωtrivial . We prove that an element of V pn (s, ωtrivial ) is completely determined by its values on any set of double coset representatives for B(Z p )\G L(2, Z p )/K n , 

where B(Z p ) =

y1 0

x y2

 5   x ∈ Z p , y1 , y2 ∈ Z× . p 

We then prove that this space of double cosets is finite. Take f ∈ V pn (s, ωtrivial ) and g ∈ G L(2, Z p ). Using the Iwasawa decomposition, Proposition 4.2.1, we write  g=

1 u 0 1



p e1 0

with e1 , e2 , ∈ Z, k ∈ G L(2, Z p ), and u =

0 p e2

 · k,

e1 −e 2 −1 =−N

(6.5.6)

u  p  , with N ∈ Z, 0 ≤

u  < p, and u −N =/ 0, unless u = 0. This expression is unique.

Theory of admissible representations of GL(2, Qp )

204

Fix a set Cn of representatives for the set of double cosets B(Z p )\G L(2, Z p )/K n .

(6.5.7)

Then k ∈ G L(2, Z p ) is uniquely expressible as k = b · ξ · k with b ∈ B(Z p ), ξ ∈ Cn , and k ∈ K n . It follows from (6.5.6) that   e   1 u p1 0

f (g) = f · b · ξ · k 0 1 0 p e2 = | p e1 |sp1 · | p e2 |sp2 · f (ξ ) = p −e1 s1 · p −e2 s2 · f (ξ ). 

Let B(Q p ) =

y1 0

x y2

 5   x ∈ Q p , y1 , y2 ∈ Q× . p 

Note that for each ξ ∈ Cn , the function     e1 ⎧ 1 u p 0 −e1 s1 −e2 s2 ⎪ · b · ξ · k , p · p , if g = ⎨ 0 p e2 0 1 f ξ (g) := ⎪ ⎩ 0, if g ∈ / B(Q p ) · ξ · K n , (6.5.8) is an element of V pn (s, ωtrivial ). It follows that the dimension of V pn (s, ωtrivial ) is equal to the number of double cosets in (6.5.7). To prove that this space of double cosets is finite, it suffices to observe that G L(2, Z p )/K n is already finite. In fact, it follows easily from the definition of K n that it is isomorphic to G L(2, Z/ p n Z). We leave it as an exercise for the reader to verify that   5   5 0 1  1 0  n−1 n 0≤c< p 0≤d< p ∪ pc 1  1 d  is a set of coset representatives, so that the number of double cosets is 1 if n = 0 and p n + p n−1 otherwise. Now consider the case when ω =/ ωtrivial . Let p m 1 , p m 2 denote the conductors of ω1 , ω2 , respectively. Recall from Definition 2.1.14 that 1 + p m i Z p is in the kernel of ωi (i = 1, 2), and m i (i = 1, 2), is the smallest positive integer with this property. Let M = max(m 1 , m 2 ). If f ∈ V pn (s, ω) then f must satisfy  f

y1 0

x y2

  g = |y1 |sp1 ω1 (y1 ) · |y2 |sp2 ω2 (y2 ) · f (g) f (gk ) = f (g),

(6.5.9) (6.5.10)

6.6 Jacquet’s map: Principal series → Whittaker functions

205

(for all x ∈ Q p , y1 , y2 ∈ Q×p , g ∈ G L(2, Q p ) and k ∈ K n ). The combination of these two conditions forces f to vanish identically when n < M. This may be seen by considering the following special case: take x = 0, and y1 , y2 ∈ Z×p , such that yi − 1 ∈ pn Z p , but ω1 (y1 )ω2 (y2 ) =/ 1. (This is possible provided y1 x

n < M.) Then 0 y2 is in K n . Take k to be an element of G L(2, Z p ). Let y x  1 k = k −1 · 0 y2 · k, which is again in K n because K n is normal in G L(2, Z p ). Combining (6.5.9) and (6.5.10), (and noting that |yi | = 1, i = 1, 2), we get  ω1 (y1 ) · ω2 (y2 ) · f (k) = f

y1 0

x y2

  k = f (kk ) = f (k).

Since ω1 (y1 ) · ω2 (y1 ) =/ 1, this forces f (k) to be zero for any k ∈ G L(2, Z p ). It then follows from the Iwasawa decomposition (6.5.6) and the definition of V p (s, ω) given in Definition 6.5.2 that f (g) = 0 for all g ∈ Q p . Consequently Vnp (s, ω) = {0} for n < M. For ω =/ ωtrivial and n ≥ M the proof that V pn (s, ω) is finite dimensional is the same as the proof (given above) in the case where ω = ωtrivial . 

6.6 Jacquet’s map: Principal series → Whittaker functions In Section 6.4 we saw that the local Whittaker function W p obtained from a Maass form, as in (6.3.1), can be obtained from one particular element of  V p (s1 , s2 ) :=

   y1  f : G L(2, Q p ) → C  f 0  ∀y1 , y2 ∈

Q×p ,

∗ y2

  g = |y1 |sp1 |y2 |sp2 · f (g), 

g ∈ G L(2, Q p ), f is locally constant ,

for the correct choice of s1 , s2 , via a certain integral, introduced by Jacquet (see Definition 6.4.2). Recall Definition 2.5.3 of an intertwining map. In this section we will show how Jacquet’s  integral can be  extended to an intertwining map Js1 ,s2 from the representation π, V p (s1 , s2 ) defined in Definition 6.5.4 to the representation (π, W p ( f )) defined  in Definition 6.3.3. This is the unramified situation since π, V p (s1 , s2 ) is always an unramified principal series representation by Definition 6.5.4. We shall define and study the map: Principal series → Whittaker functions, in the more general situation where the principal series representations may also be ramified. Accordingly, we make the following definition. Definition 6.6.1 (Jacquet’s integral) Fix a prime p, a pair of complex numbers s = (s1 , s2 ) ∈ C2 , and a pair of normalized unitary characters ω = (ω1 , ω2 )

Theory of admissible representations of GL(2, Qp )

206

of Q×p as in Definition 6.5.1. Assume (s2 − s1 ) > 1. For f ∈ V p (s, ω), as in Definition 6.5.2, we define a new function Js,ω ( f ) by the convergent integral 







Js,ω ( f ) (g) :=

f

0 1 1 0



1 0

u 1

  g e p (−u) du.

Qp

Let W p (s, ω) denote the image of the principal series V p (s, ω) under the map Js,ω . It consists of certain local Whittaker functions satisfying Definition 6.4.1. We make W p (s, ω) into a representation of G L(2, Q p ), by defining the action π to be the action of G L(2, Q p ) on W p (s, ω) by right translation. Then      π (h) . Js,ω ( f ) (g) = Js,ω π (h) . f (g), for all g, h ∈ G L(2, Q p ) and f ∈ V p (s,  that Js,ω is an intertwin ω). It follows ing operator from the representation π, V p (s, ω) to (π, W p (s, ω)) for each s1 , s2 in the domain where the integral of Definition 6.6.1 converges. We would like to be able to analytically continue Js,ω to all C2 . To do this, we will fix ω, and use a function of three variables f (g; s1 , s2 ), g ∈ G L(2, Q p ), s1 , s2 ∈ C, with the property that, f (∗; s1 , s2 ) ∈ V p (s, ω) for each s1 , s2 ∈ C2 . Definition 6.6.2 (Flat section) Fix a prime p, and a pair ω := (ω1 , ω2 ) of normalized unitary characters of Q×p as in Definition 6.5.1. A flat section is a function f : G L(2, Q p ) × C2 → C such that: • for each fixed (s1 , s2 ) ∈ C2 , the function f (∗; s1 , s2 ) : G L(2, Q p ) → C is an element of V p (s, ω); • for each fixed k ∈ G L(2, Z p ) the function f (k; ∗) : C2 → C is constant. The flat sections form a complex vector space. We denote this space by Flat p (ω). Abusing notation, we write Js,ω ( f ) for Js,ω ( f (∗; s1 , s2 )), if f ∈ Flat p (ω). Proposition 6.6.3 (Analytic continuation of Jacquet’s integral) For all flat sections f ∈ Flat p (ω), and any fixed g ∈ G L(2, Q p ), the function (Js,ω ( f )) (g) is a polynomial in p s1 , p −s1 , p s2 , and p −s2 . Further, Jacquet’s integral (Js,ω ( f )) (g) (defined in Definition 6.6.1) has analytic continuation to all s1 , s2 ∈ C, for all g ∈ G L(2, Q p ) and all flat sections f ∈ Flat p (ω). Remark Every f ∈ V p (s, ω) is a finite linear combination (with complex coefficients) of functions of the form (6.5.8) which can be viewed as flat sections.

6.6 Jacquet’s map: Principal series → Whittaker functions

207

Consequently, Proposition 6.6.3 gives the analytic continuation of Jacquet’s integral for all f ∈ V p (s, ω). Proof The first part of the proposition, that the function (Js,ω ( f )) (g) is a polynomial in p s1 , p −s1 , p s2 , and p −s2 , immediately implies the second part that (Js,ω ( f ))(g) has analytic continuation to all s1 , s2 , for all g and all flat sections f. The first part of the proposition will be deduced from Definition 6.6.4, Lemma 6.6.6, and Propositions 6.6.7, 6.6.9, and 6.6.10 below. We introduce it first in order to motivate these more technical results. It was established in Proposition 6.5.5, that (π, V p (s, ω)) is smooth. The proof of Proposition 6.5.5 is easily modified to show that for every f ∈ Flat p (ω) there exists n ∈ N such that f (gk; s1 , s2 ) = f (g; s1 , s2 ) for all g ∈ G L(2,  Q p ), s1 , s2 ∈ C, and n k ∈ K n = k ∈ G L(2, Z p ) | k − I2 ∈ p Mat(2, Z p ) , as in (6.1.3). In Definition 6.6.4, we fix a basis for the space of K n -fixed elements of 6.6.7, 6.6.9, and 6.6.10 below we compute the Flat p (ω). Then, in Propositions y x  value of Js,ω ( f ) 0 1 for each of the elements f of this basis, finding that in each case, the answer is a polynomial in p s1 , p −s1 , p s2 , and p −s2 . It follows that the same is true when f is replaced by any element of Flat p (ω). Since it’s clear that    r 0 = ω1 (r )ω2 (r )|r |sp1 +s2 · Js,ω ( f )(g), Js,ω ( f ) g · 0 r this completes the proof of Proposition 6.6.3 for g upper triangular. To handle the case of arbitrary g, we write g = bk with b upper triangular and k ∈ G L(2, Z p ), and use Lemma 6.6.6.  It is clear that if f is a flat section, and f (g; s1 , s2 ) is constant on left cosets of K n for any one value of s1 , s2 , then the same is true for all s1 , s2 , and that every flat section is constant on left cosets of some K n . For each n, we let Flat Kp n (ω) denote the space of flat sections that are constant on left cosets of K n . Now, for each n ≥ 1, we fix a basis of Flat Kp n (ω). Definition 6.6.4 (Basis for Flat Kp n (ω)) Fix a prime p, and a pair ω = (ω1 , ω2 ) of normalized unitary characters of Q×p as in Definition 6.5.1, with conductors ( p m 1 , p m 2 ) respectively, as defined in Definition 2.1.14. Let M = max(m 1 , m 2 ). Then, for any integer n ≥ M, and 0 ≤ d < p n , define     ⎧ ⎨ ω (y )|y |s1 · ω (y )|y |s2 , if g = y1 x · 0 1 ·k, 1 1 1 p 2 2 2 p n f 1,d (g; s1 , s2 ) := 1 d 0 y2 ⎩ 0, otherwise, for y1 , y2 ∈ Q×p , x ∈ Q p , and k ∈ K n where 

   K n = k ∈ G L(2, Z p )  k − I2 ∈ p n Mat 2, Z p .

Theory of admissible representations of GL(2, Qp )

208

If 0 ≤, c < p n−1 , define n (g; s1 , s2 ) f pc,1 ⎧ ⎨ ω (y )|y |s1 · ω (y )|y |s2 , 1 1 1 p 2 2 2 p := ⎩ 0,

 if g =

y1

0 otherwise,

  1 · pc y2 x

0 1

 · k,

for y1 , y2 ∈ Q×p , x ∈ Q p , and k ∈ K n . Finally, define Bn (ω) =



  

   n n f 1,d (g; s1 , s2 )  0 ≤ d < p n ∪ f pc,1 (g; s1 , s2 )  0 ≤ c < p n−1 .

Lemma 6.6.5 The set Bn (ω) is a basis of Flat Kp n (ω). Proof Exercise.



Next, we wish to compute Js,ω ( f ) for f ∈ Bn (ω). The first step is to reduce to the case when g is upper triangular. This is accomplished by the following: Lemma 6.6.6 Suppose f ∈ Bn (ω). For g ∈ G L(2, Q p ), write g = bk with b upper triangular and k ∈ G L(2, Z p ). Then f (g) = f (b) for some f ∈ Bn (ω), depending on k. Proof Let ξ be the element of   5   5 1 0  0 −1  n−1 n 0≤c< p Cn := ∪  0≤d< p , pc 1  1 d such that f is supported on B(Q p ) · ξ · K n . Since Cn is a complete set of coset representatives, it follows that there is a unique element ξ of Cn with the property that k · ξ ∈ B(Z p ) · ξ · K n . Let f denote the (unique) element of Bn (ω) which is supported on B(Q p ) · ξ · K n . Then it is easy to see that f (bk) = f (b).  Proposition 6.6.7 For x ∈ Q p , y ∈ Q×p , we have  n Js,ω ( f 1,d )

y 0

x 1



 =

e p (x − dy) · ω2 (y)|y|sp2 +1 · p −n ,

if |y| p ≤ p n ,

0,

otherwise.

Proof Clearly,  n Js,ω ( f 1,d )

y 0

x 1



 = e p (x) ·

n Js,ω ( f 1,d )

y 0

0 1

 ,

6.6 Jacquet’s map: Principal series → Whittaker functions and

  y 0 n Js,ω ( f 1,d ) 0 1     0 1 1 u y n f 1,d = 1 0 0 1 0 Qp





n f 1,d

=

1 0

Qp

=

0 y







2 ω2 (y)|y|1+s p

0 1

n f 1,d

1 0



0 1 1 u

0 1



 ; s1 , s2 e p (−u) du

y −1 u 1

1 0



209





; s1 , s2 e p (−u) du

 ; s1 , s2 e p (−yu) du.

Qp

 Now the Iwasawa decomposition of 

0 1 1u

⎧ 0 1 ⎪ , ⎪  ⎪ ⎨ 1 u 1 =   u ⎪ ⎪ u −1 1 −1 ⎪ ⎩ 0 u u −1

0 1

 is given as follows if |u| p ≤ 1, 0 1

From this we deduce that     1, 0 1 n f 1,d ; s1 , s2 = 1 u 0, Now,



 d+ pn ·Z p

e p (−uy) du =

(6.6.8)

 ,

otherwise.

if u − d ∈ p n Z p , if not.

e p (−dy) p −n ,

if |y| p ≤ p n ,

0,

otherwise.

The result follows.  n ∈ Bn (ω). Next, we will perform similar computations for the functions f pc,1 It will be helpful to have the following result in hand.

Lemma 6.6.9 Let ω0 be non-trivial normalized unitary character of Q×p , as in Definition 6.5.1 with conductor p as in Definition 2.1.14. Let Y ∈ Z, y0 ∈ Z×p . Then ⎧ p −2πi j ⎪ ω−1 (y ) ⎪  ⎨ 0 p 0 · ω0 ( j)e p , Y = , j=1 ω0 (u)e p (uy0 p −Y ) du = ⎪ ( j, p)=1 Z×p ⎪ ⎩ 0, Y =/ . Proof The assertion for arbitrary y0 follows easily from the special case y0 = 1 by a change of variables. Hence we assume y0 = 1.

Theory of admissible representations of GL(2, Qp )

210

First, suppose that Y > . Write  Z×p

ω0 (u)e p (up

−Y

) du =

p  j+ p ·Z

j=1 ( j, p)=1

=

j·(1+ p ·Z p )

ω0 (u)e p (up −Y ) du





p

ω0 ( j)

j=1 ( j, p)=1

1+ p ·Z p





=

p

p  j=1 ( j, p)=1

=

ω0 (u)e p (up −Y ) du

p

ω0 ( j)

j=1 ( j, p)=1

1+ p ·Z p

ω0 (u)e p (u j p −Y ) du

e p (u j p −Y ) du.



=

p

ω0 ( j)e p ( j p

−Y

)p

−

 Zp

j=1 ( j, p)=1

e p (u j p −Y ) du.

Since e p (u j p −Y ) is a non-trivial character of the group Z p , this vanishes. Now suppose Y < . Write  Z×p

ω0 (u)e p (up

−Y

) du =

pY  j=1 ( j, p)=1

j+ p Y ·Z p

ω0 (u)e p (up −Y ) du

Y

=

p j=1 ( j, p)=1

e p ( j p −Y )

 j·(1+ p Y ·Z p )

Y

=

p j=1 ( j, p)=1

ω0 ( j)e p ( j p −Y )

ω0 (u) du

 1+ p Y ·Z p

ω0 (u) du.

By the definition of the conductor, ω0 is a non-trivial character of the group 1 + p Y Z p . So this integral also vanishes. The case Y =  then follows from a similar computation.  Proposition 6.6.10 For x ∈ Q p , y ∈ Q×p , and c =/ 0, we have   y x n = e p (x)ω2 (y)|y|sp2 +1 · φ(y), Js,ω ( f pc,1 ) 0 1

6.6 Jacquet’s map: Principal series → Whittaker functions

211

for some locally constant function φ : Q p → C, depending on n, c and ω. There is a constant N > 0, such that φ(y) = 0 whenever |y| p > N . When ω = ωtrivial the function φ is given by φ(y) = e p (−( pc)−1 y) p −n | pc|sp1 −s2 −2 · 1 p−n ( pc)2 ·Z p (y). Proof The proof is similar to that of Proposition 6.6.7 In this case  n f pc,1

0 1 1 0



=

  1 u ; s1 , s2 0 1  −1 ω1 (u) · ω2 (u) · |u|sp2 −s1 , 0,

if u −1 − pc ∈ p n Z p , otherwise.

Let k be the positive integer such that | pc| p = p −k . It is easy to check that u −1 − pc ∈ p n Z p ⇐⇒ u − ( pc)−1 ∈ p n−2k Z p . We need to compute  ( pc)−1 + pn−2k ·Z

ω1−1 (u) · ω2 (u) · |u|sp2 −s1 e p (−uy) du.

(6.6.11)

p

If ω1 = ω2 , we find that (6.6.11) equals  =

e p (−( pc)−1 y) p −n+2k p k(s2 −s1 ) ,

if |y| p ≤ p n−2k ,

0,

otherwise.

Here, we have used the fact that |u| p is constant, and equal to p k , on the coset ( pc)−1 + p n−2k · Z p . Plugging in p k = |cp|−1 p we obtain the precise value in this case. When ω1 =/ ω2 , let  be the positive integer such that p  is the conductor of −1 ω1 ω2 . The integral (6.6.11) is equal to ω1 ( pc)ω2−1 ( pc)| pc|sp1 −s2 −1

 ·

1+ pn−k ·Z p

ω1−1 (u)

  y du. · ω2 (u) · e p −u pc

Suppose  < n − k. Then the ω1−1 (u) · ω2 (u) = 1 on the entire domain of integration, and we get ω1 ( pc)ω2−1 ( pc) times the same answer we got in the case ω1 = ω2 . On the other hand, if  > n − k, then a straightforward modification of the y = y0 p − proof of Lemma 6.6.9 shows that (6.6.11) vanishes except when pc for some y0 ∈ Z p . The result follows. 

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Theory of admissible representations of GL(2, Qp )

Proposition 6.6.12 Let x ∈ Q p , y ∈ Q×p , and let Y = − log p |y| p . Let ω1 , ω2 be normalized unitary characters of Q×p as in 6.4.1. Let p  denote the conductor of ω1−1 ω2 . Then the value of   y x n ) Js,ω ( f 0,1 0 1 is given by e p (x)ω2 (y) times  ⎧  (1− p−1 ) pn(s2 −s1 +1) 1− ps2 −s1 s2 +1 s1 ⎪ |y| + |y| s −s +1 s −s −1 p , p ⎪ 1− p 2 1 1− p 1 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ |y|sp2 +1 · (Y − n + 1) · (1 − p −1 ) − p −1 , ⎪ ⎨

if Y ≥ n − 1, ω1 = ω2 , and p s1 −s2 =/ p, if Y ≥ n − 1, ω1 = ω2 ,

and p s1 −s2 = p, ⎪ ⎪   p ⎪  − s1 ⎪ −j −1 −1 ⎪ ⎪ ω , if Y +  ≥ n, ω1 =/ ω2 , ω (y) p |y| · ω ω ( j)e 1 p 2 p ⎪ 2 1 p ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ( j, p)=1 ⎪ ⎩ 0, otherwise. Remark In the first case above (when Y ≥ n − 1, ω1 = ω2 , p s1 −s2 =/ p) we note that if p s1 −s2 = 1 then the second term vanishes and we obtain   (1 − p −1 ) p n(s2 −s1 +1) s2 +1 y x n ) = e p (x)ω2 (y) · |y| p . Js,ω ( f 0,1 0 1 1 − p s2 −s1 +1 Proof The proof is similar to that of Proposition 6.6.7. In this case      −1 n ω1 (u)ω2 (u)|u|sp2 −s1 , if |u|−1 0 1 1 u p ≥ p , n ; s1 , s2 = f 0,1 1 0 0 1 0, if not. This time, the computation required is the following. Write y = y0 p Y , with y0 ∈ Z p and Y ∈ Z. Then we find that  ω1−1 (u)ω2 (u)|u|sp2 −s1 e p (−uy) du Q p − p−n+1 Z p

=

∞  k=n

=

∞ k=n

p−k Z×p

ω1−1 (u)ω2 (u)|u|sp2 −s1 e p (−uy) du

p k(s2 −s1 +1)

 Z×p

ω1−1 (u)ω2 (u)e p (−uy) du.

Each integral over Z×p may be computed using either Proposition 1.6.5 or Lemma 6.6.9, as appropriate.

6.6 Jacquet’s map: Principal series → Whittaker functions

213

When ω1 = ω2 and Y ≥ n − 1 we obtain Y

p k(s2 −s1 +1) (1 − p −1 ) − p (Y +1) (s2 −s1 +1)−1 .

k=n

Simplifying this sum breaks into two cases depending on whether p s2 −s1 +1 = 1 or not. We leave the computations to the reader.  Theorem 6.6.13 (Description of the image of Js,ω ) Fix a prime p, a pair of complex numbers s = (s1 , s2 ) ∈ C2 , and a pair of normalized unitary characters ω = (ω1 , ω2 ) of Q×p as in Definition 6.5.1. Let f be any element of the space V p (s, ω) given in Definition 6.5.2. Then there exist constants N > ε > 0 and C1 , C2 ∈ C such that the function   y 0 (y ∈ Q×p ) Js,ω ( f ) 0 1 vanishes whenever |y| p > N and is equal to  C1 · ω1 (y)|y|sp1 + C2 · ω2 (y)|y|sp2 +1 , C1 · ω2 (y)|y|sp2 +1 log p |y| p + C2 · ω2 (y)|y|sp2 +1 ,

ω1 =/ ω2 or p s1 −s2 =/ p, ω1 = ω2 and p s1 −s2 = p,

whenever |y| p < ε. Furthermore, if ω1 = ω2 and p s1 −s2 = 1, then C1 = 0. Remark If we take f to be a fixed flat section as defined in Definition 6.6.2, and we then vary s = (s1 , s2 ), then the constants C1 and C2 are functions of s. In the case when ω1 = ω2 , the function C1 (s) vanishes whenever and p s1 −s2 = 1. Definition 6.6.14 (Model of a representation) If (π, V ) is a representation, a model of (π, V ) is another representation (π , V ) which is isomorphic to (π, V ) and used to study the properties of (π, V ). We may also think of (π, V ) and (π , V ) as different realizations of the same abstract representation. Definition 6.6.15 (Whittaker model of a principal series representation) Fix a prime p, a pair of complex numbers s = (s1 , s2 ) ∈ C2 , and a pair of normalized unitary characters ω = (ω1 , ω2 ) of Q×p as in Definition 6.5.1. Let   W p (s, ω) = Js,ω V p (s, ω) . s1 −s2 =/ 1. We Let π  denote right translation. Assume thateither ω1 =/ ω2 or p call π, W p (s, ω) the Whittaker model of π, V p (s, ω) .

At this point the usage of the term “Whittaker model” in Definition 6.6.15 is not really justified, because we have not yet actually proved that (π, V p (s, ω)) is isomorphic to (π, W p (s, ω)). The proof of Proposition 6.4.3 shows that Js,ω

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Theory of admissible representations of GL(2, Qp )

vanishes on the function f s◦1 ,s2 when ω is trivial, and p s1 −s2 = 1. It turns out that this occurs also when ω1 = ω2 is non-trivial. This is the reason for excluding these cases. In Proposition 6.8.6, we will show that in all other cases, (π, V p (s, ω)) is indeed isomorphic to (π, W p (s, ω)). It has already been shown in Proposition 6.5.5 that the principal series representations are admissible. It then follows immediately from Proposition 6.5.5 that the Whittaker model (π, W p (s, ω)) is also admissible. We also want to prove that if p s1 −s2 =/ 1, p 2 , then the principal series representation (π, V p (s, ω)) and its Whittaker model (π, W p (s, ω)) are irreducible representations of G L(2, Q p ). The proof of this fact requires another space of functions: Q×p → C, called the Kirillov space which we introduce in the next section.

6.7 The Kirillov model Langlands wrote two letters around the year 1967 [Langlands], one to Weil and one to Jacquet. In these letters he worked out the theory of local representations for G L(2) over any number field. In his comments about the letters Langlands makes the interesting remark: “The first letter did not fully deal with the nonarchimedean places. This was not possible until at some point during the year in Ankara I stumbled across, in the university library and purely by accident as I was idly thumbing through various journals, the article of Kirillov that contained the notion referred to in the notes of Jacquet-Langlands as the Kirillov model. With the Kirillov model in hand, it was possible to develop a complete local theory even at the nonarchimedean places. This is explained in the second letter.” In the second letter to Jacquet, Langlands refers to [Kirillov, 1963, 1966]. The theory appeared in book form, for the first time, in [GelfandGraev-Piatetski-Shapiro, 1969] and then later in [Jacquet-Langlands, 1970], [Godement, 1970]. Definition 6.7.1 (Kirillov representation) Fix a prime p and let K, X be two non-trivial complex vector spaces. Let (π , K) be a representation of G L(2, Q p ). If the space K consists of locally constant functions f : Q×p → X on which π operates in such a way that     a b . f (y) = e p (by) f (ay), ∀ f ∈ K, a, y ∈ Q×p , b ∈ Q p , π 0 1 then (π , K) is called a Kirillov representation. Here: e p : Q p → C, is the additive character defined in Definition 1.6.3. The vector space K is called a Kirillov space.

6.7 The Kirillov model

215

Theorem 6.7.2 (Existence and properties of the Kirillov model) Fix a prime p and let V denote a complex infinite dimensional vector space. Assume that (π, V ) is an admissible irreducible representation of G L(2, Q p ) as in Definition 6.1.4. Then (π, V ) is isomorphic to a Kirillov representation (π , K), as defined in Definition 6.7.1, which is called a Kirillov model for the representation (π, V ). Define the Bruhat-Schwartz spaces:  5   f is locally constant, and ∃N f >  f > 0 , S X (Q×p ) := f : Q×p → X  such that f (y) = 0 if |y| p <  f or |y| p > N f S(Q×p )

 :=

f :

Q×p

 5  f is locally constant, and ∃N f >  f > 0  . →C  such that f (y) = 0 if |y| p <  f or |y| p > N f

Then the Kirillov model (π , K) satisfies the following properties: (i) Every f ∈ K is locally constant and vanishes outside a compact subset of Q p . (ii) S X (Q×p ) = S(Q×p ) · X ⊂ K;   (iii) K = S X (Q×p ) + π (w0 ) . S X (Q×p ), where w0 = −10 10 . Remark Much more is true. We will prove later that every infinite dimensional, admissible, irreducible representation of G L(2, Q p ) is isomorphic to one and only one Kirillov representation whose space of functions is complex valued. Proof The proof begins with the definition of the subspace V0 ⊂ V defined by    V0 := v ∈ V   



 e p (−u) π

1 0

u 1



 . v du = 0,

for all large n .

p−n Z p

(6.7.3) Set X = V /V0 . It is convenient to introduce the following notation. Let v, v ∈ V . We say v ≡ v (mod V0 ) if v − v ∈ V0 . We define the Kirillov space of functions K as follows:   

  y 0  . v (mod V0 ), ∀y ∈ Q×p . K = f v : Q×p → V /V0  f v (y) := π 0 1 For g ∈ G L(2, Q p ) define the action of π (g) on f v ∈ K as    y 0

π (g) . f v (y) := π . (π (g) . v) (mod V0 ). 0 1

Theory of admissible representations of GL(2, Qp )

216

We will show that (π , K) is a Kirillov representation which is isomorphic to (π, V ). Define the linear map L : V → K by L(v) := f v where for every v ∈ V , the function f v ∈ K is given by     y 0 . v (mod V0 ), ∀y ∈ Q×p . (6.7.4) f v (y) = π 0 1 Following [Godement, 1970], we complete the proof in several steps.   . f v (y) = e p (by) f v (ay). Step 1: Each f v ∈ K satisfies π a0 b1 Let a, y ∈ Q×p , b ∈ Q p . We must show that 

 e p (−u) π

1 0

u 1

 :  ya . π 0

yb 1



− e p (by)π



ya 0 0 1

 ;  .v du = 0

p−n Z p

for all sufficiently large n. This is easy to prove. Choose n 0 sufficiently large so that |by| p < p n 0 and let n ≥ n 0 . Then by a simple change of variable u → u − by, it follows that        1 u ya yb e p (−u) π . π . v du 0 1 0 1 p−n Z p



 e p (−u + by) π

= p−n Z

p

1 u − by 0 1 

 e p (−u + by) π

=

ya 0

   ya . π 0

u 1

yb 1



 .v

du

 . v du.

p−n Z p

Since the integral above (at the beginning of step 1) is equal to the difference of the right and left-hand sides of this identity, we find that it is equal to zero. Step 2: Each f v ∈ K is locally constant and vanishes outside of a compact subset of Q p . First, since the action of π on V is smooth, it follows that for all v ∈ V ,   a 0 . v = v. It is an if a ∈ Q×p with |a − 1| p sufficiently small, then π 0 1 immediate consequence of (6.7.4) that f v (ay) = f v (y),

  ∀y ∈ Q×p ,

provided a ∈ Q×p with |a − 1| p sufficiently small. So f v is locally constant. there exists a sufficiently large positive integer m such that   Similarly, 1b . v = v for |b| p < p −m . Then by step 1 above, it follows that π 0 1

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217

f v (y) = e p (by) f v (y) for all y ∈ Q×p . Thus f v (y) must vanish for y ∈ Q×p outside of a compact subset of Q p . Step 3: The map L : V → K is an isomorphism of vector spaces. Step 3 requires the following lemma. It is precisely here where we use the fact that (π, V ) is infinite dimensional, irreducible, and admissible. Lemma 6.7.5 Assume (π, V ) is an infinite dimensional, irreducible,   admissi1b ble representation of G L(2, Q p ). Assume further that π . v = v for 0 1 all b ∈ Q p and some v ∈ V. Then v = 0.   1b . v = v for all b ∈ Q p . Proof of Lemma 6.7.5 Fix v ∈ V for which π 0 1 Define    H = g ∈ G L(2, Q p )  π (g) . v = c · v for some c ∈ C .   Since we are assuming that the matrices 10 b1 fix v, the subgroup of such matrices is contained in H. The map g → π (g)  . v, is locally constant. This 1 0 implies that π (m) . v = v for all matrices m = b0 1 with b0 ∈ Q p and |b0 | p   1 0 sufficiently small. Fix such a matrix b0 1 ∈ H. Then 

0 b0

−b0−1 0



 =

1 0

−b0−1 1



1 b0

0 1



1 0

−b0−1 1

0 b0

−b0−1 0

 ∈ H.

It then follows that for any x ∈ Q p 

1 x

0 1



 =

0 b0

−b0−1 0



1 −xb0−2 0 1



−1

∈ H.

H by  Lemma 6.1.6. On the other hand, Proposition Thus, S L(2, Q p ) ⊂  a 0 . v = ω(a) · v, for all a ∈ Q×p where ω is 6.1.10 tells us that π 0 a the central character associated to π. Consequently, H also contains the center of G L(2, Q p ). It easily follows that H contains the subgroup of G L(2, Q p ) consisting of those elements with square determinant. The index of this subgroup is finite. Hence, the subspace of V generated by v under the action of G L(2, Q p ) is a finite dimensional invariant subspace. Since V was taken to be both infinite dimensional and irreducible, the only such subspace is the zero subspace. Thus v must be equal to 0.  We now complete the proof of Step 3. We only need to prove that the map L : V → K given in (6.7.4) is injective. Assume there exists v ∈ V for which f v (y) = 0 for all y ∈ Q×p . We want to show that this implies v = 0. If we can

Theory of admissible representations of GL(2, Qp )   1u show that the function φ(u) := π . v (for u ∈ Q p ) is constant then 0 1 Lemma 6.7.5 implies that v = 0 and we are done. It only remains to prove that φ(u) is constant for u ∈ Q p . Under our assumptions about v, the function     1 u φn (y) := e p (−uy)π . v du = e p (−uy)φ(u) du 0 1

218

p−n Z p

p−n Z p

vanishes at y for all sufficiently large n (depending on y). Furthermore, for any compact set K ⊂ Q×p , a sufficiently large integer n can be chosen so that φn (y) = 0 for all y ∈ K . Let ψ : Q p → C be any locally constant compactly supported function. Then for sufficiently large n we must have   ψ(u)φ(u) du = ψ(u)φ(u) du. (6.7.6) p−n Z p

Qp

 (y) = By Theorem 1.6.8, the Fourier transform ψ Q p e p (−uy)ψ(u) du vanishes outside of a compact subset of Q p . If we further assume that Q p ψ(u) du = 0, then this says that ψ is orthogonal to the constant func(0) = 0, from which it follows that the support tion 1, or, equivalently, that ψ  of ψ is contained in a compact subset of Q×p . It then follows from (6.7.6) and the Fourier inversion formula that       (t) dt = 0. ψ (t)e p (tu) dt du = φn (−t)ψ ψ(u)φ(u) du = φ(u) Qp

p−n Z p

K

K

(6.7.7) Here, we take n sufficiently large so that (6.7.6) is satisfied, and φn vanishes on the support of ψ. We have shown that the integral on the left of (6.7.7) vanishes. This implies that the function φ is orthogonal to every locally constant compactly supported function ψ : Q p → C which is orthogonal to the constant function 1. It follows that φ itself must be the constant function. Step 4: The Bruhat-Schwartz space S(Q×p ) is irreducible under the action 

    a b × a ∈ Q of B1 (Q p ) := , b ∈ Q  p . p 0 1 Consider the representation π

: B1 (Q p ) → G L(S(Q×p )), with the action π given as in Definition 6.7.1, i.e.,     a b

π . f (y) = e p (by) f (ay), ∀ f ∈ S(Q×p ), a, y ∈ Q×p , b ∈ Q p . 0 1 (6.7.8) It is clear that the space S(Q×p ) is stable under the actions above.

6.7 The Kirillov model

219

An element of S(Q×p ) is a finite linear combination of characteristic func/ p n Z p . We prove that, for any a tions of cosets a + p n Z p , n ∈ Z, a ∈ Q p , a ∈ and n, the characteristic function 1a+ pn Z p is a linear combination of translates of 11+ pZ p , by the action of B1 (Q p ) given in (6.7.8). First suppose a = 1 and n = 2. For 1 ≤ i ≤ p and y = 1 + y1 p + y2 p 2 + · · · with y j ∈ {0, 1, . . . , p − 1} for j = 1, 2, . . . , we have π





1 i/ p 2 0 1



 . 11+ pZ p (y) = e p

i i · y1 + 2 p p

 · 11+ pZ p (y).

It follows that     p 1 i 1 i/ p 2 . 11+ pZ p = 11+ p2 Z p . e p − 2 · π

0 1 p i=1 p Repeating this trick we obtain 11+ pn Z p for any n > 0. Now let a and n be arbitrary, subject to the condition a ∈ / p n Z p . Write m × n a = a0 p with a0 ∈ Z p . Then m < n, and a+ p Z p = a(1 + p n−m Z p ). Since a + p n Z p iff a −1 y ∈ (1 + p n−m Z p ), it follows that  y ∈ π

a −1 0 0 1

. 11+ pn−m Z p = 1a+ pn Z p . This shows that an arbitrary element of

S(Q×p )

can be obtained, via the action of B1 (Q p ), from the single element 11+ pZ p . To complete the proof that S(Q p ) is irreducible we must also show that this process can be reversed, i.e., given an arbitrary linear combination of characteristic functions 1a+ pn Z p , we can recover 11+ pZ p via the action of B1 (Q p ). The task of passing from an arbitrary linear combination of characteristic functions back to 11+ pZ p is left as an exercise. Step 5: The Bruhat-Schwartz space S X (Q×p ) = S(Q×p ) · X ⊂ K. Fix x ∈ X. Take v ∈ V any element whose image in X = V /V0 is x. Then it follows immediately from the definition (6.7.4) that f v (1) = x. Set f 1 := f v and define     1 1 p −1

. f f f 2 := − π 1 1 . 0 1 1 − e p ( p −1 )    Then f 2 (1) = x and f 2 is supported on y ∈ Q×p  |y| p ≥ 1 . Next, take n so that the support of f 1 (and hence also of f 2 ) is contained in p −n Z p , and define ⎛

 1 1 f 3 (y) := ⎝ n π 0 p j=0 pn −1

j 1



⎞ 1 . f 2 ⎠ (y) = ⎝ n e p ( j y)⎠ f 2 (y). p j=0 ⎞

⎛

pn −1

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Theory of admissible representations of GL(2, Qp )

Then f 3 (1) = x and f 3 is supported on Z×p . Now, let N be the least positive integer such that f 3 is constant on cosets of 1 + p N Z p . Choose: ω : Z×p → C× , a normalized unitary character, as in Definition 6.5.1, which has conductor p N as in Definition 2.1.14. Then ω is constant on cosets of 1+ p N Z p in Z×p . By (2.1.15), there are precisely ϕ( p N ) = ( p−1) p N −1 such characters ω (mod p N ) and they satisfy the orthogonality relation  1, if j − 1 ∈ p N Z p , 1 ω( j) = (6.7.9) N ϕ( p ) 0, otherwise. N ω (mod p )

Now, for any such character ω (mod p N ), define    u 0 −1

. f 3 (y) d × u f ω (y) := ω(u) · π 0 1 Z×p  p 1 j −1

= ω( j) · π 0 ( p − 1) p N −1 j=1 N

0 1

 . f 3 (y).

( j, p)=1

Clearly, f ω is in K and is supported on Z×p . On the other hand f ω (a · y) = ω(a) f ω (y),

(∀ a ∈ Z×p , y ∈ Q×p ).

Thus f ω is in S(Q×p ) · f ω (1). It follows from Step 4 that the whole space S(Q×p )· f ω (1) is contained in K. On the other hand, by the orthogonality relation (6.7.9), ⎡ ⎤   pN 1 j 0 −1 ⎦

⎣ . f 3 (1) f ω (1) = ω( j) ·π 0 1 ϕ( p N ) N N ω (mod p )

ω (mod p )

j=1 ( j, p)=1

= π



1 0 0 1

 . f 3 (1)

= x. It follows from Step 4 that for every x ∈ X, the whole space S(Q×p ) · x is contained in K. This completes the proof of Step 5. Step 6: K is the sum of the subspaces S X (Q×p ) and π (ω0 ) . S X (Q×p ). Fix f ∈ S X (Q×p ) = S(Q×p ) · X. Since (π , K) is irreducible, this implies that   K is spanned by {π (g) . f | g ∈ G L(2, Q p )}. Let g = ac db be in G L(2, Q p ) but not in B(Q p ). Then we have the identity g = hw0 h where       1 ac−1 0 1 −c −d

, w0 = . h= , h = 0 1 −1 0 0 b − adc−1

6.8 The Kirillov model of the principal series representation

221

Let F := π (h ) . f. Then F ∈ S X (Q p ) because the space S X (Q×p ) is invariant under B1 (Q p ) by Step 1 and is invariant under the center (diagonal matrices in G L(2, Q p )) by Proposition 6.1.10. Consequently   π (g) . f = π (h) . π (w0 ) . F     = π (h) . π (w0 ) . F − π (w0 ) . F + π (w0 ) . F. (6.7.10)     The function π (h) . π (w0 ) . F − π (w0 ) . F (y) vanishes for all values of y such that e p (ac−1 y) = 1. It also vanishes for y outside of a compact subset of Q p by Step 2. It follows that (6.7.10) is in S X (Q×p ) + π (w0 ) . S X (Q×p ). 

6.8 The Kirillov model of the principal series representation In this section, we will explicitly construct a Kirillov model, as defined in Definition 6.7.1, for the principal series representations defined in Definition 6.5.3. In Section 6.6, we began studying the space of local Whittaker functions, as defined in Definition 6.4.1, obtained from elements of a principal series representation via Jacquet’s integral. We saw that computing the value at an upper triangular matrix is easier than for a general matrix. It will turn out that an element of  Wp (s, ω) is completely determined by its values on matrices

of the form 0y 01 with y ∈ Q×p . Very roughly, the Kirillov space we will construct is the space of restrictions of Whittaker functions to matrices of this form. To make this notion precise we introduce the following definition. Definition 6.8.1 (Restriction function) Let restr denote the linear functional {functions : G L(2, Q p ) → C} → {functions : Q×p → C} defined by

  restr( f ) (y) = f



y 0

0 1

 ,

for every f : G L(2, Q p ) → C and all y ∈ Q×p . Definition 6.8.2 (Kirillov space of the principal series representation) Let s = (s1 , s2 ) ∈ C2 , and ω = (ω1 , ω2 ) be a pair of normalized unitary characters of Q×p as in Definition 6.4.1. Let K p (s, ω) denote the space 

  restr ◦ Js,ω ( f )  f ∈ V p (s, ω) , where the map Js,ω is defined in Definition 6.6.1 and the restriction map restr is defined in Definition  6.8.1. We call this the Kirillov space of the principal series representation π, V p (s, ω) , which was defined in Definition 6.5.3.

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Theory of admissible representations of GL(2, Qp )

Lemma 6.8.3 Let s = (s1 , s2 ) ∈ C2 , and ω = (ω1 , ω2 ) be a pair of normalized unitary characters of Q×p as in Definition 6.5.1. When ω2 = ω1 and p s2 −s1 = 1, the function g → ω1 (det g)| det g|sp1 ,

(g ∈ G L(2, Q p ))

is an element of the kernel of Js,ω in V p (s, ω). Proof For s = (s1 , s2 ) ∈ C2 and ω = (ω1 , ω2 ) a pair of normalized unitary characters, let us define a function δs,ω : G L(2, Q p ) → C by setting    y1 x · k := |y1 |sp1 ω1 (y1 ) · |y2 |sp2 ω2 (y2 ). δs,ω (6.8.4) 0 y2 Clearly, δs,ω ∈ V p (s, ω). A straightforward modification of the computations done in Definition 6.4.3 shows that if ω1 = ω2 then δs,ω (as a function of s = (s1 , s2 ) ∈ C2 ) is in the kernel of the analytic continuation of Js,ω when p s1 −s2 = 1. Furthermore, it is easily seen that δs,(ω1 ,ω1 ) (g) = ω1 (det g) · | det g|sp1 . This completes the proof.  Proposition 6.8.5 (The kernels of Js,ω and restr ◦ Js,ω ) Fix a prime p, a pair of complex numbers s = (s1 , s2 ) ∈ C2 , and a pair of normalized unitary characters ω = (ω1 , ω2 ) of Q×p as in Definition 6.5.1. Let K p (s, ω) be the Kirillov space as in Definition 6.8.2, and let W p (s, ω) be the space defined in Definition 6.6.15 (which is the Whittaker model, except when ω1 = ω2 , and p s1 −s2 = 1). Then the linear mappings restr ◦ Js,ω : V p (s, ω) → K p (s, ω) Js,ω : V p (s, ω) → W p (s, ω) are vector space isomorphisms, except in the case when ω1 = ω2 , and p s1 −s2 = 1, in which case       ker restr ◦ Js,ω = ker Js,ω = span δs,ω , where δs,ω is defined in (6.8.4). Proof Let Bn (ω) be a basis for Flat Kp n (ω) as in Definition 6.6.4. We shall prove that the functions n ( f ∈ Bn (ω), f =/ f 0,1 ),

restr ◦ Js,ω ( f ), are linearly independent. Let  0 n n h 1,d (u) = f 1,d 1

1 u



 ; s1 , s2 ,

(for u ∈ Q p ),

6.8 The Kirillov model of the principal series representation

223

and define h npc,1 for c =/ 0 similarly. Then the computations which appeared in the proofs of Propositions 6.6.7 and 6.6.10 showed that these functions are all smooth, compactly supported functions with disjoint supports, and that  n  (y) = ω2 (y)|y|sp2 +1 h n1,d (y), (∀y ∈ Q×p , n ∈ N, 0 ≤ d < p n ), restr ◦ Js,ω f 1,d  n  (y) = ω2 (y)|y|sp2 +1 h npc,1 (y), (∀y ∈ Q×p , n ∈ N, 0 < c < p n−1 ). restr ◦Js,ω f pc,1 Linear independence of the functions Js,ω ( f ) (y) with f ∈ Bn (ω) follows. We may now deduce that the dimension of the kernel of restr ◦ Js,ω is less than or equal to 1 in all cases. In the case p s1 −s2 = 1, ω1 = ω2 , it follows from Lemma 6.8.3 that the kernel of Js,ω has dimension at least one. Clearly, the kernel of Js,ω is contained in that of the composite map restr ◦ Js,ω . This completes the proof in this case. s1 −s2 =/ 1, we must To complete the proof  n in  the case when ω1 =/ ω2 or p show that restr ◦ Js,ω f 0,1 is not in the span of the others. It follows from Propositions 6.6.7, 6.6.10 that for y ∈ Q×p with |y| p < p −N , and N sufficiently large, that each of the others is of the form C · ω2 (y)|y|sp2 +1 for some constant C (depending on  fn and  N ). On the other hand, by Theorem 6.6.13, it follows is of the form that restr ◦ Js,ω f 0,1  C1 ω2 (y)|y|sp2 +1 + C2 ω1 (y)|y|sp1 , ω1 =/ ω2 or p s1 −s2 =/ p, C1 |y|sp2 +1 + C2 (log p |y| p ) · |y|sp2 +1 , ω1 =/ ω2 and p s1 −s2 = p, with C2 =/ 0. Note that this is precisely the place where we need to assume that ω1 =/ ω2 or p s1 −s2 =/ 1. It follows that the kernel of restr ◦ Js,ω , and hence that of Js,ω , is trivial in this case.  Proposition 6.8.6 (Whittaker model of a principal series representation) Fix a prime p, a pair of complex numbers s = (s1 , s2 ) ∈ C2 , and a pair of normalized unitary characters ω = (ω1 , ω2 ) of Q×p as in Definition 6.5.1. Assume that either ω1 =/ ω2 or p s1 −s2 =/ 1. Let   W p (s, ω) = Js,ω V p (s, ω) .   Let π denote right translation. Then the representation π, W (s, ω) is, in p  being fact, isomorphic  to π, V p (s,  ω) with a specific isomorphism   given by Js,ω . We call π, W p (s, ω) the Whittaker model of π, V p (s, ω) . Proof This follows immediately from Proposition 6.8.5. Corollary 6.8.7 (Local representation from a Maass form is admissible)  Let π, W p ( f ) be the local representation coming from a level one, weight zero Maass form f as in Definition 6.3.3. Let s1 , s2 be related to f as in Theorem 6.4.12 and assume that p s1 −s2 =/ 1, p. Then π, W p ( f ) is admissible.

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Theory of admissible representations of GL(2, Qp )

Proof Let s1 , s2 be related to f as in Theorem 6.4.12. Then Theorem 6.4.12 shows that if p s1 −s2 =/ 1, p, then W p (g; f ) is in W p (s1 , s2 ). Since the action in both (π, V ) and W p (s1 , s2 ) is by right translation, and since W p (g; f ) generates W p ( f ), it follows that all of W p ( f ) is contained in W p (s1 , s2 ), which is isomorphic to V p (s1 , s2 ), by Proposition 6.8.6, which is admissible by Proposition 6.5.5.  Definition 6.8.8 (Action on the Kirillov space) Consider K p (s, ω), the Kirillov space as defined in Definition 6.8.2. We define an action π of G L(2, Q p ) on the Kirillov space K p (s, ω) by 6  7   π (g) . restr ◦ Js,ω ( f ) := restr ◦ Js,ω (π (g) . f ) , for all g ∈ G L(2, Q p ), f ∈ V p (s, ω). Proposition 6.8.9 (Action on the Kirillov space is well defined) The action π : G L(2, Q p ) → G L(K p (s, ω)) given in Definition 6.8.8 is well defined. The action of an upper triangular matrix is explicitly given by:   a x

. φ(y) = e p (x y)φ(ay), π 0 1   r 0 . φ(y) = ω1 (r )ω2 (r )|r |sp1 +s2 φ(y), π 0 r for all φ ∈ K p (s, ω), x ∈ Q p , and all a, r, y ∈ Q×p . Proof  To check that the action is well defined it is necessary to show that ker restr ◦ Js,ω is an invariant subspace of π. This follows easily from Proposition 6.8.5. We leave the explicit computation of the action of an upper triangular matrix to the reader.  Proposition 6.8.10 (Kirillov model of the principal series representation) The Kirillov space K p (s, ω), as defined in Definition 6.8.2, together with the explicit action π as given in Definition 6.8.8 is a Kirillov representation as in Definition 6.7.1. If we are in the situation where p s1 −s2 =/ 1 or ω1 =/ ω2 , this is the Kirillov model of the principal series representation. Proof This follows from Definitions 6.7.1, 6.8.2 and Proposition 6.8.9. It is only necessary to show that every function in K p (s, ω) is locally constant. We leave this exercise to the reader.  For the next theorem, it is necessary to recall the Bruhat-Schwartz space:  5   f is locally constant, and ∃N f >  f > 0 such that × ×  . S(Q p ) := f : Q p → C  f (y) = 0 if |y| p <  f or |y| p > N f

6.8 The Kirillov model of the principal series representation

225

Theorem 6.8.11 (Characterization of the Kirillov space) Fix a prime p, a pair of complex numbers s = (s1 , s2 ) ∈ C2 , and a pair of normalized unitary characters ω = (ω1 , ω2 ) of Q×p as in Definition 6.5.1. Then the Kirillov space K p (s, ω), defined in Definition 6.8.2, is equal to the space of all functions of the form  ⎧ s1 s2 +1 C ·1Z p (y) + f (y), if p s1 −s2 =/ 1, p ω (y)|y| + C ω (y)|y| ⎪ 1 1 2 2 p p ⎪ ⎪ ⎪ ⎪ ⎪ or ω1 =/ ω2 , ⎪ ⎪ ⎪ ⎨ C ω (y)|y|s2 +1 · 1 (y) + f (y), s1 −s2 if p = 1, 2 2 Zp p φ(y) = ⎪ and ω1 = ω2 , ⎪ ⎪   ⎪ ⎪ ⎪ ω2 (y)|y|s2 +1 · C1 + C2 log |y| p ·1Z (y) + f (y), if p s1 −s2 = p, ⎪ p p ⎪ ⎪ ⎩ and ω1 = ω2 , (6.8.12)   for C1 , C2 ∈ C, y ∈ Q×p , and some f ∈ S Q×p . Furthermore, any function φ of the form (6.8.12), with corresponding coefficients C1 , C2 =/ 0, generates K p (s, ω), i.e., the elements π (g) . φ, with g ∈ G L(2, Q p ) span K p (s, ω). Proof We first consider the case when p s1 −s2 =/ 1, p or ω1 =/ ω2 . Let   s2 +1 s1 V ω2 | | p , ω1 | | p , denote the space of functions φ : Q×p → C of the form   C1 ω1 (y)|y|sp1 + C2 ω2 (y)|y|sp2 +1 · 1Z p (y) + f (y),   with C1 , C2 ∈ C, y ∈ Q×p , and f ∈ S Q×p . We have already proved, in   Theorem 6.6.13, that K p (s, ω) ⊂ V ω2 | |sp2 +1 , ω1 | |sp1 .   The subspace of V ω2 | |sp2 +1 , ω1 | |sp1 consisting of elements such that C1 and C2 are both zero is equal to the space S(Q×p ) of locally constant, compactly supported functions on Q×p . It is clear from Proposition 6.8.9 that S(Q×p ) is   a B(Q p )-invariant subspace of V ω2 | |sp2 +1 , ω1 | |sp1 . We next prove that this subspace is contained in K p (s, ω). By Step 4 of the proof of Theorem 6.7.2, it suffices to show that K p (s, ω) contains a non-zero element of S(Q×p ). Let φ 0. Choose be any non-zero element of K p (s, ω). Choose y0 such that φ(y  0 ) =/

x0 such that e p (x0 y0 ) =/ 1. Consider the function φ := π 10 x10 . φ − φ. Clearly φ ∈ K p (s, ω). Moreover, φ (y) = 0 whenever |y| p ≤ |x0 |−1 p , because this forces e p (x0 y) to equal 1. On the other hand φ (y0 ) =/ 0. It follows from Propositions 6.6.7, 6.6.10, and 6.6.12 that the Kirillov space K p (s, ω) is contained in V(ω2 | |sp2 +1 , ω1 | |sp1 ) whenever p s1 −s2 =/ p, and that it maps surjectively onto the two-dimensional quotient space +  S(Q×p ), (6.8.13) V ω2 | |sp2 +1 , ω1 | |sp1

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Theory of admissible representations of GL(2, Qp )

except in the case when p s1 −s2 = 1. This completes the proof that K p (s, ω) = V(ω2 | |sp2 +1 , ω1 | |sp1 ). To prove the second part of the theorem in the case when p s1 −s2 =/ 1, p, or ω1 =/ ω2 , it suffices to prove that the two-dimensional quotient (6.8.13) is spanned by the translates of any element of V(ω2 | |sp2 +1 , ω1 | |sp1 ) with C1 , C2 =/ 0. Take   φ(y) := C1 ω1 (y)|y|sp1 + C2 ω2 (y)|y|sp2 +1 · 1Z p (y) + f (y),   and take a such that ω2 (a)|a|sp2 +1 =/ ω1 (a)|a|sp1 . Then π a0 01 . φ(y) is equal to   C1 ω1 (a)|a|sp1 ω1 (y)|y|sp1 + C2 ω2 (a)|a|sp2 +1 ω2 (y)|y|sp2 +1 · 1Z p (ay) + f (ay). These two elements map to linearly independent elements of the quotient (6.8.13). When ω1 = ω2 and p s1 −s2 = 1, the only difference is that the coefficient of |y|sp1 in Proposition 6.6.12 vanishes, so that the Kirillov space actually lies inside the subspace V(ω2 | |sp2 +1 ). We leave the last case as an exercise for the reader. In this connection, see the remark following Proposition 6.6.12.  Theorem 6.8.14 (The Kirillov model of the principal series is irreducible) Fix a prime p, a pair of complex numbers s = (s1 , s2 ) ∈ C2 , and a pair of normalized unitary characters ω = (ω1 , ω2 ) of Q×p as in Definition 6.5.1. Assume that either ω1 =/ ω2 or p s1 −s2 =/ 1, p 2 . Then the Kirillov model (π , K p (s, ω)) as defined in Proposition 6.8.10 is an irreducible representation of G L(2, Q p ). Corollary 6.8.15 (The principal series representation is irreducible) Fix a prime p, a pair of complex numbers s = (s1 , s2 ) ∈ C2 , and a pair of normalized unitary characters ω = (ω1 , ω2 ) of Q×p as in Definition 6.5.1. Assume that either ω1 =/ ω2 or p s1 −s2 =/ 1, p 2 . Then the principal series representation (π, V p (s, ω)), as defined in Definition 6.5.4, is an irreducible representation of G L(2, Q p ). Corollary 6.8.16 (The local representation coming from a level one Maass form is irreducible) Let f be a Maass form for S L(2, Z) as in (6.4.10). Let √ α p ∈ C be given by (6.4.11). Let s1 , s2 ∈ C be defined by p s1 = p · α p , and s2 = −s1 . Assume that p s1 −s2 =/ 1, p 2 . Then the local representation (π, W p ( f )), as defined in Definition 6.3.3, is an irreducible representation of G L(2, Q p ). Remarks (1) Note the new condition p s1 −s2 = p 2 , which appears in Definition 6.8.14, and Corollaries 6.8.15, and 6.8.16. (2) Corollary 6.8.16 can be generalized to the local representation coming from a Maass form of arbitrary level N > 1, provided p  |N .

6.8 The Kirillov model of the principal series representation

227

Proof of Definition 6.8.14, and Corollaries 6.8.15, 6.8.16 Corollary 6.8.15 is an immediate consequence of Theorem 6.8.14, because the Kirillov model is, by construction, isomorphic to the principal series representation of which it is a model (see Definition  6.6.14 of a model). Similarly, by Proposition 6.8.6, the Whittaker model π, W p (s, ω) is isomorphic to both π, V p (s, ω)  6.8.7, the local repand π , K p(s, ω) . As noted in the proof of Corollary  resentation π, W p ( f ) is a subrepresentation of π, W p (s1 , s2 ) , for the from f as in Theorem 6.4.12. Now that we know values   of s1 , s2 obtained (s , s ) is irreducible, it follows that they are equal, and that that π, W  p 1 2 π, W p ( f ) is irreducible. We now turn to the proof of Theorem 6.8.14. For simplicity, we shall, for the most part, leave the case p s1 −s2 = p, ω1 = ω2 to the reader. The only difference in this case is that |y|sp1 must be replaced by |y|sp2 +1 log |y| p . To show that K p (s, ω) is irreducible, it suffices to show that K p (s, ω) is generated by an arbitrary element φ ∈ K p (s, ω), in the sense that the functions π (g) . φ, with g ∈ G L(2, Q p ) span K p (s, ω). We have shown, in Theorem 6.8.11 that this is the case whenever the constants C1 , C2 which occur when φ is written in the form (6.8.12) are both non-zero. By the same method used in Step 5 in the proof of Theorem 6.7.2, the subspace of K p (s, ω) generated by any element contains S(Q×p ). Thus, it suffices to exhibit one single element φ of S(Q×p ), and one single element g of G L(2, Q p ) such that   π (g) . φ (y) = C1 · ω2 (y)|y|sp2 +1 · 1Z p (y) + C2 · ω1 (y)|y|sp1 · 1Z p (y) + f 0 (y), with f 0 ∈ S(Q×p ) and C1 , C2 ∈ C which are both non-zero.  1  n 1 , where f 1,d We will show that the choice φ = restr ◦ Js,ω f 1,0 − f 1,1 is   0 1 defined as in Definition 6.6.4 and g = −1 0 do what we want. By Proposition 6.6.7, we see at once that   φ(y) = p −1 ω2 (y)|y|sp2 +1 1 − e p (−y) · 1 p−1 Z p −Z p (y) ∈ S(Q×p ). Observethat if K 1 = {k  ∈ GL(2, Z p ) | k ≡ I2 (mod p)}, as in Definition 0 1 = K 1 . It follows that, for k ∈ G L(2, Z p ), 6.6.4, then −1 0 · K 1 · 01 −1 0     0 −1 0 −1 k· ∈ · K1 1 0 1 0 1 1 if and only if k ∈ K 1 . Hence, by the definition of f 1,0 and f 0,1 given in   0 −1 1 1 Definition 6.6.4, we deduce that π . f 1,0 = f 0,1 . Next, observe 1 0 that :   ;−1    −1 −1 1 0 −1 0 −1 0 −1 · · · ∈ K1. 0 −1 1 p−1 1 1 1 0

Theory of admissible representations of GL(2, Qp )         0 −1 −1 1 0 −1 It follows that k· 01 −1 ∈ ·K · ·k1 , if and only if k= 1 1 p−1 0 −1 0 1 1   0 −1 1 1 . f 1,1 = f 1, p−1 . with k1 ∈ K 1 . From this we deduce that π 1 0 Consequently    1  0 −1

. φ(y) = restr ◦ Js,ω f 0,1 − f 1,1 p−1 (y). π 1 0

228

It follows from Propositions  6.6.7 and 6.6.12 that for y ∈ Q p with |y| p 0 −1

. φ(y) is given by sufficiently small π 1 0 ⎧ C1 (s1 , s2 )ω1 (y)|y|sp2 +1 + C2 (s1 , s2 )ω1 (y)|y|sp1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ −2 p −1 ω2 (y)|y|sp2 +1 + ( p −1 − 1)ω2 (y)|y|sp2 +1 log p |y| p , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ − p −1 ω2 (y)|y|sp2 +1 + G(ω1−1 ω2 ) p −s1 ω1 (y)|y|sp1 ,

if ω1 = ω2 , p s1 −s2 =/ p, if ω1 = ω2 , p s1 −s2 = p, if ω1 =/ ω2 ,

where p  is the conductor of ω1−1 ω2 , 

G(ω1−1 ω2 ) :=

p j=1 ( j, p)=1

ω1−1 ω2 ( j)e p



−j p

 ,

is the Gauss sum, and the constants C1 and C2 are given by   1 − p −1 p s2 −s1 +1 − p −1 , C1 (s1 , s2 ) = 1 − p s2 −s1 +1 1 − p s2 −s1 C2 (s1 , s2 ) = . 1 − p s1 −s2 −1 The fact that G(ω1−1 ω2 ) =/ 0 follows from (2.1.15) (which relates the character ω1−1 ω2 to a Dirichlet character (mod p  )) and the computation of Gauss sums in Chapter 9 of [Davenport, 2000]. The constant C1 (s1 , s2 ) vanishes if and only if p s1 −s2 = p 2 . Thus, we have exhibited an element φ as required, except when ω1 = ω2 and p s1 −s2 = p 2 . 

6.9 Haar measure on G L(2, Q p ) In this section, we define a translation invariant measure on G L(2, Q p ), similar to the Haar measure on Q p defined in Section 1.5. This will allow us to integrate locally constant functions on G L(2, Q p ). Recalling (6.1.3), every compact open subset of G L(2, Q p ) is a finite disjoint union of cosets (right or left) of the subgroup

6.9 Haar measure on GL(2, Q p )

229

   K n = k ∈ G L(2, Z p )  k − I2 ∈ p n · Mat (2, Z p ) ,

(n = 1, 2, 3, . . . ) (6.9.1) for some n. It, therefore, suffices to define our measure on these cosets. Definition 6.9.2 (Invariant measure on G L(2, Q p )) Let K n be given as in (6.9.1). A measure μ on the set of all compact open subsets of G L(2, Q p ) is said to be invariant if it assigns to every coset of K n the same measure, for every n. By this we mean that there is a constant cn which depends only on n such that μ(g1 · K n ) = μ(K n · g2 ) = cn ,

(∀g1 , g2 ∈ G L(2, Q p )).

Let μ be an invariant measure on G L(2, Q p ). It follows easily that the corresponding integral on locally constant functions satisfies    f (gx) dμ(x) = f (xg) dμ(x) = f (x) dμ(x), (6.9.3) G L(2,Q p )

G L(2,Q p )

G L(2,Q p )

for all g ∈ G L(2, Q p ), and all locally constant functions f : G L(2, Q p ) → C such that any of the integrals in (6.9.3) converge. It is possible to extend this to a measure on the σ -algebra of all Borel subsets of G L(2, Q p ) with the same invariance properties. Such a measure is called a “Haar measure.” While we do not need this much detail, we will use the term “Haar measure” for our invariant integral. Lemma 6.9.4 (Existence and uniqueness of Haar measure) There is a nontrivial invariant measure μ on the set of compact open subsets of G L(2, Q p ). An invariant measure μ on the set of compact open subsets of G L(2, Q p ) is uniquely determined by μ(G L(2, Z p )). In particular, any two non-trivial invariant measures are scalar multiples of one another. Proof It follows easily from the definitions that μ (g · K n ) =

  1 · μ G L(2, Z p ) [G L(2, Z p ) : K n ]

for all g ∈ G L(2, Q p ), any positive integer n, and any invariant measure μ. As noted above, specifying the measure of the coset g · K n for each g and n completely determines μ. It follows that if μ1 and μ2 are any two invariant measures, then μ2 (S) =

μ2 (G L(2, Z p )) · μ1 (S) μ1 (G L(2, Z p ))

for every compact open subset S.

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230

Now, it is clear that we can define a measure μ on the set of compact open subsets of G L(2, Q p ) simply by fixing a positive constant c and declaring that 1 · c for all g ∈ G L(2, Q p ) and all n ≥ 1. By construcμ (g · K n ) = [G L(2,Z p ):K n ] tion, this measure is left invariant: μ(g · K ) = μ(K ) for any compact open set K . To prove that the measure thus obtained will be right invariant, we must 1 · c for all g ∈ G L(2, Q p ) verify that μ(K n · g) is also equal to [G L(2,Z p ):K n ] and all n ≥ 1. By left invariance, this is equivalent to showing that the group g −1 K n g has the same measure as K n for all g ∈ G L(2, Q p ). To prove this, consider the commensurator subgroup K n ∩ g −1 K n g. It is open, and hence of finite index in both K n and g −1 K n g (each of which is compact). We claim that the two indices are the same. Using the p-adic Cartan decomposition given in Proposition 4.2.3 we may express g as   e 0 p1 · k2 , k1 · 0 p e2 with e1 , e2 integers such that e1 ≤ e2 , and k1 , k2 ∈ G L(2, Z p ). Using the fact that K n is normal in G L(2, Z p ), we reduce to the case when k1 = k2 = I2 . This case is easily checked by hand. See Exercise 6.22 for more details.  Definition 6.9.5 (Normalized Haar measure on G L(2, Q p )) We shall denote by  f (g) d × g G L(2,Q p )

the integral corresponding to the unique invariant measure which assigns G L(2, Z p ) a measure of 1. For A a subset of G L(2, Q p ) which is both closed and open, we define   f (g) d × g = 1 A (g) · f (g) d × g, A

G L(2,Q p )

where 1 A is the characteristic function of A. Remark The requirement that A is simultaneously open and closed is necessary and sufficient to ensure that 1 A (g) · f (g) is again a locally constant function. Proposition 6.9.6 (Factorization of Haar measure) For every locally constant function f : G L(2, Q p ) → C, we have the identity  f (g) d × g G L(2,Q p )



=



Q×p

 Q×p



 f

Qp

G L(2,Z p )

a1 0

0 a2

  1 · 0

x 1



 ·k

d × k d x d × a1 d × a2 ,

6.9 Haar measure on GL(2, Q p )

231

where d × k is the normalized Haar measure on G L(2, Q p ) (defined in Definition 6.9.5), restricted to the open compact subgroup G L(2, Z p ), d x is the additive Haar measure on Q p , given in Example 1.5.4 and d × ai (i = 1, 2) is the multiplicative Haar measure on Q×p , given in Definition 1.5.7. Proof Both sides of the identity of Proposition 6.9.6 are certainly linear in f, so it is enough to verify that the identity holds for f = 1g·K n , where, for n ∈ N, the subgroup K n is given by (6.9.1), and g ∈ G L(2, Q p ). This computation is left to the reader.  Lemma 6.9.7 Let ϕ : G L(2, Q p ) → C be a locally constant, compactly supported function, and let  ϕ0 (g) :=



Q×p





Q×p

Qp

ϕ

a1 0

0 a2

  1 · 0

x 1



 ·g

d x d × a1 d × a2 . (6.9.8)

Then ϕ0 (g) satisfies  ϕ0

b1 0

0 b2

  1 · 0

u 1



 ·g

   b1  =   · ϕ0 (g), b

(6.9.9)

2 p

for all b1 , b2 ∈ Q×p , u ∈ Q p , and g ∈ G L(2, Q p ). Furthermore, every locally constant function F : G L(2, Q p ) → C, which satisfies (6.9.9) for all b1 , b2 ∈ Q×p , u ∈ Q p , and g ∈ G L(2, Q p ) is equal to ϕ0 for some ϕ. Proof The first statement follows from a simple change of variables. 

ϕ0

b1 0  

=

Q×p

 =

Q×p

 Q×p

Q×p

    1 u · ·g 0 1   1 a1 0 ϕ 0 0 a 2 Qp

0 b2 



 Qp

ϕ

a1 b1 0

x 1



  0 1 · a2 b2 0

b1 0 b2 x b1

0 b2 +u 1



1 u 0 1



 ·g



 ·g

· d x d × a1 d × a2 d x d × a1 d × a2 .

The factor |b1 /b2 | p appears from the change of variables x → x · bb12 , by (1.5.6). To verify the second part, take F : G L(2, Q p ) → C, which satisfies (6.9.9), and let  F(g), g ∈ G L(2, Z p ), ϕ(g) = 0, otherwise.

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232

Then, for k ∈ G L(2, Z p ), we have  ϕ0 (k) :=  =  =



Q×p

Q×p



Z×p



Z×p



 Qp



Z×p



Z×p

Zp

Zp

ϕ 

ϕ

a1 0

a1 0

    1 x 0 · · k d x d × a1 d × a2 a2 0 1     1 x 0 · · k d x d × a1 d × a2 a2 0 1

ϕ (k) d x d × a1 d × a2

= ϕ(k) = F(k). It then follows by (6.9.9) that ϕ0 and F agree on all of G L(2, Q p ).  Corollary 6.9.10 Let F : G L(2, Q p ) → C be a locally constant function which satisfies  F

b1 0

0 b2

  1 · 0

y 1



 ·g

   b1  =   · F(g), b2 p

(6.9.11)

for all b1 , b2 ∈ Q×p , y ∈ Q p , and g ∈ G L(2, Q p ). Then 

F(kg) d × k = G L(2,Z p )



F(k) d × k, G L(2,Z p )

for all g ∈ G L(2, Q p ). Proof By Lemma 6.9.7, we may choose a locally constant, compactly supported function ϕ : G L(2, Q p ) → C, such that the function ϕ0 given in (6.9.9) is equal to F. It then follows immediately from (6.9.9) and Proposition 6.9.6, that   F(kg) d × k = ϕ(hg) d × h. G L(2,Z p )

G L(2,Q p )

Corollary 6.9.10 is now an immediate consequence of the invariance of the Haar measure d × h, as in (6.9.3). 

6.10 The special representations Fix a prime p, a pair of complex numbers s = (s1 , s2 ) ∈ C2 , and a pair of normalized unitary characters ω = (ω1 , ω2 ) of Q×p as in Definition 6.5.1. Assume that either ω1 =/ ω2 or p s1 −s2 =/ 1, p 2 . We have shown in Corollary 6.8.15, that under these assumptions, the principal series representation (π, V p (s, ω)), as defined in Definition 6.5.4, is an irreducible representation of G L(2, Q p ). In

6.10 The special representations

233

the special case that ω1 = ω2 and p s1 −s2 = 1 or p 2 , then the principal series representation will be reducible. There are two cases to consider. When ω1 = ω2 and p s1 −s2 = 1, we will show that there is a one-dimensional invariant subspace of V p (s, ω) having the property that the quotient of V p (s, ω) by that subspace will be an infinite dimensional irreducible representation of G L(2, Q p ) which is called a “special representation.” When ω1 = ω2 and p s1 −s2 = p 2 , we’ll show that there is an infinite dimensional subspace of V p (s, ω) which is an irreducible representation of G L(2, Q p ) and is also called a “special representation.” In this case, the quotient is a one-dimensional irreducible representation of G L(2, Q p ). The special representation for ω1 = ω2 and p s1 −s2 = 1 : Let s = (s1 , s2 ) and ω = (ω1 , ω2 ) as before. We shall assume we are in the case where ω1 = ω2 and p s1 −s2 = 1. Define the function δs,ω (g) := ω1 (det g) · | det g|sp1 for g ∈ G L(2, Q p ). We proved in Proposition 6.8.5 that the map Js,ω (which maps principal series to Whittaker functions) has a one-dimensional kernel which we denote as ker(Js,ω ), where    (6.10.1) ker(Js,ω ) = c · δs,ω  c ∈ C . It follows that the image of Js,ω is no longer a model for the principal series, but rather is a model for the quotient V p (s, ω) ker(Js,ω ). Recall that the principal series (π, V p (s, ω)) is a representation of G L(2, Q p ) with action π given by   π (h) . f (g) = f (gh), f ∈ V p (s, ω), g, h ∈ G L(2, Q p ) .

To define a special representation with space V p (s, ω) ker(Js,ω ), we must show that

the action π is well defined on the quotient vector space V p (s, ω) ker(Js,ω ). This is easily accomplished by defining an action of π on cosets by letting   π (h) . f (g) + δs,ω (g) · C = f (gh) + δs,ω (g) · C,   ∀ f ∈ V p (s, ω), g, h ∈ G L(2, Q p ) . (6.10.2) This is well defined because the vector space, ker(Js,ω ) = δs,ω · C, is invariant under the action of π. Indeed, for fixed h ∈ G L(2, Q p ) and all g ∈ G L(2, Q p ), we have π (h) . δs,ω (g) · C = δs,ω (gh) · C = δs,ω (g) · ω1 (det h)| det h|sp1 · C = δs,ω (g) · C.

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Theory of admissible representations of GL(2, Qp )

Definition 6.10.3 (Special representation for ω1 = ω2 and p s1 −s2 = 1) Fix a prime p, a pair of complex numbers s = (s1 , s2 ) ∈ C2 , and a pair of normalized unitary characters ω = (ω1 , ω2 ) of Q×p as in Definition 6.5.1. Let V p (s, ω) be the complex vector space of principal series defined in Definition 6.5.2. For ω1 = ω2 and p s1 −s2 = 1, let ker(Js,ω ) be as  in (6.10.1). Let π denote  the action of right translation as in (6.10.2). Then π, V p (s, ω) ker(Js,ω ) is defined to be a special representation for ω1 = ω2 and p s1 −s2 = 1. Proposition 6.10.4 The special representation for ω1 = ω2 and p s1 −s2 = 1, as defined in Definition 6.10.3, is irreducible.   Proof Let W p (s, ω) denote the space Js,ω V p (s, ω) which is well defined even when ω1 = ω2 and p s1 −s2 = 1. In this case, it is a space of local Whittaker functions as in Definition 6.4.1 which is isomorphic to the special representation for ω1 = ω2 and p s1 −s2 = 1. It follows from Proposition 6.8.5 that restr is an isomorphism from W p (s, ω) to K p (s, ω). If ω1 = ω2 and p s1 −s2 = 1, then W p (s, ω) and K p (s, ω) are the Whittaker and Kirillov models of the special representation for ω1 = ω2 and p s1 −s2 = 1. Furthermore, we see from Theorem 6.6.13 that in the case ω1 = ω2 and s1 −s2 = 1, the Kirillov space K p (s, ω) is contained in the space V(|y|sp2 +1 ) p given by 

   φ : G L(2, Q p ) → Cφ(y) = c · ω2 (y) |y|sp2 +1 + f (y), c ∈ C, f ∈ S(Q×p ) . (6.10.5) A straightforward adaptation of the proof of Proposition 6.6.12 shows that, in fact, K p (s, ω) is equal to this space of functions when ω1 = ω2 and p s1 −s2 = 1. Furthermore, by adapting the proof of Theorem 6.6.15, we show that the space is generated (as a representation) by any element φ in (6.10.5) with c =/ 0. Next, we need to exhibit an example of a function f ∈ V p (s, ω), and an element h of G L(2, Q p ) such that restr ◦ Js,ω ( f ) is a non-zero element of S(Q×p ), and a representation of the form   restr ◦ Js,ω π (h) . f (y) = c · ω2 (y)|y|sp2 +1 + f 1 (y), (∀y ∈ Q×p ),   with f 1 ∈ S Q×p and c =/ 0. Here f 1 , c may depend on h. The same example used in the proof of Theorem 6.6.15 works.  The special representation for p s1 −s2 = p 2 : We now consider the case when ω1 = ω2 , and p s1 −s2 = p 2 . Observe that in this case |a|sp1 = |a|sp2 +2 for all a ∈ Q×p . Definition 6.10.6 (Invariant linear functional on V p (s, ω)) Fix a prime p, a pair of complex numbers s = (s1 , s2 ) ∈ C2 , and a pair of normalized unitary

6.10 The special representations

235

characters ω = (ω1 , ω2 ) of Q×p as in Definition 6.5.1. Let V p (s, ω) be the complex vector space in Definition 6.5.2. For f in the space V p (s, ω), define  L s,ω ( f ) := f (k) ω1−1 (det k) d × k. G L(2,Z p )

As suggested by its name, this functional satisfies a certain invariance property with respect to the action of G L(2, Z p ) given in Definition 6.5.3. Proposition 6.10.7 (Invariance property of L s,ω ) Assume p s1 −s2 = p 2 , and ω1 = ω2 . Let L s,ω : V p (s, ω) → C be the linear functional given in Defintion 6.10.6. Then L s,ω satisfies L s,ω (π (g) . f ) = ω1 (det g)| det g|sp1 −1 · L s,ω ( f ). 1 ω1−1 (det g). Then Proof For g ∈ G L(2, Q p ), let F(g) := f (g)| det g|1−s p    a b F ·g 0 d    a b 1 ω1−1 (ad · det g) = f · g · |ad · det g|1−s p 0 d 1 = f (g) · ω1 (a)|a|sp1 · ω2 (d)|d|s2 · |ad · det g|1−s ω1−1 (ad · det g) p a    1 =   · f (g) · | det g|1−s ω1−1 (det g) p d p a    =   · F (g) , d p

for all a, d ∈ Q×p , b ∈ Q p , and g ∈ G L(2, Q p ). Now,   f (k)ω1−1 (det k) d × k = L s,ω ( f ) = G L(2,Z p )

and

 L s,ω (π (g) . f ) = 

F(k) d × k,

G L(2,Z p )

G L(2,Z p )

= G L(2,Z p )

(π (g) . f )(k) · ω1−1 (det k) d × k f (kg) · ω1−1 (det k) d × k

= ω1 (det g)| det g|sp1 −1 ·



F(kg) d × k. G L(2,Z p )

Proposition 6.10.7 now follows from Corollary 6.9.10



Corollary 6.10.8 Let ker(L s,ω ) be the kernel of the invariant linear functional L s,ω defined in Definition 6.10.6. Then ker(L s,ω ) is an invariant subspace of V p (s, ω).

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Theory of admissible representations of GL(2, Qp )

Definition 6.10.9 (Special representation for ω1 = ω2 and p s1 −s2 = p 2 ) Fix a prime p, a pair of complex numbers s = (s1 , s2 ) ∈ C2 , and a pair of normalized unitary characters ω = (ω1 , ω2 ) of Q×p as in Definition 6.5.1. Let V p (s, ω) be the complex vector space of principal series defined in Definition 6.5.2. For ω1 = ω2 and p s1 −s2 = p 2 , let ker (L s,ω ) be the kernel of the invariant linear functional L s,ω defined in Definition 6.10.6, i.e.,  f (k) ω1−1 (det k) d × k. L s,ω ( f ) := G L(2,Z p )

  Let π denote the action of right translation. Then π, ker (L s,ω ) is defined to be a special representation for ω1 = ω2 and p s1 −s2 = p 2 . Since the action π of G L(2, Q p ) on V p (s, ω) leaves ker (L s,ω ) invariant, it follows that (π, ker (L s,ω )) is itself a representation of G L(2, Q p ). It is clear that ker (L s,ω ) is non-trivial. It is not obvious that ker (L s,ω ) is a proper subspace of V p (s, ω), i.e., that it is not the whole space V p (s, ω). However, it is easily verified that the integral which defines L s,ω ( f ) does not vanish when L s,ω is taken to be one of the elements of the basis Bn (ω) given in Definition 6.6.4. Proposition 6.10.10 The special representation for ω1 = ω2 and p s1 −s2 = p 2 , as defined in Definition 6.10.9, is irreducible. Proof The arguments given in the proof of Theorem 6.8.14 show that, even when ω1 = ω2 and p s1 −s2 = p 2 , the subspace of K p (s, ω) generated by an arbitrary element φ contains the whole Bruhat-Schwartz space S(Q×p ), and at least one function which is not in this space. It follows that   restr ◦ Js,ω ker (L s,ω ) contains S(Q×p ) and that the dimension of the quo  tient restr ◦ Js,ω ker (L s,ω ) /S(Q×p ) is at least 1. But ker (L s,ω ) =/ V p (s, ω), so   restr ◦ Js,ω ker (L s,ω ) =/ K p (s, ω). It follows that the dimension of the quotient restr ◦ Js,ω ker (L s,ω ) /S(Q×p ) is exactly 1, and that restr ◦ Js,ω ker (L s,ω ) is generated by each of its elements, i.e., is an irreducible representation. To deduce that ker (L s,ω ) is an irreducible representation as well, we recall that by Proposition 6.8.5, the map restr ◦ Js,ω : V p (s, ω) → K p (s, ω) is an isomorphism when ω1 = ω2 and p s1 −s2 = p 2 . 

6.11 Jacquet modules Definition 6.11.1 (Smooth H -module) Let V be a complex vector space. Let H be a subgroup of G L(2, Q p ) and let π : H → G L(V ) be a (smooth) representation of H . We shall refer to V as a (smooth) H -module. Definition 6.11.2 (The vector spaces V (N ) and VN ) Let (π, V ) be a smooth representation of G L(2, Q p ) as in Definition 6.1.1. Define

6.11 Jacquet modules 237 5    1 b  b ∈ Q N := p , 0 1   

 V (N ) := Span π (η) . v − v  η ∈ N , v ∈ V , and VN := V V (N ). Proposition 6.11.3 (Characterization of the space V (N )) Let (π, V ) be a smooth representation of G L(2, Q p ) as in Definition 6.1.1. Let V (N ) be as in Definition 6.11.2. Then v ∈ V (N ) if and only if    1 u π . v du = 0 0 1 p−n Z p for some integer n. Proof If v ∈ V (N ), then it has a representation as a finite combination of the form:    1 bi  . vi − vi v= ci π 0 1 i

with bi ∈ Q p , vi ∈ v, and ci ∈ C. Choose n sufficiently large so that all bi appearing in the linear combination are in p −n Z p . Then ,      1 u 1 u + bi . vi du π . v du = ci π 0 1 0 1 p−n Z p p−n Z p i    1 u π . vi du , − 0 1 p−n Z p which clearly vanishes after making the change of variables u + bi → u in the first integral on the right above. Conversely, suppose that    1 u π . v du = 0 0 1 p−n Z p for some v ∈ V and some integer Since π is smooth it follows there exists  n. 1b . v = v with b ∈ p −m Z p . Let S denote an integer m ≤ n such that π 0 1

a set of representatives of p −n Z p p −m Z p . Then       1 u 1 u+s π . v du = π . v du 0= 0 1 0 1 s∈S

p−n Z p

=



 π

s∈S

=

, s∈S

1 0

 π

1 0

p−m Z p

s 1



s 1

⎡

⎢ .⎣





 π

1 u 0 1

p−m Z p

-



.v ·

du. p−m Z p



⎤ ⎥ . v du ⎦

Theory of admissible representations of GL(2, Qp )      1 s  1 s 1 Consequently . v = 0, and v = v − π .v , π  01 01

238

s∈S

s∈S

where  denotes the number of elements in S. Thus v ∈ V (N ).  Definition 6.11.4 (The torus) The group of diagonal elements   5  t1 0  × t T := , t ∈ Q p 0 t2  1 2 is called the torus subgroup of G L(2, Q p ). Proposition 6.11.5 (V (N ) is invariant under the action of the torus) Let (π, V ) be a smooth representation of G L(2, Q p ) as in Definition 6.1.1. Let V (N ) be as in Definition 6.11.2. Then, for any t ∈ T , the space V (N ) is invariant under the action of π (t). Proof Let t ∈ T and η ∈ N . Then for every v ∈ V , we have   π (t) . π (η) . v − v = π (tηt −1 ) . v − v ∈ V (N ), provided v = π (t) . v. 

Let VN = V V (N ) as in Definition 6.11.2. We want to define an action π N : T → G L(VN ). This is easily accomplished by letting   π N (t) . v + V (N ) := π N (t) . v, (∀t ∈ T, v ∈ V ), (6.11.6) which is well defined by Proposition 6.11.3. Definition 6.11.7 (The Jacquet module) Let (π, V ) be a smooth representation of G L(2, Q p ) as in Definition 6.1.1. Let V (N ), VN be as in Definition 6.11.2, and let (π N , VN ) be the smooth representation of the torus T as in (6.11.6). Following Definition 6.11.1, we define VN to be the Jacquet module of V .

6.12 Induced representations and parabolic induction The main goal in this chapter is to obtain the classification of the irreducible admissible representations of G L(2, Q p ). A powerful technique to achieve this is the method of parabolic induction. If we start with a smooth representawe may restrict it to a representation of the parabolic tion of G L(2, Qp ), then ∗ ∗ subgroup B := 0 ∗ of G L(2, Q p ), consisting of upper triangular matrices. This gives a smooth representation of B. On the other hand, a smooth representation of B may be induced up to a smooth representation of G L(2, Q p ). We now define the smooth restricted and induced representations of parabolic subgroups of G = G L(2, Q p ). See [Bernstein-Zelevinsky, 1977] for a more general treatment of the theory of induced representations.

6.12 Induced representations and parabolic induction

239

Definition 6.12.1 (Restricted representation) Let G = G L(2, Q p ) and let H be a closed subgroup of G. For a complex vector space V , let (π, V ) be a smooth representation of G as in Definition 6.1.1. The restriction of π to H , denoted π H = ResGH (π ) is the representation of H on the same vector space V with the same operations π H (h) . v = π (h) . v,

(∀v ∈ V, ∀h ∈ H ).

The restricted representation is denoted (π H , VH ) where V = VH as a vector space, while VH denotes the fact that action is restricted to H . Remark Take H to be closed so each of the subgroups K n ∩ H is compact. This ensures that H is a locally compact group, and (π H , VH ) is a smooth representation. Definition 6.12.2 (Induced representation) Let G = G L(2, Q p ), and let H be a closed subgroup of G. For a complex vector space W , let (π, W ) be a smooth representation of H as in Definition 6.1.1. Define the vector space    W G := f : G → W  f (hgk) = π (h) . f (g), (∀h ∈ H, g ∈ G, k ∈ K ) , where K is a compact subgroup of G (which may depend on f ). Further, define a homomorphism π G : G → G L(W ) where π G (g ) . f (g) := f (gg ),

(∀g, g ∈ G).

Then (π G , W G ) is a smooth representation of G called the induced representation. The induced representation π G is also denoted as π G := IndGH (π ). Proposition 6.12.3 (Frobenius reciprocity) Let H be a closed subgroup of G = G L(2, Q p ). Let (π, W ) be a smooth representation of H and let (π , V ) be a smooth representation of G as in Definition 6.1.1. Then HomG (V, W G ) ∼ = Hom H (VH , W ), where HomG (V, W G ), Hom H (VH , W ) denote the set of intertwining operators V → W G , VH → W , respectively, as defined in Definition 2.5.3. Proof First, take T ∈ HomG (V, W G ). Thus, for each v ∈ V, T (v) is a function: G → W, and for all v ∈ V and g1 , g2 ∈ G we have T (π (g1 ) . v) (g2 ) = T (v) (g2 g1 ). Given such a map T, we define a linear map TH : V → W by TH (v) = T (v)(I2 ). Here I2 is the 2 × 2 identity matrix. Clearly, for any h ∈ H, v ∈ V we have TH (π (h) . v) = T (π (h) . v) (I2 ) = T (v) (h).

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Theory of admissible representations of GL(2, Qp )

Making use of the fact that the T (v) is, by definition, in the space W G , we find that     T (v)(h) = π (h) . T (v) (I2 ) = π (h) . TH (v) . It follows at once that TH ∈ Hom H (VH , W ). Now take T ∈ Hom H (VH , W ). We wish to define an element of HomG (V, W G ) using T. Thus, for each element of V we need to define a function T G (v) : G → W such that T G (v)(hgk) = π (h) . T G (v) (g) for all g ∈ G, h ∈ H, and k in some open compact subgroup, depending on v. The definition is T G (v)(g) = T (π (g) . v). The fact that T G (v)(hg) = π (h) . T G (v) (g) for all g ∈ G, h ∈ H follows immediately from the fact that T ∈ Hom H (VH , W ). Clearly, T G (v)(gk) = T G (v)(g) for k in any open compact subgroup which fixes v.  The most important examples of induced representations are the principal series representations introduced earlier. Indeed, suppose that we take H to be the group consisting of all upper triangular invertible matrices, take W to be C, and take the representation π to be given by  π

a 0

b d

 . z = |a|s1 ω1 (a) · |d|s2 ω2 (d) · z

for a, d ∈ Q×p , b ∈ Q p , and z ∈ C, for some fixed complex numbers s1 , s2 and normalized unitary characters ω1 , ω2 . Then the induced representation (π G , W G ) defined in Definition 6.12.2 is precisely the principal series representation defined in Definition 6.5.3.

6.13 The supercuspidal representations of G L(2, Q p ) We have studied the principal series representations and their associated special representations in Section 6.10. The remaining and most elusive local representations of G L(2, Q p ) which may occur are the supercuspidal representations. Here is the definition. Definition 6.13.1 (Supercuspidal representation) Fix a prime p. Let (π, V ) be an irreducible admissible representation of G L(2, Q p ) as in Definition 6.1.4. We shall say that (π, V ) is supercuspidal if for every v ∈ V there exists an integer n such that 

 π p−n Z p

1 0

u 1

 . v du = 0.

6.13 The supercuspidal representations of GL(2, Q p )

241

Alternatively, (π, V ) is supercuspidal if its Jacquet module (defined in Definition 6.11.7) is trivial, i.e., VN = {0}, or equivalently, V = V (N ). The following proposition is easy to establish. Proposition 6.13.2 Fix a prime p and a supercuspidal representation (π, V ) of G L(2, Q p ) as in Definition 6.13.1. Then for every v ∈ V , there exists an integer n so that    1 u π . v du = 0 0 1 p−m Z p

for all m ≥ n. Proof Since (π, V ) is supercuspidal we know that for each v ∈ V there exists n ∈ Z such that    1 u π . v du = 0. 0 1 p−n Z p

For m ≥ n, let S denote a set of representatives of p −n Z p in p −m Z p . Then 

 π p−m Z p

=

1 0



u 1

. v du 



π

s∈S p−n Z p

=



 π

s∈S

= 0.

1 0

1 s 0 1

u+s 1



⎛ ⎜ .⎝

 . v du 

 π

1 u 0 1



⎞ ⎟ . v du ⎠

p−n Z p



The next proposition tells us that supercuspidal representations must be infinite dimensional. Proposition 6.13.3 Fix a prime p and a supercuspidal representation (π, V ) of G L(2, Q p ) as in Definition 6.13.1. Then V must be infinite dimensional. Proof We proved in Theorem 6.1.7 that if (π, V ) is finite dimensional, smooth, and irreducible, then it is one-dimensional and every element of S L(2, Q p ) acts trivially. It then follows that for v ∈ V, v =/ 0 

 π p−n Z p

1 u 0 1

 . v du = p n · v =/ 0. 

242

Theory of admissible representations of GL(2, Qp )

The next theorem is very important. It allows us to immediately conclude that every irreducible admissible representation of G L(2, Q p ) is isomorphic to either a principal series representation, a special representation, or a supercuspidal representation. Theorem 6.13.4 (Smooth irreducible representations of G L(2, Q p ) must be either principal series, special, or supercuspidal) Let (π, V ) denote an infinite dimensional irreducible and admissible representation of G L(2, Q p ) as in Definition 6.1.4. If (π, V ) is not a supercuspidal representation, then it is isomorphic to either a principal series representation (as defined in Definition 6.5.3) or a special representation (as defined in Definition 6.10). Proof Suppose (π, V ) is not supercuspidal. Then by Definition 6.13.1 we have VN =/ {0}. Since V is irreducible, it is generated (as in Definition 6.1.15) by any nonzero element v ∈ V . Fix such a v. Then by smoothness (Definition 6.1.2) there exists a non-negative integer n such that π (k) . v = v for all k ∈ K n , where    K n = k ∈ G L(2, Z p )  k − I2 ∈ p n · Mat(2, Z p ) , (n = 1, 2, 3, . . . ). Now K n has finite index in G L(2, Z p ). By the Iwasawa decomposition (Proposition 4.2.1), we have  decomposition G L(2,  Q p ) = B K where

 the  a b × K = G L(2, Q p ) and B :=  a, b ∈ Q p , d ∈ Q p . It follows that V 0 d is finitely generated (see Definition 6.1.15) as a B-module. Let v1 , v2 , . . . , vr be a set of generators as a B-module. Then their images in VN = V /V (N ) generate VN as a T -module. It immediately follows from Proposition 6.1.17 that the representation (π N , VN ) has an irreducible quotient, i.e., there exists V ⊂ VN (with V invariant under π N ) such that VN /V is irreducible. Then π N acts on VN /V by the rule π N (t) . (v + V ) := π (t) . v + V ,

(∀v ∈ VN , t ∈ T ).

Consequently, for any fixed t ∈ T , the linear map π N (t) : VN /V → VN /V is an intertwining map. By Schur’s Lemma 6.1.8, every such π N (t) acts by a scalar. Since VN /V is irreducible, this implies VN /V is one-dimensional. Hence (π N , VN /V ) is the one-dimensional representation defined by a character of the torus T . It follows that there exist two multiplicative characters χ1 , χ2 : Q×p → C and a linear form (linear map from a vector space to its field of scalars) θ : VN /V → C satisfying :   ; t1 b θ πN . v = χ1 (t1 )χ2 (t2 ) θ (v), 0 t2   ∀t1 , t2 ∈ Q×p , b ∈ Q p , v ∈ VN /V .

6.14 The uniqueness of the Kirillov model

243

Now consider the linear maps V −→ VN −→ VN /V . By composing θ with the above maps, we have a linear form ! : V → C satisfying  ; :    t1 b . v = χ1 (t1 )χ2 (t2 ) θ (v), ∀t1 , t2 ∈ Q×p , b ∈ Q p , v ∈ V . ! πN 0 t2 (6.13.5) This is the same thing as an element of Hom B (VB , W ), where B is the group of upper triangular matrices in G L(2, Q p ) and W is just C regarded as a B-module with action by   t1 b . z = χ1 (t1 )χ2 (t2 ) · z. π 0 t2 Every character χ : Q×p → C× can be uniquely written in the form χ (y) = ω(y)y s ,

(∀y ∈ Q×p )

for some s ∈ C and some normalized unitary character ω : Q×p → C as in Definition 6.5.1. We say χ is determined by s and ω. Let χ1 , χ2 , appearing in (6.13.5), be determined by a pair of complex numbers s = (s1 , s2 ) ∈ C2 and a pair of normalized unitary characters ω = (ω1 , ω2 ). The existence of a non-trivial intertwining map from V to the principal series representation V p (s, ω) defined in Definition 6.5.3 now follows immediately from the existence of θ and Frobenius reciprocity, Proposition 6.12.3. Since V is irreducible, this non-trivial intertwining map will be an isomorphism of V onto its image, which is infinite dimensional and contained in V p (s, ω). If ω1 =/ ω2 or p s1 −s2 =/ 1, p 2 , then it follows directly from Corollary 6.8.15, that (π, V ) ∼ = π, V p (s, ω) . If ω1 = ω2 and p s1 −s2 = 1 or p 2 , then it follows directly from Definitions 6.10.3 and 6.10.9, and Propositions 6.10.4 and 6.10.10 that (π, V ) is isomorphic to a special representation.  Remark In the case ω1 = ω2 and p s1 −s2 = 1, the special representation was defined as a quotient. We have not proved this yet, but it will turn out that in this case the special representation can not be embedded into π, V p (s, ω) as a subspace. Thus, this case does not actually arise.

6.14 The uniqueness of the Kirillov model It was shown in Theorem 6.7.2 that every infinite dimensional, irreducible, admissible representation of G L(2, Q p ) has a Kirillov model, i.e, is isomorphic to a representation (π , K) where the space K consists of locally constant

244

Theory of admissible representations of GL(2, Qp )

functions f : Q×p → X (here X is a non-trivial complex vector space) on which π operates in such a way that     a b

π ∀ f ∈ K, a, y ∈ Q×p , b ∈ Q p . . f (y) = e p (by) f (ay), 0 1 (6.14.1) We shall now show that we may take X = C. This leads to the following uniqueness theorem [Kirillov, 1966], [Jacquet-Langlands, 1970], [Godement, 1970] which is the key ingredient in the original proof in [Jacquet-Langlands, 1970] of the classification of all the admissible irreducible representations of G L(2, Q p ). Note that we have already obtained the classification by Theorem 6.13.4. Nevertheless we feel that the approach via the Kirillov model is of independent interest and we include it for completeness. Theorem 6.14.2 (Uniqueness of the Kirillov model) Fix a prime p and let V denote a complex infinite dimensional vector space. Assume that (π, V ) is an admissible irreducible representation of G L(2, Q p ) as in Definition 6.1.4. Then there exists one and only one space K of functions f : Q×p → C, and one and only one representation π : G L(2, Q p ) → G L(K) satisfying the following conditions: (1) (π, V ) is isomorphic to (π , K). (2) π acts on K as in (6.14.1). This space K also satisfies the following additional conditions. (3) Each f ∈ K is locally constant and vanishes outside a compact subset of Q p . (4) The space S(Q×p ), as defined in Theorem 6.7.2, is a subspace of K. (5) The space S(Q×p ) has finite codimension in K. That is, conditions (1) and (2) determine the space K uniquely, and conditions (3),(4) and (5) may then be proved. Proof of theorem 6.14.2 As we remarked in the beginning of this section, we have already proved in Theorem 6.7.2 that (π, V ) is isomorphic to a Kirillov model (π , K) where        1 u  V0 := v ∈ V  e p (−u) π . v du = 0, for all large n , 0 1  p−n Z p

X = V /V0 ,

v ≡ v (mod V0 ) ⇐⇒ v − v ∈ V0 (∀v, v ∈ V ),



     K = f v : Q×p → X = V /V0  f v (y) := π 0y 01 .v (mod V0 ), ∀y ∈ Q× , (6.14.3)

6.14 The uniqueness of the Kirillov model

245

and π acts as in (6.14.1). It was also proved in Theorem 6.7.2 that each f ∈ K is locally constant and vanishes outside a compact subset of Q p and that the space S(Q×p ) · X ⊂ K. It remains to prove that X = C and that statement (5) holds. The main step in the proof will be to show that X = V /V0 has dimension one. To do this we need to introduce a linear operator on the vector space X . Definition 6.14.5 (The operator J (t, ω) : X → X ) Let X be the C-vector space as in (6.14.3). Fix t ∈ Q× and fix a unitary multiplicative character ω : Z×p → C× , as in Definition 6.5.1. Then we define the linear operator J (t, ω) : X → X by the action: J (t, ω) . x   6   7 0 1

= π . ω(t −1 y ) · 1Z× t −1 y · x −1 0 p

    

,

(x ∈ X ).

y =1

    Remark Fix x ∈ X. For y ∈ Q×p , the function y → ω t −1 y · 1Z× t −1 y · x p

is in S X (Q×p ) which is contained in K by Theorem 6.7.2 (ii). Therefore, the action of π is well  defined  on this function. The operator Jπ (t, ω) is defined to this function and then setting y = 1 afterward. by applying π −10 10 Lemma 6.14.6 Let t ∈ Q×p , f ∈ S X (Q×p ), and ωπ be the central character of π as in Proposition 6.1.10. Let p N be the conductor of f which is defined in a similar fashion as in Definition 2.8.3. Then for every y ∈ Q×p , we have    0 1 . f (y). J (t y, ψ) . f (t) d × t = π ωπ (y) −1 0 Q×p N ψ (mod p )

Proof The lemma follows directly from the orthogonality relation (6.7.9) and Definition 6.14.5. In fact  ωπ (y) J (t y, ψ) . f (t) d × t ψ (mod p N )



= ωπ (y)

Q×p

Q×p

ψ (mod

π y

pN )



 6 7     01 · ψ((t y)−1 y )·1Z× (t y)−1 y · f (t)  d × t. −1 0 p  y =1



Here, we have included a subscript y in the notation for the action π to indicate that the expression in brackets above is to be viewed as a function of the variable y (as opposed to t or y) when computing this action. That is, π is

acting “in the variable  y .”  01 is linear, as is evaluation at y = 1. Consequently, The operator π y −1 0 the integral above is equal to

Theory of admissible representations of GL(2, Qp ) ⎡    0 1 ⎢

·⎣ ωπ (y)·π y ψ((t y)−1 y ) −1 0 N

246

Q×p

ψ (mod p )

·1Z× p

= ωπ (y) · π y



⎤   (t y)−1 y · f (t) d × t ⎦

·11+ p N Z p

= ωπ (y) · π y



y =1

 , 1 . ϕ( p N ) × 0 Qp

0 −1

  (t y)−1 y · f (t) d × t

·11+ p N Z p (t) · f (t y = ωπ (y) · π y





0 1 −1 0

-      y =1

 , 1 . ϕ( p N ) 0 Q×p

0 −1

    

-    y)d t  

−1

×

y =1

@  ? . f (y −1 y )  .

(6.14.7)

y =1

Here, we have used the fact that the volume of 1 + p N Z p is ϕ( p N )−1 . Now, f (y −1 y ) = π y



y −1 0

0 1



. f (y ),

and 

0 −1

1 0

  −1 y · 0

0 1



 =

y −1 0

0 y −1

  y · 0

0 1

  0 · −1

1 0

 .

Combining the above identities with (6.14.7) yields ωπ (y) ·

π y



0 1 −1 0



? @  . f (y −1 y ) 

= ωπ (y) · π y = π y



y 0



0 1

y =1 −1

y 0



0 y −1

0 1 −1 0





y 0

0 1



 

 . f (y )  . y =1

0 −1

1 0



  . f (y )  y =1

6.14 The uniqueness of the Kirillov model   Now, suppose h = π −10 10 . f. Then the above expression equals   0 

h(yy )  = h(y) = π −1

1 0

y =1

This completes the proof.

247

 . f (y).



Lemma 6.14.8 If T ∈ End (X ) commutes with all the operators J (t, ω), where t ∈ Q×p and ω, ω : Z×p → C× (unitary multiplicative characters as in Definition 6.5.1) then T acts by a scalar on all of X . Proof Let F X (Q×p ) denote the space of all functions f : Q×p → X. Any linear operator T : X → X induces an operator T : F X (Q×p ) → F X (Q×p ),

f ∈ F X (Q×p ) → T . f,

provided we define (T . f )(y) = T . f (y),

(y ∈ Q×p ).

Here T . x denotes the linear action of T on x ∈ X. Note that T maps S X (Q×p ) to S X (Q×p ). Let f ∈ S X (Q×p ). Since S X (Q×p ) ⊂ K by theorem 6.7.2(ii), it immediately follows from Lemma 6.14.6 that for every y ∈ Q×p we have π



0 1 −1 0



= ωπ (y) ⎡

. (T . f )(y) 

ψ (mod p N )

= T . π



Q×p





⎢ = T . ⎣ωπ (y) :

  J (t y, ψ) . T . f (t) d × t

ψ (mod p N )

0 −1

1 0

⎤ ⎥ J (t y, ψ) . f (t) d × t ⎦

Q×p



;

. f (y)

= T .

We have thus proved that T commutes with π S X (Q×p ).





π



01 −1 0

0 1 −1 0





 .f

(y).

when T is consid-

It immediately follows that K is an invariant ered as an operator on

subspace of T . Indeed, if f ∈ K, then we may write f = f1 + π



0 1 −1 0

 . f2

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Theory of admissible representations of GL(2, Qp )

with f 1 , f 2 ∈ S X (Q×p ). Then    0 1 0 . f2 = T . f1 + π T . f = T . f1 + T . π −1 0 −1   0 1 . S X (Q×p ) ⊂ K. ∈ S X (Q×p ) + π −1 0

1 0



. T . f2

Next we show that T : K → K commutes withany π (g) where g ∈ G L(2, Q p ). We have already shown this for g = −10 10 . Consider now   g = a0 b1 . Then for any f ∈ K and y ∈ Q p , we have ? @ T . π (g) . f (y) = T . e p (by) f (ay) = e p (by)(T . f )(ay) = π (g) . (T . f )(y). (6.14.9)   a b

. Also, it follows from PropoConsequently T commutes with π 0 1   d 0 sition 6.1.10 that T commutes with π for any d ∈ Q×p because 0 d       d 0 acts by a scalar (central character). The matrices −10 10 , a0 b1 , π 0 d   d 0 with a, d ∈ Q×p and b ∈ Q p generate G L(2, Q p ). Thus T commutes 0 d with all π (g), with g ∈ G L(2, Q p ), as stated. It follows that T , and hence, T acts as a scalar.  Lemma 6.14.10 (Commutativity of the operators J (t, ω)) For all t, t ∈ Q×p , and all ω, ω : Z×p → C× (unitary multiplicative characters as in Definition 6.5.1), we have J (t, ω) . J (t , ω ) = J (t , ω ) . J (t, ω). Proof We begin with the matrix identity       0 1 1 t 0 −1 1 0 = −1 0 0 1 1 0 −t 1      2 −1 −1 1 −t t 0 1 t 0 = 0 1 0 t −1 −1 0 0

(6.14.11)  −t . 1

Let f ∈ S X (Q×p ). For t, y ∈ Q×p set f t (y) = π



t2 0

−t 1



t −1 0

0

t −1



. f (y) = e p (−t y)ωπ (t −1 ) · f (t 2 y).

(6.14.12) Then it follows from Lemmas 6.14.6 and 6.14.9 that for all y ∈ Q×p    −1   2  0 1 t 1 −t −1 t 0 −t

. f (y) π 0 1 −1 0 0 t −1 0 1     0 1 1 −t −1 . f t (y) = π 0 1 −1 0

6.14 The uniqueness of the Kirillov model 249   0 1 . f t (y) = e p (−t −1 y) · π −1 0  = e p (−t −1 y) · ωπ (y) J (ay, ψ) . f t (a) d × a, ψ (mod p Nt )

Q×p

where Nt is the conductor of f t . Plugging in (6.14.12), we obtain π



1 −t −1 0 1



0 1 −1 0



t −1 0

= e p (−t −1 y) · ωπ (t −1 y) 



= ωπ (t −1 y)

ψ (mod p Nt )

S X (Q×p ). π

Q×p

t2 0

−t 1

 . f (y)

? @ J (ay, ψ) . e p (−ta) · f (t 2 a) d × a

 a + y · J (ay/t 2 , ψ) . f (a) d × a. ep − t Q×p

(6.14.11). A difficulty arises because 



t −1 

ψ (mod p Nt )

Next we would like to compute π be in

0



(6.14.13)



10 . f using −t 1     0 −1 1 t

π 1 0 01

the left hand side of will not, in general,

To get around this, we use the identity    0 1 1 t 0 −1 . f (y) −1 0 0 1 1 0

 :    1 t 0 −1

. π . f (y) =π 0 1 1 0   ; 0 −1 . f (y) + f (y). − π 1 0 (6.14.14)       0 −1 Note that π 10 1t . f (y) − π 01 −10 . f (y) is in S X (Q×p ). 1 0



0 −1

1 0

Let p N be the conductor of f . We define      1 t 0 −1 0 −1

. f (y) − π . f (y) f t (y) := π 0 1 1 0 1 0    = e p (t y) − 1 ωπ (−y) J (ay, ψ ) . f (a) d × a. ψ (mod p N )

Q×p

Let p Nt be the conductor of f t . Then, using (6.14.14),     0 1 1 t 0 −1

. f (y) π −1 0 0 1 1 0   0 1 . f t (y) + f (y) = π −1 0

250

Theory of admissible representations of GL(2, Qp )  = ωπ (y) J (zy, ψ

) . f t (z) d × z + f (y) ×   Q p N ψ

mod p

t

= ωπ (y)

,





Q×p

ψ (mod p N )



ψ

mod

N p t



  e p (t z) − 1 ωπ (−z) -





·

J (zy, ψ ) .



Q×p

×

J (az, ψ ) . f (a) d a d × z + f (y). (6.14.15)

Q×p

We may define two functions K i : ×  a + y K 1 (a, y, t) := ωπ (t −1 )e p − t K 2 (a, y, t) :=





ψ (mod p N ) ψ

mod p

 N

Q×p

t

Q×p

Q×p

ψ (mod

p Nt )

×

→ End (X ), by J (ay/t 2 , ψ),

(6.14.16)

e p (t z)ωπ (−z) · J (zy, ψ ) . J (az, ψ

) d × z. (6.14.17)

By the identity (6.14.11), the expressions (6.14.13) and (6.14.15) are the same. Consequently,  (K 1 (a, y, t1 ) − K 1 (a, y, t2 )) . f (a) d × a Q×p

 =

Q×p

(K 2 (a, y, t1 ) − K 2 (a, y, t2 )) . f (a) d × a,

for all f ∈ S X (Q×p ) and all y, t1 , t2 ∈ Q×p . It follows that (K 1 (a, y, t1 ) − K 1 (a, y, t2 )) = (K 2 (a, y, t1 ) − K 2 (a, y, t2 )) ,

(6.14.18)

Q×p .

for all a, y, t1 , t2 ∈ By (6.14.16), the function K 1 (a, y, t) = K 1 (y, a, t) for all a, y, t ∈ Q×p . Hence, the left hand side of (6.14.18) is symmetric in a and y. We deduce that the right hand side is also symmetric. Now, fix t1 , t2 ∈ Q×p , and let N = max(Nt 1 , Nt 2 ). Define H1 (a, y, ψ )    e p (t1 z) − e p (t2 z) ωπ (−z) · J (zy, ψ ) . J (az, ψ

) d × z, := × Qp ψ

(mod p N ) and let H2 (a, y, ψ )    e p (t1 z) − e p (t2 z) ωπ (−z) · J (az, ψ

) . J (zy, ψ ) d × z. := × Qp ψ

(mod p N )

6.14 The uniqueness of the Kirillov model

251

Then

H1 (a, y, ψ ) = (K 2 (a, y, t1 ) − K 2 (a, y, t2 ))

ψ (mod p N )



= (K 2 (y, a, t1 ) − K 2 (y, a, t2 )) = ψ

(mod

H2 (a, y, ψ ). pN )

It follows immediately from the definition of J (t, ψ) that J (tu, ψ) = ψ(u) · J (t, ψ), for all t ∈ Q×p , u ∈ Z×p , and all characters ψ of Z×p . It follows that H1 (a, uy, ψ ) = ψ (u)· H1 (a, y, ψ ),

H2 (a, uy, ψ ) = ψ (u)· H2 (a, y, ψ ),

for all a, y ∈ Q×p , u ∈ Z×p , and all characters ψ of Z×p . Thus H1 (a, y, ψ ) =

 Z×p

 =

Z×p

ψ (u) · (K 2 (a, uy, t1 ) − K 2 (a, uy, t2 )) d × u ψ (u) · (K 2 (y, a, t1 ) − K 2 (y, a, t2 )) d × u = H2 (a, y, ψ ),

for all a, y ∈ Q×p , and each individual character ψ (mod p N ). In the exact same way, we can isolate individual terms in the sums over ψ

and deduce that    e p (t1 z) − e p (t2 z) ωπ (−z) · J (zy, ψ ) . J (az, ψ

) d × z Q×p

 =

  e p (t1 z) − e p (t2 z) ωπ (−z) · J (az, ψ

) . J (zy, ψ ) d × z, Q×p (6.14.19)

for all t1 , t2 , a, y ∈ Q×p , and all characters ψ (mod p N ), ψ

(mod N ). The identity (6.14.19) holds for all t1 , t2 ∈ Q p . This proves that the function   z → ωπ (−z) . J (zy, ψ ) . J (za, ψ

) − J (za, ψ

) . J (zy, ψ ) is orthogonal to all functions in S(Q×p ) and hence must vanish. Therefore, J (y, ψ ) . J (a, ψ

) = J (a, ψ

) . J (y, ψ ), which establishes the commutativity of the family of operators J (t, ω) given in Lemma 6.14.10. 

252

Theory of admissible representations of GL(2, Qp )

The dimension of X is one. We are now in a position to prove that the dimension of X is one. By Lemma 6.14.8, the family of operators J (t, ω) is commutative. It immediately follows from Lemma 6.14.9 that each J (t, ω) is a scalar operator. Consequently, any T ∈ End (X ) commutes with every J (t, ω). It again follows from Lemma 6.14.9 that every linear operator T ∈ End (X ) must be a scalar operator. This forces the space X to be one-dimensional.

6.15 The Kirillov model of a supercuspidal representation We shall now explicitly identify the Kirillov model of an arbitrary supercuspidal representation of G L(2, Q p ). Proposition 6.15.1 (Kirillov model of a supercuspidal representation) Fix a prime p and let (π, V ) denote an infinite dimensional irreducible admissible representation of G L(2, Q p ) as in Definition 6.1.4. Let (π , K) be the Kirillov model of (π, V ) as in Theorem 6.14.2. Let S(Q×p ) be the Bruhat-Schwartz space as defined in Theorem 6.7.2. Then K = S(Q×p ) if and only if (π, V ) is supercuspidal. Proof For f ∈ K, y ∈ Q×p , and n ∈ Z, define  f n (y) := p−n Z p

= f (y) ·

π 



1 u 0 1



 . f (y) du = p−n Z p

e p (uy) f (y) du (6.15.2)

p , if y ∈ p Z p , n

0,

n

otherwise.

Now, if (π , K) is supercuspidal then for f ∈ K, we may choose n ∈ Z such that f n (y) = 0. It immediately follows from (6.15.2) that f (y) vanishes for y ∈ p n Z p . Consequently f ∈ S(Q×p ). Conversely, suppose that S(Q×p ) = K, and consider f ∈ K. Then there exists an integer n such that f (y) = 0 for y ∈ p n Z p . It follows from (6.15.2) that f n (y) = 0 for all y ∈ Q×p . Thus (π , K) is supercuspidal. 

6.16 The classification of the irreducible and admissible representations of G L(2, Q p ) The classification of the irreducible admissible representations of G L(2, Q p ) will be obtained, via the theory of the Kirillov model, as an immediate consequence of Theorem 6.13.4 and Proposition 6.15.1. It gives the complete classification of the irreducible admissible representations of G L(2, Q p ).

Exercises for Chapter 6

253

Theorem 6.16.1 (The classification of the irreducible and admissible representations of G L(2, Q p ) Fix a prime p. Let (π, V ) be an irreducible and admissible representation of G L(2, Q p ) as in Definition 6.1.4. Then (1) If dimC (V ) < ∞, then dimC (V ) = 1 and π (g) . v = ω(det(g)) · v for all v ∈ V, g ∈ G L(2, Q p ) and some character ω : Q×p → C× . (2) If dimC (V ) = ∞, let (π , K) be the  Kirillov model of (π, V ) as in Theorem 6.14.2. Define δπ := dim K/S(Q×p ) to be the codimension of S(Q×p ) in K. Then 0 ≤ δπ ≤ 2. The representation is supercuspidal when δπ = 0, special when δπ = 1, principal series when δπ = 2.

Exercises for Chapter 6 Fix a prime number p for the duration of these exercises. 6.1 For each integer n ≥ 1, recall that    K n = k ∈ G L(2, Z p )  k − I2 ∈ p n · Mat (2, Z p ) . Prove that K n is a compact open subgroup of G L(2, Q p ), and that the collection {K n }n≥1 constitutes a base of open neighborhoods of the identity. 6.2 Fix an integer n ≥ 1 and consider the “reduction (mod p n )” map:     a b a b n . → red : G L(2, Z p ) → G L(2, Z/ p Z), c d c d (a) Show that red is well-defined in the sense that red(g) is an invertible matrix (over Z/ p n Z). (b) Show that red is a surjective homomorphism with kernel K n as in the previous exercise. (Note that this implies K n is normal.) (c) Compute the index of K n in G L(2, Z p ). 6.3 For integers n > m, show that K n is a normal subgroup of K m and compute the cardinality of the group K m /K n . 

6.4 Define B(Z p ) =

y1 0

x y2

 5   x ∈ Z p , y1 , y2 ∈ Z× . p 

Prove that the following is a complete list of double coset representatives for B(Z p )\G L(2, Z p )/K n :

254

Theory of admissible representations of GL(2, Qp )    5    5 1 0  0 1  n−1 n 0 ≤ c < p 0 ≤ d < p ∪ . pc 0  1 d 

6.5 Prove that Definitions 6.1.1 and 6.1.2 are equivalent. 6.6 Suppose (π, V ) is a smooth representation of G L(2, Q p ).  (a) Prove that V K m ⊂ V K n if m < n, and that V = n≥1 V K n . (b) Show that V K n is G L(2, Z p )-invariant for each integer n ≥ 1. (c) Conclude that if (π, V ) is admissible, then every v ∈ V lies in a finite-dimensional G L(2, Z p )-invariant subspace of V . (This is the reason we do not require a topology on V in order to study representations of G L(2, Q p ).) 6.7 Prove that S L(2, Q p ) is the commutator subgroup of G L(2, Q p ). (Recall that for a group G, the commutator subgroup [G, G] is the subgroup of G generated by the elements of the form aba −1 b−1 for a, b ∈ G.) See also Exercise 14.1. 6.8 In this exercise, we complete the proof of Theorem 6.1.7. (a) Suppose V is a complex vector space of dimension 1 and (π, V ) is a representation of G L(2, Q p ). Prove there exists a homomorphism λ : G L(2, Q p ) → C× such that π (g).v = λg .v for all g ∈ G L(2, Q p ) and v ∈ V . (b) Suppose further that (π, V ) is smooth and that S L(2, Q p ) is contained in the kernel of π . Prove that there is a (continuous) character ω : Q×p → C× such that λg = ω (det(g)). 6.9 Give a counterexample to show that the converse to Theorem 6.1.11 is false.   6.10 Let j : Q×p → G L(2, Q p ) be given by j(a) = a0 a0 . Prove that j is continuous. Conclude that the central character of an irreducible representation is continuous. 6.11 Let V be an n-dimensional complex vector space. Prove that a continuous representation (π, V ) of G L(2, Q p ) is smooth. (A continuous representation is one for which the homomorphism π : G L(2, Q p ) → G L(V ) is continuous, where G L(V ) is given the canonical topology obtained by choosing a basis and embedding it in Mat (n, C).) 6.12 Suppose (π, V ) is an irreducible smooth representation of G L(2, Q p ). Let ωπ be the associated central character. Prove that if π is unramified as in Definition 6.2.1, then ωπ is unramified in the sense of Definition 2.1.12. 6.13 Prove Theorem 6.4.12.

Exercises for Chapter 6

255

6.14 Let s = (s1 , s2 ) be a pair of complex numbers and ω = (ω1 , ω2 ) a pair of normalized unitary characters. Prove that if the principal series representation V p (s, ω) is unramified in the sense of Definition 6.2.1, then ω = ωtrivial . 6.15 Suppose s = (s1 , s2 ) is a pair of complex numbers and let f ∈ V p (s1 , s2 ) be an element of the unramified principal series representation. If f is fixed by G L(2, Z p ), show that f is a scalar multiple of the function f s◦1 ,s2 as in Definition 6.4.2. 6.16 Suppose the principal series representation V p (s, ω) admits a 1-dimensional subrepresentation of G L(2, Q p ). Prove that ω1 = ω2 and s2 = s1 + 2πi/ log p for some integer . That is, p s1 −s2 = 1. 6.17 Prove Lemma 6.6.5. 6.18 Complete the proof of Proposition 6.6.12. 6.19 Fix a pair of normalized unitary characters ω = (ω1 , ω2 ) and let f ∈ Flat p (ω) be a flat section. Prove there exists f 0 ∈ V p (0, ω) such that    y1 x k; s1 , s2 = |y1 |sp1 ω1 (y1 )|y2 |sp2 ω2 (y2 ) f 0 (k) f 0 y2 for all y1 , y2 ∈ Q×p , x ∈ Q p , k ∈ G L(2, Z p ) and s1 , s2 ∈ C. (This helps to explain the terminology “flat” section. There is no variation in the “V p -direction.”) 6.20 Using the p-adic Iwasawa decomposition, we can write any y1 x  g ∈ G L(2, Q p ) in the form g = 0 y2 k for some y1 , y2 ∈ Q×p , x ∈ Q p and k ∈ G L(2, Z p ). Prove that the quantities |y1 | p and |y2 | p depend only on g, and not on the choice of decomposition.   6.21 Suppose g = ac d0 ∈ G L(2, Q p ). By the Iwasawa decomposition, we  y1 x  can write g = 0 y2 k with y1 , y2 ∈ Q×p , x ∈ Q p and k ∈ G L(2, Z p ). Prove that  if |c| p ≤ |d| p , |a| p , |y1 | p = |ad/c| p , if |c| p > |d| p ,  |y2 | p =

|d| p , if |c| p ≤ |d| p , |c| p ,

if |c| p > |d| p .

(Note that these quantities are well-defined by the previous exercise.)

256

Theory of admissible representations of GL(2, Qp )

6.22 In this exercise, we complete the proof of Lemma 6.9.4. Recall that c > 0 is fixed and that μ is a left invariant measure on the collection of compact open subsets of G L(2, Q p ) defined by 1 · c, (n ≥ 1, g ∈ G L(2, Q p )). [G L(2, Z p ) : K n ]   e1 p 0 (a) Suppose δ = 0 pe2 , where e1 ≤ e2 are integers. For each n ≥ 1, prove the equality of indices μ(g · K n ) =

[K n : δ −1 K n δ ∩ K n ] = p e2 −e1 = [δ −1 K n δ : δ −1 K n δ ∩ K n ]. (b) Suppose g ∈ G L(2, Q p ) and n ≥ 1. Use the Cartan decomposition (Proposition 4.2.3) and part (a) to prove the equality of indices [K n : g −1 K n g ∩ K n ] = [g −1 K n g : g −1 K n g ∩ K n ]. (c) Let I be the common index from part (b). Prove that μ(K n ) = I · μ(g −1 K n g ∩ K n ) = μ(g −1 K n g) for all g ∈ G L(2, Q p ). Deduce that μ is right invariant. 6.23 Complete the proof of Proposition 6.9.6. 6.24 Complete step 4 of the proof of Theorem 6.7.2. We have already shown    that any subrepresentation of B1 (Q p ) = a0 b1  a ∈ Q×p , b ∈ Q p on

the Bruhat-Schwartz space S(Q×p ) that contains 11+ pZ p is in fact equal to S(Q×p ). Now show that if W ⊂ S(Q×p ) is a non-zero subrepresentation, then W contains 11+ pZ p .

6.25 Let (π , K) be the Kirillov model of an admissible irreducible representation (π, V ) of G L(2, Q p ) as given by Theorem 6.7.2. Recall that V (N ) is the kernel of the projection to the Jacquet module V → VN . If we identify V (N ) with its image in the Kirillov space K, prove that V (N ) = S X (Q×p ). 6.26 Let (π, V ) be a complex representation of a group G, and let W be a subspace of V that is invariant under the action of G. (So W is a subrepresentation.) It is desirable to find a complementary invariant subspace W such that V = W ⊕ W . For example, this is always possible if V is finite dimensional and G is a finite or compact group. By way of contrast, this exercise shows that things are not so nice for G L(2, Q p ). Let s = (s1 , s2 ) be a pair of complex numbers such that

Exercises for Chapter 6

257

p s1 −s2 = p 2 , and let ω = (ω1 , ω1 ) for some normalized unitary character ω1 . Then the principal series representation V p (s, ω) is not irreducible, and we may consider the special representation W = ker(L s,ω ), where L s,ω is the invariant functional of Definition 6.10.9. Prove that there does not exist a G L(2, Q p )-invariant subspace W ⊂ V p (s, ω) such that V p (s, ω) = W ⊕ W . 6.27* In the representation theory of finite groups, we have the following definition of the induced representation. Let G be a group, H a subgroup, and (π, W ) a complex representation of H . We can view W as a (left) module over the group ring C[H ]. Then the induced representation is defined to be IndGH (W ) := C[G] ⊗C[H ] W, and the action of C[G] is given by left multiplication. (For definitions and a more detailed discussion, see Sections 6.1 and 7.1 of [Serre, 1998].) (a) Define a complex vector space of functions V = { f : G → W | f (hg) = π (h). f (g) ∀h ∈ H, g ∈ G}. We can turn V into a representation of G by setting (ρ(s). f )(g) = f (gs) for g, s ∈ G. Prove that IndGH (W ) is isomorphic to the subrepresentation of V consisting of functions supported on finitely many cosets of H \G. Suppose now that G = G L(2, Q p ), H = B(Q p ) is the group of upper triangular matrices, and W = C. For fixed complex numbers s = (s1 , s2 ) and normalized unitary characters ω = (ω1 , ω2 ), we define an action of H on W as in Section 6.12 by   y1 x .z = |y1 |sp1 ω1 (y1 ) |y2 |sp2 ω2 (y2 ) z, π 0 y2 (yi ∈ Q×p , x ∈ Q p , z ∈ W ). Then IndGH (W ) = C[G] ⊗C[H ] W is a representation of G = G L(2, Q p ). (b) Is IndGH (W ) as defined above isomorphic to the principal series representation V p (s, ω)? That is, does the notion of induced representation in this exercise coincide with the one from Definition 6.12.2? 6.28 Let s = (s1 , s2 ) be a pair of complex numbers and let ω = (ω1 , ω2 ) be a pair of normalized unitary characters of G L(2, Q p ). (a) Suppose that ω1 = ω2 and that p s1 −s2 = p 2 . Prove that the special representation of G L(2, Q p ) as in Definition 6.10.9 is ramified.

258

Theory of admissible representations of GL(2, Qp ) (b) Suppose that ω1 = ω2 and that p s1 −s2 = 1. Prove that the special representation of G L(2, Q p ) as in Definition 6.10.3 is ramified. Hint: If not, there exists f ∈ V p (s, ω) such that for each k ∈ G L(2, Z p ) there is a constant c(k) for which π (k). f = f + c(k)δs,ω . Now integrate over G L(2, Z p ).

7 Theory of admissible (g, K ∞ ) modules for G L(2, R)

7.1 Admissible (g, K ∞ )-modules Recall the definition of a (g, K ∞ )-module as given in Definition 5.1.4. Definition 7.1.1 ((g, K ∞ )-module) Let g = gl(2, C), K ∞ = O(2, R), and U (g) denote the universal enveloping algebra as in Section 4.5. 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

(7.1.2)

for all α ∈ g, Dα given by Lemma 4.5.4, and k ∈ K ∞ . Further, we require that  1 π K∞ (exp(tα)) . v − v t→0 t

πg (Dα ) . v = lim

(7.1.3)

for all v ∈ V and α in the Lie algebra k of K ∞ , as given by Theorem 4.5.8. It is good to have a concrete example in mind when thinking about (g, K ∞ )modules. Example 5.2.12 will be our motivating example. It consists of all linear combinations of the images of a newform for a congruence subgroup of S L(2, Z) under the two possible actions: • action by differential operators, • action by right translation by K ∞ = O(2, R). 259

260

Theory of admissible (g, K∞ ) modules for GL(2, R)

Definition 7.1.4 (Admissible (g, K ∞ )-module) A (g, K ∞ )-module V , as defined in Definition 7.1.1, is admissible if for each integer m the space  5     cos θ sin θ imθ  . v = e · v, (∀θ ∈ R) v ∈ V  πK∞ − sin θ cos θ is finite dimensional.

7.2 Ramified versus unramified We would like to classify the admissible (g, K ∞ )-modules as defined in Definition 7.1.4. We first introduce the important distinction between ramified and unramified admissible (g, K ∞ )-modules. Here are the definitions. Definition 7.2.1 (Unramified (g, K ∞ )-module) A (g, K ∞ )-module V as defined in Definition 7.1.4, is termed unramified if there exists a non-zero K ∞ -fixed vector v ◦ ∈ V . Definition 7.2.2 (Ramified (g, K ∞ )-module) A (g, K ∞ )-module V , as defined in Definition 7.1.4, is termed ramified if there does not exist a non-zero K ∞ fixed vector v ◦ ∈ V . Let us try to motivate the above definitions. In Example 5.2.12, we constructed a (g, K ∞ )-module associated to the adelic lift f adelic of a Maass form f of arbitrary weight, level, and character. If the classical form f is an even Maass form of weight zero and level one, then the (g, K ∞ )-module is unramified. On the other hand, it will turn out that if the classical form f is odd of weight 0, or of odd weight, or corresponds to a non-constant holomorphic modular form, then the above construction will not give an unramified (g, K ∞ )-module. In these cases, one may show that there will be no K ∞ -fixed vector in the space.

7.3 Jacquet’s local Whittaker function Local and global Whittaker functions arose naturally when computing the Fourier coefficients of adelic automorphic forms as in Section 4.10. Here is a formal definition following (4.10.16) for the case of G L(2, R). Note the analogy with Section 6.3. Definition 7.3.1 (Local Whittaker function) A local Whittaker function for the group G L(2, R) is a smooth function w∞ : G L(2, R) → C, of moderate growth, which satisfies    1 u g = e∞ (u)w∞ (g), w∞ 0 1

7.3 Jacquet’s local Whittaker function

261

for all u ∈ R, g ∈ G L(2, R). Here e∞ (u) = e2πiu is the additive character on R. In his thesis [Jacquet, 1967], Jacquet showed how to explicitly construct local Whittaker functions. The construction for G L(2, R) is given by a simple integral. Definition 7.3.2 (Jacquet’s local K ∞ -invariant Whittaker function) Fix s1 , s2 ∈ C, with (s1 − s2 ) > 1. Let f s◦1 ,s2 : G L(2, R)/K ∞ → C be defined by f s◦1 ,s2



1 0

x 1



y 0

0 1



r 0 0 r



 ·k

= |y|s∞1 · |r |s∞1 +s2

for x ∈ R, y, r ∈ R× , and k ∈ K ∞ . We define Jacquet’s local Whittaker function  W∞ (g; s1 , s2 ) :=

f s◦1 ,s2



0 1 1 0



1 u 0 1

  g e∞ (−u) du,

R

for all g ∈ G L(2, R), and where e∞ is the additive character on R as in Definition 7.3.1. Remarks We must show that this integral is convergent. This is the reason for the restriction on (s1 − s2 ). We will shortly show that the function W∞ (∗; s1 , s2 ) can be holomorphically continued to all s1 , s2 ∈ C, and satisfies a functional equation. Proposition 7.3.3 (Computation of Jacquet’s local Whittaker function) Fix s1 , s2 ∈ C such that (s1 − s2 ) > 1, and let W∞ (∗; s1 , s2 ) be Jacquet’s local Whittaker function as defined in Definition 7.3.2. Then      1 x y 0 r 0 · k; s1 , s2 W∞ 0 1 0 1 0 r s1 −s2   s1 +s2 +1 2π 2 s1 +s2 = e∞ (x) |r |∞ ·  s1 −s2  |y|∞ 2 K s1 −s2 −1 2π |y|∞ 2  2 for x ∈ R, y, r ∈ R× , and k ∈ K ∞ . Here  1 ∞ − 1 y(u+1/u) s du K s (y) := , e 2 u 2 0 u

(s ∈ C, y > 0),

is the K -Bessel function. Remark Note that W∞ (g; s1 , s2 ) is invariant under the center of G L(2, R) if and only if s2 = −s1 .

Theory of admissible (g, K∞ ) modules for GL(2, R)

262

Proof Clearly,  1 W∞ 0

x 1



y 0

0 1



r 0 0 r



 · k; s1 , s2 

= e∞ (x)|r |s∞1 +s2 W∞

y 0

0 1



 ; s1 , s2 .

Furthermore,    y 0 W∞ ; s1 , s2 0 1      0 1 1 u y 0 e∞ (−u) du = f s◦1 ,s2 1 0 0 1 0 1 R      0 1 y 0 1 uy −1 e∞ (−u) du f s◦1 ,s2 = 1 0 0 1 0 1 R      0 1 y 0 1 u e∞ (−uy) du f s◦1 ,s2 = |y|∞ 1 0 0 1 0 1 R      1 0 0 1 1 u e∞ (−uy) du f s◦1 ,s2 = |y|∞ 0 y 1 0 0 1 R     0 1 1 u ◦ 2 e∞ (−uy) du. f = |y|1+s ∞ s1 ,s2 1 0 0 1 R Plugging in 

0 1

1 0

we obtain W∞





1 0

y 0

u 1

0 1

 √

 =

1+u 2

−1 √

0



· k,

(for some k ∈ K ∞ ),

1+u 2

 ; s1 , s2



∗

 2 = |y|1+s ∞

(1 + u 2 ) R

s2 −s1 2

e∞ (−uy) du.

Proposition 7.3.3 immediately follows from the above identity and formula (3.1.9) in [Goldfeld, 2006].  Corollary 7.3.4 (Functional equation of Jacquet’s Whittaker function) Fix s1 , s2 ∈ C, and let W∞ (∗; s1 , s2 ) be Jacquet’s local Whittaker function as defined in Definition 7.3.2. Then W∞ (∗; s1 , s2 ) has a holomorphic continuation to all (s1 , s2 ) ∈ C × C and satisfies the functional equation W∞ (g; s2 + 1, s1 − 1) = for all g ∈ G L(2, R).



 s1 −s2 

π s2 −s1 +1  s2 −s1 +2  W∞ (g; s1 , s2 ),

2



2

7.4 Principal series representations

263

Proof The meromorphic continuation is immediate from Proposition 7.3.3. To see the functional equation, let   s2 −s1 s1 − s2 ∗ 2 W∞ (g; s1 , s2 ).  W∞ (g; s1 , s2 ) = π 2 Then the functional equation may be stated as ∗ ∗ (g; s2 + 1, s1 − 1) = W∞ (g; s1 , s2 ). W∞

This follows immediately from Proposition 7.3.3, and the symmetry K ν (t) = K −ν (t) of the K -Bessel function.  Corollary 7.3.5 Let f be an even Maass form of type ν for S L(2, Z) as in Definition 4.8.1. Then we have the equality W∞ (g; ν, −ν) =

πν W∞ (g; f ), (ν)

(∀g ∈ G L(2, R)),

where W∞ (∗; s1 , s2 ) is Jacquet’s Whittaker function as defined in Definition 7.3.2, and W∞ (∗; f ) is the Whittaker function coming from f as given in (4.10.14).

7.4 Principal series representations By analogy with Definition 6.5.1 and referring back to (2.1.11), we shall define a normalized unitary character of R× as follows. Definition 7.4.1 (Normalized unitary character of R× ) A normalized unitary character of R× is, by definition, either the trivial character (which always takes the value one) or the sign character  1, if t > 0, sign(t) = −1, if t < 0. Let s = (s1 , s2 ) ∈ C2 and let ω = (ω1 , ω2 ) where ω1 , ω2 are normalized unitary characters of R× as in Definition 7.4.1. Following Section 6.5, we shall define a vector space V∞ (s, ω) consisting of functions : G L(2, R) → C. Definition 7.4.2 (The vector space V∞ (s, ω)) Fix s = (s1 , s2 ) ∈ C2 and fix ω = (ω1 , ω2 ) where ω1 , ω2 are normalized unitary characters of R× as in Definition 7.4.1. Define V∞ (s, ω) to be equal to       y1 ∗  g = ω1 (y1 )|y1 |s∞1 ω2 (y2 )|y2 |s∞2 · f (g), f : G L(2, R) → C  f 0 y2   ∀y1 , y2 ∈ R× , g ∈ G L(2, R), f is smooth and K ∞ -finite .

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Here, a function f : G L(2, R) → C is termed K ∞ -finite if the set { f (gk) | k ∈ K ∞ } of all right translates of f (g) (g ∈ G L(2, R)) generates a finite dimensional vector space. Note that a smooth function f : G L(2, R) → C can be expanded in a Fourier series in the variable θ since by Proposition 4.1.1 (Iwasawa decomposition), we may write g ∈ G L(2, R) in the form      1 x y 0 r 0 cos θ sin θ g= 0 1 0 1 0 r − sin θ cos θ with x ∈ R, r > 0, y ∈ R× , 0 ≤ θ < 2π. An equivalent formulation of K ∞ -finiteness is that f has a Fourier expansion of the form f (g) = cn (x, y, r )einθ , cn (x, y, r ) ∈ C, n∈Z

where all but finitely many of the cn (x, y, r ) vanish. Definition 7.4.3 (Principal series representation) Fix s = (s1 , s2 ) ∈ C2 and fix ω = (ω1 , ω2 ) where ω1 , ω2 are normalized unitary characters of R× as in Definition 7.4.1. Let V∞ (s, ω) be given as in Definition 7.4.2, and define π = (πg , π K∞ ), where πg denotes the action of U (g) on V∞ (s, ω) given by differential operators, and π K∞ is the action of K ∞ on V∞ (s, ω) given by right translation. We define the principal series representation of G L(2, R), associated to (s, ω), to be the (g, K ∞ )-module (see Definition 5.1.4) given by the vector πg , π K∞ . We denote this principal series space V∞ (s, ω) with   the two actions representation as π, V∞ (s, ω) . Proposition 7.4.4 (Principal series representations are admissible) Fix unitary s = (s1 , s2 ) ∈ C2 and fix ω = (ω1 , ω2 ) where ω1 , ω2 are normalized  characters of R× as in Definition 7.4.1. The representation π, V∞ (s, ω) given in Definition 7.4.3 is admissible as in Definition 7.1.4. Proof Fix an integer m. Let V∞,m (s, ω) denote the space of smooth and K ∞ -finite functions f : G L(2, R) → C, which satisfy    y1 x g = ω1 (y1 )|y1 |s∞1 · ω2 (y2 )|y2 |s∞2 · f (g), f 0 y2 (7.4.5)    cos θ sin θ f g· = eimθ · f (g), − sin θ cos θ (for all y1 , y2 ∈ R× , x, θ ∈ R, g ∈ G L(2, R)). We need to prove that V∞,m (s, ω) is finite dimensional. Actually, we will prove the much stronger assertion that the dimension of V∞,m (s, ω) is at most one.

7.4 Principal series representations

265

It immediately follows from (7.4.5)   for f ∈ V∞,m (s, ω), we must   that cos θ sin θ 10 imθ = e . By Proposition 4.1.1 (Iwasawa have f · f − sin θ cos θ 01 decomposition), every element of g ∈ G L(2, R) has a unique expression of the form     y1 x cos θ sin θ g= · , (x ∈ R, y1 ∈ R× , y2 > 0, 0 ≤ θ < 2π ). 0 y2 − sin θ cos θ It follows that for f ∈ V∞,m (s, ω), and any g ∈ G L(2, R) as above, we have   1 0 . f (g) = ω1 (y1 )|y1 |s∞1 · ω2 (y2 )|y2 |s∞2 · eimθ f 0 1 This shows that V∞,m (s, ω) is at most one-dimensional. By comparing the above identity at the values θ and θ + π , one easily deduces that the dimension of V∞,m (s, ω) is 1 if (−1)m = ω1 (−1)ω2 (−1), and 0 otherwise.  Definition 7.4.6 (Algebraic direct sum of vector spaces) Let {V1 , V2 , V3 , . . . } be a set (possibly infinite) of vector spaces over C. The algebraic direct sum of the vector spaces Vi (i = 1, 2, 3, . . . ) is the vector space consisting of all sums ∞

ci vi ,

(ci ∈ C, vi ∈ Vi )

i=1

where all but finitely many of the ci (i = 1, 2, 3, . . . ) vanish. Definition 7.4.7 (Basis for V∞ (s, ω)) Fix s = (s1 , s2 ) ∈ C2 and fix ω = (ω1 , ω2 ) where ω1 , ω2 are normalized unitary characters of R× as in Definition 7.4.1. Fix m ∈ Z such that ω1 (−1)ω2 (−1) = (−1)m . Define a function f m : G L(2, R) × C2 → C by     y1 x cos θ sin θ ; s1 , s2 = ω1 (y1 )|y1 |s∞1 ·ω2 (y2 )|y2 |s∞2 eimθ , fm 0 y2 − sin θ cos θ for all x ∈ R, y1 , y2 ∈ R× , y2 > 0, and 0 ≤ θ < 2π. By Proposition 7.4.4, the space V∞,m (s, ω) is just the one-dimensional space f m · C. Furthermore, V∞ (s, ω) is the algebraic direct sum (as in Definition 7.4.6) of the spaces V∞,m (s, ω) with m ∈ Z. Remark Definition 7.1.1 of a (g, K ∞ )-module requires that the subspace generated by K ∞ acting on any element of V∞ (s, ω) be finite dimensional. This is the reason why V∞ (s, ω) is taken to be the algebraic direct sum of the spaces V∞,m (s, ω) with m ∈ Z. Note that the K ∞ smoothness condition does not hold for an element of the form m∈Z cm f m with infinitely many cm =/ 0 for m ∈ Z.

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Theory of admissible (g, K∞ ) modules for GL(2, R)

Proposition 7.4.8 (Explicit action on a basis for V∞ (s, ω)) Let         1 0 1 0 0 1 0 1 Z= , X= , Y = , H= , 0 1 0 −1 1 0 −1 0 as in (5.2.1), and define the raising and lowering operators 1 1 L = (D X − i DY ), (D X + i DY ), 2 2 as in Definition 5.2.5. Also, let  denote the degree two Casimir operator for gl(2, R), which we define, as on p. 50 of [Goldfeld, 2006]. Finally, let     cos θ sin θ −1 0 κθ = ∈ S O(2, R), δ1 = ∈ O(2, R). − sin θ cos θ 0 1 R=

For m ∈ Z, let f m be given as in Definition 7.4.7. Then the action of U (g) is completely determined by the action of H, Z , L and R which is as follows: πg (D Z ) . f m = (s1 + s2 ) f m , πg (D H ) . f m = (im) f m , s1 − s2 + m ) f m+2 , πg (R) . f m = ( 2 s1 − s2 − m πg (L) . f m = ( ) f m−2 . 2 Furthermore, we have   πg () . f m = s12 + s22 − s1 + s2 f m . The group K ∞ is generated by δ1 and the elements κθ with θ ∈ [0, 2π ). The action of these elements is as follows: π K∞ (κθ ) . f m = eimθ f m , π K∞ (δ1 ) . f m = ω1 (−1) f −m . Proof The first two equations follow easily from the definitions, given that     t cos t sin t e 0 , exp(t H ) = . exp(t Z ) = − sin t cos t 0 et The next two may be deduced from the expressions for L and R in coordinates proved in [Bump, Proposition 2.2.5] and introduced previously during the Interlude in Chapter 5:   ∂ ∂ 1 ∂ r ∂ R=e iy +y + − , ∂x ∂ y 2i ∂θ 2 ∂r   ∂ ∂ 1 ∂ r ∂ L = e−2iθ −i y +y − − . ∂x ∂y 2i ∂θ 2 ∂r 2iθ

7.4 Principal series representations In terms of these coordinates,    y x r 0 cos θ fm 0 1 0 r − sin θ

sin θ cos θ



267

 ; s 1 , s2

= r s1 +s2 y s1 ,

for all x, y, r, θ ∈ R, r, y > 0. The fifth equation then follows from Lemma 5.2.6. The sixth is immediate from the definition. Finally, a simple matrix multiplication shows that       cos θ sin θ cos θ − sin θ −y1 x y1 x · δ1 = , 0 y2 − sin θ cos θ 0 y2 sin θ cos θ and the final equation then follows from the definition of f m .  Corollary 7.4.9 (Reducibility of principal series) • The principal series representation (π, V∞ (s, ω)) is irreducible, except when s1 − s2 is an integer m which satisfies (−1)m = ω1 (−1)ω2 (−1). • If s1 − s2 = m is a positive integer, and (−1)m = ω1 (−1)ω2 (−1), then Span ({ f i | i ≥ m})

and

Span ({ f i | i ≤ −m})

are irreducible sub-U (g)-modules of V∞ (s, ω), and their direct sum is an irreducible sub-(g, K ∞ )-module, and V∞ (s, ω) does not contain any other proper non-trivial sub-(g, K ∞ )-modules. • If s1 − s2 = m is a nonpositive integer, and (−1)m = ω1 (−1)ω2 (−1), then Span ({ f i | |i| ≤ −m}) is an irreducible sub-(g, K ∞ )-module of V∞ (s, ω), and V∞ (s, ω) does not contain any other proper non-trivial sub-(g, K ∞ )-modules. Proof First, let s be arbitrary, and suppose h ∈ V∞ (s, ω), non-zero. Then by the K ∞ -finiteness property of (g, K ∞ )-modules, we may write h=

N

c h  ,

=M

where M, N ∈ Z, and for each , the function h  ∈ V∞ (s, ω) satisfies   cos θ sin θ . h  = eiθ h  . πK∞ − sin θ cos θ It follows from the proof of Proposition 7.4.4 that h  must be a scalar multiple of f  , defined as in Definition 7.4.7. Adjusting c if necessary, we may assume h  = f  for each .

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Theory of admissible (g, K∞ ) modules for GL(2, R)

Next we make use of Proposition 7.4.8. Using the identity πg (D H ) . f  = i · f  , we may easily recover f  as a linear combination of U (g)-translates of h for any  such that c =/ 0. In short, we may assume that h = f  for some . / Z. Then applying either L or R an appropriate Now suppose that s1 − s2 ∈ number of times, we recover a non-zero multiple of f i for any other i ∈ Z such that i ≡  (mod 2). The assertion regarding this case then follows. Now suppose s1 − s2 is an integer, m, such that (−1)m = ω1 (−1)ω2 (−1). Then πg (L) . f m = πg (R) . f −m = 0, and these are the only instances of either L or R having a kernel in V∞ (s, ω). We investigate what portion of V∞ (s, ω) is generated by the action of R and L on f  for each  ∈ Z. If m > 0, then f  generates ⎧ if  ≥ m, ⎪ ⎨ Span ({ f i | i ≥ m}) , Span ({ f i | i ≤ −m}) if  ≤ −m, ⎪ ⎩ V∞ (s, ω) if || < m, while if m ≤ 0, then f  generates ⎧ ⎪ ⎨ Span ({ f i | i ≥ m}) , Span ({ f i | i ≤ −m}) ⎪ ⎩ Span ({ f i | |i| ≤ −m})

if  > −m, if  < m, if || ≤ −m.

Next, we note that π K∞ (δ1 ) . f i = ± f −i . From this we deduce that for any sub-(g, K ∞ )-module U of V∞ (s, ω), the set {i ∈ Z | f i ∈ U} is symmetric about 0, and the result follows.  Definition 7.4.10 (Discrete series, limits of discrete series) Fix a pair ω = (ω1 , ω2 ) of normalized unitary characters of R× as in Definition 7.4.1, and a positive integer m such that ω1 (−1)ω2 (−1) = (−1)m . Take s2 ∈ iR. Then the irreducible (g, K ∞ )-module      m m ⊂ V∞ s2 + , s2 − , ω , Span f i  i ≥ m, or i ≤ −m 2 2 is called a discrete series representation if m ≥ 2, and a limit of discrete series representation if m = 1. Remark The distinction between “discrete series” and “limit of discrete series” is motivated by considerations in the general theory of representations of Lie groups. It may be shown that for m ≥ 2 the representation in question has a model consisting of functions in L 2 (Z (G L(2, R))\G L(2, R)) (with action by right translation), while if m = 1 it does not. For this reason, discrete series representations are also known as “square integrable representations.” The representations obtained by allowing s2 to have non-trivial real part are sometimes called “essentially discrete series” (or “essentially square integrable”).

7.5 Classification of irreducible admissible (g, K∞ )-modules

269

7.5 Classification of irreducible admissible (g, K ∞ )-modules In this section, we begin with an admissible (g, K ∞ )-module which we assume is irreducible. We finish by concluding that it is isomorphic to one of the irreducible admissible (g, K ∞ )-modules already constructed. We first prove that each of the elements of the center Z (U (g)) (introduced in Section 4.6) of the universal enveloping algebra U (g) (introduced in Section 4.5) acts by a fixed scalar. Lemma 7.5.1 (Elements of Z (U (g)) act by scalars) Let V be an irreducible admissible (g, K ∞ )-module as in Definition 7.1.1, with action πg as in (7.1.3). Let D be an element of Z (U (g)). Then there exists λ ∈ C such that πg (D) . v = λ · v for all v ∈ V. Proof As explained in Section 5.1 for the action by right translation on smooth functions, the identity π K∞ (k) ◦ πg (Dα ) = πg (Dkαk −1 ) ◦ π K∞ (k),

(∀ k ∈ K ∞ , α ∈ g)

yields the identity π K∞ (k) ◦ πg (D) = πg (D) ◦ π K∞ (k),

(∀ k ∈ K ∞ ) ,

(7.5.2a)

for D ∈ Z (U (g)). Thus, the operator πg (D) commutes with all of the operators acting on V. Clearly, then, πg (D) preserves the subspace     5  cos θ sin θ imθ  .v = e v , (7.5.2b) Vm := v ∈ V  π K∞ − sin θ cos θ for each m ∈ Z. It follows from the finiteness condition in Definition 7.1.1 that at least one of the spaces Vm =/ {0} for some m ∈ Z. But Vm is a finitedimensional space, because (π, V ) is admissible. It follows that πg (D) has an eigenvector v # in Vm . Let λ be the complex number such that πg (D) . v # = λ·v # . Because πg (D) commutes with all of the operators acting on V, we deduce that πg (D) . v = λ · v for all vectors v in the orbit of v for the action of these operators, or, indeed for all vectors v in the span of this orbit. Since V was assumed to be irreducible, this span is all of V.  Recall that Z (U (g)) is a polynomial algebra generated by D Z and . By Lemma 7.5.1, each of these generators acts by a scalar, and these two scalars determine the action of all of Z (U (g)). In this section, we will show that an irreducible, admissible (g, K ∞ )-module is completely determined, up to isomorphism, by a few simple invariants. The first of these is the pair of complex numbers (λ, μ) satisfying πg () . v = λ · v,

and

πg (D Z ) . v = μ · v,

(∀ v ∈ V ).

(7.5.3)

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Theory of admissible (g, K∞ ) modules for GL(2, R)

As mentioned before, as a byproduct of our analysis, we will find that every irreducible admissible (g, K ∞ )-module is isomorphic either to an irreducible principal series representation, or to an irreducible subrepresentation of a reducible principal series representation. We saw in Proposition 7.4.8 above that for V = V∞ (s, ω), the eigenvalues λ, μ are given by λ = s12 + s22 − s1 + s2 ,

μ = s1 + s2 .

Here s = (s1 , s2 ) ∈ C and ω is a pair of normalized unitary characters of R× as in Definition 7.4.1. Motivated by this, we record the following: 2

Lemma 7.5.4 For all λ, μ ∈ C, the set of solutions of the system of equations s12 + s22 − s1 + s2 = λ,

s1 + s2 = μ,

(s1 , s2 ) ∈ C2 ,

(7.5.5)

is an orbit of the involution (s1 , s2 ) → (s2 +1, s1 −1) of C . That is, either there are two solutions and this involution reverses them, or there is one solution which it leaves stable. 2

Proof Exercise.



The previous lemma implies that every (g, K ∞ )-module has the same pair of eigenvalues as some principal series representation. Next, we introduce the next invariant which we attach to a (g, K ∞ )-module. Definition 7.5.6 (Set of K ∞ -types of an admissible (g, K ∞ )-module) Let V be an admissible (g, K ∞ )-module as in Definition 7.1.1 The set of K ∞ -types of V is the set of integers m such that the space Vm defined in (7.5.2b) is non-trivial. Next, we show that the set of K ∞ -types of any irreducible (g, K ∞ )module matches the set obtained from one of the irreducible (g, K ∞ )-modules constructed in the previous section. Proposition 7.5.7 Let V be an irreducible admissible (g, K ∞ )-module. Let S be the set of K ∞ -types of V. Then S is one of the following sets:    k ∈ Z  k ≡ 0 (mod 2) ,    k ∈ Z  k ≡ 1 (mod 2) ,    (some m ∈ Z, m ≥ 0), k ∈ Z  |k| ≤ m, k ≡ m (mod 2) ,     (some m ∈ Z, m ≥ 0). k ∈ Z |k| > m, k ≡ m (mod 2) , Furthermore, in the last two cases, the integer m appearing in the description of S satisfies m2 μ2 +m =λ− , (7.5.8) 2 2 where λ, μ are the eigenvalues given in (7.5.3).

7.5 Classification of irreducible admissible (g, K∞ )-modules

271

Remark The set of integers satisfying (7.5.8) is either empty, equal to {−1}, or equal to {m, −(m + 2)} for some unique non-negative integer, m. Proof Since V is irreducible, it is spanned by the translates of any one fixed non-zero vector. The operators  πg (D Z ),

πg (D H ),

πK∞

cos θ − sin θ

sin θ cos θ

 (∀ θ ∈ R),

all act by scalars on each of the subspaces Vk . Thus it is enough to examine the effect of the operators π K∞ (δ1 ),

πg (R),

πg (L),

because together with those listed above these generate everything. The operator π K∞ (δ1 ) is invertible and maps Vk to V−k for each k. It follows that −k ∈ S whenever k ∈ S. The operator πg (R) maps Vk to Vk+2 , while πg (L) maps Vk to Vk−2 . Thus the set S is symmetric about zero and all of its elements have the same parity. It follows from Lemma 5.2.7 that πg (L) ◦ πg (R) acts on Vk by the scalar 2 2 1 (λ− k2 − μ2 −k). Thus, if πg (R) has a non-trivial kernel in Vk , or if πg (L) has a 2 kernel in Vk+2 , then this scalar must be zero. It follows that the operator πg (R) is injective on Vk except when k = −m − 2 or m, where m satisfies (7.5.8). Similarly πg (L) is injective except on Vm+2 and V−m in the same situation. Combined with the symmetry obtained from π K∞ (δ1 ) this yields the result.  In light of this result, it is convenient to define the parity of an irreducible admissible (g, K ∞ )-module to match that of the integers in its set of K ∞ -types. Definition 7.5.9 (Parity of an irreducible admissible (g, K ∞ )-module) In view of Proposition 7.5.7, the integers appearing in the set of K ∞ -types of an irreducible admissible (g, K ∞ )-module are either all odd or all even. We say that the (g, K ∞ )-module is odd or even accordingly. We will find that for some (g, K ∞ )-modules, the eigenvalues λ and μ along with the set of K ∞ -types completely determine the isomorphism class. For others, however, we need one more invariant to determine it completely. First suppose that V0 is non-trivial. The operator π K∞ (δ1 ) restricts to an automorphism of V0 of order 2, so the only possible eigenvalues are +1 and −1. We will show that in fact for irreducible admissible (g, K ∞ )-modules, V0 is one-dimensional (when it is non-trivial). However, the two possibilities for the eigenvalue of π K∞ (δ1 ) on this one-dimensional space still give rise to two different isomorphism classes of (g, K ∞ )-modules with the given pair of eigenvectors (λ, μ) and set of K ∞ -types.

272

Theory of admissible (g, K∞ ) modules for GL(2, R)

The other case when we still need one more invariant is when V1 is nontrivial, and the kernel of πg (L) in V1 is trivial. In this case π K∞ (δ1 ) ◦ πg (L) is an automorphism of V1 , which, in the same way, will turn out to be one-dimensional. Furthermore, π K∞ (δ1 ) ◦ πg (L) ◦ π K∞ (δ1 ) ◦ πg (L) = πg (R ◦ L),   2 which acts on V1 by the scalar 12 λ − μ2 + 12 . Hence π K∞ (δ1 ) ◦ πg (L) must act by one of the square roots of this complex number. Note that if (s1 , s2 ) is either of the solutions of (7.5.5), then     μ2 1 s1 − s2 − 1 2 1 λ− + = . 2 2 2 2 Theorem 7.5.10 (Classification of irreducible admissible (g, K ∞ )-modules) Let V be an irreducible admissible (g, K ∞ )-module as in Definition 7.1.1. Let λ and μ be given by (7.5.3). Let S denote the set of K ∞ -types of V as defined in Definition 7.5.6. i) Suppose πg (R) has a non-trivial kernel in Vm for some m ≥ 0. Then m 2 2 satisfies m2 + m = λ − μ2 ,    S = k  |k| ≤ m, k ≡ m (mod 2) , and V is isomorphic to the unique irreducible subrepresentation of V∞ (s, ω), (which is the span of the vectors f i for |i| ≤ m, as in , μ+m ), and ω = (ω1 , ω2 ) such that Corollary 7.4.9), for s = ( μ−m 2 2 m ω1 (−1)ω2 (−1) = (−1) , with ω1 determined in the even case by the condition that ω1 (−1) is equal to the eigenvalue of π K∞ (δ1 ) on V0 , and in the odd case by the condition that the eigenvalue of π K∞ (δ1 ) ◦ πg (L) on V1 is ω1 (−1) −m−1 . This representation is finite dimensional. (To be 2 precise, the dimension is m + 1.) ii) Suppose that πg (L) has a non-trivial kernel in Vm for some m > 0. 2 2 Then −m satisfies m2 + m = λ − μ2 ,    S = k  |k| ≥ m, k ≡ m (mod 2) , and V is isomorphic to the unique irreducible subrepresentation of V∞ (s, ω), (which is the span of the vectors f i for |i| ≥ m, as in , μ−m ), and either value of ω = (ω1 , ω2 ) Corollary 7.4.9), for s = ( μ+m 2 2 such that ω1 (−1)ω2 (−1) = (−1)m . This representation is discrete series (m > 1) or limit of discrete series (m = 1).

7.5 Classification of irreducible admissible (g, K∞ )-modules

273

iii) If there is an integer m in the set S of K ∞ -types of V which satisfies (7.5.8), then exactly one of the conditions of i), ii) is satisfied. iv) If there is no integer m in the set S of K ∞ -types of V which satisfies (7.5.8), then each of the two solutions s = (s1 , s2 ), of (7.5.5) uniquely determines a pair ω = (ω1 , ω2 ) such that ω1 (−1)ω2 (−1) = (−1)m for any/all m ∈ S, and ω1 (−1) is equal to the eigenvalue of π K∞ (δ1 ) on V0 in the even case, while in the odd case the eigenvalue of π K∞ (δ1 ) ◦ πg (L) on V1 is ω1 (−1) s1 −s22 −1 . The two principal series representations V∞ (s, ω) obtained are irreducible and isomorphic to one another and to V. Proof Once again, because V is irreducible, it is spanned by the translates of any one fixed non-zero vector. By the finiteness condition, at least one of the spaces Vm is non-trivial, and they are all finite dimensional. Take v # ∈ Vm , non-zero for some m. Then the following operators act by scalars on v # :   cos θ sin θ πg (D Z ), πg (D H ), , πK∞ (∀ θ ∈ R). − sin θ cos θ From Proposition 5.2.10, and (7.1.2), it is enough to consider translates of the form (7.5.11) πg (R  ) . v # , and πg (R  ) . π K∞ (δ1 ) . v ∗ , for  ∈ Z. Furthermore, πg (R  ) . v # ∈ Vm+2 ,

and

πg (R  ) . π K∞ (δ1 ) . v ∗ ∈ V−m+2 .

First, suppose that we are in case ii), i.e., that πg (L) has a non-trivial kernel in Vm for some m > 0. Then πg (R) has a non-trivial kernel in V−m , so −m satisfies (7.5.8). Take v # in Vm for this m. Then, half of the translates listed in (7.5.11) are trivial, so that V is spanned by πg (R  ) . v # , ( ≥ 0)

and

πg (R  ) . π K∞ (δ1 ) . v ∗ ( ≤ 0).

All of these translates are non-zero, and they all lie in distinct eigenspaces for the action of S O(2, R), so they are linearly independent. Thus, we have a basis for V. Explicitly, take vm+2 := πg (R  ) . v #

and

v−m−2 := πg (L  ) . π K∞ (δ1 ) . v ∗ ( ≥ 0).

We now show that the action of U (g) and K ∞ on this basis is completely determined by λ and μ. Indeed, for each n ∈ S, we have   cos θ sin θ . vn = einθ · vn , πK∞ − sin θ cos θ π K∞ (δ1 ) . vn = v−n , πg (D H ) . vn = in · vn , πg (D Z ) . vn = μ · vn ,

274

Theory of admissible (g, K∞ ) modules for GL(2, R) ⎧ v , n ≥ m, ⎪ ⎨ n+2 0, n = −m, πg (R) . vn =   ⎪ ⎩ 1 λ − μ2 − n 2 + n · v n < −m n+2 2 2 2 ⎧ v , n ≤ −m, ⎪ ⎨ n−2 n = m, πg (L) . vn = 0,  ⎪ ⎩ 1 λ − μ2 − n 2 − n · v , n > m. n−2 2 2 2

The first two equations completely describe the action of K ∞ . The last four completely determine the action of U (g), since every element of U (g) is a linear combination of words in the generating set R, L , D H , D Z . It follows that any two (g, K ∞ )-modules with the same pair of eigenvalues (λ, μ) which satisfy the condition of ii) with the same value of m are isomorphic. Suppose next that V0 is non-trivial. Take v # to be an eigenvector of π K∞ (δ1 ) in V0 . Then, as before, we eliminate half of the translates in (7.5.11) from consideration, leaving as set πg (R  ) . π K∞ (δ1 ) . v # ,  ∈ Z, which all lie in distinct spaces Vn . Since these translates span V, it follows that Vn is at most one-dimensional for each n, and from this we deduce that if πg (R) has a nontrivial kernel in Vm , then Vn is trivial for all n > m. Whether or not there exists an m with this property, we obtain an explicit basis for V, and can describe the action of K ∞ and a generating set for U (g) on this basis explicitly in terms of λ, μ and , say, where  = ±1 such that π K∞ (δ1 ) . v # =  · v # . Now suppose that neither of the above cases applies. Take v # to be an eigenvector of the operator π K∞ (δ1 ) ◦ πg (L) in the subspace V1 . Denote the eigenvalue by α. As noted to the statement of the theorem, it is one of the  prior μ2 1 two square roots of 2 λ − 2 + 12 . Then we find that π K∞ (δ1 ) . v # = α · πg (L) . v # , and this again allows us to eliminate half of the translates in (7.5.11) from consideration, leaving a set which are all contained in distinct spaces Vn . As before we deduce that each Vn is at most one-dimensional, and we have a basis on which the action is completely described in terms of λ, μ and α. To deduce iii), note that whenever m satisfies (7.5.8) the operator πg (L) ◦ πg (R) annhilates the space Vm . Suppose Vm is trivial. Then by proposition 7.5.7 we deduce that S = {k ∈ Z | |k| > m}, and hence that πg (L) has non-trivial kernel in Vm+2 . Furthermore, our analysis of case ii) above showed that whenever πg (L) has non-trivial kernel in Vm+2 , the space Vm is trivial. Hence, if Vm is not trivial, then the kernel of πg (L) in Vm+2 is trivial. It follows that in fact πg (R) annhilates Vm . 

Exercises for Chapter 7

275

Exercises for Chapter 7 7.1 Let (π, V ) be an arbitrary (g, K ∞ )-module. Prove that V decomposes as a direct sum of S O(2, R)-invariant subspaces B Vm , V = m∈Z

 θ sin θ  where Vm = v ∈ V | π K ∞ −cos .v = eimθ v (∀θ ) . Recall that sin θ cos θ this means every element v ∈ V can be written uniquely as a finite sum v = vm for some elements vm ∈ Vm . (Compare with Exercise 6.6.) A 7.2 Let (π, V ) be an arbitrary (g, K ∞ )-module, and write V = m∈Z Vm as in the previous exercise. Prove that if v ∈ V is a K ∞ -fixed vector, then v ∈ V0 . Conclude that V is ramified if V0 = 0. 



7.3 Which irreducible admissible (g, K ∞ )-modules are unramified in the classification of Theorem 7.5.10? 7.4 Using the Iwasawa decomposition, we can write any g ∈ G L(2, R) in the y1 x  form g = 0 y2 k for some y1 , y2 ∈ R× , x ∈ R and k ∈ K ∞ . Prove that the quantities |y1 |∞ and |y2 |∞ depend only on g, and not on the choice of decomposition. (Compare with Exercise 6.20.) 7.5 A smooth function f : G L(2, R) → C has a Fourier expansion in the variable θ using the Iwasawa decomposition:      1 x y 0 r 0 cos θ sin θ cn (x, y, r )einθ , f = 0 1 0 1 0 r − sin θ cos θ n∈Z

where x > 0, y = 0, 0 ≤ θ < 2π and the Fourier coefficients cn depend smoothly on x, y, r . If f is K ∞ -finite, prove that cn ≡ 0 for all but finitely many n. 7.6 Prove Lemma 7.5.4. 7.7 Let (π, V ) be a (g, K ∞ )-module, and write V = cise 7.1.

A m∈Z

Vm as in Exer-

(a) Prove that πg (D H ) acts on Vm by the scalar im for each m ∈ Z. Hint: Use (7.1.3) and compute exp(t H ) explicitly. (b) Prove that π K ∞ (δ1 ) maps Vm to V−m for every m ∈ Z. (c) Prove that πg (R) maps Vm to Vm+2 for each m ∈ Z, and that πg (L) maps Vm to Vm−2 . 7.8 Let det : G L(2, AQ ) → G L(1, AQ ) be the adelic determinant map: det ({g∞ , g2 , g3 , . . . }) = {det(g∞ ), det(g2 ), det(g3 ), . . . } .

276

Theory of admissible (g, K∞ ) modules for GL(2, R) Suppose ω : A× Q → C is a unitary Hecke character as in Definition 2.1.2, and define φ = ω ◦ det to be the induced function on G L(2, AQ ). In Exercise 4.16 we proved this is an adelic automorphic form with central character ω2 . Let V = C.φ be the space of all complex multiples of φ. Prove that V is a (g, K ∞ )-module and compute the invariants of this representation as in Section 7.5. That is, find the constants λ, μ, the set of K ∞ -types, the parity, and the sign of the action of π K ∞ (δ1 ) on V0 .

7.9 Suppose f is a classical Maass form of weight k, level N and character χ as in Definition 3.5.7, and let (π, V f ) be the (g, K ∞ )-module associated to f adelic as in Example 5.2.12. Prove the following statements. (a) If f is even of weight zero and level 1, then (π, V f ) is unramified. (b) If f is odd of weight zero, then (π, V f ) is ramified. (c) If f is of odd weight, then (π, V f ) is ramified. (d) If f corresponds to a non-constant holomorphic modular form, then (π, V f ) is ramified. 7.10 For each of the (g, K ∞ )-modules in the previous exercise, compute the invariants of the representation as in Section 7.5. That is, find the constants λ, μ, the set of K ∞ -types, the parity, and the sign of the action of π K ∞ (δ1 ) on V0 (resp. π K ∞ (δ1 ) ◦ πg (L) on V1 ). 7.11 Proposition 5.2.10 asserts that any element of U (g) can be written as a linear combination of terms of the form R  ◦ D aH ◦ b ◦ D cZ with a, b, c non-negative integers and  ∈ Z. Prove that such a representation is unique provided we collect together all terms with the same ordered set of exponents (, a, b, c). Hint: Use the explicit action of U (g) on various principal series V∞ (s, ω) as given by Proposition 7.4.8.

8 The contragredient representation for G L(2)

8.1 The contragredient representation for G L(2, Q p ) Definition 8.1.1 (Dual representation) Fix a prime p. Let V be a complex vector space and let (π, V ) be a representation of G L(2, Q p ). Let V ∗ := { : V → C |  is a linear map}

(8.1.2)

be the vector space dual to V . The dual representation is defined to be (π ∗ , V ∗ ) where the action π ∗ : G L(2, Q p ) → G L(V ∗ ) is defined by   π ∗ (g) . (v) =  π (g −1 ) . v ,

(∀g ∈ G L(2, Q p ), v ∈ V,  ∈ V ∗ ).

Definition 8.1.3 (Smooth linear map) Fix a prime p and a representation (π, V ) of G L(2, Q p ). A linear map  ∈ V ∗ (the dual vector space given in (8.1.2)) is smooth relative to π if there exists an open set K ⊂ G L(2, Q p ) such that π ∗ (k) .  =  for all k ∈ K . Observe that if  ∈ V ∗ is smooth relative to π then π ∗ (g) .  is also smooth for every fixed g ∈ G L(2, Q p ). With this observation, we can make the following important definition. Definition 8.1.4 (Contragredient representation) Fix a prime p. Let V be a complex vector space and let (π, V ) be a smooth representation of G L(2, Q p ). Let ( V denote the space of smooth (relative to π ) linear maps  : V → C as in V by Definition 8.1.3. Define an action ( π of G L(2, Q p ) on (   ( π (g) . (v) =  π (g −1 ) . v ,

(∀g ∈ G L(2, Q p ), v ∈ V,  ∈ ( V ).

Then (( π, ( V ) is termed the contragredient representation of (π, V ). Remark Let (π, V ) be an irreducible admissible representation of G L(2, Q p ). The dual representation is, in general, not smooth, much less admissible. 277

278

The contragredient representation for GL(2)

Proposition 8.1.5 (Contragredient of admissible is admissible) Fix a prime p. Let (π, V ) be an admissible representation of G L(2, Q p ). Then the contragredient representation (( π, ( V ), as defined in Definition 8.1.4, is also admissible. Proof It follows immediately from the definition that the contragredient of any representation of G L(2, Q p ) is a smooth representation. We turn to the matter of admissibility. For n ∈ N, let    K n = k ∈ G L(2, Z p )  k − I2 ∈ p n Mat(2, Z p ) . (8.1.6) Since V is admissible, we know that the subspace V K n := {v ∈ V | π (k) . v = v,

∀k ∈ K n }

is finite dimensional for each n, and we need to prove the same about ( V K n := { ∈ ( V |( π (k) .  = ,

∀k ∈ K n }.

In fact, we shall prove that dim ( V K n = dim V K n .

(8.1.7)

To do this, we define a projection Projn : V → V K n by Projn (v) :=

1 π (κ) . v, [K n : K m ] κ

where m is chosen sufficiently large so that v is fixed by K m , and κ is summed over any set of coset representatives for K n /K m . Suppose that  ∈ ( V K n . Then, for all v ∈ V,    Projn (v) =

1 1 (π (κ) . v) = ( π (κ −1 ) . (v) [K n : K m ] κ [K n : K m ] κ 1 = (v) = (v). [K n : K m ] κ

Thus, every element of ( V K n factors through Projn . On the other hand, it follows easily from the definition of Projn that Projn (π (k) . v) = Projn (v) for all v ∈ V V K n is precisely equal to the set of all linear functionals and all k ∈ K n . Thus ( on V which are of the form   (v) = l Projn (v) , (∀v ∈ V ), for some l in the dual of the finite dimensional vector space V K n . This proves (8.1.7), from which it follows that (( π, ( V ) is admissible. 

8.1 The contragredient representation for GL(2, Qp )

279

Proposition 8.1.8 (The contragredient of an irreducible is irreducible) Fix a prime p. Let (π, V ) be an irreducible admissible representation of π, ( V ), as defined in G L(2, Q p ). Then the contragredient representation (( Definition 8.1.4, is also irreducible. ( be a proper non-trivial invariant subspace of ( Proof Suppose not. Let W V. ( )}. We will prove that W is a proper Let W = {v ∈ V | (v) = 0, (∀  ∈ W non-trivial invariant subspace of V, contradicting the irreducibility of V. It is clear that W is invariant, and that it is not all of V. Choose n such ( ∩( ( K n := W V K n is a proper non-trivial subspace of ( V K n . Then there that W Kn Kn ( exists v =/ 0 in V , which is killed by every element of W . Now, let  be ( . Then  is fixed by K m for some m ≥ n, and an arbitrary element of W   1 1 (v) =  π (κ) . v = ( π (κ) .  (v) , (8.1.9) [K n : K m ] κ [K n : K m ] κ where κ is summed over any set of coset representatives for K n /K m . The ( K n , which proves that (8.1.9) π (κ) .  is an element of W functional [K n 1:K m ] κ ( ( , i.e., is a non-trivial vanishes. It follows that v is killed by any element of W element of W, which completes the proof.  Proposition 8.1.10 (Constructing a model for the contragredient from a pairing) Fix a prime p. Let (π1 , V1 ) and (π2 , V2 ) be two irreducible admissible representations of G L(2, Q p ). Suppose that there is a non-trivial bilinear form  ,  : V1 × V2 → C which is invariant, in the sense that it satisfies   ∀ v1 ∈ V1 , v2 ∈ V2 , g ∈ G L(2, Q p ) . π1 (g) . v1 , π2 (g) . v2  = v1 , v2 , Then (π2 , V2 ) is isomorphic to the contragredient representation of (π1 , V1 ). Proof For v2 ∈ V2 we define a linear map L v2 : V1 → C by the formula L v2 (v1 ) := v1 , v2 . Clearly, v2 → L v2 is a linear mapping V2 → V1∗ . We need to show that it actually maps into the subspace ( V1 ⊂ V1∗ of smooth linear functionals, and that it intertwines between the two actions of G L(2, Q p ). π1 (g) on L v2 is For any v1 ∈ V1 , v2 ∈ V2 , and g ∈ G L(2, Q p ) the action of ( defined by   ( π1 (g) . L v2 (v1 ) := L v2 π1 (g −1 ) . v1 = π1 (g −1 ) . v1 , v2  = v1 , π2 (g) . v2  = L π2 (g) . v2 (v1 ).

280

The contragredient representation for GL(2)

This proves that v2 → L v2 is an intertwining map. It then follows that L v2 is fixed by any open compact subgroup which fixes v2 , and hence lies in ( V1 ⊂ V1∗ . π1 , ( V1 ) are both irreducible, it follows that any nonSince (π2 , V2 ) and (( trivial intertwining map between them has to be an isomorphism. This completes the proof.  Remark Even if (π1 , V1 ) and (π2 , V2 ) are not assumed to be irreducible we π1 , ( V1 ). obtain a non-trivial map (π2 , V2 ) → (( The next corollary follows immediately from Proposition 8.1.10, and Dixmier’s Lemma, 6.1.8. Corollary 8.1.11 (Uniqueness of invariant pairings) Fix a prime p. Let (π1 , V1 ), (π2 , V2 ), be two irreducible admissible representations of G L(2, Q p ) and  , 1 : V1 × V2 → C,

 , 2 : V1 × V2 → C,

be two non-trivial invariant bilinear 8.1.10. Then there 9forms as 8in Proposition 9 8 exists c ∈ C, c =/ 0, such that v1 , v2 1 = c · v1 , v2 2 for all v1 ∈ V1 , v2 ∈ V2 . In light of Proposition 8.1.10, it is natural to ask whether there is a canonical bilinear form between any admissible representation of G L(2, Q p ) and its contragredient. The answer is yes, as described below. Definition 8.1.12 (The canonical bilinear form on V × ( V .) Fix a prime p. π, ( V ) the conLet (π, V ) be an admissible representation of G L(2, Q p ) and (( ( tragredient as defined in Definition 8.1.4. For v ∈ V and  ∈ V , we define 8 9 v,  = (v). It follows immediately from the definitions that this is an invariant bilinear form V × ( V → C. With this definition made, the following is now also an immediate consequence of Proposition 8.1.10. Corollary 8.1.13 (Contragredient of the contragredient is the original) Fix a prime p. Let (π, V ) be an irreducible, admissible representation of π, ( V ) its contragredient as in Definition 8.1.4. Then the G L(2, Q p ), and (( contragredient of (( π, ( V ) is isomorphic to (π, V ). A linear map from a finite dimensional vector space to itself can be realized as a matrix. The coefficients of this matrix play an important role in ´ linear algebra. Matrix coefficients of Lie groups were first studied by Elie Cartan and Israel Gelfand and have played a very important role in the theory of automorphic representations (see [Gelfand-Graev-Piatetski-Shapiro, 1969], [Godement-Jacquet, 1972]). Here is the definition.

8.2 The contragredient representation of a principal series

281

Definition 8.1.14 (Matrix coefficient for G L(2, Q p )) Fix a prime p. Let (π, V ) be an irreducible admissible representation of G L(2, Q p ), with contragredient representation (( π, ( V ). Let  ,  : V × ( V → C be the canonical invari( ant bilinear form V × V → C given in Definition 8.1.12. Fix v ∈ V, ( v∈( V. Then the map 8 9 g → π (g) . v, ( v, (g ∈ G L(2, Q p )), is called a matrix coefficient of the representation (π, V ). It is frequently convenient to work not with the contragredient itself, as defined in Definition 8.1.4, but rather with a model of it constructed via a bilinear form as in Proposition 8.1.10. Lemma 8.1.15 (Matrix coefficients may be computed using any model) Fix a prime p. Let (π1 , V1 ) and (π2 , V2 ) be two irreducible admissible representations of G L(2, Q p ) and  ,  : V1 × V2 → C a non-trivial invariant bilinear form. Then for any v1 ∈ V1 and v2 ∈ V2 , the function g → π1 (g) . v1 , v2  is a matrix coefficient as defined in Definition 8.1.14, and every matrix coefficient is of this form for suitable v1 , v2 . Proof Let T : V2 → ( V1 be the isomorphism constructed in Proposition 8.1.10. To avoid confusion, write  , V1 ×V2 for the bilinear form on V1 × V2 posited in the statement of this lemma, and write  , V1 ×V˜ 1 for the canonical bilinear V1 as in Definition 8.1.12. Clearly, v1 , T (v2 )V1 ×V˜ 1 is a nonform on V1 × ( trivial invariant bilinear form on V1 × V2 . By Corollary 8.1.11, it is a multiple of  , V1 ×V2 . The result follows easily. 

8.2 The contragredient representation of a principal series representation of G L(2, Q p ) In this section we will use Proposition 8.1.10 to prove that the contragredient of an irreducible principal series representation of G L(2, Q p ) is isomorphic to another principal series representation. In order to do so, we construct an explicit bilinear pairing on the two spaces of functions using an integral. To make the relation between a principal series representation and its contragredient as symmetric as possible, it is convenient to introduce the following definitions. Definition 8.2.1 (The vector space B p (χ1 , χ2 )) Fix a prime p. Let χ1 , χ2 be characters of Q×p . We define B p (χ1 , χ2 ) to be the space      a 1  a b  2  g =   χ1 (a)χ2 (d) f (g), f : G L(2, Q p ) → C  f 0 d  d p  ∀a, d ∈ Q×p , b ∈ Q p , g ∈ G L(2, Q p ), f is locally constant .

282

The contragredient representation for GL(2)

Let s1 , s2 ∈ C and ω = (ω1 , ω2 ) be normalized unitary characters of Q×p as in Definition 6.5.1. If we choose χ1 (y) = ω1 (y)|y|sp1 , then

χ2 (y) = ω2 (y)|y|sp2 ,

(∀y ∈ Q×p ),

  B p (χ1 , χ2 ) = V p (s1 + 1/2, s2 − 1/2) , ω .

Furthermore, any character χ : Q×p → C may be expressed as χ (y) = ω(y)·|y|sp , for all y ∈ Q×p , for some unique normalized unitary character ω, (as in Definition 6.5.1), and some complex number s, which is unique up to addition of an 2πi , so Definition 8.2.1 simply gives an alternate notation integral multiple of log p for the exact same collection of vector spaces introduced in Definition 6.5.2. As in Definition 6.5.3, we define a principal series representation to be such a vector space equipped with the action of G L(2, Q p ) by right translation. Definition 8.2.2 (Principal series representation for G L(2, Q p )) Fix a prime p and two characters χ1 , χ2 : Q×p → C× . The principal series representation of  G L(2, Q p ), associated to (χ1 , χ2 ), is the representation  π, B p (χ1 , χ2 ) where π is the action by right translation, and B p (χ1 , χ2 ) is the vector space defined in Definition 8.2.1. Thus   π (h) . f (g) = f (gh) for all g, h ∈ G L(2, Q p ), and f ∈ B p (χ1 , χ2 ). Proposition 8.2.3 (Invariant bilinear form on principal series  of G L(2, Q p ))  Fix a prime p and two characters χ1 , χ2 : Q×p → C× . Let π, B p (χ1 , χ2 ) be the principal series representation 8 9 as in 8.2.2. Then there exists a nonzero invariant bilinear form, , : B p (χ1 , χ2 ) × B p (χ1−1 , χ2−1 ) → C, as in Proposition 8.1.10, which is given by 

f 1 (k) · f 2 (k) d × k,

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

for f 1 ∈ B p (χ1 , χ2 ), f 2 ∈ B p (χ1−1 , χ2−1 ), where d × k is the normalized Haar measure on G L(2, Q p ), restricted to the open compact subgroup G L(2, Z p ). Proof It is enough to show that 8 9 π (g) . f 1 , π (g) . f 2 =  f 1 , f 2  for all g ∈ G L(2, Q p ) and all f 1 ∈ B p (χ1 , χ2 ), f 2 ∈ B p (χ1−1 , χ2−1 ).

8.3 Contragredient of a special representation of GL(2, Qp )

283

Let F(g) = f 1 (g) · f 2 (g). Then it follows from the definition of B p (χ1 , χ2 ) given in Definition 8.2.1, that the function F satisfies (6.9.11). Corollary 6.9.10 immediately implies   9 8 × F(kg) d k = F(k) d × k π (g) . f 1 , π (g) . f 2 = G L(2,Z p )

G L(2,Z p )

=  f 1 , f 2 .  Proposition 8.2.4 (Contragredient of a principal series representation) Fix  × × a prime p and two characters χ1 , χ2 : Q p → C . Let π, B p (χ1 , χ2 ) denote an irreducible principal series representation as defined  in Definition  Cp (χ1 , χ2 ) is isomorphic 8.2.1. Then the contragredient representation ( π, B   to π, B p (χ1−1 , χ2−1 ) . Proof Let χ1 (y) = ω1 (y)|y|sp1 ,

χ2 (y) = ω2 (y)|y|sp2 ,

(∀y ∈ Q×p ),

where ω = (ω1 , ω2 ) is a pair of normalized unitary characters of Q×p as in Definition 6.5.1. Then   B p (χ1 , χ2 ) = V p (s1 + 1/2, s2 − 1/2) , ω .    By Corollary 6.8.15, the representation π, V p (s1 + 1/2, s2 − 1/2) ω is irreducible if and ω1 =/ ω2or p s1 −s2 +1 =/ 1, p 2 . Under this assumption,  only if −1 it follows that π, B p (χ1 , χ2−1 ) is also irreducible. Admissibility of these representations was proved in Proposition 6.5.5, and an invariant bilinear form was given in Proposition 8.2.3. Thus, Proposition 8.1.10 may be applied to them, yielding the result.      Cp (χ1 , χ2 ) and π, B p (χ −1 , χ −1 ) are isomorRemark It turns out that ( π, B 1 2 phic when they are reducible as well.

8.3 Contragredient of a special representation of G L(2, Q p ) In Section 6.10, we showed that there are two types of special representations for G L(2, Q p ). We will now prove that the contragredient of any of these special representations is isomorphic to another special representation of the other type. As in Section 8.2, we do this by constructing an explicit invariant bilinear form on the spaces of the special representations. In the next proposition, we shall write χ · | |sp for the character of Q×p given in terms of a given character χ of Q×p by χ (g)|g|sp for all g ∈ Q×p . Referring to the relationship between the B p (χ1 , χ2 ) notation for principal series and the

284

The contragredient representation for GL(2)

V p (s, ω) notation employed in Chapter 6, we see from Section 6.10 that special representations occur when χ2 = χ1 · | |±1 p . Proposition 8.3.1 (The contragredient of a special representation) Fix a prime p and a character χ : Q×p → C× . Let (π, V ) denote the special    1 −1 representation which is contained in π, B p χ · | | p2 , χ · | | p 2 , and let (π , V ) be the special representation which is a quotient of   1  −1 π, B p χ · | | p 2 , χ · | | p2 . Then the contragredient of (π , V ) is isomorphic to (π, V ) and vice versa. Proof By Proposition 8.2.3, an invariant bilinear form,    1 1 −1 −1  ,  : B p χ · | | p2 , χ · | | p 2 × B p χ · | | p 2 , χ · | | p2 → C, is given by



f 1 (k) · f 2 (k) d × k,

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

   1 1 −1 −1 for f 1 ∈ B p χ · | | p2 , χ · | | p 2 , f 2 ∈ B p χ · | | p 2 , χ · | | p2 . The representation  1 −1 B p χ · | | p 2 , χ · | | p2 contains a one-dimensional invariant subspace spanned by the function δχ (g) = χ1 (det g),

(∀g ∈ G L(2, Q p )),

(8.3.2)

and the special representation (π , V ) is given by right translation on the quotient  1 + −1 δχ · C. B p χ · | | p 2 , χ · | | p2   1 −1 By Definition 6.10.9, the space B p χ ·| | p2 , χ ·| | p 2 contains an infinite dimen  1   −1 sional invariant subspace ker L χ , where L χ : B p χ · | | p2 , χ · | | p 2 → C is given by     1 −1 L χ ( f ) := f (k) δχ (k) d × k, f ∈ B p χ · | | p2 , χ · | | p 2 . G L(2,Z p )

The special representation (π, V ) is given by the action of right  translation on  1   − 12  2 the subspace ker L χ . Thus V = f ∈ B p χ · | | p , χ · | | p   f, δχ  = 0

8.4 Contragredient of a supercuspidal representation

285

is just theorthogonal complement of δχ · C. It follows that, for any f 1 in the  space ker L χ , the function    1 −1 f 2 →  f 1 , f 2 , f 1 ∈ B p χ · | | p2 , χ · | | p 2  1  −1 is a well defined function: B p χ · | | p 2 , χ · | | p2 δχ · C → C. This defines an invariant bilinear form on the two special representations. We leave it to the reader to verify that this form is non-zero. Proposition 8.1.10 yields the result. 

8.4 Contragredient of a supercuspidal representation We have seen in Corollary 8.1.13 that the map π → ( π establishes a one-to-one correspondence between the set of isomorphism classes of irreducible admissible representations of G L(2, Q p ) and itself. We have also seen in Section 8.2 that irreducible principal series representations correspond to other irreducible principle series representations, and in Section 8.3 that special representations correspond to other special representations. It is clear from the definitions that the contragredient of a one-dimensional representation is one-dimensional. It then immediately follows from the classification Theorem 6.13.4 that the contragredient of a supercuspidal representation is again supercuspidal. A model for the contragredient can be constructed from a pairing as we have shown in Proposition 8.1.10. It is convenient to explicitly construct the pairing using the Kirillov model as in Theorem 6.7.2 (see also Theorem 6.14.2 and Proposition 6.15.1). Definition 8.4.1 (Bilinear form on S(Q×p ) corresponding to a character) Fix a prime p. Let S(Q×p ) denote the Bruhat-Schwartz space of functions Q×p → C as in Theorem 6.7.2. Let ω : Q×p → C× be a fixed character. Associated to ω, we may define a bilinear form  , ω : S(Q×p ) × S(Q×p ) → C, by requiring  f 1 (y) f 2 (−y) ω−1 (y) d × y,  f 1 , f 2 ω := Q×p

for all f 1 , f 2 ∈ S(Q×p ). Proposition 8.4.2 (Invariance of the bilinear form on S(Q×p ) corresponding to  character) Fix a prime p and a supercuspidal representation  a central π , S(Q×p ) of G L(2, Q p ) with central character ωπ as in Proposition 6.1.10. Then for all functions f 1 , f 2 ∈ S(Q×p ), and all g ∈ G L(2, Q p ), we have D E  π (g) . f 1 , π (g) . ωπ−1 (det g) . f 2 =  f 1 , f 2 ωπ , (8.4.3) ωπ

S(Q×p )

S(Q×p )

where  , ωπ : × → C, is the bilinear form corresponding to the central character ωπ of π , as in Definition 8.4.1.

286

The contragredient representation for GL(2)   Proof Clearly (8.4.3) holds when g = r0 r0 with r ∈ Q×p . To prove that (8.4.3) holds for allg ∈G L(2,  Qp ) it isenough  to show that it holds for matrices g 01 −1 0

of the form 10 b1 , a0 01 , and matrices generate   G L(2, Q p ). 1b 0 1

For g = F π



1 b 0 1

, we have

 . f1 ,  Qp

 F π



a 0

a 0 0 1

0 1

π



1 b 0 1

     G 1 b −1 . ωπ det . f2 0 1 ω 8

=

For g =

, with a ∈ Q×p , b ∈ Q p , since these

e p (by) f 1 (y)e p (−by) f 2 (−y) ωπ−1 (y) d × y = f 1 , f 2

9

π

ωπ

.

 , we have

 . f1 , π 



a 0

0 1

    a −1 . ωπ det 0 8

= Qp

f 1 (ay) f 2 (−ay) ωπ−1 (ay) d × y = f 1 , f 2

0 1



9 ωπ

G . f2

ωπ

.

  The case g = −10 01 is more difficult to deal with. We will prove the following equivalent formulation of (8.4.3): D 8 9 E  π (g) . f 1 , f 2 ω = f 1 , π (g −1 ) . ωπ (detg) . f 2 . (8.4.4) ωπ

π

Recall Lemma 6.14.6 which says   0 1

. f (y) = ωπ (y) π −1 0

ψ (mod p N )

 Q×p

J (t y, ψ) . f (t) d × t (8.4.5)

for any integer N where p N is larger than the conductor of f . It follows from (8.4.5) and the definition of the action of J (t y, ψ) given in Definition 6.14.5 that G F   0 1 π . f1 , f2 −1 0 ωπ ⎤ ⎡   ⎣ωπ (y) = J (t y, ψ) . f 1 (t) d × t ⎦ · f 2 (−y) ωπ−1 (y) d × y Q×p

 =

Q×p

⎡ ⎣

ψ (mod p N )

ψ (mod

pN )

 Q×p

Q×p

⎤

J (t y, ψ) . f 1 (t) d × t ⎦ · f 2 (−y) d × y.

(8.4.6)

8.4 Contragredient of a supercuspidal representation

287

On the other hand, in the proof of Theorem 6.14.2 (uniqueness of the Kirillov model) it was shown that J (t y, ψ) acts by a scalar. It immediately follows from (8.4.6) that G F   0 1 π . f1 , f2 −1 0 ωπ ⎡ ⎤   = f 1 (t) ⎣ J (−t y, ψ) . f 2 (y) d × y ⎦ d × t Q×p

 =

Q×p

ψ (mod p N )

⎡

Q×p

f 1 (t)ωπ (−1) · ⎣ωπ (−t)

ψ (mod p N )

 Q×p

⎤ J (−t y, ψ) . f 2 (y) d × y ⎦ · ωπ−1 (t) d × t.

ByLemma is equal  6.14.6, the expression in brackets above   to 01 01

. f 2 (−t). Using the fact that ωπ (−1) · π = π −1 0  −1 0  0 −1

π , and comparing with the definition of  , ωπ , we obtain 1 0 (8.4.4).  Proposition 8.4.7 of a supercuspidal representation) Fix   (Contragredient a prime p. Let π , S(Q×p ) be a supercuspidal representation of G L(2, Q p ) with central character ωπ as in Proposition 6.1.10. Let π ⊗ ωπ−1 ◦ det : G L(2, Q p ) → G L(S(Q×p )) be defined by π ⊗ ωπ−1 ◦ det(g) = ωπ−1 (det g) · π (g). Then the contragredient   representation of π , S(Q×p ) is isomorphic to π ⊗ ωπ−1 ◦ det, S(Q×p ) . Proof This follows immediately from Propositions 8.1.10 and 8.4.2.



Remark It can be shown that π and π ⊗ ωπ−1 are isomorphic in general, not only when π is supercuspidal. The pairings in Propositions 8.2.3 and 8.4.1, together with the classification Theorem 6.13.4, allow one to give an interesting characterization of supercuspidal representations of G L(2, Q p ) in terms of their matrix coefficients. Definition 8.4.8 (Compactly supported modulo the center) Fix a prime p. Let 5   r 0  × ∼ r ∈ Q Z (G L(2, Q p )) = = Q×p p 0 r  denote the center of G L(2, Q p ). A function f : G L(2, Q p ) → C is termed compactly supported modulo the center if there exists a compact set U ⊂ G L(2, Q p ) such that the support of f is contained in U · Z (G L(2, Q p )).

288

The contragredient representation for GL(2)

It will turn out that the matrix coefficients of a representation are compactly supported modulo the center of G L(2, Q p ) if and only if the representation is supercuspidal. We present the first part of this in the next proposition. Proposition 8.4.9 (Matrix coefficients of supercuspidal representations are compactly supported modulo the center) Fix a prime p. Let (π, V ) be a supercuspidal irreducible admissible representation of G L(2, Q p ) with contragredient (( π, ( V ). Let  ,  : V × ( V → C be the canonical invariant bilinear form given in Definition 8.1.12. Fix v ∈ V , ( v ∈ ( V . Then the matrix coefficient 8 9 (8.4.10) g → π (g) . v, ( v, (g ∈ G L(2, Q p )), defined in Definition 8.1.14, is compactly supported modulo the center as in Definition 8.4.8. Proof If (π, V ) is a supercuspidal representation then its Kirillov model is of  the form π , S(Q×p ) and we have the pairing of Definition 8.4.1. By Lemma  8.1.15, we may compute matrix coefficients using the model π ⊗ ωπ−1 ◦  det, S(Q×p ) for (( π, ( V ) and the invariant pairing  f 1 (y) f 2 (−y) · ωπ−1 (y) d × y, ( f 1 , f 2 ∈ S(Q×p )).  f1 , f2  = Q×p

By the p-adic Cartan decomposition,   every element of G L(2, Q p ) is equiv-

alent modulo the center to k1 ·

t 0 01

· k2 for some k1 , k2 ∈ G L(2, Z p ) and

t ∈ Q×p . For fixed f 1 , f 2 ∈ S(Q×p ) and fixed k1 , k2 ∈ G L(2, Z p ), the set of t such that  G   F  t 0

· k2 . f 1 , f 2 π k1 · 0 1  π (k2 ) . f 1 (t y)π (k1−1 ). f 2 (−y) d × y =/ 0 = Q×p

is clearly compact, since π (k2 ) . f 1 and π (k1−1 ) . f 2 are compactly supported. Now f 1 is fixed by a subgroup of G L(2, Z p ) which is open, and hence of finite index. Hence, as k1 ranges over G L(2, Z p ), the function π (k2 ) . f 1 only ranges over a finite set of functions in S(Q×p ). The same applies to π (k1 ) . f 2 Hence the set of t such that  G F      t 0 for some k1 , k2 ∈ G L(2, Z p ) , π k1 · · k2 . f 1 , f 2 =/ 0, 0 1 is compact. It immediately follows that (8.4.10) has compact support modulo the center of G L(2, Q p ). 

8.5 The contragredient representation for GL(2, R)

289

8.5 The contragredient representation for G L(2, R) Recall the definition of a (g, K ∞ )-module as given in Definition 5.1.4. Definition 8.5.1 ((g, K ∞ )-module) Let g = gl(2, C) and K ∞ = O(2, R). 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 for all α ∈ g, Dα given by Definition 4.5.5, and 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 ∞ . Remark Let π = (π K∞ , πg ). We denote the (g, K ∞ )-module in Definition 8.5.1 as (π, V ). Definition 8.5.2 (Dual of a (g, K ∞ )-module) Let V be a (g, K ∞ )-module as in Definition 8.5.1 and let V ∗ := { : V → C |  = linear map} be the dual space. We define the dual of the (g, K ∞ )-module V to be V ∗ with actions π K∗∞ : K ∞ → G L(V ∗ ), which are defined by   π K∗∞ (k) . (v) :=  π K ∞ (k −1 ) . v ,

πg∗ : gl(2, R) → End(V ∗ ),   πg∗ (Dα ) . (v) := − πg (Dα ) . v ,

for all k ∈ K ∞ , v ∈ V, and α ∈ g, with Dα given by Definition 4.5.5. Warning: The dual of a (g, K ∞ )-module is usually not a (g, K ∞ )-module! This is because the subspace of V ∗ spanned by {π K∗∞ (k) .  | k ∈ K ∞ } may not be finite dimensional. To illustrate the warning, consider the example where V is a principal series representation with basis { f n | n ∈ 2Z} and f n given by Definition 7.4.7. For n ∈ Z, let n ∈ V ∗ be the linear functional given by  1, n = m, n ( f m ) = 0, n =/ m. 

Then ∗

V =

∞ n=−∞

    cn n  cn ∈ C ,

290

The contragredient representation for GL(2)

  ∞   is finite if and only if  = cn n , and dim span π K∗∞ (k) .   k ∈ K ∞ with only finitely many non-zero coefficients cn .

n=−∞

Definition 8.5.3 (The space ( V ) Let (π, V ) be a (g, K ∞ )-module as in Definition 8.5.1. Let V ∗ be the dual as in Definition 8.5.2. Define   

 cos θ sin θ  . v = einθ · v, (∀θ ∈ R) , Vn := v ∈ V  πk∞ − sin θ cos θ    ∗  ( (8.5.4) V :=  ∈ V |Vn ≡ 0, for all but finitely many n ∈ Z . Here |Vn means  restricted to Vn . Lemma 8.5.5 (The space ( V is preserved by the two actions) Let ( V denote ∗ ∗ the subspace of V defined in Definition 8.5.3. Then π K∞ (k) .  and πg∗ (D) .  are in ( V , for all k ∈ K ∞ , D ∈ U (g) and  ∈ ( V. Proof Each of the spaces Vn is preserved by the action of S O(2, R). It follows that if k ∈ S O(2, R) and  vanishes on Vn , then π ∗ (k) .  also vanishes on Vn . Now suppose k ∈ O(2, R), but k ∈ S O(2, R). Then for all k ∈ S O(2, R), we have k · k · k = (k )−1 . It follows that π K∞ (k) maps Vn to V−n . Hence π ∗ (k) .  vanishes on Vn provided that  vanishes on V−n . Finally, we need to show that for each D ∈ U (g) there is a finite set S, / S. By depending on  and D, such that πg∗ (D) .  vanishes on Vn for n ∈ linearity, it is enough to handle the case when  vanishes on Vn except for one value of n, and when D is a word in R, L , D Z and D H , defined as Section 5.2. It may be shown that D Z and D H preserve each of the subspaces Vn , while R maps Vn to Vn+2 and L maps Vn to Vn−2 . Indeed, this was done in Section 5.2 under the assumption that the (g, K ∞ )-module under consideration was obtained from right translation on a space of smooth functions. The extension is straightforward and is proved in the exercises of Chapter 7. It follows that if  is supported on Vn 0 and D is a word in R, L , D Z and D H , then πg (D) .  is supported on Vn 0 −2n 1 +2n 2 , where n 1 is the number of instances of L in the word D, and n 2 is the number of instances of R. This completes the proof.  V denote the Definition 8.5.6 (Contragredient of a (g, K ∞ )-module) Let ( V with actions πg∗ and π K∗ ∞ as subspace of V ∗ defined in Definition 8.5.3. Then ( in Definition 8.5.2 is defined to be the contragredient of the (g, K ∞ )-module V . π, ( V ) the contragredient We will also use the notation ( π = (πk∗∞ , πg∗ ) and call (( representation of the (g, K ∞ )-module V . Proposition 8.5.7 (Contragredient of admissible is admissible) Let (π, V ) be an admissible (g, K ∞ )-module as in Definition 7.1.4. Then the contragrediπ, ( V ) as defined in Definition 8.5.6 is also admissible. ent (g, K ∞ )-module ((

8.5 The contragredient representation for GL(2, R)

291

Proof It is enough to check that for each integer n the space      cos θ sin θ ( .(v) V  π K∗∞ Vn :=  ∈ ( − sin θ cos θ

5

= einθ · (v), (∀θ ∈ R, ∀v ∈ V )

(8.5.8)

is finite dimensional. In fact, we shall prove the much stronger statement that Vn ) where Vn is given by (8.5.4), which is finite dimensional dim(Vn ) = dim(( by Definition 7.1.4. To do this we define, for each n ∈ Z, a projection operator, Projn : V → Vn , given by    2π 1 cos θ sin θ −inθ e ·π K∞ Projn (v) := . v dθ, (∀v ∈ V ). − sin θ cos θ 2π 0 To see that Projn (v) ∈ Vn for all v ∈ V , we compute for any θ ∈ [0, 2π ),   cos θ sin θ . Projn (v) πK∞ − sin θ cos θ 1 = 2π

2π

e−inθ · π K∞



cos(θ + θ ) sin(θ + θ ) − sin(θ + θ ) cos(θ + θ )

 .v dθ

0

=e

inθ

· Projn (v),

after making the change of variable θ → θ − θ . Suppose that  ∈ ( Vn . Then for all v ∈ V , we have      2π 1 cos θ sin θ . v dθ  (Projn (v)) = e−inθ ·  π K∞ − sin θ cos θ 2π 0    2π 1 cos θ − sin θ −inθ = . (v) dθ e ·( πK∞ sin θ cos θ 2π 0  2π 1 = e−inθ einθ · (v) dθ 2π 0 = (v). Thus, every element of ( Vn factors through Projn . On the other  hand, it follows easily from the definition of Projn that Projn (v) π K∞ (k) . v = Projn (v) for all v ∈ Vn and all k ∈ K ∞ . Therefore, ( Vn is precisely equal to the set of all linear functionals on V which are of the form (v) =  (Projn (v)) for some  in the dual of the finite dimensional vector space Vn . This proves that dim(Vn ) = π, ( V ) is dim(( Vn ) from which it follows that the contragredient representation (( admissible. 

292

The contragredient representation for GL(2)

Proposition 8.5.9 (Contragredient of an irreducible is irreducible) Let (π, V ) be an irreducible and admissible (g, K ∞ )-module as in Definition 7.1.4. π, ( V ), as defined in Definition 8.5.6, Then the contragredient (g, K ∞ )-module (( is also irreducible. ( be a proper non-trivial invariant subspace of ( Proof Suppose not. Let W V . Let ( )}. We will prove that W is a proper W = {v ∈ V | (v) = 0, (∀  ∈ W non-trivial invariant subspace of V, contradicting the irreducibility of V. Vn It is clear that W is invariant, and that it is not all of V. For n ∈ Z, let Vn , ( (n := W ( ∩( be given by (8.5.4), (8.5.8), respectively. Choose n such that W Vn is a proper non-trivial subspace of ( Vn . Then there exists v =/ 0 in Vn , which is ( . Then (n . Now, let  be an arbitrary element of W killed by every element of W      2π 1 cos θ sin θ (v) =  . v dθ e−inθ · π K∞ − sin θ cos θ 2π 0    2π 1 cos θ − sin θ −inθ . (v) dθ. e ·( πK∞ = sin θ cos θ 2π 0 (8.5.10) The functional 1 2π



2π

e

−inθ

0

 ·( πK∞

cos θ sin θ

− sin θ cos θ

 .  dθ

(n , which proves that (8.5.10) vanishes. It follows that v is an element of W ( , i.e., is a non-trivial element of W, which is killed by any element of W completes the proof.  Lemma 8.5.11 (Schur’s lemma for irreducible admissible (g, K ∞ )modules) Let (π, V ) be an irreducible admissible (g, K ∞ )-module as in Definition 7.1.4. Let T : V → V be an intertwining map as in Definition 5.1.9. Then there exists a constant c ∈ C such that T . v = c · v for all v ∈ V. Proof For n ∈ Z recall that  

 Vn := v ∈ V  πk∞

cos θ − sin θ

sin θ cos θ

 . v = einθ · v,

 (∀θ ∈ R) .

Clearly, v ∈ Vn implies that T . v ∈ Vn . Since V is assumed to be admissible, it follows that dim(Vn ) < ∞ for all n and there exists an integer n ≥ 0 such that Vn =/ {0}. Therefore, the linear map T has an eigenvector v0 ∈ Vn . Let λ be the eigenvalue and define the eigenspace    Vλ := v ∈ V  T . v = λv =/ {0}. Then, for all k ∈ O(2, R), we have       T . π K ∞ (k) . v = π K ∞ (k) . T . v = π K ∞ (k) . λv = λ π K ∞ (k) . v.

8.5 The contragredient representation for GL(2, R)

293

  Similarly, T . πg (Dα ) . v = λ · πg (Dα ) . v for all α in g. It follows that Vλ is a non-trivial invariant subspace of V . Since V is irreducible, this implies that Vλ = V.  Proposition 8.5.12 (Constructing a model of the contragredient from a pairing) Let (π, V ) and (π , V ) be two irreducible admissible (g, K ∞ )modules as in Definition 7.1.4. Suppose that there exists a non-trivial bilinear form  ,  : V × V → C which is invariant, in the sense that it satisfies 8 9 π K∞ (k) . v, π K ∞ (k) . v = v, v , (8.5.13) 8 9 8 9

πg (Dα ) . v, v = − v, πg (Dα ) . v , for all v ∈ V, v ∈ V , k ∈ K ∞ , and α ∈ g. Then (π , V ) is isomorphic to the contragredient of (π, V ). Proof For v ∈ V we may define a linear map v : V → C by the formula v (v) := v, v . Clearly, v → v is a linear mapping V → V ∗ . We need to show that it actually maps into the subspace ( V ⊂ V ∗ (defined in (8.5.4)), and V. that it is an intertwining map. We first prove that v ∈ ( For n ∈ Z recall that   

 cos θ sin θ  . v = einθ · v, (∀θ ∈ R) Vn := v ∈ V  πk∞ − sin θ cos θ Define Vn similarly. Assume v ∈ Vn , v ∈ Vm , with m =/ − n. Then v, v  = 0 by (8.5.13) because ei(n−m)θ =/ 1. By Definition 7.1.1, for any v ∈ V , there is a finite set of integers S such that v ∈ ⊕ Vn . One sees at once that v |Vm ≡ 0 if −m ∈ S. n∈S

Here v |Vm means v restricted to Vm . It immediately follows from (8.5.4) that V. v ∈ ( π = (( πK∞ , ( πg ) on For any v ∈ V, v ∈ V , k ∈ K ∞ and α ∈ g the action of ( v is defined by   ( π K∞ (k) . v (v) := v π K∞ (k −1 ) . v = π K∞ (k −1 ) . v1 , v2  = v1 , π K ∞ (k) . v  = π

K∞

(k) . v

(v).

By similar steps one also obtains ( πg (Dα ) . v (v) = π (Dα ) . v (v). g

Since V, ( V are irreducible it follows that V is isomorphic to ( V.



The next result follows immediately from Proposition 8.5.12 and Lemma 8.5.11.

294

The contragredient representation for GL(2)

Corollary 8.5.14 (Uniqueness of invariant pairings) Let (π1 , V1 ), (π2 , V2 ), be two irreducible admissible (g, K ∞ )-modules as in Definition 7.1.4. Let  , 1 : V1 × V2 → C,

 , 2 : V1 × V2 → C,

be two non-trivial invariant bilinear 8.5.12. Then there 8 9forms as 8in Proposition 9 exists c ∈ C, c =/ 0, such that v1 , v2 1 = c · v1 , v2 2 for all v1 ∈ V1 , v2 ∈ V2 . As in the p-adic case, we have a canonical invariant bilinear form between a (g, K ∞ )-module and its contragredient. Definition 8.5.15 (Canonical bilinear form on V × ( V ) Let (π, V ) denote an π, ( V ) the contragrediadmissible (g, K ∞ )-module as in Definition 7.1.4, and (( ( ent as in Definition 8.5.6. For v ∈ V and  ∈ V , we define 8 9 v,  = (v). It follows immediately from the definitions that this is an invariant bilinear form V × ( V → C. Corollary 8.5.16 (Contragredient of the contragredient is the original) Let (π, V ) be an irreducible, admissible (g, K ∞ )-module as in Definition 7.1.4, and (( π, ( V ) the contragredient as in Definition 8.5.6. Then the contragredient of (( π, ( V ) is isomorphic to (π, V ). Proof Follows from Proposition 8.5.12, in view of Proposition 8.5.9.



We would also like to have a suitable notion of matrix coefficients for G L(2, R). This turns out to be somewhat tricky. We shall defer this until the end of the next section, and exploit Theorem 7.5.10 to make a definition which relies on embedding (π, V ) into a principal series representation.

8.6 The contragredient representation of a principal series representation of G L(2, R) It is a consequence of the classification Theorem 7.5.10 that every irreducible admissible (g, K ∞ )-module is isomorphic to a principal series representation or a subrepresentation of a principal series representation. Hence, the contragredient is again a principal series representation or a subrepresentation of a principal series representation. To explicitly find the contragredient of an irreducible admissible (g, K ∞ )-module, we will construct a pairing satisfying the conditions in Proposition 8.5.12. The construction of the pairing is very similar to the construction of the pairing for G L(2, Q p ) we gave in Section 8.2. Accordingly, we start by proving an analogue of Lemma 6.9.7.

8.6 The contragredient representation of a principal series

295

Proposition 8.6.1 (Factorization of integrals over G L(2, R)) Assume that   α β  : G L(2, R) → C is continuous with compact support. For g = γ δ ∈ G L(2, R) let dα dβ dγ dδ d × g := |αδ − βγ |2∞ be the invariant measure. Then  ∞ ∞ ∞ 2π ,  r 0 y (g) d g =  0 r 0



×

G L(2,R)

+ 



x=−∞ y=0 r =0 θ=0

r 0 0 r



y 0

x 1



cos θ − sin θ

Proof First of all, note that   (g) d × g =



(g) d × g +

G L(2,R)+

G L(2,R)

sin θ cos θ



:

=

x 1



−1 0 0 1



cos θ − sin θ

sin θ cos θ



 dr dy d x. dθ r y2

(g) d × g

G L(2,R)−

  ; −1 0 (g) +  g d × g, 0 1

G L(2,R)+

where G L(2, R)+ is the set of 2×2 real matrices of positive determinant (which is a subgroup of G L(2, R)) and G L(2, R)− is the set of 2 × 2 real matrices of negative determinant (which is not; these are the connected components of G L(2, R)). Next, make the change of coordinates: α = r (y cos θ − x sin θ ), β = r (y sin θ + x cos θ ), γ = −r sin θ, δ = r cos θ. Computing the Jacobian of this transformation gives: dα dβ dγ dδ = r 3 d x dr dθ. The result follows.  dy dθ dr. Consequently d × g = d x dy y2 r Lemma 8.6.2 Let ϕ : G L(2, R) → C be a continuous function of compact support. Define      ∞  ∞  ∞  d x d y dr y x r 0 . (8.6.3) ϕ · ·g ϕ0 (g) := 0 1 0 r y 2 |r | −∞ −∞ −∞ Then ϕ0 (g) satisfies

 ϕ0

a 0

b d



 ·g

a    =   · ϕ0 (g), d ∞

(8.6.4)

for all a, d ∈ R× , b ∈ R, and g ∈ G L(2, R). Furthermore, every smooth function F : G L(2, R) → C, satisfying (8.6.4) for all a, d ∈ R× , b ∈ R, and g ∈ G L(2, R) is given by (8.6.3) for some ϕ.

296

The contragredient representation for GL(2)

Proof The proof is similar to the proof of Lemma 6.9.7. However, the method of constructing ϕ such that ϕ0 is equal to a specified F is somewhat different. The Iwasawa decomposition from Proposition 4.1.1 amounts to a set of global coordinates on the manifold G L(2, R) (more properly, on each of its two connected components). Thus we may define a continuous, compactly supported function on G L(2, R) by the formula  ϕ

1 0

x 1



y 0

0 1



cos θ − sin θ

sin θ cos θ



= h 1 (x)h 2 (y)h 3 (r ) · F

±1 0 

0 1



cos θ − sin θ

r 0 0 r

sin θ cos θ

 

±1 0

0 1

 ,

with x ∈ R, y > 0, 0 ≤ θ < 2π, r > 0, and h 1 , any continuous function of compact support on R, and h 2 , h 3 any two continuous functions of compact support on (0, ∞). If we take h 1 , h 2 , and h 3 to each have integral 1, then ϕ0 will equal F. We leave the details to the reader.  Definition 8.6.5 (The vector space B∞ (χ1 , χ2 )) Let χ1 , χ2 be characters of R× . We define B∞ (χ1 , χ2 ) to be the space 

   f : G L(2, R) → C  

 f

a 0

b d

  a 1  2 g =   χ1 (a)χ2 (d) f (g), d p



×

∀ a, d ∈ R , b ∈ R, g ∈ G L(2, R), f is K ∞ -finite . It is left to the reader to verify that B∞ (χ1 , χ2 ) = V∞ (s, ω) s1 − 12

(8.6.6) s2 + 12

⇐⇒ χ1 (t) = |t|∞ ω1 (t) and χ2 (t) = |t|∞ ω2 (t), for s = (s1 , s2 ) ∈ C2 , ω = (ω1 , ω2 ) a pair of normalized unitary characters of R× as in Definition 7.4.1, and χ1 , χ2 characters of R× . Definition 8.6.7 (Principal series representation of G L(2, R)) Let χ1 , χ2 be representation characters of R× . The principal series   of G L(2, R), associated to (χ1 , χ2 ), is the (g, K ∞ )-module π, B∞ (χ1 , χ2 ) where π = (πg , π K∞ ), with π K∞ being the action by right translation, and πg the usual action by differential operators, and B∞ (χ1 , χ2 ) is the vector space defined in Definition 8.6.5. Proposition 8.6.8 (Invariant bilinear form on principal  series of G L(2, R)) Let χ1 , χ2 be characters of R× , and let π, B∞ (χ1 , χ2 ) be a principal series representation as in Theorem 8.6.10. Then there exists a non-zero

8.6 The contragredient representation of a principal series

297

8 9 invariant bilinear form, , : B∞ (χ1 , χ2 ) × B∞ (χ1−1 , χ2−1 ) → C, as in Proposition 8.5.12, given by     2π  cos θ sin θ cos θ sin θ · f2 dθ, f1  f 1 , f 2  := − sin θ cos θ − sin θ cos θ 0 for f 1 ∈ B∞ (χ1 , χ2 ), f 2 ∈ B∞ (χ1−1 , χ2−1 ). Proof We must check that 9 of (8.5.13) are satisfied. 8 the invariance conditions =  f 1 , f 2 ,9 for all k ∈ K ∞ . (k) . f , π (k) . f First of all, it is clear that π 1 2 k∞ 9 k∞ 8 8 We want to show that πg (Dα ) . f 1 , f 2 = − f 1 , πg (Dα ) . f 2 , for all α ∈ g. For g ∈ G L(2, R), define F(g) := f 1 (g) f 2 (g). Then    a  a b   F g =   F(g) 0 d d ∞ for all a, d ∈ R× , b ∈ R, and g ∈ G L(2, R). By Lemma 8.6.2, we can choose a continuous compactly supported function  : G L(2, R) → C such that the function 0 , given in (8.6.3), is equal to F. Then, for all g ∈ G L(2, R), we have    2π  8 9 cos θ sin θ π (g) . f 1 , π (g) . f 2 = F g dθ − sin θ cos θ 0         2π  d x d y dr y x r 0 = ϕ · · kg 0 1 0 r × × y 2 |r | R R R 0 ϕ(hg) d × h. = G L(2,R)

by Proposition 8.6.1. Using the invariance of the measure d × h, we obtain  8 9 π (g) . f 1 , π (g) . f 2 = ϕ(h) d × h =  f 1 , f 2 , (8.6.9) G L(2,R)

for all g ∈ G L(2, R). It follows from (8.6.9) that for all α ∈ g that    2π  ∂ cos θ sin θ F exp(tα) dθ = 0, − sin θ cos θ ∂t 0 because the integral of exp(tα).   is independent cos θ sin θ Let k = k(θ ) = − sin θ cos θ . By the product rule, we have       ∂ F (k exp(tα))  = f 1 (k) · πg (Dα ) . f 2 (k) + πg (Dα ) . f 1 (k) · f 2 (k). ∂t t=0 8 9 8 9 This immediately implies that πg (Dα ) . f 1 , f 2 = − f 1 , πg (Dα ) . f 2 . 

298

The contragredient representation for GL(2)

Theorem 8.6.10 (Contragredients of irreducible (g, K ∞ )-modules)   (1) The principal series representation π, B∞ (χ1 , χ2 ) is irreducible except when there exists m ∈ Z, m =/ 1 such that tm χ1 (t) = , (∀ t ∈ R× ). (8.6.11) χ2 (t) |t|∞    (2) The contragredient of π, B∞ (χ1 , χ2 ) is isomorphic to π,  B∞ (χ1−1 , χ2−1 ) .     (3) If one of the two representations π, B∞ (χ1 , χ2 ) or π, B∞ (χ2 , χ1 ) is irreducible, then both are irreducible and the two are isomorphic. (4) If (8.6.11) is satisfied for some positive integer m > 1, then π, B∞ (χ1 , χ2 ) has an infinite dimensional irreducible subrepresentation, and a finite dimensional irreducible quotient. The subrepresentation is discrete series if χ1 χ2 is a unitary character, and essentially discrete series otherwise. (5) If  (8.6.11) is satisfied for some nonpositive integer m, then π, B∞ (χ1 , χ2 ) has a finite dimensional irreducible subrepresentation, and an infinite-dimensional irreducible quotient. This quotient is isomorphic to (a) the contragredient of the discrete series or essentially discrete  series representation contained in π, B∞ (χ1−1 , χ2−1 ) , (b) the discrete series or essentially discrete series representation  contained in π, B∞ (χ2 , χ1 ) . (6) The only isomorphisms among principal series representations and their subquotients are those indicated in parts (3) and (5b). (7) The limit of discrete series representations introduced in Definition 7.4.10 are equal to the irreducible principal series representations  π, B∞ (χ1 , χ2 ) occurring in the case when (8.6.11) is satisfied with If (8.6.11)is satisfied with m = 1, m = 1, provided that χ1 χ2 is unitary.  but χ1 χ2 is nonunitary, then π, B∞ (χ1 , χ2 ) may be described as essentially limit of discrete series. Proof Let s = (s1 , s2 ) ∈ C2 and ω = (ω1 , ω2 ) be a pair of normalized unitary characters as in Definition 7.4.1.  so that (8.6.6) is satisfied. According to Corollary 7.4.9, the representation π, B∞ (χ2 , χ1 ) is reducible if and only if s1 − s2 is an integer m such that (−1)m = ω1 (−1)ω2 (−1). Now, for t ∈ R× , χ1 (t) |t|s∞1 −s2 ω1 ω2−1 (t) = . χ2 (t) |t|∞ It follows easily from Definition 7.4.1 that ω1 (t)ω2−1 (t) depends only on the sign of t and that if s1 − s2 is an integer m such that (−1)m = ω1 (−1)ω2 (−1),

8.6 The contragredient representation of a principal series

299

7.4.9 then |t|s∞1 −s2 ω1 ω2−1 (t) = t m for all t ∈ R× . This proves (1). Corollary  also describes the structure of the representation π, B∞ (χ2 , χ1 ) in the cases when it is reducible. This proves part of (4) and also part of (5). The rest of (4) follows from the definitions of discrete series and essentially discrete series given in Definition 7.4.10 and the remark thereafter, and the same is true of (7). Assertions (3), (5b), and (6), characterizing the conditions under which two principal series representations or their subquotients are equal follow from Theorem 7.5.10. Finally, (2) follows immediately from Proposition 8.6.8 in view of Proposition 8.5.12, while the proof of (5a) is an easy adaptation of that of Proposition 8.3.1.  Definition 8.6.12 (Matrix coefficient of a (g, K ∞ )-module) Let (π, V ) be an π, ( V ) the irreducible, admissible (g, K ∞ ) module as in Definition 7.1.4, and (( contragredient as in Definition 8.5.6. By Theorem 7.5.10 and Definition 8.6.7, we may assume that (π, V ) is equal either to an irreducible principal series representation (π, B∞ (χ1 , χ2 ), or to an invariant subspace of a reducible one. In either case take f ∈ V and take ( f in the principal series representation −1 −1 B∞ (χ1 , χ2 ), and define      2π  cos θ sin θ cos θ sin θ ( β f, ( f · g ·f dθ f (g) = − sin θ cos θ − sin θ cos θ 0 (8.6.13) for g ∈ G L(2, R). The function β f,( f : G L(2, R) → C is called a matrix coefficient. Remark Note that the function h → f (hg) is not, in general, K ∞ -finite! Thus our functions are defined using not the invariant pairing  ,  : V × ( V →C itself, but an extension of it to a larger space of functions which is closed under right translation. Next, it is desirable to have another expression for the invariant bilinear form and matrix coefficients. This will be based on a new decomposition of G L(2, R), called the Bruhat decomposition. Definition 8.6.14 (Bruhat decomposition for G L(2, R)) The group G L(2, R) is a disjoint union of double cosets     0 −1 B(R) ∪ B(R) · · N (R) , 1 0 where B(R) denotes the group of all upper triangular invertible matrices, and N (R) is the subgroup consisting of those with 1’s on the diagonal. Proposition 8.6.15 (Haar measure on G L(2, R) in terms of the Bruhat decomposition) Let f : G L(2, R) be an integrable function, with respect to the invariant measure d × g given in Proposition 8.6.1. Then

300

The contragredient representation for GL(2) 

×



∞   ∞

f (g) d g =

f −∞ R× R× −∞

G L(2,R)



·

t1 0

0 t2



1 0

x1 1

0 −1 1 0





1 0

x2 1

 · d x1

dt1 dt2 d x2 . |t1 |2

Proof The proof is very similar to the proof of Proposition 8.6.1. Using the Bruhat decomposition, we replace the integral over G L(2, R) by an integral   0 −1 over B(R) · 1 0 N (R). (It is clear that B(R) has measure zero.) Now 

1 0

x1 1



t1 0

0 t2



0 1

−1 0



1 0

x2 1



 =

x 1 t2 t2

−t1 + x1 x2 t2 x 2 t2

 .

The series of changes of variable x1 → γ −1 α,

t2 → γ ,

x2 → γ −1 δ,

t1 → −β +

transforms d x1 |tdt|12 dt2 d x2 into the invariant measure 1 ∞

dαdβdγ dδ . |αδ−βγ |2∞

αδ γ



Proposition 8.6.16 (Invariant bilinear form on principal series of G L(2, R) in terms of Bruhat  decomposition) Let χ1 , χ2 be characters of R× , and let π, B∞ (χ1 , χ2 ) be a principal series representation as in Defini8 9 tion 8.6.7. Then the invariant bilinear form, , : B∞ (χ1 , χ2 )×B∞ (χ1−1 , χ2−1 ) given in Proposition 8.6.8 may also be expressed in terms of the Bruhat decomposition as     ∞  0 1 0 1 ( f  f, ( f = 2· f d x. (8.6.17) 1 x 1 x −∞ Proof The function F(g) := f (g) · ( f (g) satisfies      a b a  (∀a, d ∈ R× , b ∈ R, g ∈ G L(2, R)). F · g =   F(g), 0 d d ∞ (8.6.18) Now   √ x    1 √−1 2 − 12 0 −1 + x ) (1 + x 2 )− 2 −x(1 2 2 1+x 1+x √ = . (8.6.19) √1 √x 1 x 0 1 + x2 2 2 1+x

It follows that

   0 F  1

1 x

   

C

<

1+x

c , 1 + x2

where c is a constant corresponding to the maximum value of F on the compact group O(2, R). Absolute convergence of (8.6.17) follows easily.

8.6 The contragredient representation of a principal series

301

It follows from Lemma 8.6.2 and Proposition 8.6.15 that the integral on the right side of equation (8.6.17) is an invariant bilinear form, and hence a scalar f are chosen so multiple of  , . To see that the scalar is 12 , note that if f and ( that F is constantly equal to 1 on S O(2, R), then the integral on the right hand side of (8.6.17) is simply 

∞

1 d x = π, 1 + x2

−∞



while the integral in Proposition 8.6.8 is clearly 2π.

Note that neither the proof of Proposition 8.6.8 nor that of Proposition 8.6.16 makes use of the K ∞ -finiteness of the elements of B∞ (χ1 , χ2 ) and B∞ (χ1−1 , χ2−1 ). This observation will be crucial for application to matrix coefficients, which are defined using translates of elements of B∞ (χ1 , χ2 ) which are not, in general, K ∞ -finite. The following proposition will be applicable to this. Proposition 8.6.20 Let F : G L(2, R) → C be any smooth function which satisfies (8.6.18). Then 



2π

F 0

cos θ − sin θ

sin θ cos θ



 dθ = 2 ·



∞

F −∞

0 1

1 x

 d x.

Proof This proposition follows from Lemma 8.6.2 and a comparison of the two expressions for the Haar measure on G L(2, R). It can also be proved directly, using the substitution x = cot θ.  Proposition 8.6.21 (Alternate expression for matrix coefficients) Let (π, V ) be an irreducible, admissible (g, K ∞ )-module as in Definition 7.1.4, and let (( π, ( V ) denote the contragredient as in Definition 8.5.6. As in Definition 8.6.12, assume that (π, V ) is equal either to an irreducible principal series representation B∞ (χ1 , χ2 ), or to an invariant subspace of a reducible one. Then the matrix coefficient (8.6.13) of (π, V ) may also be expressed as  2·



∞

f −∞

0 1

1 x

   0 g ( f 1

1 x



 d x,

Proof Immediate from Proposition 8.6.20.

 g ∈ G L(2, R) . (8.6.22)



There is one more matter which must be addressed. In Definition 8.6.12, we have defined matrix coefficients for an irreducible admissible (g, K ∞ )-module (π, V ) by realizing (π, V ) either as an irreducible principal series (g, K ∞ )module, or as a subrepresentation of a reducible one. Such a realization always

302

The contragredient representation for GL(2)

exists by Theorem 7.5.10, but what Theorem 7.5.10 actually says is that every irreducible admissible infinite dimensional (g, K ∞ )-module can be realized, either as an irreducible principal series (g, K ∞ )-module, or as a sub-(g, K ∞ )module of a reducible principal series (g, K ∞ )-module, in two different ways! This motivates the next result. Proposition 8.6.23 (The set of matrix coefficients of a (g, K ∞ )-module is independent of the realization) Let (π, V ) and (π , V ) be any two isomorphic irreducible admissible (g, K ∞ )-modules with V ⊆ B∞ (χ1 , χ2 ) and V ⊆ B∞ (χ3 , χ4 ), where χi is a character R× → C for i = 1, 2, 3, 4. Then for every f ∈ V and ( f ∈ B∞ (χ1−1 , χ2−1 ) there exist f ∈ V and −1 −1

( f ∈ B∞ (χ3 , χ4 ) such that the matrix coefficients β f,˜v and β f ,˜v , defined as in Definition 8.6.12, are equal. Proof Take f ∈ V and ( f ∈ B∞ (χ1−1 , χ2−1 ), and let L : V → V be an isomorphism. Define f = L( f ). Using explicit bases as in Definition 7.4.7, f  = one easily writes down an element ( f of B∞ (χ1−1 , χ2−1 ), such that L(h), (

f will be unique only if V is actually h, f  for all h ∈ V. (This element ( equal to B∞ (χ1 , χ2 ).) It is immediate from the definitions that β f, ( f (g) = β f ,( f (g)

(8.6.24)

holds if g lies in the product of O(2, R) and the center of G L(2, R). To show that (8.6.24) holds for all g ∈ G L(2, R), note that each side of the equation (8.6.24) is a smooth function of g ∈ G L(2, R). The next lemma and its proof are adapted from [Bump, 1996]. Lemma 8.6.25 Let h : G L(2, R) → C be a smooth function. Then for α ∈ gl(2, R), ∞ 1 n D h(g). (8.6.26) h(g · exp(α)) = n! α n=0 Proof Fix g ∈ G L(2, R), α ∈ gl(2, R), and define a function H : R × R → C by      H (t1 , t2 ) = exp πg (Dt1 α ) . h g · exp(t2 α) . Then

     ∂ ∂ H (t1 , t2 ) = H (t1 , t2 ) = πg (Dα ) . exp πg (Dt1 α ) . h g · exp(t2 α) . ∂t1 ∂t2

Now ∂t∂1 H (t1 , t2 ) = ∂t∂2 H (t1 , t2 ) implies that H is constant on the line t1 + t2 = c for any constant c. This proves (8.6.26), since the right hand side is H (1, 0) and the left hand side is H (0, 1). 

8.7 Global contragredients for GL(2, AQ )

303

Completion of the proof of Theorem 8.6.23: Recall that α ∈ gl(2, R) acts by the differential operator Dα on both V and V . Further L(Dαn f ) = Dαn (L( f )) = Dαn f . Each side of (8.6.24) is defined by an integral of a smooth function over a compact set, so we may differentiate under the integral sign, and deduce n that Dαn β f,( f (h) = Dα β f , f˜ (h) for all α ∈ g, n ∈ N, and h ∈ O(2, R) · Z (G L(2, R)). It follows from Lemma 8.6.25 that (8.6.24) holds whenever g = h · exp(α), with h ∈ O(2, R) · Z (G L(2, R)) and α ∈ gl(2, R). It is left to the reader to verify that every element of G L(2, R) is of this form. 

8.7 Global contragredients for G L(2, AQ ) In Section 5.1 we introduced automorphic representations for G L(2, AQ ) and then described in some detail the sorts of objects that they are: (g, K ∞ ) × G L(2, Afinite )-modules as defined in Definition 5.1.5. A little further in Definition 5.5.4 we introduced the notion of an admissible (g, K ∞ ) × G L(2, Afinite )-module and then showed that irreducible, cuspidal automorphic representations, which were defined in Definition 5.1 as (g, K ∞ ) × G L(2, Afinite )-modules, are admissible. In this section, we will describe the contragredient of a (g, K ∞ ) × G L(2, Afinite )-module. We will show that the contragredient of an irreducible and admissible automorphic representation can be explicitly obtained from an invariant pairing. We will then construct such a pairing in a very explicit manner. This will then lead to a very concrete realization of the contragredient representation of an irreducible admissible cuspidal automorphic representation. Definition 8.7.1 (Dual of a (g, K∞ ) × G L(2, Afinite  )-module) Let V denote a complex vector space. Let π = (πg , π K∞ ), πfinite as in Definition 5.1.5 and let (π, V ) be a (g, K ∞ ) × G L(2, Afinite )-module as in Definition 5.1.5. Define V ∗ := { : V → C |  is a linear map} to be the dual of the vector space V . The dual of (π, V ) is defined to be  ∗ is defined by (π ∗ , V ∗ ) where the action π ∗ = (πg∗ , π K∗∞ ), πfinite       ∗ −1 .v , ∀ afinite ∈ Afinite , πfinite (afinite ) . (v) :=  πfinite afinite       π K∗∞ (k) . (v) :=  π K∞ k −1 . v , ∀ k = {k∞ , I2 , I2 , . . . }, k∞ ∈ K ∞ ,     πg∗ (Dα ) . (v) := − πg (Dα ) . v , ∀α ∈ g , for all v ∈ V and for all  ∈ V ∗ . Recall the embedding at ∞, denoted i ∞ : G L(2, R) → G L(2, AQ ), defined by        5 1 0 1 0 1 0 , , , · · · , (∀g ∈ G L(2, R)). i ∞ (g) = g, 0 1 0 1 0 1

304

The contragredient representation for GL(2)

Definition 8.7.2 (Contragredient of a (g, K ∞ ) × G L(2, Afinite )-module) Fix a (g, K ∞ ) × G L(2, Afinite )-module (π, V ) as in Definition 8.7.1. Let V ∗ be the dual as in Definition 8.7.1. Define   

 Vn := v ∈ V  πk∞ i ∞

cos θ − sin θ

sin θ cos θ

 . v = einθ · v,

 (∀θ ∈ R) ,

and let ( V denote the set of all  ∈ V ∗ which are fixed by some open compact subgroup K ⊂ G L(2, Afinite ) and such that |Vn ≡ 0 for all but finitely many n ∈ Z. Here |Vn means  restricted to Vn . V , and hence The actions π ∗ as in Definition 8.7.1 preserve the subspace ( ( give well defined actions on that subspace, and the space V with actions π ∗ as in Definition 8.7.1 is defined to be the contragredient of (π, V ). We set ∗ and call (( π, ( V ) the contragredient representation. ( π = (πg∗ , π K∗ ∞ ), πfinite Lemma 8.7.3 (Schur’s lemma for irreducible and admissible (g, K ∞ ) × G L(2, Afinite )-modules) Let (π, V ) denote an irreducible and admissible (g, K ∞ ) × G L(2, Afinite )-module as in Definition 5.1.5. Let T : V → V be any non-trivial intertwining map as in Lemma 5.1.7. Then there exists a non-zero constant c ∈ C such that T . v = c · v for all v ∈ V. Proof For each open compact subgroup K ⊂ G L(2, Afinite ) and each n ∈ Z, the intertwining map T maps the space

VnK = {v ∈ Vn | πfinite (k) . v = v,

(∀ k ∈ K )}

into itself. By admissibility, this space is finite dimensional. So T has an eigen vector and there exists c ∈ C and v0 ∈ VnK such that T . v0 = c · v0 . The eigenspace {v ∈ V | T . v = c · v} is an invariant subspace, so by irreducibility, it must be everything.  Proposition 8.7.4 (Constructing a model for the contragredient from a pairing) Let (π, V ), (π , V ) be two irreducible admissible (g, K ∞ ) × G L(2, Afinite )-modules as in Definition 5.1.5. Suppose that there exists a bilinear pairing  ,  : V × V −→ C which is invariant in the sense that 8 9 8 9 (i) πg (Dα ) . v, v = − v, πg (Dα ) . v , (∀ α ∈ g), E D (∀ k ∈ K ∞ ), (ii) π K∞ (k) . v, π K ∞ (k) . v = v, v , E   D

(g) . v = v, v , ∀ g ∈ G L(2, Afinite ) , (iii) πfinite (g) . v, πfinite for all v ∈ V, v ∈ V . Then (π , V ) is isomorphic to the contragredient of (π, V ).

8.7 Global contragredients for GL(2, AQ )

305

Proof For v ∈ V , define a linear map v : V → C by v (v) = v, v , for v ∈ V. Then the map v → v is a linear map V → V ∗ . Furthermore E D E D   [π (Dα ) . v ] (v) = v, πg (Dα ) . v = − πg (Dα ) . v, v = −v πg (Dα ) . v g   = πg∗ (Dα ) . v (v), for all α ∈ g, v ∈ V, v ∈ V . In a completely analogous fashion we have E D E D   ?π (k) . v @ (v) = v, π K ∞(k) . v = π K∞(k −1 ) . v, v = v π K∞(k −1 ) . v K∞   = π K∗∞(k) . v (v), for all k ∈ K ∞ , v ∈ V, v ∈ V . Similarly,   ∗ (g) . v (v), ?π (g) . v @ (v) = πfinite finite

for all g ∈ G L(2, Afinite ), v ∈ V, v ∈ V . This gives an intertwining map V → V ∗ . It remains to show this intertwining map is surjective onto the contragredient ( V. V for every v ∈ V . By Definition 8.7.2, this We need to prove that v ∈ ( is the same as showing that  v Vn ≡ 0, for all but finitely many n ∈ Z, and that v is fixed by some open compact subgroup K ⊂ G L(2, Afinite ).

(g) . v , it follows immediately that v is fixed by Since π ∗ (g) . v = πfinite any compact open subgroup K that fixes v . Such a subgroup exists because our representation is smooth. Furthermore, there is a finite set of integers S such that v ∈ ⊕ Vm . We m∈S  will show that v Vm ≡ 0 unless −m ∈ S. Since v → v is linear, it suffices to treat the case when S = {m}, i.e., contains a single integer m. Then    cos θ sin θ

. v = eimθ v , (∀ θ ∈ R). πK∞ i ∞ − sin θ cos θ If v ∈ Vn with n =/ − m, then the invariance of  ,  implies that       G F cos θ sin θ cos θ sin θ

. v, π K∞ i ∞ . v v, v  = π K∞ i ∞ − sin θ cos θ − sin θ cos θ = ei(n+m)θ v, v , which implies that v, v  = 0.



306

The contragredient representation for GL(2)

Corollary 8.7.5 (Uniqueness of the invariant pairing) Let (π, V ), (π , V ) be two irreducible admissible (g, K ∞ )×G L(2, Afinite )-modules as in Definition 5.1.5. Suppose that  , 1 ,  , 2 : V × V → C are two non-degenerate invariant bilinear forms as in Proposition 8.7.4. Then there exists a non-zero constant c ∈ C such that v, v 1 = c · v, v 2 for all v ∈ V, v ∈ V . Proof This follows immediately from Proposition 8.7.4 and Lemma 8.7.3.



8.8 Integration on G L(2, AQ ) In order to complete our discussion of contragredients, we need to define integrals on G L(2, AQ ) and its quotients Z (AQ )\G L(2, AQ ) and Z (AQ )G L(2, Q)\G L(2, AQ ). The purpose of this section is to introduce the requisite integration theory. As in the case of G L(1) discussed in Chapters 1 and 2, we shall not introduce the full theory of Haar measure. Rather we shall adopt a hybrid approach wherein the full force of Lebesgue integration on manifolds is used at infinity, while at the finite places a measure defined only on compact open sets is used. This is suitable for integrating smooth functions, such as those that arise in automorphic forms. Definition 8.8.1 (Integral on G L(2, AQ )) As in Definition 2.2.3, we first  define the integral of a factorizable function, f = v f v : G L(2, AQ ) → C such that • f ∞ is an integrable function G L(2, R) → C, • f p is a locally constant compactly supported function for each finite prime p, • f p is the characteristic function of G L(2, Z p ) for almost all primes p. For such a function, we define   f (g) d × g = G L(2, AQ )



f v (gv ) d × gv ,

v∈S GL(2, Q ) v

where S is a finite set containing ∞ and all the primes such that f p =/ 1G L(2,Z p ) , and where d × gv denotes the normalized Haar measure given in Definition 6.9.5 for a finite prime, or invariant measure on G L(2, R) introduced in Proposition 8.6.1, if v = ∞. We extend this to finite linear combinations of such functions by linearity, and to infinite linear combinations provided convergence is uniform. We then define   × f (g) d g = 1 E (g) f (g) d × g E

G L(2, AQ )

8.8 Integration on GL(2, AQ )

307

for any set E and function f such that the integral of 1 E · f has been defined previously. Now, relatively few of the functions which arise in the theory of automorphic forms may be integrated over G L(2, AQ ). The reason for this is essentially that no function which is invariant by the center or has a central character may  over the center of G L(2, AQ ) which we denote by

 beintegrated  r 0 × Z (AQ ) :=  r ∈ AQ . (See Exercise 8.9.) For this reason, we wish to 0r introduce an invariant integral over Z (AQ )\G L(2, AQ ). Before explaining how to do this, let us introduce the useful notion of a “locally integrable function.” In practice, we could restrict ourselves to smooth functions as these are the sorts of functions which arise in our applications, but the present discussion is a bit more transparent if we work with a “niceness” condition on our functions which is more relevant to integration. Definition 8.8.2 (Locally integrablefunction) A function f : G L(2, AQ ) → C is said to be locally integrable if E f (g) d × g is defined for each compact subset E of G L(2, AQ ). Definition 8.8.3 (Integral on Z (AQ )\G L(2, AQ )) Let E ⊂ G L(2, AQ ) be given by  ⎫  | det g p | p = 1 or p, ∀ p < ∞, ⎬  g = {g∞ , g2 , . . . , g p . . . } ∈ G L(2, AQ )  . ⎩ ⎭  1 ≤ | det g∞ |∞ ≤ e ⎧ ⎨

Here e = 2.71828 . . . is the base of the natural logarithm. Let f : G L(2, AQ ) → C, be a locally integrable function satisfying f (zg) = f (g) for all z ∈ Z (AQ ) and all g ∈ G L(2, AQ ). If the function f · 1 E , may be integrated using Definition 8.8.1, then we define   f (g) d × g =

f (g) d × g.

E

Z (AQ )\G L(2,AQ )

If not, then we say that f is not integrable on Z (AQ )\G L(2, AQ ). Remarks The key properties of the set E are as follows: it contains an open set, hence is not of measure zero; every element of G L(2, AQ ) is equal to an element of the center times an element of E (though not uniquely); and 

 A× Q

1E

r 0 0 r



d ×r = 1.

308

The contragredient representation for GL(2)

 We could have defined Z (AQ )\G L(2,AQ ) using any other reasonably “nice” set with these key properties, and the functional thus defined on the space of smooth Z (AQ )-invariant functions G L(2, AQ ) → C would be the same. Proposition 8.8.4 (Invariance of integral on Z (AQ )\G L(2, AQ )) Let f : G L(2, AQ ) → C be a smooth function such that f (zg) = f (g) for all z ∈ Z (AQ ) and g ∈ G L(2, AQ ). Then for all h ∈ G L(2, AQ ), we have    × × f (gh) d g = f (hg) d g = f (g) d × g. Z (AQ )\G L(2,AQ )

Z (AQ )\G L(2,AQ )

Z (AQ )\G L(2,AQ )

(This equation should be interpreted in the strong sense: if any one of the three functions is integrable, then they all are integrable and their integrals are equal.) Proof By linearity, it is enough to consider the case when f is a factorizable  function f = v f v and f p = 1G L(2,Z p ) for all but finitely many primes p. Let us make this assumption and introduce the notation    g p ∈ G L(2, Q p )  | det g p | p = 1 or p , if v = p < ∞,   Ev =  g∞ ∈ G L(2, R)  1 ≤ | det g∞ |∞ ≤ e , if v = ∞. Then we will show that   f v (gv h v ) d × gv = Ev

f v (h v gv ) d × gv = Ev



f v (gv ) d × gv Ev

for each place v, for all h v ∈ G L(2, Qv ). First suppose v = ∞. Then h v may be expressed uniquely as z v · h v with z v ∈ Z (R) and | det h v |∞ = 1. Then f (gv h v ) = f (gv h v ), so that   × f v (gv h v ) d gv = f v (gv h v ) d × gv . Ev

Ev

On the other hand the map gv → gv · h v −1 sends E v to itself. It follows that  

× f v (gv h v ) d gv = f v (gv ) d × gv . Ev

Ev

The proof with h v gv replacing gv h v is the same. This completes the real case. Now suppose that v = p < ∞. Then we write h p = z p h p with z p in the center of G L(2, Q p ) and | det h p | p = 1 or p. If | det h p | p = 1, then

8.8 Integration on GL(2, AQ )

309

we are in the same situation as in the real case. If | det h p | p = p, then we break E p up into E 1p := {g p ∈ G L(2, Q p ) | | det g p | p = 1} and E 2p := {g p ∈ G L(2, Q p ) | | det g p | p = p}, and we easily show that  

× f (g p h p ) d g p = f (g p ) d × g p , E 1p

while  E 2p

f (gh p )

×



f p (g p ) d × g p   g p ∈G L(2,Q p )  | det g p |= p2    −1  0 p × d f p gp · g = f p (g p ) d × g p .  p 0 p −1 E 1p

d gp = 

= E 1p

E 2p



Finally, we wish to introduce an integral on functions which are invariant by Z (AQ ), and by G L(2, Q) on the left. This integral should satisfy the invariance property   × f (gh) d g = f (g) d × g. Z (AQ )G L(2,Q)\G L(2,AQ )

Z (AQ )G L(2,Q)\G L(2,AQ )

This can be accomplished with the following definition. Definition 8.8.5 (The integral on Z (AQ )G L(2, Q)\G L(2, AQ )) Define D∞ to be a fundamental domain for G L(2, Z)\G L(2, R), which is also a Borel = {g∞ ∈ D∞ | 1 ≤ | det g∞ |∞ ≤ e}. measurable subset of G L(2, R). Let D∞ Then for any function f : G L(2, AQ ) → C which is locally integrable and satisfies   f (γ gz) = f (g), ∀g ∈ G L(2, AQ ), γ ∈ G L(2, Q), z ∈ Z (AQ ) , we define



f (g) d g = Z (AQ )G L(2,Q)\G L(2,AQ )



×



D∞ ·

p

f (g) d × g,

G L(2,Z p )

provided that the right hand side is defined. If the right hand side is not defined then we say that f is not integrable on Z (AQ )G L(2, Q)\G L(2, AQ ). The omission of D∞ from the notation is justified by the following. Lemma 8.8.6 (Integral on Z (AQ )G L(2, Q)\G L(2, AQ ) is independent of D∞ ) Take a locally integrable function f : G L(2, AQ ) → C satisfying   f (γ gz) = f (g), ∀g ∈ G L(2, AQ ), γ ∈ G L(2, Q), z ∈ Z (AQ ) .

310

The contragredient representation for GL(2)

Let D∞ , F∞ be two Borel-measurable fundamental domains for

as in Definition 8.8.5 and F∞ analogously. G L(2, Z)\G L(2, R). Define D∞ Then   f (g) d × g = f (g) d × g.  

F∞ ·

p

D∞ ·

G L(2,Z p )

p

G L(2,Z p )

This equality may be interpreted in the strong sense: if one side is defined then so is the other and they have the same value.

∩ D∞ is Borel. The set D∞ is Proof For each γ ∈ G L(2, Z) the set γ F∞

is the disjoint union of their translates the disjoint union of these sets and F ∞  

∩ D∞ = F∞ ∩ γ −1 D∞ . For each fixed γ it follows from the γ −1 · γ F∞ invariance of the Haar measure that   f (g) d × g = f (g) d × g.   −1 (γ F∞ ∩D∞ )·

p

(F∞ ∩γ

G L(2,Z p )

Summing over γ , we obtain the lemma.

D∞ )·

p

G L(2,Z p )



Next, we show that the integral just defined has the desired invariance property. Lemma 8.8.7 (Invariance of the integral on Z (AQ )G L(2, Q)\G L(2, AQ )) Let f : G L(2, AQ ) → C be a locally integrable function satisfying   ∀g ∈ G L(2, AQ ), γ ∈ G L(2, Q), z ∈ Z (AQ ) .

f (γ gz) = f (g), Then 



×

f (gh) d g = Z (AQ )G L(2,Q)\G L(2,AQ )

f (g) d × g,

Z (AQ )G L(2,Q)\G L(2,AQ )

for all h ∈ G L(2, AQ ).  Proof Let | |A := v≤∞ | |v denote the adelic absolute value. First assume 

· p G L(2, Z p ), where D∞ is defined as in that | det h|A = 1. Let D = D∞

Definition 8.8.5. It follows that D is a fundamental domain for the action of G L(2, Q) on the set {g ∈ G L(2, A) | 1 ≤ | det g|A ≤ e}, and so is D · h. What we must prove is that   f (g) d × g = f (g) d × g. D

D ·h

The proof is the same as in Lemma 8.8.6.

8.9 The contragredient representation of a cuspidal automorphic

311

If | det h|A =/ 1, then write h = h · z with z ∈ Z (A) and | det h |A = 1. Then we have 



×

f (gh) d g = Z (AQ )G L(2,Q)

×



f (gh z) d g = Z (AQ )G L(2,Q)

f (gh ) d × g,

Z (AQ )G L(2,Q)

simply because f (gh z) = f (gh ) for each value of g ∈ G L(2, A). This reduces the case | det h|A =/ 1 to the case | det h|A = 1, and completes the proof. 

8.9 The contragredient representation of a cuspidal automorphic representation of G L(2, AQ ) We now complete the discussion of global contragredients which was begun in Section 8.7. The invariant integral defined in Section 8.8 will be used to define an invariant bilinear form between two spaces of cusp forms, giving an automorphic realization of the contragredient of any cuspidal automorphic representation. Definition 8.9.1 (Bilinear form on the vector spaces of adelic cusp forms) Let Scusp, ω denote the C-vector space of all adelic cusp forms with central character ω, as in Definition 4.7.7. We define a bilinear form  ,  : Scusp, ω × Scusp, ω−1 → C as follows. Let φ1 ∈ Scusp, ω and φ2 ∈ Scusp, ω−1 . Then we set  φ1 (g) φ2 (g) d × g,

φ1 , φ2  := Z (AQ )·G L(2,Q)\G L(2,AQ )

where Z (AQ ) denotes the center of G L(2, AQ ). It is clear that the function φ1 · φ2 is invariant by Z (A) and by G L(2, Q) on the left. So the integral above is certainly well-defined. What is not clear is that it is convergent. However, this follows from growth properties of cusp forms as we now explain. Proposition 8.9.2 (Adelic cusp forms are of rapid decay) Let φ be an adelic cusp form as in Definition 4.7.7. Fix g = {g∞ , g2 , . . . , g p . . . } ∈ G L(2, A), and M ∈ N. Then there is a constant cg,M > 0 such that     y φ i ∞  0

0 1



  · g 

C

≤ y −M · cg,M ,

( as y → ∞).

Furthermore, the constant cg,M can be made uniform over g in a compact set.

312

The contragredient representation for GL(2)

Proof Let us begin by reformulating the statements to be proved slightly. For M an integer, let P(M) be the property ∀g ∈ G L(2, AQ ), ∃cg ∈ R so that        y 0 φ i ∞ ·g  ≤ y −M · cg ,  0 1 C

( as y → ∞).

(P(M))

Our goal is to prove that an adelic cusp form satisfies P(M) for every M, and that the constants occurring can be made uniform over compact sets. It follows easily from the definitions of automorphic form (4.7.6) and moderate growth (4.7.3) that every automorphic form satisfies P(M) for some M (possibly negative), with a constant which can be made uniform over compact sets. Next, we need to review the proof that π (α) . φ is of moderate growth for any automorphic form φ and any α ∈ g. Recall that this was deduced from the following result of Harish-Chandra: Theorem 8.9.3 Let ψ : S L(2, R) → C be a smooth function which is both S O(2, R)-finite and Z (U (sl(2, C)))-finite. Then there exists a smooth, compactly supported function f : S L(2, R) → C, which satisfies 

ψ(h) f (h −1 h ) d × h, ψ(h ) = SL(2,R)

for all h ∈ S L(2, R). Proof See [Bump, 1997], Section 2.9.



Fix g ∈ G L(2, AQ ). Applying Theorem 8.9.3 with ψg (h ) := φ(g · i ∞ (h )),

  for g ∈ G L(2, AQ ), h ∈ S L(2, R) ,

we find that for any α ∈ sl(2, R), (i.e., any α with trace zero),    φ(gi ∞ (h))Dα f (h −1 ) d × h. π (α) . ψg (h )  = π (α) . φ(g) = h =I2

SL(2,R)

Take K a compact set, and suppose we wish to bound        y 0 π (α) . φ i ∞ · g   0 1 C for g ∈ K . Then take K to be a larger compact set such that g∈K

and

h −1 ∈ Supp( f ) =⇒ gi ∞ (h) ∈ K .

8.9 The contragredient representation of a cuspidal automorphic Let c be a constant such that        y 0 φ i ∞ · g  ≤ cy −M ,  0 1 C Then     y π (α) . φ i ∞  0

0 1



313

(∀y > 0, g ∈ K ).

  · g  ≤ c · max(Dα f ), C

(∀y > 0, g ∈ K ).

Here max(Dα f ) denotes the maximum value of the smooth compactly supported function f. Thus, if φ satisfies P(M), then so does π (α) . φ for every α ∈ sl(2, R), and if the constant cg in P(M) can be made uniform on compact sets for φ, then it can be made uniform for π (α) . φ as well. Furthermore, the bound may be made uniform when α is allowed to vary over a compact subset of sl(2, R). This follows easily from the fact that the map α → Dα is linear. Next, we will show that if π (α) . φ satisfies P(M) (for suitable α), then φ satisfies P(M +1). It will then follow that in fact φ satisfies P(M) for every M. Consider the function      1 x y 0 ·g , (x ∈ R). (x) := φ i ∞ 0 1 0 1 We first show that this function is periodic.  Let K = p K p be an open compact subgroup of G L(2, Afinite ) such that φ(hk ) = φ(h) for all h ∈ G L(2, AQ ) and k ∈ K . For each prime p < ∞ let f p denote the smallest integer such that p pf | n =⇒ g −1 p ·



1 n 0 1



· g p ∈ K p .

For all but finitely many p, we have K p = G L(2, Z p ), and g p ∈ G L(2, Z p ),  so that f p = 0. Define N = p p f p , which is actually a finite product. If N |n, then   1 (x) = φ i diag 0  1 x = φ 0

n 1



+n 1

 i∞ 

y 0

   x y 0 ·g 1 0 1    0 1 n · g∞ , g2 , 1 0 1

1 0

... ,



1 n 0 1

5

 gp, . . .

314   = φ

1 0



1 0 = (x + n). = φ

The contragredient representation for GL(2)     1 n x +n y 0 g2 , · g∞ , g2 g2−1 0 1 1 0 1 5   1 n , . . . . . . , g p g −1 g p p 0 1   5 x +n y 0 · g∞ , g2 , . . . , g p , . . . 1 0 1

This proves that  is periodic, of period N . Hence m  1  N  m  (x) = x · e∞ (u)e∞ − u du. N N 0 N m∈Z

It follows from the cuspidality of φ that the term corresponding to m = 0 vanishes. For m =/ 0, we may integrate by parts:  N  N  m   m  N (u)e∞ − u du =  (u)e∞ − u du. N 2πim 0 N 0 Now,           1 u ∂ y 0 φ i ∞ exp t · E 1,2 · g  0 1 0 1 ∂t t=0        −1  ∂ 1 u y 0 · exp t y · E 1,2 · g  φ i∞ = 0 1 0 1 ∂t t=0      1 u y 0 ·g , = y −1 π (α) . φ i ∞ 0 1 0 1

 (u) =

−1 where α = g∞ · E 1,2 · g∞ ∈ g. It follows that           N  y 0 y  φ i ∞ π (α) . φ i∞ ·g = |(0)| ≤ C   0 1 0 2π y  C

0 1



  ·g  . C

This completes the proof that π (α) . φ satisfies P(M) =⇒ φ satisfies P(M + 1). Clearly, both α and N depend on g in a continuous fashion, so that constant cg in the bound can be made uniform over a compact set. This completes the proof of Theorem 8.9.2.  Remark bound given in Proposition 8.9.2 is not sharp. We have shown that  The y 0 φ i ∞ 0 1 · g vanishes more rapidly than any power of y as y → ∞. In fact, it can be shown that it vanishes exponentially.

8.9 The contragredient representation of a cuspidal automorphic

315

Corollary 8.9.4 (The integral in definition 8.9.1 is convergent) Let φ1 and φ2 denote adelic cusp forms such that the central character of φ2 is the inverse of that of φ1 as in Definition 8.9.1. Then the integral  φ1 (g) φ2 (g) d × g, Z (AQ )·G L(2,Q)\G L(2,AQ )

which appears in Definition 8.9.1 is convergent. Proof We define the integral over Z (AQ ) · G L(2, Q)\G L(2, AQ ) as in Lemma 8.8.6 using the fundamental domain D∞ from Theorem 4.4.4. Then after plugging in the invariant measures from Definition 6.9.5 and Proposition 8.6.1, we find that all variables are integrated over compact sets except for the coordinate y in G L(2, R) (cf. 8.6.1). Absolute convergence of the integral in y is assured by Proposition 8.9.2.  Remark For convergence it is enough to assume that one of φ1 , φ2 is cuspidal. Proposition 8.9.5 (The bilinear form above is invariant) The bilinear form introduced in Definition 8.9.1 is an invariant pairing satisfying (i), (ii), (iii) of Proposition 8.7.4. Proof A simple change of variables shows that the pairing defined in Definition 8.9.1 satisfies conditions (ii), (iii) of Proposition 8.7.4. It remains to show (i), which is equivalent to     d  φ2 (g) d × g φ1 g · exp(tα)  t=0 dt Z (AQ )·G L(2,Q)\G L(2,AQ )

=



 −

φ1 (g)

  d  d×g φ2 g · exp(tα)  t=0 dt

Z (AQ )·G L(2,Q)\G L(2,AQ )

This can be proved by the change of variables g → g · exp(−tα).  The explicit invariant bilinear pairing constructed in Definition 8.9.1, in conjunction with Proposition 8.7.4, allows us to concretely identify the contragredient representation of an irreducible admissible cuspidal automorphic representation. This is the thrust of the final proposition of this chapter. Proposition 8.9.6 (Contragredient of an irreducible admissible cuspidal automorphic representation) Let (π, V ) be an irreducible admissible cuspidal automorphic representation of G L(2, AQ ) as in Definition 5.1.14 with  π = (πg , π K∞ ), πfinite as in Definition 5.1.5. Assume that V is a space of cusp forms, as opposed to a more general subquotient. (We will prove in Proposition 9.5.8 that there is no loss of generality.)

316

The contragredient representation for GL(2)

V := {φ | φ ∈ V }, where φ denotes complex conjugation. Let Define  π = (π g , π K∞ ), π finite denote the triple of actions on V defined by π finite (g) . φ = πfinite (g) . φ,

(∀ g ∈ G L(2, Afinite )),

π g (D) . φ = πg (D) . φ,

(∀ D ∈ U (g)),

π K∞ (k) . φ = π K∞ (k) . φ,

(∀ k ∈ K ∞ ),

for all φ ∈ V. Then (π , V ) is isomorphic to the contragredient representation (( π, ( V ) as defined in Definition 8.7.2. Proof The restriction of the invariant bilinear pairing  ,  to V × V is nondegenerate, so this follows from Proposition 8.7.4. 

8.10 Growth of matrix coefficients In Proposition 8.4.9 it was shown that the matrix coefficients of supercuspidal representations of G L(2, Q p ) have compact support modulo the center. It will turn out that this property characterizes supercuspidal representations. The process of characterizing and classifying representations according to the growth properties of their matrix coefficients is very typical in representation theory. In this section, we shall study the growth properties of matrix coefficients in the remaining cases: principal series and special. It will be shown that all matrix coefficients of special representations are not compactly supported modulo the center (as in Definition 8.4.8). In the case of special representations with unitary central character, it will be shown that the matrix coefficients are square integrable modulo the center. This relies on the fact that if the central character is unitary, then the norm-square of a matrix coefficient is well defined as a function, G L(2, Q p )/Q×p → C, and means simply that this function is integrable. (See Definition 8.10.18 for the precise definition.) Finally, it will be shown that matrix coefficients of principal series representations are not square integrable modulo the center (and hence not compactly supported). These results will be proved twice, by two different methods. The first method is a simple and elegant argument which was shown to us by Herv´e Jacquet. We thank him for allowing us to include it here. The second method is based on [Casselman]. It relies on the careful study of Jacquet modules, defined as in Definition 6.11.7, and will be presented in Section 12. Before we begin, we present a proposition which explains the uniformity of behavior among matrix coefficients of a fixed irreducible representation. Corollary 8.10.2 demonstrates that all the different matrix coefficients of a fixed irreducible representation have the same coarse growth properties.

8.10 Growth of matrix coefficients

317

Proposition 8.10.1 (Every irreducible representation of G L(2, Q p ) may be realized as right translation on its own space of matrix coefficients) Fix a prime p. Let (π, V ) be an irreducible admissible representation of G L(2, Q p ), with contragredient (( π, ( V ). Let  ,  : V × ( V → C be the canonical invariant bilinear form given in Definition 8.1.12. For v ∈ V , ( v ∈ ( V , and g ∈ G L(2, Q p ), define the matrix coefficient v. βv,˜v (g) := π (g) . v, ( Let ρ denote the action of G L(2, Q p ) on the space of all functions f : G L(2, Q p ) → C given by right translation, i.e., ρ(g) . f (h) = f (hg), v∈( V , the set of functions (∀g, h ∈ G L(2, Q p )). Then, for any non-zero ( $v˜ := {βv,˜v | v ∈ V } is a vector space which is closed under the action of ρ, and the representation (ρ, $v˜ ) is isomorphic to (π, V ). Proof Fix non-zero ( v∈( V and define $v˜ := {βv,˜v | v ∈ V }. Clearly, the map v → βv,˜v is a non-trivial linear map V → $v˜ . Further, for any v ∈ V, and any g, h ∈ G L(2, Q p ), we have   βπ(g) . v,˜v (h) = π (h) . π (g) . v,( v = βv,˜v (hg) = ρ(g) . βv,˜v (h). It follows that v → βv,˜v is a non-trivial intertwining map. Since (π, V ) is irreducible, it follows at once that (π, V ) is isomorphic to its image (ρ, $v˜ ).  Corollary 8.10.2 (Uniformity of behavior of matrix coefficients) Fix a prime p. Let (π, V ) be an irreducible admissible representation of G L(2, Q p ), v0 ∈ ( V such that the matrix with contragredient (( π, ( V ). If there exist v0 ∈ V, ( coefficient βv0 ,˜v0 is compactly supported modulo the center, then βv,˜v is compactly supported modulo the center for every v ∈ V, ( v∈( V . Similarly, if there v0 ∈ ( V such that the matrix coefficient βv0 ,˜v0 is square integrable exist v0 ∈ V, ( modulo the center, then βv,˜v is square integrable modulo the center for every v ∈ V, ( v∈( V. Proof The vector space V of all locally constant functions G L(2, Q p ) → C which are compactly supported modulo the center is clearly closed under right translation. It follows that for each ( v∈( V , the space $v˜ considered in Proposition 8.10.1 is either contained in or disjoint from V. In particular, if βv0 ,˜v0 ∈ V, V is irreducible. Hence, for then $v˜ 0 is contained in V. By Proposition 8.1.8, ( any ( v∈( V , we may write ( v=

N i=1

ci · π (gi ) .( v0

318

The contragredient representation for GL(2)

for some N ∈ N, and some c1 , . . . , c N ∈ C, g1 , . . . , g N ∈ G L(2, Q p ). Then, for each v ∈ V, βv,˜v (g) =

N

ci · π (g) . v, ( π (gi ) .( v0  =

i=1

N

ci · βv,v0 (gi−1 g),

i=1

which is again compactly supported modulo the center. If we replace “compactly supported” by “square integrable” throughout, the proof remains valid.  In order to study the properties of matrix coefficients of principal series and special representations it will be convenient to have a different expression for the bilinear form on B p (χ1 , χ2 ) × B p (χ1−1 , χ2−1 ) which was constructed in Proposition 8.2.3. The construction of  , ,  which was given in Proposition 8.2.3 is based on an expression for the Haar measure on G L(2, Q p ) which is related to the p-adic Iwasawa decomposition given in Proposition 4.2.1. A second expression for the Haar measure is given in Proposition 8.10.3. A third expression for the Haar measure, based on another decomposition, called the Bruhat decomposition, will be obtained in Proposition 8.10.7. Proposition 8.10.3 (Haar measure in matrix coordinates for G L(2, Q p )) Fix a prime p. Consider a locally constant function f : G L(2, Q p ) → C such  that the integral G L(2,Q p ) f (g) d × g, as defined in Definition 6.9.5, converges absolutely. Then   α β     f  γ δ p f (g) d × g = 2 dα dβ dγ dδ, ( p − 1)( p − 1) |αδ − βγ |2p G L(2,Q p )

Qp Qp Qp Qp

(8.10.4) where the integrals on the right hand side are all defined as in Definition 1.5.3, using the Haar measure μHaar given in Example 1.5.4. Proof Fix α0 , β0 , γ0 , δ0 ∈ G L(2, Q p ). It is clear that the integral on the right in (8.10.4) corresponds to integration against a measure on G L(2, Q p ) which assigns the set    α β  α ∈ α0 + p k Z p , β ∈ β0 + p  Z p , γ ∈ γ0 + p m Z p , γ δ  5 (8.10.5) δ ∈ δ0 + p n Z p a volume of         μHaar p k Z p · μHaar p  Z p · μHaar p m Z p · μHaar p n Z p p −k−−m−n = , 2 |α0 δ0 − β0 γ0 | p |α0 δ0 − β0 γ0 |2p

8.10 Growth of matrix coefficients

319

whenever k, , m, n are chosen sufficiently small (relative to α0 , β0 , γ0 , δ0 ) to ensure that the function |αδ − βγ | p is constant on the set (8.10.5). Invariance of this measure is easy to check. It follows that the two measures p , are equal up to a constant. To check that the constant is equal to ( p2 −1)( p−1) we note that the normalized Haar measure assigns the group K 1 defined as in (6.9.1) a volume equal to 1 over its index in G L(2, Z p ). This is equal to the cardinality of G L(2, Z/ pZ), which is ( p 2 − 1)( p 2 − p). On the other hand, the measure used on the right hand side of (8.10.4) assigns K 1 a measure of p −2 . This completes the proof.  Proposition 8.10.6 (The Bruhat decomposition for G L(2, Q p )) Fix a prime p. The group G L(2, Q p ) is the disjoint union of double cosets     0 1 B(Q p ) ∪ B(Q p ) · · N (Q p ) , 1 0 where B(Q p ) is the group of all invertible upper triangular matrices with entries in Q p , and N (Q p ) is the group of all upper triangular matrices with entries in Q p and 1’s on the diagonal. The two double cosets appearing in this decomposition are called “Bruhat cells.” More specifically, the coset   double 01 B(Q p ) is called the “little cell,” and the double coset B(Q p ) · 1 0 · N (Q p ) is   called the “big cell.” The little cell consists of all matrices ac db ∈ G L(2, Q p ) such that c = 0, and the big cell consists of all elements such that c =/ 0. Proof See Exercise 8.14.



Remark Since N (Q p ) ⊂ B(Q p ), the group B(Q p ) is indeed a double coset B(Q p ) · I2 · N (Q p ) where I2 denotes the 2 × 2 identity matrix. Proposition 8.10.7 (Expression for Haar measure for G L(2, Q p ) based on the Bruhat decomposition) Fix a prime p. Let d × g be the normalized Haar measure defined in Definition 6.9.5, and let f : G L(2, Q p ) → C be a locally constant compactly supported function. Then  G L(2,Q p )

p · f (g) d g = ( p − 1)( p 2 − 1) ×

 ·

f

Q×p Q p Q p Q×p

0 1 1 0



   



1 0

x2 1



1 0

x1 1



t1 0

0 t2



dt1 d x1 d x2 dt2 . |t1 |2 (8.10.8)

Proof Let Mat(2, Z p ) denote the set of all two by two matrices with coefficients in Z p . Fix g0 ∈ G L(2, Q p ) and n ∈ Z sufficiently large to ensure

320

The contragredient representation for GL(2)

that the function g → | det g| p is constant on the set g0 + p n · Mat(2, Z p ). It suffices to prove (8.10.8) when f is equal to the characteristic function of g0 + p n · Mat(2, Z p ) for all such g0 , n. It follows from Proposition 8.10.3 that the integral of this characteristic function with respect to the normalized Haar p1−4n . measure is ( p2 −1)( p−1)| det g0 |2 In order to evaluate the integral on the right hand side of (8.10.8) when f is the characteristic function of g0 + p n · Mat(2, Z p ), it is necessary to analyze the set of quadruples (x1 , x2 , t1 , t2 ) ∈ Q p × Q p × Q×p × Q×p such that 

1 0

x1 1



t1 0

0 t2



0 1

1 0



1 0

x2 1



 =

x 1 t2 t2

t1 + x 1 x 2 t2 x 2 t2



lies in the set g0 + p n· Mat(2, Z p ). a b Write g0 = c00 d00 . Suppose that 

x 1 t2 t2

t1 + x 1 x 2 t2 x 2 t2

 ∈ g0 + p n · Mat(2, Z p ).

/ pn · Z p . Then clearly, t2 ∈ c0 + p n · Z p . First assume that c0 ∈ −k Let k be the integer such that |c0 | p = p . In other words, c0 is equal to p k times some element of Z×p . Then clearly |t2 | p is also equal to p −k , so that t2 = p k τ2 with τ2 ∈ Z×p . From this expression for t2 , it may be deduced that x1 t2 − a0 ∈ p n Z p ⇐⇒ x1 − a0 t2−1 ∈ p n−k Z p , and that, x2 t2 − d0 ∈ p n Z p ⇐⇒ x2 − d0 t2−1 ∈ p n−k Z p . Finally, t1 + x1 x2 t2 − b0 ∈ p n Z p ⇐⇒ t1 ∈ b0 − x1 x2 t2 + p n Z p . It follows that      x 1 t2 1g0 + pn Mat(2,Z p ) t2 Q×p Q p Q p Q×p





t1 + x 1 x 2 t2 x 2 t2 

 ·

1 dt1 d x1 d x2 dt2 |t1 |2



= c0 + pn ·Z p a0 t −1 + pn−k ·Z p d0 t −1 + pn−k ·Z p 2 2

b0 −x1 x2 t2 + pn ·Z p

1 dt1 d x1 d x2 dt2 . |t1 |2 (8.10.9)

Next, observe that |t1 | = |a0 d0 − b0 c0 | p /|t2 | p for all t1 , t2 , x1 , x2 in the domain of the integral on the right. To see this, it is convenient to write x ≡ y (mod p n ) for x − y ∈ p n Z p . With this notation for t1 , t2 , x1 , x2 in the domain of integration, one has: t2 ≡ c0 (mod p n ), x1 t2 ≡ a0 (mod p n ), x2 t2 ≡ d0 (mod p n ), and finally t1 ≡ b0 − x1 x2 t2 (mod p n ). Combining these yields t1 t2 ≡ (b0 c0 − a0 d0 ) (mod p n ). It follows that (8.10.9) is equal to

1 |a0 d0 − b0 c0 |2p

8.10 Growth of matrix coefficients    

321 |t2 |2 dt1 d x1 d x2 dt2

c0 + pn ·Z p a0 t −1 + pn−k ·Z p d0 t −1 + pn−k ·Z p b0 −x1 x2 t2 + pn ·Z p 2 2

1 = |a0 d0 − b0 c0 |2p =





|t2 | c0 + pn ·Z p

p −n d x1 d x2 dt2

a0 t2−1 + pn−k ·Z p d0 t2−1 + pn−k ·Z p

1 |a0 d0 − b0 c0 |2p =



2





p −2n+k d x2 dt2

|t2 |2 c0 + pn ·Z p

a0 t2−1 + pn−k ·Z p

1 |a0 d0 − b0 c0 |2p



|t2 |2 p −3n+2k dt2 .

c0 + pn ·Z p

(8.10.10)

But as noted above the condition on t2 implies that |t2 | = p −k , so that (8.10.10) is equal to  p −3n p −4n = dt = 2 |a0 d0 − b0 c0 |2p |a0 d0 − b0 c0 |2p c0 + pn ·Z p

This completes the proof in the case when c0 ∈ / p n · Z p . When c0 ∈ p n · Z p the situation is a bit more complicated. Thereason for this is that the image of the  x1 t2 t1 +x1 x2 t2 is the big cell (as defined in Definifunction (x1 , x2 , t1 , t2 ) → t2 x2 t2 tion 8.10.6), and if c0 ∈ p n · Z p then the set g0 + p n · Mat(2, Z p ) intersects the little cell. To prove that the identity (8.10.8) holds when f is the characteristic function of a set that intersects the little cell, one must, in effect argue that the intersection with the little cell is of measure zero. To make this precise it is convenient to introduce the notation μ for the measure on the set of compact subsets of G L(2, Q p ) such that the integral of f with respect to μ is equal to the right hand side of (8.10.8). It has been shown that   p −4n p · μ g0 + p n · Mat(2, Z p ) = 2 ( p − 1)( p − 1) | det g0 | for all g0 ∈ G L(2, Q p ) and n ∈ Z such that | det g| is constant on g0 + p n · / p n · Z p . Now suppose that c0 ∈ p n · Z p . Partition g0 + p n · Mat(2, Z p ) and c0 ∈ Mat(2, Z p ) into p 4 additive cosets h i + p n+1 · Mat(2, Z p ), 1 ≤ i ≤ 4n. Let ci denote the lower left entry of h i . Then the number of values of i such that ci ∈ p n+1 ·Z p is p 3 . For each such index we partition h i + p n+1 ·Mat(2, Z p ) into p 4 additive cosets of p n+2 ·Mat(2, Z p ) and so on. In short, we obtain a partition     0 1 g0 + p n · Mat(2, Z p ) ∩ B(Q p ) · · N (Q p ) 1 0 ( p−1) ∞ p   3i

=

i=1

j=1

h i, j + p n+i · Mat(2, Z p ).

322

The contragredient representation for GL(2)

It follows that      x 1 t2 1g0 + pn ·Mat(2,Z p ) t2

Q×p Q p Q p Q×p

( p−1) ∞ p 3i

=

i=1

j=1

t1 + x 1 x 2 t2 x 2 t2





( p−1) ∞ p 3i

dμ =

i=1

h i, j + pn+i ·Mat(2,Z p )

=

j=1

·

1 dt1 d x1 d x2 dt2 |t1 |2

p p −4(n+i) · 2 ( p − 1)( p − 1) | det g0 |

∞ p p −4n ( p − 1) p −i . · · ( p − 1)( p 2 − 1) | det g0 | i=1

The final sum telescopes, yielding the result.  Corollary 8.10.11 (The Bruhat pairing on principal series of G L(2, Q p )) Fix a prime p. Let χ1 , χ2 be characters of Q×p and let (π, B p (χ1 , χ2 )) denote the principal series representation given in Definition 8.2.2 with invariant bilinear form  ,  introduced in Proposition 8.2.3. Then the integral        0 1 1 x 0 1 1 x f2 dx f1 1 0 0 1 1 0 0 1 Qp   is absolutely convergent for all f 1 ∈ B p (χ1 , χ2 ) and f 2 ∈ B p χ1−1 , χ2−1 and  −1 −1  defines an invariant bilinear form  , Bruhat : B p (χ1 , χ2 )×B p χ1 , χ2 → C, which satisfies  f 1 , f 2 Bruhat =

( p − 1)( p 2 − 1)  f 1 , f 2 , p    ∀ f 1 ∈ B p (χ1 , χ2 ), f 2 ∈ B p χ1−1 , χ2−1 .

Remarks The pairing  , Bruhat is based on the Bruhat decomposition given in Proposition 8.10.6. It is clear that  , Bruhat may be used in place of  ,  to identify B p (χ1−1 , χ2−1 ) with the contragredient of B p (χ1 , χ2 ) and compute matrix coefficients. The special representations occur as subspaces when χ1 (a) = χ2 (a)|a| p , for all a ∈ Q×p , or as quotients when χ1 (a) = χ2 (a)|a|−1 p . Proof As in the proof of Proposition 8.2.3, we write F(g) = f 1 (g) · f 2 (g), and note that F is locally constant and satisfies (6.9.9). We first address the issue of convergence of the integral        0 1 1 x 0 1 1 x f2 d x. (8.10.12) f1 1 0 0 1 1 0 0 1 Qp Since locally constant, there is an integer m such that the function  F is 0 −1 is constant on {x ∈ Q p | |x| p > p m }. This is important because F 1 x −1

 F

0 1

8.10 Growth of matrix coefficients      −1 1 0 −1 1 1 x x =F 0 x 1 x −1 0 0 1   0 −1 = |x|−2 · F . p 1 x −1

323

Consequently, Q p may be partitioned into the sets {x ∈ Q p | |x| p > p m } and {x ∈ Q p | |x| p ≤ p m }. The latter set is compact, so it suffices to address convergence of the integral (8.10.12) over the first set. But this amounts to con ∞ pi · (1 − p −1 ) p −2i , which is clear. This completes vergence of the sum i=m+1 the proof of absolute convergence. Next, it follows easily from Lemma 6.9.7 that        d x1 dt1 dt2 t1 0 1 x1 g ϕ , F(g) = 0 1 0 t2 |t1 |2p Q×p Q×p Q p

for some locally constant compactly supported function ϕ : G L(2, Q p ) → C. If we plug this into (8.10.12), then the definition of  f 1 , f 2 Bruhat matches exactly the right hand side of (8.10.8), except for the normalization factor, ( p−1)(pp2 −1) . In other words, it is the integral of ϕ with respect to an invariant measure which has not been normalized. This ensures invariance. That  , Bruhat is a scalar multiple of  ,  can be deduced from the uniqueness of the invariant bilinear form or of the invariant measure. Either way the exact value of the scalar follows from (8.10.8).  Theorem 8.10.13 (Asymptotics of matrix coefficients for G L(2, Q p ) via the Bruhat pairing) Fix a prime p. Let χ1 , χ2 be two characters of Q×p and let B p (χ1 , χ2 ) denote the principal series representation as defined in Definition 8.2.2. Let  , Bruhat be the Bruhat pairing defined in Definition 8.10.11. f ∈ B p (χ1−1 , χ2−1 ). Then there exist constants Take f ∈ B p (χ1 , χ2 ), and ( c1 , c2 ∈ C and sufficiently small ε > 0 such that F  a π 0

0 d



G . f, ( f Bruhat

a 1 a 1  2  2 = c1 χ1 (a)χ2 (d)   + c2 χ2 (a)χ1 (d)   d p d p (8.10.14)

for all a, d ∈ Q p such that |a/d| p < ε. Furthermore, if |χ1 ( p)/χ2 ( p)|C > 1, then c1 and c2 are given by the convergent integrals        −1 0 0 1 0 1 ( c1 = f f −( f χ1−1 (x)χ2 (x)|x|−1 p d x, 0 1 1 x 1 0 Qp

    0 1 0 ( · f c2 = f 1 0 1 Qp

1 x

 d x.

324

The contragredient representation for GL(2)

In the case when B p (χ1 , χ2 ) contains a special representation as a subrepresentation, the function f is in this subrepresentation if and only if c2 = 0 for all ( f. Proof It is not difficult to show that the integral formula for c1 is absolutely convergent if and only if |χ1 ( p)/χ2 ( p)|C > 1. In fact, the proof is essentially identical to the proof of absolute convergence of the integral defining Jacquet’s invariant Whittaker function in Proposition 6.4.3. Assume that |χ1 ( p)/χ2 ( p)|C > 1. By definition, F   G a 0 ( π . f, f 0 d Bruhat       0 1 a 0 0 1 = f ·( f dx 1 x 0 d 1 x Qp       0 1 0 1 d 0 ( ·f dx = f 1 da x 1 x 0 a Qp      a − 1  0 1 0 1   2 ·( f d x. = χ2 (a)χ1 (d)   · f d 1 ax 1 x d p Qp (8.10.15) Now, because f and ( f are locally constant, there exist constants ε , M > 0 such that     0 1 0 1 ( f =( f , (∀x ∈ Q p , such that |x| p < ε ), 1 x 1 0     −1 0 −1 0 = f , (∀y ∈ Q p , such that |y| p > M). f 0 1 y −1 1 This second condition implies that for y ∈ Q p with |y| p > M,    −1    0 1 y 1 −1 0 f = f · 1 y 0 y y −1 1  −1 −1 −1 = χ1 (y )χ2 (y)|y| p · f 0

0 1

 .

Now, we simply choose ε sufficiently small to ensure that   a  d    |x| p > ε , and   < ε =⇒  x  > M, d p a p     0 1 01 ( which implies that for all x ∈ Q p , either ( − f = 0 or f 10 1 x  f

0 1

1 d x a



 = f

−1 0 0 1



χ1−1



    −1 d  d d x χ2 x  x  . a a a p (8.10.16)

8.10 Growth of matrix coefficients

325

Now, the integral which appears in the third line of (8.10.15) is equal to      0 1 0 1 ( ·f dx (8.10.17) f 1 da x 1 0 Qp         0 1 0 1 0 1 ( ( + · f f − f d x. 1 da x 1 x 1 0 Qp In the first integral above we make the change of variables x → da x.   0 1 The second integral in (8.10.17) may be taken over the support of ( f − 1 x   01 ( f , which is contained in the region |x| p ≥ ε , where (8.10.16) holds. 10 It then follows from (8.10.16) that (8.10.17) is equal to     a   0 1 0 1   f ·( f dx   1 x 1 0 d p Qp   a  −1 0   + χ1 (a)χ2−1 (a)χ1−1 (d)χ2 (d)   f 0 1 d p       0 1 0 1 ( · f −( f χ1−1 (x)χ2 (x)|x|−1 p d x. 1 x 1 0 Qp  − 1 Incorporating the factor of χ2 (a)χ1 (d)  da  p 2 from (8.10.15) completes the proof of (8.10.14). If |χ1 ( p)/χ2 ( p)|C ≤ 1, then the integral formulae for c1 and c2 are no longer absolutely convergent. However, if s1 and s2 are complex numbers, and ω1 and ω2 are normalized unitary characters such that χi (a) = ωi (a)|a|spi for i = 1, 2 and a ∈ Q×p , then each integral converges to a rational function in p −s1 and p −s2 whenever (s1 − s2 ) > 0. This gives meromorphic continuation to all values of s. Now assume that χ1 (t) = χ2 (t)|t| p for all t ∈ Q p . This is the case when B p (χ1 , χ2 ) contains a special representation (π, V ) as a subspace. Proposition 8.3.1 gave an explicit description for the subspace V , as the orthogonal complement in B p (χ1 , χ2 ) of a certain element of B p (χ1−1 , χ2−1 ) (see (8.3.2)). 1

−1 2 In the present notation this function is given   by δ(g) = χ2 (det g)| det g| p . 01 · 10 x1 | x ∈ Q p . Thus f lies in Clearly, this function is constant on 10 the special representation if and only if    0 1 f d x = 0. 1 x Qp

This completes the proof.



Theorem 8.10.13 will imply an alternate description of each of the three main types of infinite dimensional irreducible admissible representations of

326

The contragredient representation for GL(2)

G L(2, Q p ): principal series, special, and supercuspidal, in terms of growth properties of their matrix coefficients. We have already defined the notion of “compactly supported modulo the center,” in Definition 8.4.8. We now give a precise definition of “square integrable modulo the center.” Definition 8.10.18 (Square integrable modulo the center) Fix a prime p. A locally constant function f : G L(2, Q p ) → C is said to be square integrable modulo the center if the function g → | f (g)|2C is invariant by the center, i.e.,    2   r 0 · g  = | f (g)|2C , for all g ∈ G L(2, Q p ), r ∈ Q×p , and the integral f 0r C



D

 is convergent, where D =

 k1 ·

| f (g)|2C d × g pn 0

0 1



 5   · k2  k1 , k2 ∈ G L(2, Z p ), n ≥ 0 .

Remarks (1) If | f |2C is not invariant by the center, it does not make sense to talk about whether f is square integrable modulo the center. (2) It follows easily from the Cartan decomposition given in Proposition 4.2.3 that every element of G L(2, Q p ) is equal to an element of D times an element of the center. The set D is not a fundamental )\G L(2, to Definition 1.4.1, because domain for Z (Q p   Q p ) according  r 0 | r ∈ Z×p is non-trivial. The advantage of D ∩ Z (Q p ) = 0r D over a fundamental domain is that any fundamental domain for Z (Q p )\G L(2, Q p ) would be of Haar measure zero. The set D may be viewed as a fundamental

 domain for Z (Q p )\G L(2, Q p ) which has

been “fattened up” by

r 0 0r

| r ∈ Z×p to have positive measure.

Theorem 8.10.19 (Characterization of representations of G L(2, Q p ) via the growth of matrix coefficients) Fix a prime p and let (π, V ) denote an infinite dimensional irreducible admissible representation of G L(2, Q p ) whose contragredient is (( π, ( V ). Assume that the central character of (π, V ) is unitary. Let  ,  : V × ( V → C be the invariant bilinear form given in Definition 8.1.12. Fix v ∈ V, ( v∈( V . Then the matrix coefficient 8 9 (8.10.20a) g → π (g) . v, ( v , (g ∈ G L(2, Q p )), defined in Definition 8.1.14, is • compactly supported modulo the center, if (π, V ) is supercuspidal, • square integrable modulo the center, but not compactly supported modulo the center, if (π, V ) is special, • not square integrable modulo the center, if (π, V ) is principal series.

8.10 Growth of matrix coefficients

327

Proof It was shown in Proposition 8.4.9 that every matrix coefficient of a supercuspidal is compactly supported modulo the center. f ∈ Let χ1 , χ2 be characters of Q×p , and choose f ∈ B p (χ1 , χ2 ) and ( f Bruhat , as in Corollary B p (χ1−1 , χ2−1 ). For g ∈ G L(2, Q p ), let β(g) = π (g) . f, ( 8.10.11. Next, consider  D

|β(g)|2C d × g,

(8.10.20b)

with D as in Definition 8.10.18. Each g in D may be expressed as k1 · with k1 , k2 ∈ G L(2, Z p ) and n ≥ 0. Then F  n p β(g) = π 0

0 1



G   . π (k2 ) . f, ( f π k1−1 . (



pn 0 0 1



·k2

. Bruhat

Since each of the functions f, ( f is fixed by a subgroup of G L(2, Z p ) which is open, and hence of finite index, it follows  as k1 and k2 range over  −1that f each range over only a G L(2, Z p ), the functions π (k2 ) . f and π k1 . ( finite set of functions. First assume that χ1 (t) = χ2 (t)|t| p for all t ∈ Q×p . This is the case when B p (χ1 , χ2 ) contains a special representation (π, V ) as a subspace. Assume that f lies in the subspace V. Then π (k2 ) . f also lies in V for all k2 . Using Theorem 8.10.13, we may, therefore, choose constants c and ε such that      n     β k1 · p 0 · k2  ≤ c · χ1 ( p n ) p − n2 , (8.10.21)   0 1 for all k1 , k2 ∈ G L(2, Z p ) and all n such that p −n < ε. Now, the central character of B p (χ1 , χ2 ), and of its subrepresentation (π, V ), is   r 0 = χ1 (r )χ2 (r ), (∀r ∈ Q×p ). ωπ 0 r In the present case, this is equal to |r | p · χ22 (r ), which is unitary if and only −1/2 and χ1 (r ) = if there is a unitary character χ0 such that χ2 (r ) = χ0 (r )|r | p 1/2 χ0 (r )|r | p . The bound (8.10.21) then becomes      n   β k1 · p 0 · k2  ≤ c · p −n .   0 1 To prove the absolute convergence of (8.10.20a), then, it suffices to prove that of    n  ∞ p 0 −2n · G L(2, Z p ) . p μHaar G L(2, Z p ) · (8.10.22) 0 1 n=1

328

The contragredient representation for GL(2)

 n  To do this, we need to find the measure of the set G L(2, Z p ) · p0 10 · G L(2, Z p ), which amounts to the same thing as decomposingit into  dispn 0 tinct left G L(2, Z p )-cosets and counting the number. Clearly, k1 · 0 1 · k2 =    n   n  p 0 −1 pn 0 k . Elements k3 · p0 01 · k4 if and only if k1−1 k3 = k 2 4 0 1 0 1

k2 and k4 may holds if and only if k1−1 k3 lies in   nbechosen so that this −n G L(2, Z p ) ∩ p0 10 · G L(2, Z p ) · p0 10 . This intersection is precisely the Iwahori subgroup  5  a b  n × a, d ∈ Z K0( p ) p = . , b, c ∈ Z p p pn c d   n  Thus the number of distinct left G L(2, Z p )-cosets in G L(2, Z p ) · p0 10 · G L(2, Z p ) is equal to the index [G L(2, Z p ) : K 0 ( p n ) p ] = [S L(2, Z) : 0 ( p n )] = p n + p n−1 . Convergence of (8.10.22) is now clear. It is possible to show that every special representation which appears as a quotient as in Definition 6.10.3 is isomorphic to a special representation which appears as a subrepresentation as in Definition 6.10.9. Thus, the preceding paragraphs actually prove the square integrability modulo the center of matrix coefficients of all the special representations with unitary central character. However, we have not yet shown that every special representation which appears as a quotient is isomorphic to a special representation which appears as a subrepresentation, and it is possible to prove that the matrix coefficients of those which appear as quotients are also square integrable modulo the center in the following manner. It has been shown in Proposition 8.3.1 that a special representation which appears as a quotient is isomorphic to the contragredient of the special representation which appears as a subrepresentation. Furthermore, if (π, V ) is any admissible representation of G L(2, Q p ) with contragredient (( π, ( V ), then a choice of v ∈ V and ( v∈( V determines a matrix coefficient βv,˜v of V and a matrix coefficient βv˜ ,v of ( V which are related by the equation βv,˜v (g) = π (g) . v,( v = v, ( π (g −1 ) .( v = βv˜ ,v (g −1 ). It is easily verified that for any compact set D,   μHaar (D) = μHaar d −1 | d ∈ D . It follows that



 f (x) d x =

G L(2,Q p )

f (x −1 ) d x

G L(2,Q p )|

whenever either side is defined. Thus, the square integrability of βv,˜v follows directly from that of β v˜ ,v .

8.10 Growth of matrix coefficients

329

Now consider a principal series representation B p (χ1 , χ2 ). Assume as before that the central character χ1 χ2 is unitary. We want to show that the matrix coefficient β f, f˜ is never square integrable for any f ∈ B p (χ1 , χ2 ) and ( f ∈ B p (χ1−1 , χ2−1 ). By Corollary 8.10.2, it suffices to exhibit particuf ∈ B p (χ1−1 , χ2−1 ) such that β f, f˜ is not lar choices of f ∈ B p (χ1 , χ2 ) and ( square integrable. Exploiting the relation between the matrix coefficients of a representation and that of its contragredient, it is permissible to assume that |χ1 (a)| ≥ 1 for all a ∈ Z p . f ∈ B p (χ1−1 , χ2−1 ) so that c1 =/ 0 and c2 = 0, Choose f ∈ B p (χ1 , χ2 ) and ( where c1 and c2 are the two constants appearing in Theorem 8.10.13. Let m be an integer such that f and ( f are both fixed by the subgroup K m , defined as in (8.1.6). Then  



|β(g)|2C d × g ≥



n G L(2,Z p )· p 0 ·G L(2,Z p ) 0 1



|β(g)|2C d × g



n K m · p 0 ·K m 0 1

2 −n

= |c1 | |χ1 ( p )| p 2

n

· μHaar



 Km ·

pn 0

0 1





· Km .

Furthermore, by the same reasoning above  μHaar

 Km ·

pn 0

0 1



 · Km 

Now, it is easily verified that

= [K m : K m ∩ K 0 ( p n ) p ] · μHaar (K m ) .

a pm b pm c d

  a, , pm c, −1



pm b, ,∈ d,

−1

K m lie in the same

left K m ∩ K 0 ( p ) p -coset if and only if d c − (d ) c ∈ p n−m Z p . It follows that the index [K m: K m ∩ K 0 ( p n ) p ] = p n−m grows like p n as n → ∞. Divergence of D |β(g)|2 d × g follows easily. This completes the proof of Theorem 8.10.19.  n

Theorem 8.10.19 motivates the following definition. Definition 8.10.23 (Square integrable representation of G L(2, Q p )) Fix a prime p. An admissible representation (π, V ) of G L(2, Q p ) is said to be square integrable if its matrix coefficients are square integrable modulo the center. Remarks Irreducible square integrable representations are also called discrete series representations. If the central character of (π, V ) is not unitary, then the norm-squares of its matrix coefficients are not invariant by the center, and hence the representation cannot possibly be square integrable. It is desirable to have a generalization of the concept of a square integrable representation which holds for representations with central characters which are not unitary. This can be accomplished through the concept of twisting.

330

The contragredient representation for GL(2)

Definition 8.10.24 (Twist of a G L(2, Q p ) representation by a character) Fix a prime p. Let (π, V ) be an admissible representation of G L(2, Q p ), and let χ be a character of Q×p . We define the twist π ⊗ χ of V by χ to be the representation of G L(2, Q p ) on the same space V, with a new action defined by π ⊗ χ (g) . v = χ (det g) · π (g) . v,

(∀g ∈ G L(2, Q p ), v ∈ V ).

Lemma 8.10.25 (Every representation is a twist of a representation with unitary central character) Fix a prime p. Let (π, V ) be an irreducible admissible representation of G L(2, Q p ), with central character ωπ . Then there is a character χ and there is an irreducible admissible representation (π , V ) of G L(2, Q p ), with unitary central character ωπ , such that π = π ⊗ χ.   Proof The function r0 r0 → |ωπ (r )|C is a continuous homomorphism

from Q p into the positive reals. The image of Z×p must be both compact and a subgroup, and the group of positive reals contains no compact subgroups other than {1}. It follows that every continuous homomorphism from Q p into the positive reals is trivial on Z×p , and hence factors through | | p . It follows that there is a real number σ such that |ωπ (r )|C = |r |σp for all r ∈ Q×p . It is easy to see that the central character of π ⊗ χ is ωπ · χ 2 . Take σ/2 χ (r ) = |r | p . Then the twist π = π ⊗ χ −1 has unitary central character, and

π = π ⊗ χ.  Definition 8.10.26 (Essentially square integrable representation) Fix a prime p. An irreducible admissible representation (π, V ) of G L(2, Q p ) is said to be essentially square integrable if it is a twist of a square integrable representation. Clearly, square integrable representations are essentially square integrable. Also supercuspidal representations are essentially square integrable (and square integrable if their central characters are unitary). It follows easily from Theorem 8.10.19 that an infinite dimensional irreducible admissible representation of G L(2, Q p ) which is not supercuspidal is essentially square integrable if and only if it is a special representation.

8.11 Asymptotics of matrix coefficients of (g, K ∞ )-modules In this section we shall prove the analogues of Theorems 8.10.13 and 8.10.19 in the case of a (g, K ∞ )-module. First, it is convenient to introduce the action of G L(2, R) on a larger space of functions.

8.11 Asymptotics of matrix coefficients of (g, K∞ )-modules

331

smooth (χ1 , χ2 )) Let χ1 , χ2 be characters of Definition 8.11.1 (The space B∞ smooth (χ1 , χ2 ) to be the space of all smooth functions: R× . We define B∞ G L(2, R) → C satisfying    a 1 a b  2 f g =   χ1 (a)χ2 (d) f (g) 0 d d p

for all a, d ∈ R× , b ∈ R, g ∈ G L(2, R). smooth (χ1 , χ2 ) is the set of Recall from Definition 8.6.5 that B∞ (χ1 , χ2 ) ⊂ B∞ smooth smooth (χ1 , χ2 ) that are K ∞ -finite. Clearly B∞ (χ1 , χ2 ) confunctions in B∞ tains the space B∞ (χ1 , χ2 ) as a proper subspace, and is closed under the action of G L(2, R) by right translation, unlike B∞ (χ1 , χ2 ).

Definition 8.11.2 (Smooth principal series representation for G L(2, R)) Let representation of χ1 , χ2 be characters of R× . The principal series   G L(2, R), smooth (χ1 , χ2 ) where π is associated to (χ1 , χ2 ), is the representation π, B∞ smooth (χ1 , χ2 ) is the vector space defined the action by right translation, and B∞ in Definition 8.11.1. Thus   π (h) . f (g) = f (gh) smooth (χ1 , χ2 ). for all g, h ∈ G L(2, R), and f ∈ B∞

Finally, there is the matter of the invariant bilinear form  , . Proposition 8.11.3 (Invariant bilinear form for the action of G L(2, R)) Let smooth , χ 2 ) be the vector space defined in χ1 , χ2 be characters of R× . Let B∞  −1 (χ1−1 smooth χ1 , χ2 be defined in the same manner. Definition 8.11.1, and let B∞ Then the formula     2π  cos θ sin θ cos θ sin θ · f2 dθ,  f 1 , f 2  := f1 − sin θ cos θ − sin θ cos θ 0 (8.11.4)  −1 −1  smooth smooth for f 1 ∈ B∞ χ1 , χ2 , gives a bilinear form, (χ1 , χ2 ), f 2 ∈ B∞ 8 9  −1 −1  smooth smooth , : B∞ χ 1 , χ2 −→ C, (χ1 , χ2 ) × B∞ which is invariant with respect to the action of G L(2, R) given in Definition 8.11.2, i.e., which satisfies π (g) . f 1 , π (g) . f 2  =  f 1 , f 2  for all  smooth smooth χ1−1 , χ2−1 . (χ1 , χ2 ), f 2 ∈ B∞ g ∈ G L(2, R) and for all f 1 ∈ B∞ Furthermore, this bilinear form has an alternate expression as   f1 , f2  = 2 ·



∞ −∞

f1

0 1

1 x



 f2

0 1

1 x

 d x.

(8.11.5)

332

The contragredient representation for GL(2)

Remark By Theorem 8.6.10, the contragredient of B∞ (χ1 , χ2 ) is isomorphic to B∞ χ1−1 , χ2−1 . The proof of this result rests on Proposition 8.6.8, which constructs an invariant bilinear form on the two representations. An alternate expression for this form was given in Proposition 8.6.16. Proposmooth (χ1 , χ2 ) and sition 8.11.3 extends these results to the larger spaces B∞   −1 −1 smooth B∞ χ 1 , χ2 . Proof Bilinearity is obvious. As was previously remarked prior to Proposition 8.6.20, K ∞ -finiteness is not used at all in the proof of Proposition 8.6.8 or Proposition 8.6.16, so that both remain if f 1 and f 2 are taken in the  valid smooth smooth χ1−1 , χ2−1 , respectively. Invariance (χ1 , χ2 ) and B∞ larger spaces B∞ is given by equation (8.6.9). The validity of the second expression for  f 1 , f 2  given in (8.11.5) is an immediate consequence of Proposition 8.6.20.  The proofs in the remainder of this section were shown to us by Herv´e Jacquet. We thank him for allowing us to include them here. Proposition 8.11.6 Let χ1 and χ2 be characters of R× . Then the principal series representation B∞ (χ1 , χ2 ) has a subspace invariant under the actions of U (g) and K ∞ , if and only if there is an integer m such that χ1 χ2−1 (x) = |x|∞ · x m ,

(∀x ∈ R× ),

(8.11.7)

When this is the case, the invariant subspace of B∞ (χ1 , χ2 ) is unique. If (8.11.7) holds with m ≥ 0, then the space spanned by the functions   1 a b 2 ck d m−k , f = χ1−1 (ad − bc)|ad − bc|∞ c d     a b ∀ ∈ G L(2, R) , (8.11.8) c d smooth for 0 ≤ k ≤ m, is a G L(2, R)-invariant subspace of B∞ (χ1−1 , χ2−1 ). The orthogonal complement of this space, relative to the invariant bilinear form of smooth (χ1 , χ2 ) and Proposition 8.11.3, is a G L(2, R)-invariant subspace of B∞ smooth (χ1 , χ2 ) is the unique (g, K ∞ )its intersection with B∞ (χ1 , χ2 ) ⊂ B∞ smooth (χ1 , χ2 ) lies in this invariant subspace of B∞ (χ1 , χ2 ). An element f ∈ B∞ subspace if and only if   ∞  0 1 f · x j d x = 0, (0 ≤ j ≤ m). (8.11.9) 1 x 0

Remark Clearly, a space which is invariant by G L(2, R) is invariant by g and K ∞ , but a space which is invariant by g and K ∞ might not be G L(2, R)invariant. Proof The criterion for reducibility is a reformulation of Theorem 8.6.11(i). (See also Corollary 7.4.9.) It is easily verified that the space spanned by the

8.11 Asymptotics of matrix coefficients of (g, K∞ )-modules

333

functions (8.11.8) is invariant under the action of G L(2, R) by right translation. Being G L(2, R)-invariant, it is also invariant by g and K ∞ , so it is the invariant subspace described in Theorem 8.6.11, which can also be checked by direct computation. It follows immediately from the invariance of the bilinear form in Proposition 8.11.3 that the orthogonal complement of an invariant subspace is invariant. The characterization of the orthogonal complement given in (8.11.9) follows from combining (8.11.8) and (8.11.5).  Recall that when χ1 χ2−1 (x) = x m |x|, with m ≥ 0, then the invariant subspace of B∞ (χ1 , χ2 ) is called an “essentially discrete series representation” (see Definition 7.4.10 and the remarks following it). Abusing language, we smooth (χ1 , χ2 ) in the same way. shall refer to the invariant subspace of B∞ Theorem 8.11.10 (Asymptotics of matrix coefficients of a (g, K ∞ ) × smooth (χ1 , χ2 ) be module) Fix two characters χ1 , χ2 of R . Let π, B∞ the principal series representation as in Definition 8.11.2. Take f ∈  −1 −1  smooth smooth χ1 , χ2 . Then there exist smooth functions (χ1 , χ2 ) and ( f ∈ B∞ B∞ 2 → C such that F1 , F2 : R × K ∞ F   G   a 0 π k1−1 · f = (8.11.11) · k2 . f, ( 0 1 1 1 ⎧ 2 2 χ1 (a) |a|∞ F1 (a, k1 , k2 ) log |a|∞ + χ2 (a) |a|∞ F2 (a, k1 , k2 ) , ⎪ ⎪ ⎪ ⎪ −1 m × ⎪ ⎪ if χ1 χ2 (x) = x (∀x ∈ R ), some m ∈ Z, m ≥ 0, ⎪ ⎨ 1

1

2 2 χ1 (a) |a|∞ F1 (a, k1 , k2 ) + χ2 (a) |a|∞ F2 (a, k1 , k2 ) log |a|∞ , ⎪ ⎪ ⎪ −1 m ⎪ if χ1 χ2 (x) = x (∀x ∈ R× ), some m ∈ Z, m ≤ 0, ⎪ ⎪ ⎪ ⎩ 1 1 2 2 χ1 (a) |a|∞ F1 (a, k1 , k2 ) + χ2 (a) |a|∞ F2 (a, k1 , k2 ) , otherwise.

Further, the following formulae are valid, provided that the integrals in them are interpreted literally where convergent and by analytic continuation elsewhere:  ∞      −1 0 0 1 ( · k2 f · k1 χ1−1 χ2 (x) d x F1 (0, k1 , k2 ) = 2 f 0 1 1 x −∞

χ1 χ2−1 (x)

= x (∀x ∈ R× ), some m ∈ Z, m ≥ 0, unless  ∞      0 1 0 1 ( F2 (0, k1 , k2 ) = 2 f · k1 f · k2 χ1−1 χ2 (x) d x 1 0 1 x m

−∞

unless χ1 χ2−1 (x) = x m (∀x ∈ R× ), some m ∈ Z, m ≥ 0. (8.11.12)

334

The contragredient representation for GL(2)

Finally, in the case when χ1 χ2−1 (x) = x m |x|∞ (∀x ∈ R× ), and f is an element smooth (χ1 , χ2 ) of the essentially discrete series representation contained in B∞ (see Definition 7.4.10, and the remarks just above), the function F2 vanishes to order m at 0. Proof For k ∈ K ∞ and x =/ 0,     −1 0 1 x f ·k = f 1 x 0

  −1 0 ·k x −1 1    −1 0 −1 −1 ·k . = χ1 (x) χ2 (x)|x|∞ f x −1 1 1 x



(8.11.13)

Take θ : R → C, to be a smooth function of compact support such that θ (0) = 1, and define θ c : R → C by  1 − θ (y −1 ), y =/ 0 c θ (y) = 1, y = 0. Then θ c is also a smooth function of compact support. Further, if    0 1 φ1 (x, k) := f · k · θ (x), 1 x    −1 0 · k · θ c (y), (∀x ∈ R, k ∈ K ∞ ), φ2 (y, k) := f y 1 then φ1 and φ2 are smooth functions of compact support R × K ∞ → C, and    0 1 −1 f · k = φ1 (x, k) + χ1 (x)−1 χ2 (x)|x|−1 ∞ φ2 (x , k), 1 x (∀x ∈ R, k ∈ K ∞ ). In like fashion, define ( φ1 , ( φ2 : R × K ∞ → C such that    0 1 ( ( −1 f ·k =( φ1 (x, k) + χ1 (x)χ2 (x)−1 |x|−1 ∞ φ2 (x , k), 1 x (∀x ∈ R, k ∈ K ∞ ). Write F   a π k1−1 0  =2·

0 1



∞

f −∞



 k2

0 1

G F  a . f, ( f = π 0 1 x



a 0

0 1



0 1



  0 k2 ( f 1

 k2

1 x

G . f, ( π (k1 ) . ( f 

 k1

dx

8.11 Asymptotics of matrix coefficients of (g, K∞ )-modules

=

−1 2|a|∞2 χ2 (a)

 ·

f −∞



− 12



∞

0 1

1 x/a

x

∞



  0 k2 ( f 1

1 x



335

 k1

dx (8.11.14)



φ1 (x, k1 ) d x φ1 , k2 ( a  ∞   x −1 ( −1 , k2 χ1 (x)χ2 (x)−1 |x|−1 φ1 + 2|a|∞2 χ2 (a) · ∞ φ2 (x , k1 ) d x a −∞  ∞ a  1 2 , k2 ( φ1 (x, k1 ) d x χ1 (x)−1 χ2 (x)|x|−1 + 2|a|∞ χ1 (a) · ∞ φ2 x −∞  ∞   1 a 2 ( −1 , k2 |x|−2 χ1 (a) · φ2 + 2|a|∞ ∞ φ2 (x , k1 ) d x. x −∞

= 2|a|∞ χ2 (a) ·

−∞

Three of the four integrals in (8.11.14) are easily handled. Indeed, |a|−1 ∞



∞ −∞

φ1

x a

  , k2 ( φ1 (x, k1 ) d x =

∞ −∞

φ1 (x, k2 ) ( φ1 (ax, k1 ) d x

converges absolutely for all values of a to a smooth function H1 (a; k1 , k2 ). Next, 

∞ −∞

φ1

x a

 ( −1 , k2 χ1 (x)χ2 (x)−1 |x|−1 ∞ φ2 (x , k1 ) d x

(8.11.15)

vanishes for all a sufficiently small (i.e., small enough that x −1 lies outside the support of ( φ2 whenever ax lies in the support of φ1 ). Hence, we may set 2 (8.11.15) equal to |a|−1 ∞ · H2 (a; k1 , k2 ), where H2 : R × K ∞ → C is a smooth function, which vanishes in a neighborhood of 0. To evaluate 1

2 χ1 (a) · 2|a|∞



∞

−∞

φ2

a x

 ( −1 , k2 |x|−2 ∞ φ2 (x , k1 ) d x,

note that the multiplicative Haar measure d x/|x|∞ on R× is invariant under the change of variables x → x −1 . Hence, 

∞

φ2

−∞

a



 dx , k2 |x −1 |∞( φ2 (x −1 , k1 ) x x |x| ∞ −∞  ∞  ∞ dx φ2 (x, k1 ) = φ2 (ax, k2 ) |x|∞( = φ2 (ax, k2 ) ( φ2 (x, k1 ) d x, |x|∞ −∞ −∞ , k2



( −1 |x|−2 ∞ φ2 (x , k1 )

dx =

∞

φ2

a

which converges absolutely for all values of a to a smooth function H4 (a; k1 , k2 ).

336

The contragredient representation for GL(2)

This leaves the difficult term. It is clear that  ∞ a  , k2 ( φ1 (x, k1 ) d x χ1 (x)−1 χ2 (x)|x|−1 ∞ φ2 x −∞ converges absolutely for all values of a, k1 , and k2 , to a function which we 2 for some constant denote H3 (a; k1 , k2 ), which is supported on [−C, C] × K ∞ φ1 on [−b, b] × K ∞ , C. Further, if φ2 is supported on [−d, d] × K ∞ and ( then  b   χ1 (x)−1 χ2 (x) d x H3 (a; k1 , k2 ) ! C |x| ∞ |a|/d    −1 (8.11.16) ! max χ1 (a)χ2 (a)C , − log |a|∞ , as a → 0. (Here ! is read “is less than a constant times.”) In order to analyze H3 (a; k1 , k2 ), we make use of Fourier analysis on the group R× . This is essentially equivalent to the theory of the classical Mellin transform. Definition 8.11.17 (Mellin transform) Let  : [0, ∞) → C be a piecewise continuous function such that (y) ! |y|b∞ as y → ∞ and (y) ! |y|c∞ as y → 0 for real numbers b, c with c < b. The Mellin transform of  is defined for s ∈ C with c < −(s) < b by the convergent integral  dy . (y)y s M(s; ) := × y R It is very often possible to extend the definition of M(s; ) to a larger domain in C by analytic continuation. A key property of the Mellin transform is the Mellin inversion formula. Theorem 8.11.18 (Mellin inversion) Let  : [0, ∞) → C be a piecewise continuous function such that (y) ! |y|b∞ as y → ∞ and (y) ! |y|c∞ as y → 0 for real numbers b, c with c < b. Let M(s; ) denote the Mellin transform of  as in Definition 8.11.17. Then, for any fixed real number σ ∈ (c, b), we have σ+i∞ M(s; ) · y −s dt. (y) = σ −i∞

Definition 8.11.19 (Extended Mellin transform) Let  : R× → C be a piecewise continuous function such that (y) ! |y|b∞ as |y|∞ → ∞ and (y) ! |y|c∞ as y → 0 for real numbers b, c with c < b. Define the extended Mellin transform Mextended (s; ) to be the pair of complex-valued functions, defined for s ∈ C with c < −(s) < b by the convergent integrals

8.11 Asymptotics of matrix coefficients of (g, K∞ )-modules 337    dy dy , (y)|y|s∞ , (y)|y|s∞ sign(y) Mextended (s; ) := |y|∞ |y|∞ R× R× where sign(a) = a/|a|∞ is the sign character of R× . Remarks If  is obtained by restricting a Schwartz function: R → C to R× , then the integrals in Definition 8.11.19 are local zeta integrals for G L(1) as in Definition 2.3.1. Further, if we define the odd and even parts of a function  : R× → C as odd (x) =

1 ((x) − (−x)) , 2

even (x) =

1 ((x) + (−x)) , 2

then Mextended (s; ) =

  2 · M(s; even ), 2 · M(s; odd ) .

(8.11.20)

Theorem 8.11.21 (Injectivity of the extended Mellin transform) Let 1 and 2 be two functions as in Definition 8.11.19, with Mextended (s, 1 ) and Mextended (s, 2 ) defined in strips c1 < −(s) < b1 and c2 < −(s) < b2 which overlap non-trivially. Then 1 = 2 . Proof This follows from Theorem 8.11.18 and (8.11.20).  Theorem 8.11.22 (Image of Schwartz space under the extended Mellin transform) Suppose that  : R → C is Schwartz. Then (i) Mextended(s, ) is defined on (s) > 0, and has continuation to C as  a pair M1 (s, ), M2 (s, ) of meromorphic functions satisfying the growth estimate Mi (σ + it, ) ! |t|CN

as |t|C → ∞,

(8.11.23)

for all σ, N ∈ R, i = 1, 2. (ii) The function M1 (s, ) has at most simple poles at 0, −2, −4, . . . , while M2 (s, ) has at most simple poles at −1, −3, −5, . . . . (iii) If (m 1 (s), m 2 (s)) is any pair of meromorphic functions C → C satisfying the growth estimate (8.11.23), and such that m 1 (s) has at most simple poles at 0, −2, −4, . . . , while m 2 (s) has at most simple poles at −1, −3, −5, . . . , then there exists a unique Schwartz function  such that (m 1 (s), m 2 (s)) = Mextended (s; ). (2n) (iv) The residue of M1 (s, ) at s = −2n is 2 (2n)!(0) . The residue of (0) M2 (s, ) at s = −2n + 1 is 2 (2n+1)! . (2n+1)

Proof Convergence for (s) > 0 follows easily from the definition of a Schwartz function. Meromorphic continuation and the location of poles can

338

The contragredient representation for GL(2)

be deduced from results of Chapter 2, or from the arguments given below to prove (iv), and the growth estimate (8.11.23). Write (n) for the n th derivative of . It follows from the RiemannLebesgue lemma that Mi (s; (n) ) → 0 as t → ±∞, for (s) > 0, (i = 1, 2). This (together with continuity) implies that Mi (s; (n) ) is a bounded function of t ∈ R, (i = 1, 2). It follows from integration by parts that s · Mi (s, (n) ) = −M3−i (s + 1; (n+1) ),

((s) > 0, i = 1, 2), (8.11.24)

and this proves part (i). To prove part (iv), fix a positive integer N and consider the function  N (y) := (y) − 1[−1,1] (y)

N (n) (0)

n!

n=0

yN ,

(y ∈ R).

Because  is C ∞ , it follows readily that  N (y) ! y N +1 as y → 0, for each N . Consequently, Mextended (s;  N ) is holomorphic in the half plane (s) > −N −1. On the other hand, it follows from linearity of the integral that for (s) > 0, M1 (s, ) = M1 (s,  N ) +

 N (n) (0) n!

n=1

Further



1

y

N

−1

|y|s∞

dy = y



2 , s+N

0

1

−1

y N |y|s∞

dy . y

N even, N odd.

This proves the first half of (iv) and the second half is proved in the same way. Part (iii) follows from Mellin inversion, Theorem 8.11.18. Indeed, for fixed σ > 0 and j = 1, 2 define  j (y) =

1 2πi



σ +it σ −it

1 m j (s)y −s ds, 2

(y > 0).

The functions obtained are independent of the choice of σ, because m 1 (s) and m 2 (s) are holomorphic in (s) > 0 and satisfy (8.11.23), and one may deduce that  j , ( j = 1, 2) vanishes faster than any power at ∞. Further, one may differentiate under the integral sign, deducing that  j , ( j = 1, 2) is a smooth function: [0, ∞) → C. Then one may use (8.11.24) to show that every derivative of  j , ( j = 1, 2) also vanishes faster than any power at ∞. By shifting the contour of integration to the left, one obtains an expression for  j (y) in the form N an y n +  j,N (y), (8.11.25)  j (y) = n=0

8.11 Asymptotics of matrix coefficients of (g, K∞ )-modules

339

where a0 , . . . , an are constants and  j,N is a continuous function which satisfies lim y→0 y N  j,N (y) = 0. To be precise, the constant an is the residue of m j (s) at s = −n, and the function  j,N is the integral of m j on a vertical line which passes between −N and −N − 1. It follows that an is zero for all odd n when j = 1, and for all even n when j = 2. Now define  (x) =

1 (x) + 2 (x),

x ≥ 0,

1 (x) − 2 (x),

x < 0.

Then it follows from the expressions (8.11.25) that  is C ∞ at 0. Hence,  is Schwartz, and it follows from (8.11.20) that Mextended (s; ) = (m 1 (s), m 2 (s)).  Completion of proof of Theorem 8.11.10 In order to complete the proof of Theorem 8.11.10, we take the extended Mellin transform of H3 (a; k1 , k2 ). In the domain of convergence, we compute  da H3 (a; k1 , k2 ) · |a|s∞ |a|∞ R×   a  da dx , k2 ( φ1 (x, k1 ) = χ1 (x)−1 χ2 (x)φ2 · |a|s∞ x |x|∞ |a|∞ R× R×   da d x = χ1 (x)−1 χ2 (x)φ2 (a, k2 ) ( φ1 (x, k1 ) · |ax|s∞ |a|∞ |x|∞ R× R×   da dx φ1 (x, k1 ) = χ1 (x)−1 χ2 (x) · |x|s∞( φ2 (a, k2 ) · |a|s∞ |x|∞ |a|∞ R× R×     φ1 (∗, k1 ) , χ1−1 χ2 Z ∞ s, φ2 (∗, k2 ) , 1 . = Z ∞ s, ( (8.11.26)   (The 1 in Z ∞ s, φ2 (∗, k2 ) , 1 indicates the trivial character of R× , i.e., the constant function with value 1.) Similarly,  da H3 (a; k1 , k2 ) · |a|s∞ · sign(a) |a|∞ R×     φ1 (∗, k1 ) , χ1−1 χ2 · sign Z ∞ s, φ2 (∗, k2 ) , sign . (8.11.27) = Z ∞ s, ( To complete the proof of Theorem 8.11.10, we must find a function in the form (8.11.11) which has the same extended Mellin transform as H2 (a; k1 , k2 ). It will then immediately follow from Theorem 8.11.21 that the two functions are the same. To do this, we shall use Theorem 8.11.22, together with the next lemma.

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The contragredient representation for GL(2)

Lemma 8.11.28 (Derivatives of a Schwartz function at zero can be 2 → C chosen arbitrarily) Fix r > 0. For n = 0, 1, 2, . . . , let an : K ∞ be a smooth function. Then there is a smooth compactly supported function 2 → C such that (n) (0; k1 , k2 ) = an (k1 , k2 ) for all k ∈ K ∞ and  : R × K∞ n = 0, 1, 2, . . . . Proof See [H¨ormander, 1983], Theorem 1.2.6.



Write χi (x) = |x|ν∞i (sign(x))δi (i = 1, 2), where νi ∈ C and δi ∈ {0, 1}. Then     φ1 (∗, k1 ) , χ1−1 χ2 = Z ∞ s + ν2 − ν1 , ( φ1 (∗, k1 ) , signδ1 +δ2 Z ∞ s, (     Z ∞ s, ( φ1 (∗, k1 ) , χ1−1 χ2 · sign = Z ∞ s + ν2 − ν1 , ( φ1 (∗, k1 ) , signδ1 +δ2 +1 (8.11.29)   It is now easily verified M s; M3 (∗; k1 , k2 ) has double poles if and only if there is an integer m such that χ1 χ2−1 (x) = x m for all x ∈ R. Suppose first that this is not the case. By Theorem 8.11.22 (ii), (iv) and Lemma 8.11.28, there  exists 1 : R × 2 K ∞ → C, smooth and of compact support, so that M s; H3 (∗; k1 , k2 ) −   M s; 1 (∗; k1 , k2 ) is analytic at integer values. (See Exercise 8.17 for more detail.) For the remainder of this proof, we shall refer to a function  : R × K ∞ × K ∞ → C as “Schwartz” if it is smooth, and (∗; k1 , k2 ) : R → C is Schwartz for each fixed k1 , k2 . Let us now suppose that δ1 + δ2 is even. Then, it follows from (8.11.26), (8.11.27), (8.11.29), and Theorem 8.11.22 (iii) that there is a Schwartz function 2 (a; k1 , k2 ) such that       M s; H3 (∗; k1 , k2 ) − M s; 1 (∗; k1 , k2 ) = M s + ν2 − ν1 ; 2 (∗; k1 , k2 ) . (See Exercise 8.18 for more detail.) But then, by Theorem 8.11.21, H3 (a; k1 , k2 ) = 1 (a; k1 , k2 ) + |a|ν2 −ν1 2 (a; k1 , k2 ) = 1 (a; k1 , k2 ) + χ1−1 χ2 (a) · 2 (a; k1 , k2 ). If δ1 +δ2 = 1, then (8.11.26), (8.11.27), (8.11.29), and Theorem 8.11.22 (iii) imply that there is a Schwartz function 2 (a; k1 , k2 ) such that     M s; H3 (∗; k1 , k2 ) − M s; 1 (∗; k1 , k2 )      = M2 s + ν2 − ν1 ; 2 (∗; k1 , k2 ) , M1 s + ν2 − ν1 ; 2 (∗; k1 , k2 )

8.11 Asymptotics of matrix coefficients of (g, K∞ )-modules

341

(the extended Mellin transform of 2 with the order of the two functions reversed). Then, by Theorem 8.11.21 H3 (a; k1 , k2 ) = 1 (a; k1 , k2 ) + |a|ν2 −ν1 · sign(a) · 2 (a; k1 , k2 ) = 1 (a; k1 , k2 ) + χ1−1 χ2 (a) · 2 (a; k1 , k2 ). Now suppose that χ1 χ2−1 (x) = x m for all x ∈ R. In this case we have double poles to contend with. To treat these, we require the following lemma. Lemma 8.11.30 Let 0 : R → C be a Schwartz function, and define (x) = log |x|∞ · 0 (x) for x ∈ R× . Then the extended Mellin transform Mextended (s, ) = M1 (s, ), M2 (s, ) converges for (s) > 0, and has meromorphic continuation to C with only the following possible poles: • for each non-negative integer n the function M1 (s, ) has a pole at −2n if and only if (2n) / 0, in which case, it is a double pole with Laurent 0 (0) = expansion −2 · (2n) 0 0 (0)/(2n)! + + ··· 2 (s + 2n) (s + 2n) • for each non-negative integer n the function M2 (s, ) has a pole at −2n −1 (0) =/ 0, in which case, it is a double pole with Laurent if and only if (2n+1) 0 expansion (0)/(2n + 1)! −2 · (2n+1) 0 0 + ··· + (s + 2n + 1)2 (s + 2n + 1) Proof The proof is essentially the same as that of Theorem 8.11.22 (iv). One will need to use the identity 

1

−1

a n log |a|∞ · |a|s∞ · sign(a)δ

da 1 + (−1)n+δ =− .  |a|∞ (s + n)2

With Lemma 8.11.30 in hand we can handle the remaining case. Suppose that χ1 χ2−1 (a) = a m (∀a ∈ R× ), where m ∈ Z. Then (8.11.26) and (8.11.27) imply that   Mextended s; H (∗; k1 , k2 )  = Z ∞ (s − m, ( φ1 (∗, k1 ), signm )Z ∞ (s, φ2 (∗, k2 ), 1),

 Z ∞ (s − m, ( φ1 (∗, k1 ), signm+1 )Z ∞ (s, φ2 (∗, k2 ), sign) .

342

The contragredient representation for GL(2)

Hence, the possible poles start at s = max(0, m), while the possible double poles start at s = min(0, m). Using Lemmas8.11.30 and 8.11.28, one may choose a Schwartz function 1 so that Mextended s; H (∗; k1 , k2 )−1 (∗; k1 , k2 )·  log | |∞ has only simple poles. Here 1 (∗; k1 , k2 ) · log | |∞ indicates the function which maps a ∈ R× to 1 (a; k1 , k2 ) · log |a|∞ . If m ≤ 0, then 1 will vanish to order −m − 1 at 0, so that one may write 1 (a; k1 , k2 ) = a −m 3 (a;  k1 , k2 ), say. Further, in this case it follows  from Theorem 8.11.22 that Mextended s; H (∗; k1 , k2 ) − 1 (∗; k1 , k2 ) · log | |∞ is the extended Mellin transform of a Schwartz function 2 (a; k1 , k2 ). It follows that H3 (a; k1 , k2 ) = 2 (a; k1 , k2 ) + a −m log |a|∞ 3 (a; k1 , k2 ) = 2 (a; k1 , k2 ) + χ1−1 χ2 (a) log |a|∞ 3 (a; k1 , k2 ). Plugging this in to (8.11.14) and combining, we obtain the stated result in this case.   If m ≥ 0, then Mextended s; H (∗; k1 , k2 ) − 1 (∗; k1 , k2 ) · log | |∞ is the extended Mellin transform of a function of the form a → a −m 2 (a; k1 , k2 ), with 2 Schwartz. (This is deduced using Theorem 8.11.22 after shifting by m.) Hence H3 (a; k1 , k2 ) = a −m 2 (a; k1 , k2 ) + log |a|∞ 1 (a; k1 , k2 ) = χ1−1 χ2 (a)2 (a; k1 , k2 ) + log |a|∞ 1 (a; k1 , k2 ). Finally, we come to the case when χ1 χ2−1 (a) = a m |a|∞ (∀a ∈ R× ), and f is taken from the essentially discrete series representation which is embedsmooth (χ1 , χ2 ) as a subrepresentation. We remind the reader that this ded into B∞ means that   ∞  0 1 f · x j d x = 0, (0 ≤ j ≤ m). (8.11.31) 1 x −∞ Reviewing the steps of the proof, we find that the smooth function F2 appearing in (8.11.11) may be expressed as F2 (a; k1 , k2 ) = H1 (a; k1 , k2 ) + H2 (a; k1 , k2 ) + 2 (a; k1 , k2 ), where H2 vanishes identically in a neighborhood of zero, and  ∞ φ1 (ax, k1 ) d x. H1 (a; k1 , k2 ) = φ1 (x, k2 ) ( −∞

For n = 0, 1, 2, . . . , the value of (n) (0) is given by Theorem 8.11.22 (iv):   (n) (0) = ( φ1(n) (0, k1 ) · Z ∞ m + 1 − n, φ2 (∗, k2 ), signn+m .

8.12 Matrix coefficients of GL(2, Qp ) via the Jacquet module

343

For n ≤ m, this can be expressed as  dx (n) (0) = ( φ1(n) (0, k1 ) · φ2 (x, k2 )|x|m+1−n signm−n (x) ∞ |x|∞ (8.11.32) R×  dx =( φ1(n) (0, k1 ) · φ2 (x −1 , k2 )|x|−m−1+n signm−n (x) ∞ × |x| ∞ R   n (n) −1 −m −2 φ2 (x , k2 )x |x|∞ · x d x. =( φ1 (0, k1 ) · R×

A simple differentiation under the integral sign proves that  ∞ (n) H1 (0; k1 , k2 ) = φ1 (x, k2 ) ( φ1(n) (0, k1 ) · x n d x.

(8.11.33)

−∞

Further, −1 −1 −1 φ1 (x, k2 ) + φ2 (x −1 , k2 )x −m |x|−1 ∞ = φ1 (x, k2 ) + φ2 (x , k2 )χ1 (x)χ2 (x)|x|∞    0 1 (8.11.34) = f · k2 . 1 x

Combining (8.11.34) with (8.11.32) and (8.11.33) yields    ∞  0 1 F2(n) (0; k1 , k2 ) = ( φ1(n) (0, k1 ) · f · k1 · x n d x, 1 x −∞ which, by (8.11.31), is zero. This completes the proof of Theorem 8.11.10, except for the proof of (8.11.12) which is left to the reader. 

8.12 Matrix coefficients of G L(2, Q p ) via the Jacquet module In this section we shall given a second proof of Theorem 8.10.13 using Jacquet modules which were introduced in Definition 6.11.7. Fix a prime p. The groups    (8.12.1) K n = k ∈ G L(2, Z p )  k − I2 ∈ p n · Mat(2, Z p ) will play an important role. Lemma 8.12.2 (Iwahori factorization of K n ) Fix a prime p and n ≥ 1. Let K n be the subgroup given by (8.12.1). Define subgroups: 5   1 b  n b ∈ p Zp , Un = 0 1  5   a 0  n a, d ∈ 1 + p Z p , Dn = 0 d  5   1 0  − n c ∈ p Zp . Un = c 1 

n ∈ Z,

(8.12.3) (8.12.4) (8.12.5)

344

The contragredient representation for GL(2)

Then every element of K n is uniquely expressible as u · d · u − with u ∈ Un , d ∈ Dn , and u − in Un− . Furthermore, if f : K n → C is any locally constant function then  f (k) d × k Kn         Vol(K n ) 1 β α 0 1 0 f dα dβ dγ dδ. = 0 1 0 δ γ 1 p −4n 1+ pn Z p pn Z p pn Z p 1+ pn Z p

Proof Take

 k=

Clearly,    1 β α · 0 1 0

0 δ



1 γ

α γ

0 1

β δ





 =

∈ Kn .

α + βγ δ γδ

βδ δ



 =

α γ

β δ



has the unique solution δ = δ ,

γ = γ /δ ,

β = β /δ ,

α = α − βγ δ = α − β γ /δ .

The factorization for the Haar measure follows from the expression for the Haar measure as a scalar multiple of dα dβ dγ dδ |α δ − β γ | p and a series of changes of variable, together with the fact that for matrices n) ensures the in K n we have |δ | p = |α δ − β γ | p = 1. The constant V ol(K p−4n normalization of the measure.  Definition 8.12.6 (K n -symmetrization of a vector) Fix a prime p. Let (π, V ) be an admissible representation of G L(2, Q p ), and let v be an element of V. Let Vol(K n ) denote the total measure of K n with respect to the normalized Haar measure on G L(2, Q p ). Define the K n -symmetrization of v to be  1 π (k) . v d × k. Projn (v) := Vol(K n ) K n Recall that the admissibility of (π, V ) ensures that v is fixed by K m for some m ≥ n. The integral in Definition 8.12.6 is thus really a finite sum over the finite set K n /K m . It is clear that the K n -symmetrization is an element of the subspace V K n ⊂ V of K n -fixed vectors, and that the map Projn is a projection onto this subspace. The projection operator Projn was considered previously in Section 8.1.

8.12 Matrix coefficients of GL(2, Qp ) via the Jacquet module

345

Lemma 8.12.7 Fix a prime p. Let (π, V ) be an admissible representation of G L(2, Q p ), and let v be an element of v. Suppose that v is fixed by the action of Dn and Un− . Then    1 β n Projn (v) = p · π . v dβ. 0 1 pn Z p Proof Because v is fixed by Dn Un− we have      α 0 1 v = p 3n π 0 δ γ

0 1

 . v dβ dα dδ.

1+ pn Z p 1+ pn Z p pn Z p

Plugging this in, the result follows from Lemma 8.12.2.



We are now ready to begin our study of Jacquet modules in earnest and will take up again some notation from Section 6.11 where these modules were initially introduced. In particular, N will denote the subgroup of G L(2, Q p ) consisting of upper triangular matrices with 1’s on the diagonal, and T will denote the group of diagonal matrices in G L(2, Q p ). Lemma 8.12.8 Fix a prime p. Let (π, V ) be an admissible representation of G L(2, Q p ). Fix n ∈ Z with n ≥ 1. There exists m ∈ Z such that    1 β π . v dβ = 0 (8.12.9) 0 1 pm Z p for all v ∈ V K n ∩ V (N ). Here V (N ), as defined in Definition 6.11.2, is the subspace such that the Jacquet module is V /V (N ). Proof A vector v is in V (N ) if and only if (8.12.9) holds for some m, in which case it also holds for all m < m. By admissibility, the space V K n is finite dimensional. Consequently, we may choose m uniformly for all v ∈ V K n .  Proposition 8.12.10 Fix a prime p. Let (π, V ) be an admissible representation of G L(2, Q p ). Fix n ∈ Z with n ≥ 1, and take m to be determined by n and (π, V ) as in Lemma 8.12.8. Take v, v ∈ V K n such that the images of v and v in the Jacquet module of V are the same. Then for all a, d ∈ Q p such that |a/d| p ≤ p n−m , we have         a 0 a 0

. v = Projn π .v . Projn π 0 d 0 d Proof v, v  are fixed   by K n and |a/d| ≤ 1 it follows that  Because a 0 a 0 . v and π . v are fixed by Dn and Un− . Thus we may π 0 d 0 d

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The contragredient representation for GL(2)

replace the integral over K n in the definition of Projn with an integral over Un as in Lemma 8.12.7. Now, 

1 0

b 1



a 0

0 d



 =

a 0

0 d





bda −1 1

1 0

,

and as b ranges over p n Z p , b := bda −1 ranges over da −1 · p n · Z p . If |a/d| p ≤ p n−m , then da −1 · p n · Z p ⊃ p m . It follows from the definition of m that 

 π

a 0

0 d



1 b 0 1



. (v − v ) db = 0,

da −1 · pn ·Z p

because v and v were chosen so that v − v is in V (N ).  Corollary 8.12.11 With notation as in Proposition 8.12.10, F  a π 0

0 d



G F  a . v, ( v = π 0

0 d



G .v , ( v ,

for all ( v∈( V K n , and all a, d ∈ Q×p satisfying |a/d| p < p n−m . Proof For any u ∈ V and ( V ∈( V Kn ,  1 u,( v = u, ( π (k −1 ) .( v d × k Vol(K n ) K n  8 9 1 = π (k) . u, ( v d × k = Projn (u),( v, Vol(K n ) K n so this follows immediately from Proposition 8.12.10.  Proposition 8.12.12 Fix a prime p. Let (π, V ) be an admissible representation of G L(2, Q p ). Let (π N , VN ) denote the Jacquet module. The space

  linear functionals L : V → C  L(v) = L(π (n) . v),   ∀n ∈ N (Q p ), v ∈ V

(8.12.13)

is canonically identified with the space of all linear functionals: VN → C. In particular if (8.12.13) is of finite dimension then VN is of the same dimension. Proof A linear functional L : V → C vanishes on the subspace V (N ) appearing in the definition of VN if and only if L(v) = L(π (n) . v) for all v ∈ V and n ∈ N (Q p ). Thus an element of the space (8.12.13) gives a well defined linear functional VN → C. On the other hand, given any linear functional

8.12 Matrix coefficients of GL(2, Qp ) via the Jacquet module

347

 : VN → C, we can define an element of the space (8.12.13) by the formula L(v) := (v + V (N )). In short, L ↔ ,



such that L(v) = (v + V (N )),

(∀v ∈ V )



is a one-to-one correspondence between (8.12.13) and the space VN∗ of all linear functionals VN∗ → C. In the case when VN∗ is finite dimensional, VN has the same dimension.  Proposition 8.12.14 Fix a prime p. Let χ1 and χ2 be characters of Q×p . Let (π, B p (χ1 , χ2 )) be the principal series representation. Let L : B p (χ1 , χ2 ) → C be a functional which satisfies L (π (n) . f ) = L( f ), for all n ∈ N (Q p ). Then there exist two constants c1 , c2 ∈ C such that 

 L( f ) = c1 · f (I2 ) + c2 ·

f Qp

0 1

1 x

 (∀ f ∈ B p (χ1 , χ2 )).

d x,

Proof For any locally constant function h : Q p → C of compact support define f h ∈ B p (χ1 , χ2 ) by the formula ⎧ ⎨ χ (a)χ (d)  a  12 · h(x), 1 2 d p f h (g) = ⎩ 0,

   1 0 1 · 0 1 0 0 d if g is in the little Bruhat cell. 

if g =

a

b

x 1

 ,

The mapping h → f h is an injective linear map from the space S(Q p ) of all locally constant compactly supported functions: Q p → C into B p (χ1 , χ2 ). Its image is the subspace of B p (χ1 , χ2 ) consisting of those elements which vanish at the identity. Let L be any N -invariant functional B p (χ1 , χ2 ). Then h → L( f h ) is a translation-invariant functional on S(Q p ). As such, it must be a scalar multiple of    0 1 f d x. 1 x Qp





0 1 1 x



d x = 0 for all f ∈   0 1  d x is a B p (χ1 , χ2 ) such that f (I2 ) = 0. It follows that L( f ) − f 1 x Thus there exists c2 such that L( f ) − c2

f

Qp

scalar multiple of the functional f → f (I2 ). 

Qp

Theorem 8.12.15 (Jacquet module of a principal series representation for G L(2, Q p ) is two dimensional) Fix a prime p. Then the Jacquet module (as defined in Definition 6.11.7) of the principal series representation

348

The contragredient representation for GL(2)

(π, B p (χ1 , χ2 )), is two-dimensional, and is isomorphic (as a T -module) to the direct sum of two one-dimensional representations (μ1 , V1 ), (μ2 , V2 ) of T, with actions given by   1 −1 a 0 . v1 = χ1 (a)|a| p2 · χ2 (d)|d| p 2 · v1 , (v1 ∈ V1 ), μ1 0 d   1 −1 a 0 . v2 = χ2 (a)|a| p2 · χ1 (d)|d| p 2 · v2 , (∀a, d ∈ Q×p , v2 ∈ V2 ). μ2 0 d (8.12.16) Proof For f ∈ B p (χ1 , χ2 ) let L 1 ( f ) = f (I2 ) and    0 1 f d x. L 2( f ) = 1 x Qp

By Proposition 8.12.14, {L 1 , L 2 } is a basis for  

 linear functionals L : V → C  L(v) = L(π (n) . v), (∀n ∈ N (Q p ), v ∈ V ) . By Proposition 8.12.12, this space is canonically identified with the dual space to the Jacquet module B p (χ1 , χ2 ) N . Let i denote the linear map VN → C corresponding to L i for i = 1, 2, and let {v1 , v2 } denote the basis of VN dual to {1 , 2 }, that is, defined by the condition i (v j ) = δi, j (Kronecker delta). Thus, for any f ∈ B p (χ1 , χ2 ), the image of f in B p (χ1 , χ2 ) N is equal to c1 ( f )v1 + c2 ( f )v2 for some constants c1 ( f ), c2 ( f ) ∈ C. Furthermore, c1 ( f ) = 1 (c1 ( f )v1 + c2 ( f )v2 ) = L 1 ( f ) = f (I2 ). Likewise    0 1 c2 ( f ) = f d x. 1 x Qp

It is clear that for any f ∈ B p (χ1 , χ2 ) and any a, d ∈ Q×p , the identity :   ; a  a 0   π . f (I2 ) = χ1 (a)χ2 (d)   · f (I2 ) 0 d d p   a 0 holds. It follows that π N . v1 = χ1 (a)χ2 (d) |a/d| p · v1 . Likewise, 0 d  ;        :  a 0 0 1 0 1 a 0 π .f dx = dx 0 d 1 x 1 x 0 d Qp

 

=

d 0

0 a



0 1

Qp

1 d x a



 a  1   0  2 = χ1 (d)χ2 (a)   1 d p It follows that π N



Qp

a 0 0 d



Qp

  1   d 2 0 d x = χ1 (d)χ2 (a)   1 a p 1 x

Qp

 d x.

. v2 = χ1 (d)χ2 (a) |a/d| p · v2 . 

1 d x a

 dx

8.12 Matrix coefficients of GL(2, Qp ) via the Jacquet module

349

Theorem 8.12.17 Fix a prime p and two characters χ1 , χ2 of Q×p . Let B p (χ1 , χ2 ) be the corresponding principal series representation, as in Definition 8.2.2, and let μ1 and μ2 denote the characters of the group T ⊂ G L(2, Q p ) of diagonal matrices defined by (8.12.16). Take f ∈ B p (χ1 , χ2 ) such that • f is fixed by K n , and • the image of f in the Jacquet module B p (χ1 , χ2 ) N lies in the onedimensional space on which T acts by μ1 . Let m ∈ Z be determined by n and (π, B p (χ1 , χ2 )) as in Lemma 8.12.8. Then F   G   8 9 a 0 a 0 ( π . f, f = μ1 · f, ( f , 0 d 0 d for all a, d ∈ Q×p such that |a/d| p ≤ min( p n−m , 1), and all ( f ∈ B p (χ1−1 , χ2−1 ) K n . The same is true if μ1 is replaced by μ2 . Finally, if the image of f in the Jacquet module is trivial, then F   G a 0 π . f, ( f = 0, 0 d for all a, d ∈ Q×p such that |a/d| p ( f ∈ B p (χ1−1 , χ2−1 ) K n .

≤

min( p n−m , 1), and all

Proof For f ∈ K n and a, d ∈ Q×p satisfying |a/d| p ≤ min( p n−m , 1), the function     a 0 a 0 · f π . f − μ1 0 d 0 d is fixed by Dn Un− , and its image in the Jacquet module is zero. It follows from Corollary 8.12.11 that this function is orthogonal to every element of ( f ∈ B p (χ1−1 , χ2−1 ) K n , which completes the proof of the first assertion. The same. If the image of f in the Jacquet proof when μ1 is replaced by μ2 is the a 0 module is zero, then the function π . f itself is orthogonal to 0 d every element of ( f ∈ B p (χ1−1 , χ2−1 ) K n , by the same reasoning.  The primary goal of this section is to produce a second proof of Theorem 8.10.13. This proof is not as simple in the case of G L(2), but has the advantage that it fits into a conceptual framework which allows for natural generalization to G L(n). The first step is to explain how an intertwining map of T -modules is induced by any given intertwining map of G-modules. Definition 8.12.18 (Induced map between Jacquet modules of G L(2, Q p )) Fix a prime p. Let (π, V ) and (π , V ) be smooth representations of G L(2, Q p ). Let (π N , VN ) and (π N , VN ) be the corresponding Jacquet modules,

350

The contragredient representation for GL(2)

defined as in Definition 6.11.6. Let L : V → V be an intertwining map. Then the induced map L N : VN → VN is defined by L N (v + V (N )) = L(v) + V (N ). Remarks It is clear from the definition of V (N ), V (N ) that L(u) ∈ V (N ) for any u ∈ V (N ). This assures that L N is well-defined. It is clear from the definition of π N and π N that L N is an intertwining map of T -modules. Next, we would like to define the important Jacquet functor. For the basics of categories and functors, we refer the reader to [Lang, 1984]. Definition 8.12.19 (Jacquet functor for G L(2, Q p )) Fix a prime p. Let SR(G L(2, Q p )) denote the category of smooth representations of G L(2, Q p ), with morphisms being intertwining maps. Recall that   5  t1 0  × t , t ∈ Q T := p . 0 t2  1 2 Let SR(T ) denote the category of smooth representations of T, with morphisms being T -intertwining maps. The Jacquet functor J : SR(G L(2, Q p )) → SR(T ) is defined as follows. For any smooth representation (π, V ) of G L(2, Q p ), we define J((π, V )) to be the Jacquet module (π N , VN ). For any intertwining map L : V → V , with (π, V ) and (π , V ) smooth representations of G L(2, Q p ), we define J(L) to be the induced map L N as defined in Definition 8.12.18. Proposition 8.12.20 (Jacquet functor is exact) Take three smooth G-modules: (π, V ), (π , V ), (π

, V

), and let L : V → V and L

: V → V

be intertwining maps of G-modules. Assume that the image of L is equal to the kernel of L

. Then the image of the induced map L N : VN → VN is equal to the kernel of the induced map L

N : VN → VN

. Proof What has to be shown is that L

(v) ∈ V

(N ) ⇐⇒ ∃ v ∈ V such that v − L(v ) ∈ V (N ). By Proposition 6.11.3, L

(v) ∈ V (N ) implies that    1 u . L

(v) du = 0, π

0 1 p−n Z p for some n ∈ Z. It immediately follows (since L

is an intertwining map and the integral is really a finite sum) that    1 u π . v du (8.12.21) 0 1 p−n Z p

8.12 Matrix coefficients of GL(2, Qp ) via the Jacquet module

351

lies in the kernel of L

, which is the image of L . Thus (8.12.21) is equal to L (v ) for some v ∈ V . Then    1 u π . (v − L (v )) du 0 1 p−n Z p          1 u 1 u 1 u

= v du du π .v − π π 0 1 0 1 0 1 p−n Z p p−n Z p        1 u 1 u

v du du π .v − π = 0 1 0 1 p−n Z p p−n Z p       1 u 1 u v du π . v du − π = 0 1 0 1 −n −n p Zp p Zp = 0. It follows from Proposition 6.11.3 that v − L(v ) ∈ V (N ). This completes the proof.  In the next theorem, we shall write χ | |sp for the character of Q×p given in terms of a given character χ of Q×p by χ (t)|t|sp for all t ∈ Q×p . Recall that special , representations occur when χ2 = χ1 | |±1 p . When χ2 = χ1 | | p the representation B p (χ1 , χ2 ) contains a one-dimensional invariant subspace spanned by the function 1 (∀g ∈ G L(2, Q p )). δχ1 (g) = | det g| p2 χ1 (g), The quotient is a special representation. When χ2 = χ1 | |−1 p , the representation B p (χ1 , χ2 ) contains an infinite dimensional special representation as a subrepresentation. The quotient is one-dimensional. Theorem 8.12.22 (Jacquet modules of special representations for the group G L(2, Q p )) Fix a prime p. Let χ be a character of Q×p . Let (π,V ) denote the  special representation which is contained in  1

−1

, and let (π , V ) denote the special representation   −1 1  which is a quotient of π, B p χ | | p 2 , χ | | p2 . Then the Jacquet module   1 −1 VN ⊂ B p χ | | p2 , χ | | p 2 of (π, V ) is the one-dimensional subspace on   N a 0 acts by the scalar χ (ad)|a/d| p , while VN is the which π N 0 d    −1 1 a 0 2 2 one-dimensional subspace of B p χ | | p , χ | | p on which π N 0 d N acts by the scalar χ (ad)|a/d| p .  −1 1 Proof By Theorem 8.12.15, the Jacquet module of B p χ | | p 2 , χ | | p2 is two  a 0 acts by χ (ad) dimensional, and is spanned by one vector on which π N 0 d π, B p χ | | p2 , χ | | p 2

352

The contragredient representation for GL(2)   a a 0   . The represenand another one on which π N acts by χ (ad) d p 0 d  −1 1 tation B p χ | | p 2 , χ | | p2 contains an invariant one-dimensional subspace V0 spanned by the function χ ◦ det . V0 (N  ) is trivial It is clear that the group N acts trivially on V0 . It follows that a 0 acting and that the Jacquet module of V0 is one-dimensional with π N 0 d 1

by χ (ad)|ad| p2 . This corresponds to the subspace spanned by the vector v1 in Theorem 8.12.15. It follows immediately from Proposition 8.12.20, that the Jacquet module of the special representation (π , V )must be the span of v2 .  1

−1

Likewise the Jacquet module of B p χ | | p2 , χ | | p 2 is two-dimensional. The   1/2 −1/2 one-dimensional quotient B p χ | | p , χ | | p V is isomorphic to the con 1 −1 tragredient of the one-dimensional subrepresentation of B p χ −1 | | p 2 , χ −1 | | p2 spanned by χ −1 ◦ det . Clearly, G L(2, Q p ) acts on the space spanned by χ −1 ◦ det by the character χ −1 ◦ det itself. Hence, it acts on the contragredient by χ  ◦ det . Thus, the Jacquet module of B p (χ1 , χ2 )/V is one-dimensional

a 0 acting by χ (ad). This time, this corresponds to v2 in Theowith π N 0 d rem 8.12.15, so that the Jacquet module of the special representation is spanned by v1 . 

As a final application of the computation of Jacquet modules given in Theorem 8.12.15 and Theorem 8.12.22, it is possible to give a complete accounting of isomorphisms between principal series and special representations. Theorem 8.12.23 (Isomorphisms between principal series and special representations of G L(2, Q p )) Fix a prime p. Let χ1 , χ2 , χ3 , χ4 be characters of Q×p with (χ1 , χ2 ) =/ (χ3 , χ4 ). Then the principal series representations B p (χ1 , χ2 ) and B p (χ3 , χ4 ) are isomorphic if and only if χ3 = χ2 , χ4 = χ1 , and χ1 =/ χ2 | |±1 p . (This last condition is equivalent to the requirement that B p (χ1 , χ2 ) be irreducible.) Let (π, V ) and (π , V ) be two special representations which are not identical. Then (π, V ) and (π , V ) are isomorphic if and only if there is a character χ of Q×p such that (π, V ) is the subrepresentation of    −1 1 1 −1 B p χ | | p2 , χ | | p 2 and (π , V ) is the quotient of B p χ | | p 2 , χ | | p2 or vice versa. Proof By considering the special case of Proposition 8.12.20 when V

= {0}, we see that if an intertwining map L is surjective, then so is the induced map L N . In a similar manner, the special case when V is trivial implies that L N is injective whenever L is. It follows that isomorphic representations have isomorphic Jacquet modules. This immediately implies that the only case when two distinct special representations can be isomorphic is the one described.

Exercises for Chapter 8

353

Now assume that (π , V ) is indeed the irreducible quotient of  1 −1 B p χ | | p 2 , χ | | p2 . Extend the action of T on VN to an action of the group B of all upper triangular invertible p-adic matrices by letting the group N act trivially. Then VN is one-dimensional with an action of B by  μ2

a 0

b d

 . v2 = χ (ad)|a/d| p · v2 ,

  ∀a, d ∈ Q×p , v2 ∈ V2 .

The natural projection from V to VN is a non-zero B-intertwining map. It follows that Frobenius reciprocity (6.12.3) implies the existence of a non-zero G-intertwining map V → IndGB (μ2 ). This induced representation is exactly   1 −1 B p χ | | p2 , χ | | p 2 . The image must, therefore,   be an infinite dimensional irreducible repre−1

1

sentation of B p χ | | p2 , χ | | p 2 . It follows that it is the special representation   1 −1 (π, V ) which is a subrepresentation of B p χ | | p2 , χ | | p 2 , and that (π, V ) ∼ =

(π , V ). Similarly, Frobenius reciprocity and Theorem 8.12.15 imply that there exists a non-zero intertwining map B p (χ1 , χ2 ) → B p (χ2 , χ1 ). When these representations are irreducible, this implies that they are isomorphic. Theorem 8.12.15 also implies that if B p (χ3 , χ4 ) =/ B p (χ1 , χ2 ), B p (χ2 , χ1 ), then  = B p (χ1 , χ2 ), because the two representations do not have B p (χ3 , χ4 ) ∼ isomorphic Jacquetmodules.    1

−1

−1

1

The fact that B p χ | | p2 , χ | | p 2 and B p χ | | p 2 , χ | | p2 are not isomorphic can be deduced from the detailed analysis of these spaces done in Section 6.10. But it may also be deduced from Theorem 8.12.22. Indeed, it follows from Theorem 8.12.22 and Frobenius reciprocity that thespecial representa 1

−1

tion may be embedded as a subrepresentation only into B p χ | | p2 , χ | | p 2 and  −1 1 not into B p χ | | p 2 , χ | | p2 . It follows immediately that the two spaces are not isomorphic. 

Exercises for Chapter 8 8.1 Let (π, V ) be an admissible representation of G L(2, Q p ). As in the proof of Proposition 8.1.5, for each n ≥ 1 we can define a projection homomorphism Projn : V → V K n by Projn (v) =

1 π (κ).v, [K n : K m ] κ

354

The contragredient representation for GL(2)

where m is chosen sufficiently large so that v is fixed by K m , and κ varies over any set of coset representatives for K n /K m . Prove that Projn (v) is independent of the choice of integer m, and so is well-defined. 8.2 Prove Corollary 8.1.11 and Corollary 8.5.14. 8.3 Let  , ω be the bilinear form of Definition 8.4.1. Prove that if  f 1 , f 2 ω = 0 for all f 2 ∈ S(Q×p ), then f 1 = 0.        0 1 8.4 Prove that matrices of the form r0 r0 , a0 01 , 10 b1 , −1 with a, r ∈ Q×p 0 and b ∈ Q p generate G L(2, Q p ). 8.5 There are two places in the proof of Proposition 8.5.7 at which one interchanges an integral and a linear map. Justify these operations. 8.6 Fix a pair of complex numbers s = (s1 , s2 ) and a pair of normalized unitary characters ω = (ω1 , ω2 ). Suppose the principal series representation V p (s, ω) has a subrepresentation V such that the quotient V p (s, ω)/V is 1-dimensional. Prove that ω1 = ω2 and p s1 −s2 = p 2 . (Compare with Exercise 6.16.) 8.7 Let (π, V ) be an admissible representation of G L(2, Q p ). Prove that (π, V ) is unramified if and only if its contragredient (( π, ( V ) is unramified. Use this fact along with Exercise 6.28(a) to give another proof of Exercise 6.28(b). α β 8.8 Write g ∈ G L(2, R) as g = γ δ . Prove that the measure d×g =

dα dβ dγ dδ |αδ − βγ |2∞

is left and right translation invariant. That is, if f : G L(2, R) → C is any Schwartz function as in Definition 8.6.1, show that 



×

f (hg) d × g

f (gh) d g = G L(2,R)



G L(2,R)

f (g) d × g, (h ∈ G L(2, R)).

= G L(2,R)

8.9 Suppose U ⊂ G L(2, AQ ) is open. Show that Z (AQ ) · U contains infinitely many disjoint translates of a compact set with positive measure. Conclude that the characteristic function 1 Z (AQ )·U is not integrable on G L(2, AQ ). 8.10 In this exercise, we explore when an irreducible admissible representation of G L(2, Q p ) is isomorphic to its contragredient.

Exercises for Chapter 8

355

(a) Let (χ1 , χ2 ) be a pair of continuous characters of Q×p . Find necessary and sufficient conditions on these characters so that the principal series representation (π, B p (χ1 , χ2 )) is irreducible and so that Cp (χ1 , χ2 )). π, B (π, B p (χ1 , χ2 )) ∼ = (( (b) Suppose (π, V ) is a special representation. By Theorem 8.12.22, there exists a continuous character χ of Q×p so that (π, V ) is isomorphic to the special subrepresentation of B p (χ | | p , χ ). Determine necessary and sufficient conditions on χ so that (π, V ) ∼ π, ( V ). = (( (c) Finally, suppose (π, V ) is a supercuspidal representation that is isomorphic to its contragredient. Prove that the central character ωπ takes values in {±1}. 8.11 Repeat the previous exercise for principal series representations of G L(2, R). 8.12 In this problem we give a fundamental domain for Z (Q p )\G L(2, Q p ). Fix a finite prime p and recall the notation 

  E p = g ∈ G L(2, Q p )  | det(g)| p = 1 or p . (a) Show that any coset of Z (Q p )\G L(2, Q p ) is represented by some element of E p . Then show that two elements g, g ∈ E p represent the same coset if and only if g g −1 ∈ Z (Z p ), where Z (Z p ) = Z (Q p ) ∩ G L(2, Z p ). In particular, we have a canonical bijection of coset spaces Z (Z p )\E p ≈ Z (Q p )\G L(2, Q p ). (b) Let p > 2. Write (Z×p )2 for the subgroup of Z×p consisting of squares. Prove that Z×p /(Z×p )2 is a group with two elements. Fix a non-square  ∈ Z×p . Show that the set  

 D p = g ∈ G L(2, Q p )  det(g) = 1, , p −1 or p −1 is a fundamental domain for Z (Q p )\G L(2, Q p ). × 2 ∼ (c) Show that Z× = Z/2Z × Z/2Z. Let 1 , 2 , 3 , 4 ∈ Z×p 2 /(Z2 ) represent the four distinct classes modulo squares. Show that 

  D2 = g ∈ G L(2, Q2 ) det(g) = i or i /2 for some 1 ≤ i ≤ 4 is a fundamental domain for Z (Q2 )\G L(2, Q2 ).

356

The contragredient representation for GL(2)

8.13 Prove that the following set is a fundamental domain for the action of Z (R) on G L(2, R):   −1 0 S L(2, R). D∞ = S L(2, R) ∪ 0 1 8.14 Prove the existence of the Bruhat decomposition for G L(2, Q p ) as in   Proposition 8.10.6. More precisely, show that if g = ac db , then either c = 0, or else there exist y1 , y2 ∈ Q×p and x, z ∈ Q p such that 

a c

b d



 =

y1 0

x y2



0 1 1 0



1 z 0 1

 .

8.15 Fix a prime number p. Consider the following two sets:   n    D = k1 p0 10 k2  k1 , k2 ∈ G L(2, Z p ) and n ∈ Z ,    E p = g ∈ G L(2, Q p ) | det(g)| p = 1 or p . Let f : G L(2, Q p ) → C be any integrable function that is invariant by the center — i.e., f (zg) = f (g) for all z ∈ Z (Q p ) and g ∈ G L(2, Q p ). Prove that   f (g) d × g = f (g) d × g. D

Ep

8.16 In this exercise we construct intertwining maps between principal series representations using explicit integrals. We apply these integrals to study isomorphisms between principal series and special representations. Let χ1 , χ2 be characters of Q×p . Let B p (χ1 , χ2 ) denote the principal series representation defined in Definition 8.2.2. Let s = (s1 , s2 ) be a pair of complex numbers, and ω = (ω1 , ω2 ) a pair of normalized unitary charsi + 12

acters as in Definition 6.5.1, and assume that χi (t) = ωi (t)|t| p t ∈ Q×p , and i = 1, 2. Assume that (s1 − s2 ) > 0. (a) Prove that the integral 

 f Qp

0 1

1 0



1 u 0 1

for all

  g du

is absolutely convergent for all f ∈ B p (χ1 , χ2 ) and g ∈ G L(2, Q p ). (b) The function h f : G L(2, Q p ) → C defined by 

 h f (g) :=

f Qp

is an element of B p (χ2 , χ1 ).

0 1 1 0



1 0

u 1

  g du

Exercises for Chapter 8

357

(c) Show that the map M : B p (χ1 , χ2 ) → B p (χ2 , χ1 ) defined by M( f ) := h f is a non-zero intertwining map.   (d) Deduce that the principal series representations π, B p (χ1 , χ2 )   and π, B p (χ2 , χ1 ) are isomorphic, except when χ1 · χ2−1 = || p . (Compare with Theorem 8.12.22.) 8.17 In the proof of Theorem 8.11.10, the application rests,   of Lemma 8.11.28 implicitly, on the fact that the residues of M s; H3 (∗; k1 , k2 ) are smooth functions of k1 and k2 . Prove this. 8.18 At several points in the proof of Theorem 8.11.10, Theorem 8.11.22 (iii) is invoked to show the existence of a Schwartz function R× K ∞ × K ∞ → C. However, Theorem 8.11.22 (iii) as stated only guarantees the existence of a Schwartz function R → C for each fixed k1 , k2 ∈ K ∞ . Show that the functions R× K ∞ × K ∞ → C obtained by piecing together these Schwartz functions are indeed smooth. 8.19 Express the functions F1 and F2 appearing in (8.11.11) in terms of the functions H1 , H2 , H4 , 1 , 2 and 3 which appear in the proof of Theorem 8.11.10. 8.20 Prove the formulae (8.11.12).

9 Unitary representations of G L(2)

9.1 Unitary representations of G L(2, Q p ) Definition 9.1.1 (Positive definite Hermitian form) Let V be a complex vector space. A Hermitian form on V is a function ( , ) : V × V → C which is linear in the first variable, conjugate linear in the second, and satisfies (w, v) = (v, w),

(∀v, w ∈ V ).

Such a form is said to be positive definite if (v, v) > 0 for all non-zero v ∈ V. Definition 9.1.2 (Unitary representation) Let V be a complex vector space equipped with a positive definite Hermitian form ( , ), and let G be a group. A unitary representation of G on V is a homomorphism π : G → G L(V ), such that (π (g) . v, π(g) . w) = (v, w),

(∀g ∈ G, v, w ∈ V ).

Clearly, the definition of unitary is made relative to a choice of Hermitian form on V. If V does not come equipped with a Hermitian form which makes (π, V ) into a unitary representation, it is natural to ask whether it would be possible to make (π, V ) unitary by choosing a suitable Hermitian form. This leads to the following definition. Definition 9.1.3 (Unitarizable representation) Let V be a complex vector space, and let G be a group. A representation π : G → G L(V ), of G on V is said to be unitarizable if there exists a positive definite Hermitian form ( , ) such that (π (g) . v, π (g) . w) = (v, w),

(∀g ∈ G, v, w ∈ V ).

Proposition 9.1.4 (The contragredient of a unitary representation of G L(2, Q p ) is unitarizable) Fix a prime p and let (π, V ) be an irreducible smooth representation of G L(2, Q p ) such that the space V is equipped 358

9.1 Unitary representations of GL(2, Qp )

359

with a positive definite Hermitian form ( , )V ×V , where the representation (π, V ) is unitary with respect to this form. Then the contragredient (( π, ( V ) is unitarizable. Proof We define a conjugate linear map T : V → ( V by   T (v) (w) := (w, v)V ×V . That is, T (v) is the smooth linear functional: V → C, whose value at w ∈ V is (w, v)V ×V . Then, for all g ∈ G L(2, Q p ), and all v, w ∈ V,   T (π (g) . v) (w) = (w, π (g) . v)V ×V = (π (g −1 ) . w, v)V ×V     = T (v) (π (g −1 ) . w) = ( π (g) . T (v) (w). It follows that the image of T is a non-trivial invariant subspace of ( V . Because V was assumed to be irreducible, it follows from Proposition 8.1.8 that ( V is also irreducible, and hence that T is actually an R-linear isomorphism. We :( V ×( V → C by may then define a positive definite Hermitian form ( , ) V ×( V ( the formula   (( v, w () := T −1 (( w ), T −1 (( v) .  V ×( V ( V ×V Lemma 9.1.5 (The central character of a smooth irreducible unitary representation of G L(2, Q p ) must be unitary) Fix a prime p and let (π, V ) be a smooth and irreducible unitary representation of G L(2, Q p ). Then its central character, ωπ , (as defined in Definition 6.1.10) is unitary, i.e., satisfies |ωπ (y)| = 1 for all y ∈ Q×p . Proof Let ( , ) be the positive definite invariant Hermitian form associated to π . Then for all r ∈ Q×p we have       r 0 r 0 π . v, π . w = |ωπ (r )|2 (v, w), 0 r 0 r

(∀v, w ∈ V ).

This contradicts Definition 9.1.2 if ωπ is not of absolute value one.



Lemma 9.1.6 (Matrix coefficients of unitary representations of G L(2, Q p ) are bounded) Fix a prime p and let (π, V ) be a smooth irreducible representation of G L(2, Q p ) which is unitary. Then each matrix coefficient of π, as defined in Definition 8.1.14, is a bounded function from G L(2, Q p ) to C. Proof Let ( V be the contragredient of V as in Definition 8.1.4. Let ( , )Her : V × V → C be the positive definite Hermitian form associated to π as in DefiV → C be the canonical invariant bilinear nitions 9.1.1, 9.1.2. Let  , Bln : V × (

360

Unitary representations of GL(2)

form associated to π as in Definition 8.1.12. The main idea of the proof is that matrix coefficients of π may be expressed using ( , )Her instead of  , Bln . To pass between the two different pairings, we utilize the map T : V → ( V introduced in the proof of Proposition 9.1.4. We may then realize every matrix coefficient of π as a function g → (π (g) . v, v )Her for some fixed v, v ∈ V. It follows from the Cauchy-Schwartz inequality that             (π (g) . v, v )Her  ≤ π (g) . v  · v  = v  · v , where ||v|| =

4

(v, v)Her , for all v ∈ V . 

9.2 Unitary principal series representations of G L(2, Q p ) Next, we wish to consider the problem of unitarizability of irreducible principal series representations. We first give a sufficient condition. Proposition 9.2.1 (Principal series representations of G L(2, Q p ) induced from unitary characters are unitarizable) Fix a prime p and let χ1 and   χ2 be unitary characters of Q×p . Let π, B p (χ1 , χ2 ) be the principal series representation defined in Definition 8.2.2 Then B p (χ1 , χ2 ) is unitarizable. Proof Let  , Bln denote the bilinear form B p (χ1 , χ2 ) × B p (χ1−1 , χ2−1 ) → C which was constructed in Proposition 8.2.3. Note that for each f ∈ B p (χ1 , χ2 ), the function f is in B p (χ 1 , χ 2 ). Because χ1 and χ2 are unitary, this is the same as B p (χ1−1 , χ2−1 ). We define ( , )Her : B p (χ1 , χ2 ) × B p (χ1 , χ2 ) → C by



f 1 (k) f 2 (k) d × k.

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

This is clearly a positive definite Hermitian form, and its invariance under the action of π follows immediately from that of  , Bln .  In the previous argument, we constructed a Hermitian pairing V × V → C from the bilinear pairing V × ( V → C. It is also possible to reverse this process, and deduce the following. Proposition 9.2.2 Fix a prime p and let χ1 and χ2 be two characters of Q×p .   Let π, B p (χ1 , χ2 ) be the principal series representation   defined in Definition 8.2.2. Assume that the principal series representation π, B p (χ1 , χ2 ) is both     irreducible and unitary. Then π, B p (χ 1 , χ 2 ) ∼ = π, B p (χ1−1 , χ2−1 ) .

9.2 Unitary principal series representations of GL(2, Qp )

361

  −1 −1 Proof We saw in Proposition 8.2.4 that  π, B p (χ1 , χ2 ) is isomorphic to  the contragredient of π, B p (χ1 , χ2 ) . To complete the proof of the present

result, we prove that the same is true of π, B p (χ 1 , χ 2 ) . Let ( , )Her denote the positive definite Hermitian pairing on B p (χ1 , χ2 ). B p (χ1 , χ2 ) be defined as in the proof of Proposition Let T : B p (χ1 , χ2 ) → ( 9.1.4. Then T is an R-linear isomorphism, and T (π (g) . f ) = ( π (g) . T ( f ), but T is not an intertwining operator, because it is conjugate linear, rather than B p (χ1 , χ2 ) given by f → T ( f ) linear. However, the map B p (χ 1 , χ 2 ) → ( is an intertwining operator, as well as a C-linear isomorphism. Thus it is an isomorphism of representations. This completes the proof. 

Corollary 9.2.3 Suppose that χ1 , χ2 are two characters Q×p → C× , and that at least one of them is nonunitary. Suppose further that the principal series representation B p (χ1 , χ2 ) defined in Definition 8.2.2 is irreducible and unitary. Then χ2 = χ −1 1 . Equivalently, let s = (s1 , s2 ) ∈ C2 and let ω = (ω1 , ω2 ) be a pair of normalized unitary characters of Q×p as in Definition 6.5.1. Suppose that either (s1 ) =/ 12 or (s2 ) =/ − 12 , or both. Suppose further that the principal series representation V p (s, ω) defined in Definition 6.5.3 is irreducible and unitary. Then #(s1 ) = #(s2 ), (s1 ) = − (s2 ), and ω1 = ω2 . Remark The second paragraph of the statement of Corollary 9.2.3 is simply a reformulation of the first paragraph in a different notation. Indeed V p (s, ω) and B p (χ1 , χ2 ) are two notations for the same space precisely when s1 + 12

χ1 (t) = ω1 (t)|t| p

s2 − 12

and χ2 (t) = ω2 (t)|t| p

for all t ∈ Q×p .

Proof This follows immediately from Proposition 9.2.2, in view of Theorem 8.12.25.  Lemmas 9.1.5 and 9.1.6 will allow us to extract some information about which principal series representations have the possibility of being unitarizable. The proof of Proposition 9.2.4 is due to Garrett. Proposition 9.2.4 (Bound on the real parts of the complex parameters associated to a unitary principal series representation of G L(2, Q p )) Fix a prime p, a pair of complex numbers s = (s1 , s2 ), and ω = (ω1 , ω2 ), a pair of normalized unitary characters of Q×p as in Definition 6.5.1. Let π, V p (s, ω) be series representation as in Definition 6.5.3. Suppose that   the principal π, V p (s, ω) is unitarizable. Then (s1 + s2 ) = 0 and 0 ≤ (s1 ) ≤ 1. Proof It follows from Lemma 9.1.5 that the central character of π must be unitary. This is equivalent to (s1 +s2 ) = 0. For the second part, we use Lemma 9.1.6 which says that the matrix coefficients must be bounded. This condition

362

Unitary representations of GL(2)

will give us the inequality 0 ≤ (s1 ) ≤ 1. To see this, we use the explicit invariant bilinear form given in Proposition 8.2.3. s1 − 1

s2 + 12

Set χ1 (t) = ω1 (t)|t| p 2 and χ2 (t) = ω2 (t)|t| p B p (χ1 , χ2 ) = V p (s, ω) given by 

⎧ ⎨ ω (a)|a|s1 ω (d)|d|s2 , 1 p 2 p f 1 (g) = ⎩ 0,

a

b

. Let f 1 be the element of



· k, with k ∈ K 0 ( p) p , 0 d if g is not of this form,

if g =

and let f 2 be the analogous element of B p (χ1−1 , χ2−1 ). Here

 K 0 ( p) p :=

α β γ δ



   ∈ G L(2, Z p ) γ ∈ pZ p ,

is the Iwahori subgroup introduced in Definition 4.11.2.  originally  α β Note that for any γ δ ∈ K 0 ( p) p and any a, d ∈ Q×p such that | da | p ≤ 1, we have    αδ−βγ      α β a 0 a dδ β 0 δ , · · = a γ δ 0 d 0 d γ δ d and

 αδ−βγ  a d

δ

γ

0 δ

is again in K 0 ( p) p . It follows that   a f1 k · 0

0 d

 = ω1 (a)|a|sp1 ω2 (d)|d|sp2

for any k ∈ K 0 ( p) p and any a, d ∈ Q×p such that |a/d| p ≤ 1. We may now compute         a 0 a 0 f 2 (k) d × k = d ×k f 1 k· f 1 k· 0 d 0 d G L(2,Z p ) K 0 ( p) p   = Vol K 0 ( p) p · ω1 (a) |a|sp1 ω2 (d) |d|sp2 , for all a, d ∈ Q×p such that |a/d| p ≤ 1. By Lemma 9.1.6 this must be a bounded function of a and d in this domain. Since     1) 2) 1) |d|(s = |a/d|(s , ω1 (a) |a|sp1 ω2 (d) |d|sp2  = |a|(s p p p C

this is the case if and only if (s1 ) ≥ 0. (Here we have used that (s2 ) = −(s1 ), by the first part of this proposition.) the same To deduce that (s1 ) ≤ 1, we  use Proposition 9.1.3, and apply  reasoning to B p (χ1−1 , χ2−1 ) = V p (1 − s1 , −1 − s2 ), (ω1−1 , ω2−1 ) . 

9.2 Unitary principal series representations of GL(2, Qp )

363

Miraculously, Proposition 9.2.4, in conjunction with Corollary 9.2.3, turns out to be sharp. That is: every principal series representation which is not shown to be nonunitary by either of these results actually turns out to be unitary. We will not prove this result, but we state it for the record. A proof is given in [Bump, 1997]. Theorem 9.2.5 (Characterization of irreducible unitary principal series representations of G L(2, Q p )) Fix a prime p and let χ1 , χ2 be two characters Q×p → C× . The principal series representation B p (χ1 , χ2 ), as defined in Definition 8.2.2, is irreducible and unitary if and only if one of the following conditions holds: • χ1 and χ2 are unitary characters 2σ • χ1 = χ −1 2 , and χ1 (t)/χ2 (t) = |t|∞ with σ ∈ (−1/2, 1/2). Equivalently, let s = (s1 , s2 ) ∈ C2 and let ω = (ω1 , ω2 ) be a pair of normalized unitary characters of Q×p as in Definition 6.5.1. Then the principal series representation V p (s, ω) defined in Definition 6.5.3 is irreducible and unitary, if and only if one of the following conditions holds • (s1 ) = 12 and (s2 ) = − 12 , • ω1 = ω2 , s2 = −s1 , and (s1 ) ∈ (0, 1). Before ending this section, we define two important classes of representations. Definition 9.2.6 (Complementary series representation of G L(2, Q p )) Fix a prime p. The set   5   1 1 B p (χ , χ −1 )  χ : Q×p → C× a character, |χ (t)|C = |t|σp , 0 =/ σ ∈ − , 2 2 is called the complementary series of G L(2, Q p ). The representations in it are called complementary series representations. Definition 9.2.7 (Tempered representation of G L(2, Q p )) Fix a prime p. An admissible representation (π, V ) of G L(2, Q p ) is said to be tempered if it has a unitary central character and  |β(g)|2+ε d×g < ∞ C G L(2,Q p )/Q×p

for all matrix coefficients β of (π, V ) and all ε > 0. Proposition 9.2.8 (The characterization of the irreducible tempered representations of G L(2, Q p )) Fix a prime p. The following representations of G L(2, Q p ) are tempered:

364

Unitary representations of GL(2)

• supercuspidal representations with unitary central character, • special representations with unitary central character, • principal series representations B p (χ1 , χ2 ) where χ1 and χ2 are unitary. Furthermore, every irreducible admissible tempered representation of G L(2, Q p ) is isomorphic to a representation of precisely one of these types. Proof It was shown in Proposition 8.4.9 that matrix coefficients of supercuspidals are compactly supported modulo the center. This is clearly stronger than being L2+ε . It was shown in Theorem 8.10.19 that matrix coefficients of special representations are square integrable modulo the center, which, again, is stronger than being L2+ε . The proof that B p (χ1 , χ2 ) is tempered if and only if χ1 and χ2 are unitary is left to the reader. It requires the asymptotics of matrix coefficients,  n  given in Theorem 8.10.13, and the fact that the volume of G L(2, Z p ) · p0 10 · G L(2, Z p ) is equal to p n + p n−1 , which was shown during the proof of Theorem 8.10.19. The fact that every irreducible tempered representation of G L(2, Q p ) is isomorphic to a representation of precisely one of these types follows from the classification Theorem 6.13.4 

9.3 Unitary and irreducible special or supercuspidal representations of G L(2, Q p ) Next, we consider the problem of unitarizability of irreducible supercuspidal and special representations of G L(2, Q p ). Proposition 9.3.1 Fix a prime p. Let (π, V ) be either an irreducible special or supercuspidal representation of G L(2, Q p ) with central character ωπ . Then (π, V ) is unitarizable if and only if ωπ is a unitary character. Proof It is clear that if ωπ is not a unitary character, then no Hermitian form is preserved by the action of the center of G L(2, Q p ), much less the whole group. Hence (π, V ) can not possibly be unitarizable in this case. Now assume that ωπ is unitary. We will construct a model of π as action by right translation on a space of L2 functions, and thus equip V with a positive definite Hermitian form inherited from the L2 inner product. In fact the functions which we use are matrix coefficients of the representation π.   Let L2ωπ Q×p \G L(2, Q p ) denote the space of functions f : G L(2, Q p ) → C which satisfy  f

r 0 0 r



 ·g

= ωπ (r ) · f (g),

  ∀g ∈ G L(2, Q p ), r ∈ Q×p ,

and are square integrable modulo the center, as defined in Definition 8.10.18.

9.4 Unitary (g, K∞ )-modules

365

It was shown in Proposition 8.10.1 that (π, V ) may be realized as right translation on its own space of matrix coefficients. Further, by Theorem 8.10.19, the matrix coefficients of special and supercuspidal representations  are elements of L2ωπ Q×p \G L(2, Q p ) . We  may define a positive definite Hermitian form on the whole space 2 Lωπ Q×p \G L(2, Q p ) , by the formula 

f 1 (g) f 2 (g) d × g,

( f 1 , f 2 ) := Ep

where



  ∀ f 1 , f 2 ∈ L2ωπ Q×p \G L(2, Q p )

   E p := g ∈ G L(2, Q p )  where | det g| p = 1 or p ,

and the integral is with respect to the Haar measure on G L(2, Q p ). The invariance of the above Hermitian form is left as an exercise to the reader.  We conclude this section with a characterization of the unitary special representations of G L(2, Q p ). For characters χ1 , χ2 : Q×p → C, recall the space B p (χ1 , χ2 ) defined in Definition 8.2.1. Fix a character χ : Q×p → C. It is convenient to adopt the notation   1 −1 (9.3.2) B p χ · | | p2 , χ · | | p 2 = B p (χ1 , χ2 ) −1

for χ1 (y) = χ (y) · |y| 2 and χ2 (y) = χ (y) · |y| p 2 and all y ∈ Q×p . 1

Proposition 9.3.3 (Characterization of unitary special representations of G L(2, Q p )) Fix a prime p. Let (π, V ) be a unitary special representation of G L(2, Q p ) as in Section 6.10 and Definition 9.1.2. Then there exists a unitary character χ : Q×p → C such that V is isomorphic to a subrepresentation of   1 1 B p χ · | | 2 , χ · | |− 2 as defined in (9.3.2) and Definition 8.2.2. Further,    1  1 −1 −1 B p χ · | | p2 , χ · | | p 2 ∼ = χ ⊗ B p | | p2 , | | p 2 as in Definition 8.10.24.

  1 −1 Remark If χ is not assumed to be unitary, then B p χ · | | p2 , χ · | | p 2 is not a unitary representation, but it still contains a special representation. Every special representation is isomorphic to one of this form.

9.4 Unitary (g, K ∞ )-modules In this section we shall review results similar to those given in Sections 9.1 and 9.3, but for (g, K ∞ )-modules instead of representations of G L(2, Q p ).

366

Unitary representations of GL(2)

Definition 9.4.1 (Unitary (g, K ∞ )-module) Let V be a complex vector space equipped with a positive definite Hermitian form ( , ). Let π = (πg , π K∞ ) be a pair of actions πg : U (g) → End(V ),

π K∞ : O(2, R) → G L(V ),

such that (π, V ) is a (g, K ∞ )-module as in Definition 5.1.4. Then (π, V ) is said to be unitary if ( , ) is invariant with respect to each action in the sense which is appropriate to that action. Specifically (∀k ∈ K ∞ , v, w ∈ V ), (π K∞ (k) . v, π K∞ (k) . w) = (v, w),   (∀α ∈ g, v, w ∈ V ). (πg (Dα ) . v, w) = − v, πg (Dα ) . w , Remark The condition (πg (Dα ) . v, w) = −(v, πg (D α ) . w),

(∀α ∈ g, v, w ∈ V ),

is equivalent to the condition (πg (Dα ) . v, w) = −(v, πg (Dα ) . w),

(∀α ∈ gl(2, R), v, w ∈ V ),

because the map α → Dα is linear (Proposition 4.5.2, Definition 4.5.5). As with representations of G L(2, Q p ), the term “unitary” is defined relative to a choice of Hermitian form, and a (g, K ∞ ) is said to be unitarizable if there exists a Hermitian form with respect to which it is unitary. Proposition 9.4.2 (The contragredient of a unitary (g, K ∞ )-module is unitarizable) Let (π, V ) be an irreducible admissible (g, K ∞ )-module. Assume that the vector space V is equipped with a positive definite Hermitian form ( , )V ×V , and that the representation (π, V ) is unitary with respect to this form. Then the contragredient (( π, ( V ) is unitarizable. Proof The proof is the same as in the p-adic  case.  We define an R-linear isomorphism T : V → ( V by the formula T (v) (w) := (w, v)V ×V . Then define a positive definite Hermitian form ( , ) : ( V ×( V → C by the ( V ×( V formula   (( v, w () := T −1 (( w ), T −1 (( v) . V ×( V ( V ×V Invariance of this form (in this case, invariance with respect to O(2, R) and also with respect to U (g)) follows from that of the original form on V and implies that (( π, ( V ) is unitarizable.  Next, we wish to consider the problem of unitarizability of the irreducible principal series representations of G L(2, R), as defined in Definition 8.6.7. We first give the analogues of Proposition 9.2.1, 9.2.2. The proofs are exactly the same as in the p-adic case.

9.4 Unitary (g, K∞ )-modules

367

Proposition 9.4.3 (Principal series representations of G L(2, R) induced from unitary characters are unitarizable) Let χ1 and χ2 be unitary   × characters of R . Let π, B∞ (χ1 , χ2 ) be the principal series representation defined in 8.6.7. Then B∞ (χ1 , χ2 ) is unitarizable. Proposition 9.4.4 Let χ1 and χ2 be two characters of R× , and let   π, B∞ (χ1 , χ2 ) be the representation defined in Definition   principal series 8.6.7. Assume that π, B∞ (χ1 , χ2 ) is both irreducible and unitary. Then     π, B∞ (χ 1 , χ 2 ) ∼ = π, B∞ (χ1−1 , χ2−1 ) .   As in the p-adic case, it can be proved that if π, B∞ (χ1−1 , χ2−1 ) is unitary 1 1 2σ and χ1 , χ2 are not, then χ2 = χ −1 1 and χ1 (x)/χ2 (x) = |x|∞ with σ ∈ (− 2 , 2 ), and it can also be shown that these necessary conditions also suffice. These conditions can be put in terms of the V∞ (s, ω) notation. Proposition 9.4.5 (Characterization of the unitary principal series representations of G L(2, R)) Let χ1 and χ2 be two characters of R× . Let  B∞ (χ1 , χ2 ) = V∞

  1 1 s1 + , s2 − , ω 2 2

be a principal series representation  of G L(2, R) as previously defined in Definitions 8.6.7 and 7.4.3. Then π, B∞ (χ1 , χ2 ) is unitary if and only if one of the following holds: (t1 , t2 ∈ R), (a) s1 = it1 , s2 = it2 ,   t ∈ R, − 12 < σ < 12 . (b) ω1 = ω2 and s1 = σ + it, s2 = −σ + it,   Moreover, π, B∞ (χ1 , χ2 ) is irreducible in both of the above cases. We will not prove these results, leaving them instead for Exercises 9.10 and 9.11. We will, however, prove that every discrete series representation is unitarizable. This will be accomplished using a global-to-local argument, so we defer it until after we have covered the relevant global results. Before closing this section, we extend the definitions of “complementary series” and “tempered” to the real case. Definition 9.4.6 (Complementary series representation) The set of (g, K ∞ )-modules 

  B∞ (χ , χ −1 )  χ : R → C× a character, |χ (t)|C = |t|σ∞ , 0 =/ σ ∈ (− 12 , 12 )

is called the complementary series of G L(2, R). The representations in it are called complementary series representations.

368

Unitary representations of GL(2)

Definition 9.4.7 (Tempered representation) An admissible (g, K ∞ )-module (π, V ) is said to be tempered if it has a unitary central character and  |β(g)|2+ε d×g < ∞ C G L(2,R)/R×

for all matrix coefficients β of (π, V ) and all ε > 0. Proposition 9.4.8 (Characterization of irreducible tempered (g, K ∞ )modules) The following (g, K ∞ )-modules are tempered: • discrete series representations • principal series representations B∞ (χ1 , χ2 ) where χ1 and χ2 are unitary. Furthermore, every irreducible admissible tempered (g, K ∞ )-module is isomorphic to a representation of precisely one of these types.

9.5 Unitary (g, K ∞ ) × G L(2, Afinite )-modules In this section we first review the definition of and basic results on unitary (g, K ∞ ) × G L(2, Afinite )-modules. Since this is completely analogous to the cases of G L(2, Q p ) and (g, K ∞ )-modules already covered, proofs will be omitted. We then prove the important fact that the vector space of adelic cusp forms is a (reducible) unitary (g, K ∞ ) × G L(2, Afinite )-module. We will then use this result to deduce complete reducibility of cuspidal automorphic representations, thus removing a technical hypothesis from Theorem 5.5.15. We will also deduce unitarizability of certain local representations. Definition 9.5.1 (Unitary (g, K ∞ ) × G L(2, Afinite )-module) Let V be a complex vector space equipped with a positive definite Hermitian form ( , ), and let G be a group. Let π = ((πg , π K∞ ), πfinite ) where πg : U (g) →End(V ),

π K∞ : O(2, R) → G L(V ),

πfinite : G L(2, Afinite ) → G L(V ), are actions such that (π, V ) is a (g, K ∞ )×G L(2, Afinite )-module as in Definition 5.1.5. Then (π, V ) is said to be unitary if ( , ) is invariant with respect to each action in the sense which is appropriate to that action. Specifically   π K∞ (k) . v, π K∞ (k) . w = (v, w),

(∀k ∈ K ∞ , v, w ∈ V ),

  πg (Dα ) . v, w = −(v, πg (Dα ) . w), (∀α ∈ g, v, w ∈ V ),   πfinite (afinite ) . v, πfinite (afinite ) . w = (v, w), (∀afinite ∈ G L(2, Afinite ), v, w ∈ V ).

9.5 Unitary (g, K∞ )× GL(2, Afinite )-modules

369

Proposition 9.5.2 (The contragredient of a unitary (g, K ∞ ) × G L(2, Afinite )module is unitarizable) Suppose that (π, V ) is an irreducible admissible (g, K ∞ ) × G L(2, Afinite )-module. Assume that the vector space V is equipped with a positive definite Hermitian form ( , )V ×V , and that the representation (π, V ) is unitary with respect to this form. Then the contragredient (( π, ( V ) is unitarizable. Next, we turn our attention to those (g, K ∞ ) × G L(2, Afinite )-modules which are of primary interest to us: cuspidal automorphic representations. We begin by defining an invariant positive definite Hermitian form on the space of all adelic cusp forms of a fixed central character. Definition 9.5.3 (Hermitian form on adelic cusp forms) Let Scusp, ω denote the C-vector space of all adelic cusp forms with central character ω, as in Definition 4.7.7. We define a positive definite Hermitian form ( , ) : Scusp, ω × Scusp, ω → C as follows. Let φ1 , φ2 ∈ Scusp, ω . Then we set  (φ1 , φ2 ) := φ1 (g) φ2 (g) d × g, Z (AQ )·G L(2,Q)\G L(2,AQ )

where Z (AQ ) denotes the center of G L(2, AQ ). It follows easily from Proposition 8.9.2 (cf. Corollary 8.9.4) that the integral in Definition 9.5.3 is convergent. In fact, this integral is simply the ordinary L2 inner product on the adelic cusp forms φ1 , φ2 , which, by Proposition 8.9.2, are L2 . Lemma 9.5.4 (The Hermitian form in definition 9.5.3 is invariant) The Hermitian form ( , ) : Scusp, ω × Scusp, ω → C given in Definition 9.5.3 is invariant, as in Definition 9.5.1. Proof The proof is essentially the same as that of Proposition 8.9.5.  It follows immediately from Lemma 9.5.4 that every (g, K ∞ ) × G L(2, Afinite )-module which is isomorphic to a subspace of Scusp, ω is unitarizable. Recall that the definition of a cuspidal automorphic representation given in Definition 5.1.14 uses the notion of a subquotient, which was developed in the discussion before Definition 5.1.12. It turns out, however, that because we have an invariant Hermitian form on the entire space Scusp, ω , we are able to replace subquotient with the simpler notion of a subrepresentation. Roughly, a subquotient is a quotient W/W of two subspaces W ⊂ W ⊂ Scusp, ω . Because our invariant inner product is defined on all of Scusp, ω , we can take the orthogonal complement W

of W in W. It then turns out that the subquotient W/W is isomorphic to the subspace W

. The proof that W/W is isomorphic to W

requires some care. It reduces to the statement that for any w0 ∈ W, there exists w0 ∈ W such that

370

Unitary representations of GL(2) (w , w0 ) = (w , w0 ),

(∀w ∈ W ).

We shall eventually deduce this statement from the corresponding fact about finite dimensional vector spaces, after reducing to consideration of the finitedimensional space of cusp forms with a fixed weight and Laplace eigenvalue which are invariant by a fixed compact open subgroup of G L(2, Afinite ). (See the proof that irreducible cuspidal representations are admissible in Theorem 5.5.15 for this finite-dimensionality result.) The first step is to prove that it is enough to consider eigenfunctions of the Laplacian. This will be deduced from the following important fact. Lemma 9.5.6 ( is self-adjoint) Let  be the Casimir element of U (g) defined as in [Goldfeld, 2006; p. 50] (see also Lemma 5.2.6). Then for all unitary Hecke characters ω and for all φ1 , φ2 ∈ Scusp, ω (πg () . φ1 , φ2 ) = (φ1 , πg () . φ2 ). Proof This follows easily from the expression for  as  = D1,1 ◦ D1,1 + D1,2 ◦ D2,1 + D2,1 ◦ D1,2 + D2,2 ◦ D2,2 . and the invariance property of ( , ).  Lemma 9.5.7 (Every cusp form is a finite sum of eigenfunctions of ) Let ω be a unitary Hecke character and let φ be a cusp form with central character ω as in Definition 4.7.7. Then there exist an integer N and cusp forms φ1 , . . . , φ N , each with central character ω, and each an eigenfunction of , such that φ = φ1 + . . . + φ N . Proof Since φ is Z (U (g))-finite, {π (D) . φ | D ∈ Z (U (g))} spans a finitedimensional vector space, V, which is evidently preserved by the action of π (). It follows that we may choose a basis for V consisting of generalized eigenfunctions of π ()—i.e., a basis so that the matrix of the operator π () with respect to this basis is in Jordan canonical form. What follows from Lemma 9.5.6 is that, in fact, each of these generalized eigenfunctions is simply an eigenfunction—i.e., none of the Jordan blocks is non-trivial. This completes the proof.  Now we are ready to show that every subquotient of Scusp, ω is isomorphic to a subspace of Scusp, ω . Proposition 9.5.8 (For cusp forms, subquotient =⇒ subspace) Let ω be a unitary Hecke character, and let Scusp, ω denote the C-vector space of all adelic cusp forms with central character ω, as in Definition 4.7.7. Let (π, V ) be a subquotient of (π, Scusp, ω ) where π denotes the triple of actions defined in Definition 5.1.11. Then (π, V ) is isomorphic to a subrepresentation of (π, Scusp, ω ).

9.5 Unitary (g, K∞ )× GL(2, Afinite )-modules

371

Proof By hypothesis, V = W/W where W ⊂ W ⊂ Scusp, ω are two invariant subspaces. Define    W

= φ ∈ W  (φ, φ ) = 0, ∀φ ∈ W . Here ( , ) denotes the bilinear Hermitian form given in Definition 9.5.3. Then W

is an invariant subspace of W : this follows from the invariance of W and of the pairing ( , ). It is clear that W ∩W

= {0}. We claim that W = W ⊕W

. This amounts to the claim that for any φ0 ∈ W, there exists φ0 ∈ W such that (φ , φ0 ) = (φ , φ0 )

(9.5.9)

for all φ ∈ W . It is enough to prove this under the additional hypothesis that φ0 is an eigenfunction of  with eigenvalue λ, and that    cos θ sin θ = eikθ φ0 (g), (∀θ ∈ [0, 2π )), φ0 g · i ∞ − sin θ cos θ for some k ∈ Z, because a general φ0 will be a finite sum of forms satisfying these additional hypotheses. Furthermore, there is a minimal positive integer N such that φ0 is fixed by the compact open subgroup K (N ) of G L(2, Afinite ) defined as in (5.3.7). Let W (λ, k, N ) denote the space of all φ ∈ W such that    cos θ sin θ = eikθ φ(g), (∀θ ∈ [0, 2π )), φ g · i∞ − sin θ cos θ (∀kfinite ∈ K (N )), φ(g · kfinite ) = φ(g), φ = λ · φ, and define W (λ, k, N ) analogously. It was shown, in the proof of Theorem 5.5.15, that each of these spaces is finite dimensional. Note that W (λ, k, N ) ⊆ W (λ, k, N ). Define φ0 to be the orthogonal projection of φ0 to W (λ, k, N ). Then (9.5.9) (9.5.9) holds for holds for all φ ∈ W (λ, k, N ) by definition. Furthermore,  cos θ sin θ

all φ on which  acts with a different eigenvalue or − sin θ cos θ acts by a different character, because, for such φ , both sides of (9.5.9) vanish for trivial reasons. Now suppose φ ∈ W (λ, k, N ) for some integer N . We may assume N |N , by enlarging N , if necessary. Then W (λ, k, N ) ⊆ W (λ, k, N ). Define Proj N : W (λ, k, N ) → W (λ, k, N ) to be the orthogonal projection. Then since φ ∈ W (λ, k, N ) ⊆ W (λ, k, N ), we have   Proj N φ ∈ W (λ, k, N ) . φ = (Proj N φ ) + (φ − Proj N φ ),

372

Unitary representations of GL(2)

It follows that (9.5.9) holds for Proj N φ and φ − Proj N φ ∈ W (λ, k, N )⊥ , so that     φ − Proj N , φ0 = 0 = φ − Proj N , φ0 . Since every element of W is in the direct sum of a finite number of the subspaces W (λ , k , N ) for various values of λ ∈ C, k ∈ Z, N ∈ Z, N > 0, it follows at once that (9.5.9) holds for all elements of W , as required.  Corollary 9.5.10 (Cuspidal automorphic representations are unitarizable) Let (π, V ) be a cuspidal automorphic representation of G L(2, AQ ) as in Definition 5.1.14. Then there exists a positive definite Hermitian inner product on V with respect to which π is unitary. Proof From the definition of a cuspidal automorphic representation, we know that the representation (π, V ) is isomorphic to a subquotient of Scusp,ω for some unitary Hecke character ω. By Proposition 9.5.8 it is, in fact, isomorphic to a subrepresentation of Scusp,ω . An invariant positive definite Hermitian on V is then obtained directly from this isomorphism and the invariant positive definite inner product on Scusp,ω .  Next, we will use the preceding global results to prove a purely local result: namely, that each of the discrete series “representations” (which are really (g, K ∞ )-modules) constructed in chapter 7 (see especially Definition 7.4.10) are unitarizable. Corollary 9.5.11 (Discrete series representations are unitarizable) Suppose that (π, V ) is a discrete series representation as defined in Definition 7.4.10. Then (π, V ) is a unitary (g, K ∞ )-module, as defined in Definition 9.4.1. Proof Let f 0 be a holomorphic modular form of weight k and character χ for 0 (N ) as in Definition 3.3.4, and let f = y k/2 f 0 . Then, as shown in Proposition 3.10.2, the function f is a Maass form of weight k and character χ as in Definition 3.5.7, which lies in the kernel of the Maass lowering operator defined in Definition 3.9.2. Assume further that f is a Maass cusp form. This forces the weight k to be positive. Let f adelic denote the adelic lift of f as in Definition 4.12.13. Then f adelic is killed by the operator L ∈ U (g), defined in Definition 5.2.5 (see the interlude in Section 5.2). Let V f denote the space of functions considered in Example 5.2.12: the span of all the images of f adelic under the action of O(2, R) and U (g). The analysis of this space in Section 5.2 reveals that the dimension of the space of weight m elements of V f is ≤ 1 if m ≡ k (mod 2), and m ≥ k or m ≤ −k, and zero for all other values of m. Comparing this with the analysis of a general (g, K ∞ )-module in Section 7.5, we find at once that V f is an irreducible (g, K ∞ )-module isomorphic to the

9.5 Unitary (g, K∞ )× GL(2, Afinite )-modules

373

  discrete series representation contained in V∞ (s2 + k2 , s2 − k2 ), ω for suitable s2 ∈ C and ω = (ω1 , ω2 ), with ω1 , ω2 normalized unitary characters of R× . We wish next to discern the correct values of s2 and ω from f 0 . The central character of f adelic is the idelic lift χidelic of the character χ of f 0 and f. This unitary Hecke character is trivial on the positive reals (embedded into the ideles at ∞). It follows that s2 = 0. Furthermore ω1 (−1)ω2 (−1) = χidelic (−1) = (−1)k . There are two pairs ω = (ω1 , ω2 ), which satisfy this condition, and it was proved in Theorem 7.5.10 that the corresponding discrete series representations are isomorphic to one another. Now, V f is a subspace of Scusp,χidelic , on which there is an invariant Hermitian inner This  kproduct.  proves that the discrete series representation contained in k V∞ ( 2 , − 2 ), ω is unitarizable for all values of k, ω such that there is a nonzero holomorphic cusp form of weight k (and any level and character). The existence of such a form for each k ≥ 2 may be deduced from Theorem 2.24 (k even) or 2.25 (k odd) of [Shimura,  1971]. k k To treat V∞ (s2 + 2 , s2 − 2 ), ω , with s2 ∈ iR non-zero, observe that every   element f ∈ V∞ (s2 + k2 , s2 − k2 ), ω , is uniquely expressible as h(g) · | det g|s∞2  k  for some h ∈ V∞ ( 2 , − k2 ), ω .     Let T : V∞ (s2 + k2 , s2 − k2 ), ω → V∞ ( k2 , − k2 ), ω be defined by 2 T ( f )(g) = f (g) · | det g|−s ∞ . Clearly,   −s2 2 π K∞ (k) . T ( f )(g) = f (gk)·| det gk|−s ∞ = f (gk)·| det g|∞ = T π K ∞ (k) . f (g), and for α ∈ gl(2, R),  d −s2  2 f (g exp(tα)) · | det g|−s ∞ | det exp(tα)|∞ t=0 dt   d 2 −t·Tr(α)·s2  f (g exp(tα)) · | det g|−s = e ∞  dt t=0

πg (Dα ) . T ( f )(g) =

−s2 2 = πg (α) . f (g) · | det g|−s ∞ − s2 · Tr(α) · f (g) · | det g|∞   = T πg (α) . f (g) − s2 · Tr(α) · T ( f )(g),

where Trdenotes the trace of a matrix. Let   ( , ) be a positive definite Hermitian k k k k form V∞ ( 2 , − 2 ), ω ×V∞ ( 2 , − 2 ), ω → C which is invariant with respect to both actions as in Definition 9.4.1. Define a positive definite Hermitian form       k k k k

s2 + , s2 − s2 + , s2 − , ω × V∞ , ω → C, ( , ) : V∞ 2 2 2 2 by the formula   ( f 1 , f 2 ) := T ( f 1 ), T ( f 2 ) ,

    k k s2 + , s2 − , ω . ∀ f 1 , f 2 ∈ V∞ 2 2

Then, keeping in mind that Definition 7.4.10 requires s2 to be pure imaginary, it is easily verified that ( , ) is invariant as in Definition 9.4.1. 

374

Unitary representations of GL(2)

A final application of the construction of ( , ) is that it enables us to define matrix coefficients for cuspidal automorphic representations. Recall that matrix coefficients of an irreducible admissible representation of G L(2, Q p ) were defined in Definition 8.1.14 as functions of the form π (g) . v, v˜ , (g ∈ V , where  ,  is G L(2, Q p )), for some v ∈ V and v˜ in the contragredient ( the canonical bilinear form on V × ( V . We have already seen during the proof of Lemma 9.1.6 that for unitary representations, they may be defined equivalently as those functions of the form (π (g) . v1 , v2 ) (g ∈ G L(2, Q p )) for some v1 , v2 ∈ V, where ( , ) is the invariant Hermitian form. This motivates the following definition. Definition 9.5.12 (The matrix coefficient of a cuspidal automorphic representation) Let (π, V ) be a cuspidal automorphic representation of G L(2, AQ ) as in Definition 5.1.14. By Proposition 9.5.8, we may assume that V is an invariant subspace of Scusp, ω for some ω with actions via right translation and differential operators as in Section 5.1. Fix φ1 , φ2 ∈ V. Then the map  g → φ1 (hg) φ2 (h) d × h, (g ∈ G L(2, AQ )) Z (AQ )G L(2,Q)\G L(2,AQ )

is called a matrix coefficient of (π, V ). Remark The function h → φ1 (hg) is not, in general, an automorphic form, because it is not, in general, K ∞ -finite. However, it inherits from φ1 the rapid decay property proved in Proposition 8.9.2, and this ensures that the integral in Definition 9.5.12 is absolutely convergent. The situation may be described thus: there is a natural Hermitian inner product ( , ) on the space of all functions G L(2, AQ ) → C which satisfy    r 0 φ g = ω(r )φ(g), 0 r and the L2 condition 

|φ(g)|2 d × g < ∞.

Z (AQ )G L(2,Q)\G L(2,AQ )

We showed that Scusp, ω is contained in this space, and obtained an inner product on Scusp, ω . Now, Scusp, ω is not closed under the action of G L(2, AQ ) by right translation, but the larger L2 space is.

Exercises for Chapter 9 9.1 Let (π, V ) be a representation of a group G. Suppose there exists a positive definite Hermitian form ( , ) on V such that

Exercises for Chapter 9   π (g).v, π (g).v = (v, v),

375

(v ∈ V, g ∈ G).

Prove that ( , ) is an invariant Hermitian form for (π, V ), so that (π, V ) is unitary with respect to ( , ). 9.2 Let (π, V ) be a unitary (g, K ∞ )-module, and write V = ⊕m∈Z Vm as in Exercise 7.1. Prove that this is an orthogonal decomposition of V . That is, show that the subspaces V and Vm are orthogonal if  =/ m. 9.3 Let (π, V ) be an infinite-dimensional admissible unitary (g, K ∞ )module. Prove that the operator πg (D H ) is not a bounded operator on V with respect to the given positive definite Hermitian form. (Recall that a linear operator L : V → V is bounded if there is a constant C > 0 such that %L(v)% ≤ C%v% for all v ∈ V .) 9.4 Let (π, V ) be an admissible unitarizable representation of G L(2, Q p ) or (g, K ∞ )-module, and suppose W ⊂ V is an invariant subspace. Prove that there exists an invariant subspace W such that V = W ⊕ W . 9.5 Let (π, V ) and (π , W ) be representations of a group G. Suppose T : V → W is a conjugate linear bijection such that T (π (g).v) = π (g).T (v) for all g ∈ G and v ∈ V . (Recall that T is conjugate linear if T (λv + v ) = λT (v) + T (v ) for all v, v ∈ V and λ ∈ C.) (a) Prove that T −1 : W → V is also a conjugate linear bijection satisfying T −1 (π (g).w) = π (g).T −1 (w) for all g ∈ G and w ∈ W . (b) Suppose ( , )Her is a Hermitian form on V for which (π, V ) is unitary. Prove that the pairing  ,  : V × W → C defined by   v, w = v, T −1 (w) Her is bilinear and invariant. (c) Assume now that (π, V ) is an irreducible admissible unitary repreπ, ( V ). sentation of G L(2, Q p ). Prove that (π , W ) ∼ = (( 9.6 Translate Propositions 9.2.1 and 9.2.2 into statements about the principal series representations V p (s, ω). 9.7 Verify the remark following Definition 9.4.1. That is, if (π, V ) is a (g, K ∞ )-module and ( , ) is a positive definite Hermitian form, prove that the condition (πg (Dα ) . v, w) = −(v, πg (Dα ) . w),

(α ∈ g, v, w ∈ V ),

is equivalent to the condition (πg (Dα ) . v, w) = −(v, πg (Dα ) . w),

(α ∈ gl(2, R), v, w ∈ V ).

376

Unitary representations of GL(2)

9.8 Suppose (π, V ) is an irreducible admissible unitary representation of G L(2, Q p ). Prove that a positive definite invariant Hermitian form on V is unique up to a positive scalar. 9.9 Suppose (π, V ) is an irreducible admissible unitary (g, K ∞ )-module. Prove that a positive definite invariant Hermitian form on V is unique up to a positive scalar. 9.10* Let s = (s1 , s2 ) be a pair of complex numbers and let ω = (ω1 , ω2 ) be a pair of normalized unitary characters on R× . The goal of this exercise is to prove that the principal series representation V∞ (s, ω) is unitarizable if and only if one of the following is true: (1) s = ( 12 + it1 , − 12 + it2 ) for some real numbers t1 , t2 , or (2) V∞ (s, ω) is of even parity (i.e., ω1 = ω2 ) and s = ( 21 + iζ , − 12 + iζ ) for some complex number ζ such that 0 < |Im(ζ )| < 12 . (a) Suppose that V∞ (s, ω) is unitarizable. Prove that either (1) or (2) above must hold. Hint: Use the explicit basis given by definition 7.4.7 and the explicit actions on this basis given by Proposition 7.4.8. (b) If s is given by case (1) above, prove that V∞ (s, ω) = B∞ (χ1 , χ2 ) for some unitary characters χ1 , χ2 . Conclude that V∞ (s, ω) is unitarizable. (c) Suppose ζ is a complex number such that 0 < |Im(ζ )| < 12 . Prove that for all integers m, the following inequality holds: 1 − i(ζ − ζ ) + 2m 1 + i(ζ − ζ ) + 2m

> 0.

(d) Suppose V∞ (s, ω) is given by case (2) above. For each integer m, define f 2m : G L(2, R) → C by    y1 x cos θ sin θ f 2m 0 y2 − sin θ cos θ = ω1 (y1 ) |y1 |s∞1 ω2 (y2 ) |y2 |s∞2 e2imθ for all x ∈ R, y1 , y2 ∈ R× , y2 > 0, and 0 ≤ θ < 2π . (See Definition 7.4.7.) Recursively define a pairing on the collection { f 2m }m∈Z by  ( f 2m , f 2n ) = ( f 2(m+1) , f 2(m+1) ) =

1 if m = n = 0,

0 if m =/ n, 1 − i(ζ − ζ ) + 2m 1 + i(ζ − ζ ) + 2m

· ( f 2m , f 2m ),

(m ∈ Z).

Exercises for Chapter 9

377

Prove that this pairing extends to an invariant positive definite Hermitian form on all of V∞ (s, ω). Hint: Use the explicit actions given by Proposition 7.4.8. (e) Deduce the first statement of Proposition 9.4.5. 9.11 Prove that if V∞ (s, ω) is a unitarizable principal series representation of G L(2, R), then it is irreducible. Hint: Use Exercise 9.10. 9.12 Let (π, V ) be a unitary special representation of G L(2, Q p ). (a) Suppose V ⊂ B p (χ1 , χ2 ) for some characters χ1 , χ2 : Q×p → C× . −1/2 1/2 Prove that χ1 | | p and χ2 | | p are unitary. (b) Suppose V is a quotient of B p (χ1 , χ2 ) for some characters χ1 , χ2 : 1/2 −1/2 Q×p → C× . Prove that χ1 | | p and χ2 | | p are unitary.

10 Tensor products of local representations

10.1 Euler products In Section 2.5 we introduced the restricted tensor product and in Proposition 2.5.7 it was shown that all automorphic representations for G L(1, AQ ) are factorizable as an infinite product of local representations. This may be viewed as a representation theoretic version of the fact, first noticed by Euler in the year 1737, that ∞  −1 1 − p −s n −s = . (10.1.1) ζ (s) = p

n=1

Of course, the factorization (10.1.1) extends to Dirichlet L-functions L(s, χ ) =

 −1 1 − χ ( p) p −s ,

(10.1.2)

p

and, hence, applies to all L-functions associated to automorphic representations of G L(1, AQ ). The factorization of an automorphic representation of G L(1, AQ ) is also equivalent to the factorization of the global integrals (2.2.6), (2.2.16), that appear in Tate’s thesis. A vast generalization of the Euler product (10.1.2) was obtained by Erich Hecke (see [Hecke, 1937]) who gave a criterion for when the L-function of an automorphic form on G L(2) has an Euler product factorization. Hecke explicitly constructed certain operators (now called Hecke operators) which map automorphic forms to automorphic forms. He proved that if an automorphic form is an eigenfunction of all the Hecke operators then its L-function is Eulerian, i.e., has an Euler factorization similar to (10.1.2). The modern theory of Euler products is based on the adelic theory of automorphic representations. It was first proved in [Gelfand-Graev-PyatetskiiShapiro, 1969] that an irreducible admissible automorphic representation for G L(2, AQ ) has a factorization into an infinite tensor product of local representations. Proofs of this fundamental result also appeared a little later in 378

10.2 Tensor product of (g, K∞ )-modules and representations

379

[Jacquet-Langlands, 1970], [Godement, 1970]. A vast generalization was later obtained in [Flath, 1979]. The main goal of this chapter is to motivate and present the proofs of these results.

10.2 Tensor product of (g, K ∞ )-modules and representations In Section 2.5 the tensor product of two group representations was introduced. Let’s recall the definition. Definition 10.2.1 (Tensor product of 2 group representations) Let G, G be two groups. Let (π, V ) and (π , V ) be two group representations of the groups G, G , respectively, as in Definition 2.5.1. We may define a map π ⊗ π : G × G → G L(V ⊗ V ) by the rule        π ⊗ π (g, g ) . v ⊗ v := π (g) . v ⊗ π (g ) . v for (g, g ) ∈ G × G and v ⊗ v ∈ V ⊗ V . Then the action of    π ⊗ π (g, g ) on elements of V ⊗ V which are not pure tensors is determined by linearity, 

and π ⊗ π , V ⊗ V is a representation of G × G with space V ⊗ V . It is useful to keep in mind that an action of G × G is essentially the same thing as commuting actions of G and G . More precisely, we have the following. Lemma 10.2.2 Let G, G and H be groups. Then there is a natural one-to-one correspondence between homomorphisms  : G × G → H on the one hand and pairs of homomorphisms φ : G → H and φ : G → H with commuting images on the other, which is given as follows. • Given  : G × G → H, define φ(g) = (g, e) and φ (g ) = (e, g ), where e, e are the identity elements of G, G , respectively. • Given φ : G → H and φ : G → H such that φ(g)φ (g ) = φ (g )φ(g) for all g ∈ G and g ∈ G , define ((g, g )) = φ(g) · φ (g ), where · denotes multiplication in H. The proof is an elementary exercise. The case which is of primary interest to us is when H is the group of linear automorphisms of some vector space V. For example, in Definition 10.2.1, we defined a single homomorphism π ⊗π : G × G → G L(V ⊗ V ). But we could just as well have defined two homomorphisms of G and of G such that the two images in G L(V ⊗ V ) commute.

380

Tensor products of local representations

This latter approach makes the definition of a tensor product of a representation with a (g, K ∞ )-module (as defined in Definition 5.1.4) somewhat more transparent. Next, we consider the tensor product of a (g, K ∞ )-module (as defined in Definition 5.1.4) and a local representation of G L(2, Q p ). Recall that a (g, K ∞ )-module is a complex vector space V∞ with actions   πg : U (g) → End(V∞ ) = set of all linear maps V∞ → V∞ ,   π K∞ : K ∞ → G L(V∞ ) = set of all invertible linear maps V∞ → V∞ . (10.2.3) By abuse of language let us refer to the (g, K ∞ )-module as the local representation π∞ where π∞ = (πg , π K∞ ). For a finite prime p, let V p be a complex vector space and let (π p , V p ) be a representation of G L(2, Q p ) as defined in Section 6.1. Let us call this local representation π p . We want to define the product π∞ ⊗ π p . A natural way to do this is as follows. Definition 10.2.4 (The tensor product of a (g, K ∞ )-module and a local representation of G L(2, Q p )) Let (π∞ , V∞ ) be a (g, K ∞ )-module. In other words, let V∞ be a vector space, and let π∞ = (πg , π K∞ ) be a pair of actions as in (10.2.3), satisfying the conditions of Definition 5.1.4. Let p be a finite prime and let (π p , V p ) be a local representation of G L(2, Q p ) as in Section 6.1. To define the product of (π∞ , V∞ ) and (π p , V p ), we equip the tensor product vector space V∞ ⊗ V p with three actions πg : U (g) → End(V∞ ⊗ V p ), π K ∞ : K ∞ → G L(V∞ ⊗ V p ), π p : G L(2, Q p ) → G L(V∞ ⊗ V p ), given on pure tensors by πg (D) . (v∞ ⊗ v p ) = (πg (D) . v∞ ) ⊗ v p , π K ∞ (k) . (v∞ ⊗ v p ) = (π K∞ (k) . v∞ ) ⊗ v p ,

(∀D ∈ U (g), v∞ ∈ V∞ , v p ∈ V p ), (∀k ∈ K ∞ , v∞ ∈ V∞ , v p ∈ V p ),

π p (g p ) . (v∞ ⊗v p ) = v∞ ⊗(π p (g p )·v p ), (∀g p ∈ G L(2, Q p ), v∞ ∈ V∞ , v p ∈ V p ), and on  of V∞ ⊗ V p by linearity. We shall write π∞ ⊗ π p for the  other elements pair (πg , π K ∞ ), π p .

10.3 Infinite tensor products of local representations

381

10.3 Infinite tensor products of local representations We now pass to the more complex construction of infinite tensor products of group representations. Our main goal is to show that an irreducible admissible automorphic representation for G L(2, AQ ) factors into an infinite tensor product of local representations which are of the type studied in Chapters 6 and 7. This gives a vast generalization of the notion of an Euler product. A sticky point in the theory is that we do not have a group representation at ∞, as we do at each finite prime. Instead, we have a (g, K ∞ )-module as in Definition 5.1.4. It will turn out that an irreducible admissible automorphic representation will factor as an infinite tensor product of local group representations of G L(2, Q p ) (with p < ∞) times a (g, K ∞ )-module. In this section we shall develop the theory needed to study such factorizations.   Definition 10.3.1 (Restricted tensor product of vector spaces) Let Vv v≤∞ be a family of vector spaces indexed by the primes v ≤ ∞. Let S be a finite set of primes. For v ∈ / S, letξv◦ be  a distinguished non-zero   element of Vv . The is the space of all restricted tensor product of Vv v≤∞ with respect to ξv◦ v∈S / finite linear combinations of vectors ξ =

K

ξv

v≤∞

where ξv ∈ Vv for all v, and ξv = ξv◦ for all but finitely many v. Remark It is important to keep in mind that the space obtained actually  and not only on the collection of depends on the choice of vectors ξv◦ v∈S /   spaces Vv v≤∞ . Consider a (g, K ∞ ) × G L(2, Afinite )-module as in Definition 5.1.5. It will be a complex vector space V with actions πg : U (g) → End(V ),

π K∞ : K ∞ → G L(V ),

πfinite : G L(2, Afinite ) → G L(V ), as in Definition 5.1.5. Let π denote this (g, K ∞ ) × G L(2, Afinite )-module. We shall define a factorization of π as an isomorphism with another (g, K ∞ ) × G L(2, Afinite )-module which is realized on a vector space which is an infinite restricted tensor product. In order to define such a (g, K ∞ ) × G L(2, Afinite )module, we require the existence of the following data: (i) A (g, K ∞ )-module (π∞ , V∞ ). (ii) A local representation (π p , V p ) of G L(2, Q p ) for each finite prime p.

382

Tensor products of local representations

(iii) A finite set of primes S containing ∞. / S) where (iv) A distinguished non-zero element ξ p◦ ∈ V p (for p ∈ ξ p◦ is fixed by G L(2, Z p ). (10.3.2) Given this data, we are able to define a (g, K ∞ ) × G L(2, Afinite )-module which is the “product” of the local representations (πv , Vv ) as follows. Definition 10.3.3 (Restricted tensor product of local representations) Let (π∞ , V∞ ) be a (g, K ∞ )-module. Let S be a finite set of primes containing ∞. For each prime p ∈ / S, let (π p , V p ) be a representation of G L(2, Q p ) G L(2,Z p ) such that the space V p of G L(2, Z p )-fixed vectors is non-zero, and let G L(2,Z p ) ◦ ξ p ∈ Vp be a choice of non-zero G L(2, Z p )-fixed vector. For each finite L prime p ∈ S, let (π p , V p ) be a representation of G L(2, Q p ). Let V := v Vv be the restricted tensor product of the vector spaces Vv , as in Definition 10.3.1. Define actions πg : U (g) → End(V ),

π K ∞ : K ∞ → G L(V ),

πfinite : G L(2, Afinite ) → G L(V ) L L as follows. For D ∈ U (g), and v ξv ∈ v Vv , we define     K K ξv = πg (D) . ξ∞ ⊗ ξv . πg (D) .

For k ∈ K ∞ , and

L

v v ξv

π K ∞ (k) .

∈

L

v

K v

v<∞

Vv , we define

    K ξv = π K∞ (k) . ξ∞ ⊗ ξv .

Finally, for afinite ∈ G L(2, Afinite ), and

πfinite (afinite ) .

K v

L

v ξv

ξv = ξ∞ ⊗



∈

L

K

v

v<∞

Vv , we define 

(π p (a p ) . ξ p ) .

p<∞

). Then (π , V ) is a (g, K ∞ ) × G L(2, Afinite )-module Let π = ((πg , π K ∞ ), πfinite as in Definition 5.1.5. This (g, K ∞ ) × G L(2, Afinite )-module is often denoted L v πv .

Remarks Note that, if T is the set consisting of ∞ and all primes p < ∞ such / G L(2, Z p ) then T is finite and, for all p not in the finite set S ∪ T that a p ∈ we have π p (a p ) . ξ p = π p (a p ) . ξ p◦ = ξ p◦ .

10.4 The factorization of unramified irreducible admissible

383

L L

(afinite ) . v ξv is again in v Vv for afinite ∈ G L(2, Afinite ). It is This is why πfinite easy to see that πfinite is a homomorphism of groups G L(2, Afinite ) → G L(V ). We would like to give conditions under which a global representation π factors into a tensor product of local representations πv : K

π∼ =

πv .

(10.3.4)

v≤∞

Let Vv (with v ≤ ∞) be the vector spaces defined in (10.3.2) (i), (ii), (iii) / S) be the distinguished elements in (10.3.2) above, and let ξv◦ (with v ∈ L Vv denote the restricted tensor product of the vector (iv) above. We let v≤∞

spaces Vv (with v ≤ ∞) as in Definition 10.3.1. Assume there exists a linear isomorphism K

Vv → V (10.3.5) L: v≤∞

which satisfies       K K L πg (D) . ξ∞ ⊗ ξv = πg (D) . L ξv , v<∞

v≤∞

      K K ξv = π K∞ (k) . L ξv , L π K∞ (k) . ξ∞ ⊗ v<∞

v≤∞

⎛ ⎞     K K L ⎝ π p (g p ) . ξ p ⊗ ξv ⎠ = πfinite (i p (g p )) . L ξv , v/ =p

v≤∞

(10.3.6) for all

L

ξv ∈

v≤∞

L v≤∞

Vv , for all g p ∈ G L(2, Q p ) ( p < ∞), and for all k ∈ K ∞

and D ∈ U (g). (Here i p : G L(2, Q p ) → G L(2, Afinite ) denotes the homomorphism given by inclusion at the pth place.) One immediately sees that this gives an isomorphism of modules as in Definition 5.1.7. Thus, the existence of the linear isomorphism (10.3.5) which satisfies (10.3.6) establishes the factorization (10.3.4).

10.4 The factorization of unramified irreducible admissible cuspidal automorphic representations A cuspidal automorphic representation (π, V ), as in Definition 5.1.14, determines a local representation at each prime p < ∞ as in Section 6.3. At ∞, the cuspidal automorphic representation (π, V ) determines a (g, K ∞ )-module

384

Tensor products of local representations

in a very similar manner (see also Example 5.2.12). By abuse of language, we may refer to this (g, K ∞ )-module as a local “representation” at ∞. If this local representation is unramified for all v ≤ ∞ as in Definitions 6.2.1, 7.2.1, then we say that (π, V ) is unramified. We shall now consider the factorization of an unramified irreducible admissible cuspidal automorphic representation of G L(2, AQ ). The simplest example is given by the adelic lift f adelic of an even Maass form f of level 1 and weight 0 for S L(2, Z). Define the vector space   N     m c R f adelic g · h   N ∈ N, m  ∈ Z, c ∈ C , (10.4.1) V f :=  =1

where g ∈ G L(2, AQ ) and h  ∈ G L(2, Afinite ) · i ∞ (K ∞ ). The differential operator R  is defined by (5.2.11) and the action of R  on f adelic (g) with g = {g∞ , g2 , . . . } is given by the differential operator acting on g∞ . By Theorem 5.4.2 we see that V f together with the actions (defined in Section 5.1) πg : U (g) → End(V f ) (action by differential operators) π K∞ : K ∞ → G L(V f ) (action by right translation) πfinite : G L(2, Afinite ) → G L(V f ) (action by right translation) (10.4.2) define a cuspidal automorphic representation of G L(2, AQ ). We shall denote this representation as π f , and our goal is to show that π f factors as an infinite tensor product of local representations πf ∼ =

K

πv .

(10.4.3)

v≤∞

We shall now give a simple direct proof of (10.4.3) for the case when f is an even Maass form of level one and weight zero for S L(2, Z), which is an eigenfunction of all the Hecke operators. The first step is to replace the vector space V f by the vector space W f where  W f :=

N =1

c R

m

    W g · h  ; f  N ∈ N, m  ∈ Z, c ∈ C ,  

where g ∈ G L(2, AQ ), h  ∈ G L(2, Afinite ) · i ∞ (K ∞ ), and     1 u g e(−u) du W (g; f ) = f adelic 0 1 Q\AQ

(10.4.4)

10.4 The factorization of unramified irreducible admissible

385

is the global Whittaker function as in Definition 4.10.9. The actions (10.4.2) can be transferred naturally to actions on the Whittaker function. One immediately sees that W f defines a smooth (g, K ∞ ) × G L(2, Afinite )-module (as in Definition 5.1.5). Proposition 10.4.5 The module V f defined in (10.4.1) is isomorphic to the module W f defined in (10.4.4). Here module means (g, K ∞ ) × G L(2, Afinite )module as in Definition 5.1.4. Proof Consider the linear map L : V f → W f given by     1 u φ → W (g; φ) := φ g e(−u) du 0 1 Q\AQ

for φ ∈ V f and g ∈ G L(2, AQ ). If we apply each of the three actions (10.4.2) we see that         1 u πg (Dα ) φ W g; πg (Dα )φ = g e(−u) du 0 1 Q\AQ

  ∀α ∈ g, g ∈ G L(2, AQ ) ,

= πg (Dα )W (g; φ),   W g; π K∞ (k∞ )φ =



   1 π K∞ (k∞ ) φ 0

Q\AQ





φ

=

1 u 0 1



= π K∞ (k∞ )W (g; φ), 



W g; πfinite (h finite )φ =





φ

=



 ∀k∞ ∈ K ∞ , g ∈ G L(2, AQ ) ,

   1 πfinite (h finite ) φ 0

Q\AQ

1 u 0 1



  g e(−u) du

 g · k∞ e(−u) du

Q\AQ



u 1

u 1

  g e(−u) du

 g · h finite e(−u) du

Q\AQ

= πfinite (h finite ) W (g; φ),   ∀h finite ∈ G L(2, Afinite ), g ∈ G L(2, AQ ) . (The first identity consists of differentiating under the integral sign, which is valid because the adelic integral is actually a finite sum of ordinary Riemann integrals of smooth functions over compact sets. The latter two identities are

386

Tensor products of local representations

clear.) Thus L is an intertwining map as in Definition 5.1.7. To complete the proof of Proposition 10.4.5 it is enough to show that L is invertible so that it is, in fact, an isomorphism. By Fourier inversion on Q\AQ as in Theorem 1.8.10 (see also Proposition 4.9.5) we may recover any φ ∈ V f as an infinite linear combination of Whittaker functions:     1 u φ(g) = φ g e(−αu) du. (10.4.6) 0 1 × α∈Q

Q\AQ

Furthermore, 

 φ Q\AQ

1 u 0 1 

 φ

=

  g e(−αu) du

α 0

Q\AQ



=W

α 0

0 1

0 1 



1 α −1 u 0 1

 g; φ .

  g e(−u) du

(10.4.7)

Now, assume that there exist φ1 , φ2 ∈ V f where L(φ1 ) = L(φ2 ). It follows that W (g; φ1 ) = W (g; φ2 ) for all g ∈ G L(2, AQ ), and by (10.4.6), (10.4.7) we see that φ1 = φ2 .  In order to establish a factorization of the type (10.4.3), it is necessary to specify local representations (πv , Wv ) for each prime v ≤ ∞. If v = p is a finite prime, let      yp 0 1 xp rp 0 · k p; f Wp 0 1 0 1 0 rp  1   |y p | p2 · A |y p |−1 e p (x p ), if y p ∈ Z p , p = 0, otherwise, (10.4.8) denote the local Whittaker function as in (4.10.14) for x p ∈ Q p , y p , r p ∈ Q×p , and k p ∈ G L(2, Z p ). Following Definition 6.3.2, we define W p to be the complex vector space of all linear combinations of right translates of W p (g; f ) by elements in G L(2, Q p ). Let (π p , W p ) denote the local representation as in Definition 6.3.3. For v = ∞ we let W∞ (g∞ ; f ) denote the local Whittaker function as in (4.10.14), and define W∞ to be the complex vector space consisting of all finite linear combinations of images of W∞ (g∞ ; f ) under the two actions, of K ∞ by right translation and of U (g) by differential operators. Then we define π∞ = (πg , π K∞ ), where π K∞ is right translation and

10.4 The factorization of unramified irreducible admissible

387

πg is given by differential operators, and we easily check that (π∞ , W∞ ) is a (g, K ∞ )-module. Since the Maass form f is even, of weight zero and of level 1, it follows that f adelic is also invariant on the right by the group  K = K ∞ p G L(2, Z p ). Consequently, each local vector space Wv (for v ≤ ∞) has a distinguished right K v -invariant vector given by Wv (gv ; f ) with gv ∈ G L(2, Qv ). Fix S = {∞} to consist only of the prime at infinity. The reason S is so simple is because we started with a Maass form f of level one. We may then form the restricted tensor product K

Wv (10.4.9) Wtensor := v≤∞

of the vector spaces Wv (v ≤ ∞) as in Definition 10.3.1. By routine computations one may easily show that Wtensor is a (g, K ∞ ) × G L(2, Afinite )-module. The key point we want to highlight immediately is that the factorization of the global Whittaker function  Wv (gv ; f ) (10.4.10) W (g; f ) = v≤∞

given in Theorem 4.10.13 will lead directly to the factorization πf ∼ =

K

πv .

(10.4.11)

v≤∞

In fact, the factorization (10.4.11) is a consequence of the following proposition. Proposition 10.4.12 The module Wtensor defined in (10.4.9) is isomorphic to the module W f defined in (10.4.4). Here module means (g, K ∞ ) × G L(2, Afinite )module as in Definition 5.1.4. Proof We define a linear map L : Wtensor → W f by K v≤∞

where

 v

Wv →



Wv ,

v

Wv : G L(2, A) → C is the function defined by       Wv {gv }v≤∞ := Wv (gv ). v

v≤∞

(Keep in mind that for any fixed g = {gv }v≤∞ , all but finitely many terms in the “infinite” product on the right hand side are equal to one.) The map L is defined on elements of Wtensor which are not pure tensors by linearity. It is clear

388

Tensor products of local representations

that this gives a well defined map, that the image is (all of) W f , and that the conditions of (10.3.6) are satisfied. However, we must prove that the map L is injective. Suppose that it is not. Let W ∈ Wtensor be a non-zero finite sum of pure tensors which is in the kernel of L . Let us say W =

N

W (i) ,

W (i) =

K v≤∞

i=1

Wv(i) , (i = 1, . . . , N ).

There is a finite set SW such that the span of the functions Wv(i) , (i = 1, . . . N ) is one-dimensional whenever v ∈ / S. We assume that W (in the kernel of L) is chosen so that the number of elements in SW is minimal. ) , . . . , Bv(M) for the span of Wv(1) , . . . , Wv(N Fix v0 ∈ S. Choose a basis Bv(1) 0 0 0 0 and rewrite W as ⎛ ⎞ M N K Bv(i)0 ⊗ ⎝ ci, j Wv( j) ⎠ . i=1

j=1

v =/ v0

For all g = {gv }v ∈ G L(2, A) we have ⎛ ⎞ M N  Bv(i)0 (gv0 ) · ⎝ ci, j Wv( j) (gv )⎠ = 0. i=1

j=1

v =/ v0

Varying gv0 while keeping gv (with v =/ v0 ) fixed, we deduce that the function ⎛ ⎞ M N  ⎝ ci, j Wv( j) (gv )⎠ · Bv(i)0 : G L(2, Qv0 ) −→ C i=1

j=1

v =/ v0

is the zero function. By linear independence of the functions Bv(i)0 we deduce   N  ( j) for each i, that ci, j Wv (gv ) = 0 for all g ∈ G L(2, A). This gives j=1

v =/ v0

another element W in the kernel of L with a smaller set SW , contradicting minimality. 

10.5 Decomposition of representations of locally compact groups into finite tensor products In order to address the question of when a global representation factors into infinitely many local representations, we first consider the much simpler problem of decomposing a representation of a locally compact totally disconnected group into a finite tensor product of representations. In this section we will

10.5 Decomposition of representations of locally compact groups 389 prove the following Theorem 10.5.1 (see [Flath, 1979]). We would like to thank Herv´e Jacquet for providing us with a simple direct proof. This section is written in a somewhat different style than previous chapters of this book. In particular, it makes greater use of very general theoretical results whose proofs go beyond the scope of this book, but the reader should have no trouble following the presentation by thinking of specific examples. Recall that a locally compact group is a topological group which is also locally compact as a topological space. It can then be characterized by the fact that the identity element in the group has a compact neighborhood. Representations of locally compact groups were introduced in Definition 2.5.1. A topological space is totally disconnected if the only connected subsets are the empty set and singleton sets. Our applications will be to the groups G L(2, Q p ), where p is a finite prime, and to finite Cartesian products of these groups. It is not difficult to show that these groups are indeed totally disconnected. Assume that each of the totally disconnected groups under consideration has an open maximal compact subgroup, a condition that holds for G L(2, Q p ). The notions of “smooth” and “admissible” representation are defined for arbitrary locally compact totally disconnected groups in the same manner as for G L(2, Q p ): a representation is smooth if it is the union of the fixed spaces of open compact subgroups, and admissible if each of these fixed spaces is finite dimensional. Observe that (π, V ) ∼ = (π , V ) defines an equivalence relation on any set of representations of a group G (or on any set of (g, K ∞ )-modules, etc.). We shall say that two representations are “equivalent” if they are isomorphic to one another. Theorem 10.5.1 (Decomposition theorem for two locally compact totally disconnected groups) Let G 1 , G 2 be locally compact totally disconnected groups with open maximal compact subgroups K 1 , K 2 , respectively. (i) Fix two complex vector spaces V1 , V2 . Let (π1 , V1 ), (π2 , V2 ) be  representations of G 1 , G 2 , respectively. Then admissible irreducible π1 ⊗ π2 , V1 ⊗ V2 is an admissible irreducible representation of G1 × G2. (ii) Fix a complex vector space V . Let (π, V ) be an admissible irreducible representation of G 1 × G 2 . Then there exist two complex vector spaces V1 , V2 and two admissible irreducible representations (π1 , V1 ), (π2 , V2 ) of G 1 , G 2 , respectively, such that π is isomorphic to the tensor product π1 ⊗ π2 . Further, up to isomorphism, π1 and π2 are each uniquely determined.

390

Tensor products of local representations

Proof of Theorem 10.5.1 part (ii) Since (π, V ) is admissible, this implies that if U1 , U2 are open subgroups of K 1 , K 2 , respectively, then    V U1 ×U2 = v ∈ V  π (u) . v = v, (∀u ∈ U1 × U2 ) is finite dimensional. We want to prove that π is a tensor product. Recall that the direct product G 1 × G 2 is just the set of pairs (g1 , g2 ) with gi ∈ G i (i = 1, 2) and product defined componentwise by (g1 , g2 )(g1 , g2 ) = (g1 g1 , g2 g2 ),

  ∀gi , gi ∈ G i (i = 1, 2) .

Define π1 , π2 to be the restriction of the representation π to the groups G 1 , G 2 , respectively. Then if e1 , e2 denote the identity elements in G 1 , G 2 , respectively we may realize π1 , π2 as follows.   π1 (g1 ) := π (g1 , e2 ) ,

  π2 (g2 ) := π (e1 , g2 ) ,

(∀ g1 ∈ G 1 , g2 ∈ G 2 ). (10.5.2) Clearly, the images of π1 and π2 commute with each other and the space V is irreducible in the sense that there is no proper irreducible subspace invariant under π1 and π2 . Note that if U1 , U2 are open subgroups of K 1 , K 2 , respectively, then   V U1 := v ∈ V  π1 (u 1 ) . v = v,

(∀ u 1 ∈ U1 )



is invariant under π2 and 

V U1

U2

= V U1 ×U2 .

We now introduce the important projection operators which form a key ingredient in this proof. To do so, we make use of the fact that any compact topological group has a translation-invariant measure, which is unique up to scalar multiple and called the Haar measure. For G L(2, Z p ), we have introduced this measure, at least on compact open subsets, which is sufficient for integration of locally constant functions. Definition 10.5.3 (Projection operators) Let G 1 , G 2 be locally compact totally disconnected groups with open maximal compact subgroups K 1 , K 2 , respectively. Let (π, V ) be a representation of G 1 ×G 2 and let π1 , π2 be defined by (10.5.2). Let U1 , U2 be any open subgroups of K 1 , K 2 , respectively. Then we define projection operators PUi : V → V Ui (i = 1, 2) by  PUi . v =

Ui

πi (u i ) . v d × u i  , × Ui d u i

(∀ v ∈ V ).

10.5 Decomposition of representations of locally compact groups 391 The integrals in Definition 10.5.3 are as in Definition 1.5.3, except that we use Haar measure on Ui (i = 1, 2) instead of on the locally compact group Q p . It is easy to show that   V U1 ×U2 = PU2 . PU1 . V .

V Ui = PUi . V (i = 1, 2),

Lemma 10.5.4 Let π1 be defined by (10.5.2). Suppose V0 =/ {0} is a subspace of V which is invariant under the action of π1 . Suppose further that V U1 =/ {0}. Then V0U1 =/ {0}. Proof Because of the irreducibility, we have V =



π2 (g2 ) . V0 .

g2 ∈G 2

On the other hand, V U1 = PU1 . V =



  π2 (g2 ) . PU1 . V0 .

g2 ∈G 2

Since the left hand side of the above is non-zero, we must have PU1 . V0 =/ {0}. Thus V0U1 =/ {0}.  Lemma 10.5.5 Let π1 be defined by (10.5.2). Then the space V contains a non-trivial subspace V0 which is invariant and irreducible under the action of π1 , i.e., π1 (g) . V0 ⊆ V0 for all g ∈ G 1 , and no proper non-trivial subspace of V0 has the same property. Proof The proof requires the space U1 HU1 1 of all compactly supported functions f : G 1 → C which are bi-U1 -invariant, i.e., which satisfy f (u 1 g1 u 1 ) = f (g1 ), Now if f ∈ given by

U1

(∀u 1 ∈ U1 , g1 ∈ G 1 ).

HU1 1 , then we may define an operator π1 ( f ) : V → V 

π1 (g1 ) . v f (g1 ) d × g1 ,

π1 ( f ) . v :=

(∀ v ∈ V ).

G1

Then π1 ( f ) acts on V and leaves V U1 ×U2 invariant. By our original assumption, that the representation (π, V ) is admissible, it immediately follows that V U1 ×U2 is a finite dimensional space. In fact, we choose U1 , U2 so that V U1 ×U2 =/ {0}, which we may do because our representation (π, V ) is smooth. Hence, there is a non-trivial subspace W ⊂ V U1 ×U2 which is invariant and irreducible under

392

Tensor products of local representations

HU1 1 , meaning that π1 ( f ) . W ⊆ W ∀ f ∈ non-trivial subset of W has this property. Define

U1

V0 :=



U1

HU1 1 and that no proper

π1 (g1 ) . W.

g1 ∈G 1

Clearly, V0 is invariant under the action of π1 . We first claim that V0U1 = W. Indeed, any vector v ∈ V0 can be written as a finite sum v=

m

π1 (g1i ) . wi

i=1

with wi ∈ W, g1i ∈ G 1 (i = 1, . . . , m). Suppose that v is invariant under U1 . Since wi = PU1 . wi it follows that v = PU1 . v =

m

   PU1 . π1 (g1i ) . PU1 . wi .

i=1

  But PU1 . π1 (g1i ) . PU1 = π1 ( f 1i ) for some f 1i ∈ U1 HU1 1 . (To be precise, f 1i is equal to the characteristic function of the double coset U1 · g1i ·U1 times a scalar which depends on the normalization of the Haar measure.) It follows that the vector v is in W . Now suppose that V1 ⊆ V0 with V1 =/ {0} is invariant under π1 . By Lemma 10.5.4, we have V1U1 =/ {0}. By the observation we just made, V1U1 ⊆ W. Thus V1 contains a vector w ∈ W. Since W is irreducible under U1 U1 H1 it follows that W ⊆ V1 . Then by the definition of V0 we have V0 ⊆ V1 . Hence V1 = V0 .  If V ⊆ V is any subspace which is invariant under the action of π1 , then for each g ∈ G 1 the operator π1 (g) is well defined as an operator on V . In other words, we get a well defined homomorphism G 1 → G L(V ), which we denote again by π1 . Lemma 10.5.6 Let π1 be defined by (10.5.2). There exists an irreducible smooth representation π1 of G 1 on a vector space V1 ⊆ V (with V1 =/ {0}) which satisfies the following property. We have V =

B

Vα ,

(A is an indexing set),

α∈A

where each vector space Vα (with α ∈ A) is invariant under the action of π1 and the representation π1 : G 1 → G L(Vα ) is equivalent to π1 for every α ∈ A.

10.5 Decomposition of representations of locally compact groups 393 Proof It follows from Lemma 10.5.5 that there exists V1 ⊆ V , V1 =/ {0}, such that V1 is invariant and irreducible under the action of π1 . Then the action π1 : G 1 → G L(V1 ) determines a representation of G 1 on the vector space V1 . Let us call this representation π1 . By the irreducibility of V and the fact that π1 , π2 commute it follows that V = π2 (g2 ) . V1 . g2 ∈G 2

Now each subspace π2 (g2 ) . V1 is invariant under the action of π1 , and for each g2 ∈ G 2 , the representation π1 : G 1 → G L (π2 (g2 ) . V1 ) is equivalent to π1 . We have thus proved that we can write V as a sum

Vα

α∈A

where A is a certain indexing set and each Vα is invariant under π1 , and in addition, the representation of G 1 on Vα is equivalent to π1 . Next we show that we can choose a subset A ⊂ A such that the sum α∈A Vα is direct and V =

B

Vα .

α∈A

To show this, consider the set A of all A ⊆ A such that the sum of Vα , (α ∈ A) is direct. The set A is not empty since it contains the singletons {α}. We order it by inclusion. It is clearly an inductive set. Thus it contains a maximal element A. Consider the corresponding direct sum V =

B

Vα .

α∈A

We claim that V = V. Suppose not. Then there is at least one such α0 such that Vα0 is not contained in V . Thus V ∩ Vα0 =/ Vα0 . Since V ∩ Vα0 is invariant and Vα0 is irreducible, we get V ∩ Vα0 = {0}, and we have a direct sum  B

 Vα ⊕ Vα0

α∈A

which contradicts the maximality of A. Thus V = V as claimed.



Conclusion of the proof of theorem 10.5.1 part (ii) Let V2 denote the vector space of all linear maps  : V1 → V such that π1 (g1 ) . (v1 ) = (π1 (g1 ) . v1 )

394

Tensor products of local representations

for all g1 ∈ G 1 and v1 ∈ V1 . Because π1 and π2 commute, the formula (π2 (g2 ) . )(v1 ) = π2 (g2 ) . ((v)) gives a well-defined homomorphism π2 : G 2 → G L(V2 ). We will show that (π, V ) is isomorphic to (π1 ⊗ π2 , V1 ⊗ V2 ). We define a linear map L : V1 ⊗ V2 → V. By the formula L( ⊗ v1 ) = (v1 ) on pure tensors and by linearity elsewhere. It is clear from the definitions that this is an interwining map. To prove that it is a vector space isomorphism, we need to study V2 more closely. For each α in the index set A of Lemma 10.5.6, the representations (π1 , V1 ) and (π1 , Vα ) are isomorphic. Fix one particular isomorphism α : V1 → Vα for each α ∈ A. Schur’s lemma states that the vector space of all intertwining maps V1 → Vα is C · α . It was proved for smooth representations of G L(2, Q p ) in Lemma 6.1.8. The proof extends to any group which has a compact subgroup of countable index. (See also Exercise 10.19.) We will prove A that V2 = α∈A C·α . Fix  ∈ V2 . For each α ∈ A let pα : V → Vα denote the natural projection, coming from the direct sum decomposition given in Lemma 10.5.6. Then pα ◦  = cα · α for some cα ∈ C. We need to prove that cα = 0 for all but a finite number of α. This follows easily from the fact that, for any fixed v1 ∈ V1 , the vector (v1 ) is contained in the sum of a finite number of the vector spaces Vα . Let A

be the finite set such that cα =/ 0 ⇐⇒ α ∈ A

and write  = α∈A

cα α . Then pα ◦ ( −  ) = 0 for all α ∈ A. It follows at once A that  =  . This completes the proof that V2 = α∈A C · α . The fact that L is a vector space isomorphism follows immediately. Irreducibility of (π2 , V2 ) follows easily from that of (π, V ). We need to prove that (π2 , V2 ) is smooth and admissible. Take  ∈ V2 and v1 ∈ V1 . Then (v1 ) ∈ V, so (v1 ) is fixed by U1 × U2 for some open compact subgroups U1 ⊆ G 1 , U2 ⊆ G 2 . But then (v1 ) is fixed by the same U2 for every other v1 ∈ V1 , because the images of π1 and π2 in G L(V ) commute, and (π1 , V1 ) is irreducible. This allows us to deduce smoothness of (π2 , V2 ) from that of (π, V ). On the other hand, if 1 , . . . , k are linearly independent elements of (V2 )U2 for some U2 , and v1 ∈ V1U1 for some U1 , then 1 (v1 ), . . . , k (v1 ) are linearly independent elements of V. This allows us to deduce admissibility of (π2 , V2 ) from that of (π, V ). Suppose that (π1 , V1 ) is any other irreducible representation of G 1 which has a non-trivial intertwining map  into (π, V ). Then pα ◦  : V1 → Vα is non-trivial for some α. But then pα ◦  must be an isomorphism because

10.5 Decomposition of representations of locally compact groups 395 V1 and Vα are irreducible. This proves that the first factor (π1 , V1 ) is unique up to isomorphism. Uniqueness of the second factor follows by symmetry.  Proof of Theorem 10.5.1 part (i) We first prove that (π1 ⊗ π2 , V1 ⊗ V2 ) is admissible. The fact that it is smooth follows easily from the smoothness of π1 and π2 . Fix open subgroups U1 , U2 of the maximal compact subgroups K 1 , K 2 , respectively. We must prove that V U1 ×U2 is of finite dimension, where V := V1 ⊗ V2 . We shall accomplish this by proving that V U1 ×U2 = V1U1 ⊗ V2U2 . That V1U1 ⊗ V2U2 ⊆ V U1 ×U2 is obvious. Take v ∈ V U1 ⊗U2 , and write v = v11 ⊗ v21 + · · · + v1k ⊗ v2k , with v11 , . . . , v1k ∈ V1 and linearly independent, and v21 , . . . , v2k ∈ V2 . Because v is fixed by the action of U2 , and the vectors v1i are linearly independent, it follows that v2i ∈ V2U2 for each i. Now rewrite v as v = w11 ⊗ w21 + · · · + w1m ⊗ w2m , with w1i ∈ V1 and w2i ∈ V2U2 , and linearly independent. Then the fact that v is fixed by U1 now implies that w1i ∈ V1U1 for each i. This completes the proof that V U1 ×U2 = V1U1 ⊗ V2U2 . Admissibility of (π1 ⊗ π2 , V ) follows. Let v1 ∈ V1 (with v1 =/ 0) and v2 ∈ V2 (with v2 =/ 0). By irreducibility of π1 , the space V1 is generated by v1 under the action of π1 . That is, any vector in V1 can be written as   ci π1 (g1i ) . v1 , g1i ∈ G 1 , ci ∈ C, for all i ∈ I , i∈I

where I is some indexing set. We conclude that V1 ⊗ V2 is generated by the pure tensor v1 ⊗ v2 under the action of π1 ⊗ π2 . This does not quite suffice to prove that the tensor product space is irreducible because not every vector in the tensor product space is a pure tensor. At any rate, V is finitely generated. It follows as in Proposition 6.1.16 (using Zorn’s lemma) that V has an irreducible quotient, V , say. Write π 1 for the action of G 1 on this quotient. By Theorem 10.5.1 part (ii) we have that V can be written as a tensor product π1 ⊗ π2 where (π1 , V1 ) and (π2 , V2 ) are irreducible representations of G 1 and G 2 respectively. Now we claim that π1 is isomorphic to π1 . Indeed, it follows as in Lemma 10.5.6 that V is a direct sum of subspaces Vα such that (π 1 , Vα ) ∼ = (π1 , V1 ). On the other hand, for any fixed v2 ∈ V2 , the map v1 → v1 ⊗ v2 is an intertwining operator of representations of G 1 , from V1 into V. Because pure tensors generate, the image of the composite map V1 → V → V is non-trivial. Hence, for some α, the composite map V1 → V → V → Vα is non-trivial.

396

Tensor products of local representations

This gives a non-zero morphism of π1 into π1 which, by irreducibility, is then an isomorphism. Likewise π2 is isomorphic to π2 . Finally V ∼ = V . Since

V is irreducible, we are done. 

10.6 The spherical Hecke algebra for G L(2, Q p ) It is a classical theorem of Hecke (see [Goldfeld, 2006, Theorem 3.12.8]) that if an automorphic form f for S L(2, Z) is an eigenfunction of certain averaging operators (now called Hecke operators) then the L-function associated to f has an Euler product. We shall now study the Hecke operators in the context of adelic automorphic forms. Accordingly we will define for each prime p an algebra of Hecke operators associated to G L(2, Q p ) which is generated by elements which have properties that are very similar to the classical Hecke operators. The infinite tensor product of these algebras will give rise to the global Hecke algebra which will play a crucial role in proving that every cuspidal automorphic admissible irreducible representation of G L(2, AQ ) factors as an infinite tensor product of local representations. Definition 10.6.1 (The space K H K of bi K -invariant functions) Recall that K = G L(2, Z p ). Let K H K denote the C-vector space of all locally constant compactly supported bi K -invariant functions f : G L(2, Q p ) → C which satisfy f (k1 gk2 ) = f (g),

(∀ g ∈ G L(2, Q p ), k1 , k2 ∈ K = G L(2, Z p )).

The space K H K is just a vector space over C. We would like to define a way of multiplying two vectors in this space so that K H K becomes an associative algebra. Accordingly we define the convolution product (denoted ∗) where  f 1 ∗ f 2 (g) :=

f 1 (gh −1 ) f 2 (h) d × h,

G L(2,Q p )

 ∀ f1 , f2 ∈

K

 H K , g ∈ G L(2, Q p ) . (10.6.2)

Proposition 10.6.3 (The space K H K is a convolution algebra) The space K K H , defined in Definition 10.6.1, is an algebra under convolution as defined in (10.6.2). Proof We must check that if f 1 , f 2 ∈ K H K then f 1 ∗ f 2 ∈ K H K . This is easily done by the following simple computation. Let g ∈ G L(2, Q p ) and k1 , k2 ∈ K . Then

10.6 The spherical Hecke algebra for GL(2, Qp )  f 1 ∗ f 2 (k1 gk2 ) = f 1 (k1 gk2 h −1 ) f 2 (h) d × h

397

G L(2,Q p )



f 1 (k1 gh −1 ) f 2 (hk2 ) d × h

= G L(2,Q p )



f 1 (gh −1 ) f 2 (h) d × h = f 1 ∗ f 2 (g).

= G L(2,Q p )

Next we wish to prove that convolution is associative, i.e., that ( f1 ∗ f2 ) ∗ f3 = f1 ∗ ( f2 ∗ f3 ) for all f 1 , f 2 , f 3 ∈

K

H K . This is another simple computation:

( f 1 ∗ f 2 ) ∗ f 3 (g)  × ( f 1 ∗ f 2 )(gh −1 = 1 ) f 3 (h 1 ) d h 1 

G L(2,Q p )



 −1 × f 1 (gh −1 1 h 2 ) f 2 (h 2 ) d h 2

= 

G L(2,Q p )



G L(2,Q p )

 −1 f 1 (gh −1 3 ) f 2 (h 3 h 1 )

= G L(2,Q p )

 = 

G L(2,Q p )

= G L(2,Q p )

G L(2,Q p )

f 1 (gh −1 3 )·

· f 3 (h 1 ) d × h 1

×

d h3

· f 3 (h 1 ) d × h 1

(where h 3 = h 2 h 1 ) 

 G L(2,Q p )

× f 2 (h 3 h −1 1 ) f 3 (h 1 ) d h 1

d ×h3

× f 1 (gh −1 3 ) · ( f 2 ∗ f 3 )(h 3 ) d h 3 = f 1 ∗ ( f 2 ∗ f 3 )(g).

Further, we must exhibit an identity element in the algebra. Accordingly, we consider the characteristic function of K , i.e., the function  1 K (g) := One easily checks that 1 K ∈ K K H . 

K

1,

if g ∈ K ,

0,

otherwise.

H K and f ∗ 1 K = 1 K ∗ f = f for all f ∈

Definition 10.6.4 (The spherical Hecke algebra for G L(2, Q p )) The spherical Hecke algebra for G L(2, Q p ) is defined to be the convolution algebra K H K introduced in Proposition 10.6.3.

398

Tensor products of local representations

Remark The algebra K H K is called spherical because in the real case (for the group G L(2, R)) a function f : G L(2, R) → C which is bi K -invariant with respect to K = O(2, R) is called a radially symmetric function (spherical function). The most important property of the spherical Hecke algebra for G L(2, Q p ) is that it is a commutative algebra. This observation is central in the work of Hecke [Hecke, 1937a,b]. The proof of Theorem 10.6.5 follows [Gelfand, 1950]. Theorem 10.6.5 (The spherical Hecke algebra is commutative) The Hecke algebra K H K , as defined in Definition 10.6.5, is a commutative algebra. Proof We begin by defining an involution ι : K H K → ι

f (g) := f ( t g),

(∀ f ∈

K

K

H K where

H K , g ∈ G L(2, Q p )),

and where t g denotes the transpose of the matrix g. Since t (g1 · g2 ) = t g2 · t g1 for all g1 , g2 ∈ G L(2, Q p ) it follows that for all f 1 , f 2 ∈ K H K , and all g ∈ G L(2, Q p ), we have

ι

( f 1 ∗ f 2 )(g) = f 1 ∗ f 2

t  g =

 f1

t

 g · h −1 f 2 (h) d × h

t

 h −1 · g f 2 (h) d × h

G L(2,Q p )



ι

=

f1

G L(2,Q p )



ι

=

f 1 (h · g) f 2

t

 h −1 d × h

G L(2,Q p )



=

ι

  f 1 (h · g) · ι f 2 h −1 d × h

G L(2,Q p ) = ι f 2 ∗ ι f 1 (g).

(10.6.6)

On the other hand, by the Cartan decomposition for G L(2, Q p ) (see Proposition 4.2.3) a basis for the space K H K of bi K -invariant functions may be given  m cosets  m by the locally constant functions with support on the double p 0 p 0 K 0 pn K . It is easily verified that every double coset K 0 pn K is invariant under the action of transposition of matrices from which one sees at once that ι f = f for all f ∈ K H K . It then follows from (10.6.6) that

10.6 The spherical Hecke algebra for GL(2, Qp ) f 1 ∗ f 2 = ι ( f 1 ∗ f 2 ) = ι f 2 ∗ ι f 1 = f 2 ∗ f 1 . This proves that commutative algebra. 

399 K

H K is a

Given a smooth representation (π, V ) of G L(2, Q p ) as in Definition 6.1.1, we will construct an action of the spherical Hecke algebra on the K -fixed vectors (10.6.7) V K = {v ∈ V | π (k) . v = v}, which is induced from the action of π on V . If (π, V ) is an irreducible representation then the action of the spherical Hecke algebra on V K will also turn out to be irreducible, and then using the commutativity of the spherical Hecke algebra it will follow that V K is at most a one-dimensional space. This conclusion will play a vital role in the proof that a global admissible irreducible cuspidal representation factors as an infinite restricted tensor product of local representations. Definition 10.6.8 (The linear maps π ( f )) Let (π, V ) be a smooth representation of G L(2, Q p ) as in Definition 6.1.1. Let K H K be the spherical Hecke algebra as defined in Definition 10.6.4. Then for all f ∈ K H K we define a linear map: π ( f ) : V → V , given by  f (g) π (g) . v d × g, (∀ v ∈ V ). π ( f ) . v := G L(2,Q p )

Remark Because f is locally constant and compactly supported, it is actually a finite linear combination of characteristic functions. This, together with the fact that π is smooth, allows one to easily show that the integral we consider above is actually a finite sum. We hope there is no confusion between the notation π ( f ) and π (g). It should always be clear from the context that π ( f ) refers to the action of the Hecke algebra while π (g) refers to the action of the group G L(2, Q p ). Proposition 10.6.9 (π ( f ) maps V → V K ) Fix a smooth representation (π, V ) of G L(2, Q p ). For f ∈ K H K , let π ( f ) be the linear map defined in Definition 10.6.8. Then for all v ∈ V we have π ( f ) . v ∈ V K , with V K as in (10.6.7).   Proof We must show that π (k) . π ( f ) . v = π ( f ) . v for all k ∈ K , v ∈ V. This follows immediately from the computation:    f (g) π (kg) . v d × g π (k) . π ( f ) . v = G L(2,Q p )



= G L(2,Q p )

  f k −1 g π (g) . v d × g =



G L(2,Q p )

f (g) π (g) . v d × g.

400

Tensor products of local representations

Note that all we used here is the fact that f (kg) = f (g) for all k ∈ K . We did not need the K -invariance on the right.  Next, we show that the set of linear maps   π K H K := {π ( f ) | f ∈

K

HK }

(10.6.10)

is itself an algebra which acts on V K . The key point is that for f 1 , f 2 ∈ K H K we have   π ( f1 ∗ f2 ) . v = π ( f1 ) . π ( f2) . v , (∀ v ∈ V ). (10.6.11) The proof of (10.6.11) is immediate from the following computation:  π ( f1 ∗ f2) . v = f 1 ∗ f 2 (g) · π (g) . v d × g G L(2,Q p )



,

G L(2,Q p )

,

−1

f 1 (gh ) f 2 (h) d h · π (g) . v d × g

= 

G L(2,Q p )

×

f 1 (g) f 2 (h) d h · π (gh) . v d × g

= G L(2,Q p )

G L(2,Q p )

,



×

f 2 (h) · π (h) . v d h d × g.

f 1 (g) π (g) .

=

×

G L(2,Q p )

G L(2,Q p )

The identity (10.6.11) shows that the mapping f → π ( f ) is an action of the spherical Hecke algebra K H K on V (i.e., it is a ring homomorphism K H K → End(V )). Theorem 10.6.12 (Unramified admissible irreducible representations of G L(2, Q p ) have a one-dimensional space of K -fixed vectors) Let (π, V ) be an unramified admissible irreducible representation of G L(2, Q p ) as in Section 6.1. Set K = GL(2, Z p ) and let V K be the space of K -fixed vectors as in (10.6.7). Then dim V K = 1. The spherical Hecke algebra K H K , defined in Definition 10.6.4, acts on V K by scalar multiplication and determines a character ξ : K H K → C satisfying π ( f ) . v ◦ = ξ ( f )v ◦ ,

H K , v ◦ ∈ V K ),   ∀ f1 , f2 ∈ K HK , ξ (1 K ) = 1, ξ ( f 1 ∗ f 2 ) = ξ ( f 1 ) · ξ ( f 2 ),   ∀ f 1 , f 2 ∈ K H K , c1 , c2 ∈ C . ξ (c1 f 1 + c2 f 2 ) = c1 ξ ( f 1 ) + c2 ξ ( f 2 ),   Remark Theorem 10.6.12 tells us that dim V K = 1. Hence (up to a constant multiple) there is a unique non-zero K fixed vector v ◦ ∈ V K , cf. Exercise 6.15. (∀ f ∈

K

10.6 The spherical Hecke algebra for GL(2, Qp )

401

Proof Since the representation (π, V ) is assumed to be unramified, it follows from 6.2.1 that there exists v ◦ ∈ V K with v ◦ =/ 0. The algebra   K Definition K π H defined in (10.6.1), (10.6.2), (10.6.10), acts on V K by Proposition 10.6.9. Claim: For every v ∈ V K , there exists f ∈ that π ( f ) . v ◦ = v.

K

H K , depending on v, such

Fix v ◦ and let v ∈ V K be arbitrary. Since π is irreducible, it follows that there exist g i ∈ G L(2, Q p ), ci ∈ C (i = 1, 2, . . . , r ) such that r

ci π (g i ) . v ◦ = v.

i=1

Next, for i = 1, 2, . . . , r, define  −1  , Vol K g i K i f (g) := 0,

if g ∈ K g i K , otherwise.

The claim follows from the following computation: r

ci π ( f i ) . v ◦ =

i=1

r i=1

=

r i=1

 k∈K



=

π (k1 g i k2 ) . v ◦ d × k1 d × k2

ci K K

π (g) . v ◦ d × g

g∈K g i K

 

π (k) .

=



ci Vol(K g i K )

,

r

ci π (g ) . v i

◦

d ×k

i=1

π (k) . v d × k = v.

k∈K

We have, therefore, shown that the space V K is invariant and irreducible  under the action of the spherical Hecke algebra π K H K . K Now, π is admissible, so by Definition  K K 6.1.4 the space V is finiteK dimensional. It follows that π ( f ) ∈ π H has an eigenvector in V . Since the claim holds for any non-zero v ◦ ∈ V K , we may now assume that π ( f ) . v ◦ = λv ◦ , for some λ ∈ C. Because π K H K is commutative, it follows that the λ eigenspace of π ( f ) is preserved by the action of π (F), for every F ∈ K H K . We deduce that the λ eigenspace of π ( f ) is all of V K , because the action of K H K on V K is irreducible. As f ∈ K H K was arbitrary, we have shown that every element of K H K acts on V K by a scalar. Since the

402

Tensor products of local representations

action is irreducible we see that V K must be one-dimensional. This determines a character ξ : K H K → C which is defined by π ( f ) . v ◦ = ξ ( f )v ◦ for all f ∈ K HK .  Definition 10.6.13 (Spherical representation of G L(2, Q p )) An admissible irreducible unramified representation of G L(2, Q p ), as in Section 6.1, is frequently called a spherical representation. Definition 10.6.14 (The spherical Hecke character which is associated to a spherical representation of G L(2, Q p )) Let (π, V ) be a spherical representation as in Definition 10.6.13. Then the character: ξ : K H K → C, introduced in Theorem 10.6.12 is termed the spherical Hecke character associated to the representation π. We shall now prove that a spherical representation of G L(2, Q p ) is uniquely determined by its spherical Hecke character. Proposition 10.6.15 (Spherical representations of G L(2, Q p ) are uniquely determined by their spherical Hecke characters) Let (π1 , V1 ), (π2 , V2 ) be spherical representations of G L(2, Q p ) as in Definition 10.6.13. Assume that π1 , π2 have the same spherical Hecke character as in Definition 10.6.14. Then π1 ∼ = π2 . Proof The proof has two main steps. First we show how the values of a particular matrix coefficient can be computed from the spherical Hecke character. Then we appeal to Proposition 8.1.10, which states that any irreducible admissible representation of G L(2, Q p ) may be realized as right translation on its own space of matrix coefficients. First, arguing as in the proof of (8.1.7) we show that the contragredient, V1 ) of (π1 , V1 ) is also spherical. More precisely, let P K : V1 → V1K be the (( π1 , ( projection operator defined by P K . v1 = K π (k) . v1 d × k for all v ∈ V. (A V1 valued integral which is actually a finite sum.) Fix a non-zero G L(2, Z p )-fixed vector v1◦ ∈ V1 . By Theorem 10.6.12, V1K = C · v1◦ . Let  : C · v1◦ → C be the function c · v1◦ → c, (∀c ∈ C). Then  ◦ P K is a linear function V1 → C. Furthermore, it follows from a simple change of variables that V1 , and that ( V1 is  ◦ P K is fixed by K . It follows that  is an element of ( spherical. By Theorem 10.6.12, the spherical Hecke algebra given in Definition 10.6.4 acts on v1◦ by the spherical Hecke character, introduced in Theorem 10.6.12 (see also definition 10.6.14). That is π ( f ) . v1◦ = ξ ( f ) · v1◦ for all f ∈ K H K . Let β(g) := π1 (g) . v1◦ , .

10.7 Initial decomposition of admissible modules

403

Note that β(I2 ) = (v1◦ ) = 1. (Here I2 indicates the 2 × 2 identity matrix.) For any g ∈ G, define f g ∈ K H K by  f g (h) =

Vol(K · g · K )−1 ,

if h ∈ K · g · K ,

0,

otherwise.

Consider the linear map π1 ( f g ) : V1 → V1 defined in Definition 10.6.8. For any v1 ∈ V1 , it is given by   1 π1 ( f g ) . v1 = π (h) . v d × h = π (k1 gk2 ) . v1 d × k1 d × k2 . Vol(K · g · K ) K ·g·K K K

Hence   ξ ( f g ) = π1 ( f g ) . v1 ,  =

8 9 π1 (k1 gk2 ) . v1◦ ,  d × k1 d × k2

K K

 =

8 9 π1 (g) . v1◦ , ( π1 (k1−1 ) d × k1 = β(g).

K

Now let W denote the space of all matrix coefficients βv1 , , (v1 ∈ V1 ), where, for each v1 ∈ V1 , the function βv1 , is given by βv1 , (g) = π (g) . v1 , . It was shown in Proposition 8.1.10 that (π1 , V1 ) is isomorphic to (π , W ), where π is action by right translation, i.e., π (g)βv1 , (h) := βv1 , (hg) ∀g, h ∈ G L(2, Q p ), v1 ∈ V1 . Since (π1 , V1 ), is irreducible, (π , W ) is also irreducible. It follows that W may also be expressed as  5   ci β(ggi )  gi ∈ G L(2, Q p ) . W := i

Repeating the same arguments with (π1 , V1 ) replaced by (π2 , V2 ), we find that (π2 , V2 ) is also isomorphic to (π , W ). This completes the proof. 

10.7 Initial decomposition of admissible (g, K ∞ ) × G L(2, Afinite )-modules Our main goal in this chapter is to show that an arbitrary irreducible admissible (g, K ∞ ) × G L(2, Afinite )-module is isomorphic to an infinite tensor product as in definition 10.3.3. This, in effect, splits the (g, K ∞ ) × G L(2, Afinite )-module into “factors” indexed by the primes v ≤ ∞. The factor corresponding to v = ∞ is a little different from the others. It is clear that Theorem 10.5.1 will be extremely useful for pulling out a “factor” corresponding to some finite prime p. In this section we prove an analogous result which will allow us to

404

Tensor products of local representations

pull out the factor corresponding to v = ∞. In order to set up notation, we introduce the following notions. Definition 10.7.1 (Smooth or admissible representation of G L(2, Afinite )) A representation πfinite : G L(2, Afinite ) → G L(V ) of G L(2, Afinite ) on a vector space V is said to be smooth if it is the union of the subspaces

V K := {v ∈ V | π (k ) . v = v, ∀k ∈ K } as K ranges over compact open subgroups of G L(2, Afinite ). Such a represen tation is said to be admissible if, in addition, V K is finite dimensional for each compact open subgroup K of G L(2, Afinite ). Definition 10.7.2 (The tensor product of a (g, K ∞ )–module and a representation of G L(2, Afinite )) Let (π∞ , V∞ ) be a (g, K ∞ )-module, as in Definition 5.1.4, where π∞ = (πg , π K∞ ). Let (πfinite , Vfinite ) be a representation of G L(2, Afinite ). Let V = V∞ ⊗ Vfinite . Define actions πg : U (g) → End(V∞ ⊗ Vfinite ),

π K ∞ : K ∞ → G L(V∞ ⊗ Vfinite ),

πfinite : G L(2, Afinite ) → G L(V∞ ⊗ Vfinite ),

given on pure tensors by πg (D) . (v∞ ⊗ vfinite ) = (πg (D) . v∞ ) ⊗ vfinite , π K ∞ (k) . (v∞ ⊗ vfinite ) = (π K∞ (k) . v∞ ) ⊗ vfinite ,

πfinite (gfinite ) . (v∞ ⊗ vfinite ) = v∞ ⊗ (πfinite (gfinite ) . vfinite ),

for all D ∈ U (g), k ∈ K ∞ , gfinite ∈ G L(2, Afinite ), v∞ ∈ V∞ , vfinite ∈ Vfinite , and    on other elements of V∞ ⊗ Vfinite by linearity. Then πg , π K∞ , πfinite , V is a (g, K ∞ ) × G L(2, Afinite )-module as in definition 5.1.5. Theorem 10.7.3 (Initial factorization of a (g, K ∞ ) × G L(2, Afinite )module) Let (π, V ) be a (g, K ∞ ) × G L(2, Afinite )-module (as in Definition 5.1.5), which is both irreducible and admissible (see Definitions 5.1.6,

, V∞ ), (Definition 5.5.4). Then there exists an irreducible (g, K ∞ )-module (π∞

5.1.4) where π∞ = (πg , π K∞ ) and an irreducible admissible representation (πfinite , Vfinite ) of G L(2, Afinite ), such that (π, V ) is isomorphic to the (g, K ∞ ) ×

, V∞ ), and G L(2, Afinite )-module defined in Definition 10.7.2. The factors (π∞ (πfinite , Vfinite ) are unique up to isomorphism. Conversely, the product of an irreducible (g, K ∞ )-module and an irreducible admissible representation of G L(2, Afinite ) is an irreducible admissible (g, K ∞ ) × G L(2, Afinite )-module.

10.7 Initial decomposition of admissible modules

405

Proof The proof is accomplished by mirroring the steps of the proof of Theorem 10.5.1.  We  first prove that there is a subspace V0 ⊆ V such that  πg , π K∞ , V0 is an irreducible (g, K ∞ )-module. This is much easier to show in this case. By repeating arguments from Chapter 7, we easily show that the subspace generated by a vector v under the actions of K ∞ and U (g) is an irreducible (g, K ∞ )-module if any of the following conditions occurs: • v is in the kernel of πg (R) or πg (L), • K ∞ = O(2, R) acts on v trivially, or via det,   cos θ sin θ . v = eiθ · v for all θ ∈ R and v is an eigenvector of • πK∞ − sin θ cos θ π K∞ (δ1 ) ◦ πg (L), and that every irreducible admissible (g, K ∞ ) × G L(2, Afinite )-module contains a vector satisfying one of these conditions. Next, we prove that B Vα , (A is an indexing set), V = α∈A

where ((πg , π K∞ ), Vα ) (with α ∈ A) is an irreducible (g, K ∞ )-module for every α ∈ A, and they are all isomorphic to each other. The proof is exactly as in Lemma 10.5.6.

, V∞ ) to be any one of the spaces Vα and we take Vfinite to be Now we take (π∞

the space of all intertwining maps of (g, K ∞ )-modules (π∞ , V∞ ) → (π∞ , V ). A

can be realized as α∈A C · α , where α : V∞ → Vα As a vector space, Vfinite is any particular choice of isomorphism. This follows as before from Schur’s lemma, which was proved for (g, K ∞ )-modules in Lemma 8.5.11. Further may be equipped with an action of G L(2, Afinite ) via more, the space Vfinite

(gfinite ) . )(v∞ ) = πfinite (gfinite ) . ((v∞ )). (πfinite

Then

→ (v∞ )  ⊗ v∞

⊗ extends to an isomorphism of (g, K ∞ ) × G L(2, Afinite )-modules V∞

Vfinite → V.

, Vfinite ) is smooth and admissible. The proofs are We need to prove that (πfinite the same as in Section 10.5, except that we consider vectors on which S O(2, R)   cos θ sin θ . v = eikθ · v instead of vectors which are fixed by acts by π K∞ − sin θ cos θ a compact open subgroup U1 .

, V∞ ) is unique up to isomorphism is as in Section 10.5: The proof that (π∞



if (π∞ , V∞ ) is any other irreducible (g, K ∞ )-module which maps injectively into (π, V ), then composition with the projection pα for a suitable α ∈ A

, V∞ ). However, we can no longer deduce yields an isomorphism with (π∞

406

Tensor products of local representations

, Vfinite ) “by symmetry,” because our situation here is not uniqueness of (πfinite symmetric. On the other hand, recall from Chapter 7 that in an irreducible (g, K ∞ )-module, the space       cos θ sin θ

 ikθ . v = e · v, (∀ θ ∈ [0, 2π )) V∞,n = v ∈ V∞  π K∞ − sin θ cos θ 

is actually one-dimensional, when it is non-trivial. Let vn be some particular

with generator. Then vfinite → vn ⊗ vfinite defines an isomorphism of Vfinite       cos θ sin θ  . v = eikθ · v, (∀ θ ∈ [0, 2π )) . Vn := v ∈ V  π K∞ − sin θ cos θ  (10.7.4)

, Vfinite ) is unique up to isomorphism. This proves that (πfinite



, V∞ ) and (πfinite , Vfinite ), two irreducible and Now, suppose we start with (π∞ admissible representations, and let (π, V ) denote the product as in Definition 10.7.2. Then (π, V ) is easily seen to be the direct sum of the spaces Vn as

, V∞ ), as defined in (10.7.4), where n ranges over the set of K ∞ -types of (π∞ in Definition 7.5.6. Furthermore, each of the spaces Vn is isomorphic to Vfinite , as a representation of G L(2, Afinite ). This proves that any invariant subspace which intersects the space Vn in a non-zero vector contains it. On the other hand, if vn ∈ Vn is non-zero, then it is a simple matter to write down a word in πg (R), πg (L) and π K∞ (δ1 ) (see Proposition 7.4.8) which maps vn to a non-zero

, V∞ ). Thus an element of Vn for any other n in the set of K ∞ -types of (π∞ invariant subspace which contains one Vn must contain them all. This proves that the product is irreducible. 

10.8 The tensor product theorem Let (π, V ) be an irreducible admissible automorphic representation of G L(2, AQ ) as in Definitions 5.1.12, 5.5.4. The tensor product theorem states that π can be factored as a restricted infinite tensor product of local representations as in Definition 10.3.1. To state this in a precise form, it is necessary to define the notions of ramified and unramified for automorphic representations. Let i p : G L(2, Q p ) → G L(2, AQ ) denote the embedding at the prime p defined by

 (10.8.1) i p (g p ) := I2 , . . . , I2 , g p , I2 , I2 , . . . , !"# position p

for g p ∈ G L(2, Q p ), and where I2 denotes the two-by-two identity matrix. Definition 10.8.2 (Ramified or unramified at p) Let (π, V ) be an irreducible, admissible (g, K ∞ ) × G L(2, Afinite )-module, as in Definitions 5.1.5,

10.8 The tensor product theorem

407

5.1.6, 5.5.4. For every finite prime p, let K p = i p (G L(2, Z p )). The representation π is termed unramified at p if there exists a non-zero 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 V K p = {0}. Definition 10.8.3 (Unramified (g, K ∞ ) × G L(2, Afinite )-module) Let (π, V ) be an irreducible, admissible (g, K ∞ ) × G L(2, Afinite )-module, as in Definitions 5.1.5, 5.1.6, 5.5.4. The representation (π, V ) is termed unramified if V K =/ {0},  where K = K ∞ · p G L(2, Z p ). This is equivalent to the definition given previously in Section 10.4. Lemma 10.8.4 An admissible (g, K ∞ ) × G L(2, Afinite )-module as in Definition 5.5.4, can only be ramified, as in Definition 10.8.2, at finitely many primes p. With these preliminaries in place, we can now state and prove the tensor product theorem. Theorem 10.8.5 (Tensor product theorem) Let (π, V ) denote an irreducible admissible (g, K ∞ )×G L(2, Afinite )-module, as in Definitions 5.1.5, 5.1.6, 5.5.4. Let {q1 , . . . , qm } be the finite set of primes where π is ramified as in Definition 10.8.2, and let S = {∞, q1 , . . . , qm }. There exists • an irreducible admissible (g, K ∞ )-module (π∞ , V∞ ) as in (10.2.3), • an irreducible admissible representation (π p , V p ) of G L(2, Q p ) for each finite prime p, • a non-zero G L(2, Z p ) fixed vector v ◦p ∈ V p for each finite prime p ∈ S, such that π ∼ =

K

πv ,

v≤∞

the restricted tensor product with respect to {vv◦ }v∈S , as in Definition 10.3.3. Proof of Theorem 10.8.5 By Theorem 10.7.3, we have an isomorphism between V and V∞ ⊗ Vfinite for some admissible representation (πfinite , Vfinite ) of G L(2, Afinite ). We need to show that (πfinite , Vfinite ) is isomorphic to an infinite restricted tensor product over the finite primes.  Define K S := p∈ S K p , where K p = i p (G L(2, Z p )) as in Definition 10.8.2. KS Define a projection operator P K S : Vfinite → Vfinite , by

408

Tensor products of local representations  S P K . v = πfinite (k) . v d × k, S

KS

K is invariant under the action of G L(2, Qqi ) for for all v ∈ Vfinite . Clearly, Vfinite i = 1, 2, . . . , m, since the action of G L(2, Qqi ), (i = 1, 2, . . . , m) commutes with the action of G L(2, Q p ) with p ∈ S. We adopt the notation

S

G =



G L(2, Q p ),

p∈ S

GS =

m 

G L(2, Qqi ),

i=1

 where denotes the restricted product with respect to the groups G L(2, Z p ) (the same type of product used to define the adeles, see Section 1.3). Each of these groups has a natural embedding into G L(2, Afinite ) generated by the embeddings i p of the individual groups G L(2, Q p ). Definition 10.8.6 (S-spherical Hecke algebra) Let S be a set of primes which contains ∞. We define the S-spherical Hecke algebra to be the C-vector space KS KS H , defined as  ⎧ ⎫  f is locally constant and of compact support, ⎬ ⎨  f : G S → C   ⎭,  ⎩  f (k1 gk2 ) = f (g), ∀ k1 , k2 ∈ K S , g ∈ G S which forms an algebra under convolution (denoted ∗), where    f 1 (gh −1 ) f 2 (h) d × h, ∀g ∈ G S . f 1 ∗ f 2 (g) := GS S

S

The proof that K H K forms an algebra under convolution is the same as the proof of Proposition 10.6.3 and is left to the reader. S S KS → For each f ∈ K H K , we may also define a linear map πfinite ( f ) : Vfinite S K Vfinite by    KS πfinite ( f )(g) = . f (g) πfinite (g) . v d × g, ∀v ∈ Vfinite GS



 S S S S We define πfinite K H K to be the set of all πfinite ( f ) with f ∈ K H K . Then, as in the proof of (10.6.11), we have   S S (10.8.7) ∀ f1 , f2 ∈ K HK . πfinite ( f 1 ∗ f 2 ) = πfinite ( f 1 ) . πfinite ( f 2 ), S

K Lemma 10.8.8 No proper non-trivial subspace of Vfinite is fixed by both G S , KS KS and H .

10.8 The tensor product theorem

409

K K , with v ◦ =/ 0. Let v ∈ Vfinite be arbitrary. Proof of Lemma 10.8.8 Fix v ◦ ∈ Vfinite By the irreducibility of the global representation, it follows that we may choose r ∈ N, g1 , . . . , gr ∈ G S , and g1 , . . . , gr ∈ G S such that r πfinite (gi ) . πfinite (gi ) . v ◦ v= S

i=1

=P

,

KS

.

r

S

πfinite (gi ) . πfinite (gi ) .

 P

KS

.v

◦



i=1

=

r i=1

πfinite (gi ) . P K . πfinite (gi ) . P K . v ◦ . !" #  S S

Lemma 10.8.9 The algebra πfinite

S

S

∈ πfinite



KS

HK

K

S



HK

 is commutative.

Proof of Lemma 10.8.9: This will follow from the commutativity of the local spherical Hecke algebras K H K proved in Theorem 10.6.5. For each prime p ∈ / S the inclusion i pS : G L(2, Q p ) %→ G S induces an S S algebra homomorphism K p H K p → K H K . The characteristic function of K p · g p · K p maps to the characteristic function of K S · i pS (g p ) · K S , etc. Denote this map by I pS .  S  S Clearly, πfinite K H K is spanned by the operators which are associated to the characteristic functions of individual double cosets K S · g · K S , g ∈ G S . For any fixed g ∈ G S there is a finite set S of primes, disjoint from S such / S . Then it is easily seen that πfinite (1 K S ·g·K S ) is the that g p ∈ K p for p ∈ (convolution) product of the operators I pS (1 K p ·g p ·K p ) with p ranging over the elements of S and the product taken in any order. Thus the images of the various local spherical Hecke algebras generate the S-spherical Hecke algebra. Further, each local algebra is commutative, and they clearly commute with each other. This completes the proof of Lemma 10.8.9. 

K K is the union of the spaces Vfinite where K ranges over all The space Vfinite the compact open subgroups of G L(2, Afinite ) containing K S . Each of the subS S K spaces Vfinite is finite dimensional, and is preserved by the action of K H K . It  S  S is clear that πfinite K H K commutes with the action of G L(2, Qqi ), (i = 1, 2, . . . , m). By Lemma 10.8.9 it also commutes with itself. It follows KS that Vfinite contains a non-zero simultaneous eigenspace of all of the operaS S tors in K H K , and that this eigenspace is also preserved by the action of KS , G L(2, Qqi ), (i = 1, 2, . . . , m). It follows that this eigenspace is all of Vfinite  S  K KS acts by scalar multiplication. H i.e., that πfinite S

410

Tensor products of local representations S

K is irreducible as a representaFrom this we may deduce that the space Vfinite S S tion of the finite product G S , without considering the action of K H K . AdmisKS as a representation of G S follows easily from admissibility of sibility of Vfinite Vfinite as a representation of G L(2, Afinite ). By Theorem 10.5.1, there exist complex Vq1 , Vq2 , . .. , Vqm , and    vector spaces admissible irreducible representations πq1 , Vq1 , πq2 , Vq2 , . . . , πqm , Vqm of the locally compact groups G L(2, Qq1 ), G L(2, Qq2 ), . . . G L(2, Qqm ), respectively, such that   m K KS ∼ Vqi , Vfinite = i=1

as representations of G S . Now let p be a prime which is not in S. We need to determine the factor at p in our factorization of Vfinite . This may be accomplished by repeating the arguments above with S replaced by T := S ∪ { p}. We obtain KT

Vfinite

 m K

∼ =

 Vq i

⊗ Vp ,

(10.8.10)

i=1

for some irreducible admissible representations (πq i , Vq i ) of the primes qi , i = 1 to m, and some irreducible admissible representation (π p , V p ) of G L(2, Q p ). Observe that  T K p KS K = Vfinite . Vfinite And that the K p -invariant subspace of the product on the right hand side of (10.8.10) is   m K K

Vqi ⊗ V p p . i=1 K

S

K =/ {0}. Hence V p p =/ {0}. By Theorem 10.6.12, This is non-zero, because Vfinite K it is one-dimensional. Fix a non-zero element v ◦p of V p p . Then v → v ⊗ v ◦p defines an isomorphism of representations of G S

 m K i=1

 Vq i

→

 m K

 Vq i

⊗

K Vp p

KS ∼ ∼ = Vfinite =

i=1

 m K

 Vqi

.

i=1

The uniqueness part of Theorem 10.5.1 now implies that (πqi , Vqi ) ∼ = (πq i , Vq i ) for each i. Now define U to be the infinite tensor product U =

K

v

Vv ,

10.8 The tensor product theorem

411

defined with respect to the vectors {v ◦p } p∈S / chosen above. We may define an infinite product representation πU of G L(2, Afinite ) on U as in Definition 10.3.3 with the place ∞ omitted. We wish to define an isomorphism  : U → Vfinite of representations of the S KS group G L(2, Afinite ). We begin with an isomorphism  S : U K → Vfinite given as S K follows. The space U is equal to ⎞ ⎛ ⎞ ⎛ K Kp K

⎝ Vq ⎠ ⊗ ⎝ Vp ⎠ p∈S /

q∈S

⎫⎞ ⎛⎧ ⎛ ⎞ ⎛ ⎞  ⎨ K ⎬ K  = Span ⎝ ⎝ vq ⎠ ⊗ ⎝ v ◦p ⎠  vq ∈ Vq ∀ q ∈ S ⎠ . ⎩ ⎭  p∈S /

q∈S

 KS

We know that Vfinite

∼ =

L

 Vq . Fix some particular isomorphism

q∈S

⎛  : ⎝

K

⎞ K Vq ⎠ → Vfinite , S

q∈S

and define

⎛⎛ ⎞ ⎛ ⎞⎞ ⎛ ⎞ K K K  S ⎝⎝ vq ⎠ ⊗ ⎝ v ◦p ⎠⎠ =  ⎝ vq ⎠ . p∈S /

q∈S

q∈S

Now fix a set T of primes which contains S. Arguing as above, we  see that  L T T K Vfinite and U K are both isomorphic, as representations of G T , to Vp . p∈T

Hence they are isomorphic to each other. Since each is irreducible, the isomorT phism is unique up to scalar. Thus there is a unique isomorphism T : U K → S KT such that the restriction of T to U K is  S . It follows at once that if Vfinite T S ⊆ T ⊆ T then the restriction of T to U K is T . Observe that U=



UK

T

S⊂T

and

Vfinite =



T

K Vfinite .

S⊂T

Hence we may define a linear isomorphism  : U → Vfinite by letting (u) := T (u) for T large enough. (This is well defined by the last sentence of the previous paragraph.) We claim that (πU (g) . u) = πfinite (g) . (u),

(∀g ∈ G L(2, Afinite ), u ∈ U ).

(10.8.11)

412

Tensor products of local representations

Given any g ∈ G L(2, Afinite ), and any u ∈ U, we may choose T such that T u ∈ U K and g ∈ G T · K T . Condition (10.8.11) then follows, because  is T given by T on U K , and T is an isomorphism of representations G T , and K T acts trivially on both sides. This completes the proof.  Remarks Note that we obtain an isomorphism with the restricted tensor ◦ product defined relative to {v ◦p } p∈S / for any choice of vectors v p . In many appli◦ cations, there is a clear choice for v p . For example, the isomorphism between an infinite product of local Whittaker models and a global Whittaker model is only given by pointwise multiplication of functions if we take W p◦ to be normalized so that its value on K p is 1. Next we want two complements to this result. Theorem 10.8.12 (Uniqueness of factors) Suppose that (π∞ , V∞ ) and

, V∞ ) are two irreducible (g, K ∞ )-modules and that, for each finite prime (π∞ p, (π p , V p ) and (π p , V p ) are two irreducible admissible representations of G L(2, Q p ). Suppose further that there exist finite sets S, S containing ∞ such that K / S), (V p ) K p =/ {0}, ( p ∈ / S ), V p p =/ {0}, ( p ∈ and that the infinite product (g, K ∞ ) × G L(2, Afinite )-modules K

Vv

K

and

v≤∞

v≤∞

Vv ,

K

/ S and v p ,◦ ∈ (V p ) K p , p ∈ / (defined relative to some choices of v ◦p ∈ V p p , p ∈

S ) are irreducible and isomorphic to one another. Then (πv , Vv ) ∼ = (πv , Vv ) for each v.

, V∞ ) and that the Proof It follows from Theorem 10.7.3 that (π∞ , V∞ ) ∼ = (π∞ representations of G L(2, Afinite ) on the spaces

K

Vv

K

and

v<∞

v<∞

Vv ,

are isomorphic. Fix T which contains both S and S . Then the isomorphism K

Vv ∼ =

v<∞

K v<∞

Vv ,

of representations of G L(2, Afinite ) induces an isomorphism K p∈T

 Vp ∼ =

K

v<∞

K T Vv

 ∼ =

K

v<∞

K T Vv

∼ =

K p∈T

V p ,

10.9 The Ramanujan and Selberg conjectures for GL(2, AQ )

413

of representations of G T . The fact that (π p , V p ) ∼ = (π p , V p ) for all p ∈ T now follows from Theorem 10.5.1 (ii). Since T was an arbitrary set containing S ∪ S , it may be chosen to contain any particular prime p, so that (π p , V p ) ∼ = (π p , V p ) for all p.  The previous two theorems establish a global analogue of Theorem 10.5.1 (ii). We would also like the global analogue of Theorem 10.5.1 (i). This is given by the next theorem. Theorem 10.8.13 (All products of irreducibles are irreducible) Suppose that (π∞ , V∞ ) is an irreducible (g, K ∞ )-module and that, for each finite prime p, (π p , V p ) is an irreducible admissible representation of G L(2, Q p ). Suppose further that there exists a finite set S containing ∞ such that Kp

Vp

=/ {0},

(p ∈ / S),

K

and, for each p ∈ / S, fix v ◦p =/ 0 in V p p . Then the infinite product (g, K ∞ ) × G L(2, Afinite )-module defined in Definition 10.3.3 is irreducible. Proof By Theorem 10.7.3, this reduces to showing that the representation of L V p is irreducible. G L(2, Afinite ) on the restricted tensor product Vfinite := Take v, v ∈ Vfinite . We need to show that v ∈ πfinite (g) . v.

p<∞

(10.8.14)

g∈G L(2,Afinite )

Choose T such that v and v both lie in ⎛ ⎞ ⎛ ⎞ K K Kp T K Vfinite =⎝ Vq ⎠ ⊗ ⎝ Vp ⎠ . p∈S /

q∈T Kp

are all one-dimensional by Theorem  10.6.12. It follows that L is isomorphic to the finite tensor product Vq as a representation of

The spaces V p T

K Vfinite

q∈T

the group G T . This is an irreducible representation of G T by Theorem 10.5.1 (i). Hence, (10.8.14) holds, with the sum taken only over the subgroup G T ⊂ G L(2, Afinite ). 

10.9 The Ramanujan and Selberg conjectures for G L(2, AQ ) In [Ramanujan, 1916] it was conjectured that the Fourier coefficients τ (n) of the cusp form of weight 12: (z) = e2πi z

∞ ∞   24 = τ (n)e2πinz , 1 − e2πinz n=1

n=1

(z ∈ C, #(z) > 0),

414

Tensor products of local representations

satisfy the bound |τ ( p)| ≤ 2 p 11/2 for any prime p. This is now known as the Ramanujan conjecture. Ramanujan’s conjecture was further generalized by Petersson to arbitrary holomorphic cusp forms for congruence subgroups of S L(2, Z), i.e., of any weight and level. The Ramanujan-Petersson conjecture (for holomorphic cusp forms) has been proved by Deligne (see [Deligne, 1971]) as a consequence of his proof of the Riemann hypothesis for zeta functions associated to algebraic varieties [Deligne, 1974]. The Ramanujan conjecture is still unproved for non-holomorphic Maass cusp forms. The Selberg conjecture [Selberg, 1965], [Sarnak, 1995] states that a Maass form of weight zero, and arbitrary level, and character, as in Definition 3.5.7, has Laplace eigenvalue greater than or equal to 1/4. This conjecture is still unproved. It was originally thought that these two conjectures of Ramanujan and Selberg had nothing to do with each other. We now know that this is false. The modern theory of automorphic representations shows that, in fact, the two conjectures are essentially the same once viewed in the proper light. The Ramanujan conjecture is really a conjecture about p-adic representations which are local components of a global adelic cuspidal automorphic representation. The Selberg conjecture is really a conjecture about (g, K ∞ )modules which are local components of a global adelic cuspidal automorphic representation. Conjecture 10.9.1 (Ramanujan conjecture for G L(2, AQ )) Let π denote an automorphic cuspidal representation of G L(2, AQ ), as defined in Definition 5.1.14, where K πv , π ∼ = v≤∞

as in the tensor product Theorem 10.8.5. Then for each v ≤ ∞ the representation πv is tempered, as in Definitions 9.2.7, 9.4.7. The connection between the original Ramanujan conjecture going back to 1916 and the modern version above can be established by relating the adelic lift of classical Maass forms of arbitrary weight, level and character as given in Section 4.12 with Proposition 9.2.8 (the characterization of the irreducible tempered representations of G L(2, Q p )). In a similar manner, the connection between the original Selberg eigenvalue conjecture and Conjecture 10.9.1 can be established via Proposition 9.4.10 (the characterization of irreducible tempered (g, K ∞ )-modules). We shall return to this topic when we discuss local L-functions in Chapter 11.

Exercises for Chapter 10

415

Exercises for Chapter 10 10.1 Prove Lemma 10.2.2. 10.2 Give an example of an adelic automorphic form that cannot be factored as a product of local functions. 10.3 Prove that G L(2, Q p ) is a totally disconnected locally compact topological group. Also, show that G L(2, Z p ) is a maximal compact subgroup, and that it is open. Does your proof work for G L(n, Q p )? 10.4 Give an example of a totally disconnected locally compact topological group with no maximal compact subgroup. 10.5 Fix prime numbers , p (not necessarily distinct). This exercise guides you through a direct proof of Theorem 10.5.1 in the special case G 1 = × Q×  and G 2 = Q p . (a) Let ψ : Q×p → C× be a continuous character. Prove that the map π : Q×p → G L(1, C) defined by π (g).v = ψ(g)v yields an irreducible admissible representation of Q×p on C. (b) Let (π, V ) be an irreducible admissible representation of Q×p . Prove that V is 1-dimensional. (Compare with Exercise 10.6.) Conclude that the irreducible admissible representations of Q×p are in one-toone correspondence with the continuous characters of Q×p . × (c) Prove that the irreducible admissible representations of Q×  ×Q p are in one-to-one correspondence with pairs (ψ , ψ p ), where ψ (resp. × ψ p ) is a continuous character of Q×  (resp. of Q p ). 10.6* Let p be a finite prime. Show that every smooth irreducible representation of Q×p is 1-dimensional, and hence admissible. (Compare with Theorem 6.1.11.) 10.7 For each g ∈ G L(2, Q p ), prove that the double coset K g K is compact open. 10.8 Let f ∈ K H K . Prove that there exist complex constants c1 , . . . , cn and n ci 1 K gi K . elements g1 , . . . , gn ∈ G L(2, Q p ) such that f = i=1 10.9 If f 1 , f 2 ∈ supported.

K

H K , show that the convolution f 1 ∗ f 2 is compactly

10.10 Prove that an element of the spherical Hecke algebra K H K is deter pm 0  mined by its values on matrices of the form 0 pn for integers m ≤ n.

416

Tensor products of local representations

10.11 Proposition 10.6.9 shows that a smooth representation (π, V ) of G L(2, Q p ) induces a map π : K H K → End(V K ) given by f → π ( f ). Prove the following assertions about this map: (a) For all c ∈ C and f 1 , f 2 ∈ K H K , we have π (c f 1 + f 2 ) = c π ( f 1 ) + π ( f 2 ). (b) For all f 1 , f 2 ∈ K H K , we have π ( f 1 ∗ f 2 ) = π ( f 1 ) · π ( f 2 ). (c) If idV K is the identity operator on V K , then π (1 K ) = idV K . Conclude that π is a homomorphism of C-algebras with unit. (d) Assume (π, V ) is irreducible and admissible. Then the map π on K K H is always surjective, but never injective. 10.12 Let (π, V ) be a spherical representation of G L(2, Q p ) with spherical character ξ . Define β : G L(2, Q p ) → C by β(g) =

ξ (1 K g K ) Vol(K g K )

(g ∈ G L(2, Q p )).

Prove that β is locally constant and bi K -invariant. Show also that  ξ( f ) = f (g)β(g)d × g ( f ∈ K H K ). G L(2,Q p )

(Compare with the proof of Proposition 10.6.15.) 10.13 Let (s1 , s2 ) be a pair of complex numbers and let V p (s1 , s2 ) be the associated unramified principal series representation of G L(2, Q p ) as in Definition 6.5.4. Prove that the function β given in Exercise 10.12 is not compactly supported. (See the proof of Proposition 10.6.15 for other descriptions of β.) 10.14 Let (π, V f ) be the automorphic representation arising from a Maass newform f of type ν, weight k, and character χ for 0 (N ) as in (5.4.1). (See Definition 3.11.9 as well.) Prove that π is ramified precisely at the primes dividing N . In particular (π, V f ) is unramified if and only if f is of level 1. 10.15 Prove Lemma 10.8.4: An irreducible admissible (g, K ∞ ) × G L(2, Afinite )-module can be ramified at only finitely many primes p. 10.16 Let G 1 and G 2 be locally compact totally disconnected groups with open maximal compact subgroups K 1 and K 2 , respectively. Suppose (π1 , V1 ) and (π2 , V2 ) are admissible irreducible representations of G 1 and G 2 , respectively. Prove that the tensor product representation (π1 ⊗ π2 , V1 ⊗ V2 ) has a non-zero K 1 × K 2 -fixed vector if and only if (πi , Vi ) has a K i -fixed vector for each i = 1, 2.

Exercises for Chapter 10

417

10.17 Let (π, V ) be an irreducible admissible (g, K ∞ ) × G L(2, Afinite )module. Prove that (π, V ) is unramified at every finite prime p if and  only if V K finite = 0, where K finite = p G L(2, Z p ). (The point is to verify the agreement of the local and global definitions of unramified.) 10.18 In this exercise, we look at how the restricted tensor product depends subtly on the choices of distinguished vectors. Let {Vv }v≤∞ be a family of vector spaces indexed by the primes. Fix finite sets of primes S and T , each of which contains v = ∞. For each v ∈ S (resp. v ∈ T ) L choose a non-zero vector ζv◦ ∈ Vv (resp. τv◦ ∈ Vv ). Define V = Vv L (resp. W = Vv ) to be the restricted tensor product with respect to the collection {ζv◦ }v∈ S (resp. {τv◦ }v∈T ) as in Definition 10.3.1. (a) Show that V and W are isomorphic as abstract vector spaces. (b) If ζv◦ = τv◦ for all but finitely many v ∈ S ∪ T , show that V = W . That is, prove that the two restricted tensor products yield the same L subspace of the full tensor product Vv . (c) Construct an example to show that V need not equal W in general. 10.19 Let G be a locally compact, totally disconnected group with an open maximal compact subgroup. Prove that any intertwining map V → V, where V is an admissible representation of G, is given by multiplication by a scalar.

11 The Godement-Jacquet L-function for G L(2, AQ )

11.1 Historical remarks The analytic theory of L-functions associated to modular forms was developed by Hecke (see [Hecke, 1936]), and later extended to non-holomorphic automorphic forms in [Maass, 1946, 1949]. The analytic continuation and functional equation of such an L-function was obtained by taking the Mellin transform of an automorphic form f (z) and applying the modular relation z → −z −1 as in [Riemann, 1859]. This approach was further generalized to the adelic representation theoretic setting in [Gelfand-Graev-Pyatetski-Shapiro, 1969], [Jacquet-Langlands, 1970]. A simple algebra is an algebra A which contains no non-trivial two-sided ideals and for which there exists elements a, a ∈ A for which aa =/ 0. It was proved in [Wedderburn, 1907] that every simple algebra of finite degree n is isomorphic to the algebra of n × n matrices with entries in a division ring. Since the foundational papers of [Tate, 1950], [Iwasawa, 1952], on the analytic continuation and functional equation of L-functions for G L(1, AQ ) (see Section 2.2, 2.3), it has been expected that there should be a generalization to L-functions of simple algebras over Q. This was done in [Fujisaki, 1958, 1962] for L-functions with abelian characters. The work of Fujisaki includes earlier constructions of zeta functions in [Hey, 1929] and [Eichler, 1938]. A theory of Iwasawa-Tate type for non-abelian characters was laid out in [Godement, 1958/1959] and extended in [Tamagawa, 1963] who also obtained the theory of the Euler product by showing that the global L-function is a product of local L-functions. In [Godement-Jacquet, 1972], [Jacquet, 1979], the earlier theory in [Godement, 1958/1959] and [Tamagawa, 1963] was further generalized and elegantly recast in the modern setting of automorphic representations. The analytic continuation and functional equation of an L-function associated to 418

11.2 The Poisson summation formula for GL(2, AQ )

419

an automorphic form on G L(n, AQ ) was derived directly from the Poisson summation formula for G L(n, AQ ) exactly as in Tate’s thesis. The GodementJacquet method [Godement-Jacquet, 1972], [Jacquet, 1979], is based on the theory of matrix coefficients of automorphic representations (see Chapter 8) and the tensor product Theorem 10.8.5. In this chapter we will present the Godement-Jacquet proof of the holomorphic continuation and functional equation of an L-function associated to an adelic cusp form on G L(2, AQ ). Our exposition follows [Jacquet, 1996].

11.2 The Poisson summation formula for G L(2, AQ ) The aim of this section is to generalize the adelic Poisson summation formula given in (1.9.1). To state and prove the Poisson summation formula for G L(2, AQ ) we require some definitions. In order to motivate these definitions, note that the adelic Poisson summation formula is really a result about the additive group of the ring AQ . It is then used to obtain information about the multiplicative group G L(1, AQ ) = A× Q of this same ring. To develop an analogous theory for G L(2), it is necessary to find a ring whose multiplicative group is G L(2, AQ ). It is clear that the desired ring is simply the ring of all 2 × 2 adelic matrices. Definition 11.2.1 (The ring Mat(2, R)) For any ring R, define Mat(2, R) to be the ring of all 2 × 2 matrices with entries in R, under the usual addition and multiplication of matrices. If a = {a∞ , a2 , . . . , a p . . . }, b = {b  ∞ , b2 , . . . }, c = {c∞ , c2 , . . . }, and a b d = {d∞ , d2 , . . . } are adeles, then c d ∈ Mat(2, AQ ) may be identified with       5 a∞ b∞ a2 b2 ap bp , , ... , ... , c∞ d∞ c2 d2 cp dp which is in the restricted product of the groups {Mat(2, Qv ) | v ≤ ∞} with respect to the compact subgroups {Mat(2, Z p ) | p a finite prime }. The ring Mat(2, Q) will also be identified with its image under the diagonal embedding into Mat(2, AQ ). That is: 5      a b  Mat(2, Q) := i diag  a, b, c, d ∈ Q , c d          a b where i diag ac db := , ac db , ac db , . . . . c d Definition 11.2.2 (Bruhat-Schwartz function on Mat(2, AQ )) A complex valued function  : Mat(2, AQ ) → C is said to be a Bruhat-Schwartz function if it is a finite sum of functions of the form    (i) (m ), ∀ m = {m , m , . . . , m , . . . } ∈ Mat(2, A ) , (i) (m) = v ∞ 2 p Q v v

The Godement-Jacquet L-function for GL(2, AQ )

420

where, for each i = 1, 2, 3, . . . , (i) has rapid • (i) ∞ : Mat(2, R) → C is smooth and  (g∞ )   decay at ±∞ in a∞ b∞ each of the variables a∞ , b∞ , c∞ , d∞ , for g∞ = c∞ d∞ ∈ Mat(2, R);

• (i) p : Mat(2, Q p ) → C is locally constant and compactly supported for each finite prime p; • (i) p is the characteristic function of Mat(2, Z p ) for almost all primes p. Theorem 11.2.3 (The Poisson summation formula for Mat(2, AQ )) Let  be a Bruhat-Schwartz function as in Definition 11.2.2. Let e : AQ → C be the additive character as in Definition 1.7.1, and let Mat(2, Q) be as in Definition 11.2.1. Then we have the identity



(ξ ) =

ξ ∈Mat(2,Q)

(ξ ) 

ξ ∈Mat(2,Q)

 is given by where the Fourier transform   



α γ

β δ





    

:=

p r

q s



  e − pα − qγ − rβ − sδ

AQ AQ AQ AQ

· dp dq dr ds,    α β for all γ δ ∈ Mat(2, AQ ). In the above, each integral AQ is an adelic integral as in Definition 1.7.5.     p q α β = pα + qγ + rβ + sδ. Remark Note that Tr r s γ δ Proof Define  1

a c

b d





 

=

p c

b d

 · e(− pa) d x.

AQ

  α β Write ξ ∈ Mat(2, Q) as γ δ . Then, for each fixed β, γ , δ ∈ Q, the function   x β  γ δ , (x ∈ AQ ) is an adelic Bruhat-Schwartz function as defined in Definition 1.7.4, and its Fourier transform, defined as in Definition 1.7.7, is   y β , (y ∈ AQ ). By the Poisson summation formula (1.9.1), it follows 1 γ δ that    α β  α β .  = 1 γ δ γ δ α∈Q

α∈Q

11.2 The Poisson summation formula for GL(2, AQ )

421

Summing over β, γ , and δ yields



(ξ ) =

ξ ∈Mat(2,Q)

1 (ξ ).

ξ ∈Mat(2,Q)

Now, the formula which defines the function 1 may be described as taking the Fourier transform of  “in the variable a.” Repeating these steps and taking the Fourier transform in the remaining variables b, c, and d, it may be deduced that



(ξ ) =

ξ ∈Mat(2,Q)

ξ ∈Mat(2,Q)



=



1 (ξ ) =

2 (ξ )

ξ ∈Mat(2,Q)



3 (ξ ) =

ξ ∈Mat(2,Q)

4 (ξ ),

ξ ∈Mat(2,Q)

where  2  3  4

a c a c a c

b d b d b d





  

= AQ AQ



p c



   

= AQ AQ AQ



q d



    

=

p r

 · e(−ap − bq) dp dq, q d p r

 · e(−ap − bq − cr ) dp dq dr, q s

 · e(−ap − bq − cr − ds)

AQ AQ AQ AQ

· dp dq dr ds. We hope the reader is not confused by “ds” being used in two different senses (ξ ) is simply 4 (t ξ ). This completes the in the last formula. Furthermore,  proof.  For our later purposes it is convenient to break the sums occurring on both sides of the Poisson summation formula (Theorem 11.2.3) into three pieces according as the rank of the matrix ξ ∈ Mat(2, Q) is 0, 1, or 2. Recall that the rank of a matrix ξ ∈ Mat(2, Q) is the dimension of the space {ξ · x | x ∈ Q2 }. The rank of ξ will be 2 ifandonly if ξ is an invertible matrix, i.e., ξ ∈ G L(2, Q). Only the zero matrix 00 00 will have rank 0. The matrices ξ ∈ Mat(2, Q) with rank 1 are all of the form  ξ = X

1 0

0 0

 Y,

(X, Y ∈ G L(2, Q)).

(11.2.4)

The Godement-Jacquet L-function for GL(2, AQ )

422

It follows that we may rewrite the Poisson summation formula of Theorem 11.2.3 in the form   0 0 (ξ ) + (ξ ) +  0 0 (11.2.5) ξ ∈G L(2,Q) ξ ∈Mat(2,Q) rank(ξ )=1



=

(ξ ) + 

ξ ∈G L(2,Q)



(ξ ) +   



ξ ∈Mat(2,Q) rank(ξ )=1

0 0

0 0

 .

Actually, we shall need a variant of (11.2.5) for our applications which we present in the form of a proposition. Proposition 11.2.6 (General Poisson summation formula for G L(2, AQ )) Let  be a Bruhat-Schwartz function as in Definition 11.2.2. Let e : AQ → C be given as in Definition 1.7.1. Let G 1 = {g ∈ G L(2, A) | |det(g)| = 1}, 1 and Mat(2,  Q) be as in Definition 11.2.1. Then for h 1 , h 2 ∈ G L(2, Q)\G and t 0 τ = 0 t with t > 0, we have





h −1 2 ξ τ h1

ξ ∈G L(2,Q)

+







h −1 2 ξ τ h1



 + 

ξ ∈Mat(2,Q) rank(ξ )=1

⎡ ⎢ = t −4 ⎢ ⎣





  −1  h −1  h2 + 1 ξτ

ξ ∈G L(2,Q)



0 0 0 0



−1 (h −1  h2) 1 ξτ

ξ ∈Mat(2,Q) rank(ξ )=1

 +



0 0

⎤  0 ⎥ ⎥. 0 ⎦

Proof Fix h 1 , h 2 ∈ G 1 , t > 0, and a Bruhat-Schwartz function  as in Definition 11.2.2, and define a new function 1 : Mat(2, 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 (h −1 1 (m) = t −4   h 2 ), 1 mτ

(∀m ∈ Mat(2, AQ )).

11.3 Haar measure

423

p q Let x = r s ∈ Mat(2, AQ ) and define d x := dp dq dr ds. It is left as an exercise that         (g · x · h) d x = | det g|−2 · | det h|−2 · (x) d x, AQ AQ AQ AQ

AQ AQ AQ AQ

for all g, h ∈ G L(2, AQ ), and any function  : Mat(2, AQ ) → C, such that these integrals are defined (including Bruhat-Schwartz functions as defined in Definition 11.2.2). In other words, the “Jacobian” of the change of variables x → g −1 · x · h −1 is | det g|−2 · | det h|−2 . Now, for m ∈ Mat(2, Q),         1 (m) =   h −1 2 xτ h 1 e − Tr(x · m) d x. AQ AQ AQ AQ

We make the simple change of variables x → h 2 xτ −1 h −1 1 , and exploit the fact −1 −1 −1 −1 that Tr(h 2 · x · τ h 1 m) = Tr(x · h 1 mτ · h 2 ) to obtain 1 (m) = t −4 

   

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

AQ AQ AQ AQ

The general Poisson summation formula follows immediately from (11.2.5), but with the function 1 instead of . 

11.3 Haar measure In order to discuss the global zeta integral for G L(2, AQ ), it will be useful to have an alternate expression for the matrix coefficients of cuspidal automorphic representations, defined in Definition 9.5.12. This also provides an opportunity to discuss some general notions regarding invariant measures. We have already encountered invariant measures in several specific examples: see Example 1.5.4, Definition 1.5.7, Definition 6.9.5, Proposition 8.6.1. Following [Hewitt-Ross, 1963], we define a ring of subsets of a set X as a collection C of subsets of X, such that A ∪ B and A ∩ (X − B) are in C for all A, B ∈ C. We also define a measure on X as a function μ : C → [0, ∞], where C is a ring of subsets of X, such that μ(∅) = 0 and μ

∞  i=1

 Ai

=

∞

μ(Ai ),

i=1

whenever Ai ∈ C for all positive integers i and

∞ i=1

Ai ∈ C.

424

The Godement-Jacquet L-function for GL(2, AQ )

Now suppose that the set X is equipped with an action of some group G. Then, roughly, a measure is said to be invariant if it is preserved by the action of G on X. An important example is when the set X is the group G itself. There are three main actions of a group G on itself: by left translation, by right translation, and by conjugation. We shall not deal with conjugationinvariant measures here. The Haar measures considered in Example 1.5.4, Definition 1.5.7, Definition 6.9.5, Proposition 8.6.1, etc., are left and right translation invariant measures. Note that, a priori, there is no reason why a measure which is invariant under left translation should also be invariant under right translation. Definition 11.3.1 (Ring of Borel sets) If X is a topological space the ring B(X ) of Borel sets on X is the smallest collection of subsets of X such that: • every open subset of X is in B(X ), ∞  Ai ∈ B(X ), whenever Ai ∈ (X ) for i = 1, 2, . . . , • i=1

• A ∩ (X − B) ∈ B(X ) whenever A, B ∈ B(X ). Definition 11.3.2 (Left/right Haar measure) Let G be a locally compact group. A left Haar measure, μ : B(G) → [0, ∞] is a measure, defined on the ring of Borel sets as in Definition 11.3.1, such that: • • • •

no open set has measure 0, no compact set has measure ∞, μ(U ) = sup{μ(K ) | K is compact, and U ⊃ K }) for all open sets U, μ(A) = inf({μ(U ) | U is open and U ⊃ A}), for all A ∈ B(G), and, finally, μ satisfies the invariance property μ({g · x | x ∈ S}) = μ(S),

(∀g ∈ G, S ∈ B(G)).

(11.3.3)

A right Haar measure is defined in the same way, except that instead of (11.3.3), it satisfies μ({x · g | x ∈ S}) = μ(S),

(∀g ∈ G, S ∈ B(G)).

(11.3.4)

Theorem 11.3.5 (Existence and uniqueness of Haar measures) Let G be a locally compact group. Then there exist left and right Haar measures B(G) → [0, ∞]. Any two left Haar measures on G are proportional, as are any two right Haar measures on G. Proof See [Hewitt-Ross, 1963], (15.8).



It is clear that the left and right Haar measures on a group will coincide if the group happens to be abelian. But even when G is not abelian, they may coincide. In this case, a measure on G is a left Haar measure, if and only if

11.4 The global zeta integral for GL(2, AQ )

425

it is a right Haar measure, and may be referred to simply as a Haar measure without ambiguity. Definition 11.3.6 (Unimodular group) A locally compact group G is said to be unimodular if its left Haar measures are also right Haar measures, as in Definition 11.3.2. Definition 11.3.7 (Haar measure on a unimodular group) Let G be a unimodular locally compact group. A Haar measure on G is simply a left or right Haar measure as in Definition 11.3.2. Remarks The point, of course is that in the case of a unimodular group it is not necessary to specify “left” or “right,” since any measure which satisfies one of the invariance properties (11.3.3), (11.3.4) satisfies both of them. According to Theorem 11.3.5, such a measure is unique up to scalar. It is clear that abelian groups are unimodular. It is also true that any compact group is unimodular, and this fact was exploited in Section   For an exampleof a group which

 10.5.  a b is not unimodular, one may take  a ∈ R× , b ∈ R . 0 1 To complete this section, we present a simple lemma on the factorization of Haar measure. Lemma 11.3.7 (Factorization of Haar measure on a direct product) Let G be a locally compact group. Let μG be a left Haar measure on G as in Definition 11.3.2. Let H and K be subsets of G such that the mapping (h, k) → hk is an isomorphism H × K → G. Let μ H and μ K denote left Haar measures on H and K . Then the product measure μ H × μ K is a left Haar measure on G and hence proportional to μG . Proof This follows readily from Definition 11.3.2 and Theorem 11.3.5. 

11.4 The global zeta integral for G L(2, AQ ) Following the ideas that appeared in Tate’s thesis [Tate, 1968], Godement and Jacquet introduced a global zeta integral associated to a cuspidal automorphic representation of G L(2, AQ ) by representing it as an integral transform of a matrix coefficient. The precise definition of the integral transform defining the global zeta integral will now be given. Definition 11.4.1 (Global zeta integral for G L(2, AQ )) Let (π, V ) be a cuspidal automorphic representation of G L(2, AQ ) as in Definition 5.1.14. Fix f 1 , f 2 ∈ V , and let  β(g) := f 1 (hg) f 2 (h) d × h, (g ∈ G L(2, A)), Z (AQ )G L(2,Q)\G L(2,AQ )

The Godement-Jacquet L-function for GL(2, AQ )

426

be a matrix coefficient as in Definition 9.5.12. Let  be a Bruhat-Schwartz function as in Definition 11.2.2. For s ∈ C with (s) sufficiently large, we define the global zeta integral  Z (s, , β) :=

(g) β(g) |det(g)|s+ 2 d × g, 1

G L(2,AQ )

where for g = {gv }v≤∞ ∈ G L(2, AQ ) we define d × g :=

 v≤∞

d × gv . Here

d × gv is a multiplicative Haar measure on G L(2, Qv ) for all v, and for all non-archimedean v it is normalized so that the measure of G L(2, Zv ) is one. 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 11.4.2 (Functional equation of the global zeta integral) Let (π, V ) be a cuspidal automorphic representation of G L(2, AQ ) as in Definition 5.1.14. Let Z (s, , β) denote the global zeta integral defined in Definition 11.4.1. 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 11.2.3, 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(2, AQ ) | |det(g)| = 1}. Let   Z∞ =

r 0 0 r

   1 0 , , 0 1

 ... ,

1 0

0 1

 ,

5 5   ...  r > 0 .

Then the map (g 1 , z ∞ ) → g 1 · z ∞ (g 1 ∈ G 1 , z ∞ ∈ Z ∞ ) is an isomorphism of G 1 × Z ∞ onto G L(2, AQ ). Using Lemma 11.3.7, we may express the matrix coefficient β as an integral over G L(2, Q)\G 1 .

11.4 The global zeta integral for GL(2, AQ ) We compute

⎛

 G L(2,AQ )





⎞



⎜ (g) ⎝

Z (s, , β) =

427

s+ 1 ⎟ f 1 (h 2 g) f 2 (h 2 ) d × h 2⎠ det(g) 2 d × g

G L(2,Q)\G 1

s+ 1 ×    ×  2 d g   h −1 2 g f 1 (g) f 2 (h 2 ) d h 2 det(g)

= G L(2,AQ ) G L(2,Q)\G 1

+∞ =

⎡

0





⎢ ⎣



G L(2,Q)\G 1 G L(2,Q)\G 1

   h −1 2 ξ τ h1

ξ ∈G L(2,Q)

⎤

⎥ · f 1 (h 1 ) f 2 (h 2 ) d × h 1 d × h 2 ⎦ t 2s+1 d × t. In the above we changed g → ξ τ h 1 , with τ =

0

G L(2,Q)\G 1 G L(2,Q)\G 1

· t 2s+1 d × t +∞ + 1

⎡ ⎢ ⎣



×

t 0 0 t



(t > 0), ξ ∈ G L(2, Q),

and h 1 ∈ G L(2, AQ )\G . Then d g = d h 1 d t with d × t = dtt . 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 ⎦ 1

×



×

ξ ∈G L(2,Q)





G L(2,Q)\G 1 G L(2,Q)\G 1

ξ ∈G L(2,Q)

   h −1 2 ξ τ h 1 f 1 (h 1 ) f 2 (h 2 ) ⎤

⎥ · d × h 1 d × h 2 ⎦ t 2s+1 d × t. (11.4.3) 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 11.2.6. 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(2,Q)\G 1 G L(2,Q)\G 1

ξ ∈G L(2,Q)

· t 2s+1 d × t = I1 + I2 + I3 − I4 − I5 ,

(11.4.4)

The Godement-Jacquet L-function for GL(2, AQ )

428

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

⎡

+∞ ⎢ ⎢ I2 = ⎣ 1

⎡



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

ξ ∈Mat(2,Q) rank(ξ )=1



 



0 0

0 0



· t 3−2s d × t, ⎤ ⎥ f 1 (h 1 ) f 2 (h 2 ) d × h 1 d × h 2 ⎦

G L(2,Q)\G 1 G L(2,Q)\G 1

⎡



+∞ ⎢ I4 = ⎢ ⎣

· t 3−2s d × t, ⎤





G L(2,Q)\G 1 G L(2,Q)\G 1

⎡

+∞ ⎢ I5 = ⎣ 1





⎢ ⎣

1

1



· t 3−2s d × t, ⎤

G L(2,Q)\G 1 G L(2,Q)\G 1

+∞ I3 =

ξ ∈G L(2,Q)

G L(2,Q)\G 1 G L(2,Q)\G 1



⎥   −1  h −1 h 1 f 1 (h 1 ) f 2 (h 2 )d × h 1 d × h 2⎥ 2 ξτ ⎦

ξ ∈Mat(2,Q) rank(ξ )=1



 

0 0 0 0



· t −2s−1 d × t, ⎤ ⎥ f 1 (h 1 ) f 2 (h 2 ) d × h 1 d × h 2 ⎦

G L(2,Q)\G 1 G L(2,Q)\G 1

t −2s−1 d × t. We now show that the integrals I2 , I I5 above  are allequal to zero. In 3 , I4 , 00  fact, the integrals I3 , I5 , containing  0 0 and  00 00 , vanish because the function f 1 is orthogonal to the constant function on G 1 . To deal with the remaining two integrals, fix τ and h 2 , and consider, for example,    ×  h −1  (11.4.5) 1 ξ τ h 2 f 1 (h 1 ) d h 1 . G L(2,Q)\G 1

ξ ∈Mat(2,Q) rank(ξ )=1

We can split {ξ ∈ Mat(2, Q) | rank(ξ ) = 1} up into orbits for the action of G L(2, Q) by matrix multiplication γ · ξ (γ ∈ G L(2, Q p ), ξ ∈ Mat(2, Q), rank(ξ ) = 1). It follows from (11.2.4) that each orbit contains a representative of the form   1 0 γ2 , (γ2 ∈ G L(2, Q)). (11.4.6) 0 0

11.4 The global zeta integral for GL(2, AQ )

429

Fix such a representative for each orbit. Define the groups     1 ∗ 1 ∗ P= , U= . 0 ∗ 0 1 Then the stabilizer of each of the representatives (11.4.6) is precisely P(Q). Writing   1 0 −1 γ2 , ξ = γ1 0 0 we can replace the sum over ξ in (11.4.5) by a sum over γ1 ∈ P(Q)\G L(2, Q) and a sum over the orbit representatives (11.4.6). For each fixed representative, we obtain      0 −1 −1 1  γ2 τ h 2 f 1 (h 1 ) d × h 1  h 1 γ1 0 0 G L(2,Q)\G 1



γ1 ∈P(Q)\G L(2,Q)

=

    1 0 −1  γ2 τ h 2 f 1 (h 1 ) d × h 1 .  h1 0 0

(11.4.7)

P(Q)\G 1

    10  h −1 Note that the function  γ does not change if we replace h 1 τ h 2 2 1 00 by p · h 1 with p ∈ P(AQ ). Since    M  P(Q)\G 1 = U (Q)\U (AQ ) · U (AQ ) · P(Q) G 1 , it follows from (11.4.7) that      1 0 −1 −1  γ2 τ h 2 f 1 (h 1 ) d × h 1  h 1 γ1 0 0 G L(2,Q)\G 1



γ1 ∈P(Q)\G L(2,Q)

=

  1 −1   h1 0

0 0







γ2 τ h 2

(U (AQ )·P(Q))\G 1

f 1 (uh 1 ) du d × h 1 = 0.

U (Q)\U (AQ )

The integral above is zero because f 1 is a cusp form, i.e.,



f 1 (uh 1 )

U (Q)\U (AQ )

du = 0. This concludes the proof that the last four integrals in (11.4.4) vanish. Combining this with the expression for Z (s, , β) given in (11.4.3) yields Z (s, , β) = ⎡ +∞  ⎢ ⎣ 1



G L(2,Q)\G 1 G L(2,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(2,Q)

· t 2s+1 d × t.

The Godement-Jacquet L-function for GL(2, AQ )

430 ⎡ +∞ ⎢ + ⎣ 1



⎤





G L(2,Q)\G 1 G L(2,Q)\G 1





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

ξ ∈G L(2,Q)

· t 3−2s 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 11.4.2.

β → β. 

11.5 Factorization of the global zeta integral Let (π, V ) be a cuspidal automorphic representation of G L(2, AQ ) as in Definition 5.1.14. Let β be a matrix coefficient of π as in Definition 9.5.12 and let  be a Bruhat-Schwartz function as in Definition 11.2.2. Then for s ∈ C, with (s) sufficiently large, we may attach a global zeta integral Z (s, , β) associated to π (as in Definition 11.4.1), where  1 (g) β(g) |det(g)|s+ 2 d × g. (11.5.1) Z (s, , β) = G L(2,AQ )

If the representation (π, V ) is irreducible then it will turn out that π possesses factorizable matrix coefficients 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 11.5.2 (Factorization of the global zeta integral) Let (π, V ) be an irreducible cuspidal automorphic representation of G L(2, AQ ) as in Definition 5.1.14, and let β be a matrix coefficient of π as in Definition 9.5.12, defined using two cusp forms φ1 , φ2 ∈ V, which are mapped to pure tenL πv given in Theorem 10.8.5. Then there sors under the isomorphism π ∼ = v≤∞

exist matrix coefficients βv of the local representations πv (v ≤ ∞) so that  βv . β= v≤∞

Further, if  : Mat(2, AQ ) → C is a factorizable Bruhat-Schwartz function, i.e., can be represented in the form  v (m v ), (m = (m v )v≤∞ ∈ Mat(2, AQ )), (m) = v≤∞

11.5 Factorization of the global zeta integral

431

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



Z v (s, v , βv )

v≤∞



where

s+ 12

Z v (s, v , βv ) :=

v (g) βv (g) |det(g)|v

d × g.

G L(2,Qv )

Remark In the statement above, the “representation” at ∞ is really a (g, K ∞ )module. L πv as Proof Fix a restricted tensor product (g, K ∞ )×G L(2, Afinite )-module v≤∞

in Definition 10.3.3, which is isomorphic to π. The existence of such a module is given by Theorem 10.8.5, while Theorem 10.8.12 states that the factors are unique up to isomorphism. Assume that (π∞ , V∞ ) is realized as a subspace of a principal series representation, as in Definition 8.6.7. By Theorem 7.5.10, there L Vv → V . is no loss of generality. Also fix a particular isomorphism L : v≤∞

By Lemma 8.7.3, it is unique up to scalar. An invariant positive definite Hermitian form ( , )A on the space of all cusp forms having a fixed central character was given in Definition 9.5.3. It follows from Proposition 9.5.8 that the (π, V ) is realized on a space of cusp forms, and hence is unitary, as noted in Corollary 9.5.10. We first show that each of the local representations (πv , Vv ) is unitary. To that vv = vv◦ for see this, fix non-zero vectors vv ∈ Vv for all v =/ v, such   L / S, and define a linear map L v : Vv → V by L v (vv ) = L vv ⊗ vv . v ∈ v =/ v

Then (u v , vv ) L v = (L v (u v ), L v (vv ))A is a positive definite invariant Hermitian form on Vv , which proves that Vv is unitary. Now fix an invariant positive definite Hermitian form ( , )v : Vv × Vv → C / S. Then for each place v, subject to the restriction that (vv◦ , vv◦ )v = 1 for all v ∈ {( , )v }v≤∞ determines an invariant positive definite Hermitian form ( , )tensor L on Vv given on pure tensors by v≤∞



K

v≤∞

uv ,

K v≤∞

 vv

= tensor



(u v , vv )v .

(11.5.3)

v≤∞

(Note that for all but finitely many v, the vectors u v and vv are both vv◦ . Consequently, all but finitely many of the terms in the product on the right hand side of (11.5.3) are one.)

The Godement-Jacquet L-function for GL(2, AQ )

432

It was shown in Corollary 8.7.5 that an invariant bilinear form on a pair of (g, K ∞ ) × G L(2, Afinite )-modules is unique up to scalar, when one exists. A straightforward modification of the proof gives the same result for forms which are conjugate linear in the second argument. In particular, the invariant positive definite Hermitian form on a unitary (g, K ∞ ) × G L(2, Afinite )-module is unique up to scalar, cf. Exercises 9.8, 9.9. It follows that     K K K K  L( u v ), L( vv ) = c· uv , vv = c · (u v , vv )v , v

v

A

v

v

v

tensor

L

(11.5.4) L u v and vv in

for some non-zero constant c, and for all pure tensors v v L Vv . Adjusting the choice of any one of the forms ( , )v , we may assume

v≤∞

that c = 1. Take β : G L(2, A) → C a matrix coefficient, defined as in Definition 9.5.12 using two elements φ1 , φ2 of the vector space V, and assume that φ1 and φ2 L L are the images under L of pure tensors u v and vv respectively. Define v

v

β p (g p ) = (π p (g p ) . u p , v p ) p ,

(g p ∈ G L(2, Q p )),

for all p < ∞. To define β∞ (g∞ ), choose a realization of (π∞ , V∞ ) as a subspace of a principal series representation, and use Definition 8.6.12. The function βv (gv ) is a matrix coefficient of (πv , Vv ) for each v. Indeed, we may pass from the invariant Hermitian form on Vv × Vv to an invariant Vv as in the proof of Lemma 9.1.6. bilinear form on Vv × (  βv (gv ) for all g ∈ It follows immediately from (11.5.4) that β(g) = G L(2, A) such that the function ρ(g∞ ) . u ∞ defined by ρ(g∞ ) . u ∞ (h) = u ∞ (hg∞ ),

v

(h ∈ G L(2, R)),

lies in the (g, K ∞ )-module V∞ . It is easy to see that this is the case if g∞ is in the product of O(2, R) and the center of G L(2, R). It can be shown that it is  not the case otherwise. To prove that β(g) = βv (gv ) for all g, we proceed as v

in the proof of Proposition 8.6.23, using Lemma 8.6.25. The factorization of the global zeta integral can then be easily obtained from its definition given in (11.5.1). 

11.6 The local functional equation Fix a rational prime p and let (π, V ) be an admissible irreducible repreπ the sentation of G L(2, Q p ) as in Definitions 6.1.4, 6.1.5. We denote by (

11.6 The local functional equation

433

representation contragredient to π and ( V the space on which it acts as in Definition 8.1.4. 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   (11.6.1) β(g) := π (g) . v, ( v, g ∈ G L(2, Q p ) . The matrix coefficient (11.6.1) will be the first ingredient in a local zeta integral, analogous to the global zeta integral of Definition 11.4.1. The second ingredient is a local analogue of the Bruhat-Schwartz function for Mat(2, AQ ), defined as in Definition 11.2.2. Definition 11.6.2 (The Bruhat-Schwartz space associated to Mat(2, Q p )) Let Mat(2, Q p ) denote the ring of all 2 × 2 matrices with coefficients in Q p as in Definition 11.2.1. A function  : Mat(2, 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(2, Q p ) → C is denoted S(Mat(2, Q p )). Let  : Mat(2, Q p ) → C be a Bruhat-Schwartz function, i.e., it is locally constant and compactly supported. For s ∈ C with (s) sufficiently large, the local zeta integral 

s+ 12

Z p (s, , β) =

(g) β(g) |det(g)| p

d×g

(11.6.3)

G L(2,Q p )

was introduced in Proposition 11.5.2. 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(2, Q p ). Theorem 11.6.4 (Local functional equation) Fix a rational prime p and let (π, V ) be an admissible irreducible representation of G L(2, Q p ) as in Definition 6.1.4. Let β : G L(2, Q p ) → C be a matrix coefficient as in (11.6.1) and let  : Mat(2, Q p ) → C be a locally constant and compactly supported (Bruhat-Schwartz function). Consider, Z p (s, , β), the local zeta integral defined in (11.6.3). Then we have. (1) There exists s0 ∈ R such that the integral (11.6.3) converges absolutely for (s) > s0 . (2) For each Bruhat-Schwartz function , and each matrix coefficient β, the integral (11.6.3) 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

434

The Godement-Jacquet L-function for GL(2, AQ ) 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 (11.6.3) satisfies the functional equation  ∨ , β = γ (s, π ) Z p (s, , β). Z p 1 − s,  ∨

 is the Fourier transform as defined in Theorem 1.6.8 and β is Here  ∨

defined by β (g) := β(g −1 ) for all g ∈ G L(2, 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  ∨ , β Z p 1 − s,  Z p (s, , β) = p (s, π ) . L p (1 − s, ( π) L p (s, π )

Proof The proof is spread over the next seven sections as follows. Absolute convergence as in (1) is left as an exercise, except in the case of the special representations, for which a precise domain of convergence is required. In Sections 11.7–11.9 the precise values of L p (s, π ) will be given in various cases, Z (s,,β) and it will be shown that the ratio Lp p (s,π) is identically equal to one for suitZ (s,,β)

able , β. In sections 11.11–11.13, it is shown that the ratio Lp p (s,π) is entire for all , β, and the functional equations described in Theorem 11.6.4 (3) are established.

11.7 The local L-function for G L(2, Q p ) (unramified case) Fix a rational prime p and let (π, V ) be an infinite dimensional admissible irreducible unramified representation of G L(2, Q p ) as in Definitions 6.1.4, 6.2.1. Let  : Mat(2, Q p ) → C be a Bruhat-Schwartz function of the form      a b a b  = φ1 (a)φ2 (b)φ3 (c)φ4 (d), ∈ G L(2, Q p ) , c d c d (11.7.1)

11.7 The local L-function for GL(2, Qp ) (unramified case)

435

where φi : Q p → C (i = 1, 2, 3, 4) are all locally constant compactly supported functions. Let β be a matrix coefficient for (π, V ) defined by β(g) := π (g) . v, ( v,



 g ∈ G L(2, Q p ) ,

with v ∈ V, ( v∈( V , and  ,  : V × ( V → C is an invariant bilinear form as in Chapter 8. We are interested in explicitly evaluating the local zeta integral  s+ 1 Z p (s, , β) = (g) β(g) |det(g)| p 2 d × g (11.7.2) G L(2,Q p )

introduced in Proposition 11.5.2. As in [Jacquet, 1979], we shall evaluate (11.7.2) by realizing (π, V ) as a principal series representation. That this is possible is guaranteed by the following proposition. Proposition 11.7.3 (Irreducible spherical representations of G L(2, Q p ) of infinite dimension can be realized as principal series) Let (π, V ) be an infinite dimensional admissible irreducible unramified representation of G L(2, Q p ) as in Definitions 6.1.4, 6.13.1. Then (π, V ) is isomorphic to a principal series representation of G L(2, Q p ) as in Definition 6.5.3. Proof To prove Proposition 11.7.3 we require very explicit generators for the spherical Hecke algebra defined in Definition 10.6.4. Let K = G L(2, Z p ) and consider the space K H K of bi K -invariant functions as defined in Definition 10.6.1. For integers k ≥ 0, we shall define functions T pk , R p ∈ K K H as follows:  a   1, if g ∈ K p0 p0b K , a ≥ b ≥ 0, a + b = k, T pk (g) = 0, otherwise,    1, if g ∈ K 0p 0p K , R p (g) := 0, otherwise. We will show that the Hecke algebra K H K is generated by T p , R p , R −1 p and that for all k ≥ 1 the Hecke operators T pk and R p satisfy the relation T p ∗ T pk = T pk+1 + p R p ∗ T pk−1 , where ∗ denotes convolution, which was defined in (10.6.2). Note that ⎞ ⎛         p b p 0 1 0 K⎠. K K = K ∪ ⎝ 0 1 0 1 0 p b (mod p)

(11.7.4)

(11.7.5)

436

The Godement-Jacquet L-function for GL(2, AQ )

To prove (11.7.4) we compute 

 T p ∗ T pk (g) =



  T p (h) T pk h −1 g d × h =

= T pk

1 0

0 p

−1  g +



  T pk h −1 g d × h



K p 0 K 0 1

G L(2,Q p )





b (mod p)



T pk

p 0

b 1

−1  g .

By the Iwasawa decomposition, given in Proposition 4.2.1, and the K -invariance of the functions under consideration, it is enough to prove (11.7.4) in the case when g is upper triangular. This case follows by routine computations left to the reader. Now, by the Cartan decomposition for G L(2, Q p ) (Proposition 4.2.3), a basis for the space K H K maybe given  by the locally constant functions with support on the double cosets K

pa 0 0 pb

K with

a ≥ Such are of the form R pb times a characteristic function of  functions  b.    pa−b 0 K 0 1 K which is the function T pa−b − R p ∗ T pa−b−2 . But all the functions T pk (k ≥ 0) are in the algebra generated by T p , R p by (11.7.4). This proves K K that T p , R p , R −1 p generate H . To complete the proof of Proposition 11.7.3 we use Proposition 10.6.15, which says that spherical representations of G L(2, Q p ) are uniquely determined (up to isomorphism) by their spherical Hecke characters. The spherical Hecke character of (π, V ) was defined in Definition 10.6.14, and is a C-linear ring homomorphism ξ : K H K → C. Because K H K is generated by T p , R p , R −1 p , it follows that ξ is completely determined by the two complex numbers λ := ξ (T p ) and μ := ξ (R p ). By Proposition 10.6.15, it follows that λ and μ uniquely determine the isomorphism class of (π, V ) as well. It remains to show that there exists a spherical principal series representation with the same Hecke character as (π, V ). This is easily accomplished as follows. Fix unramified characters χ1 , χ2 of Q×p , as in Definition 2.1.12 and consider the principal series representation B p (χ1 , χ2 ) as in Definition 8.2.2. This representation is spherical, with a one-dimensional space of G L(2, Z p )fixed vectors spanned by the function f χ◦1 ,χ2 : G L(2, Q p ) → C which is defined by     1 a b a 2 k =   χ1 (a)χ2 (d), f χ◦1 ,χ2 (11.7.6) 0 d d p for all a, d ∈ Q×p , b ∈ Q p , and for all k ∈ G L(2, Z p ). We will show that it is possible to choose χ1 , χ2 so that   π T p . f χ◦1 ,χ2 = λ f χ◦1 ,χ2 ,

  π R p . f χ◦1 ,χ2 = μ f χ◦1 ,χ2 .

11.7 The local L-function for GL(2, Qp ) (unramified case) To do this we use (11.7.5) to compute      1 0 = π T p . f χ◦1 ,χ2 0 1  p 0 K



f χ◦1 ,χ2

= K

=

f χ◦1 ,χ2





1 0

1 0

0 p

0 p

1 2

 ·k

f χ◦1 ,χ2 (g) d × g K p−1 

×

d k+

b=0



= p χ2 ( p) + p · p

0 1







p−1

+

f χ◦1 ,χ2

b=0 − 12

437

f χ◦1 ,χ2

K

p 0

b 1



p 0

b 1



 ·k

d ×k



1

χ1 ( p) = p 2 (χ1 ( p) + χ2 ( p)) .

In a similar manner      1 0 = π R p . f χ◦1 ,χ2 0 1   K

=

p 0 0 p

f χ◦1 ,χ2

f χ◦1 ,χ2 (g) d × g = K



p 0

0 p

 

p 0 0 p

f χ◦1 ,χ2 (g) d × g

 K

 = χ1 ( p)χ2 ( p).

We can always choose χ1 , χ2 , so that 1

λ = p 2 (χ1 ( p) + χ2 ( p)) ,

μ = χ1 ( p)χ2 ( p).   If the corresponding principal series representation π, B p (χ1 , χ2 ) is irreducible, it follows directly from Proposition 10.6.15 that it is isomorphic to the (π, V ). To complete the proof, we show that if   original representation then (π, V ) is one-dimensional. π, Bp (χ1 , χ2 ) is reducible,  If π, B p (χ1 , χ2 ) is reducible, then χ1 (t) = χ2 (t) · |t|±1 p , for all t ∈ Q p , and we may assume that χ1 (t) = χ2 (t) · |t|−1 . It follows that f χ◦1 ,χ2 (g) = p −1

χ2 (det g)| det g| p 2 for all g ∈ G L(2, Q p ). This function spans an irreducible one-dimensional spherical representation, which has the same spherical Hecke character as (π, V ). It follows from Proposition 10.6.15 that (π, V ) is also one-dimensional.  We return to the computation of the local zeta integral (11.7.2). We require explicit forms of the matrix coefficient β and the Bruhat-Schwartz function . It follows from Propositions 8.2.3, 8.2.4, and 11.7.3, that we may take   ◦ ◦ × f χ1 ,χ2 (kg) · f χ −1 ,χ −1 (k) d k = f χ◦1 ,χ2 (kg) d × k, β(g) := 1

G L(2,Z p )

2

G L(2,Z p )

(11.7.7)

The Godement-Jacquet L-function for GL(2, AQ )

438

where f χ◦1 ,χ2 is given by (11.7.6). Theorem 11.7.8 (Evaluation of the local zeta integral of an unramified representation for a special choice of test function ) Let (π, V ) denote an infinite dimensional admissible irreducible unramified representation of G L(2, Q p ) (as in Definitions 6.1.4, 6.2.1) which is isomorphic to the principal series representation with space of functions B p (χ1 , χ2 ), as defined in Definition 8.2.1. Let the matrix coefficient β associated to π be given by (11.7.7). Let  : Mat(2, Q p ) → C be the Bruhat-Schwartz function defined by  

a c

b d



 = 1Z p (a)·1Z p (b)·1Z p (c)·1Z p (d),

a c

b d



 ∈ G L(2, Q p ) ,

where 1Z p is the characteristic function of Z p . Then the local zeta integral (11.7.2) can be explicitly evaluated (for (s) > 0) as Z p (s, , β) =

1 (1 − χ1

( p) p −s ) (1

− χ2 ( p) p −s )

.

Proof We require a variation of Proposition 6.9.6. Let f : G L(2, Q p ) → C be any locally constant function. Then we have the identity         d × a1 × a1 x · k d ×k d x f (g) d × g = f d a2 , 0 a2 |a1 | p Q×p Q×p Q p G L(2,Z p )

G L(2,Q p )

(11.7.9) where d × k is the normalized Haar measure on G L(2, Q p ) (defined in Definition 6.9.5) and which is restricted to the open compact subgroup G L(2, Z p ), d x is the additive Haar measure on Q p given in Example 1.5.4, and d × ai (i = 1, 2) is the multiplicative Haar measure on Q×p given in Definition 1.5.7. The proof of (11.7.9) is left to the reader. We shall now compute Z p (s, , β) when (s) is sufficiently large.  Z p (s, , β) = G L(2,Q p )

 = G L(2,Q p )



= G L(2,Q p )

⎡ ⎢ (g) ⎣ ⎡ ⎢ (g) ⎣



⎤ s+ 1 ⎥ f χ◦1 ,χ2 (kg) d × k ⎦ |det(g)| p 2 d × g

G L(2,Z p )



⎤ s+ 1 ⎥ f χ◦1 ,χ2 (g) d × k ⎦ |det(g)| p 2 d × g

G L(2,Z p ) s+ 12

(g) f χ◦1 ,χ2 (g) |det(g)| p

d × g.

11.7 The local L-function for GL(2, Qp ) (unramified case) It follows from (11.7.9) that      a1  Z p (s, , β) = 0 Q×p Q×p Q p







=

x a2

439

 |a1 |sp · |a2 |sp · χ1 (a1 )χ2 (a2 ) d x d × a1 d × a2

|a1 |sp · |a2 |sp χ1 (a1 )χ2 (a2 ) d x d × a1 d × a2 .

Z p −{0} Z p −{0} Z p

Finally, the above integral can be computed as in Example 1.5.8. We obtain Z p (s, , β) =

 −1  1 − χ1 ( p) p −s 1 − χ2 ( p) p −s .



Definition 11.7.10 (Langlands parameters) The pair of complex numbers (χ1 ( p), χ2 ( p)) occurring in Theorem 11.7.8 are called the Langlands parameters associated to (π, V ). As (π, V ) is isomorphic to a principal series representation with space of functions B p (χ1 , χ2 ), as defined in Definition 8.2.1, it immediately follows that the Langlands parameters uniquely determine the local representation (π, V ) up to isomorphism. Warning There is a second, related usage of the term “Langlands parameter” which is perhaps more common in the literature (see [Gross-Reeder, 2006]). Following the example of Tate’s thesis (see Section 2.4), we may define local L-functions associated to representations of G L(2, Q p ). Definition 11.7.11 (The local L-function of an unramified irreducible admissible representation of G L(2, Q p )) Let (π, V ) be an infinite dimensional admissible irreducible unramified representation of G L(2, Q p ) (as in Definitions 6.1.4, 6.2.1) which is isomorphic to the principal series representation with space of functions B p (χ1 , χ2 ), as defined in Definition 8.2.1. Then the local L-function associated to π , denoted L p (s, π ), is defined as −1  −1  L p (s, π ) := 1 − χ1 ( p) p −s · 1 − χ2 ( p) p −s ,

(s ∈ C). (11.7.12)

Conjecture 11.7.13 (Ramanujan conjecture) Let (π, V ) be an unramified irreducible admissible representation of G L(2, Q p ) with local L-function (11.7.12) which is a local component of a global cuspidal automorphic representation of G L(2, AQ ) as in Conjecture 10.9.1. Then |χ1 ( p)|C = |χ2 ( p)|C = 1, where | |C denotes the complex absolute value.

The Godement-Jacquet L-function for GL(2, AQ )

440

Remarks Conjecture 11.7.13 follows immediately from Conjecture 10.9.1. In fact, Proposition 9.2.8 tells us that if an irreducible principal series representation B p (χ1 , χ2 ) is tempered then χ1 , χ2 must be unitary.

11.8 The local L-function for irreducible supercuspidal representations of G L(2, Q p ) Fix a rational prime p and let (π, V ) denote an admissible and irreducible supercuspidal representation of G L(2, Q p ) as in Definitions 6.13.1, 6.1.4. Let  : Mat(2, Q p ) → C be a Bruhat-Schwartz function as in Definition 11.6.2. It immediately follows from Theorem 8.4.9 that a matrix coefficient β : G L(2, Q p ) → C for (π, V ) has compact support modulo the center of G L(2, Q p ). Consider the local zeta integral given, as in (11.6.3), by  s+ 1 Z p (s, , β) = (g) β(g) |det(g)| p 2 d × g. (11.8.1) G L(2,Q p )

Define G 1p := {g ∈ G L(2, Q p ) | where |det(g)| p = 1}.

(11.8.2)

Then β restricted to G 1p is compactly supported. Theorem 11.8.3 (Evaluation of the local zeta integral of a supercuspidal representation for a special choice of test function ) Let (π, V ) be an irreducible admissible supercuspidal representation of G L(2, Q p ), as in Definitions 6.13.1, 6.1.4, with matrix coefficient β as in Definition 8.1.14. Let  be the Bruhat-Schwartz function defined by  β(g), if g ∈ G L(2, Q p ) and |det(g)| p = 1, (g) := 0, otherwise. Then the local zeta integral (11.8.1) is equal to a non-zero constant. Proof It follows immediately from the definition of  that the local zeta integral given in (11.8.1) takes the form  Z p (s, , β) = β(g) · β(g) d × g =/ 0.  G 1p

Definition 11.8.4 (The local L-function of an irreducible supercuspidal representation of G L(2, Q p )) Let (π, V ) denote an irreducible and admissible supercuspidal representation of G L(2, Q p ) as in Definition 6.13.1. Then the local L-function associated to π , denoted L p (s, π ), is defined to be the constant function with value one.

11.9 The local L-function for irreducible principal series

441

11.9 The local L-function for irreducible principal series representations of G L(2, Q p ) Fix a prime p and let χ1 , χ2 : Q×p → C be characters of Q×p . Let B p (χ1 , χ2 ) be the space 

   f : G L(2, Q p ) → C   ∀ a, d ∈

Q×p ,

 f

a 0

b d

  a 1  2 g =   χ1 (a)χ2 (d) f (g), d p



b ∈ Q p , g ∈ G L(2, Q p ), f is locally constant .

  Consider the principal series representation π, B p (χ1 , χ2 ) defined in Proposition 8.2.3. By Proposition 8.2.4 it follows that the contragredient rep π,( B p (χ1 , χ2 ) , is isomorphic to resentation of π, B p (χ1 , χ2 ) , denoted (   π, B p (χ1−1 , χ2−1 ) . Fix f 1 ∈ B p (χ1 , χ2 ),

  f 2 ∈ π, B p (χ1−1 , χ2−1 ) .

An immediate consequence of Proposition 8.2.3 is that a matrix coefficient β  of the principal series representation π, B p (χ1 , χ2 ) is given by 



β(g) :=

 π (g) . f 1 (k) · f 2 (k) d × k =

G L(2,Z p )



f 1 (kg) · f 2 (k) d × k. G L(2,Z p )

(11.9.1) With these preliminaries in place, we now address the problem of evaluating the local zeta integral given in (11.6.4) for a special choice of test function in the case of a principal series representation. Theorem 11.9.2 (Evaluation of the local zeta integral of a principal series representation for a special choice of test function ) Fixa prime p and let  χ1 , χ2 : Q×p → C be characters of Q×p . Let π, B p (χ1 , χ2 ) be the principal series representation as defined in Proposition 8.2.3 with matrix coefficient β given by (11.9.1). Let  : Mat(2, Q p ) → C be a Bruhat-Schwartz function satisfying  

a1 0

x a2

 = h 1 (a1 ) · h 2 (a2 ) · 1Z p (x),

  ∀ a1 , a2 ∈ Q×p , x ∈ Q p ,

where h 1 and h 2 are the test functions given in the proof of Theorem 2.3.8, so that  h i (ai )χi (ai )|ai |sp d × ai = L p (s, χi ), (i = 1, 2), Q×p

The Godement-Jacquet L-function for GL(2, AQ )

442

and 1Z p is the characteristic function of Z p . Then for a suitable choice of f 1 ∈ B p (χ1 , χ2 ), f 2 ∈ ( B p (χ1 , χ2 ), the local zeta integral (11.6.3) can be explicitly evaluated as Z p (s, , β) = L p (s, χ1 ) L p (s, χ2 ), where, L p (s, χi ) is the local L-function defined in Section 2.4. Proof It follows from the definition of the local zeta integral (11.6.3) and the definition of the matrix coefficient given in (11.9.1) that ,  s+ 12

f 1 (kg) f 2 (k) d × k (g) |det(g)| p

Z p (s, , β) =

d × g.

G L(2,Z p )

G L(2,Q p )

(11.9.3) By Theorem 8.10.13, it immediately follows that the integral in (11.9.3) converges absolutely for (s) sufficiently large. In the integral (11.9.3) make the transformation g → k −1 g and also make the substitutions (see (11.7.9))   a1 x k1 , d × g = d × a1 d × a2 d x d × k1 |a1 |−1 g= p . 0 a2 We obtain







  

Z p (s, , β) =

f1

G L(2,Z p ) G L(2,Z p ) Q p



· k

−1



a1 0

x a2

Q×p



Q×p



s− 12

k1 |a1 | p

a1 0

x a2 s+ 12

|a2 | p



 k1

f 2 (k)

d × a1 d × a2 d x · d × k d × k1 .

Since f 1 ∈ B p (χ1 , χ2 ) we have  f1

a1 0

x a2



 k1

 1  a1  2 =   χ1 (a1 ) χ2 (a2 ) f (k1 ). a2

Consequently 



  

Z p (s, , β) = G L(2,Z p ) G L(2,Z p ) Q p

Q×p

Q×p

  a1 −1 f 1 (k1 ) f 2 (k)  k 0

x a2



 k1

· χ1 (a1 ) |a1 |sp χ2 (a2 ) |a2 |sp d × a1 d × a2 d x d × k d × k1 . (11.9.4) We want to show, that for a suitable choice of the test function , and elements f 1 ∈ B p (χ1 , χ2 ), f 2 ∈ B p (χ1−1 , χ2−1 ), that the local zeta integral Z p (s, , β)

11.9 The local L-function for irreducible principal series

443

given in (11.9.4) is a multiple of L(s, χ1 ) · L(s, χ2 ) where L(s, χi ) is defined as in Section 2.4 for i = 1, 2. To do this, we simply choose the test function  in the form   a1 x = h 1 (a1 ) · h 2 (a2 ) · 1Z p (x),  0 a2 where h 1 and h 2 are the elements of S(Q×p ) given in the proof of Theorem 2.3.8, with the property that  h i (ai )χi (ai )|ai |sp d × ai = L p (s, χi ), (i = 1, 2), Q×p

and 1Z p is the characteristic function of Z p . We now show why this choice of  works. For s ∈ C with (s) sufficiently large and k, k1 ∈ G L(2, Z p ), define the function        a1 x −1 k1 χ1 (a1 ) χ2 (a2 )  k H (s, k, k1 ) := 0 a2 Q p Q×p Q×p

· |a1 a2 |sp d × a1 d × a2 d x. It immediately follows from (11.9.4) that   H (s, k, k1 ) f 1 (k1 ) f 2 (k) d × k d × k1 . (11.9.5) Z p (s, , β) = G L(2,Z p ) G L(2,Z p )

Now

Also, for

 a1 0

     1 0 1 0 H s, , = L(s, χ1 )L(s, χ2 ). 0 1 0 1   x  b1 y a2 , 0 b2 ∈ G L(2, Z p ), we have

  y k = χ1 (a1 )χ2 (a2 )χ1−1 (b1 )χ2−1 (b2 )· H (s, k, k1 ). b2 1 (11.9.6) Note that (11.9.6) easily follows by a simple change of variables after noting that |x| p , |y| p ≤ 1 and |a1 | p = |a2 | p = |b1 | p = |b2 | p = 1. We want to choose f 1 , f 2 so that   H (s, k, k1 ) f 1 (k1 ) f 2 (k) d × k d × k1   a H s, 1 0

  b x k, 1 a2 0

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

     1 0 1 0 = H s, , . 0 1 0 1

(11.9.7)

444

The Godement-Jacquet L-function for GL(2, AQ )

This will complete the proof of Theorem 11.9.2. The functions f 1 , f 2 and  are all locally constant. It follows that if the matrices u, u 1 ∈ G L(2, Q p ) are sufficiently close to the identity then H (s, ku, k1 u 1 ) = H (s, k, k1 ),

(∀ k, k1 ∈ G L(2, Z p )).

(11.9.8)

Toestablish we shall choose f 1 , f 2 supported on matrices of the   (11.9.7),  1 0 α β form 0 δ · μ 1 with α, δ ∈ Q×p , β ∈ Q p and μ ∈ Q×p with |μ| p <  for some  > 0 sufficiently and (11.9.8) combine to ensure  (11.9.6)   small.  Then 10 10 that H (s, k, k1 ) = H s, 0 1 , 0 1 whenever k is in the support of f 2 and k1 is in the support of f 1 .  Definition 11.9.9 (The local L-function of an irreducible principal series representation of G L(2, Q p )) Let (π, V ) denote an irreducible representation of G L(2, Q p ) (as in Definition 6.1.4) which is isomorphic to the principal series representation with space of functions B p (χ1 , χ2 ), as defined in Definition 8.2.1. Then the local L-function associated to π , denoted L p (s, π ), is defined as (s ∈ C), L p (s, π ) := L(s, χ1 )L(s, χ2 ), where L(s, χ1 ) and L(s, χ2 ) are defined as in Section 2.4. Remark We expect the Ramanujan Conjecture 11.7.13 to hold for L p (s, π ) above, if π is the local component at p of some irreducible cuspidal automorphic representation.

11.10 Local L-function for unitary special representations of G L(2, Q p ) To define the L-function for the unitary special representations of G L(2, Q p ), we fix a unitary character χ : Q p → C and define: 1

χ1 (a) := χ (a)|a| p2 ,

−1

χ2 (a) := χ (a)|a| p 2 ,

(a ∈ Q×p ).

Then, by Proposition 9.3.3, the space B p (χ1 , χ2 ), defined in Definition 8.2.1, contains a special representation (π, V ) where V is the orthogonal complement in B p (χ1 , χ2 ) of an element of B p χ1−1 , χ2−1 . Let β : G L(2, Q p ) → C be a matrix coefficient for the special representation and let  : Mat(2, Q p ) → C be a Bruhat-Schwartz function. That is, take  to be locally constant and compactly supported. By (11.6.3), the Godement-Jacquet integral for the special representation is defined to be  s+ 1 (g)β(g)|det(g)| p 2 d × g. (11.10.1) Z p (s, , β) := G L(2,Q p )

11.10 Local L-function for unitary special representations

445

Proposition 11.10.2 (Region of absolute convergence of the local zeta integral for a special representation) The local zeta integral, given by (11.10.1), converges absolutely for (s) > 0. Proof This will follow directly from Theorem 8.10.19 which asserts that the matrix coefficients of the special representations are square integrable modulo the center of G L(2, Q p ). Consider first the integral over the center 5       r 0  × ∼ r ∈ Q p = Q×p . Z G L(2, Q p ) = z(r ) := 0 r  For (s) sufficiently large, define    s+ 1 Fs (g) :=  z(r )g |det(g)| p 2 |r |2s+1 χ (r )2 d ×r. Q×p

Since the central character of the special representation is equal to χ 2 , it follows that β(z(r )g) = χ (r )2 β(g) for all r ∈ Q×p , and g ∈ G L(2, Q p ). Consequently  Fs (g) β(g) d × g. (11.10.3) Z p (s, , β) = Q×p \G L(2,Q p )

By Theorem 8.10.19 we know that 

|β(g)|2 d × g < ∞,

Q×p \G L(2,Q p )

so by (11.10.3) the proof of Proposition 11.10.2 will follow from the CauchySchwartz inequality if we can show that  |Fs (g)|2C d × g Q×p \G L(2,Q p )

converges absolutely for (s) > 0. This is established as follows. Let K = G L(2, Z p ). By the Cartan decomposition (Proposition 4.2.3), we have     n  ∞ p 0 2 × K |Fs (g)|C d g = Vol K 0 1 Q×p \G L(2,Q p )

n=0

     n   Fs k1 p ·  0 K

K

0 1



2  k2  d × k1 d × k2 . C

(11.10.4)

The Godement-Jacquet L-function for GL(2, AQ )

446

On the other hand,   n p Fs k1 0

0 1





= p

k2

−n(s+ 12 )

  n rp  k1 0



0 r





2 × k2 |r |2s+1 p χ (r ) d r,

Q×p

and for (s) > 0, the integral 





 k1

r pn 0

0 r





2 × k2 |r |2s+1 p χ (r ) d r

Q×p

is bounded by a constant which is independent of n, but depends on . It follows that    n      Fs k1 p 0 k2  ! p −n ((s)+ 12 ) .   0 1 C   n   It was shown in the proof of Theorem 8.10.19 that Vol K p0 10 K ! p n + p n−1 . Combining this with (11.10.4) and the inequality above yields 

×

|Fs (g)|C d g ! 2

∞ 

 1 p n + p n−1 p −2n ((s)+ 2 ) .

(11.10.5)

n=1

Q×p \G L(2,Q p )

Since the sum on the right hand side of (11.10.5) converges for Re(s) > 0, this concludes the proof of Proposition 11.10.2. 

11.11 Proof of the local functional equation for principal series representations of G L(2, Q p ) Let χ1 and χ2 be two characters of Q×p . The principal series representation (π, B p (χ1 , χ2 )), defined in Definition 8.2.2, is the space of all locally constant functions f : G L(2, Q p ) → C satisfying  f

x a2

a1 0

    12  a1  g =   χ1 (a1 )χ2 (a2 ) f (g) a2 p

(11.11.1)

for all a1 , a2 ∈ Q×p , all x ∈ Q p , and all g ∈ G L(2, Q p ). By Proposition 8.2.4, the contragredient of B p (χ1 , χ2 ), may be identified with the space B p (χ1−1 , χ2−1 ) of all functions f(: G L(2, Q p ) → C satisfying  a1 f( 0

x a2

    12  a1  g =   χ1−1 (a1 )χ2−1 (a2 ) f (g). a2 p

(11.11.2)

11.11 Proof of the local functional equation for principal series

447

Following Definition 8.1.14, Proposition 8.2.3, and Proposition 8.2.4, a matrix coefficient β for (π, B p (χ1 , χ2 )) is given by 8 9 β(g) := π (g) . f, ( f =



f (kg)( f (k) d × k,

(g ∈ G L(2, Q p )).

G L(2,Z p )

Let  : Mat(2, Q p ) → C be a Bruhat-Schwartz function. That is, take  to be locally constant and compactly supported. Set K = G L(2, Z p ). Our goal is to obtain the functional equation of Theorem 11.6.4 for the local L-function given by the Godement-Jacquet integral 

s+ 1

(g)β(g) |detg| p 2 d × g

Z p (s, , β) = 

G L(2,Q p )

:

; s+ 1 × ( f (kg) f (k) d k |detg| p 2 d × g.

(g)

= G L(2,Q p )

K

The absolute convergence of the above integral for (s) > − 12 can be shown as in Exercise 11.12. The above integral can be evaluated with the expression for d × g in terms of the Iwasawa decomposition given in (11.7.9)  g=

a1 0

x a2



k ,

× × × d × g = |a1 |−1 p d a1 d a2 d k d x.

It follows from (11.11.1) that      Z p (s, , β) =

  a1  k −1 0

Q×p Q×p Q p K K

x a2

  k f (k ) ( f (k)

· χ1 (a1 )χ2 (a2 ) |a1 a2 |sp d × a1 d × a2 d x d × k d × k. Now, the function   m → K

(k −1 mk ) f (k ) ( f (k) d × k d × k,

(m ∈ Mat(2, Q p )),

K

is a Bruhat-Schwartz function of m ∈ Mat(2, Q p ). If we define a new function f  : Q p × Q p → C by    f  (a1 , a2 ) := K

K Qp

  a1 −1  k 0

x a2

  k

f (k) d × k d × k d x, (a1 , a2 ∈ Q p ), · f (k ) (

448

The Godement-Jacquet L-function for GL(2, AQ )

then f  (a1 , a2 ) is a Bruhat-Schwartz function of a1 , a2 ∈ Q p (that is, it is locally constant and compactly supported, see Exercise 11.13). Consequently we may write our original integral in the form  

f  (a1 , a2 ) |a1 |sp χ1 (a1 ) |a2 |s χ2 (a2 ) d × a1 d × a2

Z p (s, , β) = Q×p Q×p

= P( p s , p −s ) · L p (s, χ1 ) · L p (s, χ2 ),

(11.11.3)

by the G L(1) theory (see Chapter 2). Here, P is a polynomial, and L p (s, χi ) is defined as in Section 2.4. To obtain the functional equation of Z p (s, , β), we repeat all of the pre∨

, β ) where   is the Fourier vious calculations for the function Z p (1 − s,  ∨

transform as in Theorem 1.6.8 and β is defined by ∨

−1



f (k)( f (kg) d × k,

β (g) = β(g ) = K

for g ∈ G L(2, Q p ). In order to have convergence, (s) must be less than 32 . Define        ∨ a1 x −1  k  k f  (a1 , a2 ) := 0 a2 K K Qp

· f (k) ( f (k ) d × k d × kd x, (a1 , a2 ∈ Q×p ). ∨

Note that the definition of f   is slightly different from that of f  in that the roles of f and ( f are reversed. This is consonant with the difference between β ∨ and β . The calculations leading to (11.11.3) immediately give  ∨ , β = Z p 1 − s, 

 

∨

1−s −1 1−s −1 × × f  (a1 , a2 ) |a1 | p χ1 (a1 ) |a2 | p χ2 (a2 ) d a1 d a2

Q×p Q×p

    = Q( p s , p −s ) · L p 1 − s, χ1−1 · L p 1 − s, χ2−1 , (11.11.4) for some polynomial Q. ∨

∨

 Claim: The function f   is the Fourier transform of f  , i.e., f   = f . Proof of claim Define two functions, T : Q p ×Q p → C and T  : Q p ×Q p → C, as follows:

11.11 Proof of the local functional equation for principal series 449       a1 x a1 x  d x, T d x.   (a , a ) := T (a1 , a2 ) :=  1 2 0 a2 0 a2 Qp Qp Let e p : Q p → C be the additive character given in Definition 1.6.3. Recall the definition of the Fourier transform  



a1 0

x a2





    

= Qp Qp Qp Qp

α γ

β δ



· e p (−αa1 − δa2 − γ x) dα dβ dγ dδ. It follows that   T  (a1 , a2 ) =

⎡ ⎢ ⎣

Qp Qp



    Qp Qp Qp



   

= Qp Qp Qp

α γ

β δ



⎤ ⎥ e p (−αa1 − δa2 ) dα dβ dδ ⎦

· e p (−γ x)dγ d x  α β e p (−αa1 − δa2 ) dα dβ dδ =  T (a1 , a2 ). 0 δ

Indeed, if we define 

   

F(γ ) := Qp Qp Qp

α γ

β δ

 e p (−αa1 − δa2 ) dα dβ dδ,

 then the integral in the variable γ is equal to F(x). The integral in x returns  N F(0) = F(0). This proves that T  = T . Now,   f  (a1 , a2 ) = K

K

Tk1 k2 (a1 , a2 ) f (k2 )( f (k1 ) d × k1 d × k2 ,

k1

k2 (g) := (k2−1 gk1 ).

k1 . Further, interchanging the order 1 k 2 = k 2  An easy calculation shows that k of two integrals that are in fact finite sums yields  f  (a1 , a2 ) =

  K

 

K

= K

K

k1 k2 (a1 , a2 ) f (k 2 ) ( f (k1 ) d × k1 d × k2 T k (a , a ) f (k 2 ) ( Tk2 f (k1 ) d × k1 d × k2 = 1 1 2

This completes the proof of the claim.



∨

f  (a1 , a2 ).

450

The Godement-Jacquet L-function for GL(2, AQ )

It is an immediate consequence of (11.11.3), (11.11.4), and the above claim that   f  (a1 , a2 ) |a1 |sp χ1 (a1 ) |a2 |s χ2 (a2 ) d × a1 d × a2 , Z p (s, , β) = Q×p Q×p

 ∨ , β = Z p 1− s, 

 

−1 1−s −1 × ×  f  (a1 , a2 ) |a1 |1−s p χ1 (a1 ) |a2 | p χ2 (a2 ) d a1 d a2 .

Q×p Q×p

(11.11.5) The local functional equation for G L(1) in one variable (see (2.4.6)), for an arbitrary Bruhat-Schwartz function  : Q p → C, takes the form   −1 1−s × s ×  Q×p (a)χi (a) |a| p d a Q×p (a)χi (a) |a| p d a , (i = 1, 2). )· = (s, χ   i L p (s, χi ) L p 1 − s, χi−1 Finally, it follows from the above and (11.11.5) that for − 12 < (s) < 32 , we have the functional equation  ∨ , β Z p 1 − s,  Z p (s, , β) .     = (s, χ1 )(s, χ2 ) · −1 −1 L p (s, χ1 )L p (s, χ2 ) L p 1 − s, χ1 L p 1 − s, χ2 This is obvious if f  is of the form 1 (a1 )2 (a2 ) with 1 , 2 being BruhatSchwartz functions Q p → C. But every Bruhat-Schwartz function Q p ×Q p → C is a finite linear combination of functions of this form, so the result follows in general. See Exercise 11.3. 

11.12 The local functional equation for the unitary special representations of G L(2, Q p ) Following Section 11.10, we fix a character χ : Q×p → C and define −1

1

χ1 (a) := χ (a)|a| p2 ,

χ2 (a) := χ (a)|a| p 2 ,

(a ∈ Q×p ).

Then the local functional equation for principal series takes the form  (s, χ1 )(s, χ2 ) G L(2,Q p )

s+ 1

(g) β(g) |det(g)| p 2 × d g L p (s, χ1 )L p (s, χ2 )  = G L(2,Q p )

1   2 (g) β g −1 |det(g)|1−s+  p     d×g L p 1 − s, χ1−1 L p 1 − s, χ2−1

11.12 The local functional equation for the unitary special

451

which can be written more explicitly as 

s+ 1

(g) β(g) |det(g)| p 2 × (s, χ1 )(s, χ2 )    d g  1 −1 1 −1 1 − χ ( p) p −s− 2 · 1 − χ ( p) p −s+ 2 G L(2,Q p ) 3    (g) β g −1 |det(g)| p2 −s  × =  −1  −1 d g. 3 1 − +s − +s 1 − χ −1 ( p) p 2 · 1 − χ −1 ( p) p 2 G L(2,Q p ) (11.12.1) Now, we use the algebraic identity 1−x x(1 − x) = −x = −1 1−x x −1 with x = χ −1 ( p) p s− 2 . This implies that 1

1 − χ −1 ( p) p s− 2 1

1 − χ ( p) p

1 2 −s

= −χ −1 ( p) p s− 2 . 1

It immediately follows from this that (11.12.1) can be rewritten in the form I (s) := −χ

−1

( p) p

s− 12

 (s, χ1 )(s, χ2 ) G L(2,Q p )

 = I I (s) := G L(2,Q p )

s+ 1

(g) β(g) |det(g)| p 2 ×   d g 1 −1 1 − χ ( p) p −s− 2

3   (g) β g −1 |det(g)| p2 −s  ×  −1 d g. s− 32 −1 1 − χ ( p) p

Proposition 11.10.2 implies that I (s) is holomorphic for (s) > 0. In a similar manner, replacing s by 1−s, we see that I I (s) is holomorphic for (1−s) > 0. Since I (s) and I I (s) are equal up to a holomorphic function, it follows that both I (s) and I I (s) must be holomorphic for all s. This establishes Theorem 11.6.4 for the special representations, provided we make the following definition. Definition 11.12.2 (The local L-function and -factor associated to a special representation of G L(2, Q p )) Fix any character χ : Q×p → C× , and define 1

χ1 (a) := χ (a) · |a| p2 ,

−1

χ2 (a) := χ (a) · |a| p 2 ,

(a ∈ Q×p ).

Let (π, V ) denote the special representation of G L(2, Q p ) (as in 6.10) which is contained in the representation B p (χ1 , χ2 ), as in Section 11.10. Thus, V is

452

The Godement-Jacquet L-function for GL(2, AQ )

equal to the orthogonal complement of the function g → χ −1 (det g) (which is an element of B(χ1−1 , χ2−1 ). Then the local L-function associated to π , denoted L p (s, π ), is defined as   1 −1 L p (s, π ) := 1 − χ ( p) p −s− 2 , and the -factor in the functional equation in Theorem 11.6.4 (3) is given by p (s, π ) := −χ ( p)−1  (s, χ1 )  (s, χ2 ) p s− 2 . 1

11.13 Proof of the local functional equation for the supercuspidal representations of G L(2, Q p ) Let (π, V ) denote an irreducible and admissible supercuspidal representation of G L(2, Q p ) as in Definitions 6.13.1, 6.1.4. Let  : Mat(2, Q p ) → C be a Bruhat-Schwartz function as in Definition 11.6.2. Theorem 8.4.9 tells us that a matrix coefficient β : G L(2, Q p ) → C for the supercuspidal representation (π, V ) has compact support modulo the center of G L(2, Q p ). Consider the local zeta integral given, as in (11.6.3) by  s+ 1 (g) β(g) |det(g)| p 2 d × g. (11.13.1) Z p (s, , β) = G L(2,Q p )

Proposition 11.13.2 (The local zeta integral associated to a supercuspidal representation of G L(2, Q p ) is an entire function) Let (π, V ) be an irreducible and admissible supercuspidal representation of G L(2, Q p ) as in Definitions 6.13.1, 6.1.4. Then the local zeta integral given in (11.13.1) is an entire function of s. Proof This will follow directly from Theorem 8.4.9 which says that matrix coefficients of supercuspidal representations are compactly supported modulo the center. Recall the definition of the center of G L(2, Q p ): 5       r 0  × Z G L(2, Q p ) = z(r ) :=  r ∈ Qp ∼ = Q×p . 0 r Let ωπ : Q×p → C× be the central character of π , which is characterized by the   r 0 . v = ωπ (r )v for all v ∈ V, r ∈ Q×p . It immediately property that π 0r follows that we may rewrite the integral (11.13.1) in the form       r 0 2s+1 × Z p (s, , β) =  g ωπ (r ) |r | p d r 0 r Q×p Q×p \G L(2,Q p )

s+ 1

· β(g)| det g| p 2 d × g

(11.13.3)

11.13 Proof of the local functional equation

453

By Theorem 8.4.9, the matrix coefficient β has compact support on Q×p \G L(2, Q p ), which implies that the integral (11.13.3) converges absolutely for (s) sufficiently large. If ωπ is ramified it follows from the G L(1) theory (see Section 2.3) that the inner integral on the right side of (11.13.3), given by     r 0  g ωπ (r ) |r |2s+1 d ×r p 0 r Q×p is an entire function of s. This immediately implies that Z p (s, , β) is entire if ωπ is ramified. On the other hand, if ωπ is trivial (unramified) then the inner integral (11.13.3) has a pole at s = − 12 , with residue (0), independently of g. The residue of (11.13.3) itself is therefore  s+ 1 (0) · β(g) | det g| p 2 d × g = 0, Q×p \G L(2,Q p )

    because Q p β 10 x1 g d x = 0 by the definition of supercuspidal given in Definition 6.13.1. It follows that Z p (s, , β) is always an entire function of s if π is supercuspidal.  Recall that a function Mat(2, Q p ) → C is said to be Bruhat-Schwartz if it is locally constant and compactly supported, and that the vector space of all such Bruhat-Schwartz functions: G L(2, Q p ) → C is denoted S(Mat(2, Q p )). Definition 11.13.4 (Fourier transform on Mat(2, Q p )) Given  in  : Mat(2, Q p ) → C by the S(Mat(2, Q p )), define the Fourier transform  formula   (m) = (x)e p (−Tr (mx)) d x, (∀m ∈ Mat(2, Q p )), Mat(2,Q p )

where e p is the additive character on Q p of Definition 1.6.3, and Tr(m) denotes the trace of a matrix m ∈ Mat(2, Q p ). The proof of the local functional equation makes use of certain integral identities involving integration over G L(2, Q p ). It is necessary to have a space of test functions with support on G L(2, Q p ), such that their Fourier transforms also have support on G L(2, Q p ). This motivates the following definition. Definition 11.13.5 (The space S0 (Mat(2, Q p ))) Define S0 (Mat(2, Q p )) to be the subspace of S(Mat(2, Q p )) consisting of all functions  with support on  is also supported G L(2, Q p ), with the property that the Fourier transform  on G L(2, Q p ).

The Godement-Jacquet L-function for GL(2, AQ )

454

Remark The space S0 (Mat(2, Q p )) is characterized by the property that all functions in the space and their Fourier transforms vanish on matrices in Mat(2, Q p ) with determinant zero. To prove the local functional equation of Theorem 11.6.4, the next step is to establish the following key identity. Proposition 11.13.6 (Fundamental identity for local integrals of unitary supercuspidal representations of G L(2, Q p )) Let (π, V ) denote a unitary supercuspidal representation of G L(2, Q p ) as in Definition 6.13.1 and Section   9.3. Let ( π, ( V be its contragredient. Fix vectors v ∈ V, ( v ∈ ( V , and let  ,  : V ×( V → C be an invariant bilinear pairing as in Definition 8.4.1. Take  ∈ S(Mat(2, Q p )) and  ∈ S0 (Mat(2, Q p )), defined as in Definitions 11.6.2 and 11.13.5 respectively. Then   D E s+ 1   3 −s (h) π (g) . v, ( (g) π (h) .( v det g  p 2 det h  p2 d × g d × h G L(2,Q p ) G L(2,Q p )





= G L(2,Q p ) G L(2,Q p )

D     E (g)(h) π g −1 . v, (  π h −1 .( v  3 −s  s+ 1  · detg  p2 det h  p 2 d × g d × h,

(11.13.7)

and both integrals converge absolutely for − 12 < (s) < 32 . Proof First note that if  ∈ S0 (Mat(2, Q p )), then there exists sufficiently small  > 0 such that (g) = 0 if |det(g)| p < . This is because the function g → |det(g)| p is a continuous function, and the image of a compact set, such as the support of , is a compact set. If it does not contain 0, then it is bounded away from zero. For each fixed value of h, the integral over g is of the type considered in (11.13.1), and is easily seen to converge for (s) > −1/2. On the other hand,  has compact support in G L(2, Q p ), and is locally constant, while ( v since  is fixed by an open compact subgroup of G L(2, Q p ), the integral in h may be replaced by a finite sum. It follows that the integral on the left side of (11.13.7) converges absolutely for (s) > −1/2. By a similar argument, one may show that the integral on the right side converges absolutely for (s) < 3/2. It remains to prove the equality of the two integrals for − 12 < (s) < 32 . Let I (s) denote the integral on the left hand side of (11.13.7). If we make the change of variables g → hg, it follows that ⎛ ⎞   D E p ⎜ (h) dh ⎟ π (g) . v, ( v ⎝ (hg) I (s) = 2 ⎠ ( p − 1)( p − 1) G L(2,Q p )

G L(2,Q p )

s+ 1  · det g  p 2 d × g.

11.13 Proof of the local functional equation

455

Note that the measure on the inner integral here is the additive Haar measure p dh/| det h|2 . Simidh, not the multiplicative Haar measure d × h = ( p2 −1)( p−1) larly, let I I (s) denote the integral on the right hand side of (11.13.7). In this integral, make the change of variables g → gh. Then we obtain

I I (s) =

p ( p 2 − 1)( p − 1)



D

⎛ E ⎜ π (g) . v, ( v ⎝

G L(2,Q p )



G L(2,Q p )

We leave it to the reader to check that  f 1 (h) f 2 (h) dh = G L(2,Q p )



⎞ 



 g −1 h (h) dh ⎟  ⎠ s− 3  · det g  p 2 d × g.

 f 1 (h) f 2 (h) dh

G L(2,Q p )

for any two functions f 1 ∈ S(Mat(2, Q p ), f 2 ∈ S0 (Mat(2, Q p ) as in Definition 11.13.5. We now apply the above identity with f 1 (h) = (hg) (which  2 −1  implies f 1 (h) = |det(g)|  g h ) and f 2 = . Then plugging this into I (s), I I (s), above immediately proves the fundamental identity (11.13.7).  The next step is to define operator-valued local zeta integrals. This is done as follows. Definition 11.13.8 (Operator valued local zeta integrals) Let (π, V ) denote an irreducible admissible supercuspidal representation of G L(2, Q p ) as in Definitions 6.13.1, 6.1.4. Let  : Mat(2, Q p ) → C be a Bruhat-Schwartz function as in Definition 11.6.2. Define the operator valued local zeta integral Z p (s, , π ) : V → V by the formula 

s+ 1

(g)| det g| p 2 π (g) d × g.

Z p (s, , π) = G L(2,Q p ) ∨

∨

Similarly, let π be defined by π (g) = π (g −1 ), (∀g ∈ G L(2, Q p )), and set ∨



s+ 1

(g)| det g| p 2 π (g −1 ) d × g.

Z p (s, , π ) = G L(2,Q p )

If the support of  is a compact subset of G L(2, Q p ), then these integrals converge absolutely for all values of s. It follows from Definition 11.13.8 that the left hand side of (11.13.7) is equal to D

E   , π∨ . Z p (s, , π) . v, ( Z p 1 − s,  v,

(11.13.9)

456

The Godement-Jacquet L-function for GL(2, AQ )

while the right hand side is D E   , π∨ . v, ( Z p (s, , π ) . Z p 1 − s,  v.

(11.13.10)

To prove the local functional equation of Theorem 11.6.4, we require the following three lemmas. The proofs of Lemmas 11.13.11, 11.13.12 will be deferred until later. Lemma 11.13.11 Given v ∈ V, ( v ∈ ( V , and s ∈ C, there exists  ∈ S0 (Mat(2, Q p )) such that Z p (s, , π) . w = w,( v · v for all w ∈ V. Lemma 11.13.12 Given v ∈ V, define the subset  5   ∃ ∈ S0 (Mat(2, Q p )), c = / 0, n ∈ Z  . Uv := u ∈ V  such that Z p (s, , π) . v = cp −ns u (∀s ∈ C) Then Uv spans v for all v ∈ V. Lemma 11.13.13 For all s ∈ C, there exists unique γ (s) ∈ C such that   , π∨ = γ (s) Z p (s, , π), Z p 1 − s,  for all  ∈ S0 (Mat(2, Q p )). Proof of Lemma 11.13.13 Assume there exists an operator γ (s) : V → V   , π∨ = γ (s) . Z p (s, , π), (where . denotes composition such that Z p 1 − s,  of operators) for all  ∈ S0 (Mat(2, Q p )). The first step is to show that γ (s) must be unique. This will follow from Lemma 11.13.11. In fact, if v is any element of V, then by Lemma 11.13.11, there exists  ∈ S0 (Mat(2, Q p )) such   , π∨ . v. that Z p (s, , π) . v = v. It is immediate that γ (s) . v = Z p 1 − s,  For existence of an operator γ (s), the crucial issue is the following: for given v ∈ V there may exist many choices for  satisfying Z p (s, , π) . v = v.   , π∨ . v depends on a specific choice of , then there If the value of Z p 1−s,  will be no well-defined operator γ (s). Thus, take two functions 1 , 2 such that Z p (s, 1 , π ) . v = Z p (s, 2 , π ) . v = v.     1 , π∨ . v = Z p 1 − s,  2 , π∨ . v. It is necessary to prove that Z p 1 − s,  Suppose not. Let       1 , π∨ . v − Z p 1 − s,  2 , π∨ . v = Z p 1 − s,  1 −  2 , π∨ . v. w = Z p 1 − s,  Choose  ∈ S0 (Mat(2, Q p )) such that Z p (s, , π ) . w = w. Then   1 −  2 , π∨ . v = w =/ 0. Z p (s, , π) . Z 1 − s, 

11.13 Proof of the local functional equation

457

On the other hand     , π∨ . Z p (s, 1 − 2 , π ) . v = Z p 1 − s,  , π∨ . 0 = 0. Z p 1 − s,  This is a contradiction, in view of (11.13.9), (11.13.10), and Proposition 11.13.6. As a consequence, there is a well defined function γ (s) : V → V   , π∨ . v, where  is any which is given on the vector v ∈ V by Z 1 − s,  element of S0 (Mat(2, Q p )) satisfying Z p (s, , π) . v = v. The next step is to verify that γ (s) : V → V is a linear operator. This is accomplished by generalizing the preceding argument. Suppose that α · v1 + β · v2 = v3 , where v1 , v2 , v3 ∈ V and α, β ∈ C. For each i from 1 to 3 take i ∈ S0 (Mat(2, Q p )) satisfying Z p (s, i , π ) . vi = vi . It is necessary to show that 1 , π ) . v1 + β · Z p (1 − s,  2 , π ) . v2 − Z p (1 − s,  3 , π ) . v3 = 0. α · Z p (1 − s,  This follows exactly as before. It remains to prove that the linear operator γ (s) : V → V is a scalar. For fixed h ∈ G L(2, Q p ), define the function h : Mat(2, Q p ) → C by h (m) := (hm),

(m ∈ Mat(2, Q p )).

−1 h (m) = | det h|−2  Then  p (mh ). It is obvious that h is again in the space S0 (Mat(2, Q p )). By routine computations, −s− 12

Z p (s, h , π ) . v = | det h| p

π (h −1 ) . Z p (s, , π) . v,

1     2 h , π∨ . v = | det h|−s− , π∨ . v. Z p 1 − s,  π (h −1 ) . Z p 1 − s,  p

It follows that   , π∨ . v = π (h) . γ (s) . π (h −1 ) . Z p (s, , π) . v, Z p 1 − s,  for all v ∈ V, h ∈ G L(2, Q p ), and  ∈ S0 (Mat(2, Q p ). When  is chosen so that Z p (s, , π) . v = v, this becomes γ (s) . v = π (h) . γ (s) . π (h −1 ) . v. As a consequence, γ (s) commutes with π (h) for all h ∈ G L(2, Q p ). Since π is assumed to be irreducible, it follows from Dixmier’s Lemma 6.1.8 that γ (s) is scalar. Finally, it is necessary to prove that the identity   , π∨ . v γ (s) . Z p (s, , π) . v = Z p 1 − s,  holds for all v ∈ V and  ∈ S(Mat(2, Q p )). Up to now, it is only known that this identity holds when  ∈ S0 (Mat(2, Q p ) and Z p (s, , π) . v = v.

458

The Godement-Jacquet L-function for GL(2, AQ )

For general  and v, choose  Z p (s, , π) . v = v. Then

∈

S0 (Mat(2, Q p )) such that

    , π∨ . v = Z p 1 − s,  , π∨ . Z p (s, , π ) . v Z p 1 − s,    , π∨ . v = Z p (s, , π) . γ (s) . v. = Z p s, , π) . Z p (1 − s,  Because γ (s) is scalar, this is equal to γ (s) . Z p (s, , π) . v.



Proof of Theorem 11.6.4 for supercuspidal representations The absolute convergence of the local zeta integral was already shown in Proposition 11.13.2. It was further shown in Proposition 11.13.2 that the local zeta integral has analytic continuation to C as an entire function for any  and β. It follows that 1 is a common divisor of local zeta integrals Z p (s, , β) in the sense of Theorem 11.6.4 (2). Finally, it was already shown in Theorem 11.8.3, that one may choose , β so that Z p (s, , β) = 1, which implies that 1 is, in fact, a greatest common divisor, and justifies Definition 11.8.4, which states that L p (s, π ) = 1 for π supercuspidal. ∨

If β(g) = π (g) . v, ( v is a matrix coefficient of (π, V ), define β (g) = β(g −1 ), for all g ∈ G L(2, Q p ). Then for s ∈ C with (s) > − 12 , and  ∈ S(Mat(2, Q p )), the local zeta integral Z p (s, , β) is given in terms of the operator-valued local zeta integral by the identity v. Z p (s, , β) = Z p (s, , π) . v, ( It, therefore, follows immediately from Lemma 11.13.13 that  ∨ , β = γ (s) Z p (s, , β) Z p 1 − s, 

(11.13.14)

whenever both integrals are convergent. This is the case in the strip − 12 < (s) < 32 . To complete the proof of Theorem 11.6.4, part (3), it is necessary to prove that γ (s) is a rational function of p −s . It turns out that γ (s) is actually a monomial. To see this, recall from Lemma 11.13.12 that the set  5   ∃ ∈ S0 (Mat(2, Q p )), c =/ 0, n ∈ Z  (11.13.15) Uv := u ∈ V  such that Z p (s, , π) . v = cp −ns u (∀s ∈ C) spans V for any fixed v ∈ V. Consequently, there exist  ∈ S0 (Mat(2, Q p ))  has and a matrix coefficient β such that Z p (s, , β) is just cp −ns . Because  compact support,  ∨ , β = γ (s) · Z p (s, , β). Z p 1 − s,  !" # !" #

= polynomial in p−s , ps

= cp−ns

11.13 Proof of the local functional equation

459

 ∨ , β It follows that γ (s) ∈ C[ p −s , p s ]. Now choose  so that Z p 1 − s,  is a monomial c p −ns . Then it follows that γ (s) and Z p (s, , β) are also monomials. The identity (11.13.14) gives the analytic continuation of both sides as entire functions of s, for all  ∈ S(Mat(2, Q p )) and all matrix coefficients β.  Finally, it is necessary to give the proofs of Lemmas 11.13.11 and 11.13.12. Proof of Lemma 11.13.12 Let Uv be the set defined in (11.13.15). To prove v ∈ ( V , and let β(g) = π (g) . v, ( v be the matrix that Uv is non-zero, take ( coefficient corresponding to v ∈ V and ( v∈( V . Define  (g) =

β(g), if g ∈ G L(2, Q p ) and |det(g)| p = 1, 0, otherwise.

(11.13.16)

It must be shown that  lies in the space S0 (Mat(2, Q p )). It is clear that   has the same has compact support on G L(2, Q p ). The issue is to show that  property. This will require the identity  Qp





 g1 ·

1 0

x 1



 · g2

d x = 0,

  ∀g1 , g2 ∈ G L(2, Q p .

(11.13.17) Now, (11.13.17) is obvious if |det(g1 ) · det(g2 )| p =/ 1, because  is supported on matrices of determinant 1. For any g1 , g2 ∈ G L(2, Q p ) with v. Then (11.13.17) |det(g1 ) · det(g2 )| p = 1, set v := π (g2 ) . v and v( := π (g1 ) .( takes the form  F  1 π 0

x 1



G

( . v , v d x = 0.

(11.13.18)

Qp

It follows immediately from the definition of a supercuspidal representation, given in Definition 6.13.1, that 

 p−N Z p

π

1 0

x 1



. v d x = 0

for any sufficiently large integer N other hand,E it follows from ProposiD . On  the 1 x . v , v( is compactly supported. tion 8.4.9 that the function x → π 0 1 D   E 1 x

( = 0 for all x ∈ Taking N large enough that π . v / p −N Z p , we , v 0 1 deduce (11.13.18), and hence (11.13.17).

The Godement-Jacquet L-function for GL(2, AQ )

460

(m) = 0 for all singular matrices m ∈ Mat(2, Q p ), we To prove that  observe that any such matrix may be expressed as g2−1

 ·

a 0

0 0



· g1−1

for suitable g1 , g2 ∈ G L(2, Q p ) and a = 0 or 1. Then 

(m) = 

  (m )e p −Tr(m · m) dm

Mat(2,Q p )



= | det(g1 )|2p · | det(g2 )|2p

(11.13.19)

(g1 m g2 ) Mat(2,Q p )

   a

· e p −Tr m · 0

0 0



dm .

Now,  is supported in G L(2, Q p ), so the same is true of the function m → (g1 m g2 ). Hence, the integral in m may be taken only over G L(2, Q p ). The measure is not the Haar measure for G L(2, Q p ) but the additive product measure. However, it was shown in Proposition 8.10.3 that the two are related by a simple factor of | det m |2p . Using a simple modification of Proposition 8.10.7, we may express the integral over m ∈ G L(2, Q p ) in Bruhat coordinates



m =

1 0

x1 1



t1 0

0 t2

   0 1 1 · 1 0 0

x2 1

 .

We may  regard the  integral over x2 ∈ Q p as an integral over N where

 then 1 x | x ∈ Q p , and regard the remaining integrals as one integral N = 0 1 over G L(2, Q p )/N . With these conventions in place, (11.13.19) becomes  | det(g1 g2 )|2p G L(2,Q p )/N



    a 0 (g1 ·m ·n·g2 ) e p −Tr m · n · dn dm . 0 0 N

The key point is that n · (11.13.20) takes the form



a 0 0 0



 =



a 0 0 0



(11.13.20) for all n ∈ N . As a consequence,





(g1 · m · n · g2 ) dn

| det(g1 g2 )|2p G L(2,Q p )/N

N

   a · e p −Tr m · 0

0 0



dm .

(11.13.21)

11.13 Proof of the local functional equation

461

Now (11.13.17) implies that (11.13.21) equals zero. This completes the proof that the function  given in (11.13.16) is an element of the space S0 (Mat(2, Q p )). Then D E Z p (s, , π) . v, ( v = Z p (s, , β). As was shown in Theorem 11.8.3, this is simply a non-zero constant. Consequently, if we set u = Z p (s, , π) . v, then u is a non-zero element of Uv . As noted during the proof of Lemma 11.13.13, if h is defined by h (x) = −1 (hx) then Z p (s, h , π ) = | det h|−2 p · π (h ) ◦ Z p (s, , π). It follows that −1 π (h ) . u ∈ Uv for all u ∈ Uv and h ∈ G L(2, Q p ). The span of Uv must, therefore, be V, because V is irreducible.  In order to prove Lemma 11.13.11, it is necessary to prove a version of the Schur orthogonality relations. Proposition 11.13.22 (Schur orthogonality for unitary supercuspidal representations) Let (π, V ) denote an irreducible unitary supercuspidal π, ( V ), as in Definirepresentation of G L(2, Q p ), with contragredient (( tion 6.13.1 and Proposition 8.4.7. Let  ,  denote the canonical invariant bilinear form V × ( V → C, given in Definition 8.1.12. There is a complex constant c =/ 0 such that  8 9 8 9 π (g) . v, ( v w, ( π (g) . w ( d × g = c · v, w ( · w,( v, (11.13.23) G L(2,Q p )/Q×p

for all v, w ∈ V and all ( v, w (∈( V. Proof We first claim that for each fixed ( v ∈ ( V and w ∈ V, the integral in (11.13.23) is equal to some constant (which depends on ( v and w) times the invariant bilinear form v, w (. It is clear that (11.13.23) is linear in v and w (, so it defines a bilinear form, which, by Corollary 8.1.11, will be a scalar multiple of v, w (, if we can show that it is invariant. This is easily seen, because if v is replaced by π (h) . v and w ( is replaced by ( π (h) . w ( for some h ∈ G L(2, Q p ), then the change of variables g → gh −1 in the integral on the left side of (11.13.23) returns it to the original form. It follows that the integral in (11.13.23) is equal to c(( v, w) · v, w (, where c(( v, w) is a complex constant which depends on ( v and w. On the other hand, for each fixed v and w (, the function c(( v, w) is a bilinear form in ( v and w, and a similar argument shows that it is invariant. Therefore, c(( v, w) = c · w,( v, for some c ∈ C. What is left is to show that c =/ 0. Recall that (π, V ) is assumed to be unitary. Let ( , ) denote an invariant positive definite Hermitian form on V. As in

The Godement-Jacquet L-function for GL(2, AQ )

462

Lemma 9.1.6, it is possible to express the matrix coefficients of (π, V ) using ( , ) instead of the canonical bilinear form  , . More precisely, there is a conjugate-linear map L : ( V → V, which is an R-linear isomorphism, such that π (g), ( v = (π (g), L(( v)) and w, ( π (g) . w ( = (w, π (g) . L(( w)) for all v = w ( = L −1 (v). Then the left g ∈ G L(2, Q p ). Choose w = v, and choose ( side of (11.13.23) becomes 



   π (g) . v, v v, π (g) . v d × g =

G L(2,Q p )/Q×p

which is clearly non-zero.

   π (g) . v, v 2 d × g, C

G L(2,Q p )/Q×p



Proof of Lemma 11.13.11 First, note that we may assume s = − 12 . Otherwise −s− 1

modify the function  by | det | p 2 . Define 9 8 v, ( π (g) .( v , if |det(g)| p = 1 or p, (g) = 0, otherwise. This is almost the same function which was used in the proof of Lemma 11.13.12, and the proof that this function lies in S0 (Mat(2, Q p )) is the same. For any w ∈ V and w (∈( V, D

Z p (−1/2, , π) . w,

P

O

E

w ( =  =

Q×p

×

(g)π (g) . w d g,

w (

8 98 9 v, ( π (g) .( v π (g) . w, w ( d × g, Ep (11.13.24)

where E p is the set of elements of G L(2, Q p ) of determinant 1 or p, as in Proposition 8.8.4. This agrees with our definition of the integral over G L(2, Q p )/Q×p , given in Definition 8.10.18. By Schur orthogonality, Proposition 11.13.22, the integral (11.13.24) is equal to c · w, ( vv, w (, for some non-zero constant c ∈ C. Modifying  by a factor of c−1 , we obtain, for fixed v ∈ V, ( v∈( V , that there exists a function  ∈ S0 (Mat(2, Q p )) such that D

  ∀w ∈ V, w (∈( V . (11.13.25) Given v1 , v2 ∈ V, one can easily show that v1 = v2 if and only if ( = v2 , w ( for all w ( ∈ ( V . Lemma 11.13.11 immediately follows v1 , w from (11.13.25).  Z p (−1/2, , π ) . w,

E w ( = w, ( vv, w (,

11.14 The local L-function for irreducible principal

463

11.14 The local L-function for irreducible principal series representations of G L(2, R) Let χ1 , χ2 : R× → C be characters of R× . Let B∞ (χ1 , χ2 ) be the space 

   f : G L(2, R) → C  

 f

a 0

b d

  a 1  2 g =   χ1 (a)χ2 (d) f (g), d ∞



×

∀a, d ∈ R , b ∈ R, g ∈ G L(2, R), f is O(2, R)-finite and smooth .   Consider the principal series representation π, B∞ (χ1 , χ2 ) as in Definition 8.6.7. It follows from Theorem 8.6.10 that the contragredient rep   π,( B∞ (χ1 , χ2 ) , is isomorphic to resentation of π, B∞ (χ1 , χ2 ) , denoted (   π, B∞ (χ1−1 , χ2−1 ) . Fix   f 2 ∈ π, B∞ (χ1−1 , χ2−1 ) .

f 1 ∈ B∞ (χ1 , χ2 ),

According to Definition 8.6.12,  a matrix coefficient β for the principal series  representation π, B∞ (χ1 , χ2 ) is given by 



2π

β(g) :=

f1 0

cos θ − sin θ

sin θ cos θ



  · g · f2

cos θ − sin θ

sin θ cos θ

 dθ. (11.14.1)

With these preliminaries in place, we now address the problem of evaluating the local zeta integral given in Theorem 11.5.2 for a special choice of test function in the case of a principal series representation. Theorem 11.14.2 (Evaluation of the local zeta integral of a principal series representation for a special choice of test function ) Let χ1 , χ2 : R× → C be characters of R× , given by  χ1 (t) =

t |t|∞

δ1

|t|ν∞1 ,

 χ2 (t) =

t |t|∞

δ2

|t|ν∞2 ,

  with ν1 , ν2 ∈ C and δ1 , δ2 ∈ {0, 1}. Let π, B∞ (χ1 , χ2 ) be the principal series representation as defined in Proposition 8.2.3 with matrix coefficient β given by (11.14.1). Let  : Mat(2, R) → C be a Bruhat-Schwartz function satisfying  

a c

b d



= a δ1 d δ2 · e−π (a

2

+b2 +c2 +d 2 )

.

The Godement-Jacquet L-function for GL(2, AQ )

464

  Then for a suitable choice of f 1 ∈ B∞ (χ1 , χ2 ), f 2 ∈ π, B∞ (χ1−1 , χ2−1 ) , the local zeta integral of Theorem 11.5.2 can be explicitly evaluated as     s+ν +δ s+ν +δ s + ν1 + δ1 s + ν2 + δ2 − 12 1 − 22 2   Z ∞ (s, , β) = π ·π . 2 2 Proof It will be convenient to introduce the notation   cos θ sin θ k(θ ) := . − sin θ cos θ It follows from the definition of the local zeta integral given in Theorem 11.5.2 and the definition of the matrix coefficient given in (11.14.1) that :



;

2π

Z ∞ (s, , β) =

s+ 12

f 1 (k(θ ) · g) · f 2 (k(θ )) dθ (g) |det(g)| p

d × g.

0

G L(2,R)

(11.14.3) In the integral (11.14.3) make the transformation g →  k(θ ) g and also make the substitutions (see Proposition 8.6.1)   a1 x k(θ1 ), d × g = d × a1 d × a2 d x dθ1 |a1 |−1 g= ∞. 0 a2 −1

It immediately follows that 

2π 2π ∞   Z ∞ (s, , β) =

f1 0

0

  a1 −1 ·  k(θ ) 0

−∞

x a2

R×



R×



a1 0

s− 1

x a2



 k(θ1 )

f 2 (k(θ ))

s+ 1

k(θ1 ) |a1 |∞ 2 |a2 |∞ 2 d × a1 d × a2 d x d × k d × k1 .

Since f 1 ∈ B∞ (χ1 , χ2 ) we have  f1

a1 0

x a2



 1  a1  2 k(θ1 ) =   χ1 (a1 ) χ2 (a2 ) f (k(θ1 )). a2 ∞ 

Consequently 2π 2π ∞   Z ∞ (s, , β) = 0

·

0 −∞ R× R× χ1 (a1 ) |a1 |s∞

  a f 1 (k(θ1 )) f 2 (k(θ ))  k(θ )−1 1 0

  x k(θ1 ) a2

χ2 (a2 ) |a2 |s∞ d × a1 d × a2 d x d × k d × k1 . (11.14.4)

11.14 The local L-function for irreducible principal

465

We want to show, for a suitable choice of test function , and for elements f 1 ∈ B∞ (χ1 , χ2 ), f 2 ∈ B∞ (χ1−1 , χ2−1 ), that the local zeta integral Z ∞ (s, , β) given in (11.9.4) is a multiple of L ∞ (s, χ1 ) · L ∞ (s, χ2 ) where L ∞ (s, χi ) := π

−

s+νi +δi 2



s + νi + δi  2

 ,

(i = 1, 2).

To do this, we simply choose the test function  in the form  

a c

b d



= a δ1 d δ2 e−π(a

2

+b2 +c2 +d 2 )

.

We now show why this choice of  works. An easy calculation shows that   −1 a1  k(θ ) 0

  δ  x k(θ1 ) = a1 cos θ cos θ1 − x cos θ sin θ1 + a2 sin θ sin θ1 1 a2 δ  2 2 2 · a1 sin θ sin θ1 + x sin θ cos θ1 + a2 cos θ cos θ1 2 e−π(a1 +a2 +x ) .

We show that in every case, f 1 ∈ B∞ (χ1 , χ2 ) and f 2 ∈ B∞ (χ1−1 , χ2−1 ) may be chosen so that 2π 2π 0

  a1  k(θ )−1 0

x a2



 k(θ1 )

f 1 (k(θ1 )) f 2 (k(θ ))dθ dθ1

0

= a1δ1 a2δ2 e−π(a1 +a2 +x ) . 2

2

2

(11.14.5) The key point is that f 1 (k(θ1 )) may be taken to be any finite C-linear combination of the functions eiθ1 , where  has the same parity as δ1 + δ2 . Likewise f 2 (k(θ )) is an arbitrary finite C-linear combination of the functions eiθ , subject to the same parity condition. If δ1 = δ2 = 0, then we may obtain (11.14.5) simply by choosing f 1 (k(θ1 )) and f 2 (k(θ )) to be constant. If δ1 = 1, while δ2 = 0, then we may choose f 1 (k(θ1 )) = π1 cos(θ1 ) and f 2 (k(θ )) = π1 cos(θ ). Then (11.14.5) follows from the fact that 

2π 0

 cos θ sin θ dθ = 0, 0

2π

 cos2 θ dθ =

2π

sin2 θ dθ = π.

0

If δ1 = 0 while δ2 = 1, then the same choice of f 1 and f 2 works. Finally, if δ1 = δ2 = 1, then the same identities show by routine computation that (11.14.5) once again holds when f 1 (k(θ1 )) and f 2 (k(θ )) are constant.

466

The Godement-Jacquet L-function for GL(2, AQ )

Plugging (11.14.5) into (11.14.4) yields ∞   Z ∞ (s, , β) =

a1δ1 a2δ2 e−π(a1 +a2 +x ) χ1 (a1 ) 2

2

2

−∞ R× R×

⎛ =⎝

⎞

∞ e

−π x 2

d x⎠ ·

−∞

 R×

· |a1 |s∞ χ2 (a2 ) |a2 |s∞ d × a1 d × a2 d x  −πa12 s+ν1 +δ1 × e |a1 |∞ d a1

 2 × 2 +δ2 e−πa2 |a2 |s+ν d a 2 ∞ R×     s+ν1 +δ1 s+ν2 +δ2 s + ν1 + δ1 s + ν2 + δ2 · π− 2  .  = π− 2  2 2 

·

Definition 11.14.6 (The local L-function of an irreducible principal series representation of G L(2, R)) Let (π, V ) denote an irreducible admissible (g, K ∞ )-module (as in Definition 7.1.4) which is isomorphic to the principal series representation with space of functions B∞ (χ1 , χ2 ), as defined in Definition 8.6.7, where χi (t) = |t|ν∞i sign(t)δi ,

(νi ∈ C, δi ∈ {0, 1}),

(i = 1, 2).

Then the local L-function associated to π , denoted L ∞ (s, π ), is defined as L ∞ (s, π ) := π

−

s+ν1 +δ1 2

   s+ν +δ s + ν1 + δ1 s + ν2 + δ2 − 22 2 ·π ,   2 2 

(s ∈ C).

Conjecture 11.14.7 (Ramanujan conjecture at ∞) Let (π, V ) denote an irreducible principal series (g, K ∞ )-module with local L-function given as in Definition 11.14.6. Assume that π is the local component at ∞ of a global cuspidal automorphic representation of G L(2, AQ ) as in Conjecture 10.9.1. Then (ν1 ) = (ν2 ) = 0. Remarks Conjecture 11.14.7 follows immediately from Conjecture 10.9.1. In fact, Proposition 9.4.8 tells that if an irreducible principal series representation B∞ (χ1 , χ2 ) is tempered then χ1 , χ2 must be unitary. Note that conjecture 11.14.7 is precisely the Selberg eigenvalue Conjecture in the case that π is constructed from an adelic lift of a classical automorphic form of weight zero as in Section 5.4.

11.15 Proof of the local functional equation for principal

467

11.15 Proof of the local functional equation for principal series representations of G L(2, R) In this section, the analogue of Theorem 11.6.5 for the real place will be given. Following Godement and Jacquet, we consider only those Schwartz functions of the form   2 2 2 2 a b (P a polynomial).  = e−π(a +b +c +d ) P(a, b, c, d), c d Theorem 11.15.1 (Local functional equation in the real case) Let (π, V ) be an admissible irreducible (g, K ∞ )-module as in Definitions 7.1.1, 7.1.4. Let β : G L(2, R) → C be a matrix coefficient as in (8.6.13) and let  : G L(2, R) → C be given by   2 2 2 2 a b (11.15.2)  = e−π(a +b +c +d ) P(a, b, c, d), c d where P is a polynomial. Consider, Z ∞ (s, , β), the local zeta integral defined in Theorem 11.5.2. Then we have. (1) There exists s0 ∈ R such that the local zeta integral converges absolutely for (s) > s0 . (2) For each polynomial P, and each matrix coefficient β, the local zeta integral is a finite linear combination of functions of the form     s+r1 s+r2 π − 2  s+r2 1 π − 2  s+r2 2 , for some r1 , r2 ∈ C. The set of such functions obtained as P varies admits a common divisor, L ∞ (s, π ), that does not depend on  and β and is unique up to a scalar factor. Its main property is that the ratio Z ∞ (s, , β) L ∞ (s, π ) is a polynomial function of s which is identically 1 for suitable . (3) There exists a meromorphic function (which does not depend on  or β), denoted γ (s, π ), such that  ∨ , β = γ (s, π ) Z ∞ (s, , β). Z ∞ 1 − s,   is the Fourier transform Here   



α γ

β δ

 :=

∞ ∞ ∞ ∞  p  r

−∞−∞−∞−∞

q s



  e − pα − qγ − rβ − sδ · dp dq dr ds,

The Godement-Jacquet L-function for GL(2, AQ )

468 ∨

∨

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 )

Proof The absolute convergence for (s) sufficiently large follows from Theorem 8.11.10. To prove the rest, recall that by Theorem 7.5.10, the (g, K ∞ )-module may always be realized either as an irreducible principal series representation or as a subrepresentation of a reducible one. Assume that V ⊆ B∞ (χ1 , χ2 ) where χi (t) = |t|ν∞i sign(t)δi for all t ∈ R× , where νi ∈ C, δi ∈ {0, 1} for i = 1, 2. It follows as in the proof of Theorem the form (11.14.4). Recall that we have assumed 11.14.2 that Z ∞ (s, , β) is of 2 2 2 2 a b = e−π(a +b +c +d ) P(a, b, c, d), where P is a  to be of the form  c d polynomial. It is clear that if     a1 x a b −1 k(θ1 ) = k(θ ) 0 a2 c d as in (11.14.4), then e−π(a +b +c +d ) = e−π(a1 +a2 +x ) . Further, P(a, b, c, d) is a polynomial in a1 , a2 , x, cos θ, sin θ, cos θ1 and sin θ1 . It follows that 2





2π

0

2π

0

2

2

e R×

2

2

  a1 f 1 (k(θ1 )) f 2 (k(θ )) k(θ )−1 0

is a polynomial Q(a1 , a2 , x). Now, for any positive integer n,   −πa12

2

2

χ1 (a1 )|a1 |s∞ a1n

da1 =

π− 0,

s+ν1 +n 2



x a2

 s+ν1 +n  2



 k(θ1 )

dθ dθ1

, if n ≡ δ1 (mod 2), if not.

Further, if n = 2k + δ1 then s+ν1 +n 2



s+ν1 +δ1 2



π− π−

 s+ν1 +n 

−k 2  s+ν1 +δ1  = π · (s + ν1 + δ1 ) · . . . · (s + ν1 + n − 1) . 2

The same applies to the integral in a2 . The integral in x is simply constant. This completes the proof of (2) in the case when π is an irreducible principal series representation. Furthermore, it proves that (2) actually holds for

11.15 Proof of the local functional equation for principal

469

reducible principal series representations as well. To handle the general case, consider     1 d t·Z t·Z s+ 2 t·Z ×  (g · e )| det g · e |∞ β(g · e ) d g  . (11.15.3)  dt  G L(2,R) t =0





Here Z denotes the matrix 10 01 , regarded as an element of g. On the one hand, it follows from a simple change of variable that the integral in (11.15.3) is actually independent of t, so that the derivative is zero. On the other hand, for (s) sufficiently large the integral is absolutely and uniformly convergent, permitting differentiation under the integral sign. This yields   d s+ 1 (g · et·Z )t=0 | det g|∞ 2 β(g) d × g 0= dt G L(2,R)   s+ 1 d   det g · et·Z ∞ 2 β(g · et·Z )  (g) d×g + t =0 dt G L(2,R)

 d  t(2s+1) e ωπ (et ) t = 0 dt =Z ∞ (s, D Z . , β) + [(2s + 1) + μ] · Z ∞ (s, , β), =Z ∞ (s, D Z . , β) + Z ∞ (s, , β) ·

(11.15.4)

where D Z denotes the differential operator corresponding to Z ∈ g defined as in Definition 4.5.1, ωπ denotes the central character of π, and μ is the complex constant defined by μ = dtd ωπ (et ). Now, the (g, K ∞ )-module (π, V ) has been realized as a subspace of a reducible principal series representation B∞ (χ1 , χ2 ) for some characters χ1 , χ2 : R× → C. It follows from Theorem 7.5.10 (see also Proposition 8.11.6) that one may assume m−1

χ1 (a) = |a|∞2

+ν

sign(a)δ ,

−m+1

χ2 (a) = |a|∞2

+ν

,

(∀ a ∈ R× ),

for some m ∈ Z, m > 0, ν ∈ R and δ ∈ {0, 1} with δ ≡ m (mod 2). Let G(s) equal π−

s+ν+δ m+1 2 − 4

 

   m+1 s+ν+δ m+1 s+ν s+ν m+1 + π− 2 + 4  − . 2 4 2 4

The function G(s) is essentially the local L function for the reducible principal series representation (π, B∞ (χ1 , χ2 )). Since every matrix coefficient of (π, V ) is a matrix coefficient of (π, B∞ (χ1 , χ2 )), it immediately follows that G(s)−1 · Z ∞ (s, , β) is a

470

The Godement-Jacquet L-function for GL(2, AQ )

polynomial in s for every matrix coefficient β of (π, V ) and every function  of the form (11.5.2). Let I ⊂ C[s] denote the set of polynomials 

 G(s)−1 · Z ∞ (s, , β) | β a matrix coefficient of (π, V ),  as in (11.5.2) .

Then I is actually an ideal in the ring C[s]. Indeed, it is clear that it is a subvector space, and it follows from (11.15.4) that P(s) ∈ I =⇒ s · P(s) ∈ I. But s and C generate the ring C[s], so P(s) ∈ I =⇒ Q(s) · P(s) ∈ I for all Q(s) ∈ C[s]. The ring C[s] is a principal ideal domain. We fix P(s) ∈ I which generates I, and denote P(s) · G(s) by L ∞ (s, π ). Then the ratio Z ∞ (s, , β)/L ∞ (s, π ) is a polynomial for all  and β. Of course, the requirement that P(s) generates I only determines P(s) up to a scalar. We defer the discussion of how best to normalize L ∞ (s, π ) until the next section. For now, it is enough to know that it exists. This completes the proof of (2). To prove (3), we proceed as in the p-adic case, expressing the matrix coefficient β as 

2π

β(g) :=

f (k(θ )g) ( f (k(θ )) dθ,

(g ∈ G L(2, R)).

0

Then    2π 2π Z ∞ (s, , β) = R× R× R

0

  a1 −1  k (θ ) 0

x a2



 k(θ1 )

f (k(θ1 )) ( f (k(θ ))

0

· χ1 (a1 )χ2 (a2 ) |a1 a2 |s∞ dθ1 dθ d x d × a1 d × a2 . Note that this is valid both in the case when (π, V ) is an irreducible principal series representation and in the case when it is a subrepresentation of a reducible principal series representation. Define a new function f  : R × R → C by        a1 x k f (k ) ( f  (a1 , a2 ) :=  k −1 f (k) d × k d × k d x, 0 a2 K

K

R

(a1 , a2 ∈ R). Then f  (a1 , a2 ) is equal to e−π(a1 +a2 ) · Q(a1 , a2 ), where Q is a polynomial. The proof of this is left to the reader. In particular, f  is Schwartz. Likewise, define      a1 x k f (k ) ( f (k) d x, (a1 , a2 ∈ R). T (a1 , a2 ) :=  k −1 0 a2 2

R

2

11.16 The local L-function for irreducible discrete series

471

∨

N  Then T  = T and f  is given by f   , defined exactly as in the p-adic case, by essentially the same proof.  ∨ , β For − 12 < (s) < 12 , the local zeta integrals Z ∞ (s, , β) and Z ∞ s,  are given by   f  (a1 , a2 ) |a1 |s∞ χ1 (a1 ) |a2 |s χ2 (a2 ) d × a1 d × a2 , Z ∞ (s, , β) = R× R×

 ∨ , β = Z ∞ 1 − s, 

 

∨

1−s −1 1−s −1 × × f  (a1 , a2 )|a1 |∞ χ1 (a1 ) |a2 |∞ χ2 (a2 ) d a1 d a2 .

R× R×

First assume that f  (a1 , a2 ) = 1 (a1 )2 (a2 ), which 1 and 2 being Schwartz functions R → C. Then Z ∞ (s, , β) = Z ∞ (s, 1 , χ1 ) · Z ∞ (s, 2 , χ2 ) =

1 , χ1−1 ) · Z ∞ (1 − s,  2 , χ2−1 ) L ∞ (s, χ1 )L ∞ (s, χ2 )Z ∞ (1 − s, 

∞ (s, χ1 )∞ (s, χ2 )L ∞ (1 − s, χ1−1 )L ∞ (1 − s, χ2−1 )  ∨ L ∞ (s, χ1 )L ∞ (s, χ2 ) , β . = Z ∞ 1 − s,  −1 −1 ∞ (s, χ1 )∞ (s, χ2 )L ∞ (1 − s, χ1 )L ∞ (1 − s, χ2 )

Setting γ (s, π ) = ∞ (s, χ1 )∞ (s, χ2 ) = γ (s, χ1 )γ (s, χ2 ),

L ∞ (1 − s, χ1−1 )L ∞ (1 − s, χ2−1 ) L ∞ (s, χ1 )L ∞ (s, χ2 )

and ∞ (s, π ) = ∞ (s, χ1 )∞ (s, χ2 )

L ∞ (s, π ) L ∞ (s, χ1 )L ∞ (s, χ2 ) L ∞ (1 − s, χ1−1 )L ∞ (1 − s, χ2−1 ) · , L ∞ (1 − s, ( π)

completes the proof of part (3). 

11.16 The local L-function for irreducible discrete series representations of G L(2, R) In this section we determine the local L-function associated to an irreducible discrete series representation of G L(2, R). According to Definition 7.4.10 and Theorem 7.5.10, (see also Proposition 8.11.6 and the remarks which follow

The Godement-Jacquet L-function for GL(2, AQ )

472

its proof) each such representation may be realized as a subrepresentation of either of two nonisomorphic principal series representations. This fact will be exploited in the proof. Theorem 11.16.1 Let m ≥ 0 be an integer. For x ∈ R, define χ1 (x) = m+1

+it

− m+1 +it

· sign(x)m and χ2 (x) = |x|∞ 2 , so that B∞ (χ1 , χ2 ) contains a |x|∞2 discrete series representation (π, V ) as a subspace. For s ∈ C, define     s + it + 1 m + 1 s + it m + 1 −s−it− m+2 2  . (11.16.2)  + + G(s) := π 2 4 2 4 Then G(s)−1 Z ∞ (s, , β) is a polynomial for any Schwartz function  : Mat(2, R) → C and any matrix coefficient β of (π, V ). Furthermore, there exists a Schwartz function  : Mat(2, R) → C and a matrix coefficient β of (π, V ), such that G(s)−1 Z ∞ (s, , β) = 1. Proof Let δ denote the element of {0, 1} such that δ ≡ m (mod 2). It was shown in the proof of Theorem 11.15.1 that for any Schwartz function  : Mat(2, R) → C and any matrix coefficient β of B∞ (χ1 , χ2 ), the local zeta integral Z ∞ (s, , β) is equal to a polynomial P1 (s, , β) times     s + it + δ m + 1 s + it m+1 s+it+δ m+1 s+it m+1 + − π− 2 − 4  π− 2 + 4  . 2 4 2 4 (11.16.3) Now, according to Theorem 7.5.10, it is also possible to embed the (g, K ∞ )module (π, V ) into the principal series representation (π, B∞ (χ1 · sign, χ2 · sign). By Proposition 8.6.23, every matrix coefficient β of (π, V ), is a matrix coefficient of (π, B∞ (χ1 ·sign, χ2 ·sign)), as well as (π, B∞ (χ1 , χ2 )). It follows, exactly as in the proof of Theorem 11.15.1, that Z ∞ (s, , β) is equal to some polynomial P2 (s, , β) times     s + it m + 1 s + it + δ m+1 s+it m+1 s+it+δ m+1 π− 2 − 4  + π− 2 + 4  − . 2 4 2 4 (11.16.4) By considering the set of poles, one easily finds that any meromorphic function which is a polynomial multiple of both (11.16.3) and (11.16.4) is, in fact, a polynomial multiple of (11.16.2). To complete the proof of Theorem 11.16.1, we must exhibit a Schwartz function  : Mat(2, R) → C, and a matrix coefficient β of (π, V ) such that Z ∞ (s, , β) is equal to G(s) times a non-zero constant.

11.16 The local L-function for irreducible discrete series

473

As in the proof of Theorem 11.15.1 one has    2π 2π Z ∞ (s, , β) = R× R× R

0

  a1  k −1 (θ ) 0

x a2



 k(θ1 )

f (k(θ1 )) ( f (k(θ ))

0

· χ1 (a1 )χ2 (a2 ) |a1 a2 |s∞ dθ1 dθ d x d × a1 d × a2 . 

Here k(θ ) :=

cos θ − sin θ



sin θ cos θ

,

(∀θ ∈ [0, 2π )).

  If g = ac db ∈ G L(2, R), define 0 (g) = exp(−π (a 2 + b2 + cd + d 2 )), and let (g) = 0 (g) · Q(g) where Q is a polynomial. Since 0 is invariant by S O(2, R) on both the left and the right,           a1 x a1 x a1 x −1 −1  =  k (θ ) k(θ1 ) Q k (θ ) k(θ1 ) . 0 a2 0 a2 0 a2 A simple computation shows that   ∞ G(s) =

s+ m+1 2

e−π(a1 +a2 +x ) |a1 |∞ 2

2

2

s− m+1 2

|a2 |∞

sign(a2 )·a2m+2 d x

R× R× −∞

da1 da2 . |a1 |∞ |a2 |∞

Thus, it suffices to find a polynomial Q : G L(2, R) → C, and elements f ∈ f ∈ B∞ (χ1−1 , χ2−1 ) such that V ⊂ B∞ (χ1 , χ2 ), ( 2π 2π 0

  a1 −1 Q k (θ ) 0

x a2



 k(θ1 )

f (k(θ1 )) ( f (k(θ ))dθ1 dθ = a2m+2 (11.16.5)

0

for all a1 , a2 ∈ R× , x ∈ R, θ1 , θ ∈ [0, 2π ). If g = define P1 (g) = a + ib and P2 (g) = c + id. Then P j (g · k(θ1 )) = eiθ1 · P(g),



a b c d



∈ G L(2, R), let us

(∀θ ∈ [0, 2π ), j = 1, 2).

Further, P1 and P2 form a basis for the space of all polynomial functions P : G L(2, R) → Cwhichsatisfy P (g · k(θ1 )) = eiθ1 · P(g), (∀ θ1 ∈ [0, 2π )). a x Clearly a2 = i −1 P2 01 a2 for all a1 , a2 ∈ R× , x ∈ R. Now let Q 1 = P1 +i P2 and Q 2 = P1 − i P2 . Then i −1 P2 = (Q 2 − Q 1 )/2, and so (i

−1

P2 )

m+2

−m+2

=2

· (Q 2 − Q 1 )

m+1

−m+2

=2

·

 m+2  m+2 j=0

j

j

m+2− j

(−1) j Q 1 Q 2

.

(11.16.6)

The Godement-Jacquet L-function for GL(2, AQ )

474

One checks that for all θ, θ1 ∈ [0, 2π ) and g ∈ G L(2, R), Q 1 (k −1 (θ )·g·k(θ1 )) = ei(θ+θ1 ) ·Q 1 (g),

Q 2 (k −1 (θ )·g·k(θ1 )) = ei(−θ+θ1 ) ·Q 2 (g),

and it follows immediately from (11.16.6) that for all n,  ∈ Z 



 −1  i P2 (k −1 (θ )gk(θ1 )) e−i(nθ1 +θ) dθ1 dθ 0  ⎧ 0 ⎨ π m + 2 Q (g) j Q (g)m+2− j , if n = m + 2, and  = 2 j − m − 2, 1 2 2m = j for some 0 ≤ j ≤ m + 2, ⎩ 0, if not. 2π

2π

Reviewing the results of Chapter 7, the space B(χ1−1 , χ2−1 ) contains an element i(2 j−m−2) ( , while the space V contains an f which satisfies ( f (k(θ )) = m+2 j=0 e element f which satisfies f (k(θ1 )) = e−i(m+2)θ . It is immediate that for this choice of f, ( f , and the polynomial Q := (i −1 P2 )m+2 , equation (11.16.5) is satisfied. 

Exercises for Chapter 11 11.1 Let ξ ∈ Mat(2, Q) be a matrix with 1. Show there exist matrices  1rank 0 X, Y ∈ G L(2, Q) such that ξ = X 0 0 Y .  11.2 Recall that G 1 = {g ∈ G L(2, A)  | det(g)|A = 1}. (a) Show that G L(2, Q) is a subgroup of G 1 . Is it normal? (b) Show that Z (A)· G L(2, Q) is a subgroup of G L(2, A). Is it normal? (c) Prove that the canonical inclusion G 1 %→ G L(2, A) induces a surjective map of cosets β : G L(2, Q)\G 1 → Z (A) · G L(2, Q)\G L(2, A). (d) Let α :

 p

Z×p → G L(2, Q)\G 1 be defined by

   α {u 2 , u 3 , . . . , u p , . . . } = I2 , u02

0 u2

  , u03 

0 u3

up 0

 ,... ,   0 , . . . . up

Prove that α is injective and that β ◦ α is constant with value equal to the coset Z (A) · G L(2, Q). (e) Prove that β(g) = β(g ) if and only if g −1 g lies in the image of α. (f) Use strong approximation to show that a fundamental domain for 

· p G L(2, Z p ), where D∞ ⊂ G L(2, Q)\G 1 is given by D∞ G L(2, R) is a fundamental domain for S L(2, Z)\S L(2, R).

Exercises for Chapter 11

475

11.3 Let G be a locally compact totally disconnected group (e.g., Qnp or G L n (Q p )), and let {K n }n≥1 be a basis of compact neighborhoods of the identity in G. Let f : G → C be a compactly supported and locally constant function (i.e., a Bruhat-Schwartz function on G). Prove that f is of the form N ci 1gi K ni , f = i=1

for some complex numbers ci , elements gi ∈ G, and positive integers n i . (Compare with Exercise 1.6.) 11.4 Suppose  is a Bruhat-Schwartz function on Mat(2, A). Let g, h ∈ G L(2, A). Prove that the function  (m) := (g · m · h) is also Bruhat-Schwartz. p q 11.5 Let x = r s ∈ Mat(2, AQ ) and define d x := dp dq dr ds. Prove that     (g · x · h) d x = | det g|

−2

· | det h|

−2

    ·

AQ AQ AQ AQ

(x) d x,

AQ AQ AQ AQ

for all g, h ∈ G L(2, AQ ), and any function  : Mat(2, AQ ) → C such that these integrals are defined. 11.6 Prove that the global zeta integral in Definition 11.4.1 converges absolutely for (s) sufficiently large (depending on  and β). Hint: Use the fact that cuspidal representations are unitary, and consequently they have bounded matrix coefficients by Lemma 9.1.6. 11.7* For any cuspidal automorphic form φ on G L(2, AQ ), prove that  φ(g) d × g = 0. G L(2,Q)\G 1

11.8 Let k1 , k2 ∈ G L(2, Z p ), and let  : Mat(2, Q p ) → C be a BruhatSchwartz function. Define a new function by k1 k2 (g) := (k2−1 gk1 ).  k1 . 1 k 2 = k 2  Prove that k 11.9 Set K = G L(2, Z p ). Prove the decomposition in (11.7.5), namely  K

p 0

0 1



 K =

1 0

0 p



 K

⎛ ∪ ⎝

Show also that these cosets are disjoint.

 b (mod p)



p 0

b 1



⎞ K⎠.

476

The Godement-Jacquet L-function for GL(2, AQ )

 pr 0  11.10 For each r, s ∈ Z, write K (r, s) = K 0 ps K . As in Section 11.6, define T pr = 1 K (a,b) . R ap = 1 K (a,a) , a≥b≥0 a+b=r

(a) Show that R ap ∗ R bp = R a+b p for every a, b ∈ Z. (b) For every a, r, s ∈ Z, show that 1 K (r,s) ∗ R ap = 1 K (r +a,s+a) . (c) Show that 1 K (r,0) = T pr − R p ∗ T pr −2 for every r ≥ 2. 11.11 Derive the integration formula (11.7.9). 11.12 Let χ1 , χ2 be characters of Q×p . Consider the principal series represenf ∈ B p (χ1−1 , χ2−1 ), and tation (π, B p (χ1 , χ2 )). Fix f ∈ B p (χ1 , χ2 ) and ( consider the matrix coefficient β(g) = π (g). f, ( f  as in Section 11.9. Finally, let  : Mat(2, Q p ) → C be a Bruhat-Schwartz function. Prove that |(g)β(g)|C is uniformly bounded and compactly supported on Mat(2, Q p ). Conclude that the local zeta integral Z p (s, , β) converges absolutely for (s) > −1/2. 11.13 Let  : Mat(2, Q p ) → C be a Bruhat-Schwartz function, let χ1 , χ2 : Q×p → C× be characters, and let f ∈ B p (χ1 , χ2 ) and ( f ∈ ( B p (χ1−1 , χ2−1 ) be elements of the corresponding principal series representations. Define f  : Q p × Q p → C by    f  (a1 , a2 ) =

  k

K

K Qp

−1



a1 0

x a2

  k

f (k) d x d × k d × k (a1 , a2 ∈ Q p ). · f (k ) ( (a) For u, v ∈ Z×p and a1 , a2 ∈ Q p , prove that f  (ua1 , va2 ) = χ1 (u)−1 χ2 (v)−1 f  (a1 , a2 ). (b) Since   aisb aBruhat-Schwartz function, there exists M > 0 such that  c d = 0 if |a| p , |b| p , |c| p , or |d| p exceeds M. Prove that f  (a1 , a2 ) = 0 if either of |a1 | p or |a2 | p exceeds M. Conclude that f  is compactly supported. (c) Prove that f  is locally constant on Q p × Q p . 11.14 Let A1 and A2 be positive real numbers, and suppose that f 1 , f 2 are Bruhat-Schwartz functions on Mat(2, Q p ) such that f j is supported on the set    x ∈ Mat(2, Q p )  | det(x)| p = A j ,

Exercises for Chapter 11

477

for each j = 1, 2. (Note that this set will be empty unless A j is a power of p.) Prove that 



f 1 (h)  f 2 (h) d × h = A22

A21 G L(2,Q p )

 f 1 (h) f 2 (h) d × h.

G L(2,Q p )

11.15 Let K 1 = I2 + p · Mat(2, Z p ). (Here we abused notation by writing p 0 p instead of 0 p .) Let f 1 = 1 pK 1 and f 2 = 1 K 1 be characteristic functions. Evaluate the integrals in Exercise 11.14 for this choice of f 1 and f 2 . 11.16 Let f 1 , f 2 : Mat(2, Q p ) → C be Bruhat-Schwartz functions in S0 (Mat(2, Q p )) as in Definition 11.13.5. Prove that 



f 1 (h)  f 2 (h) dh =

Mat(2,Q p )

 f 1 (h) f 2 (h) dh.

Mat(2,Q p )

(Compare with Exercises 11.14 and 11.15.) 11.17 Let  : Mat(2, R) → C be defined by  

a c

b d



= e−π(a

2

+b2 +c2 +d 2 )

P(a, b, c, d),

where P is a polynomial. Let χ1 , χ2 : R× → C be characters, and fix f ∈ B∞ (χ1−1 , χ2−1 ). Recall that k(θ ) := elements f ∈ B∞ (χ1 , χ2 ) and (  cos θ sin θ  . Define a function f  : R × R → C by − sin θ cos θ 

2π

f  (a1 , a2 ) = 0



2π 0

 R

  a1  k(θ )−1 0

x a2



k(θ )



f (k(θ )) d x dθ dθ, · f (k(θ ))(

for a1 , a2 ∈ R. Prove that f  (a1 , a2 ) = e−π(a1 +a2 ) Q(a1 , a2 ) for some polynomial Q. Hint: Do it first for the case where P is a monomial and f and ( f lie in the basis given by Definition 7.4.7. 2

2

Solutions to Selected Exercises

1.1 Use the fact that the group of non-zero elements in a finite field is cyclic to compute the absolute value of a generator. 1.2 If | · | is nonarchimedean, then for any n ≥ 1 the strong triangle inequality shows |n| ≤ max{|n − 1|, |1|} ≤ 1, by induction. The result is clear for n = 0 and the case n < 0 follows because |n| = | − n|. If conversely we have |n| ≤ 1 for all n, then for any x, y ∈ F,   n     n     i n−i  x y  ≤ (n + 1) · max |x|i |y|n−i  0 ≤ i ≤ n |x + y| =    i n

i=0

≤ (n + 1) · max{|x|, |y|}n . The second inequality follows by the usual triangle inequality and the  fact that each ni is an integer (and so has absolute value bounded by 1 by hypothesis). Now take nth roots and let n → ∞. 1.4 Use the geometric series formula from calculus to give its series expansion. 1.5 For all of these, start by assuming there exists a solution to x 2 = −1 ∞ ai pi , in Q p . Then |x|2 = | − 1| = 1, so that x ∈ Z×p . Write x = i=0 where 0 ≤ ai < p for each i and a0 =/ 0. Expand x 2 as a series, and compare it to the series −1 =

∞ i=0

478

( p − 1) · pi .

Solutions to Selected Exercises

479

A contradiction will be reached for p = 2 by reducing modulo 4 and for p = 3 by reducing modulo 3. For p = 5, the equation x 2 = −1 is soluble, and a solution can be constructed by induction. 1.6 (a) By general metric space nonsense, we can cover the open compact set K by a disjoint union of basic neighborhoods of the form a + p m Z p . These neighborhoods form an open cover of K , so by compactness we only require a finite number. (b) For each complex number c, let Bc = h −1 (c) = {a ∈ Q p | h(a) = c}. Any locally constant function is continuous; by continuity, each set Bc is closed. For c =/ 0, Bc is a closed subset of the support of h, which is a compact set; hence Bc is itself compact. As h is locally constant, each Bc is also open. Part (a) shows that for c =/ 0, Bc is a finite union of neighborhoods of the form a + p m Z p . The sets {Bc : c =/ 0} form an open cover of the support of h, and so by compactness there must be only a finite number of them. It follows that h is of the correct form. (c) The argument in (b) applies to show that the sets Bc = h −1 (c) are open and compact for each c =/ 0. The only open and compact subset of R is the empty set, so we must have B0 = R. That is, h ≡ 0. 1.7 (b) Pull back the open set V × AQ through the map i from part (a). 1.8 By Proposition 1.6.7, the Fourier transforms at finite primes are given by e p (−x p / p)  1 p−2 Z p (x p ). f p (x p ) = 1Z p (x p ) + p2  It isn’t hard to see that the infinite product  f (x) = v  f v (xv ) converges absolutely for any x ∈ AQ . Indeed, write x = {x∞ , x2 , x3 , . . . } ∈ AQ . Let p0 be a prime such that p > p0 implies x p ∈ Z p . Then 

 f p (x p ) =

p> p0

which converges because

 p> p0



1+

 e p (−x p / p) , p2

1/ p 2 converges.

1.10 Apply Strong Approximation and Lemma 1.6.4 to see that the integral is zero if α ∈ Z. Now compute the infinite part of the integral explicitly. 1.11 (c) Periodicity is trivial from the definitions. To prove smoothness, we  may assume x lies in the fundamental domain [−1/2, 1/2)· p Z p .

480

Solutions to Selected Exercises Then e(x) = 1 if and only if x∞ = 0. Compute the limit of g(x) as x∞ → 0 by L’Hˆopital’s rule to show g is continuous. It now suffices to show the function h ∞ (t + n) − h ∞ (n) t → n∈Z 1 − exp(2πit) is smooth at t = 0 in the usual sense. Replace g with its infinite part so that it suffices to prove g(t) is smooth at t = 0. Show by induction that the nth derivative of g(t) near t = 0 is of the form g (n) (t) =

G n (t) [1 − exp(2πit)]n+1

(t =/ 0), ( j)

where G n (t) is smooth in a neighborhood of t = 0 and G n (0) = 0 for j = 0, 1, . . . , n, and that g (n+1) (0) =

(−1)n+1 (2πi)n+1

:

; G (n+2) (0) G (n+1) (0) − πi . (n + 2)! n!

For example, the case n = 0 is immediate from our definitions. You will need to use Taylor’s theorem to write exp(2πit) = 1 + 2 t 2 + O(t 3 ). 2πit + (2πi) 2 2.1 For continuity, write φ(g) = exp(it log |g|A ). It now suffices to prove that g → |g|A is continuous, which is straight-forward. The kernel of φ contains {1, 1, 1, . . . }, but it does not contain any neighborhood of the form    × 1 + p m( p) Z p · Zp , (a, b) · p∈S

p∈ S

where (a, b) ⊂ R is an interval, S is a finite set of primes, and m( p) ≥ 1 for all p ∈ S. Indeed, an element g = {g∞ , g2 , g3 . . . } in this neighborhood satisfies |g|A = |g∞ |, which clearly cannot be identically 1 for all g∞ ∈ (a, b). 2.2 Evidently c = φ({1, 1, 1, . . . }). Set g = {e, 1, 1, . . . }, so that χ idelic (g) = 1, and |g|itA = exp (it log |e|∞ ) = exp(it). It follows that φ({e, 1, 1, . . . }) = c exp(it); hence t = arg

φ({e, 1, 1, . . . }) . φ({1, 1, 1, . . . })

Solutions to Selected Exercises

481

2.4 Observe that  Z v (s, , ω) =

Q× v

 =

Q× v

(x)ω(x)|x|s d × x (−x)ω(−x)| − x|s d × x

= ω(−1)Z v (s, , ω), where the second to last equality follows from the change of variables x → −x, and the final equality is a consequence of the fact that  is even. Now use the fact that ω(−1) = −1. 2.5 Choose ◦∞ (x) = xe−π x when v = ∞, and choose ◦p (x) = ω p (x)1Z×p (x) when p < ∞. 2

2.7 Choose a test function  such that Z v (s, , ω) ≡ 0. Show from the ¯ ω). Check also definition by an integral that Z v (s, , ω) = Z v (¯s , ,  (x) = (−x) for all x ∈ Qv . Then that  γ (s, ω) = =

Z v (s, , ω) , ω) Z v (1 − s,  ¯ Z v (¯s , , ω)

, ω) Z v (1 − s¯ ,  ¯ ω) Z v (¯s , , = = ω(−1)γ (¯s , ω).  ¯ ω) ω(−1)Z v (1 − s¯ , ,

Note that ω(−1) = ±1. By this calculation and Proposition 2.3.7, we see that |γ (1/2, ω)|2 = γ (1/2, ω) · γ (1/2, ω) = γ (1/2, ω) · ω(−1) γ (1/2, ω) = ω(−1) γ (1/2, ω) · γ (1 − 1/2, ω) = ω(−1)2 = 1. 2.9 If V = 0, we’re finished. Suppose V =/ 0 and let g0 ∈ G. There exists an eigenvector v with eigenvalue λ so that π (g0 ).v = λv. Let Wλ be the eigenspace of π (g0 ) associated to the eigenvalue λ. For any g ∈ G and w ∈ Wλ , we have π (g0 ). (π (g).w) = π (g0 g).w = π (g g0 ).w = π (g). (π (g0 ).w) = λπ (g).w. It follows that π (g).Wλ ⊂ Wλ , which means π (g).Wλ = Wλ . As g is arbitrary, we find that Wλ is a G-stable subset of V , which means

482

Solutions to Selected Exercises that Wλ = V by irreducibility. Consequently, the element g0 acts by scalar multiplication by λ on the whole space V . As g0 was arbitrary, we see that every g ∈ G acts by scalar multiplication. But if v ∈ V is any non-zero vector, we find that C.v is G-stable since G acts by scalar multiplication. This means v generates V , and dim(V ) = 1.

2.10 (a) First note that f (λ(gh) − λ(g) − λ(h)) =

eiλ(gh) eiλ(g) eiλ(h)

=

ω(gh) = 1, ω(g)ω(h)

since ω is a homomorphism. It follows that the image of  lies entirely inside the set 2π Z ⊂ R. But  is continuous by the continuity of λ, and since continuous maps send connected sets to connected sets, we find that  is constant. Since (1, 1) = 0, we find  is identically zero, which is tantamount to showing that λ is a homomorphism. (b) By part (a), the map x → λ(e x ) is a linear map from (the onedimensional vector space) R to itself, and so it must be of the form x → xt for some t ∈ R. For any g ∈ R+ , we have   f (g) = f (elog g ) = exp iλ(elog g ) = exp (it log g) = g it . (c) By part (b) and the fact that f (g) = ω(g) for all g ∈ R+ , we see that ω has the desired form whenever g > 0. For g < 0, we have ω(g) = ω(−1)ω(−g) = ω(−1)|g|it . 2.11 By compactness of Z×p , there is a closed interval [a, b] such that ω(Z×p ) ⊂ [a, b]. But ω(Z×p ) is a subgroup. The only compact subgroups of R× are {1} and {±1}. The image is contained in the positive real numbers, so it must be trivial, and ω is unramified. 2.12 Set s = − log ω( p)/ log p for some branch of the complex logarithm. Since ω( p) is unitary, s is purely imaginary. Now set s = it for some real number t. The ambiguity in defining the complex logarithm shows s is only well defined modulo 2πi/ log p. Now for u ∈ Z×p and r ∈ Z, we have ω(upr ) = ω( p)r = exp(−itr log p) = ( p −r )it = |upr |itp . 2.13 (b) Let B be a basis for R viewed as a Q-vector space. Choose any function f : B → R other than a scalar multiple of the identity

Solutions to Selected Exercises

483

map. Then f extends to a Q-linear map on all of R, which we also denote by f . In particular, it satisfies f (a + b) = f (a) + f (b) for all a, b ∈ R, but it is not R-linear. However, the function a → e f (a) is not continuous, because if it were, then f would also be continuous, and consequently R-linear. 3.3 (a) Using the Taylor series for the exponential, the general factor of the product looks like 1 z  z/ω+ 1 (z/ω)2 2 e = 1 − (z/ω)3 + O((z/ω)4 ). ω 3  Since the product (1 + an ) converges absolutely if and only if an does so, we find that the product defining σ converges absolutely on compact subsets of C. Note that the presence of the cubic term (z/ω)−3 is important for convergence; a quadratic term would not suffice. (c) Consider the function 

1−

1 1 σ (z + 1) σ (z) − = − σ (z + 1) σ (z) z + 1 z  1 1  1 − + + . z+1−ω z − ω ω2 ω =/ 0

ψ(z) =

Its derivative is  1 1 1  1 − + + + (z + 1)2 z 2 ω =/ 0 (z + 1 − ω)2 (z − ω)2   1 1 = − . + (z + 1 − ω)2 (z − ω)2 ω

ψ (z) = −

Evidently the terms of this series cancel in pairs, so that ψ(z) must be a constant, say η1 . Similar considerations show that there is a constant η2 such that σ (z + i) σ (z) − = η2 . σ (z + i) σ (z) Now the result follows by induction on r and s. (d) Compute d σ (z + r + si) log dz σ (z) and apply part (c) to learn that   σ (z + r + si) = Ar,s exp (r η1 + sη2 )z σ (z),

(z ∈ C)

484

Solutions to Selected Exercises for some constant Ar,s . Set z = −(r + si)/2 and use the fact that σ is odd to determine the constants.   3.4 (a) Consider the matrix ac db , where a, b, c, d are integers in the interval [0, N ). If the entry c is invertible modulo N or if c = 0, then the proof of Lemma 6.1.6 applies verbatim to show that we may write this matrix as a product of upper and lower triangular matrices in S L(2, Z/N Z). So now suppose that c is non-zero and not invertible modulo N , which is to say that (c, N ) > 1. Let x ∈ Z be unknown for the moment and consider the matrix      a b 1 0 ∗ ∗ = . c d x 1 c + dx ∗ We want to choose x so that c + d x is invertible (mod N ). This allows us to reduce to the previous case. The Chinese Remainder Theorem allows us to solve the following system of (non-)congruences modulo N :  if p | d and pr || N , x ≡ 1 (mod pr ), x ≡ −c/d (mod p), if p  d and pr || N . Now I claim that c + d x is coprime to N . For if not, then it shares a prime factor with N , say p. Note that (c, d) = 1, since otherwise the original matrix can’t have determinant coprime to N . If p | d, then the congruence c + d x ≡ 0 (mod p) shows p | c, a contradiction. Therefore p  d. But now the congruence c + d x ≡ 0 (mod p) implies x ≡ −c/d (mod p), which contradicts the construction of x. Hence we must conclude c + d x is coprime to N . (b) The kernel of π is (N ) by definition. Surjectivity is trivial by part (a). (c) The Chinese Remainder Theorem implies that  S L(2, Z/ pr Z). S L(2, Z/N Z) ∼ = pr ||N r So it suffices  a b  to count the number of elements in S L(2, Z/ p Z). Let c d be an undetermined matrix with coefficients in Z/ pr Z. The first column can have arbitrary entries, provided that not both of a, c are divisible by p. There are ( pr )2 − ( pr −1 )2 = p 2r (1 − 1/ p 2 ) such pairs a, c. Let us suppose that a and c are fixed and determine how many possibilities there are for b and d.

Solutions to Selected Exercises

485

Suppose further that p  a for the moment. Then the congruence ad − bc ≡ 1 (mod pr ) can be rewritten as d ≡ (bc + 1)/a (mod pr ), which shows that b can be arbitrary, and then d is determined by it. So for this choice of a, c, there are pr possible choices for pairs b, d. Finally, we treat the case that (a, pr ) = p s with s ≥ 1. In this case we know that p  c. Write a = p s a with (a , p) = 1. So the congruence ad − bc ≡ 1 (mod pr ) can be rewritten as a p s d ≡ 1 + bc (mod pr ). Set b = c−1 (b − 1) for some integer b , where the inverse of c is computed modulo N . The above congruence becomes a p s d ≡ b

(mod pr ),

which shows b is divisible by p s . Write b = p s b

. Finally, the congruence becomes a d ≡ b

(mod pr −s ),

which can be solved by taking d ≡ b

/a (mod pr −s ). If we choose any b

modulo pr −s , then d is determined modulo pr −s . Retracing our steps shows there are p s choices for d (mod pr ) and pr −s choices for b, so that in total we arrive at pr choices again for the pair b, d. In summary, we have shown that there are p 2r (1 − 1/ p 2 ) · pr = 3r p (1 − 1/ p 2 ) elements in S L(2, Z/ pr Z). Taking the product over all p dividing N gives #S L(2, Z/N Z) = N 3

 1 1− 2 . p p|N

(d) This is most easily accomplished by counting the number of elements in B, and by part (c) it suffices to do this inside S L(2, Z/ pr Z). Observe that the top left entry of a matrix in B must be invertible, and there are ϕ( pr ) = pr − pr −1 such elements. Once we have fixed the top left entry, the bottom right entry is also fixed since the determinant must equal 1. The upper right entry is free, and so #B = pr ( pr − pr −1 ) = p 2r (1 − 1/ p). It follows from part (c) that the index is    N 3 p|N 1 − p12  1   . 1 + = N  p 1− 1 N2 p|N p|N

p

486

Solutions to Selected Exercises (e) By part (b), the group S L(2, Z) acts transitively on the right coset space S L(2, Z/N Z)/B by reduction followed by left multiplication. The stabilizer of the coset B is precisely 0 (N ), so we get the equality of indices [S L(2, Z) : 0 (N )] = [S L(2, Z/N Z) : B]. Apply part (d).   does not preserve (f) No, because conjugation by the matrix 01 −1 0 0 (N ). 3.5 We know (2) is normal by Exercise 3.4, so the quotient makes sense. Define     1 1 0 −1 and f 2 = . f1 = 0 1 1 0 Since f 1 and f 2 generate S L(2, Z), they must generate the quotient. Now f 12 ≡ f 22 ≡ ( f 1 f 2 )3 ≡ I2 (mod (2)). Finally, observe that the transpositions (12) and (13) generate S3 and have the same relations as f 1 and f 2 . 3.6 A congruence subgroup  satisfies (N ) ⊂  ⊂ S L(2, Z) for some N ≥ 1. Define an action of S L(2, Z)/(N ) on the right coset space S L(2, Z)/ by (a.(N ))(b.) = (ab).. (It is well defined because (N ) is normal; see Exercise 3.4.) The stabilizer of  is /(N ), so that [S L(2, Z) : ] =

[S L(2, Z) : (N )] . [ : (N )]

The right side is finite by Exercise 3.4.  0  3.10 Set γ = −1 . If f ∈ A∗k,χ (0 (N )), then by definition we have 0 −1  (−1)−k f (γ z) = ( f k γ )(z) = χ (−1) f (z). But since γ z = z for all z ∈ h, we find that (−1)−k f (z) = χ (−1) f (z). If f does not vanish at some point z, then χ (−1) = (−1)−k = (−1)k . 3.14 The map f → f ∗ defined in Exercise 3.12 gives the isomorphism.   3.15 If z = x + i y, then for γ = ac db we set γ z = x + i y where x =

ac(x 2 + y 2 ) + (ad + bc)x + bd , (cx + d)2 + c2 y 2

y =

y . (cx + d)2 + c2 y 2

Solutions to Selected Exercises

487

 0 1   , it suffices to assume Since S L(2, Z) is generated by 10 11 and −1 0 γ is one of these two matrices to simplify the computations. In the former case, the Jacobian determinant is 1, and in the latter it is 1/(x 2 +

y = d xd . y 2 )2 . In either case, we find that d(yx dy )2 y2 3.17 (c) The discussion preceding Definition 3.7.3 shows √

ma 0

ga = γa Write γa =

a b c d

 ga =

√

0 m a −1



1 0

1 1

√

m a −1 0

0 √ ma



γa−1 .

and expand to get

1 − acm a −c2 m a

a2ma 1 + acm a



 =

aa ca

ba da

 .

As ga ∈ 0 ( p), we see p | c2 m a , and hence da = 1 + acm a ≡ 1 (mod p). Now χ (da ) = 1, which shows the cusp parameter satisfies μa = 0. (d) Let N = 4 and let χ be the non-trivial quadratic character modulo 4. That is, χ (3) = −1. One checks easily that the stability 1 , so that the subgroup for the cusp a = 1/2 is generated by −1 −4 3 cusp parameter is μa = 1/2. 3.18 Using the fact that L k = −(z − z¯ ) ddz¯ −

k 2

and the chain rule, we find

L k (y k/2 f ) = −2i y k/2+1

df . d z¯

It follows that L k (y k/2 f ) = 0 if and only if f is holomorphic. A similar strategy for Rk shows that ker(Rk ) = {y −k/2 f | f is anti-holomorphic}. Recall that a function f is called anti-holomorphic if ddzf = 0. For example, if g(z) is holomorphic, then g(¯z ) is anti-holomorphic.  p −1   1 p−1  p 0  1 −1  . 4.1 0 1 = 0 1 0 1 0 1 4.2 For u ∈ Z p , the matrix is already in G L(2, Z p ), and hence in Iwasawa form. For u ∈ Z p , write u = p −r v with r > 0 and v ∈ Z×p . Observe that |1/(1 + u)| p = p −r , so that we can write 2r −1 1 = ai pi + ai pi , 1+u i=r i≥2r

(ar =/ 0).

488

Solutions to Selected Exercises Define u to be the first sum and u

to be the second, so that 1/(1+u) = u + u

. Following the proof of Proposition 4.2.1 we find     r  1 0 1 u p 0 = k, u 1 0 1 0 p −r where  k=

1 0

p −2r u

1



1 0

0 v + pr



p−r 1+u u 1+u

−r

p − 1+u 1 1+u

4.4 Write g ∈ G L(2, R) in Iwasawa form as     1 x y 0 cos θ sin θ ±1 g= 0 1 0 1 − sin θ cos θ 0

 ∈ G L(2, Z p ).

0 1



r 0 0 r

 ,

with x ∈ R, y > 0, 0 ≤ θ < 2π, r > 0. Define a function f : G L(2, R) → h × S 1 × R× by f (g) = (x + i y, eiθ , ±r ), where the choice of sign is determined by the sign in the Iwasawa decomposition. The existence and uniqueness of the Iwasawa decomposition guarantee that f is a well-defined bijection. The coordinate functions of the inverse map are given by polynomials in x, y, r, sin θ , and cos θ , and the Jacobian determinant is nowhere zero. Hence it is a diffeomorphism. 4.10 (a) If H is open, then so is the union of non-trivial left cosets  g H = G  H. g∈ H

It follows that H is closed. Now suppose G is compact. If H is open, then the left cosets with respect to H constitute an open partition (cover) of G, and so there must be only finitely many of them. Hence H has finite index. Conversely, if H is closed and of finite index, and if g1 , . . . , gr are representatives of the finitely many non-trivial left cosets with respect to H , then H =G

r 

gi H.

i=1

The right side is the complement of the union of finitely many closed sets, and hence is open.

Solutions to Selected Exercises

489

(b) Mat(n, Z p ) is compact open in Mat(n, Q p ), and det : Mat(n, Z p ) → Z p is a continuous map. Hence G L(n, Z p ) = det−1 (Z×p ) is open, and hence also closed by part (a). A closed subset of a compact space is also compact. (c) In the notation of the previous part, we have S L(n, Z p ) = det−1 (1), so that S L(n, Z p ) is closed inside the compact group G L(n, Z p ). Hence it is compact. On the other hand, G L(n, Z p )/S L(n, Z p ) ∼ = Z×p by the first isomorphism theorem, and so S L(n, Z p ) is not of finite index. It follows that it cannot be open in G L(n, Z p ), or even G L(n, Q p ). (d) If we consider the determinant map det : G L(n, Q p ) → Q p , then S L(n, Q p ) = det−1 (1), which shows it is closed in G L(n, Q p ). It is not open in G L(n, Q p ) because, for example, the diagonal matrix  · In is close to the identity if  is close to 1, and yet its determinant is  n =/ 1. To see it is not compact, observe that for n > 1, the sequence (gr ) has no convergent subsequence in S L(n, Q p ), where ⎛ r ⎞ p ⎜ ⎟ p −r ⎜ ⎟ ⎟. 1 gr = ⎜ ⎜ ⎟ .. ⎝ ⎠ . 1 λ ∗  4.13 (a) Conjugating α into Jordan form, say α = β 01 λ2 β −1 , we find  λ ∗  that exp(α) = β exp 01 λ2 β −1 lies in the center of G L(2, R) if and only if the same is true of   λ   e 1 ∗ λ1 ∗ = . exp 0 λ2 0 e λ2 If α is not diagonalizable, then the upper right entry of this last matrix is non-zero. (See the computation in the solution to Exercise 4.15.) Hence exp(α) lies in the center of G L(2, R) if and only if α is diagonalizable with eλ1 = eλ2 ∈ R, from which it follows matrix. that λ1 = λ2and α is a scalar multiple  0 1  of the  i identity 1 0  −1 ; then H = −1 = β β . For any θ ∈ (b) Set β = √12 1i −i 0 0 −i R, we see   iθ 0 exp(θ H ) = β exp β −1 0 −iθ   iθ   0 e cos θ sin θ −1 β = =β ∈ S O(2, R). 0 e−iθ − sin θ cos θ

490

Solutions to Selected Exercises Conversely, suppose exp(α) ∈ S O(2, R). By Exercise 4.15, we see that det(exp(α)) = eTr(α) = 1

=⇒

Tr(α) = 0.

If α is not then it can be conjugated into Jordan form   diagonal, as α = β 00 10 β −1 . The computations for Exercise 4.15 show that   exp(α) = β 10 11 β −1 , and hence exp(α) is not diagonalizable. But S O(2, R) is an abelian group, so all of its elements are diagonalizable. This contradiction shows that α must be diagonalizable. 0  −1 β . We may We conjugate it into diagonal form as α = β λ0 −λ −1 further assume that β is a unitary matrix; i.e, β = t β. Then   λ 0 e β −1 ∈ S O(2, R). exp(α) = β 0 e−λ Since t exp(α) = exp(t α), we see that   λ  λ 0 e t −1 I2 = exp(α) · exp(α) = β β ·β e −λ 0 e 0   2(λ) 0 e β −1 . =β 0 e−2(λ)

0



e−λ

β −1

It follows that λ is purely imaginary; write λ = iθ for some 0 β −1 ∈ θ ∈ R. Now verify that if β is unitary, then α = β iθ0 −iθ Mat(2, R) if and only if α = c · H for some c ∈ R. −1 4.15 Write α in Jordan canonical form  as α = βδβ , where β is some λ1  invertible matrix in gl(2, R), δ = 0 λ2 , λ1 , λ2 are the eigenvalues of

α, and  is either 1 or 0. Now exp(tα) = β exp(tδ)β −1 . By induction, check that   n λ1 nλn−1 n 1 . δ = 0 λn2

(Do the cases  = 0, 1 separately.) Then  n n n n−1   t λ1 nt λ1 t n δn etλ1 n! exp(tδ) = = = t nn!λn2 0 n! 0 n≥0

tetλ1 etλ2

n!

Multiplicativity of the determinant shows det(exp(tα)) = det(β) det(exp(tδ)) det(β)−1 = det(exp(tδ)) = et(λ1 +λ2 ) = et Tr(α) .

 .

Solutions to Selected Exercises

491

4.16 (a) By Exercise 2.2, we can write ω(a) = χidelic (a) |a|itA for all a ∈ AQ , where χ is a Dirichlet character and t = arg ω({e, 1, 1, . . . }). If we write a = {a∞ , a f }, where a f is the finite part of the adele a, then we can define ψa f (a∞ ) := ω(a). We view ψa f as a function on R× . Similarly, for g ∈ G L(2, AQ ) write g = {g∞ , g f }. Now let α ∈ gl(2, C). By linearity of the formula we wish to prove, we may assume α ∈ gl(2, R). Then  d  ψdet(g f ) (det(g∞ esα )) s=0 ds   d

sα  det g = ψdet(g (det(g )) · e  , ∞ ∞ ) f s=0 ds

Dα · φ(g) =

by the chain rule. Now det is multiplicative, and g∞ is independent of s, so we may apply Exercise 4.15 to get  d  det(g∞ )es Tr(α)  s=0 ds

= ψdet(g (det(g )) · Tr(α) det(g ). ∞ ∞ f)

(det(g∞ )) · Dα · φ(g) = ψdet(g f)

Now observe that ψdet(g f ) (x) = ω({x, det(g f )}) = χidelic ({x, det(g f )}) |x|it∞ | det(g f )|itf . Here |a f | f denotes the product of the p-adic absolute values of the p-adic entries of a f . As χidelic ({x, det(g f )}) is locally constant as a function of x, and since ddx |x|z = z|x|z /x for any complex number z, we find

(x) = χidelic ({x, det(g f )}) it ψdet(g f)

|x|it∞ | det(g f )|itf . x

Replacing x by det(g∞ ) gives ψdet(g (x) = itφ(g)/ det(g∞ ). Comf) bining this with the above calculation of Dα · φ(g) gives the result. (b) First note that the multiplicativity of the determinant and the fact that ω is a character gives φ(gg ) = φ(g)φ(g ) for all g, g ∈ G L(2, AQ ). We verify properties (1)–(5) of Definition 4.7.6. For (1), note that φ(γ ) = ω(det(γ )) = 1 if γ ∈ G L(2, Q), since 2 det(γ ) ∈ Q× . For (2), if z ∈ A× Q , then φ(z) = ω(det(z)) = ω(z ) = ω(z)2 , from which it follows that ω2 must be the central character of φ. For (3), note that if k ∈ K , then φ(k) is a complex number, so that φ(gk) is a complex multiple of φ(g). In particular, the line generated by φ is invariant under right translation by elements

492

Solutions to Selected Exercises of K . For (4), we apply part (a) of this exercise to see that Dα · φ is a complex multiple of φ for any α ∈ Z (U (g)). Finally, φ has absolute value 1, so it is clearly of moderate growth; i.e., (5) is satisfied. (c) The zeroth Fourier coefficient of φ is exactly φ itself.

4.17 (a) Verifying (1) and (3)–(5) is trivial. Suppose (2) does hold for some character ω. Then for any g ∈ G L(2, AQ ) and z ∈ A× Q , we have ω1 (z)φ1 (g) + ω2 (z)φ2 (g) = φ(gz) = ω(z)φ(g) = ω(z)φ1 (g) + ω(z)φ2 (g). If ω = ω1 , then choosing g so that φ2 (g) =/ 0 shows ω2 = ω, contradicting the fact that ω1 and ω2 are distinct. So we may fix z 0 such that ω(z 0 ) =/ ω1 (z 0 ). Rearranging the above equation shows φ1 (g) = −

ω(z 0 ) − ω2 (z 0 ) φ2 (g) ω(z 0 ) − ω1 (z 0 )

(g ∈ G L(2, AQ )).

That is, φ1 = c φ2 for some constant c. Now choose g so that φ1 (g) =/ 0. Then for all z ∈ A× Q , we have ω1 (z)φ1 (g) = φ1 (gz) = c φ2 (gz) = c ω2 (z)φ2 (g) = ω2 (z)φ1 (g). Choosing g so that φ1 (g) =/ 0 implies ω1 = ω2 , a contradiction. × (b) (1) and (3) are trivial  |γ |A = | det(k)|A = 1 for all γ ∈ Q

 because av bv  and k ∈ K . If g = cv dv , then | det(g)|A =



|av dv − bv cv |v ⇒ %g%−1 ≤ | det(g)|A ≤ 2%g%2 .

v≤∞

Since f is assumed to have moderate growth, so does φ, proving (5). Next one uses the chain rule and Exercise 4.15 to show   Dα φ(g) = Tr(α) f | det(g)|A | det(g)|A , (g ∈ G L(2, AQ ), α ∈ gl(2, R)), which immediately gives (4). An example for which (2) does not hold is f = log. 4.20 f adelic is invariant under the action of (right) translation by elements of the center at infinity.

Solutions to Selected Exercises

493

4.21 Let’s start by computing f adelic according to the definition in Section 4.8. We assume that f : h → C is an even or odd Maass form of weight zero and level 1. For g ∈ G L(2, AQ ), write  g = i diag (γ ) ·

x∞ 1

y∞ 0



r 0 0 r



5 , I2 , . . . , I2 , . . .

as in Theorem 4.4.4. Note that k ∈ K = O(2, R)·

 p

· k,

G L(2, Z p ). Then

f adelic (g) = f (x∞ + i y∞ ) · det(k∞ ) j , where j = 0 if f is even and j = 1 if f is odd. Now we compute f adelic according to Section 4.12. Since f has level 1, it has trivial character χ . We may rearrange the above expansion for g to get g = i diag (γ ) · i ∞ (g∞ ) · kfinite , where

 g∞ =

y∞ 0

x∞ 1



r 0 0 r

 · k∞ .

Observe that k∞ ∈ S O(2, R) if and only if g∞ ∈ G L(2, R)+ . So we have two cases to consider. If g∞ ∈ G L(2, R)+ , then g∞ is in the correct form to apply Definition 4.12.8, and we find f adelic (g) = ˜f (g∞ ) = f (g∞ .i) = f (x∞ + i y∞ ). As det(k∞ ) = 1 in this case, the two definitions of f adelic agree. If on the other hand g∞ ∈ G L(2, R)+ , then we need to put it in the

∈ S O(2, R) correct form to apply Definition 4.12.8. There exists k∞  −1 0   y x  −1 0   −1 0  y −x  such that k∞ = 0 1 k∞ . Since 0 1 0 1 = 0 1 0 1 , we have g = i diag (γ ) · i ∞ (g∞ ) · kfinite       y∞ x∞ r 0 −1 0 · k · kfinite = i diag (γ ) · i ∞ 0 1 0 r 0 1 ∞        −1 0 y∞ −x∞ r 0

· k∞ · i∞ = i diag γ · 0 1 0 r 0 1   −1 0 · kfinite . · i finite 0 1

494

Solutions to Selected Exercises

Now g is in the correct form to apply Definition 4.12.8, and we find  y∞ f f adelic (g) = ( 0

−x∞ 1



r 0 0 r



· k∞



= f (−x∞ + i y∞ ) = (−1) j f (x∞ + i y∞ ), where j = 0 if f is even and j = 1 if f is odd, which is in perfect agreement with f adelic as computed by the Definition of 4.8 above. 5.1 (b) Use the integral identity in Exercise 1.10 to compute the Fourier coefficients. −1 ) is open 5.2 Since K is compact, it suffices to show that K ∩ (afinite K afinite in K . This is easily verified since the ∞-part of the intersection is precisely K ∞ . For the final conclusion, if φ is an automorphic form, then we choose k1 , . . . , kn ∈ K such that the functions g → φ(gki ) span the space of all right K -translates of φ. Let k1 , . . . , km ∈ K be a −1 ) ⊂ K . For collection of left coset representatives for K ∩ (afinite K afinite

−1

k ∈ K , we can then write k = k j afinite k afinite for some k ∈ K , so that

φ(gkafinite ) = φ(gk j afinite k ) =

n

ci φ(gk j afinite ki )

i=1

for some constants c1 , . . . , cn . That is, any right translate of g → φ(gafinite ) by an element of K must lie in the span of the maps g → φ(gk j afinite ki ), which is finite-dimensional. 5.4 Yes. 5.5 One can check that K ∞ acts trivially (obvious), that πg (Dα ).φ = Tr(α)φ (use the technique in the solution to exercise 4.16), and that πfinite (afinite ).φ = φ(afinite ) + φ. Hence the space of functions generated by these actions is V = C ⊕ C.φ. The subspace of constant functions is invariant under all of the actions, so V is not irreducible. 5.6 (a) Suppose |X | = m < n. Then the space of all complex functions on X has dimension m, a contradiction. (b) Choose a basis f 1 , . . . , f n of V . There exists x1 ∈ X such that f 1 (x1 ) =/ 0; else f 1 ≡ 0, a contradiction. Suppose we have chosen x1 , . . . , xr ∈ X so that the vectors ( f 1 (xi ), . . . , fr (xi )) ∈ Cr for 1 ≤ i ≤ r are linearly independent. Suppose that we cannot extend this statement to r + 1. Then for every x ∈ X , the vectors ( f 1 (xi ), . . . , fr +1 (xi )) for 1 ≤ i ≤ r and ( f 1 (x), . . . , fr +1 (x)) are

Solutions to Selected Exercises

495

linearly independent in Cr +1 . That is, for every x ∈ X , we have ⎛ ⎞ f 1 (xr ) f 1 (x) f 1 (x1 ) · · · ⎜ .. .. .. ⎟ .. 0 = det ⎝ . . . . ⎠ fr +1 (x1 )

⎛

⎜ = ± fr +1 (x) det ⎝

··· f 1 (x1 ) .. . fr (x1 )

fr +1 (xr ) ··· .. . ···

fr +1 (x) ⎞ f 1 (xr ) r .. ⎟ + c f (x), ⎠ i i . fr (xr )

i=1

from which we may conclude that fr +1 ∈ Span{ f 1 , . . . , fr }. But this contradicts the linear independence of the f i ’s. By induction, we may select x1 , . . . , xn ∈ X such that the vectors ( f 1 (xi ), . . . , f n (xi )) ∈ Cn ,

(i = 1, . . . , n),

are linearly independent. To complete the exercise, suppose we have f ∈ V with f (xi ) = 0 for each i = 1, . . . , n. Write f = c j f j for some scalars c1 , . . . , cn . Then 0 = f (xi ) = c j f j (xi ) (i = 1, . . . , n) ⎛ ⎞⎛ ⎞ ⎛ ⎞ f 1 (x1 ) · · · f 1 (xn ) c1 0 ⎜ .. ⎟ . . .. .. ⎠ ⎝ ... ⎠ = ⎝ ... ⎠ . ⇒⎝ . 0 cn f n (x1 ) · · · f n (xn ) This matrix is nonsingular by the previous paragraph, and so c1 = · · · = cn = 0; i.e., f ≡ 0. 5.9 S O(2, R) is abelian. If V = {π K ∞ (k).v|k ∈ K ∞ } is finite dimensional, then the action of S O(2, R) splits into 1-dimensional invariant subspaces. 5.10 Suppose that f : G L(2, R) → C is a smooth function and g ∈ G L(2, R). First we treat the case det(g) > 0. Use the Iwasawa decomposition to write     1 x y 0 r 0 g= κθ , 0 1 0 1 0 r  θ sin θ  for some where x, y, r ∈ R with y, r > 0 and κθ = −cos sin θ cos θ θ ∈ [0, 2π ). Then     t 0 1 x y 0 re κθ , g · exp(t Z ) = 0 r et 0 1 0 1

496

Solutions to Selected Exercises and so the chain rule gives      t  ∂ 0 1 x y 0 re κθ  D Z f (g) = f 0 r et 0 1 0 1 ∂t t=0  ∂(r et )  ∂ ∂ f (g) · f (g). = =r· ∂r ∂t t=0 ∂r Now we consider D H . Just as above, suppose det(g) > 0 first and write g as above. Then  g + gHt =  =

1 0

x 1

1 0

x 1

 

   1 t r 0 κθ −t 1 0 r  √  0 r 1 + t2 √0 1 0 r 1 + t2  1  t

y 0

0 1

y 0



· κθ

√

√

1+t 2 √−t 1+t 2

.

1+t 2 √1 1+t 2

Notice that the √ By choosing α(t) such that √final matrix is orthonormal. cos α(t) = 1/ 1 + t 2 and sin α(t) = t/ 1 + t 2 , we find that  g + gHt =

1 0

x 1



y 0

0 1

 √ r 1 + t2 0

√0 r 1 + t2

 κθ+α(t) .

Hence  √ r 1 + t2 0

 √0 r 1 + t2   ·κθ+α(t)  t=0 √   ∂(r 1 + t 2 )  ∂(θ + α(t))  ∂f ∂f (g) · =  + ∂θ (g) ·  . ∂r ∂t ∂t t=0 t=0

∂ f D H f (g) = ∂t



1 0

x 1



y 0

0 1

The first term vanishes, and one checks easily that ∂α(t) = 1+t1 2 . So the ∂t whole expression reduces to ∂∂θf (g). Finally, if det(g) < 0, then the above arguments proceed verbatim 0 . upon replacing g with g −1 0 1  t  5.13 For the first identity, we find exp(t Z ) = e0 e0t , which commutes with  t sin t  = kt . Hence everything. Next observe that exp(t H ) = −cos sin t cos t  1 π K ∞ (kt ).v − v , t→0 t

πg (D H ).v = lim

(v ∈ V ).

Solutions to Selected Exercises

497

The second and third identities follow easily since kt commutes with anything in S O(2, R) and since kt δ1 = δ1 k−t . 5.14 Since f adelic is constant on Z (G L(2, R)), we find that D Z = r · ∂r∂ kills it. The fact that  acts by a scalar on f adelic follows from the fact that f is a Maass form. 5.15 The module structure follows from Exercise 4.16(a) and the definition of φ. For smoothness, write ω = χadelic · | · |itA for some t ∈ R and  Dirichlet character χ (mod N ). Factor N = p f p . Let    U= (1 + p f Z p ) × Z×p , U = det−1 (U ) ∩ G L(2, Z p ). p|N

pN

p

Then ω|U = 1, so that π (a).φ = φ for all a ∈ U . Since U ⊆ G L(2, Afinite ) is compact open, this shows V is smooth. 5.16* Take f 0 : G L(2, R) → C such that    cos θ sin θ = eikθ f 0 (g), f0 g · − sin θ cos θ for some k =/ 0. Let

Let

(g ∈ G L(2, R)),

   2 0 f (g) = f 0 g · . 0 1

   −1  0 2 2 0 2 0 , ·H· = − 12 0 0 1 0 1  0 1 where H = −1 . The differential operator D H acts by ∂θ∂ in our 0 usual coordinates (Exercise 5.10), so that D H f 0 = ik f 0 . Check   that D H f = ik · f. On the other hand, H = 54 H + 34 Y , where Y = 00 11 . Thus 3 5 3 DH = DH − i R + i L . 4 4 4 Each element of an admissible (g, K ∞ ) × G L(2, Afinite )-module decomposes as a finite sum of elements from D H -eigenspaces. (See also Exercise 7.1.) Now show that no eigenvector of D H can be written as such a finite sum. The key point is that D H eigenspaces have a certain weight, and the operators R and L raise or lower this weight, respectively. H =



5.17 (a) We find that D Z commutes with all differential operators because Dα ◦ D Z − D Z ◦ Dα = D[α,Z ] = Dα Z −Z α = D0 = 0.

498

Solutions to Selected Exercises   (b) Write α = ac db = a E 1,1 + bE 1,2 + cE 2,1 + d E 2,2 , where {E i, j } is the standard basis for the space of 2 × 2 matrices. Compute  ◦ Dα − Dα ◦  by repeatedly using the identity Dβ ◦ Dγ = Dγ ◦ Dβ + D[β,γ ] . (c) Prove by induction using Lemma 5.2.7. (d) By Proposition 5.2.10, it suffices to show that if D commutes with D H , R, and L, then D = b,c αb,c b D cZ . So suppose D= α,a,b,c R  ◦ D aH ◦ b ◦ Dzc ∈ Z (U (gl(2, C))). ,a,b,c

Then if D H commutes with D, we find α,a,b,c (D H ◦ R  ) ◦ D aH ◦ b ◦ Dzc DH ◦ D = = α,a,b,c (R  ◦ D H + 2iR  ) ◦ D aH ◦ b ◦ Dzc b c = α,a,b,c R  ◦ D a+1 H ◦  ◦ Dz + 2i α,a,b,c  R  ◦ D aH ◦ b ◦ Dzc = D ◦ D H + 2i α,a,b,c  R  ◦ D aH ◦ b ◦ Dzc . By the uniqueness of the representation in Proposition 5.2.10 (which is proved in Exercise 7.11), we must have  = 0 for all terms of D in order to make the final sum cancel. So α0,a,b,c D aH ◦ b ◦ Dzc . D= a,b,c

Now observe that D aH ◦ R  = R  ◦ (D H + 2i)a . So α0,a,b,c D aH ◦ R  ◦ b ◦ D cZ D ◦ R = = α0,a,b,c R  ◦ (D H + 2i)a ◦ b ◦ D cZ = R  ◦ D + other terms in canonical form if a =/ 0. Again by the uniqueness of this representation, we have a = 0. This proves D is of the desired form.  t sin t  , this follows immediately from the last 5.18 Since exp(t H ) = −cos sin t cos t displayed equation in the definition of a (g, K ∞ )-module. 5.19 By direct computation, one sees that the commutator of K 0 (M) is contained inside K 1 (M). Conversely, if M =/ 1, let p f || M and consider the element    5 1+ pf 0 , I2 , . . . ∈ K 1 (M). g = I2 , . . . , I2 , 0 1

Solutions to Selected Exercises

499

 f  If g is a commutator of K 0 (M), then g p = 1+0p 01 is a commutator of G L(2, Q p ). But Exercise 6.7 shows g p ∈ S L(2, Q p ), which is absurd. 5.21 (c) φclassical ≡ 1 6.2 (a) For u ∈ Z p , we can truncate the series definition of u to see that u = u n +u n , where 0 ≤ u n < p n is an integer and u n ∈ p n Z p . Then   u can be represented by the integer u n . If g = ac db ∈ G L(2, Z p ), then ad − bc ∈ Z×p . Writing a = an + an etcetera as above, we find det(g) = an dn −bn cn +v, where v ∈ p n Z p . That is, p  an dn −bn cn . It follows that the reduction red(g) is an invertible matrix. α β (b) By definition, the kernel is equal to K n . For surjectivity, let γ δ be an element of G L(2, Z/ p n Z). We may represent α, β, γ , δ as α β integers in the interval [0, p n ). Now we may view γ δ as a matrix with integer entries, or since Z ⊂ Z p , as a matrix with Z p entries. The invertibility of the original matrix implies its determinant is a unit in Z/ p n Z, which is to say that αδ−βγ is not divisible by p. This clearly shows the matrix is invertible in G L(2, Z p ) as well. Hence red is surjective. (c) The index is given by counting the number of elements in  G L(2, Z/ p n Z). Let g = ac db be an element of G L(2, Z/ p n Z). We view a, b, c, d as integers. Each column must have at least one entry that is prime to p, or else the determinant will be divisible by p, causing g to be singular. The same statement must hold for the rows. Moreover, the second column cannot be equal to the first column times an integer prime to p (modulo p n ). These conditions are sufficient for invertibility as well. We consider two cases now. If the first column has no entry divisible by p, then the second column is required to have at least one entry prime to p and it cannot be a unit multiple of the first column. There are ϕ( p n )2 such choices for the first column, and then p 2n − p 2(n−1) − ϕ( p n ) choices for the second column. In the second case, in which the first column has one entry divisible by p, the second column is only required to have a unit in the position diagonally opposite the unit in the first column (so that the determinant is not divisible by p). There are 2ϕ( p n ) · ( p n − ϕ( p n )) choices for the first column, and ϕ( p n ) · p n choices for the second column. In total, we find the number of invertible matrices is ϕ( p n )2 ( p 2n − p 2(n−1) − ϕ( p n )) + 2ϕ( p n ) · ( p n − ϕ( p n )) · ϕ( p n ) · p n ? @ = p 3n−3 ( p − 1)2 p n−1 ( p 2 + 2 p − 1) − p + 1 .

500

Solutions to Selected Exercises 6.3 It follows easily from the definitions that K n is a subgroup of K m . Restrict the reduction homomorphism red : G L(2, Z p ) → G L(2, Z/ p n Z) to the subgroup K m . The kernel will still be K n , which shows K n is normal in K m . Now compute the index using the previous exercise and the isomorphism G L(2, Z p )/K n . G L(2, Z p )/K m ∼ = K m /K n 6.6 (a) Let v ∈ V . Since V is smooth, we know v is stabilized by some compact open subgroup K . Since the K n ’s form a base of open neighborhoods of the identity, we know K n ⊂ K for some n. So  v ∈ V K n , and we have V ⊂ n≥1 V K n . The other containment is trivial, and the fact that V K m ⊂ V K n if m < n is an immediate consequence of the definitions. (b) Let v ∈ V K n and k ∈ G L(2, Z p ). We must show π (k).v ∈ V K n , or equivalently, that π (k ).(π (k).v)) = π (k).v for all k ∈ K n . As K n is normal in G L(2, Z p ), there is k

∈ K n such that k k = kk

. Since v is stable under the action of K n , we have π (k ).(π (k).v) = π (k k).v = π (kk

).v = π (k).(π (k

).v) = π (k).v. 6.7 Let G = G L(2, Q p ). If g, h ∈ G, then det(ghg −1 h −1 ) = 1. Hence [G, G] ⊂ S L(2, Q p ). On the other hand, for x, y ∈ Q p , we have the identities        1 x 1 x/2 1 0 1 −x/2 1 0 = , 0 1 0 1 0 −1 0 1 0 −1 

    1 0 1 0 1 0 1 0 = . y/2 1 0 −1 −y/2 1 0 −1   1 0 This shows that 10 x1 and y 1 are commutators. Since these matrices generate S L(2, Q p )—by Lemma 6.1.6—we have shown S L(2, Q p ) ⊂ [G, G]. 1 y

0 1





6.8 (a) Since V is 1-dimensional, π must act by scalars. For each g ∈ G L(2, Q p ), define λg ∈ C× by π (g).v = λg v. One checks easily that g → λg is a homomorphism. (b) Observe that det : G L(2, Q p ) → Q×p is surjective with kernel S L(2, Q p ); hence, G L(2, Q p )/S L(2, Q p ) ∼ = Q×p . Since π : × G L(2, Q p ) → C contains S L(2, Q p ) in its kernel, so does λ. Thus it must factor through the determinant, and we get a

Solutions to Selected Exercises

501

homomorphism ω : Q×p → C× such that λg = ω(det(g)). By smoothness of π , its kernel contains a compact open subgroup K n for some n, which shows ω(det(K n )) = ω(1 + p n Z p ) = {1}. Hence ω is continuous. 6.9 Consider the direct sum of two smooth irreducible representations. By the theorem, each summand is admissible, and it is easy to see that if V = V1 ⊕ V2 , then V K n = V1K n ⊕ V2K n . Hence V is also admissible, but clearly not irreducible. 6.10 Observe that j −1 (K n ) = 1+ p n Z p , so j is continuous. Next observe that the central character of an irreducible representation (π, V ) is given by ωπ = π ◦ j, a composition of continuous maps. 6.11 Take a neighborhood U of the identity in G L(V ) that contains no non-trivial subgroup. (This is where we use the fact that V is finitedimensional.) By continuity of π , we find K n ⊂ π −1 (U ) for some positive integer n. (See Exercise 6.1.) Hence π (K n ) ⊂ U . As U contains no non-trivial subgroup, we find K n ⊂ ker(π ), which is to say that K n stabilizes every vector in V . Thus π is smooth. 6.12 Let v ∈ V be a non-zero G L(2, Z p )-fixed vector. Then for any a ∈ Z×p , we have   a 0 v = π .v = ωπ (a)v. 0 a Hence ωπ is trivial on Z×p . 6.14 Suppose f ∈ V p (s, ω) is non-zero and fixed by G L(2, Z p ). Let y 0  y1 , y2 ∈ Z×p . Then 01 y2 ∈ G L(2, Z p ), so    y1 0 y1 . f (I2 ) = f I2 . 0 y2 0 = ω1 (y1 )ω2 (y2 ) f (I2 ). 

f (I2 ) = π

0 y2



Varying y1 , y2 shows f (I2 ) = 0 or 1 =  ω2 ≡ 1. If f (I2 ) = 0, then for  yω 1 x any g ∈ G L(2, Q p ), write g = 0 y2 k with y1 , y2 ∈ Q×p , x ∈ Q p , and k ∈ G L(2, Z p ). Since π (k). f = f , we have     y1 x y1 x k = f 0 y2 0 y2 s1 s2 = |y1 | p ω1 (y1 )|y2 | p ω2 (y2 ) f (I2 ) = 0. 

f (g) = f

As g was arbitrary, we conclude f = 0, a contradiction.

502

Solutions to Selected Exercises

 y1 x  6.15 For g ∈ G L(2, Q p ), write g = 0 y2 k with y1 , y2 ∈ Q×p , x ∈ Q p , and k ∈ G L(2, Z p ). Since f is fixed by G L(2, Z p ),      y1 x y1 x f (g) = f k = f = |y1 |sp1 |y2 |sp2 f (I2 ). 0 y2 0 y2 Comparing with Definition 6.4.2, we find f = c f s◦1 ,s2 , where c = f (I2 ). 6.16 Let f be a non-zero element of the 1-dimensional subrepresentation. Then Span( f ) is a 1-dimensional smooth representation, which must be of the form π (g). f = η(det(g)) f for some character η : Q×p → C× by Theorem 6.1.7. Now for any y1 , y2 ∈ Q×p , x ∈ Q p and g ∈ G L(2, Q p ), we have      y1 x y1 x g = η(det(g)) f f 0 y2 0 y2 = η(det(g))|y1 |sp1 ω1 (y1 ) · |y2 |sp2 ω2 (y2 ) f (I2 ). This shows f (I2 ) =/ 0 since varying g ∈ G L(2, Z p ) and yi ∈ Q×p and x ∈ Q p generates all of G L(2, Q p ). Now take y = y1 = y2−1 , x = 0  −1  and g = y0 0y to find −1 2 f (I2 ) =⇒ |y|sp1 ω1 (y) f (I2 ) = η(1)|y|sp1 ω1 (y)|y|−s p ω2 (y)

= |y|sp2 ω2 (y),

(y ∈ Q×p ).

Varying y ∈ Z×p shows ω1 = ω2 , and taking y = 1/ p shows p s1 = p s2 . The remaining claim about s1 and s2 follows upon applying a branch of the complex logarithm. 6.19 Define f 0 (g) = f (g; 0, 0). Then f 0 ∈ V p (0, ω) by the definition of flat section. Now we compute the desired quantity using the fact that f (∗; s1 , s2 ) ∈ V p (s, ω) for each s, and that f (k; 0, 0) = f (k; s1 , s2 ) for every k ∈ G L(2, Z p ) and every s:    y1 x f k; s1 , s2 = |y1 |sp1 ω1 (y1 )|y2 |sp2 ω2 (y2 ) f (k; s1 , s2 ) 0 y2 = |y1 |sp1 ω1 (y1 )|y2 |sp2 ω2 (y2 ) f (k; 0, 0). 6.20 Suppose we have two decompositions g = b1 k1 = b2 k2 with bi upper triangular and ki ∈ G L(2, Z p ). Then b2−1 b1 = k2 k1−1 ∈ B(Q p ) ∩ G L(2, Z p ).

Solutions to Selected Exercises

503

u v   yi Thus b2−1 b1 = 0 u with u, u ∈ Z×p and v ∈ Z p . Writing bi = 0 we find        y2 x2 u v y2 u ∗ y1 x1 = = . 0 y1 0 y2 0 u 0 y2 u

xi yi

 ,

This shows |y1 | p = |y2 u| p = |y2 | p , and similarly |y1 | p = |y2 | p . 6.21 Note d =/ 0. If |c/d| p ≤ 1, then the Iwasawa decomposition is      a 0 a 0 1 0 = . c d 0 d c/d 1 The values of |y1 | p and |y2 | p be can read from this decomposition. If |c/d| p > 1, then we can instead use      a 0 −ad/c a − pc/(c + pd) c/(c + pd) = . c d 0 c + pd c/(c + pd) d/(c + pd) Note that |c+ pd| p = |c| p by the strong triangle inequality, since |c| p > |d| p > | pd| p . 6.22 (a) Write H = δ −1 K n δ ∩ K n and e = e2 − e1 for simplicity. We claim that    5  a b H= ∈ K n  b ≡ 0 (mod p e+n ) . c d   Indeed, ac db ∈ H if and only if it lies in K n and  δ

a c

b d



δ −1 =



a pe c

p −e b d

 ∈ Kn ,

which is equivalent to saying p −e b ∈ p n Z p . That is to say, b ≡ 0 (mod p e+n ).     α β Now suppose that ac db and γ δ are two matrices in K n . Then they determine the same right coset of H if and only if   −1   1 a b α β ∗ bα − aβ ∈ H. ∈ H ⇐⇒ c d γ δ ∗ αδ − βγ ∗ The above matrix lies in H if and only if the upper right entry satisfies bα ≡ aβ (mod p e+n ). Since b and β are divisible by p n and since a and α lie in Z×p (by definition of K n ), this amounts to saying a −1

b β ≡ α −1 n n p p

(mod p e ).

504

Solutions to Selected Exercises   So the right coset of ac db is uniquely determined by the value of a −1 pbn modulo p e , from which it follows easily that the index [K n : H ] is p e . The second equality of indices is proved similarly.  p e1 0 (b) Let H = g −1 K n g ∩ K n . Write g = k1 · 0 pe2 · k2 for some k1 , k2 ∈ G L(2, Z p ) and  e1 ≤ e2 , by the Cartan decom integers position. Write δ = gives

p e1 0 0 p e2

δ −1 K n δ = k2 (g −1 K n g)k2−1

as in (a). Then normality of K n

k2 H k2−1 = δ −1 K n δ ∩ K n .

and

Conjugation by k2 is an automorphism of G L(2, Z p ) that fixes each K n , and so [K n : H ] = [K n : k2−1 H k2 ] = [K n : δ −1 K n δ ∩ K n ] = [δ −1 K n δ : δ −1 K n δ ∩ K n ] by part (a) = [k2 (g −1 K n g)k2−1 : k2 H k2−1 ] = [g −1 K n g : H ]. (c) The first statement follows upon writing K n and g −1 K n g as disjoint unions of (compact open) left cosets for the subgroup g −1 K n g ∩ K n and then using the left invariance and finite additivity of μ. For the second statement, since μ is left invariant we have μ(K n · g) = μ(g −1 K n g) = μ(K n ). 6.23 On one hand, by translation invariance of the Haar measure μ, we have  1g K n (h)dh = μ(g K n ) = μ(K n ). G L(2,Q p )

Now define  I =

Q×p





 Q×p



 Qp



1g K n G L(2,Z p )

a1 0

0 a2



1 0

x 1

  k

· d × k d x d × a1 d × a2 .

b1 c with ξ ∈ G L(2, Z p ), bi ∈ Q×p , c ∈ Q p 0 b2 using the Iwasawa decomposition. Make the following changes of variables: a1 → a1 /b1 , a2 → a2 /b2 and x → x − a2 c/a1 b2 . The measures d × ai and d x are invariant under these types of changes, which Write g −1 = ξ

Solutions to Selected Exercises shows  I =



Q×p

 Q×p

Qp

505



 a1 /b1 0 1g K n 0 a2 /b2 G L(2,Z p )    1 x − a2 c/a1 b2 k d × k d x d × a1 d × a2 . · 0 1



Now the integrand vanishes unless the displayed matrix product is in g K n , or equivalently    a1 /b1 1 x − a2 c/a1 b2 0 k g −1 0 1 0 a2 /b2    a1 /b1 0 b1 c = ξ 0 b2 0 a2 /b2   1 x − a2 c/a1 b2 k · 0 1   a1 a1 x k = ξ 0 a2    1 x a1 0 k ∈ Kn = ξ 0 a2 0 1    1 x a1 0 k ∈ ξ −1 K n ⊆ G L(2, Z p ). ⇔ 0 1 0 a2 This last condition fails if ai ∈ Z×p or x ∈ Z p , so our integral becomes         a1 0 1 x k 1ξ −1 K n I = 0 1 0 a2 Z×p Z×p Z p G L(2,Z p )

Now make the change of variables k ξ −1 k to get

 I =

Z×p

 Z×p

 Zp

 G L(2,Z p )

·d × k d x d × a1 d × a2 .   1 −x 1/a1 → 0 0 1

0 1/a2



1ξ −1 K n (k) d × k d x d × a1 d × a2

= μ(ξ −1 K n ) = μ(K n ). 6.25 For f ∈ K, the Kirillov model of (π, V ), Theorem 6.7.2 (Step 2) shows f vanishes outside of some compact subset of Q p . The proof of Proposition 6.15.1 gives    1 u

. f du = p n · f · 1 pn Z p , π (n ∈ Z). 0 1 p−n Z p

506

Solutions to Selected Exercises Now this last function vanishes identically for all n sufficiently large if and only if f vanishes on p n Z p  {0} for n sufficiently large — i.e., if f ∈ S X (Q×p ).

6.26 If such a W existed, we would have an exact sequence of representations L s,ω 0 → W → W ⊕ W → C → 0. This implies W ∼ = C is a 1-dimensional subrep= (W ⊕ W )/W ∼ resentation of V p (s, ω). By Exercise 6.16, this means p s1 −s2 = 1, a contradiction. 6.27* (a) Let {gα }α∈A be a collection of right coset representatives for the subgroup H . Then {gα−1 }α∈A is a collection of left coset representatives. A basis for C[G] ⊗C[H ] W as a complex vector space is given by {gα−1 ⊗ 1}α∈A . Define a map L : C[G] ⊗C[H ] W → V as follows. A vector v ∈ C[G] ⊗C[H ] W can be written uniquely in the form v = α gα−1 ⊗ λα . Then L v : G → W is given by L v (hgα ) = π (h).λα ,

(h ∈ H, α ∈ A).

One checks easily that L v ∈ V , and that it is linear and injective. To see that it is an intertwining map, choose s ∈ G. For each α, there exists a unique h α ∈ H and gβ such that gα s −1 = h α gβ . Equivalently, we have h −1 α gα = gβ s. Note that left multiplication by s permutes the left cosets in G/H . Then (sgα−1 ) ⊗ λα = (h α gβ )−1 ⊗ λα = gβ−1 ⊗ π (h −1 s.v = α ).λα α

=⇒ L s.v (gβ ) =

β

π (h −1 α ).λα

β

=

L v (h −1 α gα )

= L v (gβ s) = ρ(s).L v (gβ ).

Clearly L v is supported on the finite set of right cosets {H gα  λα =/ 0}, and any such function lies in the image of L. (b) The image of the map L from part (a) has a basis consisting of characteristic functions of a single coset in H \G. In our case, one can check that B(Q p )\G L(2, Q p ) is uncountable. Indeed, a collection of coset representatives is given by       0 1 1 0  .  y ∈ Qp ∪ 1 0 y 1  Thus C[G] ⊗C[H ] W has uncountable dimension. The principal series representation V p (s, ω) is admissible, and so it has countable dimension by exercise 6.6.

Solutions to Selected Exercises

507

6.28 (a) We know by Exercise 6.14 that V p (s, ω) contains a G L(2, Z p )fixed vector if and only if ω = ωtrivial , in which case the vector is f = f s◦1 ,s2 . But then f ∈ ker(L s,ω ) because  L s,ω ( f ) = G L(2,Z p )

f (k)ω1−1 (det(k)) d × k = 1.

(b) If f ∈ V p (s, ω) is G L(2, Z p )-invariant in the special representation, this means that for each k ∈ G L(2, Z p ) there exists a constant c(k) ∈ C for which the following formula holds: (π (k). f )(g) = f (g) + c(k)δs,ω (g),

(g ∈ G L(2, Q p )).

Now we integrate along G L(2, Z p ) to obtain a new function  F(g) := (π (k). f )(g) d × k = f (g) + Cδs,ω (g), G L(2,Z p )

(g ∈ G L(2, Q p )),  where C = G L(2,Z p ) c(k) d × k. (Note that since f is smooth, c(k) is a locally constant function, and hence integrable.) One checks easily that F ∈ V p (s, ω) and that it is G L(2, Z p )-invariant, using the defining feature of the Haar measure. We have two cases to consider. If F ≡ 0, then f is a scalar multiple of δs,ω , and so f vanishes in the quotient V p (s, ω)/C.δs,ω . If F ≡ 0, then it is a non-trivial G L(2, Z p )-fixed vector. By Exercises 6.14 and 6.15, we see that ω = ωtrivial and that F is a scalar multiple of f s◦1 ,s2 . A simple computation then shows δs,ω = f s◦1 ,s2 . So f = F − Cδs,ω lies in the span of δs,ω , which means it vanishes in the quotient V p (s, ω)/C.δs,ω . 7.1 Clearly Vm is an S O(2, R)-invariant subspace for each m ∈ Z. Now suppose v ∈ V . By the alternative formulation of K ∞ -finiteness (see the remark following Definition 5.1.4), we have integers M < N , vec tors v ∈ V , and scalars c for M ≤  ≤ N such that v =  c v , which is the type of decomposition we seek. It remains to show that this decomposition of v is unique, and without loss of generality we may suppose v = 0. Moreover, we may replace c v by v . To summa rize, we have 0 =  v with v ∈ V for M ≤  ≤ N , and we would like to prove that each v = 0. Let us suppose otherwise, and choose a non-trivial linear combination v = 0 with the minimum possible number of non-zero terms. If it had only one term, we would immediately see that term is zero, a contradiction. So there is some index

508

Solutions to Selected Exercises j =/ 0 with | j| maximal such that v j =/ 0. For any fixed angle θ that is  θ sin θ  to the equation not a rational multiple of π , apply π K ∞ −cos sin θ cos θ iθ v = 0 to obtain e v = 0. Subtracting ei jθ times the original equation from this one gives (eiθ − ei jθ )v = (eiθ − ei jθ )v = 0. 

 =/ j

Our choice of θ guarantees none of the remaining coefficients vanish. Thus we have produced a linear combination representing zero with one fewer term than before, which is a contradiction. 7.2 Suppose v = v is a non-zero K ∞ -fixed vector, where each v ∈ V and all but finitely many are zero. If we take an angle θ that is not a rational multiple of π , then   cos θ sin θ .v = eiθ v =/ v, πK∞ − sin θ cos θ unless v = v0 ∈ V0 . 7.3 By Exercise 7.2, V must be even, and the K ∞ -fixed vectors all lie in the S O(2, R)-invariant subspace V0 . Moreover, we see that a vector in V0 is K ∞ -fixed if and only if π K ∞ (δ1 ) acts on V0 with eigenvalue 1. 7.4 Let B(R× ) be the group of 2 × 2 invertible upper triangular matrices  0  . The with real coefficients. Observe that K ∞ ∩ B(R× ) = ±1 0 ±1 solution now proceeds just as for Exercise 6.20. 7.5 Let V be the complex vector space generated by the action of K ∞ on f . By hypothesis, it’s finite-dimensional. Now S O(2, R) is a compact abelian group, so V decomposes as a direct sum of S O(2, R)-invariant subspaces on which S O(2, R) acts by characters. In particular, we have a decomposition f = nj=1 f j , where the smooth functions f j satisfy π K ∞ (kt ). f j (g) = f j (gkt ) = ei jt f j (g), (t ∈ R, g ∈ G L(2, R), j = 1, . . . , n).  t sin t  Here we have set kt = −cos . To complete the proof, it suffices to sin t cos t show that precisely one of the Fourier coefficients of each f j is nonzero. So let us suppose that cn (x, y, r )einθ , f j (gkθ ) = n∈Z

Solutions to Selected Exercises where g =

509

 1 x  y 0  r 0  . Fix x, y, r . Then 0 1 0 1 0r

cn (x, y, r )ein(θ+t) = f j (gkθ+t )

n∈Z

= π K ∞ (kt ). f j (gkθ )   = ei jt f j (gkθ ) = cn (x, y, r ) ei(nθ+ jt) ⇒



n∈Z

  cn (x, y, r ) eint − ei jt einθ = 0.

n∈Z

Fix t to be some angle that is not a rational multiple of π . If we view this last expression as a function of θ , then we have a function with vanishing Fourier expansion. That is, all of its coefficients   int i jt must be zero. If n =/ j, then we must have cn (x, y, r ) e − e cn (x, y, r ) = 0. Now vary x, y, r to see that cn ≡ 0 for n =/ j. 7.6 Let s1 = μ − s2 to get a quadratic equation in s2 with discriminant 4 + 8λ − 4μ2 . So there are exactly two solutions for s2 unless λ = (μ2 − 1)/2, in which case there is only one. On the other hand, observe that if (s1 , s2 ) is a solution, then so is the image of the involution (s1 , s2 ) → (s2 + 1, s1 − 1). So we get a second solution unless s1 = s2 + 1 ⇔ s1 =

μ2 − 1 μ+1 μ−1 , s2 = ⇔λ= . 2 2 2

Putting these two observations together shows that either the given system of equations has a single solution, which is fixed by the involution, or else there are two solutions, which are swapped by the involution.  t sin t  . If v ∈ 7.7 (a) By diagonalizing H , one sees that exp(t H ) = −cos sin t cos t Vm , then (7.1.3) shows   d 1  πg (D H ).v = lim eimt v − v = eimt  v = (im)v. t→0 t t=0 dt  θ sin θ  (b) Use the fact that kθ δ1 = δ1 k−θ , where kθ = −cos . sin θ cos θ (c) Follows immediately from Lemma 5.2.4. 7.9 (a) f adelic is K ∞ -invariant by the proof of Proposition 4.8.4, so V f is unramified in this case. (b) Lemma 5.2.4 and the relations before Lemma 5.2.6 show Vf =

 N =1

c R

m

 5   f adelic (g)  N ∈ N, m  ∈ Z, c ∈ C .

510

Solutions to Selected Exercises As f is of weight zero, we see V f is an even (g, K ∞ )-module, and we find V f,2m = span(R m f adelic ). As any K ∞ -fixed vector must lie in V f,0 by Exercise 7.2, we see it must be of the form v = c f adelic for some complex number c. But f is odd, so π K ∞ (δ1 ).v = −v. Hence V f is ramified. (c) One sees V f is an odd (g, K ∞ )-module, so by Exercise 7.2 it is ramified. (d) If φ is a holomorphic modular form of weight k (≥ 0) and character χ for 0 (N ), then f (z) = y k/2 φ(z) is a Maass form of weight k and type ν = k/2. We know L k ( f ) = 0, which shows πg (L). f adelic = 0. If k > 0, then we are in case (ii) of Theorem 7.5.10, which shows V f,0 = 0 and that V f is ramified. So suppose it has weight k = 0. Then we can compute the action π K ∞ (δ1 ) explicitly using the definition of f adelic : 



π K ∞ (δ1 ). f adelic i ∞

y x 0 1 



 



= f adelic i ∞ 

= f adelic i diag (δ1 ) · i ∞

y 0

−x 1

=( χidelic (i finite (δ1 )) f (−x + i y),

y 0 

x 1

  δ1 

· i finite (δ1 ) (x + i y ∈ h).

Since π K ∞ (δ1 ) acts by ±1 on V f,0 , it follows that f (x + i y) = ± f (−x + i y). If this equality holds with a minus sign, then f is an odd function and we find V f is ramified by part (b). If instead we have f (x + i y) = f (−x + i y) for all x + i y ∈ h, then the Cauchy/Riemann equations show f is constant. 7.11 To prove uniqueness of this type of representation for differential operators, it suffices to prove the following: If D is any differential operator of the form D=



e j R  j ◦ D Hj ◦ b j ◦ D Zj , a

c

j

for which e j ∈ Z for all j, all tuples ( j , a j , b j , c j ) are distinct, and for which D.φ = 0 for every smooth function φ : G L(2, R) → C, then e j = 0 for all j. Let V∞ (s, ω) be an irreducible principal series representation for G L(2, R), where we choose s = (s1 , s2 ) such that s1 ± s2 ∈ Z. The collection of such s constitutes an (analytic) open set in C2 , being

Solutions to Selected Exercises

511

the complement of countably many algebraic curves. Suppose f m is a non-zero basis vector given by Definition 7.4.7. (Note that we are assuming m is a K ∞ -type for V∞ (s, ω).) Applying the explicit actions of differential operators to f m as in Proposition 7.4.8 gives 0 = πg (D). f m = e j (im)a j λb j μc j πg (R  j ). f m j

=

⎡

⎤

⎥ ⎢ ⎢ ⎥ ⎢ e j (im)a j λb j μc j ⎥ πg (R  ). f m . ⎣ ⎦ ∈Z

j  j =

Write A (m, λ, μ) for the polynomial in brackets. By hypothesis on V∞ (s, ω) and on m, the elements πg (R  ). f m are linearly independent (use Proposition 7.4.8 again), and hence A (m, λ, μ) = 0 for every  ∈ Z. Now fix  ∈ Z. By fixing m and varying s, we see that the polynomial A (m, λ, μ) vanishes on an analytic open subset of pairs (λ, μ) ∈ C2 . Since it is a polynomial, we see that A (m, Y, Z ) = 0 as a polynomial in Y and Z . But this statement holds for infinitely many choices of m, and hence A (X, Y, Z ) = 0 as a polynomial in three variables. Equivalently, e j = 0 for all j as desired. 8.1 Let n < m < m be such that π (k).v = v for all k ∈ K m ⊃ K m . Since K n /K m ∼ = (K n /K m )/(K m /K m ), we see that Q Q Kn = κλK m , κ∈K n /K m λ∈K m /K m

where the disjoint unions are taken over coset representatives κ for K n /K m and coset representatives λ for K m /K m . (Compare with Exercise 6.3.) Since each λ as above has the property that π (λ).v = v, we see that 1 π (κλ).v [K n : K m ] κλ∈K /K n m   1 = π (κ). π (λ).v [K : K m ][K m : K m ] κ∈K /K λ∈K m /K m n m   1 = π (κ). [K m : K m ] v [K : K m ][K m : K m ] κ∈K /K n

1 π (κ).v. = [K : K m ] κ∈K /K n

m

m

512

Solutions to Selected Exercises 8.3 Set f 2 (y) = f 1 (−y) ω(−y). Then  f 1 , f 2 ω > 0 unless f 1 ≡ 0. 8.4 Observe that   −1 0 0 0 −1 −1

1 0



1 −b 0 1



0 −1

1 0



 =

1 b

0 1

 .

Applying Lemma 6.1.6 shows that we can generate S L(2, Q p ). If   g ∈ G L(2, Q p ) has determinant a, then g = a0 01 g for some g ∈ S L(2, Q p ). 8.6 We have an exact sequence of representations 0 → V → V p (s, ω) → V p (s, ω)/V → 0, where the final non-zero term has dimension 1. Passing to contragredient representations gives another exact sequence  ( 0 → V p (s, ω)/V → V p (s, ω) → V → 0. The middle term is isomorphic to the principal series V p ((1 − s1 , −1 − s2 ), ω−1 ) by Proposition 8.2.2, and this sequence shows it has a 1-dimensional subrepresentation. By Exercise 6.16, we find ω1 = ω2 and p (1−s1 )−(−1−s2 ) = p 2−s1 +s2 = 1, which is equivalent to what we wanted to prove. V Kn ) 8.7 The proof of Proposition 8.1.5 shows that dim(V K n ) = dim(( for all n ≥ 1. In fact, the proof also works for n = 0, which is V does. to say that V has a G L(2, Z p )-fixed vector if and only if ( Exercise 6.28(a) shows that one kind of special representation is ramified, and Proposition 8.3.1 now implies that the other is also ramified. 8.9 Let F ⊂ U be a compact set of the form   Fv × G L(2, Z p ), F= v∈S

v∈ S

where S is a finite set of primes containing ∞, F∞ is a nonempty closed interval, and Fv is a compact open set in Qv for each finite v ∈ S. Choose p ∈ S. Then for distinct integers m, n, we have   −n    −m  p p 0 0 F ∩ ip F = ∅. ip 0 p −n 0 p −m 8.10 Use Theorems 8.2.5, 8.3.1, 8.4.7, and 8.12.25.

Solutions to Selected Exercises 8.14 If c =/ 0, then

a b c d

=

513

 − det(g)/c a  0 1  1 d/c  . 0 c 10 0 1

8.15 By the Cartan decomposition, we see that   n Q :  pn p 0 K K ( K Ep = 0 p −n 0

0



p −n−1

n≥0

; K ,

and hence  Ep

⎡  ⎣  n f (g) d × g = p K

n≥0



0 K 0 p−n

f (g) d × g ⎤

 + K

=

,  n≥0

=

 n≥0

 f (g) d g +

2n K p 0 K 0 1





×



 pn



n K p 0 K 0 1

f (g) d × g =

0

0 p−n−1





f (g) d × g ⎦ K

×

 f (g) d g

2n+1 0 K K p 0 1



f (g) d × g. D

The second equality follows by invariance of f under the center. 8.16

(a) The proof of absolute convergence is essentially the same as the proof of absolute convergence of Jacquet’s local Whittaker function in Proposition 6.4.3. Observe that Jacquet’s integral in Definition 6.6.1 and the integral in (a) differ by a factor of e p (−u), which has absolute value 1, and that f can attain several values on G L(2, Z p ) for f ∈ B p (χ1 , χ2 ). This causes no material difference in the proof though. (b,c) It is clear that h π(g) . f (g1 ) = h f (g1 g) for all g, g1 ∈ G L(2, Q p ). The smoothness of h f now follows from that of f. To check that h f ∈ B(χ2 , χ1 ), we make a simple change of variables:          a b 0 1 1 u a b g = hf f g du 0 d 1 0 0 1 0 d Qp     0 1 a 0 = f 1 0 0 d Qp    1 (du + b)/a · g du 0 1      a  d 0 0 1   f =  · 0 a 1 0 d p Qp    1 u · g du 0 1

514

Solutions to Selected Exercises a  1 −1   =   · χ1 (d)|d| p2 χ2 (a)|a| p 2 d p    0 1 1 · f 1 0 0 Qp 1

u 1

  g du

−1

= χ2 (a)|a| p2 χ1 (d)|d| p 2 · h f (g). Linearity of the function M follows readily from that of the integral. The fact that M is an intertwining map then follows from the above observation that h π(g) . f (g1 ) = h f (g1 g) = π (g) .h f (g1 ),

(g, g1 ∈ G L(2, Q p )).

To prove that M does not vanish identically, take f ∈ B(χ1 , χ2 ) defined by ⎧ a 1  a b  0 1  × 2 ⎪ ⎨ χ1 (a)χ2 (d)  d  p , if g = 0 d 1 u with a, d ∈ Q p , f (g) = b ∈ Qp, u ∈ Zp, ⎪ ⎩ 0, if g is not of this form. (One must check that f is smooth!) Then     0 1 M( f )(I2 ) = f du = 1 du = 1. 1 u Qp Zp (d) It is clear that (π, B(χ1 , χ2 )) and (π, B(χ2 , χ1 )) are isomorphic when they are irreducible. It follows from Corollary 6.8.15 that this is the case except when ω1 = ω2 and p s1 −s2 = 1 or p 2 . When phrased in terms of χ1 and χ2 , this condition becomes χ1 (t) · +1 χ2−1 (t) = |t|±1 p ∀t. Since we assume (s1 − s2 ) > 0, only |t| p is ∼ possible in the present set-up. If B p (χ2 | | p , χ2 ) = B p (χ2 , χ2 | | p ), then the latter must contain a special subrepresentation, which is false. (It has a special subquotient.)   φ1 (∗, k1 ), χ1−1 χ2 as the quotient of a 8.17 Use (8.11.24) to express Z ∞ s, ( convergent integral involving some derivative of φ1 and a polynomial in s which is independent of k1 . Treat the other local zeta integrals appearing in (8.11.26) and (8.11.27) similarly. 8.18 The key point is that Theorem 8.11.22(iii) is proved using an explicit absolutely and uniformly convergent integral, so one may differentiate under the integral sign. 8.19 If χ1 χ2−1 (x) is not of the form x m , then F1 (a, k1 , k2 ) = 2 · H4 (a; k1 , k2 ) + 2 · 1 (a; k1 , k2 ) F2 (a, k1 , k2 ) = 2 · H1 (a; k1 , k2 ) + 2 · H2 (a; k1 , k2 ) + 2 · 2 (a; k1 , k2 ).

Solutions to Selected Exercises

515

If χ1 χ2−1 (x) = x m with m ≥ 0, then F1 (a, k1 , k2 ) = 2 · H4 (a; k1 , k2 ) + 2 · 1 (a; k1 , k2 )  F2 (a, k1 , k2 ) = 2 · H1 (a; k1 , k2 ) + H2 (a; k1 , k2 ) + a m

 · H4 (a; k1 , k2 ) + 2 (a; k1 , k2 ) .

If χ1 χ2−1 (x) = x m with m ≤ 0, then  F1 (a, k1 , k2 ) = 2 H4 (a; k1 , k2 ) + 2 (a; k1 , k2 ) + a m (H1 (a; k1 , k2 )  + H2 (a; k1 , k2 )) F2 (a, k1 , k2 ) = 2 · 3 (a; k1 , k2 ). 8.20 Adapt the method used to prove that F2 vanishes to order m at 0 in the discrete series case. 9.1 Immediate consequence of the polarization identity: 3  1 m i v + i m w, v + i m w . (v, w) = 4 m=0

 θ sin θ  . Since kθ acts on V (resp. Vm ) with eigenvalue 9.2 Let kθ = −cos sin θ cos θ eiθ (resp. eimθ ), the invariance of the Hermitian form gives         eimθ v , vm = v , π K ∞ (k−θ ).vm = π K ∞ (kθ ).v , vm = eiθ v , vm . When m =/ , this shows (v , vm ) = 0. 9.3 Let v ∈ V . Then πg (D H ).v = i v. Since V is admissible and infinitedimensional, we see V =/ 0 for infinitely many . Letting  → ∞ shows %πg (D H ).v% = ∞. sup %v% v =/ V,v =/ 0 9.4 Set W = W ⊥ . 9.5 (a,b) Straightforward computations. ( (c) This part   follows upon defining a map L : W → V by L(w) (v) = v, w. Then L(w) is an intertwining map, and ker(L) is an invariant subspace of W . The conjugate linear bijection forces W to be irreducible and admissible as well. Now ker(L) =/ W because ( , )Her is not identically zero, and so ker(L) = 0. But we also observe that ( V is irreducible by Proposition 8.1.8. Hence L must be surjective.

516

Solutions to Selected Exercises 9.8 For j = 1, 2, let ( , ) j : V × V → C be an invariant positive definite Hermitian form. As in the proof of Proposition 9.1.4, ( we obtain conjugate linear  isomorphisms T j : V → V satisfying T j (π (g).v) = π˜ (g). T j (v) for all g ∈ G L(2, Q p ) and v ∈ V . Now T2−1 ◦ T1 : V → V is an intertwining map, so by Schur’s Lemma (6.1.8), there is a constant c such that T2−1 ◦ T1 (v) = cv for all v ∈ V . By definition, this means T1 (v) = T2 (cv) = cT2 (v),

(v ∈ V ).

Evaluating both sides at a vector v ∈ V shows (v , v)1 = c(v , v)2 . Taking v = v =/ 0 shows c > 0. 9.9 The proof is essentially identical to that of Exercise 9.8. 9.10* First suppose that V∞ (s, ω) is unitarizable. By Exercise 7.1, we have a direct sum decomposition B V∞ (s, ω) = V∞,m (s, ω). m∈Z

Let ( , ) be a positive definite Hermitian form on V∞ (s, ω). By exercise 9.1, the above decomposition is orthogonal. Moreover, recall V∞,m (s, ω) is either 0- or 1-dimensional and spanned by an explicit basis element f m (∗; s1 , s2 ) given by Definition 7.4.7. Set s1 = 12 +iζ1 and s2 = − 12 +iζ2 for some complex numbers ζ1 , ζ2 . We want to show that the ζ j are real, or else they are conjugate complex numbers. For this purpose, we use the explicit actions described by Proposition 7.4.8. Observe that s1 + s2 = i(ζ1 + ζ2 ), 1 s12 + s22 − s1 + s2 = − − ζ12 − ζ22 . 2 If V∞,m (s, ω) =/ 0, then we have (πg (D Z ). f m , f m ) = (s1 + s2 ) ( f m , f m ) = i(ζ1 + ζ2 ) ( f m , f m ) −( f m , πg (D Z ). f m ) = −(s1 + s2 ) ( f m , f m ) = i(ζ¯1 + ζ¯2 ) ( f m , f m ). By invariance of the Hermitian form and the fact that ( f m , f m ) =/ 0, we obtain ζ1 + ζ2 = ζ¯1 + ζ¯2 , which is equivalent to saying that ζ1 + ζ2 ∈ R. We also have (πg (). f m , f m ) = (s12 + s22 − s1 + s2 ) ( f m , f m )   1 = − − ζ12 − ζ22 ( f m , f m ) 2 2 ( f m , πg (). f m ) = (s1 + s22 − s1 + s2 ) ( f m , f m )   1 2 2 = − − ζ¯1 − ζ¯2 ( f m , f m ). 2

Solutions to Selected Exercises

517

Since πg () is a symmetric operator for any invariant Hermitian form, 2 2 these two expressions must be equal, which implies ζ12 + ζ22 = ζ¯1 + ζ¯2 , or equivalently that ζ12 +ζ22 ∈ R. As we have already shown ζ1 +ζ2 ∈ R, we conclude that either ζ1 and ζ2 are both real, or else they are complex conjugates. Suppose we are in the latter case so that ζ = ζ1 = ζ¯2 is not real. Observe that s1 − s2 + m = 1 + i(ζ − ζ¯ ) + m, − (s1 − s2 − (m + 2)) = 1 − i(ζ − ζ¯ ) + m. If m is a K ∞ -type of V∞ (s, ω), then it follows (again from Proposition 7.4.8) that s − s + m  1 2 ( f m+2 , f m+2 ) (πg (R). f m , f m+2 ) = 2  1 + i(ζ − ζ¯ ) + m  ( f m+2 , f m+2 ) = 2  s − s − (m + 2)  1 2 ( fm , fm ) −( f m , πg (L). f m+2 ) = − 2  1 − i(ζ − ζ¯ ) + m  ( f m , f m ). = 2 By invariance of the Hermitian form, these last two quantities are equal. As ( f m , f m ) > 0 for all K ∞ -types, if one of 1 ± i(ζ − ζ¯ ) + m vanishes, then both are equal to zero. One concludes that ζ ∈ R in this case, which is a contradiction. Hence 1 ± i(ζ − ζ¯ ) + m =/ 0 for any K ∞ -type m. Now we find that either both 1 ± i(ζ − ζ¯ ) + m are positive, or both are negative. If both are positive, then adding the two inequalities together shows m > −1. If both 1 ± i(ζ − ζ¯ ) + m are negative, then adding the two shows m < −1. Notice that the case m = −1 is excluded, which proves that it is not a K ∞ -type of V∞ (s, ω), and in particular that this representation must have even parity. Returning to the previous line of argument, if both 1±i(ζ − ζ¯ )+m are positive, then m > −1, and setting m = 0 shows |Im(ζ )| < 1/2. Observe also that Im(ζ ) =/ 0 by our initial hypotheses on it. This completes the proof of necessity. Now we wish to show that V∞ (s, ω) is unitarizable in both of the cases presented in the statement. It is immediate in the first case by s −1/2 Proposition 9.4.3 because the associated characters χ1 = ω1 | · |∞1 s +1/2 and χ2 = ω2 | · |∞2 are both unitary. So let us assume that s = ( 21 + iζ, − 12 + i ζ¯ ) for some complex number ζ such that 0 < |Im(ζ )| < 1/2, and that V∞ (s, ω) has even parity.

518

Solutions to Selected Exercises We will construct an invariant Hermitian form using the explicit basis mentioned above. Define the following quantity: Am =

1 − i(ζ − ζ¯ ) + 2m , 1 + i(ζ − ζ¯ ) + 2m

(m ∈ Z).

Notice that the numerator and denominator of this last expression never vanish because 2 Im(ζ ) − 1 is not an integer. In fact, our hypothesis on Im(ζ ) implies that 2m < 1 ± 2 Im(ζ ) + m < 2(m + 1),

(m ∈ Z).

This shows either both are positive or both are negative, and in particular that Am > 0 for all m ∈ Z. Now define ⎧ if m = n ≥ 0, ⎪ ⎨ A0 · A1 · · · Am−1 , −1 ( f 2m , f 2n ) = (Am · Am+1 · · · A−1 ) , if m = n < 0, ⎪ ⎩ 0, if m =/ n. The above formulas describe the Hermitian form on a basis of V , and we extend it to all of V using linearity in the first variable and conjugate linearity in the second. By construction ( , ) is a positive definite Hermitian form. We must show ( , ) is invariant. We begin by showing invariance under K ∞ . By Proposition 7.4.8, for any m, n ∈ Z, we have (π K ∞ (kθ ). f 2m , π K ∞ (kθ ). f 2n ) = ei(m−n)θ ( f 2m , f 2n ) = ( f 2m , f 2n ), (π K ∞ (δ1 ). f 2m , π K ∞ (δ1 ). f 2n ) = ω1 (−1) ω1 (−1) ( f −2m , f −2n ). The first line shows invariance under S O(2, R). For the second, we observe that Aj =

1 − i(ζ − ζ¯ ) + 2 j 1 + i(ζ − ζ¯ ) − 2( j + 1) = = A−1 −( j+1) . 1 + i(ζ − ζ¯ ) + 2 j 1 − i(ζ + ζ¯ ) − 2( j + 1)

It follows easily that ( f 2m , f 2m ) = ( f −2m , f −2m ) for all m ∈ Z, so that ( , ) is invariant under the action of δ1 by the previous computation. To check that ( , ) is invariant under the action of g, it suffices to check that it is invariant under the actions of D H , D Z , , R, and L (by Proposition 5.2.10). We will use Proposition 7.4.8 again. For D H , we have (πg (D H ). f 2m , f 2n ) = (2im)( f 2m , f 2n ) = −( f 2m , 2im f 2n ) = −( f 2m , πg (D H ). f 2n ).

Solutions to Selected Exercises

519

Since s1 + s2 ∈ iR, we see that (πg (D Z ). f 2m , f 2n ) = (s1 + s2 ) ( f 2m , f 2n ) = −(s1 + s2 ) ( f 2m , f 2n ) = −( f 2m , πg (D Z ). f 2n ). Also, one sees that s12 + s22 − s1 + s2 is real, which implies that (πg (). f 2m , f 2n ) = (s12 + s22 − s1 + s2 ) ( f 2m , f 2n ) = ( f 2m , πg (). f 2n ). Finally, we check that our Hermitian form is invariant under the actions of R and L. By symmetry, it will be enough to check it for R. Observe first that s1 − s2 + 2m = 1 + i(ζ − ζ¯ ) + 2m ∈ R, − (s1 − s2 − 2(m + 1)) = 1 − i(ζ − ζ¯ ) + 2m ∈ R. For n =/ m + 1, it is easy to see that (πg (R). f 2m , f 2n ) = 0. On the other hand,  s − s + 2m  1 2 ( f 2(m+1) , f 2(m+1) ) 2  1 + i(ζ − ζ¯ ) + 2m  ( f 2(m+1) , f 2(m+1) ) = 2  1 − i(ζ − ζ¯ ) + 2m  ( f 2(m+1) , f 2(m+1) ) = A−1 m 2  s − s − 2(m + 1)  1 2 ( f 2m , f 2m ) , πg (L). f 2(m+1) ) = − 2  1 − i(ζ − ζ¯ ) + 2m  ( f 2m , f 2m ). = 2

(πg (R). f 2m , f 2(m+1) ) =

−( f 2m

It follows from the definition of the Hermitian form that ( f 2(m+1) , f 2(m+1) ) = Am ( f 2m , f 2m ),

(m ∈ Z).

This identity allows one to verify that the two quantities in the previous computation agree, which concludes the proof that ( , ) is invariant. 9.11 Suppose that V∞ (s, ω) is unitarizable, and suppose we are in case (1) of Exercise 9.10. We suppose that s = ( 21 + it1 , − 12 + it2 ) for some real numbers t1 , t2 . Using the explicit actions given in Proposition 7.4.8, one sees that 1 λ = − − t12 − t22 ; 2

μ = i(t1 + t2 ),

520

Solutions to Selected Exercises where λ, μ are the invariants attached to an irreducible admissible (g, K ∞ )-module as in Chapter 7. By the classification Theorem 7.5.10, there should be no solutions to the equation m 2 /2 + m = λ − μ2 /2, or equivalently, m 2 + 2m = −1 − t12 − t22 + 2t1 t2

⇔

(m + 1)2 = −(t1 − t2 )2

⇔

t1 = t2 , m = −1.

If t1 =/ t2 , then we have proved the irreducibility of V∞ (s, ω). If t1 = t2 , then m = −1 is a solution to m 2 /2 + m = λ − μ2 /2. This representation falls into case (ii) of the classification Theorem 7.5.10, but it is still irreducible because the set of K ∞ -types is 1 + 2Z. In case (2) of Exercise 9.10, we see that 1 ± i(ζ − ζ¯ ) + m = 0. The explicit actions given in Proposition 7.4.8 show that neither πg (L) nor πg (R) has a kernel in V∞,m (s, ω) for any m. The classification Theorem 7.5.10 implies that V∞ (s, ω) is irreducible. 9.12 (a) By the classification of special representations, we may write χ1 = 1/2 −1/2 for some character χ : Q×p → C× . χ | · | p and χ2 = χ | · | p The central character of π is unitary (Lemma 9.1.5), and an easy −1/2 computation shows it is χ 2 . Hence χ is unitary, and so are χ1 |·| p 1/2 and χ2 | · | p . (b) The contragredient of (π, V ) is a unitarizable special subrepresentation of ( B(χ1 , χ2 ) ∼ = B(χ1−1 , χ2−1 ). Hence by (a), we know −1/2 1/2 −1 −1 χ1 | · | p and χ2 | · | p are unitary. Now take inverses. 10.4 Consider G = Q p with its additive structure. Then every open ball p n Z p is a compact open subgroup, but none of them is maximal. ×

10.5 (b) By smoothness, there exists n such that V 1+ p Z p =/ 0. Let g ∈ 1 + p n Z×p . Let E 1 be the space of eigenvectors of g with eigenn × value 1. Then V 1+ p Z p ⊆ E 1 , so that E 1 =/ 0. Since Q×p is abelian, every element of Q×p preserves the subspace E 1 . By irreducibility, we must have V = E 1 . But g was arbitrary, so we have n × V = V 1+ p Z p , which is finite-dimensional by admissibility. In general, an irreducible finite-dimensional representation of an abelian group must be 1-dimensional (Exercise 2.9). Now let v ∈ V be a non-zero vector. Then we can define a character ψ : Q×p → C× by π (g).v = ψ(g)v. Moreover, since π is smooth, there is an open subgroup of Q×p in its kernel, which shows ψ is continuous. Combining this analysis with part (a) completes the proof of (b). n

Solutions to Selected Exercises

521

(c) Given any pair (ψ , ψ p ), we can define a representation of × Q×  × Q p on C by π (g , g p ).λ = ψ (g )ψ p (g p )λ,

(g ∈ Q , g p ∈ Q p , λ ∈ C).

It is clearly admissible and irreducible. Conversely, an irre× ducible admissible representation (π, V ) of Q×  × Q p must be 1-dimensional (by virtually the same argument as in (b)), and we may define characters ψ , ψ p by π ((g , 1)) .v = ψ (g )v,

  π (1, g p ) .v = ψ p (g p )v, (g ∈ Q , g p ∈ Q p , v ∈ V ).

In this way we obtain the desired one-to-one correspondence. 10.6* Let π : Q×p → G L(V ) be a smooth irreducible representation. Let S(Q×p ) be the space of Schwartz functions on Q×p . Observe that S(Q×p ) acts on V via  π ( f ).v = f (g)π (g).v d × g, ( f ∈ S(Q×p ), v ∈ V ). Q×p

One checks easily that each π ( f ) is an intertwining operator on V . The proof of Schur’s lemma (Lemma 6.1.8) carries over essentially verbatim to show than any intertwining operator of a smooth irreducible representation of Q×p must be given by multiplication by a scalar. Hence each π ( f ) acts by a scalar. On the other hand, the proof of Claim 1 in Theorem 6.1.11 carries over as well to show that the space of Schwartz functions S(Q×p ) acts transitively on V . But since each Schwartz function acts by a scalar, we must have dim(V ) = 1. 10.7 Since left cosets are open and K g K = ∪k∈K kg K , we see K g K is open. Also, the map K × K → G L(2, Q p ) given by (k, k ) → kgk is continuous. Thus K g K is the continuous image of the compact set K × K , and so is itself compact. 10.8 Let f be an element of the spherical Hecke algebra with support F. If f ≡ 0, the result is trivial, so we suppose f ≡ 0. By definition, the support F is compact, and since f is locally constant, we also see that F is open. Moreover, if g ∈ F, then K g K ⊂ F since f is bi K invariant. Sets of the form K g K for g ∈ G L(2, Q p ) cover G L(2, Q p ). n K gi K , and By compactness, there exist g1 , . . . , gn such that F = ∪i=1 we may take n minimal with this property. As f is constant on K gi K , n f (gi )1 K gi K . we see that f = i=1

522

Solutions to Selected Exercises

10.9 Let Fi be the compact open subset on which f i is supported. Then F1 · F2 is compact (being the continuous image of the compact set F1 × F2 under the multiplication map), and we claim f 1 ∗ f 2 has support inside F1 · F2 . Indeed, by definition we have  f 1 ∗ f 2 (g) = f 1 (gh −1 ) f 2 (h)d × h. G L(2,Q p )

The integrand vanishes unless h ∈ F2 and g ∈ F1 h ⊂ F1 · F2 . 10.10 Immediate consequence of the Cartan decomposition from Section 4.2. 10.11 (d) By Theorem 10.6.12, dim(V K ) ≤ 1. But K H K is a C-algebra of dimension greater than 1, so π : K H K → End(V K ) cannot be injective. If π is ramified, then V K = 0 and π : K H K → End(V K ) = 0 is trivially surjective. Suppose π is unramified. Then V K has dimension 1 by Theorem 10.6.12 and End(V K ) ∼ = C·idV K . Since π (c1 K ) = c π (1 K ) = c · idV K , we see that π is surjective. 10.12 The function β is clearly constant on left (or right) cosets of K , and hence locally constant. It is also evidently bi K -invariant. For ci 1 K gi K for some complex conf ∈ K H K , we may write f = stants ci and some elements gi ∈ G L(2, Q p ) by Exercise 10.8. Now we compute:   f (g)β(g)d × g = ci β(g)d × g G L(2,Q p )

= =



K gi K

ci β(gi )Vol(K gi K ) ci ξ (1 K gi K ) = ξ ( f ).

 n  p 0 10.13 Let n be an integer and write gn = 0 pn . Let f = f s◦1 ,s2 be the standard K -invariant element of V p (s1 , s2 ). Using the fact that gn lies in the center of G L(2, Q p ) and that Vol(K gn K ) = Vol(gn K ) = 1, we find that    π 1 K gn K . f = π (g). f d × g K gn K  n s1 n s2 d × g = p −n(s1 +s2 ) f. = | p |p | p |p f K gn K

Thus ξ (1 K gn K ) = p −n(s1 +s2 ) by definition, and then by Exercise 10.12 we see β(gn ) = p −n(s1 +s2 ) =/ 0. The set of matrices {gn }n∈Z does

Solutions to Selected Exercises

523

not have compact closure in G L(2, Q p ), and hence β does not have compact support. 10.14 It follows immediately from the definitions that this representation is unramified at all primes p such that p  N . If it is also unramified at a prime p that divides N , deduce that f must be an oldform. 10.15 Let v be any non-zero vector in V . It follows from the definition of smooth (g, K ∞ ) × G L(2, Afinite )-module that v is fixed by some compact open subgroup K of G L(2, Afinite ). By passing to a  smaller subgroup if necessary, we may assume that K = p∈S U p ·  p∈ S G L(2, Z p ) for some finite set of primes S. So v is fixed by K p for all p ∈ S, and hence (π, V ) is ramified only at primes in S (at most). 10.16 If v j ∈ V j is a non-zero K j -fixed vector for j = 1, 2, then v1 ⊗ v2 is a non-zero K 1 × K 2 -fixed vector in V1 ⊗ V2 . For the converse, write K = K 1 × K 2 . Suppose v ∈ V1 ⊗ V2 is a non-zero K -fixed vector, and write v = i vi ⊗ vi with vi ∈ V1 and vi ∈ V2 for each i. Let {e1 , . . . , em } be a basis for the span of the space of K 1 -translates of the vi , and let {e1 , . . . , en } be a basis for the span of the space of K 2 translates of the vi . We may choose these to be finite bases because of the smoothness of our representations. In this way, we may write v uniquely in the form

v =

λi, j ei ⊗ e j ,

i, j

for some complex scalars λi, j . Write π = π1 ⊗ π2 . Then for k ∈ K 1 , we have   π (k, 1).v = λi, j π1 (k).ei ⊗ e j . j

i

As v is K -invariant, we have 0 = π (k, 1).v − v =

 j

 λi, j (π1 (k).ei − ei ) ⊗ e j .

i

By linear independence of the set {e1 , . . . , en }, we find that the inner sum vanishes for each j = 1, . . . , n, or equivalently that π1 (k).

 i

 λi, j ei

=

i

λi, j ei

(k ∈ K 1 , j = 1, . . . , n).

524

Solutions to Selected Exercises Since λi, j =/ 0 for some pair of indices i, j, we have located a non-zero K 1 -invariant vector in V1 . By symmetry, one can produce a non-zero K 2 -invariant vector in V2 .

v ∈ V K finite will be a spherical vector 10.17 If V K finite =/ 0, then any non-zero  for the action of K p = i p G L(2, Z p ) . For the converse, one can use the full power of the tensor product Theorem 10.8.5. Alternatively, here is a self-contained solution. We use smoothness to find a finite set of primes S and a compact open 

subgroup K S ⊂ p∈S G L(2, Q p ) such that V K =/ 0, where K = KS ×



G L(2, Z p ) ⊂ K finite .

p∈ S

 Write K S = p∈ S G L(2, Z p ). By Theorem 10.5.1, we may find rep resentations (π1 , V1 ) of p∈S G L(2, Q p ) and (π2 , V2 ) of the restricted  product p∈ S G L(2, Q p ) such that (π, V ) is isomorphic to the tensor product (π1 ⊗ π2 , V1 ⊗ V2 ). The tensor product admits a non-zero K S × K S -invariant vector by construction, so the previous exercise implies that (π2 , V2 ) has a non-zero K S -invariant vector. We may further decompose (π1 , V1 ) into a product of local representations over all p ∈ S by repeated use of Theorem 10.5.1, each of which has a nonzero spherical vector by hypothesis. Now apply the previous exercise inductively to construct a non-zero vector that is invariant under the action of K finite . 11.1 Applying  1 0 row and column operations to ξ allows one to reduce it to the form 0 0 . But applying row (resp. column) operations is equivalent to multiplying on the left (resp. right) by an invertible matrix. 11.2 (a) Apply the product formula (Theorem 1.1.8) to the determinant of an element of G L(2, Q) to see that G L(2, Q) is a subset of G 1 . It is × with γ =/ δ and let t ∈R× , and clearly a subgroup. Let γ , δ ∈ Q γ 0 consider the elements g = 0 δ ∈ G L(2, Q) and h = i ∞ 10 1t ∈  ) . In G 1 . Then the infinite place of hgh −1 is given by γ0 t(δ−γ δ order to have hgh −1 ∈ G L(2, Q), we would need t(δ − γ ) = 0, which is false, so that G L(2, Q) is not normal. (b) Since Z (A) commutes with G L(2, Q), one checks easily that Z (A) · G L(2, Q) is a subgroup of G L(2, A). The example in the previous paragraph shows it is not normal. (c) Let g ∈ G L(2, A). By strong approximation (4.4.2), we find g = γ dh, where γ ∈ G L(2, Q), d ∈  G L(2, R), and h ∈  −1 0 −1 0 d if p G L(2, Z p ). By replacing γ by γ 0 1 and d by 0 1

Solutions to Selected Exercises

525

necessary, we may assume d ∈ G L(2, R)+ . Let t ∈ R be such  −1 0  that t 2 = det(d). Then t 0 t −1 dh ∈ G 1 , and it is equivalent to g modulo Z (A) · G(2, Q). Hence we have shown that the map G 1 → Z (A) · G L(2, Q)\G L(2, A) is surjective, which implies that β is surjective. (e) Set h = g −1 g . If β(g) = β(g ), then h ∈ Z (A) ∩ G 1 . Write   

  t∞ 0 t 0 h= ,... , p ,... , 0 t∞ 0 tp  where tv ∈ G L(2, Qv ) and v |tv |2v = 1. For each finite prime p, choose an integer r ( p) such that pr ( p) t p ∈ Z×p , and let q = q 0  r ( p) . Set γ = 0 q ∈ G L(2, Q). Then p p γh =



s∞ 0

0 s∞



 ,... ,

sp 0

0 sp



 ,... ,

where |s p | p = 1 for all finite primes p, and | det(γ h)|A = |s∞ |2∞ =  0  γ if necessary so that 1, so that s∞ = ±1. Replace γ by −1 0 −1  s∞ = 1. Then γ h ∈ p Z×p , diagonally embedded; i.e., γ h lies in the image of the map α. Since γ h is congruent to h modulo G L(2, Q), we are finished. 11.3 The proof of Exercise 1.6 carries over to this (more general) setting. 11.5 It suffices to verify this for elementary matrices. 11.7 First consider the function φ (g) =



φ(gk) d × k,

K



where K = O(2, R) · p G L(2, Z p ). Evidently φ is also a cuspidal automorphic form, and it is right K -invariant. It follows from

is a Maass form of weight 0 and level 1. Lemma 5.5.10 that φclassical By strong approximation (or Exercise 11.2(f)), we see that   φ(g) d × g = φ (g∞ , · · · , I2 , · · · ) d × g∞ G L(2,Q)\G 1

SL(2,Z)\SL(2,R)/S O(2,R)



=

φclassical (x + i y)

d xd y . y2

SL(2,Z)\h

with the This last expression is the Petersson inner product of φclassical

constant function 1. Write f = φclassical for simplicity now.

526

Solutions to Selected Exercises By properties of the Laplacian, we know that  f = 0 implies f is constant, which further shows it cannot be cuspidal. Therefore  f =/ 0. Since f can be written as a linear combination of eigenfunctions of the Laplacian, it suffices to assume  f = λ f for some non-zero scalar λ. Since  is self-adjoint for the Petersson inner product (, )h , we see that ( f, 1)h =

1 1 ( f, 1)h = ( f, (1))h = 0, λ λ

and hence the above integral vanishes as desired. 11.8 We make the change of variables h → k2−1 hk1 in the integral defining k2 k1 φ . By Exercise 11.5 (Jacobian of the change of variables), we see that     k2 k1 φ = φ(h) e(−Tr(hk1−1 gk2 )) dh AQ AQ AQ AQ

   

=

1 φ k2 . φ(k2−1 hk1 )e(−Tr(k2−1 hgk2 )) dh = kR

AQ AQ AQ AQ

The final equality follows from the definition of the Fourier transform and the fact that the trace is conjugation invariant. 11.9 Write LHS and RHS for the left and right hand sides of the identity in question. First note that the cosets in the RHS are disjoint by the p-adic Iwasawa decomposition. The identities       1 0 0 1 p 0 0 1 = , 0 p 1 0 0 1 1 0      p b 1 b p 0 = 0 1 0 1 0 1 show that RHS ⊂ LHS. For the other inclusion, take g ∈ LHS, and use the Iwasawa decomposition to write     r p 0 p up s k = k3 g = k1 0 ps 0 1 2 for some k1 , k2 , k3 ∈ G L(2, Z p ), r, s ∈ Z, and u ∈ Q p . Thus we see gk3−1 ∈ Mat(2, Z p ), and hence r, s ≥ 0 and r + s = 1 (look at the determinant). If s = 0, then r = 1, and we also see u ∈ Z p . Write u = b + pu , with u ∈ Z p and 0 ≤ b ≤ p − 1. Then

Solutions to Selected Exercises 527    p b 1 u gk3−1 = 0 1 0 1     p b =⇒ g = k3 , some k3 ∈ G L(2, Z p ) 0 1   p b =⇒ g ∈ K. 0 1  1 up  If instead we have s = 1, then r = 0, and we find gk3−1 = 0 p with up ∈ Z p . Hence      1 0 1 up 1 0 −1 =⇒ g ∈ K. gk3 = 0 p 0 1 0 p Since g was arbitrary, we find LHS ⊂ RHS.  pa 0  11.10 (a) Note K (a, a) = 0 pa K = pa K for any a ∈ Z. Then     p −b 0 a b −1 × g R p ∗ R p (g) = 1 pa K (h g) d h = 1 pa K 0 p −b pb K

= 1 pa+b K (g) = R a+b p (g). (c) Using the definition of T pr and part (b), we see that 1 K (a,b) = 1 K (r,0) + 1 K (a,b) T pr = a≥b≥0 a+b=r

a≥b≥1 a+b=r



=1 K (r,0) +

1 K (a+1,b+1)

a≥b≥0 a+b=r −2

⎞

⎛

⎟ ⎜ = 1 K (r,0) + R p ∗ ⎜ 1 K (a,b) ⎟ ⎠ ⎝ a≥b≥0 a+b=r −2

= 1 K (r,0) + R p ∗ T pr −2 . 11.11 We have that   



 f

Q×p Q×p Q p G L(2,Z p )

  





=

f Q×p Q×p Q p G L(2,Z p )

a1 0

x a2 a1 0



 ·k

0 a2



d × a1 × d a2 |a1 | p

d ×k d x 1 0

xa1−1 1

· d ×k d x



 ·k

d × a1 × d a2 . |a1 | p

528

Solutions to Selected Exercises Now make the change of variables x → a1 x, which gives d x → |a1 | p d x. The result then follows from Proposition 6.9.6.

11.12 Since  is locally constant and compactly supported, say on a compact f set F ⊂ Mat(2, Q p ), it takes only finitely many values. Since f and ( are continuous, they are also bounded on F. Hence |β| is a bounded function with support in F. So the absolute value of the integrand of s+1/2 Z p (s, , β) is bounded by a constant times | det(g)| p , and we need only integrate over the set F. The set F ∩ {| det(g)| p = p −n } is empty for n sufficiently negative. Now we see that   s+1/2 × −n(s+1/2) | deg(g)| p d g= p d×g F

n≥N

≤C



F∩{| det(g)| p = p−n }

p −n(s+1/2) ,

n≥N

 where the constant C = F d × g depends only on F. Evidently the sum converges absolutely if and only if (s) > −1/2. 11.13 (b) For k, k ∈ G L(2, Z p ), x ∈ Q p , and a1 , a2 ∈ Q×p , write a x    k −1 01 a2 k = CA DB . If |a1 a2 | p > M 2 , then taking determinants shows |AD − BC| p = |a1 a2 | p > M 2 . By the ultrametric inequality, one of A, B, C, D must have absolute value strictly larger than M, and hence f φ (a1 , a2 ) = 0 in this case.   Now we suppose that |a1 a2 | p ≤ M 2 . Write k −1 = ac db and   α β k = γ δ . Then 

A C

B D



  a1 x = k −1 k 0 a2  aa1 α + axγ + ba2 γ = ca1 α + cxγ + da2 γ

aa1 β + axδ + ba2 δ ca1 β + cxδ + da2 δ

 .

  Suppose |a1 | p > M and |a2 | p < M. We’ll show φ CA DB = 0. If   |x| p > |a1 | p , then there exists a coefficient of CA DB with absolute  A B  value equal to |x| p > M. So φ C D = 0. If |x| p ≤ |a1 | p , then there exists a term with absolute value |a1 | p . For if not, then |αa1 + γ x| p ≤ |aa1 α + axγ | p < |a1 | p and |βa1 + δx| p ≤ |aa1 β + axδ| p < |a1 | p , so that |a1 | p = |(αδ − βγ )a1 | p = |αδa1 + γ δx − βγ a1 − γ δx| p 

≤ max |δ| p · |αa1 + γ x| p , |γ | p · |βa1 + δx| p < |a1 | p .

Solutions to Selected Exercises 529   But |a1 | p > M, so it follows that φ CA DB = 0. In either case, we conclude that f φ (a1 , a2 ) = 0. If we suppose that |a1 | p < M and |a2 | p > M, then an argument similar to the one in the last paragraph gives the conclusion. (c) Let ! be a Bruhat-Schwartz function on G L(2, Q p ). First check that there exists a compact open subgroup K n (as in Section 6.1) such that !(k −1 gk) = !(g) for all k, k ∈ K n and g ∈ G. Conclude that the new  function  !(k −1 gk ) d × k d × k

(g) = K

K

is a linear combination of characteristic functions of compact open subsets, and hence locally constant and compactly supported. (By using Exercise 11.3, one may assume that ! is a linear combination of characteristic functions of compact open subsets of Mat(2, Q p ).) Finally, show that the function    a1 x dx  f (a1 , a2 ) = 0 a2 Qp is locally constant. (In fact, this holds if  is replaced by any compactly supported and locally constant function.) 11.14 We may assume that any integral over Mat(2, Q p ) is in fact an integral over G L(2, Q p ). Write G = G L(2, Q p ). We have    f 1 (h)  f 2 (h) d × h = f 1 (h) f 2 (x) e(−Tr (xh)) G

G G

  =

· d x d ×h f 1 (h) f 2 (h −1 x) e(−Tr (x)) | det(h −1 )|2

G G

  =

· d x d ×h f 1 (xh) f 2 (h −1 ) e(−Tr (x)) | det(h −1 x −1 )|2

G G

  =

· d x d ×h f 1 (x) f 2 (h −1 ) e(−Tr (xh −1 ))

G G

· | det(x −1 h −1 )|2 d x d × h      det(h) 2  f 1 (x) f 2 (h) e(−Tr (xh))  = det(x)  G G

· d x d × h.

530

Solutions to Selected Exercises In the second line, we made the change of variables x → h −1 x and used the fact that the trace is conjugation invariant. We also made use of Exercise 11.5. In the third line we made the change of variables h → xh; note that the Haar measure d × h is invariant under this operation. In the fourth line we use the change of variables x → xh −1 , and finally in the fifth line, we use the change h → h −1 . To complete the exercise, observe that f 1 (x) = 0 if | det(x)| =/ A1 and f 2 (h) = 0 if | det(h)| =/ A2 .

11.15 The first integral evaluates to 

f 1 (h)  f 2 (h) d × h = Vol(K 1 ) · Vol× (K 1 ),

G L(2,Q p )

where Vol and Vol× denote the volumes for the additive and multiplicative Haar measures, respectively. The second integral can be evaluated by Exercise 11.14. 11.16 The exercise follows immediately from the definitions and the fact that for all matrices A, B we have Tr(AB) = Tr(B A).

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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

537

538 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 ofmatrices 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

539

μ, measure 12 μHaar , dμHaar , Haar measure 13 d x, d × x14 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

540

Symbols Index

G G (π  , V ), induced 239 ( π,( V , contragredient 277, 290

 TE  π , V (2) 62 Lr

(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 L product L, 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

541

542 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

543 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

544 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

545 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

546

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

547 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

548 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

549 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

550

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