Elements of the Representation Theory of the Jacobi Group

Modern Birkh¨auser Classics Many of the original research and survey monographs in pure and applied mathematics published by Birkh¨auser in recent decades have been groundbreaking and have come to be regarded as foundational to the subject. Through the MBC Series, a select number of these modern classics, entirely uncorrected, are being re-released in paperback (and as eBooks) to ensure that these treasures remain accessible to new generations of students, scholars, and researchers.

Rolf Berndt Ralf Schmidt

Elements of the Representation Theory of the Jacobi Group

Reprint of the 1998 Edition

Rolf Berndt Department Mathematik Universität Hamburg Bundesstr. 55 20146 Hamburg Germany

Ralf Schmidt Department of Mathematics University of Oklahoma Norman, OK 73019-3103 USA

ISBN 978-3-0348-0282-6 e-ISBN 978-3-0348-0283-3 DOI 10.1007/978-3-0348-0283-3 Springer Basel Dordrecht Heidelberg London New York Library of Congress Control Number: 2011941499 Mathematics Subject Classification (2010): 11F55, 11F50, 11F70, 14K25, 22E50, 22E55

© Springer Basel AG 1998 Reprint of the 1st edition 1998 by Birkhäuser Verlag, Switzerland Originally published as volume 163 in the Progress in Mathematics series This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use, permission of the copyright owner must be obtained. Printed on acid-free paper

Springer Basel AG is part of Springer Science+Business Media (www.birkhauser-science.com)

Auffallender Weise hat eine so wichtige Function noch keinen andern Namen, als den der Transcendente Θ, nach der zuf¨ alligen Bezeichnung, mit der sie zuerst bei J a c o b i erscheint, und die Mathematiker w¨ urden nur eine Pflicht der Dankbarkeit erf¨ ullen, wenn sie sich vereinigten ihr J a c o b i s Namen beizulegen, um das Andenken des Mannes zu ehren, zu dessen sch¨ onsten Entdeckungen es geh¨ ort, die innere Natur und hohe Bedeutung dieser Transcendente zuerst erkannt zu haben. from: L. Dirichlet: Ged¨ achtnisrede auf C.G.J. Jacobi

Preface

The Jacobi group is a semidirect product of a symplectic group with a Heisenberg group. Its importance prima facie stems from the fact that it sets the frame to treat theta functions and elliptic and abelian functions. Up to now, most work concerning this group has been done for the simplest case “of degree one”, where the symplectic group is simply SL(2) and the Heisenberg group is a three parameter nilpotent group. The Jacobi group, whose theory is intensively interwoven with that of the metaplectic group, is, together with the Heisenberg group, the most evident example for a non-reductive group. This treatise is meant to show how the general theory of automorphic forms for reductive groups extends by some slight alterations to this first more general example. The reader will see that a lot of the following may easily be extended to the higher degree case of a semidirect product of a symplectic group Sp(n) with a corresponding Heisenberg group. We were tempted to do this, but as the generalizations are sometimes fairly easy on the one hand, and as the degreeone case has special features, e.g. concerning the cusp conditions, on the other hand, we restrict ourselves to this case, denoted GJ , here.

v

Contents

1

2

3

4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

The 1.1 1.2 1.3 1.4

Jacobi Group Definition of GJ . . . . . GJ as an algebraic group The Lie algebra of GJ . . GJ over the reals . . . . .

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1 3 7 9

Basic Representation Theory of the Jacobi Group 2.1 Induced representations . . . . . . . . . . . . . . . . . . 2.2 The Schr¨ odinger representation . . . . . . . . . . . . . . 2.3 Mackey’s method for semidirect products . . . . . . . . 2.4 Representations of GJ with trivial central character . . 2.5 The Schr¨ odinger-Weil representation . . . . . . . . . . . 2.6 Representations of GJ with non-trivial central character

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15 18 21 22 24 28

Local Representations: The Real Case 3.1 Representations of gJC . . . . . . . . . . . . . . . . . . . . . . 3.2 Models for infinitesimal representations and unitarizability . . 3.3 Representations induced from B J . . . . . . . . . . . . . . . . 3.4 Representations induced from K J and the automorphic factor 3.5 Differential operators on X = H × C . . . . . . . . . . . . . . ˆ J and Whittaker models . . . 3.6 Representations induced from N

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32 39 48 51 59 63

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76 83 88 94

The 4.1 4.2 4.3 4.4

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Space L2 (ΓJ \GJ (R)) and its Decomposition Jacobi forms and more general automorphic The cusp condition for GJ (R) . . . . . . . . The discrete part and the duality theorem . The continuous part . . . . . . . . . . . . . vii

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forms . . . . . . . . . . . .

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viii 5

6

7

Contents Local Representations: The p-adic Case 5.1 Smooth and admissible representations . . . . . . . . . . . 5.2 Whittaker models for the Schr¨ odinger-Weil representation 5.3 Representations of the metaplectic group . . . . . . . . . . 5.4 Induced representations . . . . . . . . . . . . . . . . . . . 5.5 Supercuspidal representations . . . . . . . . . . . . . . . . 5.6 Intertwining operators . . . . . . . . . . . . . . . . . . . . 5.7 Whittaker models . . . . . . . . . . . . . . . . . . . . . . . 5.8 Summary and Classification . . . . . . . . . . . . . . . . . 5.9 Unitary representations . . . . . . . . . . . . . . . . . . .

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105 107 112 115 119 121 127 132 135

Spherical Representations 6.1 The Hecke algebra of the Jacobi group . . . . . 6.2 Structure of the Hecke algebra in the good case 6.3 Spherical representations in the good case . . . 6.4 Spherical Whittaker functions . . . . . . . . . . 6.5 Local factors and the spherical dual . . . . . . 6.6 The Eichler-Zagier operators . . . . . . . . . .

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138 140 148 153 163 167

Global Considerations 7.1 Adelization of GJ . . . . . . . . . . . . . . . . . . . 7.2 The global Schr¨ odinger-Weil representation . . . . 7.3 Automorphic representations . . . . . . . . . . . . 7.4 Lifting of Jacobi forms . . . . . . . . . . . . . . . . 7.5 The representation corresponding to a Jacobi form

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174 176 179 183 193

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Index of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Introduction

After Pyatetski-Shapiro [PS1] and Satake [Sa1] introduced, independent of one another, an early form of the Jacobi Theory in 1969 (while not naming it as such), this theory was given a definite push by the book The Theory of Jacobi Forms by Eichler and Zagier in 1985. Now, there are some overview articles describing the developments in the theory of the Jacobi group and its automorphic forms, for instance by Skoruppa [Sk2], Berndt [Be5] and Kohnen [Ko]. We refer to these for more historical details and many more names of authors active in this theory, which stretches now from number theory and algebraic geometry to theoretical physics. But let us only briefly indicate several – sometimes very closely related – topics touched by Jacobi theory as we see it: • fields of meromorphic and rational functions on the universal elliptic curve resp. universal abelian variety • structure and projective embeddings of certain algebraic varieties and homogeneous spaces • correspondences between different kinds of modular forms • L-functions associated to different kinds of modular forms and automorphic representations • induced representations • invariant differential operators • structure of Hecke algebras • determination of generalized Kac-Moody algebras and as a final goal related to the here first mentioned • mixed Shimura varieties and mixed motives. Now, letting completely aside the arithmetical and algebraic geometrical approach to Jacobi forms developed and instrumentalized by Kramer [Kr], we ix

x

Introduction

will treat here a certain representation theoretic point of view for the Jacobi theory parallel to the theory of Jacquet-Langlands [JL] for GL(2) as reported by Godement [Go2], Gelbart [Ge1] and, recently, Bump [Bu]. Our text assembles and regroups material from several papers mainly on the real theory by the first-named author, and by the second-named author some definite ameliorations and additions to the non-archimedean and adelic theory contained in the thesis of Homrighausen [Ho]. More precisely, our aim is • to give a classification of the irreducible unitary representations π of the Jacobi group GJ over local fields, • to construct explicit models for these representations, in particular the Whittaker models, • to discuss the relation between automorphic forms for GJ , i.e. the holomorphic Jacobi forms, their skew holomorphic counter parts (first studied by Skoruppa) and possible generalizations, and the automorphic representations of GJ , and • to prepare the ground for a further discussion of automorphic L-functions. To reach these aims, we pursue the following plan. In the first chapter we present the Jacobi group GJ in some different realizations and determine its Lie algebra gJ . This gives some ideas about the structure of our non-reductive GJ , and indicates in particular the important subgroups of GJ one should look at. We take a closer look at the real points GJ (R) of GJ , where we find a sort of generalized Iwasawa decomposition GJ = N J AJ K J . Here N J is a substitute for the unipotent radical of a maximal parabolic subgroup in the reductive theory. It is characterized as the closed connected subgroup of GJ whose Lie algebra is the sum of positive root spaces. In chapter two a method of Mackey will lead us to the fundamental principle in the representation theory of GJ , which reads in our language: m ππ ˜ ⊗ πSW

Here π is a representation of GJ , π ˜ is a genuine representation of the metaplectic m group Mp, and πSW is a certain projective standard representation of GJ , called the Schr¨ odinger-Weil representation. The meaning of the above equation is that there is a 1-1 correspondence     Irreducible representations Irreducible, genuine reJ . of G with fixed non-trivial ←→ presentations of Mp central character One of our objectives in the later chapters is to make the above isomorphism explicit, thereby showing that this bijection is canonical.

xi

Introduction

As the methods to be used in the archimedean, the non-archimedean and the adelic theory are sometimes quite different, we treat these cases separately starting with the real case in the chapters 3 and 4. Here the key for obtaining explicit information is a mixture of • the induction procedure and • the infinitesimal method realizing gJ ⊗ C by differential operators. On the way we discuss covariant differential operators on the homogeneous space X = H × C = GJ (R)/SO(2) × R. and determine the (noncommutative) ring of invariant differential operators. Introducing Satake’s determination of an automorphic factor for GJ , we discuss the definition of holomorphic and more general Jacobi forms in a first step as functions on X and in a second step as functions on GJ (R) (and later on in a third and final step on the adelized group GJ (A)). Aiming at the proof of a duality theorem, tying automorphic forms of a certain type with corresponding cuspidal representations of GJ (R), we give the definition of a cusp condition using the (conjugacy classes of) the important standard unipotent group N J , and introduce a cuspidal subspace H0 = L20 (ΓJ \GJ (R)),

ΓJ = SL(2, Z)  Z2

in the standard L2 -space H, which as in the classical theory may be decomposed discretely. We observe the interesting phenomenon that in the real theory for functions φ with fixed transformation property with respect to the center Z(GJ ), i.e. with φ(κg) = e2πimκ φ(g)

for κ ∈ Z(GJ ),

the single cusp i∞ of SL(2, Z) degenerates into (up to maximally) 2m cusps. This chapter is concluded by a sketchy discussion of the continuous part, i. e. the orthocomplement of H0 in H, where the notion of a general Jacobi Eisenstein series appears. This section in particular is open to further research looking for a better way to get at the functional equation and analytic continuation of these Eisenstein series. The Chapters 5 and 6 treat the p-adic case. Here the basic ingredients are • the induction procedure, • the recourse to Waldspurger’s results on the metaplectic group Mp, and • information about the (local) Jacobi Hecke algebra.

xii

Introduction

In the non-archimedean case the natural objects to study are the admissible representations. The determination of all of these and the unitary representations is easy by some general results and Waldspurger’s results in [Wa1]. It is more difficult to analyse which of the classes obtained are equivalent. Here we derive some results on intertwining operators, which together with an analysis of the Whittaker and Kirillov models do the task. In particular, having the adelization and application to L-functions in mind, it is of importance to discuss which representations π contain a spherical vector, meaning here an element invariant under GJ (O), O the maximal order of the local field F . The determination of the spherical representations and the Hecke algebra for certain “good” cases (for the prime p in relation to the number m ruling the central character) contains the main part of Chapter 6. For an extension of these results to more general “worse” cases, we refer to the forthcoming thesis [Sch2] of R. Schmidt, University of Hamburg. Moreover, we remark that the existence of spherical vectors and of vectors of dominant weight (in the real case) allows for the definition and computation of local factors via the computation of a certain zeta integral, which is specific for our theory. The final chapter 7 shows how the local considerations can be put together to give a global theory for the adelized group GJ (A). Here again the Schr¨ odingerWeil representation πSW plays a central role. In the global context it is best to realize it as a space of theta functions ϑf corresponding to Schwartz functions f ∈ S(A). The notion of automorphic representation is introduced as in the general theory, and we establish a representation theoretic analogue of a sort m of “Shimura isomorphism”, leading via πSW to a one-to-one correspondence between genuine automorphic representations of the metaplectic Mp(A) and automorphic representations of GJ (A) with fixed non-trivial central character. Classical holomorphic Jacobi forms f ∈ Jk,m , already characterized as functions on GJ (R) in chapter 4, are now moreover lifted to functions on GJ (A), and conditions characterizing these are given. The discussion of Hecke operators from chapter 6 is now enriched by a representation theoretic version of certain involutions Wp at the bad places p|m, acting on Jk,m from the theory of Eichler and Zagier. The chapter culminates in a theorem stating that a Jacobi form cusp Jk,m , eigenform for all Hecke operators and these involutions, generates an irreducible automorphic representation πf of GJ (A), whose infinite component + is a discrete series representation πm,k , while for p  2m∞, its p-component is a spherical principal series representation πχ,m (characterized by the Hecke eigenvalue c(p) = pk−3/2 (χ(p) + χ(p)−1 ) of f ). We have tried to write this report so it may be read independently of other texts, but we understand that some knowledge of the above mentioned sources for the GL(2)-theory will be helpful. We close this introduction by thanking several people who participated sometimes even without their knowledge. The second named author learnt a lot in courses given by S. Kudla, the first one still draws on conversations with F. Shahidi a long time ago, and with M. Eichler

xiii

Introduction

and D. Zagier, still a longer time ago. We both used hints and comments given by J. Michaliˇcek, P. Slodowy and most of all by J. Dulinski. About half of the text was typeset by Mrs. D. Glasenapp, and it is our pleasure to thank her too, as well as our local TEX-adviser E. Begemann. Last not least, we appreciate the help of Mrs. C. Baer and Th. Hintermann from the Birkh¨ auser Verlag. Hamburg, November 6, 1997

R. Berndt, R. Schmidt

Hanc nostram de transformatione theoriam et, quae alia inde in analysin functionum ellipticarum redundant, iam fusius exponemus. from: C.G.J. Jacobi: Fundamenta Nova Theoriae Functionum Ellipticarum

1 The Jacobi Group

The Jacobi group is a semidirect product of a (semisimple) symplectic group with a (nilpotent) Heisenberg group. It comes along in several presentations which may be more or less appropriate for the different parts of the theory. So we will discuss here several realizations and change from one to the other from time to time. To keep track it is helpful to think of the Jacobi group as a certain subgroup of a bigger symplectic group.

1.1 Definition of GJ Let R be a commutative ring with 1. Consider the symplectic group Sp(2, R), which is by definition the group of matrices   A B ∈ GL(4, R), C D where A, B, C, D are (2 × 2)-matrices which fulfill At D − C t B = E,

At C = C t A,

B t D = Dt B.

(1.1)

We define the Jacobi group GJ (R) over R, our object of study, as the subgroup of Sp(2, R) consisting of matrices of the form ⎛ ⎞ ⎜ ⎜ ⎝

⎟ ⎟. ⎠

∗ 0 0

0 1

R. Berndt and R. Schmidt, Elements of the Representation Theory of the Jacobi Group, Modern Birkhäuser Classics, DOI 10.1007/978-3-0348-0283-3_1, © Springer Basel AG 1998

1

2

1. The Jacobi Group

An easy calculation using (1.1) yields ⎛ ⎞ a 0 b μ ⎜ λ 1 μ κ ⎟ ⎜ ⎟ ⎝ c 0 d −λ ⎠ 0 0 0 1 with ad − bc = 1

and

  ab (λ, μ) = (λ , μ ) cd 



as the most general element of GJ (R). We ⎛ a   ⎜ 0 ab ⎜ ∈ SL(2, R) with ⎝ c cd 0 and

⎛ (λ, μ, κ) ∈ H(R)

with

1 ⎜ λ ⎜ ⎝ 0 0

now identify ⎞ 0 b 0 1 0 0 ⎟ ⎟ ∈ GJ (R) 0 d 0 ⎠ 0 0 1

⎞ 0 0 μ 1 μ κ ⎟ ⎟ ∈ GJ (R). 0 1 −λ ⎠ 0 0 1

Here H(R) denotes the Heisenberg group, which is R3 as a set, and with multiplication (λ, μ, κ)(λ , μ , κ ) = (λ + λ , μ + μ , κ + κ + λμ − μλ ). If we put X = (λ, μ), X  = (λ , μ ), this can also be written as    X      (X, κ)(X , κ ) = X + X , κ + κ +    , X where | | denotes the determinant. The above identifications obviously yield injections SL(2, R) → GJ (R)

and

H(R) → GJ (R),

and it is also obvious that every element g ∈ GJ (R) can uniquely be written as g = Mh

or as

g = h M 

with M, M  ∈ SL(2, R) and h, h ∈ H(R). Projection onto the SL(2)-part is immediately recognized as a group homomorphism, yielding an exact sequence 1 −→ H(R) −→ GJ (R) −→ SL(2, R) −→ 1.

(1.2)

1.2. GJ as an algebraic group

3

This sequence splits by means of the above injection SL(2, R) → GJ (R), so that the Jacobi group becomes the semidirect product of SL(2, R) and the Heisenberg group: GJ (R) = SL(2, R)  H(R).

(1.3)

For reasons of brevity we define G := SL(2). A small calculation makes the action of G(R) on H(R) explicit; it is given by M (X, κ)M −1 = (XM −1 , κ)

for M ∈ SL(2, R), (X, κ) ∈ H(R),

where XM −1 means matrix multiplication row times matrix. So, for example, the product of g = M (X, κ) and g  = M  (X  , κ ) is given by     XM          , gg = M M XM + X , κ + κ +  (1.4) X  and the product of g = (X, κ)M and g  = (X  , κ )M  is given by     X    −1   gg = X + X M , κ + κ +   −1  M M  . XM

(1.5)

In the classic book [EZ] of Eichler and Zagier on Jacobi forms the element M h of the (real) Jacobi group is written as a pair M h = [M, h]

(M ∈ SL(2), h ∈ H)

or as M h = [M, X, κ]

(M ∈ SL(2), X = (λ, μ) ∈ R2 , κ ∈ R).

Another notation as a pair is also often used, namely hM = (M, h) = (M, Y, κ)

(M ∈ SL(2), h = (Y, κ) ∈ H).

In case of GJ over the reals this is more than just two ways of notation, namely it results in covering the manifold GJ (R) by two different charts. In order not to be disturbed by these different coordinatizations and notations, it may be helpful to keep in mind our first definition of GJ as a subgroup of Sp(2). But almost all the time the Jacobi group will be used in its realization (1.3) as a semidirect product.

1.2 GJ as an algebraic group The Jacobi group is defined by polynomial conditions as a group of matrices, and as such it can be considered as an affine algebraic group. In order to have available all the theorems and notions for general algebraic groups, we look at GJ (k), where k is an algebraically closed field of characteristic zero. But it

4

1. The Jacobi Group

should be noted that GJ and all of its relevant subgroups soon to be defined are already defined over Q, so that there is no difficulty in considering the Krational points of GJ for every field K between Q and k. Closer looks at the real and p-adic Jacobi group will be taken in subsequent chapters. From (1.3) one can see that GJ (k) is a six-dimensional closed connected subgroup of GL(4, k). Its Heisenberg part H(k) is a unipotent group; this is seen, for instance, by realizing H(k) as a group of upper triangular unipotent (3×3)matrices via ⎛ ⎞ 1 λ κ ⎝ 0 1 μ ⎠ −→ (λ, μ, 2κ − λμ). 0 0 1 In particular H(k) contains no semisimple elements. If GJ (k) would contain a two-dimensional torus, then in view of the exact sequence (1.2), the Heisenberg group would contain a nontrivial torus, which is not the case. Hence the maximal tori in GJ (k) are one-dimensional. One of them is the usual SL(2)-torus    a 0 ∗ A= : a ∈ k , 0 a−1 (regarded as a subgroup of GJ (k)) and the others are got from A by conjugation in GJ (k). Let B be the standard Borel subgroup of SL(2, k). Then it is immediate that    a x ∗ BH = (λ, μ, κ) : a ∈ k , x, λ, μ, κ ∈ k 0 a−1 is a maximal closed, connected, solvable subgroup, i.e. a Borel subgroup of GJ (k). All other Borels are conjugate to this one. The unipotent radical of BH is quickly identified as    1x (BH)u = N H = (λ, μ, κ) : x, λ, μ, κ ∈ k , 01 where N is the unipotent radical of B in SL(2, k). As usual BH = A  (BH)u . We have GJ (k)/BH  SL(2, k)/B  P1 (k), so that GJ (k) is of semisimple rank 1. In particular the Weyl group consists of two elements, the non-trivial one represented by   0 1 w= ∈ GJ (k). −1 0 The only Borel subgroups containing the maximal torus A are BH and its conjugate    a 0 ∗ (λ, μ, κ) : a ∈ k , x, λ, μ, κ ∈ k . wBHw−1 = x a−1

1.2. GJ as an algebraic group

5

The Bruhat decomposition reads GJ (k) = BH ∪ BHwBH (disjoint). Because of the semisimplicity of SL(2) the radical of GJ (k), meaning the unique maximal closed solvable normal subgroup, is given by the Heisenberg group H. Its subgroup consisting of unipotent elements, which by definition is the unipotent radical of GJ (k), is H itself, as we saw above: R(GJ ) = Ru (GJ ) = H. In particular, GJ is far from being reductive. Accordingly, its center is not a torus, but unipotent. To be more precise, a quick calculation shows that it equals the center of the Heisenberg group Z = {(0, 0, κ) : κ ∈ k}  Ga (k). It will often simply be written κ ∈ GJ (k) for κ ∈ k, meaning that κ is identified with (0, 0, κ) ∈ H ⊂ GJ (k). The above considerations show that the standard algebraic structure of GJ is strongly dominated by the SL(2) part. The nilpotent Heisenberg part appears merely as an appendix to all the usual subgroups in SL(2). Consequently the above standard notions for algebraic groups are not best suited for working with GJ . A better insight into which subgroups should instead be considered comes from determining the Lie algebra and root structure of GJ , which will be done in the following section. But before doing this we make a remark on the derived group of GJ . 1.2.1 Proposition. The Jacobi group is its own commutator group over any field K of characteristic not equal to 2, i.e. (GJ (K), GJ (K)) = GJ (K). Proof: SL(2) is semisimple, so SL(2, K) = (SL(2, K), SL(2, K)) ⊂ (GJ (K), GJ (K)).

(1.6)

Because of our hypothesis on the characteristic, the Heisenberg group is exactly 2-step nilpotent, more precisely (H(K), H(K)) = Z(H(K)) = Z(GJ (K)), where Z denotes the center. Hence Z(GJ (K)) ⊂ (GJ (K), GJ (K)).

(1.7)

6

1. The Jacobi Group

The commutator of M ∈ SL(2, K) and (X, 0) ∈ H(K) (with X ∈ K 2 ) is M (X, 0)M −1 (−X, 0) = (XM −1 , 0)(−X, 0)    XM −1 = X(M −1 − 1),  −X

   . 

This together with (1.7) shows that (GJ (K), GJ (K)) contains all elements of the form (X(M − 1), κ),

M ∈ SL(2, K), X ∈ K 2 , κ ∈ K,

and this is the whole Heisenberg group. The assertion follows in view of (1.6).  1.2.2 Corollary. The Jacobi group has no nontrivial characters . 1.2.3 Corollary. The real, complex, p-adic and adelic Jacobi groups are all unimodular . Proof: The modular character is a character.



In connection with this last corollary, we mention the following measure theoretic fact. 1.2.4 Proposition. Consider the real, complex, p-adic or adelic Jacobi group. If dM and dh denote Haar measures on SL(2) and H, respectively, then     f −→ f (hM ) dh dM = f (M h) dh dM SL(2) H

SL(2) H J

(f a suitable function on G ) defines a (biinvariant) Haar measure on GJ . Proof: We only have to show the equality of the two integrals. Abbreviating hM = M −1 hM , we trivially have     f (hM ) dh dM = f (M hM ) dh dM. SL(2) H

SL(2) H

Our claim follows once it is shown that for fixed M ∈ SL(2) and every suitable function F on H   F (h) dh = F (hM ) dh. H

H

But it is clear that the expression on the right also defines a Haar measure on H, and therefore the two integrals differ at most by a positive constant, which we denote by α(M ). The map M → α(M ) obviously is a character of SL(2). But this group, being semisimple, has no non-trivial characters, and we are done. 

1.3. The Lie algebra of GJ

7

1.3 The Lie algebra of GJ Let k be as in the last section. The Lie algebra gJ of GJ (k) is very easily determined as a subalgebra of M (4, k), because GJ (k) was originally defined as a subgroup of GL(4, k), cf. Section 1.1. We just list a natural basis of gJ and behind the six basis elements the closed connected subgroups of GJ (k) corresponding to the one-dimensional subspaces spanned by these elements. ⎛

0 ⎜ 0 ⎜ F =⎝ 0 0 ⎛ 0 ⎜ 0 ⎜ G=⎝ 1 0 ⎛ 1 ⎜ 0 H=⎜ ⎝ 0 0 ⎛ 0 ⎜ 1 P =⎜ ⎝ 0 0 ⎛ 0 ⎜ 0 Q=⎜ ⎝ 0 0 ⎛ 0 ⎜ 0 R=⎜ ⎝ 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

⎞ 0 0 ⎟ ⎟, 0 ⎠ 0 ⎞ 0 0 0 0 ⎟ ⎟, 0 0 ⎠ 0 0 ⎞ 0 0 0 0 ⎟ ⎟, −1 0 ⎠ 0 0 ⎞ 0 0 0 0 ⎟ ⎟, 0 −1 ⎠ 0 0 ⎞ 0 1 1 0 ⎟ ⎟, 0 0 ⎠ 0 0 ⎞ 0 0 0 1 ⎟ ⎟, 0 0 ⎠ 0 0 1 0 0 0







1x 01

10 x1



 : x∈k



a 0 0 a−1

 Ga (k). 

: x∈k  : a∈k

 Ga (k).

∗

  Gm (k).

{(λ, 0, 0) : λ ∈ k}  Ga (k).

{(0, μ, 0) : μ ∈ k}  Ga (k).

{(0, 0, κ) : κ ∈ k}  Ga (k).

If K is a field between Q and k, then gJ certainly has a K-structure, as is apparent by viewing the above matrices as elements of M (4, K). The real Lie algebra could equally well have been determined analytically using the exponential function. Because GJ (k) is a semidirect product of SL(2, k) and H(k), the Lie algebra gJ is a semidirect product of sl(2) and h, the Lie algebra of the Heisenberg group. In particular, sl(2) appears as a subalgebra and h as an ideal of gJ . The exact commutation relations fulfilled by the above basis elements are the following:

8

1. The Jacobi Group [F, G] = H,

[H, F ] = 2F,

[H, G] = −2G,

(1.8)

[P, Q] = 2R,

[R, P ] = 0,

[R, Q] = 0,

(1.9)

[F, P ] = −Q,

[F, Q] = 0,

[G, P ] = 0,

[G, Q] = −P,

(1.10)

[H, P ] = −P,

[H, Q] = Q,

[F, R] = [G, R] = [H, R] = 0.

(1.11)

The Heisenberg Lie algebra makes the Killing form on gJ highly degenerated. Here is its matrix in the above basis: F G H P Q R

F 0 5 0 0 0 0

G H P 5 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0

Q 0 0 0 0 0 0

R 0 0 0 0 0 0

For comparison the matrix of the Killing form of sl(2) is also given: F G H

F 0 4 0

G H 4 0 0 0 0 8

We now come to the root space decomposition of gJ . The maximal torus   ∗0 A= ⊂ GJ (k) 0∗ operates on gJ via the adjoint representation. This gives the decomposition  gJ = gJn n∈Z

where gJn =



X ∈ gJ : ∀ a ∈ k ∗ Ad



a 0 0 a−1



 X = an X .

A quick calculation yields the roots 1, −1, 2, −2. More precisely, gJ = gJ0 ⊕ gJ1 ⊕ gJ−1 ⊕ gJ2 ⊕ gJ−2 , with gJ0 = kH ⊕ kR, gJ1 = kQ,

gJ−1 = kP,

gJ2 = kF,

gJ−2 = kG.

1.4. GJ over the reals

9

The roots 1 and 2 shall be called the positive ones. The picture of our onedimensional root system is as follows: 



u

-

-

-2

-1

0

1

2

There are two subgroups of GJ (k) which will play important roles in the sequel. They are obtained by taking prominent subgroups of SL(2, k) and adequately adjoining subgroups of the Heisenberg group. The first one is    a 0 ∗ AZ = (0, 0, κ) : a ∈ k , κ ∈ k  Gm (k) × Ga (k). 0 a−1 This is the subgroup of GJ belonging to gJ0 , and it will be for GJ what a maximal torus is for a reductive group. The second one is    1x NJ = (0, μ, 0) : x, μ ∈ k  Ga (k)2 . 01 This N J can be nicely characterized as the closed connected subgroup of GJ (k) whose Lie algebra is the sum of the positive root spaces. Accordingly it will turn out to be something like the unipotent radical of a parabolic subgroup in a reductive group. For example, cusp forms on GJ will later on be characterized by integrating over conjugates of N J (see section 4.2 and Definition 7.4.4).

1.4 GJ over the reals GJ (R), though not reductive, has several special features which allow for the application of some fairly general principles. Before going into this, we will demonstrate two slightly different ways to describe the elements of GJ (R), where in the sequel EZ recalls the book [EZ] by Eichler and Zagier, and S points back to Siegel’s parametrization of the Siegel half spaces. The EZ-coordinates (x, y, θ, λ, μ, κ) of an element g ∈ GJ (R) are fixed as follows. For g = M (X, κ) = [M, X, κ] ∈ GJ (R), we take the Iwasawa decomposition SL(2) = N AK and write    1/2   1x y 0 cos θ sin θ M= 01 − sin θ cos θ 0 y −1/2         n(x) r(θ) t(y) with x ∈ R, y ∈ R>0 , θ ∈ R (θ is determined mod 2π only) and X = (λ, μ) with

λ, μ ∈ R.

10

1. The Jacobi Group

The S-coordinates (x, y, θ, p, q, κ) of g ∈ GJ (R) are given by g = (Y, κ)M = (M, Y, κ) ∈ GJ (R), where M is as above and Y = (p, q) = XM −1 ∈ R2 . Using EZ-coordinates just means to covering GJ (R) with the charts   1/2  1x y 0 (x, y, θ, λ, μ, κ) −→ r(θ)(λ, μ, κ), 01 0 y −1/2 and using S-coordinates amounts to cover GJ (R) by   1/2  1x y 0 (x, y, θ, p, q, κ) −→ (p, q, κ) r(θ) 01 0 y −1/2 (the variables in appropriate parts of Euclidean space). Sometimes it is convenient to deal with G (R) := GJ (R)/Z, J

and use here the central coordinate ζ = e(κ) ∈ S 1 ,

e(u) = exp(2πiu) for u ∈ R.

Then we have parallel to the Iwasawa decomposition SL(2) = N AK the decomposition G = N J AJ K J , J

where AJ =

  1/2   y 0 t(y, p) := (p, 0, 0) : p ∈ R, y > 0  R  R>0 , 0 y −1/2

and where K J is the compact group   K J = r(θ)ζ : θ ∈ R, ζ ∈ S 1 . Sometimes we will extend these notations slightly to cover other cases. For instance, we will write also GJ

=

N J AJ KZ,

g

=

n(x, q)t(y, p)r(θ)κ

where here

  1x n(x, q) = (0, q, 0) , 01

 1/2  y 0 t(y, p) = (p, 0, 0) . 0 y −1/2

1.4. GJ over the reals

11

The EZ- and the S-coordinates are adapted to describe an action of GJ (R) and J G (R) on H × C in the following way: We denote τ = x + iy ∈ H and z = ξ + iη = pτ + q ∈ C. Then GJ (R) and G J (R) act on H × C by  z + λτ + μ  g(τ, z) := M (τ ), , cτ + d where g = M (X, κ) = [M, X, κ] is meant in the EZ-coordinates with   aτ + b ab M (τ ) = for M = . cd cτ + d This operation looks more natural in the S-coordinates. We have ∼

GJ (R)/(SO(2) × Z(R)) −→ H × C, g = (p, q, κ)M

−→ g(i, 0) = (τ, pτ + q)

(τ = M (i)).

(The matrix M determines an elliptic curve, namely C/(Zτ +Z), and Y = (p, q) determines the point z = pτ + q on this curve.) GJ (R) may also be made to act on functions living on H × C by the aid of automorphic factors. This will be discussed in 3.4 as a side effect of the study of induced representations. For this induction procedure and the definition of theta- and zeta-transforms, we will moreover use the groups ˆ J := N J Z, N being maximal abelian in GJ , and B J := N J AZ. As already mentioned in Corollary 1.2.3, GJ is unimodular. More precisely, from Proposition 1.2.4 and the well-known form of the invariant measures on SL(2) and H, we see the following. 1.4.1 Remark. Both in the EZ- and the S-coordinates a biinvariant measure on GJ (R) is given by dg = y −2 dx dy dθ dp dq dκ. B J is not unimodular. A right-invariant measure on B J is given by dr b = dx dq

dy dκ, y

and we have 3/2

dr (b0 b) = y0 dr b.

12

1. The Jacobi Group

1.4.2 Remark. The modular function of B J (R) is in S-coordinates ΔB J (b) = y 3/2 . The complexified Lie algebra gJC and differential operators The EZ- and S-coordinates will be used to realize the real Lie algebra gJ and its complexification gJC by left-invariant differential operators. Here, we have gJC = gJ ⊗R C = Z, X± , Y± , Z0 C with Z = −i(F − G), Z0 = −iR,

1 (H ± i(F + G)) 2 1 Y± = (P ± iQ) 2

X± =

and thus from (1.8)–(1.11) the commutation relations [Z0 , gJC ] = 0 and [Z, X± ] = ±2X± , [Z, Y± ] = ±Y± ,

[X± , Y∓ ] = −Y± , [X± , Y± ] = 0,

[X+ , X− ] = Z, [Y+ , Y− ] = Z0 .

We put kJ = Z, Z0 

and

p± = X± , Y± .

For X ∈ gJ we define the left GJ (R)-invariant differential operator LX by   d LX φ(g) := φ(g exp tX) for φ ∈ C ∞ (GJ (R)), dt t=0 and we put LX = LX1 + iLX2

for X = X1 + iX2 ∈ gJC .

These operators do certainly not depend on the coordinates chosen, but look different in different charts. For completeness we give these operators in both types of coordinates: For g = (Y, ζ)M = (M, Y, κ) with the six S-coordinates (x, y, θ, p, q, κ) we have LZ0 LY±

= −i∂κ = (1/2)y −1/2 e±iθ (∂p − (x ∓ iy)∂q − (p(x ∓ iy) + q)∂κ )

LX± LZ

= ±(i/2)e±2iθ (2y(∂x ∓ i∂y ) − ∂θ ) = −i∂θ

1.4. GJ over the reals

13

and for g = M (X, ζ) = [M, X, ζ] with (x, y, θ, λ, μ, κ) LZ0

= −i∂κ

LY±

= (1/2)(∂λ ± i∂μ ± i(λ ± iμ)∂κ )

LX±

= ±(i/2)e±2iθ (2y(∂x ∓ i∂y ) − ∂θ ) + (1/2)(λ ± iμ)(∂λ ± i∂μ )

LZ

= −i∂θ − i(−μ∂λ + λ∂μ ).

On the group G (R) = GJ (R)/Z, one has to replace in these formulas ∂κ by 2πiζ∂ζ . J

GJ as a group of Harish-Chandra type As announced at the beginning of this section and as already to be seen by the existence of a generalized Iwasawa decomposition, some aspects of the general theory of reductive groups may be carried over to our case. Thus, GJ (R) is a group of Harish-Chandra type in the sense of Satake [Sa1] pp. 118-119. Satake discusses this in a far more general situation in his Example 2 on page 121. For our special case, we will give here and moreover in 3.4 some details adapted to our coordinates. Satake calls a Zariski connected R–group G with Lie algebra g of Harish– Chandra type, if it fulfills the following two conditions. (HC1) The complexification gC of g is a direct sum of vector spaces gC = p+ + kJ + p− , the “canonical decomposition”, with [kJ , p± ] ⊂ p± ,

¯ p+ = p− .

Here p± are the Lie algebras of connected unipotent C–subgroups P± contained in the complexification GC of G. (HC2) One has a holomorphic injection P+ × KC × P− −→ P+ KC P− ⊂ GC . Now, as we have in our case gJC = Z, X± , Y± , Z0 

with

[Z, Y± ] = ±Y± , [Z, X± ] = ±2X± ,

these conditions are fulfilled for p± = X± , Y±  and kJ = Z, Z0 

14

1. The Jacobi Group

and P+

=

P−

=

KCJ

=

  (1 + zX+ )(w(1, i), 0)) : z, w ∈ C ,   (1 + uX− )(v(1, −i), 0)) : u, v ∈ C ,   R(0, 0, κ) : R ∈ SO(2, C), κ ∈ C .

A second notion is very close to this: 

X = H × C = G J (R)/K J (R) as a reductive coset space Helgason introduced (see [He1] or [He2]) for his studies of invariant differential operators and eigenspace representations the notion of a reductive coset space X = G/K for groups K ⊂ G with Lie algebras k and g such that one has a decomposition g=k+m

with AdG (k)m ⊂ m

for all k ∈ K.

Here AdG (k) denotes the adjoint representation of G operating as usual on m. And this condition is fulfilled in our case again for k = F − G, R

and

m = F + G, H, P, Q,

because we have from (1.8)–(1.11) [k, m] ⊂ m.

2 Basic Representation Theory of the Jacobi Group

Depending on whether we look at the archimedean, a p-adic or the adelic case, the methods for studying representations are sometimes very different. In this chapter we will collect some general material, mainly going back to Mackey, which will be useful in all three cases. We start by explaining the induction procedure, and apply it to describe the representations of the Heisenberg group. We treat the representations of the Jacobi group GJ with trivial central character and set the way for all further discussions of the cases with non-trivial central character by introducing a certain projective representation of GJ , the Schr¨ odinger-Weil representation (others would perhaps call it the oscillator representation). This fundamental representation will later on be elaborated thoroughly in the different cases, and will allow to reduce, in a sense to be made precise later, the GJ -theory to the metaplectic theory.

2.1 Induced representations There is a general method (studied in detail by Mackey) to construct representations of a locally compact group G by an induction process starting from representations of a subgroup B. As we will apply this method later on at several occasions, we sketch here this procedure following essentially Kirillov [Ki] pp. 183-184. There are two natural realizations of an induced representation: R. Berndt and R. Schmidt, Elements of the Representation Theory of the Jacobi Group, Modern Birkhäuser Classics, DOI 10.1007/978-3-0348-0283-3_2, © Springer Basel AG 1998

15

16

2. Basic Representation Theory of the Jacobi Group

1.) in a space of vector valued functions φ on the group G that transform according to a given representation σ of B under left translations by elements of the group B, 2.) in a space of vector valued functions F on the coset space X = B\G. The transition from one model to the other is sometimes a difficult task, as we will see later on. The first realization To describe the first realization, we will consider a closed subgroup B of G and a representation σ of B in a Hilbert space V = Vσ . We denote by dr g and dr b right Haar measure on G resp. B and by ΔG (g) and ΔB (b) the modular function with dr (g0 g) = ΔG (g0 )dr g resp. correspondingly for ΔB (b). Then we induce from σ a representation π = indG Bσ of G given by right translation ζ on the space H = Hπ of measurable Vσ -valued functions φ on G with the two properties  1/2 ΔB (b) i) φ(bg) = σ(b)φ(g) for all b ∈ B and g ∈ G. ΔG (b)  ii) φ(s(x))2v dμs (x) < ∞. X

Here s : X = B\G → G is a Borel section of the projection p : G → B\G given by g → Bg. Then every g ∈ G can uniquely be written in the form g = b · s(x),

b ∈ B,

x ∈ X,

and G (as a set) can be identified with B × X. Under this identification, the Haar measure on G goes over into a measure equivalent to the product of a quasi-invariant measure on X and the Haar measure on B. More precisely, if a quasi-invariant measure μs on X is appropriately chosen, then the following equalities are valid. dr g =

ΔG (b) dμs (x)dr b ΔB (b)

and

dμs (xg) ΔB (b(x, g)) = , dμs (x) ΔG (b(x, g))

17

2.1. Induced representations

where b(x, g) ∈ B is defined by the relation s(x)g = b(x, g)s(xg). If G is unimodular, i.e. ΔG ≡ 1, and if it is possible to select a subgroup K that is complementary to B in the sense that almost every element of G can uniquely be written in the form g = b · k,

b ∈ B,

k ∈ K,

then it is natural to identify X = B\G with K and to chose s as the embedding of K in G. In this case, we have dg = ΔB (b)−1 dr b dr k = dl b dr k. If both G and B are unimodular (or more generally, if ΔG (b) and ΔB (b) coincide for b ∈ B), then there exists a G–invariant measure on X = B\G. If it is possible to extend ΔB to a multiplicative function on G, then there exists a relatively invariant measure on X which is multiplied by the factor ΔB (g)ΔG (g)−1 under translation by g. It is a fundamental fact that π = indG B σ is unitary if σ is. In this case B = Bπ is a Hilbert space with a G-invariant scalar product of the form  φ1 , φ2  = φ1 (g), φ2 (g)V dμ(g), G

where the measure μ on G is such that   φ(g)2V dμ(g) = φ(s(x))2V dμs (x) G

X

holds for all φ ∈ X. The second realization Using the section s : X → G, we associate to each φ ∈ X a function f on X defined by f (x) := φ(s(x)). Obviously φ is uniquely determined by f and we have an isomorphism of Bπ onto the space B π = L2 (X, μs , V ) of V -valued functions on X having summable square norm with respect to the measure μs . The problem now is to exhibit the representation operator corresponding to the right translation ρ on Hπ . It can be shown that we have π(g)f (x) = A(g, x)f (xg)

for

f ∈ Hπ

18

2. Basic Representation Theory of the Jacobi Group

where the operator valued function A(g, x) is defined by the equality  A(g, x) =

ΔB (g) ΔG (b)

1/2 σ(b),

in which the element b ∈ B is defined from the relation s(x)g = bs(xg).

2.2 The Schr¨ odinger representation As an example, we will discuss the Heisenberg group and its Schr¨ odinger representation. From now on, almost everything depends on the choice of some additive character of the underlying field. Thus we will now introduce the socalled additive standard characters, following [Tate], 2.2. For every prime p (including p = ∞) we can define a homomorphism of additive groups λ : Qp −→ R/Z as follows. If Qp = R, then λ(x) = −x mod 1. If p is finite, then we map a Laurent series in p to its main part: ∞    λ ai p i = i>>−∞

−1 

ai p i .

i>>−∞

If F is a finite extension of Qp , then the additive standard character ψ : F −→ S 1 is defined by ψ(x) = e−2πiλ(Tr(x)) , where Tr is the trace mapping F → Qp . Hence if F = R, then ψ(x) = e2πix , and if F = C, then ψ(x) = e4πiRe(x) . Caution: Our character is precisely the inverse of the character defined in [Tate]. We have made our choice of characters analogous to that in the papers [Be1]-[Be6] in the real case. For m ∈ F , the notation ψ m (x) = ψ(mx)

19

2.2. The Schr¨ odinger representation

will be used throughout. From [Tate] 2.2 it is known that the map m → ψ m identifies F with its own character group. It is also important to know that if F is discrete and d denotes the absolute different of F then d−1 is the greatest ideal of F , on which ψ is trivial. In particular, if F = Qp , then ψ is trivial on Zp and on no bigger ideal. Now, let F be a number field, {p} the set of places of F , and Fp the completion of F at p. We can define a global additive character ψ of the adele ring A of F by  ψ(x) = ψp (xp ) for all x = (xp )p ∈ A, p

where ψp are the local standard characters defined above. The adele ring is also self-dual via the identification A  m → ψ m (cf. [Tate] Theorem 4.1.1). The global character thus defined has the special property that ψ(x) = 1 for all x ∈ F , i.e., it is a character of A/F . Every other such character is then of the form ψ m with m ∈ F ([Tate] Theorem 4.1.4). We will always consider these characters in the global theory. Returning now to local considerations, we let F be a local field of characteristic 0, and consider G B

= =

H BH

= {h = (λ, μ, κ) : λ, μ, κ ∈ F } = {b = (0, μ, κ) : μ, κ ∈ F }.

For ψ the additive standard character of F as explained above and m ∈ F ∗ , let σ(b) = σ(0, μ, κ) = ψ m (κ) = ψ(mκ). Here we have the simplest situation, i.e. G and B are unimodular and we have the decomposition H = BH AH

with

AH = {a = (λ, 0, 0) : λ ∈ F }

and h = (λ, μ, κ) = (0, μ, κ )(λ, 0, 0)

with

κ = κ + λμ.

This already shows that the first realization of π = indG B σ is given by right translation ρ on the space Hπ of measurable C-valued functions φ on H with φ(bh) = ψ(mκ)φ(h) and

for all b ∈ BH

and h ∈ H

 |φ(λ, 0, 0)|2 dλ < ∞. F

This realization is sometimes called the Heisenberg representation.

20

2. Basic Representation Theory of the Jacobi Group

The restriction map φ → f given by f (x) = φ(x, 0, 0) intertwines this model with the usual Schr¨ odinger representation πSm on the π 2 space H = L (F ). The prescription given above for the representation operator A(g, x) here means to solve the equation s(x)h = bs(xh) for given x, i. e. s(x) = (x, 0, 0), and h = (λ, μ, κ) by b = (0, μ, κ + 2xμ + λμ). This means we have for f ∈ L2 (F ) the well known formula (πSm (λ, μ, κ)f )(x) = ψ m (κ + (2x + λ)μ)f (x + λ).

(2.1)

One can see directly that πSm is a unitary representation. In case F is nonarchimedean, it is customary to regard πSm as a representation on the space of smooth vectors of πSm , which is just the Schwartz space S(F ). The representation theory of the Heisenberg group is very simple, due to the following theorem which we give in both the real and the p-adic cases. Proofs can for instance be found in [LV], 1.3 (for the real case) and [MVW], 2.I.2, 2.I.8 (for the p-adic case). The notion of smooth representation appearing in Theorem 2.2.2 will be explained in Section 5.1. 2.2.1 Theorem. (Archimedean Stone–von Neumann theorem) i) πSm is an irreducible unitary representation of H(R) with central character ψ m , and every such is isomorphic to πSm . ii) A unitary representation of H(R) with central character ψ m decomposes into a direct sum of Schr¨ odinger representations πSm . 2.2.2 Theorem. (Non-archimedean Stone–von Neumann theorem) Let F be a p-adic field. i) The representation πSm on S(F ) is an irreducible, smooth representation of H(F ) with central character ψ m , and every such is isomorphic to πSm . ii) A smooth representation of H(R) with non-trivial central character ψ m decomposes into a direct sum of Schr¨odinger representations πSm . It is indeed the Stone–von Neumann theorem which enables much of our treatment of the representation theory of the Jacobi group.

2.3. Mackey’s method for semidirect products

21

2.3 Mackey’s method for semidirect products The aim of the present and the following sections is to compute the unitary dual of the Jacobi group over a local field or over the adeles of a number field. We will make a distinction between the representations which have trivial central character and those which do not. In the first case we impose a general method of Mackey for determining the unitary dual of certain semidirect products. The second case can be treated more directly by using only the Stone–von Neumann theorem. In this section we begin with presenting Mackey’s method in a degree of generality that suffices for our purposes. Let G be a locally compact topological group and H  a commutative closed normal subgroup, such that the exact sequence 1 −→ H  −→ G −→ G /H  −→ 1

(2.2)

splits, i.e., G is a semidirect product of G := G /H  with H  : G = G  H  . We wish to determine the unitary representations of G in terms of those of G and H  . The method to be described goes back to Mackey [Ma1] and is repeated, for instance, in [Ma2], p. 77.  is known and has been given the topology of Assume the unitary dual H uniform convergence on compact subsets. G operates on H  by conjugation,  : and this induces an operation of G on H  G × H (g, σ)

 , −→ H

−→ σ g ,

where the representation σ g is given by σ g (h) = σ(ghg −1 )

for all h ∈ H  .

Of course, if g ∈ H, then σ g is equivalent to σ. Hence H operates trivially on  , and only the action of G has to be considered. H Mackey’s theory does not work for arbitrary semidirect products. One has to  . Namely it is impose a certain smoothness condition on the orbits of G in H    demanded that for every G -orbit Ω in H and for every σ ∈ Ω with stabilizer Gσ ⊂ G the canonical bijection Gσ \G −→ Ω be a homeomorphism. If this condition is fulfilled then H  is called regularly embedded, and G = G  H  is called a regular semidirect product. The result of Mackey is now as follows.

22

2. Basic Representation Theory of the Jacobi Group

2.3.1 Theorem. Let G be a locally compact topological group and H  a closed commutative normal subgroup such that the sequence (2.2) splits. Assume that  let G the stabilizer H  is of type I and regularly embedded. For every σ ∈ H σ   , and of σ under the above action of G on H  ˇ σ = {τ ∈ G σ : τ  is a multiple of σ}. G  H

Then the induced representation 

IndG Gσ τ ˇ σ , and G  is a disjoint union is irreducible for every τ ∈ G    = ˇ G {IndG Gσ τ : τ ∈ Gσ }.  /G H

2.4 Representations of GJ with trivial central character Let R be a local field of characteristic 0 (R and C included) or the ring of adeles of a number field, and let GJ be the Jacobi group over R. In this section we determine the irreducible unitary representations of GJ which have trivial central character. These representations are obviously in 1-1 correspondence with the irreducible unitary representations of the group G := GJ /Z  G  H  ,

where H  := R2 .

Now G contains H  as an abelian normal subgroup which allows determination of its unitary dual by means of the method described in the last section. The first step is to determine the irreducible unitary representations of H  . This is very easy in our case because R is self-dual. Hence the unitary dual  identifies with R2 itself by associating with (m1 , m2 ) ∈ R2 the unitary H character R2 (λ, μ)

−→ C∗ ,

−→ ψ m1 (λ)ψ m2 (μ).

 : G operates on H  by conjugation and thus also on H  −→ H  , G×H (M, σ)

−→ (X → σ(XM ))

(XM means matrix multiplication). A small calculation shows that under the  = R2 this operation goes over to the natural action above identification H G × R2 −→ R2 , (M, Y ) −→ M Y

2.4. Representations of GJ with trivial central character

23

 (now think of Y ∈ R2 as a column vector). This makes it obvious that H decomposes into two G-orbits, one of them consisting only of the trivial representation. As a representative for the non-trivial characters we choose Ψ : H (λ, μ)

−→ C,

−→ ψ(λ)

(corresponding to the point (1, 0) ∈ R2 ). The stabilizer of the trivial representation is certainly G itself, and the stabilizer of Ψ is    10  GΨ = (λ, μ) : c, λ, μ ∈ R . c1 Theorem 2.3.1 gives the following result, where we leave it as an exercise to check the hypotheses in this theorem. 2.4.1 Proposition. The irreducible unitary representations of G are exactly the following:     i) The representations σ where σ  is trivial and σ  is an irreducible  H G unitary representation of G. 

ii) The representations IndG GΨ τ , where τ runs through the irreducible unitary representation of GΨ whose restriction to H  is a multiple of Ψ. It remains to describe more closely the representations appearing in ii). Suppose τ is an irreducible unitary representation of GΨ whose restriction to H  is a multiple of Ψ. Then an element (λ, μ) ∈ H  operates by multiplication   with 1∗ ψ(λ). Thus every subspace which is invariant under the matrices is yet 01 invariant under GΨ . Hence the restriction of τ to the matrix group must be irreducible. This group being isomorphic to R itself we see that our representation is one-dimensional and the matrices act through a unitary character of R. Conversely, given such a unitary character ψ r with r ∈ R it is immediately checked that   1c (λ, μ) −→ ψ r (c)ψ(μ) 01 defines a homomorphism GΨ → C∗ . So the representations τ from which we start our induction constitute a one-parameter family indexed by r ∈ R. Putting everything together we have the following result. 2.4.2 Theorem. The irreducible unitary representations of GJ with trivial central character are exactly the following.

24

2. Basic Representation Theory of the Jacobi Group     i) The representations σ where σ  is trivial and σ  is an irreducible uniH G tary representation of G. J

ii) The representations IndG GΨ τr , where 

τr : GΨ  1c (λ, μ) 01

−→ C∗ ,

−→ ψ(rc + μ).

2.5 The Schr¨ odinger-Weil representation It will turn out in the following section that every irreducible unitary (respectively smooth) representation π of GJ with non-trivial central character can be written as a tensor product of two representations, where one factor is a certain standard representation independent of π. The present section is devoted to introducing this so-called Schr¨ odinger-Weil representation, which is not really a representation of GJ but a projective one. The construction is standard and carried out in much greater generality in [We]. Let R be the real or complex numbers, a p-adic field, or the adele ring of a number field, and consider GJ = G  H over R. The starting point is the Schr¨ odinger representation πSm : H −→ GL(V ) with central character ψ m , m ∈ R∗ , which was discussed in Section 2.2. Now G operates on H by conjugation inside GJ in the following way: G×H

−→ H,

(M, h) −→ M hM −1 = (XM −1 , κ),

h = (X, κ), X ∈ R2 , κ ∈ R.

In particular, M leaves the central part of h untouched. Hence the irreducible unitary representation H h

−→ GL(V )

−→ πSm (M hM −1 )

has central character ψ m , just like πSm . By the Stone–von Neumann theorem, this conjugated representation must be equivalent to πSm itself, i.e., there is a unitary operator m πW (M ) : V −→ V

such that m m πSm (M hM −1 ) = πW (M )πSm (h)πW (M )−1

for all h ∈ H.

(2.3)

2.5. The Schr¨ odinger-Weil representation

25

m By Schur’s lemma, πW (M ) is determined up to nonzero scalars. We fix one m πW (M ) for each M ∈ G arbitrarily. Now for M1 , M2 ∈ G we have m m m m πW (M1 )πW (M2 )πSm (h)πW (M2 )−1 πW (M1 )−1 m m = πW (M1 M2 )πSm (h)πW (M1 M2 )−1 ,

and again by Schur’s lemma there must exist a scalar λ(M1 , M2 ) of absolute value 1 such that m m m πW (M1 M2 ) = λ(M1 , M2 )πW (M1 )πW (M2 ).

(2.4)

From the associativity law in G it follows that λ(M1 M2 , M3 )λ(M1 , M2 ) = λ(M1 , M2 M3 )λ(M2 , M3 ), which just says that λ is a 2-cocycle for the trivial G-modul S 1 . The freedom m in multiplying the operators πW (M ) with scalars of absolute value 1 amounts to changing λ by a coboundary. Hence the representation πSm we started with determines in a unique way an element λ ∈ H 2 (G, S 1 ). From [We] or [Ku1] it is known that • H 2 (G(R), S 1 ) is trivial if R = C. • H 2 (G(R), S 1 ) consists of exactly two elements if R = R or R = F a p-adic field. It is further known that λ represents the non-trivial element of H 2 (G(R), S 1 ) if R is real or p-adic. In [Ge2] a version of this cocycle can be found which has the property that   λ =1 if R is p-adic and not an extension of Q2 . O ∗ ×O ∗

We will use in all that follows this cocycle in the real or p-adic case, λ = 1 in the complex case, and the product of the corresponding local cocycles in the adelic case. Coming back to the above notations we see that m M −→ πW (M )

is a projective representation of G on V with multiplier λ. It is called the Weil m representation with character ψ m . Note that πW is an ordinary representation m exactly in the complex case. Otherwise we can make πW into an ordinary  or representation by going over to the metaplectic group Mp (also denoted G, Mp(R)), which is by definition the topological group extension of G by {±1} determined by the cocycle λ. In other words, as a set we have Mp = G × {±1},

26

2. Basic Representation Theory of the Jacobi Group

the multiplication is defined by (M, ε)(M  , ε ) = (M M  , λ(M, M  )εε ), and there is an exact sequence of topological groups 1 −→ {±1} −→ Mp −→ G −→ 1. Now the map m (M, ε) −→ πW (M )ε

obviously defines a representation of Mp in the ordinary sense. It is also called the Weil representation. We put the Schr¨ odinger and the Weil representation together and define m πSW : GJ

hM

−→ GL(V ), m

−→ πSm (h)πW (M )

for all h ∈ H, M ∈ G.

m m The defining property (2.3) of πW immediately shows that πSW is a projective representation of GJ with multiplier λ, the latter extended canonically to GJ . It is called the Schr¨ odinger-Weil representation of GJ with central character ψ m . We give the same name to the corresponding ordinary representation of J of GJ which is defined analogously to GJ . Note that the two-fold cover G there is a commutative diagram

J −−−−→ G ⏐ ⏐

 G ⏐ ⏐

GJ −−−−→ G J identifies with the semidirect product of G  with H. and that G Finally we give some explicit formulas for the Weil representation. There will be the appearence of the so-called Weil constant. This is a function γ : R∗ −→ S 1 which depends on the different cases and on the character ψ m . • If R = C then γ is the constant function 1. • If R = R then γ(a) = eπi sgn(m)sgn(a)/4 . • If R = F is a p-adic field, then !  m 2 γ(a) = lim ψ (ax ) dx | . . . |. n→∞ ω −n O

• If R = A then γ is the (well-defined) product of local Weil constants.

2.5. The Schr¨ odinger-Weil representation

27

If the dependence on the character ψ m is to be emphasized, we write γm instead of γ. Though not obvious in the non-archimedean case, the Weil constant is always an eighth root of unity (see [We] or [Sch1]). As a further ingredient to the explicit formulas below there is the (second) Hilbert symbol (·, ·) : R∗ × R∗ −→ {±1}. If R is a local field then it is defined as (a, b) = 1

⇐⇒

√ b is a norm from R( a).

In particular the Hilbert symbol is constantly 1 in the complex case. The global Hilbert symbol is defined to be the product of the local symbols. More about Hilbert symbols can be found in texts on algebraic number theory. Now we are ready to state the explicit formulas for the Weil representation. As a model for πSm the Schwartz space S(R) is used. Then the associated Weil representation acts on the same space as follows.     m 1 b πW f (x) = ψ m (bx2 )f (x). (2.5) 01     m a 0 πW f (x) = (a, −1)γ(a)γ(1)−1 |a|1/2 f (ax). (2.6) 0 a−1   0 1 m πW f = γ(1)fˆ. (2.7) −1 0 Here fˆ denotes the Fourier transformation of f ∈ S(R):  fˆ(x) = |2m|1/2 f (y)ψ m (2xy) dy. R

The factor |2m|1/2 normalizes the measure on R to make Fourier inversion hold: ˆ fˆ(x) = f (−x). It is not easy to deduce the formulas (2.5)–(2.7), but it is easy to prove them. It just has to be checked that (2.3) holds with these operators, but we will not carry this out. For the real case, see [Mum] Lemma 8.2 or [LV] Section 2.5. Assume now R to be a local field. Since the Schr¨ odinger representation is irm reducible, the Schr¨ odinger-Weil representation is also. But if we restrict πSW m to SL(2, R), i.e., we consider the Weil representation πW , then from the formulas (2.5)–(2.7) we immediately find the invariant subspaces S(F )+ and S(F )− m± consisting of even resp. odd Schwartz functions. Let πW denote the subrepresentations on these spaces. They are called the positive (resp. negative) or even (resp. odd) Weil representations.

28

2. Basic Representation Theory of the Jacobi Group

2.5.1 Proposition. The positive and negative Weil representations are irreducible, and we have m m+ m− πW = πW ⊕ πW .

Between the irreducible Weil representations there are exactly the following equivalences: 

m± m± πW  πW

⇐⇒

mF ∗2 = m F ∗2 .

Proof: It is easy to see from (2.5) and (2.7) that the isomorphism S(F ) f

−→ S(F ),

−→ (x → f (ax)), 2

m a m intertwines πW with πW , for any a ∈ F ∗ . So if R = C, we are done. The case R = R will follow from our considerations in the first part of Section 3.2. For the p-adic case, see [MVW] 2.II.1. 

2.6 Representations of GJ with non-trivial central character Let again GJ be the real or p-adic Jacobi group. In principle Mackey’s method could also be used to determine the unitary representations of GJ with nontrivial central character. Since the Heisenberg group is not commutative, one would have to check carefully the hypotheses made in [Ma1]. However, we prefer a direct method similar to the construction in [We]. The procedure is also described in Kirillov [Ki] pp. 218–219. When dealing with the real Jacobi group, we are interested in unitary representations, while for the p-adic Jacobi group, we consider smooth representations. Both cases can be treated in a very similar way. The decisive point is to have the Stone–von Neumann theorem at hand. We treat the unitary case and leave the minor changes for the p-adic case to the reader. So let π be a unitary representation of the real Jacobi group GJ on a Hilbert space V with central character ψ m , m = 0. The restriction of π to the Heisenberg group decomposes into unitary representations, each of which must be equivalent to the Schr¨ odinger representation πSm with central character ψ m , by the Stone–von Neumann theorem 2.2.1. So this restriction is isotypical, and consequently we may assume that V is a Hilbert tensor product V = V1 ⊗ V2 , where H acts trivially on V1 and where V2 is a representation space for πSm .

2.6. Representations of GJ with non-trivial central character

29

m From the defining property (2.3) of πW , which also acts on V2 , it follows easily that m m π(M −1 )(1V1 ⊗ πW (M ))π(h) = π(h)π(M −1 )(1V1 ⊗ πW (M )), m i.e., the operator π(M −1 )(1V1 ⊗ πW (M )) commutes with the action of the Heisenberg group. Hence it must be of the form m π(M −1 )(1V1 ⊗ πW (M )) = π ˜ (M ) ⊗ 1V2

with π ˜ (M ) ∈ Aut(V1 ).

As a result we were able to separate the action of G in one on V1 and one on V2 : m π(M ) = π ˜ (M ) ⊗ πW (M ).

(2.8)

More generally, for every element g = hM of the Jacobi group with M ∈ G and h ∈ H we have m π(hM ) = π ˜ (M ) ⊗ πSW (hM ), m where πSW is the Schr¨ odinger-Weil representation introduced in the last chapter. From (2.4) it follows that for M1 , M2 ∈ G

π ˜ (M1 M2 ) = λ(M1 , M2 )−1 π ˜ (M1 )˜ π (M2 ). m In other words, π ˜ and πSW are both projective representations of G resp. GJ −1 with multiplier λ resp. λ. After tensorizing the cocycles cancel and the result is an ordinary representation of GJ . Summarizing we obtain the following result.

2.6.1 Theorem. The above construction gives a 1-1 correspondence m π ˜ −→ π ˜ ⊗ πSW

between the irreducible unitary projective representations of SL(2, R) with multiplier λ and the irreducible unitary representations of GJ (R) with non-trivial central character ψ m . The corresponding non-archimedean result is as follows. 2.6.2 Theorem. Let F be a p-adic field. There is a 1-1 correspondence m π ˜ −→ π ˜ ⊗ πSW

between the irreducible smooth projective representations of SL(2, F ) with multiplier λ and the irreducible smooth representations of GJ (F ) with non-trivial central character ψ m .

30

2. Basic Representation Theory of the Jacobi Group

m The only difference in the complex case is that πSW is a representation, not a projective one. Then π ˜ will also turn out to be a representation of G, and we get the following result:

2.6.3 Theorem. The map m π ˜ −→ π ˜ ⊗ πSW

establishes a 1-1 correspondence between irreducible, unitary representation of SL(2, C) and irreducible, unitary representations of GJ (C) with central character ψ m (m ∈ C∗ ). We refer the reader to Knapp [Kn] II, §4, for a classification of the irreducible, unitary representations of SL(2, C), and thus for a classification of irreducible, unitary representations of GJ (C). Much more will be said in the following chapters about the correspondence m π ˜ → π ˜ ⊗ πSW , with specific reference to the underlying field.

3 Local Representations: The Real Case

Here we rearrange and extend material from [Be1]–[Be4] and [BeB¨ o]. By the general theory from the last section, we have as a fundamental object the m Schr¨ odinger-Weil representation πSW which is a genuine representation of the J (R) and may be identified with a projective representation metaplectic cover G m of GJ (R). If we tensorize πSW with another genuine representation π ˜ of the metaplectic cover Mp(R) (again to be identified with a projective representation of SL2 (R)) we get m π = πSW ⊗π ˜,

a representation of GJ (R) with central character ψ m , i.e. π(0, 0, κ) = ψ m (κ) = e2πimκ

for all κ ∈ R.

This way, we get all unitary representations π with m = 0 if we take all unitary representations π ˜ of Mp(R). The representations of the metaplectic group were studied to a large extent by Waldspurger [Wa1-3] and Gelbart [Ge2]. Thus, at least for the unitary representations, we easily get a rather complete picture simply by applying Mackey’s method. But, since in the real theory we also have the possibility to apply the infinitesimal method, we use it here as the starting point. Afterwards we will discuss several features of the induction procedure, coming up, among other things, with the canonical automorphic factor and invariant differential operators on H × C.

R. Berndt and R. Schmidt, Elements of the Representation Theory of the Jacobi Group, Modern Birkhäuser Classics, DOI 10.1007/978-3-0348-0283-3_3, © Springer Basel AG 1998

31

32

3. Local Representations: The Real Case

3.1 Representations of gJC We have already dealt with the complexified Lie algebra of the Jacobi group in Section 1.4. It is given by gJC = Z, X±  + X± , Z0  = kJ + p+ + p− , where p± = X± , Y± .

kJ = Z, Z0 ,

From the commutation relations given in Section 1.4, we repeat the following: [Z0 , gJC ] = 0,

[Z, Y± ] = ±Y± ,

[Z, X± ] = ±2X± .

Because of this decomposition, for each representation π ˆ of gJC the representation space V decomposes as V =



Vk

with

π ˆ (Y± )Vk ⊂ Vk±1 ,

π ˆ (Z0 )Vk = μVk ,

k∈Z

π ˆ (Z)Vk = ρk Vk ,

π ˆ (X± )Vk ⊂ Vk±2 ,

where μ and ρk are complex numbers. 3.1.1 Remark. μ = 0 will be fixed here through out. As we will only be interested in representations π ˆ being a derived representation of a (unitary) repre˜ J , μ will be thought of the form sentations π or π ˜ of GJ resp. G μ = 2πm,

m ∈ R∗ the “index” of π resp. π ˆ,

and ρk , the weight of Vk , should be an integer or a half integer. As later on we J will be interested in representations of G = GJ /Z, the real number m will then be fixed as a non-zero integer. 3.1.2 Definition. Let π ˆ be a representation of gJC with space V =

"

Vk as above.

i) π ˆ is called of lowest (highest) weight k, if there is a Vk = {0} with π ˆ (p− )Vk = {0}

resp.

π ˆ (p+ )Vk = {0}.

The elements in Vk will then be called lowest (highest) weight vectors. ii) π ˆ is called spherical (nearly spherical), if there is a Vk = {0} with ρk = 0

resp.

ρk = 1/2 or 1.

Elements v ∈ Vk will correspondingly be called spherical (nearly spherical) vectors.

3.1. Representations of gJC

33

If there is no danger of confusion, we will use abbreviations like X=π ˆ (X)

for X ∈ gJC

or X 2 = (ˆ π (X))2 , the latter being understood as the operator belonging to the element X 2 in the universal enveloping algebra U (gJC ) of gJC . In particular, we will use the elements D± := X± ± (2μ)−1 Y±2 ∈ U (gJC ). Interpreted as operators, D+ will later be recognized as the “heat operator”. From the general theory of the last chapter we know that there is a 1-1 correspondence between irreducible, unitary, genuine representations π ˜ of Mp and irreducible, unitary representations π of GJ with central character ψ m through the relation m π=π ˜ ⊗ πSW .

By differentiating, this remains true on the infinitesimal level. More generally, we have a bijection between irreducible representations π ˜ of sl2 (the Lie algebra of Mp) and irreducible representations π ˆ of gJC , given by m π ˆ=π ˜⊗π ˆSW .

(3.1)

3.1.3 Remark. By more thoroughly analyzing the infinitesimal situation, we would avoid the recourse to Mackey’s theory, and thereby also arrive at the correspondence (3.1). m m For the representation π ˆSW = dπSW of gJC we have the following result.

3.1.4 Proposition. Let m ∈ R∗ . If m > 0, the infinitesimal Schr¨ odinger-Weil m representation π ˆSW is a lowest weight representation. It operates on the space V = vj j∈N0 by Z0 vj = μvj Y+ vj = vj+1 Y− vj = −μjvj−1

 1 Zvj = j + vj 2 1 X+ vj = − vj+2 2μ μ X− vj = j(j − 1)vj−2 2

(3.2)

m where μ = 2πm and v−1 = v−2 = 0 understood. If m < 0, then πSW is a highest weight representation with space V = v−j j∈N0 , the action given by

34

3. Local Representations: The Real Case  1 Zv−j = − j + v−j 2 1 X− v−j = v−(j+2) 2μ μ X+ v−j = − j(j − 1)v−(j−2) 2

Z0 v−j = μv−j Y− v−j = v−(j+1) Y+ v−j = μjv−(j−1)

(3.3)

(with v1 = v2 = 0 understood). This will be proved in the next section. In particular, the weights (eigenvalues) of Z are given by half integers. It remains to describe the representations π ˜ . Since we are only interested in those representations π ˆ where Z has integral weights, we only have to classify those π ˜ where Z acts by half integers. These representations, which we call genuine, were thoroughly studied by Waldspurger [Wa1] (see in particular p. 22). Taking over as far as possible here his notations, we have: A) The principal series representations s ∈ C \ {Z + 1/2}, ν = ±1/2,

π ˜=π ˆs,ν , are given by  1 Zwl = l − wl , 2

X± wl =

 1 1  s+1± l− wl±2 , 2 2

acting on Ws,ν = wl ,

l ∈ 2Z + ν + 1/2.

B) The discrete series representations π ˜=π ˆk±0 ,

k0 ∈ Z + 1/2,

are given by Zw±l X± w±l

= =

X∓ w±l

=

±(k0 + l)w±l w±(l+2)  l l − k0 + − 1 w±(l−2) 2 2

acting on Wk±0 = w±l ,

l ∈ 2N0 .

3.1.5 Remark. π ˆk+0 is a lowest weight representation of lowest weight k0 , while − π ˆk0 is a highest weight representation of highest weight −k0 .

3.1. Representations of gJC

35

Tensorizing, the following types of representations of gJC already discussed in [Be3,4] and [BeB¨o] appear. We give the explicit formulas only for m > 0 and leave it to the reader to write down the other case explicitly. 3.1.6 Proposition. For any m > 0, the principal series representation m π ˆm,s,ν := π ˆSW ⊗π ˆs,ν ,

s ∈ C \ {Z + 1/2}, ν = ±1/2,

acts on Vm,s,ν := vj ⊗ wl ,

j ∈ N0 , l ∈ 2Z + ν + 1/2,

by Z0 (vj ⊗ wl ) Y+ (vj ⊗ wl )

= μvj ⊗ wl = vj+1 ⊗ wl

Y− (vj ⊗ wl ) Z(vj ⊗ wl )

= −μjvj−1 ⊗ wl = (j + l)vj ⊗ wl  1 1 1  = − vj+2 ⊗ wl + s+1+ l− vj ⊗ wl+2 2μ 2 2  1 1 1  = μj(j − 1)vj−2 ⊗ wl + s+1− l− vj ⊗ wl−2 2 2 2

X+ (vj ⊗ wl ) X− (vj ⊗ wl )

(μ = 2πm). There are similar formulas for m < 0, using the equations (3.3) from Proposition 3.1.4. This representation has as a cyclic vector the element v0 ⊗w1/2+ν characterized by Z0 (v0 ⊗ w1/2+ν ) = Z(v0 ⊗ w1/2+ν ) = Y− (v0 ⊗ w1/2+ν ) = (D− D+ )(v0 ⊗ w1/2+ν ) =

μ(v0 ⊗ w1/2+ν ) 1  + ν (v0 ⊗ w1/2+ν ) 2 0  1 2 s − (ν + 1)2 (v0 ⊗ w1/2+ν ) 4

with the “heat operators” D± = X± ±

1 2 Y 2μ ±

defined above. 3.1.7 Proposition. For any m > 0, the discrete series representation + m =π ˆSW ⊗π ˆk+0 , π ˆm,k

k = k0 + 1/2 ∈ Z,

36

3. Local Representations: The Real Case

acts on + Vm,k = vj ⊗ wl ,

j ∈ N0 , l ∈ 2N0 ,

by Z0 (vj ⊗ wl )

= μvj ⊗ wl

Y+ (vj ⊗ wl ) Y− (vj ⊗ wl )

= vj+1 ⊗ wl = −μjvj−1 ⊗ wl

Z(vj ⊗ wl ) X+ (vj ⊗ wl ) X− (vj ⊗ wl )

= (j + l + k)vj ⊗ wl 1 = − vj+2 ⊗ wl + vj ⊗ wl+2 2μ μ l 3 l = j(j − 1)vj−2 ⊗ wl − k− + vj ⊗ wl−2 2 2 2 2

and − m π ˆm,k =π ˆSW ⊗π ˆk−0 ,

k = k0 + 1/2 ∈ Z,

acts on − Vm,k = vj ⊗ w−l ,

j ∈ N0 ,

l ∈ 2N0 ,

by Z0 (vj ⊗ w−l ) =

μvj ⊗ w−l

Y+ (vj ⊗ w−l ) = Y− (vj ⊗ w−l ) =

vj+1 ⊗ w−l −μjvj−1 ⊗ w−l

Z(vj ⊗ w−l ) =

(j − l + 1 − k)vj ⊗ w−l l  1 l 1 X+ (vj ⊗ w−l ) = − vj+2 ⊗ w−l + −k− − 1 vj ⊗ w−(l−2) 2μ 2 2 2 μ X− (vj ⊗ w−l ) = j(j − 1)vj−2 ⊗ w−l + vj ⊗ w−(l+2) 2 (μ = 2πm). There are similar formulas for m < 0, using the equations (3.3) from Proposition 3.1.4. There is in both cases a cyclic vector v0 ⊗w0 of “dominant weight” characterized by π ˆ (Z0 ) = μ and π ˆ (Z)v0 ⊗ w0 = k v0 ⊗ w0 ,

π ˆ (Y− )v0 ⊗ w0 = π ˆ (X− )v0 ⊗ w0 = 0

+ for π ˆm,k resp.

π ˆ (Z)v0 ⊗ w0 = (1 − k)v0 ⊗ w0 , − for πm,k .

π ˆ (Y− )v0 ⊗ w0 = π ˆ (D+ )v0 ⊗ w0 = 0

3.1. Representations of gJC

37

3.1.8 Remark. There is a slight asymmetry in the naming of gJ -representam tions which comes from the fact that π ˆSW raises the weight by 1/2 (if m > 0): + − The distinguished vector of πˆm,k has weight k, while in π ˆm,k it has weight + 1 − k. One might therefore feel the temptation to index our π ˆm,k by another integer instead of k, for example by 1 − k or by k − 1. We have thought about this problem for hours, and come to the conclusion that the choice made offers some convincing advantages. For instance, the formula for the eigenvalue of the + Casimir operator C given below in Proposition 3.1.10 is the same for π ˆm,k and − π ˆm,k . Another, and perhaps more important, point is that, as will be seen in ∗ Section 4.1, the Jacobi forms in Jk,m resp. Jk,m correspond to representations + − πm,k resp. πm,k . Taking into account the above mentioned classification of genuine metaplectic representations, we arrive at the following classification of infinitesimal representations of GJ (which is in fact a classification of irreducible (gJC , K)-modules, K = SO(2), though we have not mentioned this terminology). 3.1.9 Theorem. Let m ∈ R∗ . The following is a complete list of the irreducible representations of gJC where Z0 acts by μ = 2πm and Z has integral weights. i) The principal series representations m π ˆm,s,ν := π ˆSW ⊗π ˆs,ν

for s ∈ C \ {Z + 1/2}, ν = ±1/2. ii) The positive discrete series representations + m π ˆm,k =π ˆSW ⊗π ˆk+0

for k = k0 + 1/2 ∈ Z. iii) The negative discrete series representations − m π ˆm,k =π ˆSW ⊗π ˆk−0

for k = k0 + 1/2 ∈ Z. The only equivalences between these representations are π ˆm,s,ν  π ˆm,−s,ν , all other representations are inequivalent.

38

3. Local Representations: The Real Case

In the following section we will decide which of these representations are unitarizable, thereby classifying the irreducible, unitary representations of GJ (R). There is another somewhat different approach to the determination of the representations of gJ proposed by Borho and exploited in [BeB¨ o]: If the universal enveloping algebra U (gJC ) is localized to U (gJC ) by dividing out the principal ideal generated by Z0 − μ, there is a Lie homomorphism 

: sl2 −→ U (gJC ) := U (gJC ) / (Z0 − μ)

given by X+ X− Z

→ D+ := X+ + (2μ)−1 Y+2

→ D− := X− − (2μ)−1 Y−2

→ Δ1 := Z + (2μ)−1 Δ0 ,

Δ0 := Y+ Y− + Y− Y+

(this can be verified by direct calculation or seen from [Bo] Lemma 3.4). As a consequence of the relation [h, sl2 ] = 0 in U (gJC ) (which is easy to check), there is an isomorphism ∼

i : U (hC ) ⊗ U (sl2 ) −→ U (gJC ) ,

U (hC ) = U (hC ) / (Z0 − μ).

which (also) gives an explanation that the representations πˆ of gJC with π ˆ (Z0 ) = μ = 2πm = 0 are of the type π = πm ⊗ π ˜ ,

V = Vm ⊗ W  ,

where hC acts as usual on Vm = ξj j∈N0 by Z0 ξj = μξj ,

Y+ ξj = ξj+1 ,

Y− ξj = −μjξj−1

and π ˜  is one of the representations given above in A) and B) below 3.1.4, but now thought of as representations of sl2 resp. as U (sl2 )-modules. In particular, this explains nicely the appearance of the “heat operators” D± and shows that C := D+ D− + D− D+ + (1/2)Δ21 is a Casimir operator for the representations π ˆ of gJC with π ˆ (Z0 ) = μ = 0. 3.1.10 Proposition. The image of the operator C lies in the center of U (gJC ) . Consequently C acts on the irreducible representations of gJC given in Theorem 3.1.9 by multiplication with a scalar λ. We have ⎧ ⎪ ⎪ ⎨ 1 (s2 − 1) for π ˆm,s,ν , 2 λ=    ⎪ 1 1 5 ⎪ ± ⎩ . k− k− for π ˆm,k 2 2 2 Proof: These are straightforward calculations.



3.2. Models for infinitesimal representations and unitarizability

39

3.2 Models for infinitesimal representations and unitarizability In this section we present models for the infinitesimal Schr¨odinger-Weil representation as well as for the principal and discrete series representations of the last section, and after that discuss the question of unitarizability of these representations. The infinitesimal Schr¨ odinger-Weil representation m We want to compute and characterize the derived representation of πSW on the J 2 Lie algebra g , acting on the space of smooth vectors S(R) ⊂ L (R). We have already used the result in Proposition 3.1.4. Similar formulas like the ones in Proposition 3.2.1 and 3.2.2 below also appear in Section 2.5 of [LV].

J is a real Lie group, and the exact sequence The two-fold cover G J −→ GJ −→ 1 1 −→ {±1} −→ G yields an isomorphism of Lie algebras ∼ J −→ g gJ .

We conclude from this that the infinitesimal representation m dπSW : gJ −→ gl(S(R)) m is really a homomorphism of Lie algebras, and the projectivity of πSW is no J longer visible on the infinitesimal level. For an element X ∈ g the operator m dπSW (X), often simply written as X, is given by  d  m (dπSW (X)f )(x) = (exp(tX)f )(x)  (f ∈ S(R), x ∈ R). dt t=0

For X = X1 + iX2 ∈ gJC with Xi ∈ gJ , we set m m m dπSW (X) = dπSW (X1 ) + i dπSW (X2 ).

3.2.1 Lemma. The infinitesimal Schr¨ odinger representation dπSm acts on S(R) by the following operators: P =

d dx

Q = 4πimx R = 2πim

1 d − 2πmx 2 dx 1 d Y− = + 2πmx 2 dx Z0 = 2πm Y+ =

(The elements P, Q, R, Y± , Z0 are defined in Sections 1.3 resp. 1.4). Proof: This is an easy exercise using the formula (2.1).



40

3. Local Representations: The Real Case

The infinitesimal Weil representation is more difficult to compute. First of all we need an explicit description of the cocycle λ defining the metaplectic group. For      ab a b  M= , M = , cd c d two elements of SL(2, R), it is given by λ(M, M  ) = (x(M ), x(M  ))(−x(M )x(M  ), x(M M  )),

(3.4)

where ( , ) denotes the Hilbert symbol and  c if c = 0, x(M ) = d if c = 0. (see [Ge2], p. 13–14). Define  −1 if t > 0, ρ(t) = 1 if t ≤ 0. Then, using the above description of λ, one can check that we have the following one-parameter subgroups R → Mp(R) corresponding to the Lie algebra elements F, G, H ∈ gJ :    1t φF (t) = ,1 , 01    10 φG (t) = , ρ(t) , t1  t   e 0 φH (t) = ,1 . 0 e−t m m From this it is very easy to calculate dπW (F ) and dπW (H), but there is a m small difficulty in determining dπW (G). The best thing is to use the Fourier transformation F . One computes m F ◦ dπW (G) ◦ F −1 = −2πimx2 ,

and derives from this the formula in the following lemma. 3.2.2 Lemma. We have the following formulas for the infinitesimal Weil repm resentation dπW acting on S(R): 1 1 d 1 d2 + x − πmx2 − 4 2 dx 16πm dx2 2 i d 1 1 d 1 d2 G= X− = + x + πmx2 + 2 8πm dx 4 2 dx 16πm dx2 2 1 d 1 d H = +x Z = 2πmx2 − 2 dx 8πm dx2 (The elements F, G, H, X± , Z are defined in Sections 1.3 resp. 1.4). F = 2πimx2

X+ =

3.2. Models for infinitesimal representations and unitarizability

41

We want to describe the infinitesimal Schr¨odinger-Weil representation in a purem J , dely algebraic way. Observe that πSW , regarded as a representation of G  which is a two-fold cover of composes over the maximal compact subgroup K, SO(2). Hence the element F − G, which spans the Lie algebra of this maximal  compact subgroup, acts on an irreducible K-module by ik, where k ∈ 12 Z. In J m  other words, Z ∈ gC acts on irreducible K-submodules of πSW by half-integers. The subspace V of K-finite vectors therefore allows a decomposition  V = Vk with Vk = {v ∈ V : Zv = kv}. (3.5) k∈ 12 Z

Moreover, every Vk is at most one-dimensional, because Zv = kv is a second order differential equation for the Schwartz function v, and at most one of its m solutions will lie in S(R). We further observe that πSW is a lowest (highest) weight representation if m > 0 (resp. m < 0), i.e., there is a vector v ∈ V such that X− v = Y− v = 0 (resp. X+ v = Y+ v = 0). The lowest (highest) weight 2 vector is given by e−2π|m|x . Now there is the following purely algebraic result. 3.2.3 Proposition. Let m ∈ R∗ . There is exactly one lowest (resp. highest) weight representation of gJC on a space V which admits a decomposition (3.5) such that dim Vk ≤ 1 for all k, and such that Z0 acts by 2πm. In the lowest weight case, this representation has the space V = vj j∈N0 and is given by Z0 vj = μvj Y+ vj = vj+1 Y− vj = −μjvj−1

 1 Zvj = j + vj 2 1 X+ vj = − vj+2 2μ μ X− vj = j(j − 1)vj−2 2

(3.6)

(v−1 = v−2 = 0 understood). In the highest weight case, this representation has the space V = v−j j∈N0 and acts by Z0 v−j = μv−j Y− v−j = v−(j+1) Y+ v−j = μjv−(j−1)

 1 Zv−j = − j + v−j 2 1 X− v−j = v−(j+2) 2μ μ X+ v−j = − j(j − 1)v−(j−2) 2

(3.7)

(with v1 = v2 = 0 understood). 3.2.4 Remark. In the lowest weight case, this representation may equivalently be characterized by

42

3. Local Representations: The Real Case a) the existence of a lowest weight vector v0 of weight 1/2, i.e. with Z0 v0 = μv0 ,

Zv0 = (1/2)v0 ,

b) the relation D+ = X+ +

1 2 2μ Y+

Y− v0 = X− v0 = 0

and

= 0.

The proof of the above proposition (to be found in [Be1]) is straightforward: Starting with a vector v0 of lowest weight k0 , one looks at vj := Y+j v0

and

j  v2j := X+ v0 ,

 and using the Lie algebra relations verifies, that v2j is a multiple of v2j if and 2 only if 2μX+ and Y+ have the same action and if k0 = 1/2 holds. 

From this proposition and the considerations before it, we see the following. 3.2.5 Corollary. m i) If m > 0, the infinitesimal Schr¨ odinger-Weil representation dπSW is given 2 by the formulas (3.6), if in the space of K-finite vectors of L (R) we set 2 v0 = e−2πmx and vj+1 := Y+ vj for j ≥ 0. m ii) If m < 0, then dπSW is given by the formulas (3.7), if in the space of 2 2 K-finite vectors of L (R) we set v0 = e2πmx and v−(j+1) := Y− v−j for j ≥ 0.

3.2.6 Remark. Note that Proposition 3.2.3 actually yields more representations than the infinitesimal Schr¨ odinger-Weil representations. But the additional ones do not come from unitary representations of the group. The right half of the formulas (3.6) and (3.7) is nothing but the infinitesimal m m Weil representation, because πW is just the restriction of πSW to sl2 . We see m that π ˆW decomposes into two irreducible components, m m+ m− π ˆW =π ˆW ⊕π ˆW , m+ where π ˆW , the positive or even Weil representation, acts on the space spanned m− by the vj with even indices, and π ˆW , the negative or odd Weil representation, acts on the space spanned by the vj with odd indices (note that the sign in m± the symbol π ˆW has nothing to do with the representation being of highest or m± lowest weight). The naming even and odd comes from the fact that if πW is 2 realized on the space of K-finite vectors in L (R), then it consists entirely of even resp. odd functions. See also Proposition 2.5.1.

3.2.7 Corollary. m+ i) Let m > 0. The infinitesimal even Weil representation π ˆW acts on the space vj j∈2N0 by the right half of the formulas (3.6). Hence it is a lowest weight representation of lowest weight 1/2. The odd Weil representation m− π ˆW acts on vj j∈2N0 +1 and is a lowest weight representation of lowest weight 3/2.

3.2. Models for infinitesimal representations and unitarizability

43

m+ ii) Let m < 0. Then π ˆW acts on the space v−j j∈2N0 by the right half of the formulas (3.7). It is a highest weight representation of highest weight m− −1/2. The odd Weil representation π ˆW acts on v−j j∈2N0 +1 , and is a highest weight representation of highest weight −3/2.

From this corollary we immediately see how the even and odd Weil representations fit into the classification of metaplectic representations given in the preceding section: 3.2.8 Corollary. i) For m > 0 we have + m+ π ˆW =π ˆ1/2 ,

+ m− π ˆW =π ˆ3/2 .

ii) For m < 0 we have − m+ π ˆW =π ˆ1/2 ,

− m− π ˆW =π ˆ3/2 .

A model for principal and discrete series representations The representations (ˆ π , V ) enumerated in the last section may be realized by the action of the left invariant differential operators   d LX φ(g) = φ(g exp tX) dt t=0 on functions φ living on GJ (R). Here the S-coordinates (x, y, θ, p, q, ζ) seem more appropriate than the EZ-coordinates. We use the notation φ(g) = φS (x, y, θ, p, q, ζ). The elements of the Lie algebra gJC may be viewed as the following differential operators on such functions φ: LZ0 LY± LX± LZ

= 2πζ∂ζ ,

  = (1/2)y −1/2 e±iθ ∂p − (x ∓ iy)∂q − (p(x ∓ iy) + q)2πiζ∂ζ ,   = ±(i/2)e±2iθ 2y(∂x ∓ i∂y ) − ∂θ , = −i∂θ .

We remind the reader that for these coordinates we have x, p, q ∈ R,

y ∈ R>0 ,

θ ∈ R/2πZ,

ζ ∈ S1.

In particular, to define a function φ on GJ (R), φS has to be periodic in θ with period 2π. We will later on come up with functions φS with period 4π in θ.

44

3. Local Representations: The Real Case

These functions then will be thought of as functions on the metaplectic cover  J (R) of GJ (R) (resp. on SL(2,  R) if only x, y, θ appear). Having this in mind, G we will often simply skip the suffix “S” and write φ(x, y, θ, p, q, ζ). Now, here is the first model. 3.2.9 Proposition. a) The space Vm,s,ν = vj ⊗ wl  for the principal series representation π ˆm,s,ν is realized by vj ⊗ wl = φm,s,j,l ,

j ∈ N0 , l ∈ 2Z + ν + 1/2

with φm,s,j,l (g) = ζ m ei(j+l)θ y (s+3/2)/2 em (pz)ψj (py 1/2 ). + + b) The space Vm,k = vj ⊗ wl  for the discrete series representation π ˆm,k is realized by

vj ⊗ wl = cl φ+ m,k,j,l

j ∈ N0 , l ∈ 2N0

with m i(k+j+l)θ k/2 m φ+ y e (pz)ψj (py 1/2 ), m,k,j,l (g) = ζ e

cl = (k − 1/2)(k + 1/2) · · · (k − 1/2 + l + 1). − − c) The space Vm,k = vj ⊗ wl  for the discrete series representation π ˆm,k is realized by

vj ⊗ wl = cl φ− m,k,j,l

j ∈ N0 , l ∈ −2N0

with m i(1−k+j+l)θ k/2 m φ− y e (pz)ψj (py 1/2 ), m,k,j,l (g) = ζ e

cl = (k − 1/2)(k − 3/2) · · · (k − 1/2 − (l − 1)). In all cases ψj (j ∈ N0 ) is a family of polynomials in one variable, say u, with ψ0 (u) = 1,

ψj+1 = (1/2)ψj − 2μuψj



and ψj − 4μuψj + 4μjψj = 0,

i.e., related to the Hermite polynomials Hj (v) by the substitution u = (2μ)−1/2 v. Proof: The functions given here arise from the construction of representations π of GJ (R) by the induction process to be described below. Beside this, a direct computation shows that application of the differential operators LX produces precisely the exact relations between the vj ⊗ wl required by Propositions 3.1.6 and 3.1.7 in the last section:

3.2. Models for infinitesimal representations and unitarizability

45

As is easily seen, the functions φ given in the proposition are products of functions vj = φ˜m,1/2,j with φ˜m,1/2,j (˜ g) = ζ m y 1/4 ei(j+1/2)θ em (pz)ψj (py 1/2 ),

j ∈ N0 ,

˜ l with and wl = Ψ ˜ l (˜ Ψ g) = y (s+1)/2 ei(l−1/2)θ ,

l ∈ 2Z + ν + 1/2,

in case a)

˜ ±l with resp. w±l = c±l Ψ ˜ ±l (˜ Ψ g) = y k/2−1/4 e±i(k−1/2+l)θ ,

l ∈ N0 ,

in cases b), c).

˜ l live on the metaplectic cover, but their Obviously the factors φ˜m,1/2,j and Ψ product is a function on GJ (R). We have, for instance, ˜ l = (1/2)cl (2k − 1 + l)Ψ ˜ l+2 LX+ Ψ and by the relations prescribed by π ˆk+ this has to be ˜ l+2 , cl+2 Ψ explaining the formula in the proposition given for the coefficients cl , l ∈ N0 . For −l ∈ N0 the computation goes the same way. In case a) there is no need of a “normalizing” constant on behalf of the symmetry of the relations for πs,ν in the + and − direction. Similarly, we get by application of the differential operator LX+ to vj = φm,1/2,j LX+ vj = (1/2)(j + 1 − 4μp2 y + py 1/2 ψj /ψj )e2iθ vj . m By the relations for πSW one has also

LX+ vj = −(2μ)−1 vj+2 . Thus, we get ψj+2 = (4μ2 p2 y − (j + 1)/2)ψj − py 1/2 ψj . And from LY+ vj = (1/2)(2μpy + (1/2)y 1/2 ψj /ψj )y −1/2 eiθ vj = vj+1 , LY+ vj = (1/2)(ψj /ψj )e−iθ vj = −μjvj−1 we deduce ψj+1 = (1/2)ψj − 2μpy 1/2 ψj

and ψj = −2μjψj−1 .

46

3. Local Representations: The Real Case

With u = py 1/2 both equations combine to 

ψj − 4μuψj + 4μjψj = 0 and this is consistent with the equation above coming from LX+ vj and the corresponding expression for LX− vj to be treated in the same way. The unitarizability question As in the general theory, we can decide here which of the given infinitesimal representations π ˆ listed in Theorem 3.1.9 may come from a unitary representation π of GJ (R). As, for instance, in [La] p. 122 one easily deduces that for a unitary (π, V ) with scalar product  ,  we have dπ(X)v, v˜ + v, dπ(X)˜ v = 0

for all v, v˜ ∈ V

and dπ(X± ) = −dπ(X∓ )∗ ,

dπ(Y± ) = −dπ(Y∓ )∗ ,

if X± and Y± are the elements of gJC as above. Using this for the representation m πSW from Proposition 3.1.4 we see that for V = vj j∈N0 to carry a scalar product  ,  we necessarily have Y+ vj , vj+1  = −vj , Y− vj+1 , i.e., with μ = 2πm, vj+1 2 = (j + 1)¯ μvj 2 . Thus we recover the following result, which may also be seen by inspection of the usual formulas for the Schr¨ odinger representation as a representation on L2 (R). m 3.2.10 Remark. The Schr¨ odinger-Weil representation π ˆSW is unitarizable.

In the same manner, we get for the principal series representation πˆs,ν of sl2 from ˆ + wl , wl+2  = −wl , X ˆ − wl+2  X the relation (s + 1 + l − 1/2)wl+2 2 = −(¯ s + 1 − (l + 2 − 1/2))wl 2 i.e. for l = 0 and s¯ = 1/2 the condition s + 1/2 >0 −¯ s + 1/2 which demands for s real and s2 < 1/4

or

s ∈ iR.

47

3.2. Models for infinitesimal representations and unitarizability

For π ˆk+0 we come up with  l + 2 l+2 wl+2 2 = k0 + − 1 wl 2 2 2 and for π ˆk−0 with

 l + 2 l+2 k0 + − 1 w−l 2 2 2 showing that in both cases we have to require k0 ≥ 1/2. All this put together gives the following result. w−(l+2) 2 =

3.2.11 Proposition. The representation π ˆm,s,ν is unitarizable for m > 0 ± and s ∈ iR or s ∈ R with s2 < 1/4, and π ˆm,k is unitarizable for m > 0 and k ≥ 1. m Proof: As π ˆSW is unitarizable by the last remark, a reasoning like in Proposition m 5.9.1 below tells us that π ˆ =π ˜⊗π ˆSW is unitarizable exactly if π ˜ is. By the way, we can see this directly: As we have by Proposition 3.1.6

ˆ + (vj ⊗ wl ) = (1/2)(s + 1 + (l − 1/2))vj ⊗ wl+2 D and ˆ − (vj ⊗ wl+2 ) = (1/2)(s + 1 − (l + 2 − 1/2))vj ⊗ wl , D we come up for l = 0 with the condition s + 1/2 > 0, −¯ s + 1/2 exactly as above for the sl2 -case. The scalar products defined for the generating elements vj ⊗ wl make the space spanned by these elements a pre-Hilbert space which may be completed to a Hilbert space.  It is to be remarked here, that π ˆm,s,ν , having infinite dimensional subspaces of fixed weight, is not admissible. We can summarize and give the following classification of irreducible unitary representations of GJ (R): 3.2.12 Theorem. An irreducible unitary representation π of GJ (R) with central character e2πimx , m ∈ R∗ , is infinitesimally equivalent to a continuous series representation π ˆm,s,ν ,

s ∈ iR,

or a complementary series representation π ˆm,s,ν ,

s ∈ R, s2
0

iff m < 0

W n,r exists if

mN > 0

mN > 0

mN < 0

C operates by

1 2 2 (s

− 1)

1 2 (k

− 12 )(k − 52 )

1 2 (k

− 12 )(k − 52 )

Table 3.1: Infinitesimal unitary representations of gJC

3.3 Representations induced from B J In the SL(2)-theory, the representations of the principal and the discrete series may be realized as representations induced from the Borel subgroup NA resp., as subrepresentations of such. Now, the right way to carry this over to the Jacobi group is to induce from the group B J = AZN J = {b = n ˜ (x, q, ζ)t(y) : x, q ∈ R, y ∈ R>0 , ζ ∈ S 1 } We denote by χm,s the character of B J given by χm,s (˜ n(x, q, ζ)t(y)) := ζ m y s/2 ,

(3.8)

3.3. Representations induced from B J

49

which is unitary exactly for s ∈ iR, and apply the machinery described in 2.1. Because of the commutation rule t(y, p)˜ n(x, q, ζ) = n ˜ (x , q  , ζ  )t(y, p) with x = xy,

q  = qy 1/2 + pxy,

ζ  = ζe(2pqy 1/2 + p2 xy),

it is easily seen that we have dr b = dx dq

dζ dy ζ y

and

ΔB J (b) = y 3/2 .

The decomposition ˆ t˜(y)ˆ g = n(x)t(y)r(θ)(p, q, ζ) = n ˜ (x, qˆ, ζ) r (θ, pˆ) with pˆ = py 1/2 ,

qˆ = a + px,

ζˆ = ζe(p(px + q)),

shows that the Borel section s used to construct the induced representation may be chosen here to be K = B J \GJ  (θ, pˆ) → r(θ, pˆ) ∈ GJ such that we come out with the quasi-invariant measure μs given by dμs = dˆ p dθ = y 1/2 dp dθ.

(3.9)

Now the prescription given in 2.1 produces the induced representation J

πm,s := indG B J χm,s given by right translation ρ on the space Hm,s of measurable functions φ on GJ with i) ii)

(s+3/2)/2

φ(b0 g) = y0 ζ0m φ(g)  φ2Hm,s = |φ(r(θ, p)| ˆ 2 dθ dˆ p < ∞. K

Hm,s is a Hilbert space with the scalar product.  φ1 , φ2  = φ1 (r(θ, pˆ))φ2 (r(θ, p)dθ ˆ dˆ p K

The decomposition above shows that these functions φ are of the type i’) ii’)

φ(g) = y (s+3/2) ζ m em (p(px + q))ϕ(θ, py 1/2 )  |ϕ(θ, v)|2 dθ dv < ∞.

(3.10)

50

3. Local Representations: The Real Case

Remembering that the space of functions ϕ with (ii’) can be spanned by functions of the type 2

ϕ(θ, v) = eilθ e−v Hj (v),

l ∈ Z,

j ∈ N0 ,

with the Hermite polynomials Hj (v) we obtain after the substitution v = (2πm)1/2 u,

Hj ((2πm)1/2 u) =: ψj ,

the following statement. 3.3.1 Proposition. For each integer m > 0 and each s ∈ C there is a representation πm,s of GJ (R) given by right translation on the Hilbert space Hm,s spanned by the family of functions φm,s,j,l (g) = ζ m y (s+3/2)/2 eilθ em (pz)ψj (py 1/2 ),

l ∈ Z,

j ∈ N0 .

J

By the general theory πm,s = indG B J χm,s is unitary if χm,s is, i.e., for s ∈ iR. The question of irreducibility will be answered by the following comparison with the infinitesimal results. As the representation of GJ on Hm,s is given by right translation πm,s (g0 )φ(g) = φ(gg0 ), its derived representation dπm,s is given by   d dπm,s (X)φ(g) = φ(g expt X)  dt t=0

i.e., produces the operators realizing gJ resp. gJC . As we can identify the functions spanning the representation spaces in Proposition 3.2.9 with those appearing here by φm,s,j,l

=

φm,s,j,j+l

, l ∈ 2Z ± 1

for π ˆm,s,ν

φ+ m,k,j,l

=

φm,k−3/2,j,k+j+l

, l ∈ 2N0

+ for π ˆm,k

φ− m,1−k,j,l

=

φm,k−3/2,j,1−k+j+l

, l ∈ 2N0

− for π ˆm,k

we may state: 3.3.2 Corollary. The family {φ } of functions above span invariant subspaces of Hm,s resp. Hm,k−3/2 and thus give models for the representations πm,s,ν resp. ± πm,k by right translations. 3.3.3 Remark. It is perhaps not without interest that by the same reasoning m the Schr¨ odinger-Weil representation πSW is realized for m ∈ N by the functions ˜ J (R). And for the φm,−1,j,−(j+1/2) , j ∈ N0 living on the metaplectic cover G m “mirror image” πSW with m < 0 we come up with a space spanned by the family of functions φ¯m,−1,j,j+1/2 , j ∈ N0 , complex conjugate to those above, and with −m ∈ N.

3.4. Representations induced from K J and the automorphic factor

51

3.4 Representations induced from K J and the automorphic factor We have the Iwasawa decomposition GJ (R) = N J (R)AJ (R)K J (R), the projection pr : GJ (R) −→ GJ (R)/K J (R) = X with x0 = (i, 0) ∈ X and τ = x + iy, z = pτ + q, given by g = ((u(x)t(y)r(θ)), (p, q, κ)) → g(x0 ) = (τ, z), and the natural section s : X −→ GJ (R), for x = (τ, z) = (x + iy, pτ + q) given by s(x) = (p, q, 0)n(x)t(y) =: gx . Then it is a very natural thing to take the representation χ∗m,k of K J (R) = {r(θ, κ) = r(θ)κ : θ, κ ∈ R} for m, k ∈ Z given by χ∗m,k (r(θ, κ)) = eikθ em (κ)

(3.11)

and to induce from here to GJ (R), i.e., to look at the representation GJ (R)

∗ πm,k := indK J (R) χ∗m,k

given by left translation λ(g0 )φ(g) = φ(g0−1 g) on the space Bχ∗m,k of measurable functions φ on GJ (R) with φ(gr(θ, κ)) = φ(g)χ∗m,k (r(θ, κ)) and

 |φ((p, q, 0)n(x)t(y))|2 dμx < ∞

for dμx = y −2 dx dy dp dq.

for all g ∈ GJ , r(θ, κ) ∈ K J ,

52

3. Local Representations: The Real Case

There is an equivalent model for this representation given by π ∗ (g0 )F (x) = F (g0−1 (x))eikθ0 em (κ0 ) for

F ∈ L2 (X, dμx ).

These considerations may be refined considerably by the following standard procedure leading to the canonical automorphic factors, which is a special case of a general discussion by Satake ([Sa1] or [Sa2] pp. 118-119) on the basis of older work (see for instance Matsushima and Murakami [MM] and Borel and Baily [BB] 1.9). As already remarked in Section 1.4, GJ (R) is a group of Harish–Chandra type. More precisely, we have the following situation. The complexified Lie algebra gJC of GJ is given by gJC = p0 ⊕ p+ ⊕ p−

with p± = X± , Y± C ,

p0 = Z, Z0 C

and (see 1.4) [p0 , p± ] ⊂ p± ,

p+ = p− ,

i.e. Satake’s condition (HC 1). And for the complex groups P± = exp p± ,

P0 = exp p0 ,

we have Satake’s condition (HC 2), i.e. GJ → P+ P0 P− → GJC = SL(2, C)  H(C).

(3.12)

This will become more transparent if we twist by a “partial Cayley transform” ˜ which in the symplectic part gives the usual bounded realization of the C, upper half plane H by the unit disc D : We take the matrices     1 1 −i i i C= and C −1 = −1 1 2i 1 i which induce a biholomorphic map C −1

H −→ D, τ

∗

−→ τ = C

(3.13) −1

τ −i (τ ) = , τ +i

and twist G = SL(2, R) into G∗

= =

C −1 GC = SU(1, 1)     αβ ∗ M = ¯ : α, β ∈ C, |α|2 − |β|2 = 1 βα ¯

with

 2α = a + d + (b − c)i,

2β = a − d − (b + c)i

for

M=

 ab . cd

3.4. Representations induced from K J and the automorphic factor

53

Moreover GJ (R) = SL(2, R)  H(R) is twisted by C˜ into     αβ ∗J −1 J ∗ ˜ ˜ G (R) = C G (R)C = g = ¯ (X ∗ , κ) βα ¯ with X ∗ = X C˜ = (λi − μ, λi + μ)

for X = (λ, μ).

In particular, K = SO(2) × R = {r(θ, κ) : θ, κ ∈ R} is twisted into  −1   γ 0 ∗J K = (0, 0, κ) = d(γ −1 , κ) : γ ∈ S 1 , κ ∈ R 0 γ J

(3.14)

with γ = e−iθ . And we get a biholomorphic map C˜ −1 : H × C → D × C given via the commutativity of the diagram J G ⏐ ⏐

H × C = GJ /K J (τ, z)

−→

∗J G ⏐ ⏐

−→ G∗J /K ∗J = D × C = DJ τ − i z 

−→ (τ ∗ , z ∗ ) = , . τ +i τ +i

We will use the notation  1 τ ∗   J N+ = (0, z ∗ , 0) = n(τ ∗ , z ∗ ) : τ ∗ , z ∗ ∈ C 0 1 and J N−

 1 0   = (v, 0, 0) = n− (u, v) : u, v ∈ C . u1

Then we have the following central fact. 3.4.1 Remark. Each g ∗ = M ∗ (Y ∗ , κ∗ ) ∈ G∗J may be uniquely decomposed as ∗ ∗ ∗ g ∗ = g+ g0 g−

with ∗ g− g0∗

= =

n− (u, v), u = c∗ /d∗ , v = λ∗ − μ∗ c∗ /d∗ d(γ −1 , κ), γ = d∗ , κ = κ∗ + μ∗ (λ∗ − μ∗ c∗ /d∗ )

∗ g+

=

n(τ ∗ , z ∗ ), τ ∗ = b∗ /d∗ , z ∗ = μ∗ /d∗ .

(3.15)

(3.16)

54

3. Local Representations: The Real Case

Since for M + ∈ SU(1, 1) we have a∗ = β and d∗ = α ¯ with |α|2 − |β|2 = 1, here is |τ ∗ | = |β/α ¯ | < 1. From this remark it may be concluded that G∗J /K ∗J g∗

∼

−→ D × C =: DJ ,

−→ (τ ∗ , z ∗ ),

is a Harish–Chandra realization of H × C as a “partially bounded domain”. For the operation of G∗ on DJ we have 3.4.2 Remark. An element g ∗ = M ∗ (X ∗ , κ∗ ) acts on (τ ∗ , z ∗ ) ∈ D by  z ∗ + λ∗ τ ∗ + μ∗  g ∗ (τ ∗ , z ∗ ) = M ∗ (τ ∗ ), . c∗ τ ∗ + d∗

(3.17)

This easily comes out by the decomposition g ∗ n(τ ∗ , z ∗ ) = (g ∗ n)+ (g ∗ n)0 (g ∗ n)− . We get (g ∗ n)+ = n(˜ τ ∗ , z˜∗ ) with a∗ τ ∗ + b ∗ τ˜∗ = M ∗ (τ ∗ ) = ∗ ∗ , c τ + d∗

z˜∗ =

z ∗ + λ∗ τ ∗ + μ∗ c∗ τ ∗ + d∗

and (g  n)0 = d(γ −1 , κ), where γ

=

κ =

(c∗ τ ∗ + d∗ ), (z ∗ + λ∗ τ ∗ + μ∗ )2 κ ∗ − c∗ + λ∗2 τ ∗ + 2λ∗ z ∗ + λ∗ μ∗ . c∗ τ ∗ + μ∗

The element (g ∗ n)0 = d(γ −1 , κ) ∈ KC∗J is called the canonical automorphic factor and denoted by J ∗ (g ∗ , (τ ∗ , z ∗ )). Its significance becomes clear if we define a unitary representation of G∗J by a method very close to the induction procedure described in 2.1, and which, as in the usual case (see for instance Knapp [Kn] p. 150), establishes the “holomorphic discrete series”: J We denote by B− the complex group     a 0 J ∗ B− = b= (λ, 0, κ) : a ∈ C , c, λ, κ ∈ C , c a−1

(3.18)

3.4. Representations induced from K J and the automorphic factor

55

and for m, k ∈ N0 by ξm,k its character given by ξm,k (b) = ak em (κ).

(3.19)

As we have J J G∗J (R) → G∗J (R)B−  DJ KcJ N− → GJ (C)

with a complex structure on the term in the middle, we get 3.4.3 Proposition. A unitary representation πm,k of G∗ J (R) is given by leftJ translation on the space Vξm,k of holomorphic functions φ living on G∗J B− with the properties J J φ(gb) = ξm,k (b)φ(b) for all g ∈ G∗J B− and b ∈ B− ,  φ2 := |φ(g ∗ )|2 dg ∗ < ∞,

i) ii)

G∗J (R)/Z

where dg is a Haar measure which will be normalized a bit further down. It is not evident but true (and can be proved by the usual method as for instance in [La] pp. 182-184) that Vξm,k is a Hilbert space and that this representation is irreducible. As in the usual procedure described in 2.1, there is a second realization of this representation given by restriction of the functions φ to DJ . Writing φ(g ∗ ) = φ(τ ∗ , z ∗ , γ, κ, u, v) for g ∗ = n(τ ∗ , z ∗ )d(γ −1 , κ)n(u, v), we put F (τ ∗ , z ∗ ) := φ(τ ∗ , z ∗ , 1, 0, 0, 0) and get a holomorphic function F defined on DJ . Obviously the map r : φ → F from Vξm,k to holomorphic functions on DJ is for g ∗ given as above inverted by r∗ : F −→ φF ,

φF (g) = γ −k em (κ)F (τ ∗ , z ∗ ).

r is an interwining operator if the action of g0∗ ∈ G∗J on F is defined by ∗ (πm,k (g0−1 )F )(τ ∗ , z ∗ ) = jk,m (g0∗ , (τ ∗ , z ∗ ))F (g0∗ (τ ∗ , z ∗ )),

where jk,m (g ∗ , (τ ∗ , z ∗ )) = ξm,k (J ∗ (g ∗ , (τ ∗ , z ∗ )))   (z ∗ + λ∗ τ ∗ + μ∗ )2 ∗2 ∗ ∗ ∗ ∗ ∗ = (c∗ τ ∗ + d∗ )−k em κ∗ − c∗ + λ τ + 2λ z + λ μ c∗ τ ∗ + d∗

56

3. Local Representations: The Real Case

r is unitary if we define the norm of F as follows. We have as a G∗J –invariant measure on DJ dμD = (1 − (x∗2 + y ∗2 ))−3 dx∗ dy ∗ dξ ∗ dy ∗ for τ ∗ = x∗ + iy ∗ , τ ∗ = ξ ∗ + iy ∗ (as can be seen for instance by pullback from the GJ -invariant measure on H × C). We normalize such that we have  ∗ dg = dμDJ dk with dk = 1. (3.20) K ∗J /Z

The decomposition ∗ ∗ G∗J  g ∗ = g(τ ∗ ,z ∗ ) k ,

∗ ∗ ∗ k ∗ ∈ K ∗J , g(τ ∗ ,z ∗ ) (0, 0) = (τ , z ),

and the unitarity of ξm,k allow to write  2 φF  = |φβ (g ∗ )|2 dg G∗J /Z



|F (g ∗ (0, 0))|2 |jk,m (g, (0, 0))|2 dg

= G∗J /Z



=

|F (τ ∗ , z ∗ )|2 |jk,m (g(τ ∗ ,z∗ ) , (0, 0))|2 dμDJ .

DJ

This expression will be taken as F 2 , and a small computation shows that we may also write   τ¯∗ z ∗2 − τ ∗ z¯∗2  F 2 = |F (τ ∗ , z ∗ )|2 e−m (1 − |τ ∗ |2 )k−3 dx∗ dy ∗ dξ ∗ dy ∗ . 1 − |τ ∗ |2 DJ

To summarize, we get another model for the representation of the last proposition. 3.4.4 Corollary. We get another realization of the representation πm,k of G∗J (R) by the action ∗ (πm,k (g0∗−1 )F )(τ ∗ , z ∗ ) = jk,m (g0∗ , (τ ∗ , z ∗ ))F (g0∗ (τ ∗ , z ∗ ))

with ∗ jk,m (g ∗ , (τ ∗ , z ∗ ))

  (z ∗ + λ∗ τ ∗ + d∗ )2 ∗2 ∗ ∗ ∗ ∗ ∗ = (c∗ τ ∗ + d∗ )−k em κ∗ − c + λ τ + 2λ z + λ μ c∗ τ ∗ + d∗

for holomorphic functions F on DJ = D × C with F  < ∞.

3.4. Representations induced from K J and the automorphic factor

57

Now, via GJ  g −→ C˜ −1 g C˜ = g ∗ ∈ G∗J the representations of G∗J may be “untwisted” to get representations of GJ . We will do this and at the same time go back by the Cayley transformation from DJ to the biholomorphic equivalent space X = H × C = GJ (R)/K J (R). Denoting as above ∗ 2 dμDJ ,m,k = |jk,m (g(τ ∗ ,z ∗ ) , (0, 0))| dμD J

and L2hol (DJ , dμDJ ,m,k ) for the space of square integrable holomorphic functions F on DJ , we associate to F a function e−1 m,k F on X given by −1 ˜ C˜ −1 (τ, z))−1 . (Cm,k F )(τ, z) = F (C˜ −1 (τ, z))jk,m (C,

(3.21)

This map is inverted by ˜ ∗ , z ∗ ))jk,m (C, ˜ (τ ∗ , z ∗ )). (Cm,k f )(τ, z ∗ ) = f (C(τ Here we have used the fact that the definition of the automorphic factor G∗J × DJ −→ C given above also works for any pair (g ∗ , (τ ∗ , z ∗ )), g ∗ ∈ GJ (C), (τ ∗ , z ∗ ) ∈ C2 −1 ∗ ∗J such that g ∗ g(τ ∗ ,z ∗ ) ∈ Kc . Now, by the maps Cm,k and Cm,k the representation ∗ ∗ ∗J πm,k , for g ∈ G given by ∗ πm,k (g ∗−1 )F ((τ ∗ , z ∗ )) = jk,m (g ∗ , (τ ∗ , z ∗ ))F (g, (τ ∗ , z ∗ )),

is intertwined to a representation πm,k of GJ given by πm,k (g −1 )f (τ, z) = jk,m (g, (τ, z))f (g(τ, z)) with ˜ C˜ −1 (τ, z)) jk,m (g, (τ, z)) = jk,m (C˜ −1 g C, ˜ C˜ −1 g(τ, z))jk,m (C, C −1 (τ, z))−1 . ·jk,m (C, Here the function jk,m : GJ × X −→ C is defined by the same prescription as j : G∗ × DJ → C which comes out if one uses the cocycle condition of the automorphic factor to reduce the right hand side.

58

3. Local Representations: The Real Case

Cm,k maps L2hol (DJ , dμDJ ,m,k ) isomorphically into the space L2hol (X, dμx,m,k ) of holomorphic functions on X = H × C, square integrable with respect to dμx,m,k

= |jk,m (g(τ,z) , (i, 0))|2 dμx = e−4πmη

2

/y k−3

y

dx dy dξ dη.

Here we use again a decomposition GJ (R)  g = g(τ,z) k where k ∈ K J and ' 1/2 −1/2  ( y xy 1/2 −1/2 −1/2 g(τ,z) = , (py , pxy + qy , 0) 0 y −1/2 is such that g(τ,z) (i, 0) = (τ, z). Summing up, we have the 3.4.5 Proposition. A unitary representation πm,k of GJ (R) is given by πm,k (g −1 )f (τ, z) = f (g, (τ, z))jk,m (g, (τ, z)) for f ∈ L2hol (X, dμx,m,k ). By the usual lifting f −→ φf

with φf (g) = f (g(i, 0))jk,m (g, (i, 0))

we get a model of this representation by functions φ living on GJ (R) : 3.4.6 Corollary. A unitary representation πm,k of GJ (R) is given by πm,k (g0−1 )φ(g) = φ(g0 g),

g0 , g ∈ GJ (R),

for functions φ of the type φ(g) = em (k)eikθ y k/2 em (pz)f (g(i, 0)), where here g = (M, p, q, κ) ∈ GJ in the S-coordinates and f ∈ L2hol (H × C, dμH×C,m,k ), or equivalently f is of the type τ − i z  τ + i k m  z  f (τ, z) = F , e − τ +i τ +i 2i τ +i with F ∈ L2hol (DJ , dμDJ ,m,k ). Obviously, we have arrived at a subrepresentation of the representation GJ (R)

∗ indK J (R) ξm,k

from the beginning of this section.

3.5. Differential operators on X = H × C

59

3.5 Differential operators on X = H × C After having provided an “explanation” of the canonical automorphic factor jk,m (g, (τ, z)) for GJ (R) we will interrupt for a moment the discussion of the models for the representations π of GJ (R) and study the algebras of in- and covariant differential operators on X = H × C. We use again the S-coordinates (τ = x + iy, z = pτ + q) ∈ H × C, and by the usual misuse of notation write f (x, y, p, q) = f (τ, τ¯, z, z¯) for infinitely differentiable C–valued functions f living on H×C. We abbreviate jk,m (g) := jk,m (g, (i, 0)) = em (κ)y k/2 eikθ em (pz). 3.5.1 Remark. Applying the GJ (R)-left invariant differential operators LX , X ∈ gJC from 1.4 to the functions φ = jk,m f living on GJ we get after a small calculation LZ0 φ = jk,m 2πmf, LY± φ =

LZ φ = jk,m kf,

jk±1,m Y±m,k f,

m,k LX± φ = jk±2,m X± f,

with Y+ f = Y+m,k f

= =

(1/2y)(fp − (x − iy)fq − 4py2πm f )   z − z¯ i fz + 4πim f τ − τ¯

Y− f = Y−m,k f

=

(1/2)(fp − (x + iy)fq ) = (1/2)(τ − τ¯)fz¯

m,k X+ f

=

i(fx − ify ) + (4πimp2 − ik/y)f     z − z¯ 2 z − z¯ 2k  i 2fτ + 2 fz + 4πim + f τ − τ¯ τ − τ¯ τ − τ¯

X+ f =

= m,k X− f = X− f

= =

−iy 2 (fx + ify ) (i/2)(τ − τ¯)((τ − τ¯)fτ¯ + (z − z¯)fz¯).

We see immediately that the holomorphy of f is equivalent to Y− f = X− f = 0.

(3.22)

60

3. Local Representations: The Real Case

The operator D+ = X+ +(4πm)−1 Y+2 from the discussion of the representations of gJC in 3.1 induces the differential operator D+ = X+ + (4πm)−1 Y+2 = ∂τ − (4πm)−1 ∂z2 + (2k − 1)(τ − τ¯)−1 which for k = 1/2 may be recognized as the heat operator. It was used in [Be2] for a characterization of the Jacobi theta function. We use again the symbol πm,k to indicate the action of GJ on C ∞ (X) given by πm,k (g −1 )f (τ, z) = (f |k,m g)(τ, z) = jk,m (g, (τ, z))f (g(τ, z)) for g ∈ GJ , f ∈ C ∞ (X). Then we have the 3.5.2 Remark. The differential operators introduced by the last remark obey the commutation rules m,k m,k πm,k±2 (g)X± = X± πm,k (g)

(3.23)

and πm,k±1 (g)Y±m,k = Y±m,k πm,k (g)

for all g ∈ GJ .

(3.24)

Proof: The operators LX , X ∈ gJC are left invariant, i.e. we have 

LX (φg ) = (LX φ)g



with



φg (g) := φ(g  g).

For φ(g) = jk,m (g)f (g(x0 )) = jk,m (g)f (x) we get 

φg (g) = jk,m (g  g)f (g  g(x0 )) = jk,m (g)jk,m (g  , x)f (g  (x)), and thus by the last remark 

k,m (LX± φg )(g) = jk±2,m (g)X± jk,m (g  , x)f (g  (x)),

as well as 

k,m (LX± φ)g (g) = jk±2,m (g  g)X± f (g  (x)) k,m = jk±2,m (g)jk±2,m (g  , x)X± f (g  (x)).

The relation for Y±m,k comes out the same way.



Now, we denote by Dm;k ,k (X) the space of differential operators D on X = H × C with πm,k ◦ D = D ◦ πm,k

(3.25)

3.5. Differential operators on X = H × C

61

and ask for the structure of Dm;k ,k (X), in particular for D(X) := D0;0,0 (X). This will be deduced from Helgason’s theory of differential operators on homogenous spaces, e.g. in [He1] or [He2], which obviously may be extended from R-valued to C-valued functions. The starting point is the fact that our space X = GJ (R)/K J (R) is a reductive coset space: As remarked in 1.4 we have gJ = kJ + mJ

with (Ad(k))mJ ⊂ mJ

for all k ∈ K J .

Thus, we may apply his results and we have by Theorem 10 in [He1] or Theorem 2.8 and Corollary 2.9 in [He2] the following description of the algebra D(X) of invariant differential operators on X : Let I(mJC ) denote the polynomials P in the symmetric algebra S(mJC ) of mJC which are invariant under AdG (K J ). Then, one has a linear bijection ∼

I(mJC ) −→ D(X), Q −→ Dλ(Q) . Here, for a basis X1 , . . . , Xn of I(mJC ) and f ∈ C ∞ (X), Dλ(Q) is given by   Dλ(Q) f (x) = Q(λt1 , . . . , λtn )φ(g exp(t1 X1 + · · · + tn Xn )

, t=0

where φ is the K J -invariant lift of f , i.e. φ = f ◦ pr, and λ indicates the symmetrization λ(Y1 , . . . , Yp ) =

1  Yσ(1) ◦ · · · ◦ Yσ(p) . p! σ∈Sp

Helgason adds the warning that the map Q →

Dλ(Q) is not in general multiplicative (even when D(X) is commutative): We have Dλ(Q1 Q2 ) = Dλ(Q1 ) Dλ(Q2 ) + Dλ(Q) where Q ∈ I(mJC ) has degree deg Q1 +deg Q2 . We reproduce a result from [Be1]. 3.5.3 Proposition. D(X) is a non-commutative algebra generated by Dλ(Pi ) (i = 1, . . . , 4) with λ(P1 ) =

(1/2)(Y+ Y− + Y− Y+ ),

λ(P2 ) = λ(P3 ) =

(1/2)(X+ X− + X− X+ ), (1/6)(Y+2 X− + Y+ X− Y+ + X− Y+2 ),

λ(P4 ) =

(1/6)(Y−2 X+ + Y− X+ Y− + X+ Y−2 ).

62

3. Local Representations: The Real Case

Proof: We have mJC = Y± , X± C and Q ∈ I(mJC ) may be unterstood as a polynomial in Y± , X± with (ad Z0 )Q = (ad Z)Q = 0. Z0 being in the center of gJC , the first equality gives no condition. To evaluate the second equality, we can deduce from the multiplication table in gJC that j l m l m ad(Z)(X+ Y−j X+ X− ) = (j − k + 2l − 2m)(Y+j Y−k X+ X− )

(3.26)

holds. This gives zero for the four “basic” combinations j = k = 1;

l = m = 1;

j = 2,

m = 1;

k = 2,

l = 1,

where only the non-zero numbers are mentioned. These four quadruples exactly lead to the terms given in the proposition, and from (3.26) it follows that all Ad(K J )-invariant elements of U (mJC ) may be algebraically combined from these four terms.  As examples, the GJ –invariant differential operators on X corresponding to the first two terms are given by 0,0 0,0 0,0 0,0 Δ0,0 0 = Y+ Y− + Y− Y+ =

1 2 (∂ − 2x∂p ∂q + (x2 + y 2 )∂q2 ) 2y p

and 0,0 0,0 0,0 0,0 2 2 2 Δ0,0 1 = X+ X− + X− X+ = (2y )(∂x + ∂y ).

3.5.4 Remark. As discussed in [Be5] and [Be6] for each c > 0 Δ = Δ0 + cΔ1 is the Laplace-Beltrami operator belonging to a GJ -invariant metric on the space X = H × C given by ds2 = y −2 (dx2 + dy 2 ) + (cy)−1 ((x2 + y 2 ) dp2 + 2xdp dq + dq 2 ) It is not difficult to extend this to the determination of Dm,k ,k (X). At first, we have a completely analogous situation for k  = k. 3.5.5 Corollary. Dm;k (X) = Dm;k,k (X) is generated by the differential operam,k tors associated to P1 , . . . , P4 realized as polynomials in X± and Y±m,k . And another look at the relation (3.26) in the proof of the proposition above shows that for instance for k  = k + 2 we have the following statement. 3.5.6 Corollary. Each element of Dm;k+2,k (X) is got by symmetrization of (m,k) 2 (m,k) (Y+ ) , X+ and elements of Dm;k (X). For later use we define the differential operators Δk,m := Y+k,m Y−k,m + Y−k,m Y+k,m .

(3.27)

ˆ J and Whittaker models 3.6. Representations induced from N

63

ˆJ 3.6 Representations induced from N and Whittaker models We start again by the Iwasawa decomposition, this time in the form ˆ J (R)AJ (R)K(R), GJ (R) = N

ˆ J := N J Z) (N

ˆ J (R) defined take the character ψ m,n,r for m, n, r ∈ Z, m = 0 and n ˆ (x, q, κ) ∈ N by ψ m,n,r (ˆ n(x, q, κ)) = e(mκ + nx + rq),

e(u) = exp(2πiu),

(3.28)

and study now the induced representation J

m,n,r π m,n,r := indG ˆJ ψ N

acting by right translation on the space W m,n,r of measurable C–valued functions W living on GJ (R) with the properties ˆ J , g ∈ GJ , W (ˆ ng) = ψ m,n,r (ˆ n)W (g) for all n ˆ∈N  2 W  := |W (t˜(y)(r(θ), (ˆ p, 0, 0)))|2 dμNˆ J \GJ < ∞.

i) ii)

ˆ J \GJ N

Here we use coordinates ˆ t˜(y)(r(θ), p, g=n ˆ (x, qˆ, ζ) ˆ 0, 0)) related to the “old” S–coordinates by qˆ = pˆ = ζˆ =

q + px py 1/2 ζe(p(px + q)).

ˆ J \GJ given by and the quasi-invariant measure on N dμNˆ J \GJ = y −5/2 dy dθ dˆ p. Then in these coordinates, an element W ∈ W m,n,r , called Whittaker function of type (m, n, r), is of the form W (g) = ψ m,n,r (ˆ n)F (y, θ, pˆ) with

 W 2 =

|F (y, θ, pˆ)|2 y −5/2 dy dˆ p dθ < ∞.

64

3. Local Representations: The Real Case

Now, we look for subspaces W m,n,r (π) such that the restriction of right translation ρ to the subspace is equivalent to a given representation π of GJ with central character ψ m . This will be called a Whittaker model of type (n, r) for π. There are several ways to get at the Whittaker models. As we have developed the infinitesimal method and the method of differential operators this far, we will use it again here. It is convenient to separate moreover the K-variable θ and take for k ∈ Z and fixed m, n, r W (g) = ck (x, qˆ, κ ˆ , , θ)ϕk (y, pˆ) with ck (x, qˆ, κ ˆ , θ) = ψ m,n,r (ˆ n)eikθ . 3.6.1 Remark. Applying the GJ –leftinvariant differential operators LX we get by a small computation LZ0 W LY± W

LZ W LX± W

= 2πmW = ck±1 ϕ± k

= =

kW ck±2 ϕk± ,

where we have for ϕ = ϕk ϕ±

=

(1/2)ϕpˆ ∓ (2πmˆ py 1/2 + πry 1/2 )ϕ

ϕ±

=

(1/2)ˆ pϕpˆ + yϕy ∓ (2π(mˆ p2 + rpˆy 1/2 + ny) − k/2)ϕ.

Guided by the form of the automorphic factor, we normalize the ϕk and put (0)

ϕk = ϕk ψk ,

± ϕ± k = ϕk ψk , (0)

(0)

ϕk± = ϕk ψ±

with 2

ϕk (y, pˆ) = y k/2 e−2π(mpˆ (0)

+r py ˆ 1/2 +ny)

resp. with μ = 2πm and N = 4mn − r2 ˆ ϕk (y, pˆ) = y k/2 e−μ(((p+(r/2m))y (0)

1/2 2

) +(N/(4m2 ))y)

I.e., now we have W (g) = e(m(κ + pz) + nτ + rz)y k/2 eikθ ψk or = jk,m (g, (i, 0))e(nτ + rz)ψk with ψ+ ψ−

= (1/2)ψpˆ − 2μ(ˆ p + (r/2m))y 1/2 ψ, = (1/2)ψpˆ,

.

ˆ J and Whittaker models 3.6. Representations induced from N

65

and ψ+ ψ−

= (1/2)ˆ pψpˆ + (k − 2μ(ˆ p2 + (r/(2m))ˆ py 1/2 + (n/(2m))y)ψ, = (1/2)ˆ pψpˆ + yψy .

From here, we immediately get a statement about the existence and uniqueness + of the Whittaker model for the discrete series representations πm,k of GJ (R). 3.6.2 Proposition. For m > 0, N = 4mn − r2 > 0 and k ≥ 2 there is exactly + one subspace W m,n,r (πm,k ) contained in W m,n,r such that the right regular + representation ρ restricted to this space is equivalent to πm,k . We have the − 2 same statement for πm,k with N = 4mn − r < 0 and k ≥ 2. + Proof: By Proposition 3.1.7 π = πm,k has a cyclic vector W0 of lowest weight characterized by

π ˆ (Y− )W0 = π ˆ (X− )W0 = 0, π ˆ (Z)W0 = kW0 , π ˆ (Z0 )W0 = μW0 . (3.29) As we have here π ˆ (X) = LX for X ∈ gJC , it is clear by the last remark and the formulae in the sequel that for ψk = 1 W (g) = jk,m (g, (i, 0))e(nτ + rz) is such a vector, and it is unique up to a constant factor. We further have ∞ ∞ 1/2 2 2 ˆ ) +(N/(4m)2 )y) −5/2 W  = y k e−2μ((p+(r/2m)y y dˆ p dy 0 −∞

= (1/2)m

−1/2

∞

e−(πN/(2m))y y k−5/2 dy < ∞

0 − for N, m > 0, k ≥ 2. Again by Proposition 3.1.7, π = πm,k has a cyclic vector W0 characterized by

π ˆ (Y− )W0 = π ˆ (D+ )W0 = 0, π ˆ (Z)W0 = (1 − k)W0 , π ˆ (Z0 ) = 2μW0 . (3.30) The action of D+ is given here by ++ (LX+ + (2μ)−1 L2Y+ )W = (W/ψ1−k )(ψ(1−k)+ + (2μ)−1 ψ1−k )

where the formulae for ψ+ and ψ + combine to D+ ψ

:=

ψ+ + (2μ)−1 ψ ++

=

(1/(8μ))ψpˆ p/2 + ry 1/2 /(2m))ψpˆ + yψy + (ay + b)ψ ˆp − (ˆ

with a = −πN/m,

b = 1/2 − k.

66

3. Local Representations: The Real Case

π ˆ (Y− )W = 0 translates into ψ − = 0, i.e. ψpˆ = 0. Thus, ψ is a function depending alone on y, and π ˆ (D+ )W = 0 is equivalent to yψy + (ay + b)ψ = 0. Hence, we have up to a constant factor ψ1−k = ψ = y −b e−ay = y k−1/2 eπN y/m and W (g) = jk,m (g, (i, 0))e(nτ + rz)y k−1/2 eπN y/m = e(m(κ + p(px + q)) + nx + r(q + px))ei(1−k)θ y k/2−1/2 ˆ ·e−2πm(p+y

1/2

r/(2m))2 +πN y/(2m)

Here the norm W  is seen to be finite for N < 0 and k ≥ 2.

. 

The treatment of the principal series is a little bit more subtle. By Proposition 3.1.6 a cyclic vector W0 for π = πm,s,ν is characterized by π ˆ (Y− )W0 = 0,

π ˆ (Z)W0 = (1/2 + ν)W0 ,

π ˆ (Z0 )W0 = μW0

(3.31)

and π ˆ (D− D+ )W0 = λW0

with λ = (1/4)(s2 − (ν + 1)2 ) (3.32)

For W (g) = e(m(κ + pz) + nτ + rz)y k/2 eikθ ψk , we have here k = 1/2 + ν, μ = 2πm, and π ˆ (Y− )W0 = 0 makes again that ψ = ψk is a function of y alone. The relation (3.32), by the formula for D+ given in the proof of the last proposition and a similar formula for D− , comes down in this case to y 2 ψyy + (ay 2 + by)ψy − (b + λ)ψ = 0, where λ is as above and we have again a = −πN/m and b = k − 1/2. Here we substitute ψ(y) = eσ(y) χ(−ay) and get for σ = −(1/2)(ay + b log y) the equation χ + (−1/4 − b/(2ay) + (−(b/2)(1 + b/2) − λ)/(ay)2 )χ = 0 i.e., substituting b and λ for k = 1/2 + ν χ + (−1/4 + ν/(−2ay) + ((1/4)(1 − s2 )/(ay)2 )χ = 0.

ˆ J and Whittaker models 3.6. Representations induced from N

67

In Whittaker-Watson [WW] p. 337 we find the equation for the confluent hypergeometric function W = WkW ,mW (z)   d2 W  1 k  1 + − + + − m2 z −2 W = 0. 2 dz 4 z 4

(3.33)

Hence, our equation may be identified with this equation for kW = ν/2 and mW = s/2, where we have replaced the letters k, m from Whittaker-Watson by kW and mW to distinguish them from the letters already used and fixed in our context. Now, in [WW] p. 337 we find as independent solutions of (3.33) for small |z| and 2mW ∈ Z W (z) = z 1/2±mW e−z/2 {1 + z ∗ } and on p. 343 for |z| large and | arg z| ≤ π − α < π W (z) = WkW ,mW (z) ∼ e−z/2 z kW {1 + Σz −n (. . . )} resp. W (z) = W−kW ,mW (−z). Putting all this together, we have for the cyclic vector W0 for πm,s,ν the form W0 (g) = e(m(κ + pz) + nτ + rz)eikθ y 1/4 eπN y/(2m) Wν/2,s/2 (πN y/m), or equivalently W0 (g) =

e(m(κ + p(px + q) + nx + r(q + px)eikθ y 1/4 ·e−2πm(p+r(2m))

2

y

Wν/2,s/2 (πN y/m).

Using the asymptotic behaviour of WkW ,mW (z) we have the finiteness of the norm exactly for one fundamental solution, and we have as a final statement the existence and uniqueness of the Whittaker model for the principal series representation πm,s,ν : 3.6.3 Proposition. For m and N > 0 (and similarly for m, N < 0) there is exactly one subspace W m,n,r (πm,s,ν ) contained in W m,n,r such that the right regular representation ρ restricted to this space is equivalent to πm,s,ν . As it is another example of beautiful analysis, it seems worthwile to present a second approach to these Whittaker models via an integral transformation. ± The models of the representations πm,k and πm,s,ν in 3.2 were constructed by the induction procedure, i.e., they are spanned by smooth functions φ living on GJ (R) and having the transformation property φ(ˆ n(x, q, κ)t˜(y)g) = em (κ)y s0 φ(g)

(3.34)

68

3. Local Representations: The Real Case

with

 s0 =

1 2

  s + 32

k−

for πm,s,ν , ± for πm,k .

3 2

As usual, we get from these functions — at least formally — functions fulfilling the functional equation of a Whittaker function of type (m, n, r) by the map I n,r φ −→ W = Wφn,r with

 Wφn,r (g) =

φ(w ˜−1 n0 g)e(−nx0 − rq0 ) dn0 ,

n0 = n(x0 , q0 ).

NJ

Later on it will become clear, that to get finite expressions here the integration over x0 has to be taken as a certain path integral then to be specified. The easily verified commutation rule w ˜−1 n(x0 , q0 )t(y, p) = (t(y −1 ), (0, p), p2 x0 − 2pq0 )w˜−1 n(˜ x0 , q˜0 ), where x ˜0 = x0 y −1 , leads to Wφ (t(y, p)) =

q˜0 = (q0 − px0 )y −1/2 , 

φ(t(y −1 ), 0, p, 0)w ˜ −1 n(˜ x0 , q˜0 ))em (p2 x0 − 2pq0 ) ·e(−n1 x0 − r1 q0 ) dn0 .

Changing variables by x0 = x ˜0 y, q0 = q˜0 y 1/2 + p˜ x0 y,

i.e.

dn0 = dx0 dq0 = y 3/2 d˜ x0 d˜ q0

and using the transformation property (3.34), we get  3/2−s0 Wφ (t(y, p)) = y φ(w˜−1 n(x0 , q0 ))e(−Ax0 − Bq0 )dx0 dq0 with A = (mp2 + rp + n)y = m(p + r/2m)2 y + N y/(4m), B = 2m(p + r/2m)y 1/2 , or w (AB) Wφ (t(y, p)) = y 3/2−s0 φ

N = 4mn − r2 ,

ˆ J and Whittaker models 3.6. Representations induced from N

69

w denotes the Fourier transform of where φ

φw (x, q) := φ(w˜−1 n(x, q)). Using again the fact that the representations π considered here are cyclic and generated by functions realizing ν0 ⊗ w0 resp. ν0 ⊗ w1/2+ν , the existence of Whittaker models may be proved by showing that the map φ → Wφ makes sense for certain functions φ = φ1 with the property (3.34) and defined by φ1 (g) = em (κ)eilθ y s0 em (pz) and has an image Wφ1 = 0. To calculate Wφ1 , we start by the nasty commutation rule w ˜−1 n(x0 , q0 ) = (n(x1 )t(y1 )r(θ1 ), (p1 , q1 , κ1 )) with x1 + iy1 =

−1 i − x0 = , i + x0 1 + x20

eiθ1 =

 x − i 1/2 x0 − i 0 = , x0 + i (1 + x20 )1/2

p1 = −q0 , q1 = 0, κ1 = 0 Hence we have 2 −(s0 +l/2) φw (x0 − i)l em (−q02 /(x0 + i)) 1 (x0 , q0 ) = (x0 + 1)

and therefore



Wφ1 (t(y, p)) = y 3/2−s0

(x20 + 1)−(s0 +l/2) (x0 − i)l · e(−mq02 /(x0 + i) − Ax0 − Bq0 ) dx0 dq0 .

The usual Fourier transformation formula  2 2 e−π(a+ib)x e−2πixy dx = (a + ib)−1/2 e−πy /(a+ib) ,

a, b ∈ R, a > 0,

R

may be applied here with x = q0 ,

a + ib = 2m(1 + ix0 )/(1 + x20 )

to give 2

Wφ1 (t(y, p)) = y 3/2−s0 (2mi)−1/2 e−πB /(2m)  · (x20 + 1)−(s0 +l/2−1/2) (x0 − i)l−1/2 · e−πiN x0 y/(2m) dx0 (3.35) Integrals such as the one appearing in this expression describe classical Whittaker functions and show up in the literature in different places. For instance

70

3. Local Representations: The Real Case

in [Wa1] pp. 23–24 Waldspurger discusses on his way to Whittaker models for the metaplectic group a function  2 2 Wn (α, η, θ) = α1−s einθ (1 + b2 )−(s+1+n)/2 (i − b)n e−2πic(bα −ηα ) dp Γc

where α, η, θ, c ∈ R, α > 0, c = 0, s ∈ C, n ∈ 1/2 + Z and Γc is a path like in the sketch.

Γc , c > 0

6 r −ηα2 + iα2  -

Γc , c < 0

 r −ηα2 − iα2 ?

i) For s = −1/2 + 2ν and n = s + 1 = 1/2 + 2ν, ν ∈ Z, he gives for  n Wn (1, 0, 0) = i (1 − ib)−(s+1) e−2πicb db

(3.36)

Γc

the expression Wn (1, 0, 0) = |c|1/2 (2πc)s e−2πc Γ(−s)2 sin(π(s + 1))in .

(3.37)

Here the factor |c|1/2 seems to us to be superfluous, but in any case the function is not zero for a half integer s. ii) For s ∈ 1/2Z and n = 1/2 Waldspurger remarks in [Wa1] on p. 94 that Wn (α, 0, 0) is not identically zero. Now, these results may be used for the discussion of the map φ → Wφ resp. the calculation of Wφ1 in the following way. At first we are led to fix the integration over x0 in the integral transform Wφ as a path integral over Γc as in the sketch above. Then, we realize case by case a lowest weight vector of the different types of representations π.

ˆ J and Whittaker models 3.6. Representations induced from N

Case

71

+ πk,m

By Corollary 3.3.2 we have to take φ1 with s0 = k/2 and l = k. Thus, the integral in (3.35) specializes to  (x0 + i)−k+1/2 e−2πicx0 dx0 , c = N y/(4m), for s = k − 3/2 (3.38) Γc i.e. essentially the integral for Wn in (3.36) for s = k − 3/2. This already shows that (3.35) gives a nontrivial representation of a lowest weight vector by a Jacobi-Whittaker function. And introducing (3.38) evaluated by (3.37) (without |c|1/2 ), we get up to trivial nonzero factors exactly the function W (t(y, p)) = y k/2 e−2πm((p+r/2m)

2

y) −πN y/(2m)

e

,

which is the specialization of the function W (g) appearing in the proof of Proposition 3.6.3. Case

− πm,k

Here we have to take φ1 (g) = em (κ)em (pz)y k/2 ei(1−k)θ , i.e. s0 = k/2 and l = 1 − k. Then (3.35) specializes to 2

Wφ1 (t(y, p)) = y 3/2−k/2 (2im)−1/2 e−2πm(p+r/(2m)) y  · (x0 − i)1/2−k e−2πi(N y/(4m))x0 dx0 . Γc

For the integral we can again take over Waldspurger’s result: Comparison with (3.36) shows that this time we have to put b = −x0 , c = −N y/(4m) and s = k − 3/2. Using (3.37) we then get up to a trivial nonzero factor the function 2

W (t(y, p)) = y k/2 e−2πm(p+r/(2m)) y eπN y/(2m) which is again a special value of the function W (g) appearing in the proof of − Proposition 3.6.3 for πm,k . Case

πm,s,ν

We start by the cyclic spherical (or nearly spherical) vector φ1 (g) = em (k + pz)y (s+3/2)/2 eilθ , i.e., we have here s0 = s/2 + 3/4,

l = ν + 1/2,

l = ν + 1/2, ν = ±1/2,

72

3. Local Representations: The Real Case

and (3.35) specializes to 2

Wφ1 (t(y, p)) = y 3/4−s/2 (2mi)−1/2 e−2πm(p+r/2m) y  · |x20 + 1|−s−1+ν (x0 + i)−ν e−2πiN yx0 /(4m) dx0 . (3.39) The integral in this expression may be identified with the integral given by Waldspurger und thus, by his result cited above in ii), we have a nontrivial Whittaker function again. But this still may be pursued a bit further. In [Ja] p. 283 Jacquet introduces the function ∞ sJ WkJ (u, sJ ) = u |t + iu|2kJ −2sJ (t + iu)−2kJ e−it dt −∞

with u = πN/(2m),

kJ = ν/2 and sJ = s/2 + 1/2.

Introducing this function into our expression, we have up to constant nonzero factors Wφ1 (t(y, p) ∼ y 1/4 Wν/2 (πN/2m, s/2 + 1/2) for Re s  0 and, using moreover [Ja] (4.2.17), we come back to the classical Whittaker function Wk,m and get 2

Wφ1 (t(y, p)) ∼ y 1/4 e−2πm(p+r/2m) y Wν/2,−s/2 (πN y/m), hence again a special value of the function from the proof of Proposition 3.6.3. The following statement summarizes the content of this section. It is of some importance for the definition of an L-factor “at infinity”. 3.6.4 Corollary. Let n, r ∈ Z and N := 4mn − r2 . We have existence and uniqueness of the Whittaker models W n,r (π) for + π = πm,k

with

mN > 0

(k ≥ 1),

− πm,k

with

mN < 0

(k ≥ 1),

with

mN > 0.

π=

π = πm,s,ν

In all three cases we have a distinguished cyclic element W0 ∈ W n,r (π), namely + for πm,k ,

W0 (g) = e(m(κ + pz) + nτ + rz)eikθ y k/2 W0 (g) = e(m(κ + pz) + nτ + rz)ei(1−k)θ y k/2 eπN y/m = e(m(κ + p(px + q)) + nx + r(q + px))ei(1−k)θ y k/2 · e−2πm(p+r/(2m))

2

y+πN y/(2m)

− for πm,k ,

ˆ J and Whittaker models 3.6. Representations induced from N

73

and with the classical Whittaker function Wm,k (z) from [WW], p. 337, W0 (g) = e(m(k + p(px + q)) + nx + r(q + px))ei(ν+1/2)θ y 1/4 2

·e−2πm(p+r/(2m)) y Wν/2,s/2 (πN y/m)

for πm,s,ν .

In all three cases W0 is the image W0 = Wφn,r =: I n,r φ0 0 of the corresponding cyclic vector φ0 ∈ Bπ from the model coming from the induction procedure, i.e., φ0 and W0 are lowest resp. dominant weight vectors ± for πm,k and spherical resp. nearly spherical vectors for πm,s,ν . For the sake of completeness, we mention that there is still another way leading to the Whittaker models, closely related to the last discussion and of great importance in the non archimedean case. Namely, for a representation π with representation space Vπ consisting of smooth functions φ we look at a Whittaker functional ln,r , defined as a continuous linear map ln,r : Vπ −→ C with the property ln,r (π(n)φ) = ψ n,r (n)ln,r (φ).

(3.40)

Then the associated Whittaker model is given by right translation upon the space W n,r = {g → ln,r (π(g)φ) : φ ∈ Vπ }. In our case the Whittaker functional is uniquely given by  n,r l (φ) = φ(w˜−1 n)ψ n,r (n) dn NJ

where the integration has to be taken carefully as explained above.

4 The Space L2(ΓJ \GJ (R)) and its Decomposition

In the last chapter the induction procedure presented in 2.1 was exploited, ˜ J . Now, another, albeit rather trivial, starting by the subgroups B J , K J and N way to use this again is to take the discrete subgroup 

Γ J = SL2 (Z)  Z2 

of G J (R) or equivalently the subgroup ΓJ = SL2 (Z)  H(Z) of GJ (R), and in each case the trivial representation id, and induce from here, i.e., to study GJ (R) the representation indΓJ (id) given (in the “second realization”) by right translation ρ on the space 



H = L2 (Γ J \G J (R)). We will collect in this chapter some material (prepared in [BeB¨ o] and [Be3]) about the decomposition of this representation into a cuspidal and a continuous part. We hope to give an impression of the theory even if we restrict to some  main points, for instance leaving aside the possibility to replace here Γ J by some, say, congruence subgroup 





Γ J (N, N ) := Γ(N )  (N Z)2 . Not striving for the same completeness as in the other chapters, we will at least introduce the standard objects showing up here, i.e., the Jacobi forms, Jacobi  Eisenstein series and more general automorphic forms. In this chapter G J will  J J J stand for G (R) and G for G (R). R. Berndt and R. Schmidt, Elements of the Representation Theory of the Jacobi Group, Modern Birkhäuser Classics, DOI 10.1007/978-3-0348-0283-3_4, © Springer Basel AG 1998

75

4. The Space L2 (ΓJ \GJ (R)) and its Decomposition

76

4.1 Jacobi forms and more general automorphic forms 

As K J = SO(2) × S 1 is a commutative compact subgroup of G J (R), we have the decomposition of ρ related to the characters χm,k of K J , namely    H = L2 (Γ J \G J ) = Hm m∈Z

with Hm = {φ ∈ H : and Hm =



φ(gζ) = ζ m φ(g)

for all ζ ∈ S 1 , g ∈ GJ }

Hm,k

k∈Z

with Hm,k = {φ ∈ Hm : φ(gr(θ)) = eikθ φ(g)

for all r(θ) ∈ SO(2), g ∈ GJ }.

By the discussion in 3.4, we have the notion of the “canonical automorphic factor” jk,m and with it the distinction of elements φ = φf ∈ Hm,k , which may be interpreted as lifts of certain holomorphic functions f living on 

X = H × C = G J /K J . We take this as a motivation to repeat here the usual definition of the Jacobi forms from [EZ] and to discuss moreover some generalizations (even if part of these won’t appear in the decomposition of H). Holomorphic Jacobi forms The canonical automorphic factor for the Jacobi group, as described in 3.4, goes back to Satake and has in the EZ-coordinates 

g = n(x)t(y)r(θ)(λ, μ, ζ) ∈ G J the form  c(z + λτ + μ)2  jk,m (g, (τ, z)) = ζ m em − + λ2 τ + 2λz + λμ (cτ + d)−k . cτ + d We take over the definition from [EZ] p. 9: 4.1.1 Definition. A Jacobi form of weight k and index m (k, m ∈ N) is a complex valued function f on H × C satisfying i) (f |k,m [γ])(τ, z) := jk,m (γ, (τ, z))f (γ(τ, z)) = f (τ, z) for all γ ∈ Γ ii) f is holomorphic

J

4.1. Jacobi forms and more general automorphic forms

77

iii) f has a Fourier development of the form f (τ, z) =



c(n, r)e(nτ + rz).

n,r∈Z 4mn−r 2 ≥0

f is called a cusp form, if it satisfies moreover iii’) c(n, r) = 0

unless

4mn > r2 .

cusp The vector spaces of all such functions f are denoted by Jk,m resp. Jk,m . They are finite dimensional by Theorem 1.1 of [EZ]. One could also define Jacobi forms for subgroups of Γ J of finite index, but we do not need this here.

As an easy consequence of the transformation law i) one has for the Fourier coefficients the following fundamental result ([EZ] Theorem 2.2) c(n, r) depends only on N = 4mn − r2 and on r mod 2m

(4.1)

As already mentioned in the introduction, there is a lot of work done using these Jacobi forms. We will here not go into this but only indicate the charac terization of Jacobi forms as functions on G J (R) (as in [BeB¨o] 5.). 4.1.2 Proposition. Jk,m is isomorphic to the space Am,k of complex functions J φ ∈ C ∞ (G ) with 

i) φ(γg) = φ(g) for all γ ∈ Γ J ii) φ(gr(θ, ζ)) = φ(g)ζ m eikθ for all r(θ, ζ) ∈ K J iii) LY− φ = LX− φ = 0 iv) for all M ∈ SL2 (Z) the function g −→ φ(g)y −k/2 is bounded in domains of type y > y0 . cusp Jk,m is isomorphic to the subspace A0m,k of Am,k with

iv’) the function g → φ(g) is bounded. Proof: As in 3.4, for each k, m ∈ N0 , the automorphic factor jk,m defines a  lifting ϕk,m from functions f living on H × C to functions φ living on G J by f ϕ−

→ φf k,m

with φf (g) = f (g(i, 0))jk,m (g, (i, 0)) = f (x, y, p, q)ζ m em (pz)eikθ y k/2 ,

4. The Space L2 (ΓJ \GJ (R)) and its Decomposition

78

where g is meant in the S–coordinates (x, y, θ, p, q, ζ) and the letter f denotes the function in the four real variables x, y, p, q, which, when holomorphic as a function of τ = x + iy

and z = pτ + q,

Γ is also denoted by f (τ, z). Now, ϕk,m identifies the space Fk,m of functions f on H × C satisfying the transformation formula i) of the Jacobi forms with the  J Γ set Fk,m (G J ) of functions φ : G → C satisfying i) and ii). The equivalence of the holomorphy of f and the equations

LY− φ = LX− φ = 0 for φ = φf are immediate from the formulae for LX− and LY− , as already remarked in 3.5. The condition iii) resp. iii’) in the definition of the Jacobi forms and the condition iv) resp. iv’) in the proposition are equivalent by the following standard fact. 4.1.3 Lemma. For a holomorphic function f : H × C → C with Fourier expansion  f (τ, z) = c(n, r)e(nτ + τ z) n,r∈Z

the condition a) c(n, r) = 0 for all n, r with N = 4mn − r2 < 0 is equivalent to b) For all positive real numbers y0 and p0 the function f (τ, z)em (pz) is bounded in domains of type {(τ, z) ∈ H × C : y ≥ y0 , |p| ≤ p0 } (where as before τ = x + iy and z = pτ + q). Proof: For all n, r and y > 0, η ∈ R we have with τ = x+iy, z = pτ +q = ξ +iη  c(n, r)e−2π(ny+rη) = f (τ, z)e−2πi(nx+rξ) dξ dx. (4.2) (R/Z)2

If N = 4mn − r2 < 0, then there is a p1 ∈ R with n + rp1 + mp21 = λ < 0.

4.1. Jacobi forms and more general automorphic forms

79

2

(4.2) for η = p1 y gives after multiplication by e−2πmyp1  2 c(n, r)e−2πyλ = f (τ, z)e−2πi(nx+rξ)−2πmyp1 dξ dx. (R/Z)2

and hence |c(n, r)|e

−2πyλ



2

|f (x + iy, ξ + ip1 y)|e−2πmyp1 dx dξ

≤ (R/Z)2

=

     f (x + iy, ξ + p1 (x + iy))em (p1 (ξ + p1 (x + iy))) dx dξ. (R/Z)2

The boundedness condition b) now implies |c(n, r)|e−2πyλ ≤ L for all y ≥ y0 , where L > 0 depends only on p1 and some y0 > 0. Since λ < 0, this implies c(n, r) = 0. Assume conversely a) to be fulfilled. It is a well known fact that the series  |c(n, r)e(nτ + rz)| n,r

converges uniformly on compact sets, and hence the definition  2 A(y, p) = |c(n, r)|e−2πy(mp +rp+n) n,r 2

= e−2πymp



|c(n, r)e(niy + rpiy)|

n,r

makes sense and defines a real-valued continuous function on R>0 × R. Now for m > 0 and 4mn − r2 ≥ 0 mp2 + rp + n ≥ 0

for all p ∈ R,

and thus A(y, p) ≤ A(y0 , p)

for all y ≥ y0 .

By continuity there is a constant L > 0 such that A(y0 , p) ≤ L

for all p ∈ R with |p| ≤ p0 .

On the set {(τ, z) ∈ H × C : y ≥ y0 , |p| ≤ p0 } we thus have the chain of inequalities |f (τ, z)em (pz)| ≤

A(y, p) ≤ A(y0 , p) ≤ L.



4. The Space L2 (ΓJ \GJ (R)) and its Decomposition

80

The conditions ii) and iii) in the last proposition show that each φ ∈ Am,k qualifies as a candidate for a lowest weight vector of a discrete series represen+ tation πm,k . Before going deeper into this, let us to some extend follow the observation that this should nourish the expectation to have a similar picture − for the other types of representations πm,k and πm,s,ν . Skoruppa’s skew-holomorphic Jacobi forms While studying certain general theta functions, Skoruppa introduced in [Sk2] ∗ p. 179 parallel to the definition of the space Jk,m a space Jk,m of skew-holomorphic Jacobi forms f of weight k and index m (k, m ∈ N) as the space of smooth functions f : H × C → C satisfying i) f |∗k,m [γ] = f for all γ ∈ Γ

J

ii) ∂z¯f = (8πim∂τ + ∂z2 )f = 0 iii) f has a Fourier development of the form  f (τ, z) = c(n, r)e(nτ + iy(r2 − 4mn)/(2m) + rz). (4.3)

n,r∈Z 4mn−r 2 ≤0

Here again one has τ ∂τ

= =

x + iy, z = pτ + q = ξ + iη (1/2)(∂x − i∂y ), ∂z = (1/2)(∂ξ − i∂η ), ∂z¯ = (1/2)(∂ξ + i∂η )

and the slash operator |∗k,m is given, slightly misusing the notation f (τ, z) = f (x, y, ξ, η), by ∗  ∗ (f  [g])(τ, z) = f (g(τ, z))jk,m (g, (τ, z)) k,m

with the automorphic factor ∗ jk,m (g, (τ, z)) = j0,m (g, (τ, z))(c¯ τ + d)−k+1 |cτ + d|−1 .

As above in Proposition 4.1.2 we can lift these functions f to the group, this time by the lifting f −→ φf

∗ with φf (g) = f (g(i, 0))jk,m (g, (i, 0))

= f (τ, z)ζ m em (pz)ei(1−k)θ y k/2 . and by a slightly more difficult but similar proof (see [Be4]) we get ∗ 4.1.4 Proposition. Jk,m is isomorphic to the space A∗m,k of complex functions  ∞ J φ ∈ C (G ) with

4.1. Jacobi forms and more general automorphic forms

81



i) φ(γg) = φ(g) for all γ ∈ Γ J ii) φ(gr(θ, ζ)) = φ(g)ζ m ei(1−k)θ for all r(θ, ζ) ∈ K J iii) LY− φ = (4πmLX+ + L2Y+ )φ = 0 iv) φ(g)y −k/2 is bounded in domains of type y > y0 . Comparing this with Proposition 3.1.7, we see that each φ ∈ A∗m,k may be − thought of as a cyclic vector for the representation πm,k . 4.1.5 Remark. In [Sk2] the Fourier development is given a form which looks symmetrical for holomorphic and skew-holomorphic Jacobi forms. Let f ∈ Jk,m have a Fourier development like in Definition 4.1.1 iii). By (4.1) the definition  r2 − Δ  C(Δ, r) := c ,r for Δ ∈ −N0 and r2 ≡ Δ mod 2m 4m (4.4) makes sense. f can therefore be written as   f (τ, z) = c(n, r)e(nτ + rz) Δ≤0

=

 Δ≤0

n,r∈Z Δ=r 2 −4mn



r∈Z Δ≡r 2 mod 4m

C(Δ, r)e

 r2 − Δ 4m

 τ + rz .

(4.5)

∗ Now, if f ∗ ∈ Jk,m is a skew-holomorphic form with Fourier development (4.3), we make a similar definition as (4.4), with only Δ ∈ −N0 replaced by Δ ∈ N0 , and arrive at  r2 − Δ    r2 + |Δ| f ∗ (τ, z) = C(Δ, r)e x+i y + rz . 4m 4m Δ≤0 r∈Z (4.6) 2 Δ≡r mod 4m

Of course, one could write Δ instead of |Δ| in this formula. The reason for prefering the absolute value is that if we change the summation from Δ ≥ 0 to Δ ≤ 0, then we get exactly the Fourier development (4.5) of f . Maaß-Jacobi forms It is tempting to try to generalize the Maaß wave forms for our case and moreover discuss the general notion of an automorphic form as for instance in Borel’s article [B]. All this should be done in such a way that Arakawa’s Eisenstein series in [Ar] and their generalizations in [Be3] fit in, and cyclic vectors for πm,s,ν appear. We resume here and generalize a bit a discussion from [Be1] and [BeB¨ o]. 

As we have developed in 3.5 the notion of a G J -invariant Laplace operator ΔH×C on H × C it is easy to propose the following

4. The Space L2 (ΓJ \GJ (R)) and its Decomposition

82

4.1.6 Definition. A smooth function f : H × C → C is called a Maaß-Jacobi form if i) f is ΓJ –invariant ii) f is a ΔH×C eigenfunction iii) f fulfills a boundedness condition, say, f is of polynomial growth. m,k A variant of this would be using the operators X± , Y±m,k commuting with |k,m and introduced in 3.5

4.1.7 Definition. f ∈ C ∞ (H × C) is called a (k, m)-Maaß-Jacobi form if i) f |k,m [γ] = f

for all γ ∈ ΓJ .

ii) Δk,m f = λf

for some λ ∈ C.

iii) f y −k/2 is bounded in domains of type y > y0 . Arakawa’s Eisenstein series in [Ar] and their generalizations in [Be3], to be discussed later on, are examples for these forms with λ = 0, and the holomorphic Jacobi forms with λ = 0. We may try to compare this with the general notion of an automorphic form as defined in Borel [B] for a reductive group. Here we have the problem that the second condition in our definitions, the analycity condition, usually is defined using the center z(gC ) of the universal enveloping algebra U (gC ). And in our case the center of U (gJC ) is too small to single out anything useful. Now, the definitions above propose to take the Laplacian of a GJ left invariant metric as the operator to single out eigenfunctions φ on GJ . But we have a better more invariant choice: For fixed m, we defined in 3.1 the localization U (gJC ) of U (gJC ) and found a Casimir operator C := D+ D− + D− D+ + (1/2)Δ21 which operates by multiplication with ⎧ ⎪ ⎪ ⎨ 1 (s2 − 1) for π ˆm,s,ν , 2 λ=    ⎪ ⎪ ± ⎩ 1 k− 1 k− 5 for π ˆm,k . 2 2 2 Realizing this C as a differential operator LC as in 1.4, we now propose 4.1.8 Definition. An automorphic form of type m, k, λ∗ is a complex function  φ ∈ C ∞ (G J ) with

4.2. The cusp condition for GJ (R)

83



i) φ(γg) = φ(g) for all γ ∈ Γ J ii) φ(gr(θ, ζ)) = φ(g)ζ m eikθ for all r(θ, ζ) ∈ K J iii) LC φ = λ∗ φ iv) φ(g) is slowly increasing, meaning it is of polynomial growth in y. Apparently the lifts of holomorphic Jacobi forms f ∈ Jk,m are automorphic in this sense of type m, k, λ∗ with λ∗ = (k − 1/2)(k − 5/2)/2 and those of the ∗ skew-holomorphic forms f ∈ Jk,m are of type m, 1 − k, λ∗ with the same λ∗ . Another important example is given by the Eisenstein series  Ek,m,λ ((τ, z), s1 ) = (fk,m,λ,s1 |k,m [γ])(τ, z) (4.7) J γ∈ΓJ N \Γ

with ΓJN = Γ ∩ N J and J

fk,m,λ,s1 (τ, z) = em (λ2 τ + 2λz)y s1 −(k−1/2)/2 ,

λ ∈ {0, . . . , 2m − 1}.

This series is absolutely convergent for Re(s1 ) > 5/4, and the vector Ek,m = (Ek,m,λ )λ has a functional equation and analytic continuation by [Ar] Theorem 3.2. This J Ek,m,λ (τ, z, s1 ) may be lifted to a function on G given by  Ek,m,λ (g, s) = φm,s,0,k ((λ, 0, 0)γg) with s = 2s1 − 1 J γ∈ΓJ N \Γ

(φm,s,0,k was defined in Proposition 3.2.9). This is an example for a form of type m, k, λ∗ with λ∗ = (s2 − 1)/2.

4.2 The cusp condition for GJ (R) In the general theory for a real reductive group G with discrete subgroup Γ, the discrete spectrum of the right regular representation ρ is separated from the rest by distinguishing as a cuspidal part H0 in H = L2 (Γ\G) the closed subspace, invariant under ρ, spanned by the φ ∈ H with  φ0N  (g) := φ(n g0 )dn = 0 for almost all g0 ∈ G and all cuspidal N  . ΓJ ∩N 

Here the notion “cuspidal N  ” is as in Lang [La] p. 219-220 (and thus a bit different from, for instance, Harish-Chandra [HC] p. 8, where the adelic case is treated) and means N  is the unipotent radical of a Q-parabolic P  ⊂ G with Γ ∩ N  \N 

compact.

4. The Space L2 (ΓJ \GJ (R)) and its Decomposition

84

Now, besides the smallness of the center of U (gJC ) already dealt with by localization in 3.1, it is one of the main features distinguishing our case from the general theory of reductive groups, that this definition doesn’t work here, as in the direct generalization N  would come out as too big. But from the end of 1.3, we have a natural way to define here the notion of a cuspidal group and a cusp in the following way (which then further down will be tied as usual to the cusp condition iv) in the definition of a Jacobi form): We take the maximal torus A of G , a root decomposition belonging to A, and get a two-parameter subgroup    1x J N = (0, q, 1) =: n(x, q) : x, q ∈ R 01 J

of G belonging to the positive roots. Obviously, J

(N



J



∩ Γ J )\N



J

 (Z\R)2

is compact, and by [GGP] p. 95 we are led to introduce the following notions. 

4.2.1 Definition. A subgroup N ∗J of G J is called horospherical if and only if    it is conjugate to N J , i.e. N ∗J = (N J )g , with g ∈ G J (R), and cuspidal (for ΓJ ) if moreover 

(N ∗J ∩ Γ J )\N ∗J

is compact.

We easily have 4.2.2 Remark. A horospherical N ∗J is cuspidal if and only if g −1 ΓJ g ∩ N



J

is a Z-lattice of rank 2, 

and if and only if (N ∗J )γ is cuspidal for γ ∈ Γ J . Moreover, by a longer but straightforward calculation we can prove: 4.2.3 Proposition. A horospherical N ∗J is cuspidal if and only if it is conjugate J J to N  by an element of G (Q). 









As B J = N J AZ normalizes N J , the ΓJ –conjugacy classes [(N J )g ] are  parametrized by the double cosets [g] of g ∈ G J in 





Γ J \G J /B J . 

Using the proposition above, one is ready to call the classes [g] with g ∈ G J (Q)  cusps for Γ J . But even independently from Proposition 4.2.3, we can easily show the following characterization of cusps resp. cuspidal groups N ∗J .

4.2. The cusp condition for GJ (R)

85





4.2.4 Proposition. The Γ J –conjugacy classes of ΓJ –cuspidal subgroups of G J are in bijection to the set gλ = (λ, 0, 1) ∈ H(R)

with λ ∈ Q/Z.

Proof: i) By Remark 4.2.2 we have to look for g = M (λ, μ, 1) such that 

(g −1 Γ J g) ∩ N



J

is a lattice of rank 2.

From the SL(2)-theory (see for instance [La] p. 220-221) we know M = 1, and  using the action of B J from the right we may restrict to g of type λ ∈ R.

g = gλ = (λ, 0, 1), For γ = M0 (p0 , q0 , 1) ∈ Γ gλ−1 γgλ ∈ N





J

the condition

J

(4.8)

asks for M0 = n(x0 ), x0 ∈ Z. Then we have gλ−1 n(x0 )(p0 , q0 , 1)gλ   = n(x0 ) p0 , q0 − λx0 , e(λ2 x0 − 2λq0 − λp0 x0 ) , (4.9) and (4.8) asks for p0 = 0. We see that gλ−1 ΓJ gλ ∩ N



J

is a Z-lattice of rank 2 exactly if e(λ2 x0 − 2λq0 ) = 1 holds, i.e., if λ ∈ Q, as x0 , q0 are given as integers. ii) The equality 

Γ J gλ B



J



= Γ J gλ B



J

is equivalent to the existence of g, g˜ ∈ B γgλ g = γ˜ gλ g˜ With γˆ := γ −1 γ˜ =

that is 

J



and γ, γ˜ ∈ Γ J with

g˜ g −1 = g−λ γ −1 γ˜ gλ .

 a ˆ ˆb (ˆ p, qˆ, 1) cˆ dˆ

the condition g˜ g −1 ∈ B cˆ = 0





J

requires at first ˆ and a ˆ = d = 1,

and then with g−λ γˆ gλ =



1 ˆb 01



 ˆˆb)) − λ + λ + pˆ, qˆ − ˆbλ , e(−λ (ˆ q + pˆˆb − λ

finally λ − λ − pˆ =

0,

that is

λ ≡ λ mod 1.



4. The Space L2 (ΓJ \GJ (R)) and its Decomposition

86

Now, we will use this to separate the discrete from the continuous part in H. 



4.2.5 Definition. The cuspidal part H0 of the space H = L2 (Γ J \G J ) is distinguished by    H0 = φ ∈ H : φ(ng0 )dn = 0 for almost all g0 ∈ G J  and all cuspidal N ∗J .

(N ∗J ∩Γ J )\N ∗J





Any Γ J –invariant function φ on G J for which the integral in the definition makes sense and which fulfills this “cuspidal condition” will be called cuspidal. For φ with φ(gζ) = ζ m φ(g), m ∈ Z, there is only a finite number of conditions to check. 

4.2.6 Proposition. φ is cuspidal exactly if for almost all g0 ∈ G J one of the following equivalent conditions holds.   i) φ(ng0 )dn = 0 for N ∗J = (N J )gλ (N ∗J ∩Γ J )\N ∗J

with gλ = (λ, 0, 0), where  λ=

r/(2m), r = 0, . . . , 2m − 1 s.t. λ2 m ∈ Z 0

if m = 0, if m = 0.

 ii)

φ(gλ ng0 ) dn = 0 (N

J

for gλ as above.

−1 J ∩gλ Γ gλ )\N  J

 φ(ng0 )ψ¯n,r (n) dn = 0,

iii)

ψ n,r (n(x, q)) = e(nx + rq)

(N  J ∩ΓJ )\N  J

with n, r, ∈ Z

such that

N = 4mn − r2 = 0.

Proof: With ΓN λ = gλ N  gλ−1 ∩ Γ , J

J

we have the isomorphism ΓN λ \N ∗J  NΓλ \N 

J

NΓλ = N  ∩ gλ−1 Γ gλ , J

J

4.2. The cusp condition for GJ (R)

87

(induced by conjugation with gλ−1 ), and the integral in the cuspidal condition may be written as   Wλ (g0 ) = φ(ng0 ) dn = φ(gλ ngλ−1 g0 ) dn ΓN λ \N ∗J



NΓλ \N  J

φ(gλ n(x, q)gλ−1 g0 ) dx dq,

= F (NΓλ )

where for a subgroup Γ0 ⊂ Γ ∩ N  a fundamental domain in N  is denoted by F (Γ0 ). Using J

J

J

gλ n(x, q)gλ−1 = n(x, q + λx)e(λ2 x + 2λq), we have

 φ(n(x, q + λx)g0 )em (λ2 x + 2λq) dx dq.

Wλ (g0 ) = F (NΓλ )

The substitution (x, q) → (x, q+λx) amounts to changing F (NΓλ ) into F (ΓN λ ), whence  Wλ (g0 ) = φ(n(x, q)g0 )em (2λq − λ2 x) dx dq. F (ΓN λ )

The decomposition  J F (ΓN λ ) = γj F (Γ N ), j

where γj ∈ Γ JN is a complete set of representatives for the finite abelian group J J ΓN λ \Γ N , and the Γ -invariance of φ lead to   Wλ (g0 ) = χλ (γj ) φ(n(x, q)g0 )χλ (x, q) dx dq, j

F (Γ J N)

where χλ denotes the character of N  J given by χλ (n(x, q)) = em (2λq − λ2 x). " J The character sum χλ (γj ) for the finite group ΓN λ \Γ N is zero if the restricj

tion of χλ to Γ N is not the trivial character. This is the case if and only if J

2λm ∈ Z

and

λ2 m ∈ Z.

Thus the cusp condition Wλ (g0 ) = 0 for almost all g0 ∈ GJ and all λ ∈ Q/Z comes down to the finitely many cases denoted in the proposition. Condition iii) comes out remembering that for 4mn − r2 = 0 we have  r 2 r  e(nx + rq) = em x+2 q .  2m 2m

4. The Space L2 (ΓJ \GJ (R)) and its Decomposition

88

It is easy to relate the cusp conditions from the last proposition to the cusp   condition for Jacobi forms. For a Γ J -invariant function φ on G J such that the integral exists, we define its (n, r)-Whittaker-Fourier coefficient by  n,r Wφ (g) = φ(ng)ψ¯n,r (n) dn. (4.10) (N  J ∩Γ J )\N  J

4.2.7 Remark. For φ = φf with φf (g) = jk,m (g, (i, 0))f (τ, z),

f ∈ Jk,m ,

we have Wφn,r (g) = jk,m (g, (i, 0))c(n, r)e(nτ + rz). This is straightforward, as we have jk,m (n, g(i, 0)) = 1 and 11 Wφn,r (g)

jk,m (ng, (i, 0))f (n(τ, z))ψ¯n,r (n(x, q)) dx dq

= 00

= jk,m (g, (i, 0))

11 

c(n, r)e(n(τ + x) + r(z + ξ)) · e(−nx − rξ) dx dξ

00

= jk,m (g, (i, 0))e(nτ + rz)c(n, r). 4.2.8 Remark. For φ = φf with ∗ φf (g) = jk,m (g, (i, 0))f (τ, z),

∗ f (τ, z) ∈ Jk,m ,

we have as well ∗ Wφn,r (g) = jk,m (g, (i, 0))c(n, r)e(nτ + rz + iy(r2 − 4mn)/(2m)).

4.3 The discrete part and the duality theorem 



0 We denote by Hm the closure of the subspace of H = L2 (Γ J \G J ) spanned by all φ ∈ H with

φ(gζ) = ζ m φ(g) for all ζ ∈ S 1 , and with the cusp condition   φ(gλ ∩ g0 ) dn = 0 for almost all g0 ∈ G J and all λ = r/(2m), (N  J ∩Γ J )\N  J

r = 0, . . . , 2m − 1 with λ2 m ∈ Z.

As in the general theory (see for instance Godement [Go1] or Lang [La] p. 234) we have a discrete decomposition.

89

4.3. The discrete part and the duality theorem 

0 4.3.1 Theorem. The representation ρ of G J given by right translation on Hm is completely reducible, and each irreducible component occurs only a finite number of times in it.

As H has the ρ-invariant decomposition H = ⊕Hm , the same result holds for 0 H0 = ⊕Hm . There is a proof of this theorem in [BeB¨ o] 8 which follows the lines prescribed in Lang’s book and Godement’s article. We won’t reproduce this proof entirely, but only indicate some steps, hoping that someone will find a more elegant proof. 

0 I. For functions ϕ ∈ Cc∞ (G J ) one introduces an operator T (ϕ) on Hm by  0 T (ϕ)φ(g1 ) = φ(g1 g2 )ϕ(g2 ) dg2 for φ ∈ Hm . G J

By general theorems ([La] p. 234) the assertion of the theorem follows if it is 0 shown that there exists a number Cϕ such that for all φ ∈ Hm T (ϕ)φ ≤ Cϕ φ2 holds, where   ist the sup norm. 



II. With the G J –biinvariant measure on G J given by dg = y −2 dx dy dθ dp dg

dζ ζ

for

g = (n(x)t(y)r(θ), p, q, ζ) 



we have by the periodicity of φ with ΓJ∞ = N J ∩ Γ J   T (ϕ)φ(g1 ) = ϕ(g1−1 n(λ, μ)g2 )φ(g2 ) dg2 . J ΓJ ∞ \G

λ,μ∈Z

With ϕg1 ,g2 (λ, μ) = ϕ(g1−1 n(λ, μ)g2 ), the kernel Kϕ (g1 , g2 ) =



ϕg1 ,g2 (λ, μ)

ϕ,μ∈Z

may be expressed by the Poisson formula as  Kϕ (g1 , g2 ) = ϕˆg1 ,g2 (λ, μ) (ˆ = Fourier transform) ϕ,μ∈Z

=

Kϕ0 (g1 , g2 ) + Kϕ1 (g1 , g2 )

4. The Space L2 (ΓJ \GJ (R)) and its Decomposition

90 where



Kϕ0 (g1 , g2 ) =

ϕˆg1 ,g2 (λ, μ),

4mλ+μ2 =0



Kϕ1 (g1 , g2 ) =

ϕˆg1 ,g2 (λ, μ).

4mλ+μ2 =0

From now on we restrict to the case m = 0. The case m = 0 may be treated similarly (see [BeB¨o] p. 41). III. By a routine calculation the cusp condition leads to Kϕ0 (g1 , g2 ) = 0. IV. We are left with



T (ϕ)φ(g1 ) = J ΓJ /S 1 ∞ \G





ϕˆg1 ,g2 ,ζ (λ, μ)ζ2m−1 φ(g2 ) dζ2 dg2 ,

4mλ+μ2 =0 S 1

where ϕg1 ,g2 ,ζ (u, q) := ϕ(g1−1 n(u, q)g2 ζ). V. For the Siegel set 

S(G J ) = {g = (x, y, θ, p, q, ζ) :

0 ≤ x ≤ 1, y ≥ 1/2, θ ∈ [0, 2π], ) 0 ≤ p ≤ 1/2, 0 ≤ q ≤ 1, ζ ∈ S 1

one has 





G J = Γ J S(G J ). 

We use the symbol ΩG to denote a compact subset of a subgroup G of G J and abbreviate ωg1 = g1−1 t1 ,

ωg1 ,g2 = t−1 1 g2

for gi = ni ti ri ,

i = 1, 2,

to get ϕg1 ,g ,ζ (u, q) = ϕ(ωg1 t−1 ˜ t1 ωg1 ,g ζ2 ). 1 n 2

2

Then, we can prove by some juggling around with compact sets: 

Remark 1: There is a compact subset ΩGJ of G J such that for g1 ∈ S(GJ ) one has ωg1 ∈ ΩG J . 

Remark 2: If g1 ∈ S(G J ) and ϕ(g1−1 nΓ g2 ) = 0 for some nΓ ∈ ΓJ∞ , we may assume that modulo changes of g2 on the left by an element of ΓJ∞ we have ωg1 ,g2 = t−1 1 g 2 ∈ ΩG  J .

91

4.3. The discrete part and the duality theorem

VI. Using this, we can easily modify the expression from part IV to get with ζ ∗ = ζ2 e(p21 x − 2qp1 )     3/2 T (ϕ)φ(g1 ) = y1 ϕω −1 ,ωg ,g ,ζ∗ (˜ x, q˜) 4mλ+μ2 =0

J ΓJ /S1 ∞ \G

S 1 R2

g1

1

2

·e((2mp1 − μ)y 1/2 q˜ + (mp21 − (λ + μp1 ))y1 x ˜) d˜ x d˜ q ζ ∗m−1 dζ ∗ φ(g2 ) dg2 In the inner integral there is a finite C ∞ -function ϕ of x ˜ and q˜ depending for  g1 ∈ S(G J ) on parameters in compact sets (see remarks 1 and 2 in V.). Thus after d1 partial integrations in q˜ and d2 partial integrations in x ˜ we obtain for this integral with 1/2

A = Aμ = |2mp1 − μ|y1 and B = Bλ,μ = |mp21 − (λ + μp1 )|y1     x, q˜)e(A˜ q + Bx ˜) d˜ q d˜ x ≤ CA−d1 B −d2 ,  ϕωg−1 ,ωg g ,ζ ∗ (˜ R2

1

1 2

where C is a constant depending on ϕ, d1 , d2 and S. Here, at least, A or B has to be non–zero, because A = B = 0 is equivalent to 4mλ + μ2 = 0. And this case is excluded in the sum, because it has been provided for with the cusp  condition at the beginning. So, we have for g1 ∈ S(G J )   3/2 −d1 −d2 |T (ϕ)φ(g1 )| ≤ y1 C(ϕ, d1 , d2 , S)|A| |B| |φ(g2 )| dg2 . 4mλ+μ2 =0

J ΓJ ∞ \G g2 ∈t1 ΓG

By Schwarz’s inequality, the last integral may be estimated by  |φ(g2 )| dg2 ≤ (vol (t1 ΩG ))1/2 φ2 . Thus, the assertion of the theorem will be proved, if one can show that by chosing d1 , d2 appropriately, there is an estimation of the series on the right J hand side which is uniform in p1 , y1 for g1 ∈ S(G ), i.e., for 0 ≤ p1 ≤ 1/2, y1 ≥ 1/2. VII. This estimation is given by the following 4.3.2 Lemma. One may choose d1 = d1 (p, λ, μ) and d2 = d2 (p, λ, μ) in such a way (depending on p, λ, μ) that one gets for fixed m = 0 a uniform bound for  R(y, p) := y 3/2−d1 /2−d2 |2mp − μ|−d1 |mp2 − μp − λ|−d2 4mλ+μ2 =0

((y, p) varying in a set of type y ≥ y0 > 0, p in a compact set).

4. The Space L2 (ΓJ \GJ (R)) and its Decomposition

92

For a proof of this lemma we refer to [BeB¨ o] p. 39–40 where the set of (λ, μ) is divided into three subsets and for each of these d1 and d2 are suitably chosen. 0 We will study the decomposition of H0 resp. Hm more closely. The discussion of the cusp condition has the following consequence.

4.3.3 Remark. We have for all m, k ∈ N0 0 A0m,k ⊂ Hm .

Proof: By Proposition 4.1.2 the function 2

|φf (g)| = |f (τ, z)|y k/2 e−2πmp

y

is bounded for all g ∈ G and k, m ∈ N0 . Since a fundamental domain for the   operation of Γ J on G J has finite measure with respect to the biinvariant dg, the assertion follows.  J

The discussion of the skew–holomorphic forms in Proposition 4.1.4 leads to the ∗ same result for A∗0 m,k , defined as the subspace of bounded φ in Am,k . cusp As the lift φf of a Jacobi form f ∈ Jk,m is a lowest weight vector for the + representation πm,k we expect here the same duality relation as in the SL(2)theory between the dimension of the space of holomorphic cusp forms and the multiplicity of the representation in the right regular representation ρ:

4.3.4 Theorem (duality theorem). For m, k ∈ N the multiplicity + m+ m,k = mult (πm,k , ρ) 

0 in the right regular representation ρ of G J on Hm equals the dimension of the space of cusp forms of weight k and index m: cusp m+ m,k = dim Jk,m .

Proof: The finiteness of m+ m,k is already contained in Theorem 4.3.1. i) Let 0 Hm = ⊕ Hm,n n

be a decomposition of Hm into irreducible subspaces under the operation of + right translation. In each Hm,n which is isomorphic to the space Hπ+ of πm,k m,k there exists (up to scalars) exactly one analytic lowest weight vector φ = φnk,m satisfying φnk,m (gr(θ, ζ)) = φnk,m (g)ζ m eikθ

for all θ, ζ.

93

4.3. The discrete part and the duality theorem

By the discussion in 3.1 and 3.2 this is equivalent to LZ0 φ = 2πmφ,

LZ φ = kφ,

(4.11)

and LY− φ = LX− φ = 0.

(4.12)

From Proposition 4.1.2 we then know that φ is the lift of a cusp form, i.e. cusp m∗m,k ≤ dim Jk,m . cups ii) To obtain the other inequality, let f be an element of Jk,m . Its lifting φ = φf + is a lowest weight vector for πm,k which fulfills the conditions just stated. From this we have that φ is an eigenfunction for the Casimir operator C from Section 3.1:

LC φ = λk φ,

λk = k 2 − 3k + 5/4.

A given decomposition 0 Hm = ⊕ Hm,n

induces

n

φ=

(4.13) 

φn

with φn ∈ Hm,n .

n

Since φ satisfies (4.13), each φn satisfies these relations too. We want to show that all φn needed in the decomposition of φ are vectors of lowest weight for a + representation equivalent to πm,k . This will be clear, if each Hm,n containing a component φn = 0 is equivalent to the space Hπ+ . Since S 1 as a subgroup 

m,k



of G J commutes with G J , each ρ(r(0, ζ)) operates as a scalar on Hm,n . The first equation in ii) of Proposition 4.1.2 shows that this scalar is ζ m , and this already fixes the type m of the representation belonging to Hm,n . But in a representation where Z0 operates as a scalar, C commutes with all representation operators, and, therefore, operates as a scalar too, say λn . Moreover by the same reasoning as in the discussion of the unitarizability we have for a vector space V with scalar product  ,  D+ φ, Ψ = −φ, D− Ψ

for φ, Ψ ∈ V,

and hence D+ D− φ, Ψ = φ, D+ D− Ψ and thus the symmetry of C = D+ D− + D− D+ + (1/2)Δ21 . Then we have by (4.13) above for V = H and smooth functions λn (φn , φ) = (LC φn , φ) = (φn , LC φ) = λk (φn , φ)

i.e.

λn = λk .

From the table of the possible eigenvalues of C in Proposition 3.1.10 we con+ .  clude π = πm,k

4. The Space L2 (ΓJ \GJ (R)) and its Decomposition

94

Apparently, the considerations leading to the proof of the duality theorem show that Definition 4.1.8 indeed may be specialized to another characterization of Jacobi cusp forms. 

cusp 4.3.5 Corollary. Jk,m is isomorphic to the space of smooth functions φ on G J with

i) φ(γgr(θ, ζ)) = φ(g)ζ m eikθ



for all γ ∈ Γ J and r(θ, ζ) ∈ K J

ii) LC φ = (k 2 − 3k + 5/4)φ iii) φ is bounded (and ergo cuspidal). Moreover all this reasoning goes through also for the other types of representations to give analogous statements, for instance the duality theorem ∗,cusp m− k,m = dim Jk,m .

(4.14)

4.4 The continuous part After the discussion of the discrete decomposition  0 Hm = Hm,n , n

where the Hm,n are equivalent to the representation spaces of the representa± tions πm,k , πm,s,ν discussed above, we now turn to the orthogonal complement c 0 Hm of Hm in Hm and expect a continuous decomposition as in the GL(2)theory, described for instance in [Ge1] p. 161–162, [Ku2] p. 75 f, or [La] p. 239, i.e. something like   c Hm = Hm,s,ν ds, Hm,s,ν the space of πm,s,ν . ν=±1/2 Re s=0 Im s>0

c To make this more precise, the starting point is to describe as usual Hm by incomplete theta series or as in [La] p. 240 by a theta transform. As already done in the last section, in the sequel we only give a sketch, following closely the exposition in [Be3], which we refer to for most of the proofs.

The theta transform and its adjoint The space 



Y := N J \G J 

has the right G J –invariant measure dμY = y −2 dy dθ dp dζ/ζ.

95

4.4. The continuous part

Using this measure, we define 



Lm = L2 (N J \G J )m 

as the space of functions ϕ on G J with for n ˆ = n(x1 , q1 , ζ1 ) ∈ N J Z

ϕ(ˆ ng) = ζ1m ϕ(g) and

 |ϕ|2 dμY < ∞. Y

As above, we use with λ ∈ Λ,

gλ = (λ, 0, 1) where

Λ = {λ = r/(2m) : r = 0, 1, . . . , 2m − 1}. Moreover, as in 4.2, we will use the subset Λ0 = {λ ∈ Λ : λ2 m ∈ Z} and the group 

ΓJN = Γ J ∩ N



J

with its conjugate subgroups 



∼

ΓN λ = Γ J ∩ gλ N J gλ−1 −→ NΓλ = N



J

−1

∩ gλ−1 ΓJ gλ = (ΓN λ )gλ .

Here, and in several steps in the sequel, we work with the fundamental “commutation relation” gλ n(x, q) = n(x, q + λx)gλ e(2λq + λ2 x).

(4.15)

For appropriate functions ϕ ∈ Lm and λ ∈ Λ, we define the theta transform ϑλ ϕ by  ϑλ ϕ(g) := ϕ(gλ γg). γ∈ΓN λ \Γ J

This is well defined, because for γ0 = gγ ngλ−1 ∈ ΓN λ we have ϕ(gλ γg) = ϕ(ngλ γg) = ϕ(gλ (gλ−1 ngλ )γg). As to the convergence of ϑλ ϕ one has as in the usual SL(2)-theory (see for instance [La] p. 240) • ϑλ ϕ is convergent for a Schwartz function ϕ. • ϑλ ϕ has for fixed g only finitely many terms if supp(ϕ) is compact. 





• ϑλ ϕ ∈ Cc (Γ J \G J ) for ϕ ∈ Cc (N J \GJ )m .

4. The Space L2 (ΓJ \GJ (R)) and its Decomposition

96

An adjoint operator ϑ∗λ to ϑλ is given by an integral already used in the discussion of the cusp condition in Proposition 4.2.6. For φ ∈ Hm and λ ∈ Λ we define  ϑ∗λ φ(g) := φ(gλ−1 ng) dn. Npλ \N  J

4.4.1 Proposition. Wherever the two operators ϑλ and ϑ∗λ make sense, they are adjoint, i.e., we have for λ ∈ Λ ϑλ ϕ, φHm = ϕ, ϑ∗λ ϕLm . Proof: From the definitions we get immediately   ¯ dg = ϑλ ϕ, φHm = ϕ(gλ γg)φ(g) Γ J \G J

and ϕ, ϑ∗λ φLm

J

ΓN λ \Γ

 = 

=

¯ dg ϕ(gλ g)φ(g) ΓN λ \G J



¯ −1 ng) φ(g ˙ dn dg˙ λ

ϕ(g) ˙ N  J \G J



NΓλ \N J

¯ −1 g) dg. ϕ(g)φ(g λ

NΓλ \G J

Both expressions are equal, because the substitution gλ−1 g → g corresponds to the conjugation NΓλ → ΓN λ .  c 4.4.2 Definition. We denote by Hm the space of all incomplete theta series, i.e.,   the closure in Hm of the space spanned by all ϑλ ϕ, ϕ ∈ Cc (N J \G J )m and λ ∈ Λ. 0 c 4.4.3 Corollary. Hm is the orthogonal complement of Hm in Hm .

Proof: For φ ∈ Hm ϑλ ϕ, φHm = 0





for all ϕ ∈ Cc (N J \G J )m and λ ∈ Λ

is equivalent to ϕ, ϑ∗λ φLm = 0 

for all ϕ and λ.



As Cc (N J \G J )m is dense in Lm , this leads to ϑ∗λ φ = 0 for all λ, i.e., φ is cuspidal.  As in the cusp condition only the λ ∈ Λ0 are essential, we are led to the following additional calculation.

97

4.4. The continuous part

4.4.4 Remark. For λ ∈ Λ0 we may define  ϑ◦λ ϕ(g) = ϕ(gλ γg),

(4.16)

J γ∈ΓJ N \Γ

and we have with cm := {ΓN λ \ΓJN } ϑλ ϕ(g) = cm ϑ◦λ ϕ(g). This comes out, as we here have em (±2qλ ± xλ2 ) = 1 for all q, x ∈ Z and hence for γ0 = n(l, r) l, r ∈ Z ϕ(gλ γ0 γg) = ϕ(n(l, r + λl)gλ γg)em (2rλ + lλ2 ) = ϕ(gλ γg). If γj ∈ ΓJN (j ∈ J) is a family representing ΓN λ \ΓJN , we get   ϑλ ϕ(g) = ϕ(gλ γj γg) = cm ϑ◦λ ϕ(g). J J γj ∈ΓN λ \ΓJ N γ∈ΓN \Γ

Similarly, we have for λ ∈ Λ0 using the fundamental commutation relation (4.15) from above  ϑ∗λ φ(g) = φ(gλ−1 ng) dn NΓλ \N  J



φ(n(x, q − λx)gλ−1 g)em (−2qλ + xλ2 ) dn

= NΓλ \N  J



φ(n(x, q)gλ−1 g)em (−2qλ − xλ2 ) dn.

= NΓλ \N  J

Defining ϑ◦∗ λ φ(g)

 :=

φ(ngλ−1 g)χλ (n) dn,

χλ (n) = em (2qλ + xλ2 )

ΓN \N  J

we arrive at ϑ∗λ φ(g) = cm ϑ◦∗ λ φ(g), and we can refine Proposition 4.4.1 4.4.5 Corollary. Only the ϑ◦λ for λ ∈ Λ0 are essential, and for these λ we have ◦ ϑ◦∗ λ is adjoint to ϑλ .

4. The Space L2 (ΓJ \GJ (R)) and its Decomposition

98

The zeta transform and Eisenstein series c Now, we look for a relation between the elements of Hm and the spaces Hm,s,ν of the principal series representations πm,s,ν . We recall from the discussion in 3.3 that the space Hm,s of functions φ on GJ with (s+3/2)/2

φ(b0 g) = y0 and

for b0 = n(x0 , q0 )t(y0 )ζ0 ∈ B J

ζ0 φ(g)

 φ2 =

|φ(r(θ, p))| ˆ 2 dθ dˆ p 0, and is entire in s for Cc (N J \G J )m . The standard formula ∞ ϕ(y0 y)y0−s−1 dy0 = y s Γ(l − s) for ϕ(y) = e−y y l 0

shows that the system (φ) of functions spanning Hm,s is up to factors the image of the zeta transform of ϕm,l,j (g) = ζ m em (pz)eilθ e−y Hj (v),

v = (2πm)1/2 py 1/2 .

Combining the zeta transform with the theta transform produces Eisenstein series Em,λ (ϕ, g, s) := =

ϑ◦λ Z(ϕ, g, s)  Z(ϕ, gλ γg, s).

(λ ∈ Λ◦ )

(4.18)

J γ∈ΓJ N \Γ 



These series converge absolutely for Re(s) > 3/2 and ϕ ∈ S(N J \G J )m by the same reasoning which is used for the Eisenstein series discussed by Arakawa [Ar] (generalizing a notion by Eichler and Zagier [EZ], p. 17) and already mentioned in 4.1:

99

4.4. The continuous part

We put for λ ∈ Λ◦ fκ,m,λ,s1 (τ, z) = em (λ2 τ + 2λz)y s1 −κ , and



Ek,m,λ ((τ, z), s1 ) =

κ = (k − 1/2)/2

( fκ,m,λ,s1 |k,m [γ])(τ, z).

J γ∈ΓJ N \Γ

This series is absolutely convergent for Re(s1 ) > 5/4, and if k > 3 and s1 = κ coincides with the holomorphic Eisenstein series of Eichler and Zagier. It is seen to be a special case of the Eisenstein series introduced above: 

We may lift it to a function living on G J , A Ek,m,λ = ϕk,m (Ek,m,λ ),

which may be recognized (using the notation from Proposition 3.3.1) as A Ek,m,λ (g, s1 ) = ϑ◦λ φm,s,0,k (g)

with s1 = (s + 1)/2.

A bit more general, we put as well Ek,m,j,λ (g, s ) = ϑ◦λ φm,s,j,k (g),

s = (s + 1)/2 + 1/4,

and take these together as column vectors Em,k,j (g, s) = (Eκ,m,j,λ (g, s))λ∈Λ◦κ , where Λ◦k corresponds bijectively to Arakawa’s set Rknull which he chose to get a set of linear independent series. 4.4.6 Remark. As the elements of the system (φ) spanning Hm,s are related by the ladder operators LX± , LY± , we may in the same way reduce everything to a “cyclic” Eisenstein series Em,0,0 . For instance we have Em,k,j = LjY+ Em,k,0 . We have another kind of reduction principle. 4.4.7 Remark. As the space Lm may be spanned by functions of the type ϕ(g) = ζ m em (pz)eikθ Hj (v)ϕ1 (y), and we have, using the usual Mellin transform ∞ Lϕ1 (s) =

ϕ1 (y)y −s−1 dy,

Z(ϕ, g, s) =

Lϕ1 (s)φm,s,j,k (g),

0

k ∈ Z, j ∈ N0 ,

(4.19)

4. The Space L2 (ΓJ \GJ (R)) and its Decomposition

100

we can say that everything concerning the column vector Em (ϕ, g, s) = (Em,λ (ϕ, g, s))λ∈Λ◦k , ϕ ∈ Lm may be reduced to the fundamental vectors k ∈ Z, j ∈ N0 ,

Lϕ1 (s)Em,k,j (g, s),

and thus (by the previous remark) to Arakawa’s Eisenstein series. The functional equation for the Eisenstein series In [Be3] p. 240 this reduction principle is used to prove the following statement by going back to a functional equation shown by Arakawa for his Eisenstein series by again going back to results about the metaplectic case. 4.4.8 Theorem. For ϕ ∈ L◦m and Re(s) = σ > 3/2 one has a function ϕˆ ∈ L◦m such that the functional equation Em (ϕ, ˆ g, s) = Λ(s)Em (ϕ, ˆ g, 3/2 − s) holds with a matrix Λ(s) meromorphic in s with Λ(s)Λ(3/2 − s) = 1. 

Here L◦m is a space of “nice” functions ϕ on G J , i.e., with ϕ(ˆ ng) = ζ m ϕ(g)

for n ˆ∈N



J

S1,

ϕ(−1g) = ϕ(g), ϕ ∈ C∞

and has compact support mod N



J

For details of the proof we refer here to the paper [Be3] but we would like to encourage further research to get a better proof, e.g. perhaps by coming down from results from the higher dimensional symplectic group. The inner product formula This formula is a first step in the direction of the aim to establish a Plancherel formula generalizing the one of the SL(2)-theory given in [Ku2] p. 85 or [La] p. 260, that is, to express the norm in Hm of a theta transform of ϕ by an integral over its “components” in the representation spaces Hm,s . We recall that functions φ in Hm,s essentially only depend on the variables θ and pˆ = py 1/2 , that is, they are fixed by their dependance on the space K with elements rˆ = r(θ, p) ˆ and measure dμ(ˆ r ) = dˆ p dθ. 4.4.9 Lemma. For ϕ, ψ ∈ L◦m , λ, λ ∈ Λ◦κ and Re s = σ > 5/4 we have   1 ϑ◦λ ϕ, ϑ◦λ ψHm = Z(ϕλ,λ , rˆ, 3/2 − s¯)Z(ψ, rˆ, s) ds dμ(ˆ r) 2πi K

Re s=σ

101

4.4. The continuous part

Proof: Denoting the elements ◦ ϕλ,λ := ϑ◦∗ λ ϑλ ϕ

of the constant term matrix ϕ, we get by the adjointness relation in the Proposition 4.4.1 ϑ◦λ ϕ, ϑ◦λ ψHm

ϕλ,λ , ψLm  ϕλ,λ (g)ψ( ˙ g) ˙ dg˙

= =

N  J \G J

with dg˙ = y −2 dy dθ dp

for g˙ = t(y)r(θ, py 1/2 ).

Putting f1 (y, θ, pˆ) = ϕλ,λ (t(y)r(θ, pˆ)) and f2 (y, θ, pˆ) = ψ(t(y)r(θ, pˆ)) and using the formula (7.2.1.) from [Ku2] p. 78, ∞

f1 (y)y −S1 f2 (y)y −S2

dy 1 = y 2π

0

∞ Lf1 (S1 − it)Lf2 (S2 − it) dt, −∞

with S1 = 3/2 − σ and S2 = σ, we get ϑ◦λ ϕ, ϑ◦λ ψHm

 ∞ =

f1 (y, θ, pˆ)y −3/2 f2 (y, θ, pˆ)

K 0

=

1 2π

dy dθ dˆ p y

 ∞ Lf1 (3/2 − σ + it)Lf2 (σ + it) dt dθ dˆ p.

K −∞

Resubstituting f1 and f2 and remembering ∞ Lf (s) =

f (y, θ, pˆ)y 0

we arrive at the assertion.

−s dy

y

∞ =

ϕ(t(y)ˆ r )y0−s

dy0 = Z(ϕ, rˆ, s) y0

0



The representations πm,s,ν are unitary for s ∈ iR. As Hm,s is spanned by functions φ with y–dependence given by y s/2+3/4 , this amounts here to the intention to shift the integration in the S–plane to the line Re(s) = σ = 3/4. The reason that this may be done is the meromorphic continuation of the Eisenstein series following from its functional equation. This argument is also inherent in the following assertion whose full significance will become clear immediately in the proof of the main theorem.

4. The Space L2 (ΓJ \GJ (R)) and its Decomposition

102

4.4.10 Lemma. For ϕ, ψ ∈ L◦m and Re s = σ = 3/4 we have  Z(ϕ, rˆ, s¯)Z(ψ, rˆ, s¯) dμ(ˆ r) K

 ˆ rˆ, s)t Λ(s) dμ(ˆ Z(ϕ, rˆ, s)Z(ψ, r ).

= K

Proof: As in the preceding Lemma, we get for the inner product ϑ◦λ ϕ, Eλ (ψ, ·, s¯)Hm = =

ϕλ,λ , Z(ψ, ·, s¯)Lm ∞  ϕλ,λ (t(y)ˆ r )Z(ψ, t(y)ˆ r , s¯) dμ(ˆ r )y −5/2 dy 0 K

and by the transformation properly of Z “of type y s ”  = Z(ϕλ,λ rˆ, 3/2 − s)Z(ψ, rˆ, s¯) dμ(ˆ r ). K

ˆ 3/2 − s) for (ψ, s) and multiplying with Λ(¯ ¯ s) leads to Substituting here (ψ,  ˆ ·, 3/2 − s¯)H ¯ λ ,μ (¯ ϑ◦λ ϕ, Λ s)Eμ , (ψ, m   ˆ rˆ, s) dμ(ˆ = Z(ϕλ,μ , rˆ, s¯)Λλ μ (¯ s)Z(ψ, r ). μ K

The left sides of both last equations are equal by the functional equation of the Eisenstein series. Thus, with 3/2 − s = s¯ for Re(s) = 3/4 we get the assertion of the Lemma.  c Now everything is ready for the final formula comparing inner products in Hm with those in the unitary representation spaces Hm,s .

4.4.11 Theorem. For ϕ, ψ ∈ L◦m and s = σ + it with σ = 3/4 we have the matrix equation (ϑ◦λ ϕ, ϑ◦λ ψHm )λ,λ ∈Λ◦k

=

1 2π

∞ Z(ϕ, ·, s)η −1 Z(t ψ, ·, s)Hm,s dt 0

with η = (ηλ,λ )λ,λ ∈Λ◦k ,

ηλ,λ := δλ,λ + δ1−λ,λ

(4.20)

103

4.4. The continuous part

Proof: By some further consideration we may take Lemma 4.4.9 and write ϑ◦λ ϕ, ϑ◦λ ψHm  1 = 2πi

 Z(ϕλ,λ , rˆ, s)Z(ψ, rˆ, s) dμ(r) ds

Re s=σ

=

=

1 2π 1 2π

∞ 

K

Z(ϕλ,λ , rˆ, σ + it)Z(ψ, rˆ, σ + it) dμ(r) dt 0 K

∞ 

Z(ϕλ,λ , rˆ, σ − it)Z(ψ, rˆ, σ − it) dμ(r) dt, 0 K

By Lemma 4.4.10 this amounts to < ϑ◦λ ϕ, ϑ◦λ ψ >Hm ∞   1 = Z(ϕλ,μ , rˆ, σ + it) × 2π  0

μ

K

ˆ rˆ, σ + it)) dμ(ˆ ×(δμ ,λ Z(ψ, rˆ, σ + it) + Λλ ,μ (σ − it)Z(ψ, r ) dt. Now, the assertion follows from the matrix relation (E is here the unit matrix) ˆ g, s)Λ(3/2 − s) = t η −1 Z(t ψ, g, s). Z(ϕ, g, s)E + Z(ψ, This relation is the result of some tedious calculation of the constant term matrices given in [Be3] Section 4.7. and 8.

5 Local Representations: The p-adic Case

In this chapter we turn to the local non-archimedean case and study the representation theory of GJ (F ), where F is a finite extension of some Qp . We will reach the goal of classifying all irreducible, admissible representations of m GJ (F ) by using the fundamental relation π  π ˜ ⊗ πSW and the classification of representations of the metaplectic group given by Waldspurger in [Wa1]. Roughly speaking, all non-supercuspidal irreducible representations of Mp are obtained by parabolic induction. This result can be taken over to the Jacobi m group by making the isomorphism π  π ˜ ⊗ πSW explicit in the context of standard models for induced representations (Theorem 5.4.2). The importance of this explicit isomorphism will also become apparent while discussing intertwining operators in Section 5.6. We will see there how the naturally defined intertwining integrals correspond on both sides, and how the analytic continuation can be established. The discussion of Whittaker models in Section 5.7 will fill the last gap for a complete classification of irreducible, admissible representations. Finally, for global applications, it is important to single out the unitary representations.

5.1 Smooth and admissible representations Let F be a p-adic field, i.e., a finite extension of some Qp , where the prime number p is the characteristic of the residue field of F . Let further be • q = pf the number of elements of the residue field. • O the ring of integers of F . • p the maximal ideal of O. R. Berndt and R. Schmidt, Elements of the Representation Theory of the Jacobi Group, Modern Birkhäuser Classics, DOI 10.1007/978-3-0348-0283-3_5, © Springer Basel AG 1998

105

106

5. Local Representations: The p-adic Case

• v = vF the normalized valuation of F . • ω ∈ F an element with v(ω) = 1. This section deals with the Jacobi group over F . So let GJ , G, H, . . . be the F -rational points of the corresponding algebraic groups (as before G = SL(2)). We stick to the usual notions for representations of p-adic groups. In particular, we call a representation of GJ on some complex vector space V smooth if V is covered by the subspaces V K = {v ∈ V : gv = v for all g ∈ K} where K runs through the open and compact subgroups of GJ . If, moreover, every V K is finite dimensional, we call the representation admissible. More details on the general representation theory of p-adic groups can for instance be found in [Ca]. In Section 2.6 the fundamental fact was already stated that every smooth representation π of GJ with non-trivial central character is of the form m π=π ˜ ⊗ πSW

with m ∈ F ∗ and π ˜ a smooth representation of Mp. The standard model for both the Schr¨ odinger and the Weil representation is the Schwartz space S(F ) of locally constant compactly supported functions F → C. The explicit formulas for these representations were already given in Sections 2.2 resp. 2.5, but we recall them here to have them at hand:     πSm (λ, μ, κ)f (x) = ψ m(κ + (2x + λ)μ) f (x + λ).     m 1 b πW f (x) = ψ m (bx2 )f (x). 01     m a 0 πW f (x) = δm (a)|a|1/2 f (ax). 0 a−1      0 1 m πW f (x) = γ(1)fˆ(x) = γm (1) ψ(2mxy)f (y) dy. −1 0 F

(Throughout the chapter ψ denotes the additive standard character defined in Section 2.2.) Here we abbreviate δm (a) = (a, −1)γ(a)γ(1)−1 . This function is called the Weil character and will be discussed more closely in Section 5.3. The Weil constant γ = γm was introduced in Section 2.5. We will mainly be concerned with admissible representations, for which there are the following results. 5.1.1 Lemma. The Schr¨ odinger and the Schr¨ odinger-Weil representations are admissible.

5.2. Whittaker models for the Schr¨ odinger-Weil representation

107

Proof: The second assertion certainly follows from the first. Let K be an open compact subgroup of H. It is clear that K contains a set of the form ωαO × ωβ O × ωγ O

with integers α, β, γ.

The invariance of f ∈ S(F ) under (λ, 0, 0) with λ ∈ ω α O means that f is invariant under additive translations by ω α O. The invariance under (0, μ, 0) with μ ∈ ω β O easily implies that f has support in (2mω β )−1 O. These two facts together show that S(F )K is finite-dimensional.  5.1.2 Proposition. π is admissible if and only if π ˜ is admissible. Proof: Assume π ˜ : Mp → GL(V ), also regarded as a projective representation of SL(2), is admissible and let K ⊂ GJ be a compact open subgroup. Given Φ ∈ V ⊗ S(F )K we can write it as  Φ= ϕi ⊗ fi i

with linearly independent ϕi ∈ V and linearly independent fi ∈ S(F ) (simply take the lowest possible number of summands). The intersection K1 := K ∩ H is a compact open subgroup of H. From  Φ = π(k1 )Φ = ϕi ⊗ πSm (k1 )fi for all k1 ∈ K1 i

and the linear independence of the ϕi it follows that fi ∈ S(F )K1 for each i. m Choose a small enough compact open subgroup K0 of K ∩ G such that πW (K0 ) K1 fixes each element of the finite-dimensional space S(F ) (Lemma 5.1.1). Then from π(k0 )Φ = Φ for all k0 ∈ K0 and the linear independence of the fi it follows that ϕi ∈ V K0 . What we have shown is that (V ⊗ S(F ))K ⊂ V K0 ⊗ S(F )K1 , and the admissibility of π follows from that of π ˜ . It is easy to see that the converse is true. 

5.2 Whittaker models for the Schr¨ odinger-Weil representation In this section we establish the existence and uniqueness of Whittaker models for the Schr¨ odinger and the Schr¨ odinger-Weil representations. As a commutative subgroup of H we take NH = {(0, μ, 0) : μ ∈ F }  F,

108

5. Local Representations: The p-adic Case

and we are at first interested in Whittaker models for πSm with respect to this subgroup and the character ψ r of NH , where r ∈ F . That is, we are looking for a space of locally constant functions W : H → C with the property W ((0, μ, 0)h) = ψ r (μ)W (h)

for all μ ∈ F, h ∈ H,

such that right translation on this space defines a representation of H equivalent to πSm . The existence and uniqueness of such a space is equivalent to the fact that the space of linear functionals l : S(f ) → C with the property   l πSm (0, μ, 0)f = ψ r (μ)f for all μ ∈ F, f ∈ S(F ), is one-dimensional. Such a functional is called a ψ m,r -Whittaker functional, and the associated Whittaker model shall be denoted WSm,r . 5.2.1 Theorem. For any r ∈ F there exists a unique Whittaker model WSm,r for πSm . The associated Whittaker functional is given by lS : S(F ) f

−→ C,  r 

−→ f . 2m

The proof is taken from [Ho], which in turn goes back to [Wa1] and [Be6]. First we need a lemma. 5.2.2 Lemma. For a functional l : S(F ) → C and a point ξ ∈ F the following statements are equivalent. i) There exists c ∈ C such that l(f ) = cf (ξ) for all f ∈ S(F ). ii) l(f ) = 0 for all f ∈ F with f (ξ) = 0. Proof: Certainly ii) follows from i). Conversely, assume f (ξ) = 0 implies l(f ) = 0. Let θ be the characteristic function of ξ + O. Then for arbitrary f ∈ S(F ) the function f − f (ξ)θ vanishes at ξ, hence 0 = l(f − f (ξ)θ) = l(f ) − l(θ)f (ξ). So i) follows with c = l(θ).



Proof of Theorem 5.2.1: A very simple calculation shows that lS is indeed a non-trivial ψ r -Whittaker functional. It remains to show that lS is unique up to scalars. So let l be another functional of this type. By the lemma, it is enough to show that l(f ) = 0 for f ∈ S(F ) with f (r/2m) = 0. Now, by Fourier inversion, there exists fˆ ∈ S(F ) such that  f (x) = fˆ(y)ψ (xy) dy. F

5.2. Whittaker models for the Schr¨ odinger-Weil representation

109

Since fˆ is locally constant with compact support this may be written as a finite sum:  f (x) = fˆ(y)ψ (xy) y∈ω a O/ω b O

with suitable a, b ∈ Z, a < b. We multiply this by the characteristic function 1ωk O of ω k O, where k is chosen small enough such that ω k O contains the support of f , and obtain  y   f= fˆ(y)πSm 0, , 0 1ω k O . 2m a b y∈ω O/ω O

Since f (r/2m) = 0 this can also be written as    y   ry   f= fˆ(y) πSm 0, , 0 1ω k O − ψ 1ω k O . 2m 2m a b y∈ω O/ω O

Application of our Whittaker functional l on both sides yields l(f ) = 0, and we are done.  5.2.3 Corollary. For every M ∈ G there is exactly one non-trivial space WM of locally constant functions f : H(F ) → C with the following properties: i) WM is stable under right translations. ii) The representation on WM defined by right translation is isomorphic to πSm . iii) Every f ∈ WM satisfies   f ((0, μ)M, κ)h = ψ m (κ)f (h)

for all μ, κ ∈ F, h ∈ H.

Proof: For M = 1 this is just the theorem with r = 0. Let L(M ) be the space of linear functionals l : S(F ) → C such that   l πSm ((0, μ)M, 0)f = l(f ) for all f ∈ S(F ), μ ∈ F. Then, very similar to usual Whittaker models, the assertion is equivalent with the fact that dim(L(M )) = 1. If we extend πSm on S(F ) to the Schr¨ odinger-Weil m representation πSW , then it is easy to check that the linear map L(1) −→ L(M ),   m l −→ f → l(πSW (M )f ) is an isomorphism of vector spaces (use (2.3) in Section 2.5). By the case M = 1 already known, the proof is complete. 

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5. Local Representations: The p-adic Case

Now we discuss Whittaker models for the Schr¨ odinger-Weil representation. The subgroup under which we want to have the Whittaker transformation property is as in the real case    1x NJ = (0, μ, 0) : x, μ ∈ F  F 2 . 01 m But we have to be a bit careful because πSW is projective with cocycle λ. So what we are looking for is a space of locally constant functions W : GJ → C with the property       1x 1x W (0, μ, 0)M h = λ , M ψ ν (x)ψ r (μ)W (M h) 01 01

for all x, μ ∈ F , M ∈ G, h ∈ H, where ν and r are elements of F , and such that the following operation ρ of GJ defines a projective representation equivalent m to πSW : (ρ(g  )W )(g) = λ(g, g  )W (gg  ). m,ν,r If such a space exists, we denote it by WSW . It is again easy to see that m,ν,r the existence and uniqueness of WSW is equivalent with the existence and uniqueness (up to scalars) of a ψ ν,r -Whittaker functional, i.e. a linear map l : S(F ) → C with      1x m l πSW (0, μ, 0) f = ψ ν (x)ψ r (μ)l(f ) for x, μ ∈ F, f ∈ S(F ). 01 m,ν,r m 5.2.4 Theorem. A Whittaker model WSW for the representation πSW exists if 2 and only if ν = r /4m. In this case it is unique, and the Whittaker functional lSW is given by evaluating functions at r/2m:  r  lSW (f ) = f for all f ∈ S(F ). 2m m,ν,r m,r Because ν is determined by r, we denote WSW simply by WSW . m Proof: If a ψ ν,r -Whittaker functional lSW for πSW is given, then by the definir tions it is also a ψ -Whittaker functional for πSm . This proves the uniqueness in view of Theorem 5.2.1. The rest is easy: If we take lS for lSW , then          r  1x m m 1 x m lSW πSW (0, μ, 0) f = πW πS (0, μ, 0)f 01 01 2m  2     xr 2μr r = ψm ψm f 4m2 2m 2m  2  r = ψ x + rμ lSW (f ). 4m

So lSW is a ψ ν,r -Whittaker functional exactly for ν = r2 /4m.



5.2. Whittaker models for the Schr¨ odinger-Weil representation

111

m 5.2.5 Corollary. The Weil representation πW has a ψ ν -Whittaker model if ν ∈ ∗2 mF . m,0 We pay special attention to the Whittaker model WSW , which exists by the theorem. Using the known properties of the cocycle λ from [Ge2], one easily establishes for any W in this model the transformation formula    a x W (0, μ, κ)M h 0 a−1    a 0 = λ , M δm (a)ψ m (κ)|a|1/2 W (M h), (5.1) 0 a−1

where a ∈ F ∗ , x, μ, κ ∈ F , M ∈ G and h ∈ H. If we extend W to a function J by setting  on G W  (g, ε) = εW (g), W then this transformation property goes over to     a x   (g). W (0, μ, κ), ε g = εδm (a)ψ m (κ)|a|1/2 W 0 a−1 This may be interpreted as follows. Consider the subgroup     a x ∗ J = B (0, μ, κ), ε : a ∈ F , x, μ, κ ∈ F, ε ∈ {±1} 0 a−1 J and its character of G    a x χ (0, μ, κ), ε = εδm (a)ψ m (κ). 0 a−1

(5.2)

(5.3)

(5.4)

Then the space of the induced representation * J

indG * χ BJ J → C satisfying (5.2) and invariant on  : G consists exactly of functions W  the right by an open subgroup of GJ . The latter condition is also satisfied by m,0 m  coming from W ∈ WSW the above W , because πSW is smooth. We see that * J m,0 m,0 G  J  W → W maps WSW into indB*J χ. If we let G act on WSW by (ρ(g  , ε)W )(g) = ελ(g, g  )W (gg  ), m J , then W → W  becomes an interi.e. we view πSW as a representation of G m twining map. Since πSW is irreducible, we have proved the following. m J occurs as a 5.2.6 Proposition. The Schr¨ odinger-Weil representation πSW of G subrepresentation in the induced representation * J

indG * χ, BJ J and the character χ are given by (5.3) and (5.4). where B

112

5. Local Representations: The p-adic Case

5.3 Representations of the metaplectic group In this section we repeat the classification of the irreducible, admissible representations of the metaplectic group Mp over the p-adic field F , as was given by Waldspurger in [Wa1]. We are interested in the genuine representations, i.e., those which do not factor through G = SL(2, F ), or equivalently, those for which (1, −1) ∈ Mp operates as the negative of the identity. At first we have the following lemma ([Wa1] II, lemme 3). 5.3.1 Lemma. Every genuine irreducible admissible representation of Mp is infinite dimensional. Next, one can single out the supercuspidal representations. An irreducible admissible representation π : Mp → GL(V ) is called supercuspidal if for every v ∈ V there is an integer n such that    1x π v dx = 0. 01 ωn O

Here one should remember that the short exact sequence defining the metaplectic group splits over N , so that N can be considered a subgroup of Mp, as we did. By a familiar reasoning (cf. [Ge2] p. 96) one comes to the conclusion that any non-supercuspidal irreducible admissible representation is a subquotient of some representation induced by a character of the ‘torus’     a 0 ∗ ˜ A= , ε : a ∈ F , ε ∈ {±1} . 0 a−1 ˜ i.e., those which map (1, −1) ∈ Mp We are interested in genuine characters of A, to −1. One of them is what we call the Weil character δu . It is defined by    a 0 δu , ε = ε(a, −1)γu (a)γu (1)−1 0 a−1 where ( , ) is the Hilbert symbol and γu the Weil constant, both explained in Section 2.5. The parameter u ∈ F will later on usually be fixed to −m when dealing with representations of GJ with central character ψ m . We write δu (a) for δu (d(a), 1). The following lemma gives the basic properties of the Weil character. Proofs can be found in [Rao] and [Sch1] (see also [Wa1], lemme 1, and [GePS2], p. 150). 5.3.2 Lemma. i) δu (a)δu (a ) = (a, a )δu (aa ) for all a, a ∈ F ∗ . In other words, δu is a ˜ (genuine) character of A. ii) δu (a)−1 = (a, −1)δu (a) = δ−u (a) for all a ∈ F ∗ . iii) δu (a) = 1 for all a ∈ F ∗2 .

5.3. Representations of the metaplectic group

113

If the residue characteristic of F is not 2, then we have in addition: iv) δur (a) = (a, r)δu (a) for all a, r ∈ F ∗ . v) δu (a) = 1 for all a ∈ O∗ , if v(u) is even. vi) δu (a) = (a, u) for all a ∈ O∗ . Other genuine characters of A˜ can be built simply by adjoining characters of A: If χ is a character of A, then    a 0 , ε −→ εδu (a)χ(a) 0 a−1 ˜ These are the ones to be used now for the defines a genuine character of A. induction procedure. Namely let Bχ,u be the space of functions ϕ : Mp → C with the following two properties: i) ϕ is right invariant by some open subgroup of Mp. ii) For every a ∈ F ∗ , ε ∈ {±1}, x ∈ F and g ∈ Mp the following holds:      a 0 1x ϕ ,ε g = εδu (a)|a|χ(a)ϕ(g). 0 a−1 01 The action of Mp on Bχ,u is given by right translation and denoted by ρχ,u . Here is Waldspurger’s result concerning the irreducibility of these induced representations: 5.3.3 Theorem. i) If χ2 = | | and χ2 = | |−1 , then ρχ,u is irreducible. ii) If χ2 = | |, then one can find ξ ∈ F ∗ such that χ = | |1/2 (·, ξ), where ( , ) is the Hilbert symbol. Bχ,u contains exactly one non-trivial proper invariant subspace Bξ,u . The representation on the factor space is isomorphic to the positive (even) Weil representation with character ψ uξ . iii) If χ2 = | |−1 , then χ = | |−1/2 (·, ξ) with some ξ ∈ F ∗ . The induced representation Bχ,u contains exactly one non-trivial proper invariant subspace, and the representation on it by right translation is isomorphic to the positive Weil representation with character ψ uξ . The representation on the factor space is isomorphic to Bξ,u . Consequently we have the following complete list of irreducible, admissible, genuine, non-supercuspidal representations of Mp. 5.3.4 Definition. i) If χ2 = | | and χ2 = | |−1 , then ρχ,u is called a principal series representation . and denoted by π ˜χ,u . If m ∈ F ∗ is fixed and u = −m, then we denote it by π ˜χ .

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5. Local Representations: The p-adic Case

ii) If χ = | |1/2 (·, ξ), then the representation on Bξ,u is called a special representation and denoted by σ ˜ξ,u . For fixed m ∈ F ∗ and u = −m it is denoted by σ ˜ξ . The positive Weil representation on Bχ,u /Bξ,u with uξ+ character ψ uξ is denoted by πW . We will see later (in Sections 5.6 and 5.8) what equivalences there are between these representations. A comment on the irritating appearance of the character ψ −m will be made in Remark 5.3.6 below. To be precise, the above theorem does not appear in this form in [Wa1]. This is because Waldspurger restricts to the case |χ| = | |α with α ≥ 0, which can be done in view of the following lemma ([Wa1], lemme 4). 5.3.5 Lemma. The contragredient representation of Bχ,u is Bχ−1 (·,−1),u (where (· , ·) denotes the Hilbert symbol). (We make the usual misuse of notation and identify representations with their representation spaces). We give a proof of case iii) in the above theorem, which is not treated explicitly by Waldspurger. Let χ = | |−1/2 (·, ξ) with ξ ∈ F ∗ . By the preceding lemma we have Bχ  Bˇχ−1 (·,−1) = Bˇ| |1/2 (·,−ξ) . (we omit the u from the notation, it is fixed). It follows that the invariant subspaces of Bχ correspond to the invariant quotients of B| |1/2 (·,−ξ) . By part ii) of the theorem, there exists a unique proper non-trivial subspace Bξ of Bχ , and the following holds:  ∨  ∨ −ξu+ ξu+ Bξ  B| |1/2 (·,−ξ) /B−ξ  πW  πW . Now, it will be proved in Proposition 5.6.4 that there is always a non-zero intertwining map Bχ −→ Bχ−1 .

(5.5)

If, for our χ, this map were injective, then ξu+ πW  Bξ  Bξ = σξ

would hold, which is not the case. Hence (5.5) has the kernel Bξ , and Bχ /Bξ



Bξ .



Among the supercuspidal representations there are the negative (odd) Weil representations u− , πW

u ∈ F ∗ /F ∗2 ,

5.4. Induced representations

115

defined with the character ψ u . This is proved in [Ge2] 5.5. They are very important and should be mentioned explicitly instead of just putting them in with the supercuspidals. Indeed, they turn out to be much more important then u+ the πW , which do not appear as local factors in automorphic representations of Mp (Proposition 23 in [Wa1]). We now fix an m ∈ F ∗ , and come up with the following complete list of irreducible, admissible, genuine representations of Mp(F ). • Those supercuspidal representations which are not equal to negative Weil representations. • The principal series representations πχ with χ2 = | |±1 . • The special representations σξ with ξ ∈ F ∗ /F ∗2 . −mξ+ • The positive (even) Weil representations πW with ξ ∈ F ∗ /F ∗2 . −mξ− • The negative (odd) Weil representations πW with ξ ∈ F ∗ /F ∗2 .

5.3.6 Remark. a) One should always keep in mind that the symbols πχ and σξ depend on m ∈ F ∗ , and that everything depends on the underlying character ψ. Such a dependence on the choice of a character seems to complicate matters, but is indeed an important feature of the metaplectic theory, as becomes apparent while reading [Wa1]. For the Jacobi theory, the character has the following m significance. The Schr¨ odinger-Weil representation πSW is constructed with unm derlying character ψ . It has a certain cocycle λ, since it is projective. The metaplectic representations from above are defined with underlying character ψ −m , and consequently, they have a cocycle exactly inverse to λ. Tensorizing m πSW with such a representation thus makes the two cocycles cancel, and yields a non-projective representation of the Jacobi group. b) If ξ runs through F ∗ /F ∗2 , then −mξ certainly does also. In the above list ξ± −mξ± we could therefore simply have written πW instead of πW . But we prefer the more complicated form since it makes reference to the character ψ −m and is consistent with the symbol σξ , cf. Theorem 5.3.3 ii).

5.4 Induced representations Now we define induced representations for the Jacobi group. Let χ be a characJ ter of A, and let Bχ,m be the space of functions Φ : GJ → C with the following properties. i) Φ is right invariant by some open subgroup of GJ . ii) For every a ∈ F ∗ , x, μ, κ ∈ F and g ∈ GJ the following holds:    a x Φ (0, μ, κ)g = χ(a)ψ m (κ)|a|3/2 Φ(g). 0 a−1

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5. Local Representations: The p-adic Case

J GJ operates on Bχ,m by right translation. This is exactly the representation J of G induced by the character   a 0 (0, 0, κ) −→ χ(a)ψ m (κ) 0 a−1 J of AZ. Theorem 5.4.2 below relates Bχ,m to induced representations for the metaplectic group as they were introduced in the last section. We need a lemma for preparation.

5.4.1 Lemma. Let U be a space of locally constant functions Φ : GJ → C with Φ((0, μ, κ)g) = ψ m (κ)Φ(g)

for all μ, κ ∈ F, g ∈ GJ .

Assume further that U is invariant under right translation, and that the representation of H on U defined by right translation (with elements of H) is equivalent to πSm . Then there is a locally constant function ϕ : G → C such that every Φ ∈ U is of the form Φ(M h) = ϕ(M )W (M h) m,0 m with some W (depending on Φ) in the Whittaker model WSW for πSW with trivial character.

Proof: For any M ∈ G the map which associates to every Φ ∈ U the function h → Φ(M h) on H is an intertwining map for right translation with H on both sides. By Corollary 5.2.3, the image is either the zero space or the space WM defined in this corollary. We further have the isomorphism   m,0 WSW −→ WM , W −→ h → W (M h) , which is also an intertwining map for the action of H. Note the image is indeed WM by Corollary 5.2.3. Finally there is the isomorphism ∼

m,0 WSW −→ W1

given by restriction. Now we fix M and compare the map U −→ WM from the beginning to the composition m,0 U −→ W1 −→ WSW −→ WM .

Everything in sight intertwines the H-action, and consequently, by Schur’s lemma, these two maps differ only by a scalar, which we call ϕ(M ). Going through the definitions we see that for every Φ ∈ U and h ∈ H Φ(M h) = ϕ(M )W (M h) m,0 with some W ∈ WSW depending only on Φ. Given such W = 0 and M ∈ G, m,0 one sees by the construction of WSW , that there exists a h ∈ H such that W (M h) = 0. This proves finally that ϕ is locally constant. 

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5.4. Induced representations

5.4.2 Theorem. Let χ be a character of A and m ∈ F ∗ . Let Bχ be the representation of Mp introduced in the last section, where the reference character is J fixed to ψ −m , and Bχ,m the representation of GJ defined at the beginning of m the present section. If S(F ) is as usual taken as a model for πSW , then there is a canonical isomorphism Bχ ⊗ S(F )

J −→ Bχ,m ,  

−→ M h → ϕ(M, 1)Wf (M h) ,

ϕ⊗f

(5.6)

which commutes with the GJ -operation. Here Wf denotes the Whittaker funcm,0 tion in the model WSW corresponding to f ∈ S(F ). Proof: The functions in Bχ transform according to       a x a 0 ϕ M, 1 = λ , M χ(a)δ−m (a)|a|ϕ(M ). 0 a−1 0 a−1 m,0 For W ∈ WSW we have the transformation formula (5.1). Now using property ii) from Lemma 5.3.2 we see at once that the map (5.6) is well defined. Now we J show the surjectivity. By the Stone–von Neumann theorem Bχ,m decomposes after restriction to H into irreducible representations U isomorphic to πSm . Any such U fulfills the hypotheses of the preceding lemma. So we just have to check that with the function ϕ appearing in this lemma the function

(g, ε) −→ εϕ(g) m,0 J lies in Bχ . This is easily done using the fact that for every W ∈ WSW on G and every M ∈ G there exists an h ∈ H such that W (M h) = 0. Hence we have shown the surjectivity of our map. For the injectivity one can reason as  follows. Since π  is isotypical, there is a natural isomorphism H

∼

J J HomH (S(F ), Bχ,m ) ⊗ S(F ) −→ Bχ,m .

(5.7)

If we associate to any ϕ ∈ Bχ the H-intertwining map S(F ) f

J −→ Bχ,m ,  

−→ M h → ϕ(M, 1)Wf (h) ,

we get an obviously injective map J Bχ −→ HomH (S(F ), Bχ,m ).

Tensorizing yields from this an injection J Bχ ⊗ S(F ) −→ HomH (S(F ), Bχ,m ) ⊗ S(F ).

The composition of this injection with the isomorphism (5.7) is nothing but the map in the theorem, which is therefore injective. 

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5. Local Representations: The p-adic Case

Remark: The occurrence of Whittaker models in the canonical isomorphism mentioned in the theorem might at first glance seem strange. But one should m,0 remember that Proposition 5.2.6 identifies WSW as a representation space induced by a character of a torus, and thus this Whittaker model fits into the picture. 5.4.3 Lemma. Let V be a smooth representation of Mp and take S(F ) as the m space for πSW . Then the GJ -invariant subspaces of V ⊗ S(F ) are precisely those of the form W ⊗ S(F ) with W a Mp-invariant subspace of V . Proof: Let U ⊂ V ⊗ S(F ) be GJ -invariant. Define W = {v ∈ V : v ⊗ S(F ) ⊂ U }. It is trivial that W ⊗ S(F ) ⊂ U . But even equality holds, because of the Stone– von Neumann theorem and the interpretation of V as HomH (S(F ), V ⊗ S(F )).  Theorem 5.4.2 together with the above lemma and the classification of the induced representations of Mp in Theorem 5.3.3 yields the following classification J of the representations Bχ,m of GJ . 5.4.4 Theorem. J i) If χ2 = | | and χ2 = | |−1 , then Bχ,m is irreducible.

ii) If χ2 = | |±1 , then one can find ξ ∈ F ∗ such that χ = | |±1/2 (·, ξ), where J ( , ) is the Hilbert symbol. Bχ,m contains exactly one non-trivial proper J invariant subspace B±,ξ,m . We have + J J B∓,ξ,m  B|J|±1/2 (·,ξ),m B±,ξ,m . + J J The representation on B|J|1/2 (·,ξ),m B+,ξ,m resp. on B−,ξ,m is isomorphic −mξ+ m to πSW ⊗ πW .

5.4.5 Definition. J i) If χ2 = | | and χ2 = | |−1 , then the representation on Bχ,m is called a principal series representation and denoted by πχ,m . J ii) If χ = | |1/2 (·, ξ), then the representation on B+,ξ,m is called a special representation and denoted by σξ,m . The representation on the quotient J J Bχ,m /B+,ξ,m is called a positive Weil representation and is denoted by + σξ,m .

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5.5. Supercuspidal representations

J 5.4.6 Proposition. The contragredient representation to Bχ,m is BχJ−1 ,−m .

Proof: By Lemma 5.3.5 the representation Bχ−1 (·,−1),m is contragredient to Bχ,m , and by (ii) in Lemma 5.3.2 is identical to Bχ−1 ,−m . This means that there is a non-degenerate bilinear Mp-invariant pairing  , 0 : Bχ,m × Bχ−1 ,−m −→ C. Consider further the bilinear non-degenerate pairing  , 1 : S(F ) × S(F ) (f , f  )

−→ C, 

−→ f (x)f  (x) dx. F

m −m If GJ acts on the left factor S(F ) by πSW and on the right by πSW , then  , 1 J is G -invariant. Taking these two pairings together we get a third one on the tensor products

Bχ,m ⊗ S(F ) × Bχ−1 ,−m ⊗ S(F ) 

−→ C,



(ϕ ⊗ f , ϕ ⊗ f ) −→ ϕ, ϕ 0 f, f  1 . This is easily seen to be GJ -invariant and non-degenerate, if GJ acts on the first and the second factor S(F ) as before. The assertion follows now from Theorem 5.4.2. 

5.5 Supercuspidal representations In Section 5.3 we already defined the notion of supercuspidal representation for the metaplectic group. Now we make an analogous definition for the Jacobi group. So let π be an admissible representation of GJ (F ) on a vector space V . Then we call π supercuspidal (relative to N J ) if for every v ∈ V there exists an open compact subgroup N of N J such that  π(n)v dn = 0. N

We look at representations π of GJ of the form m π=π ˜ ⊗ πSW ,

where π ˜ is an admissible representation of Mp, considered as a projectice representation of G with cocycle λ. 5.5.1 Proposition. For π as above we have π supercuspidal

⇐⇒

π ˜ supercuspidal.

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5. Local Representations: The p-adic Case

Proof: Assume π ˜ supercuspidal. For given v ∈ V there exists an integer r such that    1x π v dx = 0. 01 ωr O m If there is further given an f ∈ S(F ), our standard model for πSW , we can find an integer s such that for every y ∈ F the integral     1 ψ 2my μ + xy f (y) dμ 2 ωs O

is independent from x ∈ ω r O, because f has compact support. We conclude that     1x m πSW (0, μ, 0) f dμ does not depend on x ∈ ω r O. 01 ωs O

Therefore  



  1x (0, μ, 0) (v ⊗ f ) dx dμ 01 ωr O ωs O            1x 1x m = π ˜ v ⊗ πSW (0, μ, 0) f dx dμ 01 01 ωr O ωs O         1x m = π ˜ v ⊗ πSW (0, μ, 0)f dx dμ 01 ωr O ωs O     1x m = π ˜ v dx ⊗ πSW (0, μ, 0)f dμ = 0. 01 m (˜ π ⊗ πSW )

ωr O

ωs O

m This shows that π = π ˜ ⊗ πSW is supercuspidal. Assume conversely this is true. Given v ∈ V , we set f = 1O and find integers r, s such that      1x m (˜ π ⊗ πSW ) (0, μ, 0) (v ⊗ f ) dx dμ = 0. 01 ωr O ωs O

This equation remains true if we shift s far into the negatives. Then the above argument shows again that the right side of the tensor is independent from x ∈ ω r O, and we arrive at     1x m π ˜ v dx ⊗ πSW (0, μ, 0)f dμ = 0. 01 ωr O

ωs O

5.6. Intertwining operators

121

But the Schwartz function  m πSW (0, μ, 0)f dμ ωs O

does not vanish at 0, hence the first integral equals zero. This proves that π ˜ is supercuspidal.  Just like in the reductive theory, we have the following subrepresentation theorem. 5.5.2 Proposition. If an irreducible, admissible representation of GJ (with central character ψ −m ) is not supercuspidal, then it is a principal series, special, J or positive Weil representation, i.e., it is a subquotient of some Bχ,m . Proof: Given π an irreducible, admissible representation of GJ , we can write m it as π = π ˜ ⊗ πSW with an irreducible, admissible representation π ˜ of Mp. If π is not supercuspidal, then π ˜ is also not. But for the metaplectic group the subrepresentation theorem holds, cf. [Ge2], Section 5.1, which means that π ˜ is a subrepresentation or a quotient of some Bχ . In view of Theorem 5.4.2, the assertion for the Jacobi group follows.  From Proposition 5.5.1 and the remarks made after Lemma 5.3.5, we see that there exist some supercuspidal representations of GJ which deserve special attention, namely the representations − m −mξ− σξ,m := πSW ⊗ πW

coming from negative Weil representations of the metaplectic group. We call them also negative Weil representations of GJ . They will appear in our final classification of the irreducible, admissible representations of GJ (F ) in Section 5.8.

5.6 Intertwining operators The results of the last two sections give an overview over the irreducible, admissible representations of GJ . For a complete classification, it remains to describe the equivalences between them. This section makes a step in this direction by introducing intertwining operators between representations which one would expect to be equivalent. After that we will show that there are no other equivalences except these obvious ones, by utilizing Whittaker and Kirillov models. First we deal with the metaplectic group and its induced representations Bχ , as they were defined in Section 5.3. From the general theory for reductive groups one expects that χ and χ−1 yield the same representation of Mp. We

122

5. Local Representations: The p-adic Case

are proving this fact by writing down an intertwining operator very similar as in the case of SL(2). Namely, for ϕ ∈ Bχ we define      1x ˜ (Iϕ)(g) = ϕ (w, 1) g dx, g ∈ Mp. (5.8) 01 F

5.6.1 Proposition. Let χ be a character of F ∗ and σ ∈ R be defined by |χ(a)| = |a|σ . If σ > 0, then the integral (5.8) converges absolutely for every g ∈ G, the ˜ lies in Bχ−1 , and the association ϕ → Iϕ ˜ is a nonzero intertwining function Iϕ map Bχ → Bχ−1 . Proof: For the question of convergence we note that                1x ϕ (w, 1) 1 x g  = ϕ w , 1 g     01 01   −1       −x 1 1 0  = ϕ , 1 g  −1 0 −x x 1        1 0 −1 −σ  = |x| |x| ϕ , 1 g  . −1 x 1 If the absolute value of x tends to infinity, then the elements    1 0 ,1 x−1 1 converge to the identity in Mp (if this should not be clear because of the strange topology in Mp, conjugate by (w, 1) and use the fact that N is contained in Mp). Therefore we have for x large enough        ϕ (w, 1) 1 x g  = |x|−1 |x|−σ |ϕ(g)| ,   01 so that the absolute convergence of (5.8) is equivalent with the convergence of  |x|−σ−1 dx (c > 0). |x|>c

But it is a standard p-adic computation to show that the latter integral is finite exactly for σ > 0. ˜ ∈ Bχ−1 . It is clear that We have to show that indeed Iϕ    1x ˜ ˜ Iϕ g = Iϕ(g) for all x ∈ F, g ∈ Mp, 01 so it remains to prove that for all a ∈ F ∗ , ε ∈ {±1} and g ∈ Mp     a 0 ϕ , ε g = εδm (a)|a|χ−1 (a)ϕ(g). 0 a−1 But this is straightforward and may therefore be left to the reader.

123

5.6. Intertwining operators

It is clear now by the definition that I˜ intertwines Bχ with Bχ−1 . Finally it has to be shown that I˜ is not the zero operator. For this purpose we utilize the Bruhat decomposition in SL(2, F ) which says SL(2, F ) = N A ∪ N AwN (disjoint). A function ϕ on Mp can therefore be defined by  ϕ

    a b 1x w , ε = εδm (a)χ(a)|a| 0 a−1 01

if a ∈ F ∗ , b ∈ F, x ∈ O,

and ϕ(g) = 0 if g is not of the indicated form. Then it is obvious that ϕ lies in ˜ does not vanish at g = 1. So I˜ is not zero. Bχ , and that Iϕ  Let us still assume that |χ| = | |σ with σ > 0. Then the intertwining operator I˜ : Bχ → Bχ−1 gives via the isomorphism ∼

J Bχ ⊗ S(F ) −→ Bχ,m

rise to an operator J I : Bχ,m −→ BχJ−1 ,m , J which we shall now compute. For this purpose let Φ ∈ Bχ,m be of the form

Φ(M h) = ϕ(M, 1)Wf (M h)

for all M ∈ G, h ∈ H,

m,0 where ϕ ∈ Bχ , f ∈ S(F ), and Wf ∈ WSW the Whittaker function corresponding to f . Then

˜ (IΦ)(M h) = (Iϕ)(M, 1)Wf (M h)      1x = ϕ (w, 1) (M, 1) Wf (M h) dx 01 F        1x 1x = λ w ,M λ ,M 01 01         1x 1x · ϕ w M, 1 Wf M h dx 01 01 F     1x = λ w ,M 01          1x 1x · ϕ w M, 1 Wf w−1 w M h dx. 01 01

(5.9)

F

Now we utilize the following lemma, which for later use is stated in greater generality than presently required.

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5. Local Representations: The p-adic Case 2

m,r r 5.6.2 Lemma. Let WSW be the ψ ν,r -Whittaker model with parameters 4m and r ∈ F for the Schr¨ odinger-Weil representation (cf. Theorem 5.2.4). Then for m,r every M ∈ G, h ∈ H and W ∈ WSW the following holds:  W (w−1 M h) = λ(w−1 , M )γm (−1) W ((μ, 0, 0)M h)ψ(−rμ) dμ. F

Proof: Let W = Wf with f ∈ S(F ). Then one computes, using explicit formulas for λ and the Schr¨ odinger-Weil representation,      r  −1 0 m W (w−1 M h) = πSW wM h f 0 −1 2m   r  m m = λ(−1, wM ) πSW (−1)πSW (wM h)f 2m   −r  m m = λ(−1, wM )λ(w, M )δm (−1) πSW (w)πSW (M h)f 2m    −1 m = λ(w , M )λ(−1, w)δm (−1)γm (1) πSW (M h)f (μ)ψ(−rμ) dμ F

= λ(w−1 , M )(−1, −1)δm(−1)γm (1)    m πSW ((μ, 0, 0)M h)f (0)ψ(−rμ) dμ = λ(w−1 , M )γm (−1)



F

W ((μ, 0, 0)M h)ψ(−rμ) dμ.



F

 Now, using this lemma for r = 0 and w

 1x M instead of M , we can go on 01

with the calculation (5.9). ˜ (IΦ)(M h) = (Iϕ)(M, 1)Wf (M h)         1x 1x = γm (−1)λ w, M λ w−1 , w M 01 01           1x 1x ϕ w M, 1 Wf (μ, 0, 0)w M h dμ dx 01 01 F F     1x = γm (−1)λ(w−1 , w)λ 1, w M 01           1x 1x ϕ w M, 1 Wf w (0, μ, 0)M h dμ dx 01 01 F F       1x = γm (−1) Φ w (0, μ, 0)M h dμ dx. 01 F F

125

5.6. Intertwining operators

Up to the factor γm (−1), this is the exact analogue to the definition (5.8), because the double integral may be interpreted as integration over N J . But it is not hard to see that this time the convergence is not absolute, hence the order of integration must not be changed. In view of Proposition 5.6.1 we have proved the following. J 5.6.3 Proposition. Let |χ| = | |σ with σ > 0. For any Φ ∈ Bχ,m and g ∈ GJ the integral       1x (IΦ)(g) = γm (−1) Φ w (0, μ, 0)g dμ dx 01 F F

is convergent, but not absolutely convergent. The map Φ → IΦ is a nonzero J intertwining map Bχ,m → BχJ−1 ,m , which coincides with the operator I˜ ⊗ id via ∼

J the isomorphism Bχ ⊗ S(F ) −→ Bχ,m . J If Bχ,m and BχJ−1 ,m are both irreducible, and if |χ| = | |σ with σ > 0, then J this proposition implies that Bχ,m  BχJ−1 ,m . The same is true if σ < 0 by interchanging the roles of χ and χ−1 . It remains to treat the case σ = 0, which indeed turns out to be the most interesting one, because these representations constitute the unitary principal series.

We can write our character χ (in a non-unique way) as χ(a) = |a|s χ0 (a)

(a ∈ F ∗ ),

where χ0 is a unitary character of F ∗ and s ∈ C. The above σ is nothing but Re(s). We can get into the region σ = 0 by viewing the intertwining integrals as functions of s in the domain Re(s) > 0, and then continue analytically. This J will be done here on both sides of the isomorphism Bχ ⊗ S(F )  Bχ,m , i.e., for the metaplectic group as well as for the Jacobi group. The metaplectic case is treated first. The Iwasawa decomposition G = BK (K = SL(2, O)) implies that any ϕ ∈ Bχ is determined by its restriction to ˜ = (K, {±1}) ⊂ Mp. This restriction satisfies K     a x ϕ , ε k = εδm (a)χ0 (a)ϕ(k) 0 a−1 for a ∈ O∗ , x ∈ O, ε ˜ constant functions K it is obvious that for function ϕs ∈ Bχ0 | |s . section.

˜ Define V to be the space of all locally ∈ {±1}, k ∈ K. → C having this property. Now if ϕ ∈ V is fixed, then every s ∈ C there exists a unique extension of ϕ to a As in [Bu] p. 350 we refer to the map s → ϕs as a flat

126

5. Local Representations: The p-adic Case

5.6.4 Proposition. Let s → ϕs be the flat section corresponding to a fixed ϕ ∈ V . For fixed g ∈ Mp the integral      ˜ s )(g) = ϕs (w, 1) 1 x g dx (Iϕ 01 F

defines a holomorphic function on the domain Re(s) > 0. This function has analytic continuation to all s except where χ = χ0 | |s = 1, and defines a nonzero intertwining operator Bχ0 | |s −→ Bχ−1 | |−s . 0

Proof: Because of     a x ˜ (Iϕs ) , ε g = ε|a|1−s χ0 (a)−1 δm (a)ϕs (g), 0 a−1 ˜ Similar to the proof of Proposition 5.6.1 we find we may assume that g ∈ K. a positive integer N ∈ N such that     1 0 ϕ , 1 g = ϕ(g) if |x| > q N . x−1 1 , , ˜ s )(g) into Then we split the integral (Iϕ and |x|>qN . The first integra|x|≤qN tion being over a compact set, analytic continuation is no problem. The second integral equals   ∞  dx −n −s n ϕ(g) |x|−s−1 χ−1 (x) dx = ϕ(g) |ω | χ (ω ) χ−1 . 0 0 0 (x) |x| n=N +1

|x|≥qN +1

O∗

If χ0 is not unramified, i.e. is not trivial on O∗ , then all the integrals over O∗ vanish, hence there is nothing to prove. Otherwise we arrive at ϕ(g)

∞  n=N +1

q

−ns

n

χ0 (ω) = ϕ(g)(q

−s

N +1

χ0 (ω))

∞ 

(q −s χ0 (ω))n ,

n=0

the multiplicative measure being suitably normalized. The last sum equals (1 − q −s χ0 (ω))−1 provided q −s χ0 (ω) = 1. This condition is equivalent to χ = 1, and the assertions about holomorphy and analytic continuation are ˜ s )(g) lies in B −1 −s , that the proved. The fact that the function g → (Iϕ χ0 | | analytically continued integral defines an intertwining operator, and that this operator is nonzero, now follow in a straightforward way from the identity theorem.  The results obtained in this proposition will now be taken over to the Jacobi J group. Every Φ ∈ Bχ,m is determined by its restriction to KH, this restriction satisfying    a x Φ (0, μ, κ)g = χ0 (a)ψ m (κ)Φ(g) 0 a−1

5.7. Whittaker models

127

for a ∈ O∗ , x, μ, κ ∈ F , g ∈ KH. If V J is the space of all such functions on KH, then for every s ∈ C any Φ ∈ V J can be extended uniquely to a function Φs ∈ Bχ0 | |s ,m by the rule    a x Φs kh = |a|3/2 χ(a)Φ(kh) for a ∈ F ∗ , x ∈ F, k ∈ K, h ∈ H. 0 a−1 The map s → Φs is again called the flat section belonging to Φ ∈ V J . 5.6.5 Proposition. Let Φ ∈ V J and s → Φs the corresponding flat section. For fixed g ∈ GJ the integral       1x (IΦs )(g) = Φs x (0, μ, 0)g dμ dx 01 F F

defines a holomorphic function in Re(s) > 0. This function can be analytically continued to all s where χ = χ0 | |s = 1, and after that the operator I defines a nonzero intertwining operator BχJ0 | |s ,m −→ BχJ−1 | |−s ,m . 0

Proof: This is an easy consequence of Propositions 5.6.3 and 5.6.4.



5.6.6 Corollary. The principal series representations for the characters χ and χ−1 are equivalent: πχ,m  πχ−1 ,m .

5.7 Whittaker models As usual, let π ˜ be an irreducible smooth representation of Mp and m π=π ˜ ⊗ πSW

be the corresponding irreducible smooth representation of GJ with central character ψ m on a space V . We would like to realize π as a space of functions W : GJ → C which transform according to    1x W (0, μ, 0)g = ψ n (x)ψ r (μ)W (g) for all x, μ ∈ F, g ∈ GJ , 01 where n, r ∈ F are parameters. If such a space exists such that right translation on it defines a representation equivalent to π, then π is said to have a ψ n,r Whittaker model (cf. the corresponding notions in the real case, Section 3.6). This model will then be denoted Wπn,r . It is said to be unique if there is only one such space in the space of all locally constant functions on GJ . If

128

5. Local Representations: The p-adic Case

v → Wv intertwines V with Wπn,r , then the corresponding Whittaker functional l : V → C, v → Wv (1), has the property      1x l π (0, μ, 0) v = ψ n (x)ψ r (μ)l(v) for all x, μ ∈ F, v ∈ V. 01 Conversely, any such functional on V yields a ψ n,r - Whittaker model via Wv (g) = l(π(g)v). The existence and uniqueness of Wπn,r is equivalent to the property that the space of ψ n,r -Whittaker functionals be one-dimensional. Very similar notions exist for representations of Mp. Here one requires the transformation property for the subgroup N of Mp. It should be clear what a ψ ν -Whittaker model or Whittaker functional for π ˜ is without stating all the details. m Whittaker models for πSW were discussed in Section 5.2. The unique ψ ν,r m Whittaker functional (ν = r2 /4m) for πSW coincides with the unique ψ r m Whittaker functional for πS and is given by  r  m,r lSW (f ) = f for all f ∈ S(F ). 2m

5.7.1 Proposition. Let π ˜ : Mp → GL(V ) be a smooth representation, and let m S(F ) be the standard space for πSW . The following are equivalent: i) There exists a ψ n,r -Whittaker functional l on V ⊗ S(F ). ii) There exists a ψ ν -Whittaker functional ˜l on V , where ν = n −

r2 4m .

In case of existence we have m,r l(v ⊗ f ) = ˜l(v)lSW (f )

for v ∈ V, f ∈ S(F ).

m So the space of ψ -Whittaker functionals for π = π ˜ ⊗ πSW is isomorphic to r2 ν the space of ψ -Whittaker functionals for π ˜ , where ν = n − 4m . In particular, if π ˜ is irreducible, then it has a unique ψ ν -Whittaker model if and only if π has a unique ψ n,r -Whittaker model. n,r

Proof: If ˜l : V → C is a ψ ν -Whittaker functional, then it is very easy to check that l : V ⊗ S(F ) v⊗f

−→ C, m,r

−→ ˜l(v)lSW (f ), 2

r defines a ψ n,r -Whittaker functional, with n = ν + 4m . Conversely, if a ψ n,r Whittaker functional l : V ⊗ S(F ) → C is given, then it is obvious that for fixed v ∈ V the linear map

S(F ) f

−→ C,

−→ l(v ⊗ f )

5.7. Whittaker models

129

defines a ψ r -Whittaker functional for πSm . By Theorem 5.2.1, this functional m,r differs from lSW (f ) only by a constant ˜l(v) depending on v, i.e. m,r l(v ⊗ f ) = ˜l(v)lSW (f ).

Now it is easy to check that v → ˜l(v) defines a ψ ν -Whittaker functional for π ˜, r2 with ν = n − 4m .  By this proposition, the existence and uniqueness question for the Whittaker models for representations of GJ is completely reduced to the metaplectic case. Concerning the induced representations Bχ of Mp (cf. Section 5.3) there are the following complete results of Waldspurger in [Wa1], where the reference character is as before fixed to ψ −m . For questions of convergence we have to ourselves to the case |χ| = | |α with α ≥ 0. This is enough in view of Proposition 5.6.4. On the space Bχ consider the functional ˜lν , ν ∈ F ∗ , given by      ˜lν (ϕ) = ϕ w 1 x , 1 ψ −ν (x) dx (ϕ ∈ Bχ ). 01 F

It is almost obvious that this is a ψ ν -Whittaker functional on Bχ provided it is non-zero. From [Wa1], Prop. 3, p. 14, one can deduce the following: • If χ2 = | | (i.e. the principal series case), then ˜lν is non-trivial for every ν ∈ F ∗. • If χ = | |1/2 (·, ξ), then the restriction of ˜lν to Bξ is non-trivial exactly for νF ∗2 = −mξF ∗2 . • If χ = | |1/2 (·, ξ), and νF ∗2 = −mξF ∗2 , then the resulting functional on Bχ /Bξ is non-trivial. In any case, l˜ν is unique up to scalars. The above proposition states that m,r l := ˜ lν ⊗ lSW defines a Whittaker functional on Bχ ⊗ S(F ). This will be J taken over now to Bχ,m (cf. Theorem 5.4.2). The calculation is only a slight generalization of (5.9), so we just state the result:       1x l(Φ) = Φ w (0, μ, 0) ψ −n (x)ψ −r (μ) dμ dx (5.10) 01 F F J for Φ ∈ Bχ,m (up to an irrelevant constant). This is exactly what one would expect. But just like in our discussion of intertwining integrals, the convergence of (5.10) is not absolute, so that the order of integration has to be observed. Waldspurger’s results, the above proposition, Theorem 5.4.2 and Corollary 5.2.5 add up to the following result.

130

5. Local Representations: The p-adic Case

5.7.2 Theorem. Let n, r ∈ F and N = 4mn − r2 . i) The principal series representations πχ,m have a ψ n,r -Whittaker model for all n, r ∈ F with N = 0. ii) The special representation σξ,m has a ψ n,r -Whittaker model if N = 0 and −N F ∗2 = ξF ∗2 . ± iii) The Weil representations σξ,m have a ψ n,r -Whittaker model if N = 0 and ∗2 ∗2 −N F = ξF .

In each case, if Wπn,r exists, it is unique. A non-trivial Whittaker functional on the standard space of the induced representation is given (resp. induced) by (5.10), provided |χ| = | |α with α ≥ 0. Let π ˜ be one of the induced representations of Mp (resp. an irreducible subquotient). Assume the Whittaker model Wπ˜ν exists for some ν ∈ F . It turns out to be important to consider the space of functions      a 0 ν K(˜ π ) := x −→ W , 1 : W ∈ Wπ˜ . 0 a−1 This space of functions is called the Kirillov model for π ˜ , although it is not a representation space for π ˜ , due to the fact that restriction of Whittaker functions to the diagonal in general is not injective. In [Wa1], Prop. 4, Waldspurger gives the following description of K(˜ π ) in terms of the even Schwartz space S(F )+ = {f ∈ S(F ) : ∀ x ∈ F f (x) = f (−x)}. • If π ˜=π ˜χ,−m is a principal series representation with χ2 = 1, then  K(˜ π ) = a → δ−m (a)|a|(χ(a)f1 (a) + χ−1 (a)f2 (a)) :  f1 , f2 ∈ S(F )+ . • If π ˜=π ˜χ,−m is a principal series representation with χ2 = 1, then  K(˜ π ) = a → δ−m (a)|a|(χ(a)f1 (a) + χ−1 (a)vF (a)f2 (a)) :  f1 , f2 ∈ S(F )+ . • If π ˜=σ ˜ξ,−m is a special representation, then   K(˜ π ) = a → δ−m (a)|a|χ(a)f (a) : f ∈ S(F )+ . −mξ+ • If π ˜ = πW is a positive Weil representation, then   K(˜ π ) = a → δ−m (a)|a|χ−1 (a)f (a) : f ∈ S(F )+ .

5.7. Whittaker models

131

If π is a representation of GJ with Whittaker model Wπn,r , then it seems reasonable to define the Kirillov model for π as      a 0 ∗ n,r K(π) := F × F  (a, λ) −→ W (λ, 0, 0) : W ∈ Wπ . 0 a−1 m We know that Whittaker functions for π = π ˜ ⊗ πSW are combined from Whitm,r m taker functions for π ˜ and those for πSW . Now for any W = Wf ∈ WSW , where f ∈ S(F ), one computes easily     ra  a 0 1/2 W f +λ (5.11) −1 (λ, 0, 0) = δm (a)|a| 0a 2m

for a ∈ F ∗ , λ ∈ F . Thus we have the following description of the Kirillov space for representations of GJ . 5.7.3 Proposition. Let π be a non-supercuspidal representation of GJ . Assuming that the ψ n,r -Whittaker model with parameters n, r ∈ F exists, the corresponding Kirillov space K(π) consists of all linear combinations of functions h : F ∗ × F → C of the following type: i) If π = πχ,m is a principal series representation with χ2 = 1, then h(a, λ) = |a|3/2 f (ra + 2mλ)(χ(a)f1 (a) + χ−1 (a)f2 (a))  (f ∈ S(F ), f1 , f2 ∈ S(F )+ . ii) If π = πχ,m is a principal series representation with χ2 = 1, then h(a, λ) = |a|3/2 f (ra + 2mλ)(χ(a)f1 (a) + χ−1 (a)vF (a)f2 (a))  (f ∈ S(F ), f1 , f2 ∈ S(F )+ . iii) If π = σξ,m is a special representation, then h(a, λ) = |a|3/2 f (ra + 2mλ)χ(a)f1 (a)  (f ∈ S(F ), f1 ∈ S(F )+ . + iv) If π = σξ,m is a positive Weil representation, then

h(a, λ) = |a|3/2 f (ra + 2mλ)χ−1 (a)f1 (a)  (f ∈ S(F ), f1 ∈ S(F )+ . Caution: The function f appearing in these formulas is not quite the Schwartz m,r function corresponding to W in the Whittaker model WSW . We have shifted the argument multiplicatively by 2m to avoid fractions, cf. (5.11).

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5. Local Representations: The p-adic Case

5.8 Summary and Classification We are almost ready to classify the irreducible, admissible representations of GJ . Only a few simple lemmas are still necessary. 5.8.1 Lemma. Let ε > 0 and χ, χ characters of F ∗ such that χ(a) = χ (a)

for all a ∈ F ∗ with |a| < ε.

Then χ = χ . Proof: Let a, a ∈ F ∗ be arbitrary. The hypotheses implies the existence of some N ∈ N such that χ(ω n a) = χ (ω n a)

for all n ≥ N.

The number (χ(ω)/χ (ω))n does not depend on n, hence χ(ω) = χ (ω). This in turn implies χ(a) = χ (a).  5.8.2 Lemma. Let c1 , c2 ∈ C, ε > 0 and χ, χ characters of F ∗ . If c1 χ(a) + c2 χ−1 (a) = χ (a)

for all a ∈ F ∗ with |a| < ε,

then χ = χ or χ = χ−1 . Proof: If χ(a) = χ (a)

for all |a| < ε,

then Lemma 5.8.1 shows χ = χ . Assume on the contrary that χ(a) = χ (a) for some a ∈ F ∗ with |a| < ε. The hypotheses implies  n  −1 n χ(a) χ (a) c1 + c =1 for all n ≥ 1. 2 χ (a) χ (a) It is a little exercise to deduce from this that c1 = 0. Then it follows quickly that χ = χ−1 .  5.8.3 Theorem. Here is a complete list of the irreducible, admissible representations of the p-adic Jacobi group GJ (F ) with non-trivial central character ψm . i) The principal series representations πχ,m . ii) The special representations σξ,m . + iii) The positive Weil representations σξ,m . − iv) The negative Weil representations σξ,m . − v) The supercuspidal representations not of the form σξ,m for some ξ ∈ F ∗ .

133

5.8. Summary and Classification

Between these representations there are exactly the following equivalences: πχ,m  πχ−1 ,m . σξ,m  σξ ,m

⇐⇒

ξF ∗2 = ξ  F ∗2 .

± σξ,m  σξ± ,m

⇐⇒

ξF ∗2 = ξ  F ∗2 .

Proof: By Theorem 5.4.4 and the subrepresentation theorem 5.5.2 the given list is complete. The indicated equivalences hold by Corollary 5.6.6 resp. the trivial fact that the Hilbert symbol (a, ξ) depends only on ξ mod F ∗2 . We have to show that there are no other equivalences. So assume first that πχ,m  πχ ,m for two principal series representations. We choose some Whittaker model for these and look at the corresponding Kirillov space. By the description in Proposition 5.7.3, there must exist c, d ∈ C such that χ (a) = cχ(a) + dχ−1 (a) for all a ∈ F ∗ with small enough absolute value. Thus by the preceding lemma, we have indeed χ = χ or χ = χ−1 . The same argument shows that a special or a Weil representation can not be equivalent to a principal series representation. It is even simpler to see that special and Weil representations are not equivalent. What remains to examine are the equivalences between two special resp. two Weil representations. So assume σξ,m  σξ ,m

for some ξ, ξ  ∈ F ∗ .

Again by looking at Kirillov models, we conclude that cχ(a) = χ (a) for a constant c ∈ C and all small enough a ∈ F ∗ . Then c is easily seen to equal 1, hence χ = χ by Lemma 5.8.1. This means (a, ξ) = (a, ξ  )

for all a ∈ F ∗ .

But the Hilbert symbol is non-degenerate in the sense that this equation implies ξF ∗2 = ξ  F ∗2 .  Now we summarize the results so far obtained on the irreducible, admissible representations of GJ . From the Stone-von Neumann theorem it follows that each smooth representation π of GJ with central character ψ m can be written as a tensor product m π=π ˜ ⊗ πSW

with a smooth representation π ˜ of the metaplectic group Mp. Table 5.1 lists the relationships between π and π ˜. By the subrepresentation theorem 5.5.2 every non-supercuspidal irreducible admissible representation (with non-trivial central character) is induced. Table 5.2 summarizes the main properties of those induced representations which are not positive Weil representations.

134

5. Local Representations: The p-adic Case π and π ˜ are simultaneously admissible irreducible supercuspidal induced pre-unitary

see Proposition 5.1.2 Lemma 5.4.3 Proposition 5.5.1 Theorem 5.4.2 Proposition 5.9.1

Table 5.1: Relationship between π and π ˜

name

principal series special representation representation

symbol

πχ,m

σξ,m −1

inducing character χ = | |, | | 2

χ = | |±1/2 (·, ξ), ξ ∈ F ∗

space

J Bχ,m

subspace of B|J|1/2 (·,ξ),m

isomorphic to

m πχ ⊗ πSW

m σξ ⊗ πSW

equivalences

πχ,m  πχ−1 ,m

σξ,m  σξa2 ,m ∀a ∈ F ∗

W n,r exists

if N = 0

if − ξN ∈ F ∗ \ F ∗2

Table 5.2: Non-archimedean principal series and special representations of GJ

name

positive and negative Weil representation

symbol

± σξ,m

isomorphic to

−mξ± m πW ⊗ πSW

equivalences

± ± ∗ σξ,m  σξa 2 ,m ∀a ∈ F

W n,r exists

if − ξN ∈ F ∗2

Table 5.3: Non-archimedean Weil representations of GJ We have not classified the supercuspidal representations, but among them we found the negative Weil representations. Table 5.3 lists the basic properties of the positive and negative Weil representations.

135

5.9. Unitary representations

5.9 Unitary representations As usual, a smooth representation π of a p-adic group G on a complex vector space V is called pre-unitary if there exists a non-degenerate positive definite hermitian form  ,  on V such that π(g)v, π(g)v   = v, v  

for all v, v  ∈ V, g ∈ G.

m 5.9.1 Proposition. Let π ˜ be a smooth representation of Mp and π = π ˜ ⊗ πSW J the corresponding smooth representation of G . Then π is pre-unitary if and only if π ˜ is pre-unitary. m Proof: Let V be the space of π ˜ and S(F ) the space of πSW . We know that the 2 ordinary L -scalar product  f, f  S(F ) = f (x)f  (x) dx (5.12) F m on S(F ) is invariant under πSW . Consequently, if π ˜ is pre-unitary with invariant positive definite form  , 0 on V , then one can define an invariant hermitian form  ,  on V ⊗ S(F ) with the property

ϕ ⊗ f, ϕ ⊗ f   = ϕ, ϕ 0 f, f  S(F ) . This form is indeed positive definite: Since S(F ) is a separable pre-Hilbert space, it has a countable orthonormal basis. Write any given element Φ = 0 of V ⊗ S(F ) in the form  Φ= ϕi ⊗ fi finite

with fi elements from such a basis and linear independent. Then obviously  Φ, Φ = ϕi , ϕi 0 > 0. Assume conversely that  ,  is an invariant, positive definite hermitian form an V ⊗ S(F ). For any fixed ϕ ∈ V the bilinear form (f, f  ) −→ ϕ ⊗ f, ϕ ⊗ f   is positive definite and H-invariant, and must consequently up to a constant coincide with the scalar product (5.12). Thus if we call this constant c(ϕ), then ϕ ⊗ f, ϕ ⊗ f   = c(ϕ)f, f  S(F ) . Now for ϕ, ϕ ∈ V define  1 ϕ, ϕ 0 := c(ϕ + ϕ ) − c(ϕ − ϕ ) + ic(ϕ + iϕ ) − ic(ϕ − iϕ ) . 4

136

5. Local Representations: The p-adic Case

A quick calculation yields ϕ, ϕ 0 f, f S(F ) = ϕ ⊗ f, ϕ ⊗ f 

for all f ∈ S(F ).

So we have indeed constructed an invariant positive definite hermitian form on V .  Using this proposition together with well-known metaplectic results, we arrive at the following theorem. 5.9.2 Theorem. Here is a complete list of all irreducible pre-unitary admissible representations of GJ with non-trivial central character ψ m : i) The principal series representations πχ,m where χ is a unitary character of F ∗ . ii) The principal series representations πχ,m where χ is a real valued character of F ∗ such that |χ| = | |σ

with

−

1 1 0 maps T J (ω α ) −→ q

3α 2

(X α + X −α ) + (1 − q −1 )q

3α 2

α−2  j=−α+2 j≡α mod 2

Xj

6.2. Structure of the Hecke algebra in the good case

147

Proof: It is clear that the indicated map is an isomorphism of vector spaces. The only thing to check is that the polynomials on the right fulfill the relations of the T J (ω α ) in Proposition 6.2.5. This is an elementary calculation.  The usual Satake isomorphism for SL(2) says H(G, K)  C[X ±1 ]W , but the connection between T J (ω α ) and   α   ω 0 α T (ω ) = char K K ∈ H(G, K) 0 ω −α is not as simple as one might think. The latter element identifies with the polynomial q α (X α + X −α ) + q α (1 − q −1 )

α−1 

Xj,

j=−α+1

and the multiplication law is T (ω)2 = T (ω 2 ) + (q − 1)T (ω) + q 2 + q, T (ω)T (ω α) = T (ω α+1 ) + (q − 1)T (ω α ) + q 2 T (ω α−1 )

(α ≥ 2).

If we identify Hecke operators with polynomials, then one can explicitly formulate a relation between Jacobi- and SL(2)-operators. But since this relation is not very enlightning and will be of no use to us, we only mention this here and leave it as an exercise to be done if necessary. We also give the rationality theorem for SL(2), ∞  α=0

T (ω α )X α =

1 + (q − 1)X − qX 2 , 1 + (q − 1 − T (ω))X + q 2 X 2

which the reader might wish to compare with Corollary 6.2.7. We can see from these formulas that the Hecke algebras of GJ and of SL(2), both being polynomial rings, are isomorphic, but not “canonical”: The basic elements T J (ω) and T (ω) correspond to different polynomials. This situation changes if we do not compare GJ with SL(2), but with PGL(2). Let G1 denote the group PGL(2, F )  SO(3, F ) and K1 its maximal compact subgroup. The Satake isomorphism for the Hecke algebra of this pair is easily computed (see [Ca]) and yields H(G1 , K1 )  C[X ±1 ]W ,

(6.7)

148

6. Spherical Representations

the same polynomial ring as before. But if now T P GL (ω) denotes the basic Hecke operator, i.e., the characteristic function of the double coset ' ( ω0 K1 K1 , 01 then the isomorphism (6.7) sends T P GL (ω) to the polynomial q 1/2 (X + X −1 ). Therefore, we have a “canonical” isomorphism ∼

H(GJ , K J , ψ −m ) −→ H(G1 , K1 ), T J (ω) −→ qT P GL (ω). Via the well known connection between characters of the Hecke algebra and spherical representations (see Proposition 6.3.1 below) there should thus exist a “canonical” correspondence between spherical representations of GJ (F ) and PGL(2, F ). This is in fact true, and is a special case of a “lifting map” between automorphic representations of GJ and PGL(2). More details on this subject can be found in [Sch2].

6.3 Spherical representations in the good case In the previous section we have completely determined the structure of the local Hecke algebra H(GJ , K J , ψ −m ) in the good case. It turned out just to be a polynomial ring. In particular, the commutativity of this algebra allows us to conclude that all of its non-trivial finite dimensional irreducible representations are one dimensional, i.e., they are just characters (algebra homomorphisms H(GJ , K J , ψ −m ) → C). Furthermore, the commutativity allows for application of the business of spherical functions, just like in [Ca], 4.3, 4.4, and the conclusion is that the arrow in (6.3) is surjective. Summing up, we have the following result. 6.3.1 Proposition. In the good case, there is a natural 1-1 correspondence ⎧ ⎫ Irreducible, admissible,⎪ ⎪ ⎪ ⎪ ⎪ ⎨spherical representa- ⎪ ⎬   J tions of G with ←→ HomAlg H(GJ , K J , ψ −m ), C . ⎪ ⎪ ⎪ ⎪ (6.8) ⎪ ⎪ ⎩non-trivial central ⎭ m character ψ Next we want to figure out which of the representations of GJ classified in Section 5.8 are spherical. This is easy for the metaplectic group, and we will use the canonical isomorphism from Theorem 5.4.2 to take the results over to GJ . To do that we need some information on whether the Schr¨ odinger-Weil representation is spherical or not. Although we are mainly interested in the good case, we state Proposition 6.3.4 below in greater generality, because it will

6.3. Spherical representations in the good case

149

also be needed at another point. So for the moment we give up our assumption n = v(m) = 0, demand only n ≥ 0, and define ⎧ ⎪ ⎪ ⎨ n if v(m) even, 2 n0 := ⎪ n−1 ⎪ ⎩ if v(m) odd. 2 6.3.2 Lemma. If S(F ) denotes the standard space for the Schr¨ odinger-Weil representation, then we have   S(F )H(O) = f ∈ S(F ) : supp(F ) ⊂ ω −v(2m) O, f is O-invariant ,   S(F )N (O) = f ∈ S(F ) : supp(F ) ⊂ ω −n0 O . Proof: We have for all λ, μ, κ ∈ O

πSm (λ, μ, κ)f = f if and only if ψ(2mxμ)f (x + λ) = f (x)

for all λ, μ ∈ O, x ∈ F.

Hence f ∈ S(F )H(O) must be O-invariant. If f (x) = 0, then for all μ ∈ O.

ψ(2mxμ) = 1

From this it follows that 2mx ∈ O, i.e. v(x) ≥ −v(2m). This proves the first assertion. The second one is treated similarly.    0 1 In the sequel w denotes as usual the matrix . −1 0 6.3.3 Lemma. For all k ∈ Z we have m πW (w)1ωk O = γm (1)q −k−v(2m)/2 1ω−k−v(2m) O

(this holds even for any m ∈ F ∗ ). Proof: If dy denotes the additive measure on F which gives O the volume 1, then Fourier transformation has to be normalized by  −v(2m)/2 ˆ ψ(2mxy)f (y) dy f (x) = q F

150

6. Spherical Representations

to make Fourier inversion hold. Hence    m πW (w)1ωk O (x) = γm (1)q −v(2m)/2 ψ(2mxy)1ωk O (y) dy F

= γm (1)q

−v(2m)/2

 ψ(2mxy) dy.

ωk O

The integral is non-zero exactly for v(2mx) ≥ −k, in which case it equals q −k .  m 6.3.4 Proposition. The Weil representation πW contains a K-invariant vector if and only if F has odd residue characteristic and v(2m) is even. If this is fulfilled, then the K-invariant vector is unique up to scalar multiples, and is even K J -invariant. In the model S(F ) it is given by

1ω−v(m)/2 O . Proof: If n = v(m) is even and 2 is a unit in F , then 1ω−n/2 O is K J -invariant by the preceding two lemmas (note that γm (1) = 1 by Lemma 5.3.2 v)). Assume conversely that f ∈ S(F ) is K-invariant. By Lemma 6.3.2, for all y, y  ∈ F with v(y  ) ≥ n0 − n we have   m f (y + y  ) = πW (w)f (y + y  )  = γψm (1) ψ(2m(y + y  )z)f (z) dz F



= γψm (1)

ψ(2myz)ψ(2my  z)f (z) dz

ω −n0 O



= γψm (1)  =

ψ(2myz)f (z) dz

ω −n0 O

 m πW (w)f (y) = f (y),

i.e. f is ω n0 −n O − invariant.

(6.9)

If n is odd, then from this condition and Lemma 6.3.2 it follows that f = 0. If n is even, then it follows from the same conditions that f is a multiple of 1ω−n/2 O . Assuming it to be non-trivial, Lemma 6.3.3 for k = n/2 yields v(2) = 0, i.e., F is not an extension of Q2 .  Now we can make the connection between spherical representations of GJ and of the metaplectic group. Here we call a smooth representation π ˜ : Mp → GL(V ) spherical if there exists a non-zero v ∈ V with π ˜ (k, 1)v = v

for all k ∈ K.

6.3. Spherical representations in the good case

151

6.3.5 Proposition. Let π ˜ : Mp → GL(V ) be an admissible representation of m Mp and π = π ˜ ⊗ πSW the corresponding admissible representation of GJ with central character ψ m . Assume the good case v(m) = 0, and assume F has odd residue characteristic. Then π is spherical if and only if π ˜ is spherical. In this case J

(V ⊗ S(F ))K = V K ⊗ C1O

(6.10)

is one-dimensional. Proof: Let  v= ϕi ⊗ fi a K J -invariant vector in V ⊗ S(F ), where we may assume the ϕi to be linearly independent. The Heisenberg group acts only on the fi , and from the linear independence of the ϕi we conclude that fi ∈ S(F )H(O) for all i. Lemma 6.3.2 then says that all fi are multiples of 1O . Thus v is a pure tensor v = ϕ ⊗ 1O . Now 1O is K J -invariant by Proposition 6.3.4. Hence letting K act on v we see that ϕ is K-invariant, i.e. π ˜ is spherical.  Now it is our intention to figure out which of the representations of Mp are spherical. Remember the definition of the induced representations Bχ,−m in Section 5.3. 6.3.6 Lemma. The induced representation Bχ,−m of Mp is spherical if and only if χ(a) = δ−m (a)

for all a ∈ O∗ .

(6.11)

If the residue characteristic of F is odd and v(m) is even, then this is equivalent to χ being unramified. If Bχ,−m is spherical, then the (up to scalars) unique K-invariant vector is given by     a x ϕ , ε (k, 1) = εδ−m (a)|a|χ(a) (6.12) 0 a−1 for a ∈ F ∗ , x ∈ F, ε ∈ {±1}, k ∈ K. Proof: This is clear because of the Iwasawa decomposition ˜ Mp = BK (we have written K for the set of all (k, 1), k ∈ K): A K-invariant function ϕ ∈ Bχ,−m can be well-defined by (6.12) if and only if the inducing character is ˜ ∩ K. This condition is expressed by (6.11). The second assertion trivial on B follows by v) in Lemma 5.3.2. 

152

6. Spherical Representations

This lemma already tells us exactly in which cases the principal series representations of Mp are spherical. We need two more lemmas to treat the special representation. 6.3.7 Lemma. Let G → GL(V ) be an admissible representation of a p-adic group G , and let K  < G be an open and compact subgroup. Then every K  -invariant subspace of V has a K  -invariant complement. Proof: Every K  -representation is completely reducible, as is well known. Hence our assertion would be clear if V were finite-dimensional. But we can reduce to this case by decomposing V into (finite-dimensional!) isotypic components.  6.3.8 Lemma. If F has odd residue characteristic and (a, ξ) = 1

for all a ∈ O∗

(6.13)

holds for some ξ ∈ F ∗ , then v(ξ) is even. Proof: It is well known that the Hilbert symbol is trivial on O∗ × O∗ if F has odd residue characteristic. Hence (6.13) is true if v(ξ) is even. If it also holds for some ξ with v(ξ) odd, then it would hold for all ξ ∈ F . This would imply O∗ ⊂ F ∗2 . But in odd residue characteristic it is also well known that O∗2 has index 2 in O∗ .  6.3.9 Proposition. Assume the residue characteristic of F is odd and that we are in the good case v(m) = 0. We use the notion of Definition 5.3.4. i) The principal series representation πχ is spherical if and only if χ is unramified. ii) The special representation σξ is never spherical. −mξ iii) The positive Weil representation πW is spherical if and only if v(ξ) is even.

Proof: i) is already contained in Lemma 6.3.6. iii) follows from Proposition 6.3.4. For the special representation σξ to be spherical, the character χ = | |1/2 ( · , ξ) must be unramified by Lemma 6.3.6, because σξ is a subrepresentation of Bχ,−m . Thus (a, ξ) = 1

for all a ∈ O∗

must necessarily hold. By Lemma 6.3.8, the valuation v(ξ) must then be even. −mξ But in this case the Weil representation πW is spherical. Lemma 6.3.7 then implies that σξ is not spherical, since Bχ,−m contains the trivial representation of K at most once. 

6.4. Spherical Whittaker functions

153

Now we are ready to state the main result of this section. 6.3.10 Theorem. Assume the good case v(m) = 0, and assume F has odd residue characteristic. Then the following is a complete list of the irreducible, admissible, spherical representations of GJ (F ) with central character ψ m . i) The principal series representation πχ,m with unramified χ. + ii) The positive Weil representation σξ,m with v(ξ) even.

Proof: In view of Propositions 6.3.5 and 6.3.9 the only thing that remains to prove is that in the good case supercuspidal representations are not spherical. This will follow from the considerations in Section 6.5 (see Corollary 6.5.6). 

6.4 Spherical Whittaker functions In Theorem 6.3.10 we have determined – under good conditions – which of the irreducible, admissible representations of GJ are spherical. Now we want to compute explicitly the (up to scalars unique) spherical vector in the Whittaker models for these representations. Before doing this we give the spherical vector in another model, namely in the induced model, where it is easy to see. ConJ sider the space Bχ,m of the induced representation. To determine the spherical vector (if it exists) in this model we take together the following four items: J • The isomorphism Bχ ⊗ S(F )  Bχ,m from Theorem 5.4.2.

• The spherical vector in Bχ from Proposition 6.3.6. m • The spherical vector for πSW in S(F ) from Proposition 6.3.4

• The equation (6.10) in Proposition 6.3.5. 6.4.1 Proposition. Assume the good case v(m) = 0 and that F is not an exJ tension of Q2 . Then the representation Bχ,m is spherical if and only if χ is unramified, and in this case a spherical vector is given by    a x Φ (λ, μ, κ) = |a|3/2 χ(a)ψ m (κ + λμ)1O (λ) 0 a−1 for all a ∈ F ∗ and x, λ, μ, κ ∈ F . Proof: In view of the above remarks there remains only to compute the Whittaker function W1O corresponding to the Schwartz function 1O in the model m,0 m WSW . By Theorem 5.2.4 we have W1O (g) = (πSW (g)1O )(0) for all g ∈ GJ . An easy computation gives    a x W1O (λ, μ, κ) = δm (a)|a|1/2 ψ m (κ + λμ)1O (λ), 0 a−1 and the assertion follows.



154

6. Spherical Representations

We could try to compute the spherical vector in a ψ n,r -Whittaker model by applying to this result the Whittaker functional given by (5.10). However, this is not so easily done. Instead we prefer another method which utilizes the Hecke algebra. Let π either be a spherical principal series representation or a spherical positive Weil representation, induced from the unramified character χ of F ∗ (Theorem 6.3.10). In all that follows we restrict ourselves to the case v(2m) = 0, i.e., F has odd residue characteristic, and we are in the good case. Assume we have a ψ n,r -Whittaker model W of π, where N := 4mn − r2 = 0, (Theorem 5.7.2) and denote by W the (up to scalars) unique non-trivial K J invariant Whittaker function. By the ‘Iwasawa decomposition’ G = N AK and the Whittaker transformation property, we know W completely if we know the values ˜ λ)), W (d(a,

a ∈ F ∗ , λ ∈ F,

where in this section we abbreviate   0 ˜ λ) := a −1 d(a, (λ, 0, 0) ∈ GJ (F ) 0a

for a ∈ F ∗ , λ ∈ F.

˜ λ)) = 0, then N a2 ∈ O and ra + 2mλ ∈ O. 6.4.2 Lemma. If W (d(a, Proof: W is K J -invariant, hence for all x, μ ∈ F     ˜ λ)) = W d(a, ˜ λ) 1 x (0, μ, 0) W (d(a, 01   2 ˜ λ)) = ψ x(na + raλ + mλ2 ) + μ(ra + 2mλ) W (d(a, ˜ λ)) = 0, then it follows that must hold. If W (d(a,   ψ x(na2 + raλ + mλ2 ) + μ(ra + 2mλ) = 1

for all x, μ ∈ O.

But our choice of ψ then forces na2 + raλ + mλ2 ∈ O

and

ra + 2mλ ∈ O.

The assertion follows from the identity (ra + 2mλ)2 = 4m(na2 + raλ + mλ2 ) − N a2 and the hypothesis v(2m) = 0.



6.4. Spherical Whittaker functions

155

From Proposition 6.3.5 and the way Whittaker models for GJ -representations m are built from Whittaker models for Mp-representations and for πSW , we know that W must be of the form ˜ λ)) = WSW (d(a, ˜ λ))F (a), W (d(a, m with WSW the spherical Whittaker function for πSW (corresponding to the Schwartz function 1O ∈ S(F )) and F a function in the Kirillov model of the Mp-representation π ˜ corresponding to π. It is easy to compute WSW : By Theorem 5.2.4 we have   r  m ˜ ˜ λ)) = WSW (d(a, πSW (d(a, λ))1O 2m     r  a 0 = πW −1 πS (λ, 0, 0)1O 0a 2m    ra  = δm (a)|a|1/2 πS (λ, 0, 0)1O 2m  ra  = δm (a)|a|1/2 1O λ + 2m = δm (a)|a|1/2 1O (ra + 2mλ).

The last step is legitimized by the fact that 2m is a unit, and is carried out merely to avoid fractions. By our description of Kirillov models in Proposition 5.7.3 we arrive at the following cases. • If π = πχ,m is a spherical principal series representation with χ2 = 1, then there exist f1 , f2 ∈ S(F )+ such that for all a ∈ F ∗ , λ ∈ F   ˜ λ)) = |a|3/2 1O (ra + 2mλ) χ(a)f1 (a) + χ−1 (a)f2 (a) . W (d(a, • If π = πχ,m is a spherical principal series representation with χ2 = 1, then there exist f1 , f2 ∈ S(F )+ such that for all a ∈ F ∗ , λ ∈ F   ˜ λ)) = |a|3/2 1O (ra + 2mλ) χ(a)f1 (a) + χ−1 (a)v(a)f2 (a) . W (d(a, + • If π = σξ,m is a spherical positive Weil representation, then there exists + f ∈ S(F ) such that for all a ∈ F ∗ , λ ∈ F

˜ λ)) = |a|3/2 1O (ra + 2mλ)χ−1 (a)f (a). W (d(a, Notice that there are exactly two spherical principal series representations πχ,m with χ2 = 1, according to the two unramified characters of F ∗ characterized by χ(ω) = 1 resp. χ(ω) = −1. By Theorem 6.3.10 there are also exactly two spherical Weil representations, because O∗2 has index 2 in O∗ . They come from the unramified characters χ given by χ(ω) = q −1/2 resp. χ(ω) = −q −1/2 .

156

6. Spherical Representations

6.4.3 Lemma. The functions f1 , f2 , f ∈ S(F )+ in the above description of W may be assumed to be multiplicatively O∗ -invariant, i.e. f (aa ) = f (a)

for all a ∈ F ∗ , a ∈ O∗ ,

and similarly for f1 , f2 . Proof: We prove this for the principal series representations with χ2 = 1. For any a ∈ F ∗ , λ ∈ F and a ∈ O∗ ˜ λ)d(a ˜  , 0)) = W (d(a, ˜ λ)), W (d(a, holds, which leads to χ(a)f1 (aa ) + χ−1 (a)f2 (aa ) = χ(a)f1 (a) + χ−1 (a)f2 (a) for a ∈ O∗ . Define for i = 1, 2  f˜i (a) := fi (aa ) d∗ a ,

(6.14)

a ∈ F,

O∗

where d∗ a denotes multiplicative Haar measure. Then f˜i is a (multiplicatively) O∗ -invariant function in S(F )+ . Because of    −1 ˜ ˜ χ(a)f1 (a) + χ (a)f2 (a) = χ(a)f1 (aa ) + χ−1 (a)f2 (aa ) d∗ a (6.14)

 

=

O∗

 χ(a)f1 (a) + χ−1 (a)f2 (a) d∗ a

O∗

=

˜ λ)), χ(a)f1 (a) + χ−1 (a)f2 (a) = W (d(a,

f1 and f2 may be replaced by f˜1 and f˜2 .



In the following we treat only the case of principal series representations with χ2 = 1, and remark that the other cases can be treated analogously. According to the preceding lemma f1 and f2 can be written in the following form:  f1 = b i 1ω i O ∗ , bi ∈ C, i∈Z

f2 =



ci 1 ω i O ∗ ,

ci ∈ C.

i∈Z

The spherical function then reads   ˜ λ)) = q −3i/2 1O (ra + 2mλ) χ(ω)i bi + χ(ω)−i ci , W (d(a,

i = v(a),

157

6.4. Spherical Whittaker functions

and it remains to determine the numbers bi and ci . Now introduce the integer ⎧ ⎨ − 1 v(N ) if v(N ) even, 2 l := (6.15) ⎩ − 1 (v(N ) − 1) if v(N ) odd. 2 ˜ λ)) vanishes if v(N a2 ) < 0, i.e. if We know from Lemma 6.4.2 that W (d(a, v(a) < l. So it may be assumed that b i = ci = 0

for i < l.

Furthermore, as f1 and f2 are constant on a neighbourhood of 0, the bi and ci become constant for large i: bi = b ∈ C

∀ i  0,

ci = c ∈ C

∀ i  0.

(6.16)

Finally, we can change finitely many of the ci arbitrarily when the corresponding bi are adjusted correctly: Just replace ci → ci + x,

bi → bi − χ(ω)−2i iε x,

x ∈ F arbitrary.

Taking these facts together, the ci may be assumed to look like this:  0 for i < l, ci = c for i ≥ l. The numbers

    r i  Wi := W d˜ ω i , − ω = q −3i/2 χ(ω)i bi + χ(ω)−i c , i ≥ l, 2m will play the key role in determining c, the bi , and the eigenvalue α of W under the Hecke operator T J (ω) ∈ H(GJ , K J , ψ −m ). This is because of the following decisive lemma. 6.4.4 Lemma. The Wi fulfill the recursion formula     −N ω 2i α− q Wi = Wi−1 + q 3 Wi+1 , ω

for all i ≥ l,

where α ∈ C is the Hecke eigenvalue of W , defined by T J (ω)W = αW, and where Wl−1 is meant to be zero. 1 2 The symbol ω· used here is defined by ⎧ if x ∈ ωO, x ⎨ 0 1 if x ∈ O∗ , x ¯ ∈ (O/ω)2 , = ⎩ ω ∗ −1 if x ∈ O , x ¯∈ / (O/ω)2

(6.17)

for x ∈ O. The proof of this lemma will be postponed to the end of the section. In the meantime we use it to compute α.

158

6. Spherical Representations

6.4.5 Lemma. The Hecke eigenvalue α of the spherical Whittaker function W ∈ W n,r is in all of the above three cases given by   α = q 3/2 χ(ω) + χ(ω)−1 . Proof: We only treat the case of the principal series representation πχ,m with χ2 = 1, the other ones being less complicated. Case 1: c = 0. Then from the recursion formula of the previous lemma it follows that   αχ(ω)i bi = q 3/2 χ(ω)i−1 bi−1 + χ(ω)i+1 bi+1 for all i ≥ l + 1. (6.18) If b (defined in (6.16)) were 0, one could conclude from this that all the bi and thus also W would vanish. So we have b = 0, and for large enough i one reads off from (6.18) that α has the desired value. Case 2: c = 0. The recursion formula for large enough i reads    α χ(ω)i b + iε χ(ω)−i c = q 3/2 χ(ω)i−1 b +

 + (i − 1)ε χ(ω)−(i−1) c + χ(ω)i+1 b + (i + 1)ε χ(ω)−(i+1) c ,

which can be rewritten as     χ(ω)2i b α − q 3/2 (χ(ω) + χ(ω)−1 = c q 3/2 (χ(ω) + χ(ω)−1 ) − α . The right side does not depend on i, thus for the left one to do the same, it is necessary that α has the desired value.  Now the final result can be stated. 6.4.6 Theorem. Assume v(2m) = 0 and let W n,r the Whittaker model with parameters n, r ∈ F for the irreducible, spherical, representation π of GJ . Let W ∈ W n,r be the (up to scalars) unique K J -invariant function. i) If π = πχ,m is a principal series representation with χ2 = | |, 1, then   ˜ λ)) = |a|3/2 charO (ra + 2mλ) χ(a)b + χ−1 (a)c 1O (N a2 ) W (d(a, for all a ∈ F ∗ , λ ∈ F , where b = q 3l/2

χ(ω)1−l − β , χ(ω) − χ(ω)−1

c = q 3l/2

β − χ(ω)l−1 . χ(ω) − χ(ω)−1

159

6.4. Spherical Whittaker functions

ii) If π = πχ,m is a principal series representation with χ2 = 1, then   ˜ λ)) = |a|3/2 charO (ra + 2mλ) χ(a)b + v(a)χ−1 (a)c 1O (N a2 ) W (d(a, for all a ∈ F ∗ , λ ∈ F , where b = q 3l/2 χ(ω)l+1 (1 − lχ(ω) − lβ) ,

c = q 3l/2 χ(ω)l+1 (χ(ω) − β)

+ iii) If π = σm,ξ is a positive Weil representation, then

˜ λ)) = |a|3/2 charO (ra + 2mλ)χ−1 (a)charO (N a2 ) W (d(a, for all a ∈ F ∗ , λ ∈ F .   2l Here we have β = −Nωω q −1/2 , N = 4mn − r2 , l is defined in (6.15), and 1·2 the symbol ω in (6.17). The Hecke eigenvalue of W is in all cases   α = q 3/2 χ(ω) + χ(ω)−1 . Proof: Again only the more difficult case of a principal series representation is treated. Assume χ2 = 1. Under our previous assumption on the ci , the recursion formula for the Wi from Lemma 6.4.4 with the value of α from Lemma 6.4.5 takes the form   χ(ω) + χ(ω)−1 bi = χ(ω)−1 bi−1 + χ(ω)bi+1 for all i ≥ l + 1. The bi becoming constant, equal to b, for large i, one deduces from this that all the bi for i ≥ l take on the same value b. So we have shown that W is indeed of the form described in the theorem, and it remains to determine the constants b and c. For this purpose we normalize W as follows:   ! 1 = Wl = q −3l/2 χ(ω)l b + χ(ω)−l c . As a second equation we use the recursion formula for i = l:       −N ω 2l α− q = q 3 Wl+1 = q 3 q −3(l+1)/2 χ(ω)l+1 b + χ(ω)−(l+1) c . ω Substituting the known value for α from Lemma 6.4.5, these two equations lead to the values for b and c given in the theorem.  There still remains to prove Lemma 6.4.4. We show the following more general statement.

160

6. Spherical Representations

6.4.7 Proposition. The action of T J (ω) on a spherical function W ∈ W n,r is given by    ˜ λ)) = W d˜ aω −1 , − r aω −1 (T J (ω)W )(d(a,   2m −N a2 ˜ λ)) + q 3 W (d(aω, ˜ + q W (d(a, λω)), ω if ra + 2mλ ∈ O and N a2 ∈ O, and ˜ λ)) = 0 (T J (ω)W )(d(a, otherwise. Proof: Let ra + 2mλ ∈ O and N a2 ∈ O; the other case is clear by Lemma 6.4.2. The operator T J (ω) by convolution with the characteristic function  acts  ω 0 of the double coset K J K J . From Lemma 6.2.4 it is easy to deduce 0 ω −1  KJ

  -  ω −1 0  ω −1 0 KJ = , λω −1 , 0, 0 K J 0 ω 0 ω λ∈O/ω  -  1 uω −1  −1 2 −1 ∪ , 0, uμω , uμ ω KJ 0 1 u∈O/ω (u,ω)=1

-

∪

μ∈O/ω

 -  ω uω −1  −1 , 0, μω , 0 KJ . 0 ω −1

u∈O/ω 2 μ∈O/ω

Hence

  −1   ω 0 ˜ λ) W d(a, , λ ω −1 , 0, 0 0 ω λ ∈O/ω      1 uω −1 ˜ λ) W d(a, , 0, uμω −1, uμ2 ω −1 0 1

˜ λ)) = (T J (ω)W )(d(a, +

 u∈O/ω



μ∈O/ω

(u,ω)=1

+





u∈O/ω 2 μ∈O/ω

=

 λ ∈O/ω

    ω uω −1 −1 ˜ λ) W d(a, , 0, μω , 0 0 ω −1

  ˜ −1 , λω −1 + λ ω −1 )) W d(aω

(6.19)

161

6.4. Spherical Whittaker functions

+



u∈O/ω (u,ω)=1

μ∈O/ω



+

 W n(uω −1 a2 , uω −1 (μa + λa))



 ˜ λ)(uω −1 (μ2 + 2λμ + λ2 )) d(a, (6.20)    ˜ W n(a2 u, a(λu + μ))d(aω, λω)(λ2 u + 2λμ) , (6.21)

u∈O/ω 2 μ∈O/ω

where we have used the abbreviation   1x n(x, μ) = (0, μ, 0) 01

for x, μ ∈ F.

The expressions (6.19), (6.20), (6.21) will be shown to equal the terms given in the proposition. First we have v(raω −1 + 2m(λω −1 + λ ω −1 )) = v(ω −1 (ra + 2mλ + 2mλ )) ≥ 0 ⇔ ra + 2mλ + 2mλ ∈ ωO  ra  ⇔ λ ≡ − λ + mod ω, 2m so that by Lemma 6.4.2 the sum (6.19) reduces to    ˜ −1 , λω −1 + λ ω −1 ) W d(aω λ ∈O/ω

   ra  −1  = W d˜ aω −1 , λω −1 − λ + ω    2m r = W d˜ aω −1 , − aω −1 . 2m For (6.20) we have by the Whittaker transformation property

(6.22)

˜ λ)) (6.20) = b(N, a)W (d(a, with b(N, a) :=





u∈O/ω (u,ω)=1

μ∈O/ω



 ψ uω −1 (na2 + raλ + mλ2 + μ(ra + 2mλ) + mμ2 ) .

(6.23)

These numbers are computed in the next lemma and give the desired values (at the moment they should perhaps be called b(n, r, m, a, λ), because it is not at all clear that they depend only on N and a). Finally we have by the Whittaker transformation property     (6.21) = ψ u(na2 + raλ + mλ2 ) + μ(ra + 2mλ) u∈O/ω 2 μ∈O/ω

˜ W (d(aω, λω)) ˜ = q 3 W (d(aω, λω)).



162

6. Spherical Representations

It remains to prove the following lemma, which is also valid without our general assumption that F has odd residue characteristic. 6.4.8 Lemma. For a ∈ F ∗ and λ ∈ F with ra + 2mλ ∈ O one has

 

b(N, a) =

and na2 + raλ + mλ2 ∈ O

−N a2 ω

 q

if q odd, if q even,

0

where N = 4mn − r2 . Proof: A simple calculation shows      1 −1 2 2 b(N, a) = ψ uω N a + (ra + 2mλ + μ) . 4m u∈O/ω (u,ω)=1

μ∈O/ω

Because of ra + 2mλ ∈ O this simplifies to      1 b(N, a) = ψ uω −1 N a2 + μ2 . 4m u∈O/ω (u,ω)=1

μ∈O/ω

First let q be odd. Case 1: N a2 ∈ ωO Then one has b(N, a) =





u∈O/ω (u,ω)=1

μ∈O/ω

= q−1+

 ψ



1 uω −1 μ2 4m



 ψ

u∈O/ω μ∈O/ω (u,ω)=1 (u,ω)=1

= q−1+





1 uω −1 4m



(−1) = 0.

μ∈O/ω (u,ω)=1

Case 2: N a2 ∈ O∗ , −N a2 ∈ / (O/ω)2 . (The bar denotes the residue class in O/ωO.) Then for all μ ∈ O we have −N a2 = μ ¯2 , hence μ2 + N a2 ∈ O∗ , and this implies  b(N, a) = (−1) = −q. μ∈O/ω

163

6.5. Local factors and the spherical dual

Case 3: N a2 ∈ O∗ , −N a2 ∈ (O/ω)2 . When μ runs through the cyclic group (O/ω)∗ , then μ2 runs exactly twice through the group (O/ω)∗2 . So we have 



b(N, a) =

ψ

u∈O/ω (u,ω)=1

+2

1 uω −1 N a2 4m 





u∈O/ω (u,ω)=1

μ∈(O/ω)∗2

ψ



 1 −1 2 uω (N a + μ) . 4m

As N a2 is a square modulo ωO, we have N a2 + μ ∈ ωO for exactly one μ ∈ (O/ω)∗2 , so that  b(N, a) = (−1) + 2

 q−1 − 1 (−1) + 2(q − 1) = q. 2

Note that case 2 can only occur if q is odd, and that in the last case this assumption was used. Now if q is even, the map μ → μ2 is the Frobenius automorphism of O/ω. Consequently one has b(N, a) =

=

=





u∈O/ω (u,ω)=1

μ∈O/ω





u∈O/ω (u,ω)=1

μ∈O/ω



=

(q − 1) +

 1 uω −1 (N a2 + μ) 4m



 1 uω −1 μ 4m

ψ

ψ(0) +

u∈O/ω (u,ω)=1

 ψ





 ψ

u∈O/ω μ∈O/ω (u,ω)=1 (μ,ω)=1



1 uω −1 4m

(−1) = 0.





μ∈O/ω (μ,ω)=1

6.5 Local factors and the spherical dual In this section we make our first attempt to attach local factors to irreducible, spherical representations of GJ . For some reductive groups such factors may be obtained as zeta integrals of spherical Whittaker functions. We have computed such functions in the previous section, hence we try this approach. The formula appearing in the following definition is a more or less natural generalization of the zeta integral from the GL(2)-theory, and is also inspired by the Mellin transform of a Jacobi form (see [Be5]), resp. an integral appearing in [Su] 4.

164

6. Spherical Representations

6.5.1 Definition. Let W be a ψ n,r -Whittaker model for the irreducible admissible representation π of GJ . Then for W ∈ W, the zeta integral is defined as   ζ(W, s) = W (d(a, λ))|a|s−3/2 dλ d∗ a, s ∈ C. F∗ F

Notice the slight difference with the zeta integral defined in [Ho] 2.2.1. Using Proposition 5.7.3, it is not hard to prove that the integral above converges for Re(s) > s0 , with s0 independent of W , and represents a holomorphic function on this right half plane ([Ho] 2.2.2). 6.5.2 Proposition. Let v(2m) = 0 and π = πm,χ a spherical representation with Whittaker model W n,r . Let W be the spherical Whittaker function normalized as in Theorem 6.4.6. i) If π is a principal series representation with χ2 = | |, 1, then ζ(W, s) = q l(3/2−s)

b + cq −s 1 − (χ(ω) + χ(ω)−1 )q −s + q −2s

with b=1−β

χ(ω)l − χ(ω)−l , χ(ω) − χ(ω)−1

c=β

χ(ω)l−1 − χ(ω)1−l . χ(ω) − χ(ω)−1

ii) If π is a principal series representation with χ2 = 1, then ζ(W, s) = q l(3/2−s)

b + cq −s (1 − χ(ω)q −s )2

with b = χ(ω)(1 − 2lβ),

c = β(2l − 1) + χ(ω) − 1.

iii) If π is a positive Weil representation, then ζ(W, s) = q −ls

χ(ω)−1 . 1 − χ(ω)−1 q −s

  2l Here we have as before β = −Nωω q −1/2 , N = 4mn − r2 , l is defined in 1·2 (6.15), and the symbol ω in (6.17).

165

6.5. Local factors and the spherical dual

Proof: These are standard p-adic computations. We only go through one of them, namely when χ2 = | |, 1. According to Theorem 6.4.6,   W (d(a, λ)) = |a|3/2 1O (ra + 2mλ) χ(a)b + χ(a)−1 c 1O (N a2 ) with b = q 3l/2

χ(ω)1−l − β , χ(ω) − χ(ω)−1

c = q 3l/2

β − χ(ω)l−1 . χ(ω) − χ(ω)−1

So we compute for Re(s) large enough     ζ(W, s) = 1O (ra + 2mλ) χ(a)b + χ(a)−1 c 1O (N a2 )|a|s dλ d∗ a F∗ F





 χ(a)b + χ(a)−1 c 1O (N a2 )|a|s d∗ a F∗     = χ(a)b + χ(a)−1 c 1O (N a2 )|a|s d∗ a =

i∈Z

=

i

∗

ω O    i≥l

= b

 χ(ω)i b + χ(ω)−i c q −is d∗ a

ωi O∗



χ(ω)q −s

i≥l

 l = b χ(ω)q −s

i +c

i  χ(ω)−1 q −s i≥l

 l 1 1 −1 −s + c χ(ω) q . −s 1 − χ(ω)q 1 − χ(ω)−1 q −s

Inserting the values of b and c gives the desired result. The cases (ii) and (iii) are treated similarly; for (ii) one uses the formula  i≥l

ixi =

lxl + (1 − l)xl+1 , (1 − x)2

instead of the geometric series.

x ∈ C, |x| < 1, 

For the proof of this proposition it is not really necessary to have the spherical Whittaker function explicitly at hand. In fact, from the results of the previous section one easily obtains  ζ(W, s) = Wi q −is . i≥l

Then the recursion formula in Lemma 6.4.4 and comparison of formal power series also gives the result. Now we take the denominators of the fractions in this proposition as our local factors attached to irreducible, spherical representations.

166

6. Spherical Representations

6.5.3 Definition. Let πχ,m be a spherical principal series representation of GJ (F ), where we still assume that v(2m) = 0. The local Euler factor L(s, π) attached to π is defined as L(s, π) :=

1 (1 −

χ(ω)q −s )(1

− χ(ω)−1 q −s )

.

6.5.4 Remark. We do not define local Euler factors for the remaining two spherical representations, which are positive Weil representations. The reason is that these representations do not appear as local components in global automorphic representations of the Jacobi group, as will follow later by the corresponding statement for the metaplectic group (Proposition 23 on p. 80 of [Wa1]) and Corollary 7.3.5. Much of our discussion in the previous chapters can be summarized in the following commutative diagram, in which all the arrows are bijections. 

Irreducible, admissible, spherical representations of GJ with central character ψ m



∼-

∼ ?

6 ∼

  HomAlg C[X ±1 ]W , C

induction

{unramified characters F ∗ → C∗ }/W

  HomAlg H(GJ , K J , ψ −m ), C

 ∼

∼ ? ⎧ ⎫ ⎨semisimple con-⎬ jugacy classes ⎩ ⎭ in SL(2, C)

We explain the objects and maps in this diagram, starting in the lower left corner. Given an unramified character χ : F ∗ → C, we associate to it the principal series representation πχ,m if χ2 = | |, resp. the Weil representation + σξ,m if χ = | |1/2 (·, ξ) or χ = | |−1/2 (·, ξ), with v(ξ) even. If χ is replaced −1 by χ , then by Theorem 5.8.3 the same representation results. Hence if the nontrivial element in the Weyl group W of GJ operates on the unramified characters of F ∗ by χ → χ−1 , then we get the arrow indexed by ‘induction’. Given an irreducible, spherical representation with central character ψ m , the Hecke algebra H(GJ , K J , ψ −m ) operates on the space of K J -invariant vectors. This space is one-dimensional, and thus a character (algebra homomorphism) of H(GJ , K J , ψ −m ) is defined. This gives the upper horizontal map. If χ is the character we started with, then H(GJ , K J , ψ −m ) → C is characterized by T (ω) → q 3/2 (χ(ω) + χ(ω)−1 ), cf. Theorem 6.4.6. The upper arrow on the left is clear by the Satake isomorphism 6.2.8.

6.6. The Eichler-Zagier operators

167

An algebra homomorphism C[X ±1 ]W → C clearly is determined by mapping X to a non-zero complex number z. Since we are dealing with polynomials which are invariant under X → X −1 , the complex numbers z and z −1 yield the same algebra  homomorphism. As a result we can associate to the conjugacy class of  z 0 in SL(2, C) the algebra homomorphism C[X ±1 ]W → C which maps 0 z −1 X + X −1 to z + z −1 , and every character of C[X ±1 ]W is thus obtained. This explains the  lower  left arrow. Finally, the lower horizontal arrow is induced by z 0 the map

→ χ, where χ(ω) = z. 0 z −1 6.5.5 Remark. The parametrization of spherical representations by semisimple conjugacy classes in the complex Lie group SL(2, C) offers another way to define local factors. As in the general reductive theory we could set  −1 L(s, π) := det 1 − gq −s , where g ∈ SL(2, C) is any element in the conjugacy class corresponding to π. It is immediate from the above diagram that this factor coincides in the case of a principal series representation with the one defined in 6.5.3. Now we can finish the proof of Theorem 6.3.10. 6.5.6 Corollary. If F has odd residue characteristic and v(m) = 0, then supercuspidal representations of GJ with central character ψ m are not spherical. Proof: This is because there are simply no characters of the Hecke algebra left: They all come from induced representations. 

6.6 The Eichler-Zagier operators Let f be a classical Jacobi form of weight k and Index m, as defined in 4.1.1. There is a general lifting mechanism which assigns to f a function Φ = Φf on the adelic Jacobi group GJ (A), where A is the adele ring of the number field Q. For the necessary global notions, see Section 7.1 below. The lifting procedure will be described in Section 7.4, but nevertheless, it seems apt to discuss the relationship between classical and adelic Hecke operators right now. The lifted form Φ turns out to be right invariant under the local groups GJ (Zp ), for all finite places p. The local Hecke algebras H(GJ (Qp ), GJ (Zp )) operate on such global functions by convolution:  (ϕ.Φ)(x) = ϕ(y)Φ(xy) dy, ϕ ∈ H(GJ (Qp ), GJ (Zp )). GJ (Qp )

168

6. Spherical Representations

If π denotes the representation by right translation, then this is just the corresponding representation of the Hecke algebra. But to fit in the context of classical Hecke operators, we let H(GJ (Qp ), GJ (Zp )) for the moment act on the right, and denote this action by ∗:  (Φ ∗ ϕ)(x) = ϕ(y)Φ(xy −1 ) dy, ϕ ∈ H(GJ (Qp ), GJ (Zp )). GJ (Qp )

It is not hard to see that if  ϕ= char(GJ (Zp )gi )

with gi ∈ GJ (Q),

i

then the corresponding action on the Jacobi form f is given by     f ϕ = f  gi . k,m

i

k,m

The operator |k,m on the right-hand side is the one defined in [EZ], Theorem 0 1.4, or here in 4.1.1. In [EZ], §4, two more Hecke operators TEZ und TEZ on classical Jacobi forms are defined:   f  TEZ (pα ) := k,m       pα(k−4) f det(M )1/2 M (X, 0) , M∈SL2 (Z)\M2 (Z)

X∈Z2 /pα Z2

k,m

det(M)=p2α gcd(M)=

  f

k,m

0 TEZ (pα ) :=





M∈SL2 (Z)\M2 (Z)

X∈Z2 /pα Z2

pα(k−4)

  f



 det(M )1/2 M (X, 0) .

k,m

det(M)=p2α gcd(M)=1

The condition gcd(M ) =  (resp. = 1) means summation over those matrices only where the greatest common divisor of all coefficients is a square number (resp. 1). These operators are now to be compared with the T J (pα ). The following lemmas are valid for every number field, so we formulate them in greater generality than necessary. 6.6.1 Lemma. Assume O to the  ring of integers of a p-adic field, and let  be −α ω 0 K J = GJ (O). For γ ∈ K J K J let 0 ωα  Eγ := K J (γ × H(O)) = K J γ(Y, 0). Y ∈O 2

169

6.6. The Eichler-Zagier operators

Then:  −α  ω mω −α γ∈ / SL(2, O) , m ∈ O =⇒ Eγ = 0 ωα  −α  ω mω −α γ ∈ SL(2, O) , m ∈ O =⇒ Eγ = 0 ωα

-

K J γ(λ, 0, 0),

λ∈O/ω α

-

K J γ(0, μ, 0).

μ∈O/ω α

Right multiplication by an element g = (X, κ) ∈ H(O), X ∈ O2 , κ ∈ O, induces a bijection Eγ → Eγ , and permutes the cosets K J \Eγ . Proof: The coset decompositions are an exercise, and the other assertions are clear.  6.6.2 Lemma. With γ and Eγ as in the previous lemma, we have   char K J γ(λ, μ, 0) = q α char(Eγ ).

 λ,μ∈O/ω α

 −α  ω mω −α Proof: Assume γ is not contained in the coset SL(2, O) (the 0 ωα other case is treated analogously). One computes    char K J γ(λ, μ, 0) λ,μ∈O/ω α

=





μ∈O/ω α λ∈O/ω α

=





  char K J γ(λ, 0, −λμ)(0, μ, 0)   char K J γ(λ, 0, 0)(0, μ, 0)

μ∈O/ω α λ∈O/ω α

=



char(Eγ ).

μ∈O/ω α



For the last step Lemma 6.6.1 was used. 6.6.3 Lemma. We have i) the coset decomposition 2α 3 ) M ∈ M2 (O) : det(M ) = ω 2α =

-

f =0 u∈O/ω

 2α−f  ω u SL(2, O) , 0 ωf f

170

6. Spherical Representations

ii) the decomposition 3

) M ∈ M2 (O) : det(M ) = ω 2α , gcd(M ) = 1  2α−f  2α ω u = SL(2, O) 0 ωf f =0

u∈O/ω f

(u,ω f ,ω 2α−f )=1

=

 −α  ω 0 ω α SL(2, O) SL(2, O), 0 ωα

iii) and finally {M ∈ M2 (Z) : det(M ) = p2α , gcd(M ) = 1}  2α−f  2α p u = SL(2, O) . 0 pf f =0

u∈Z/pf

f

(u,p ,p2α−f )=1



Proof: This is straightforward. 6.6.4 Lemma. We have

{M ∈ M2 (Z) : det(M ) = ω 2α , gcd(M ) = } [ α2 ] 2(α−2j)  (α−2j)−f  ω uω −(α−2j) α =ω SL(2, O) . 0 ω f −(α−2j) j=0

f =0

u∈O/ω f

(u,ω f ,ω 2(α−2j)−f )=1

Proof: By Lemma 6.6.3 (i), {M ∈ M2 (Z) : det(M ) = ω 2α , gcd(M ) = }  2α−f  ∞ 2α ω u = SL(2, O) 0 ωf j=0 f =0

u∈O/ω f

f

(u,ω ,ω 2α−f )=ω 2j

=

[ α2 ] 2α−2j j=0 f =2j

u∈O/ω f , ω 2j |u (u,ω f ,ω 2α−f )=ω 2j

 2α−f  ω u SL(2, O) 0 ωf

171

6.6. The Eichler-Zagier operators

[ α2 ] 2α−2j -

=

j=0 f =2j

u∈O/ω f −2j (u,ω

=

 2α−f  ω uω 2j SL(2, O) 0 ωf

-

[ α2 ] -

2(α−2j)

j=0

f =0

f −2j

,ω 2α−f −2j )=1

-

 2α−f −2j  ω uω 2j SL(2, O) 0 ω f +2j

u∈O/ω f



(u,ω f ,ω 2(α−2j)−f =1

In the following proposition, which makes the connection between T J and the Eichler-Zagier operators, we return to F = Q, p = (p), ω = p.

6.6.5 Proposition. 0 TEZ (pα ) = pk+α−4 T J (pα ).

i)

[ α2 ] 

α

ii)

TEZ (p ) =

0 p2j(k−2) TEZ (pα−2j ).

j=0

Proof: (i) It is not hard to see thatif γ runs over a complete set of repre4 pα 0 sentatives of G(Zp ) G(Zp ) G(Zp ), G = SL(2), and for every γ the 0 p−α element g runs over a set of representatives of GJ  (Zp )\(γ ×H(Zp )), then g runs 4 pα 0 over a set of representatives of GJ (Zp ) GJ (Zp ) GJ (Zp ). Hence we 0 p−α can compute   f

T J (pα ) =



γ

g∈GJ (Zp )\Eγ

k,m Lemma 6.6.2

=

 γ

Lemma 6.6.3

=

  f g



1 pα



X∈Z2 /pα Z2

1  f (γ(X, 0)) pα





M∈SL2 (Z)\M2 (Z) 2α

det(M)=p

gcd(M)=1

=

  0 α p−α p4−k f TEZ (p ).

X∈Z2 /pα Z2

   f  det(M )1/2 M (X, 0)

172

6. Spherical Representations

(ii) With the help of Lemma 6.6.4 one computes [ α2 ]    f TEZ (pα ) = pα(k−4) j=0

2(α−2j)







f =0

u∈O/ω f

X∈Z2 /pα−2j Z2

(u,ω f ,ω 2(α−2j)−f =1

 Y ∈Z2 /p2j Z

= pα(k−4)

[ α2 ]  j=0

 (α−2j)−f    p up−(α−2j) α−2j f  (X + p Y, 0) 0 pf −(α−2j) 2 

M∈SL2 (Z)\M2 (Z)

p4j



   f  p−(α−2j) M, X, 0

X∈Z2 /pα−2j Z2

det(M)=p2(α−2j) gcd(M)=1 α 2

= pα(k−4)

[ ]  j=0

    0 α−2j p4j p(α−2j)(4−k) f TEZ (p ) .



7 Global Considerations

After having classified all unitary, resp. admissible, representations of GJ (F ), where F is a local field of characteristic zero, in the preceding chapters, we are now ready to consider representations of GJ (A), where A is the adele ring of some number field. The first section of this chapter collects some basic results about the adelized Jacobi group. After that, we consider once more the Schr¨ odinger-Weil representation, this time in the global context. It will be shown how an automorphic version of m πSW is constructed by means of theta functions. One of the main results will be an explicit version in the global context of the fundamental relation m π=π ˜ ⊗ πSW ,

yielding a canonical bijection between automorphic representations of GJ (with fixed non-trivial central character) and genuine automorphic representations of the metaplectic group (see Theorem 7.3.3). Similar to the situation for some reductive groups, the first examples of automorphic representations of GJ come from classical Jacobi cusp forms on H× C. These can be lifted to the adelized group GJ (A) (A here the adeles of Q), thereby yielding an element of a certain cuspidal L2 -space. We describe this lifting procedure in Section 7.4. After that, one has to prove that the subrepresentation of the right regular representation generated by this lifted function is irreducible, provided we start with an eigenform. This will be achieved by the help of a strong multiplicityone result for the metaplectic group by Waldspurger and Gelbart, PiatetskiShapiro, carried over to the Jacobi group by means of the above mentioned explicit isomorphism. R. Berndt and R. Schmidt, Elements of the Representation Theory of the Jacobi Group, Modern Birkhäuser Classics, DOI 10.1007/978-3-0348-0283-3_7, © Springer Basel AG 1998

173

174

7. Global Considerations

7.1 Adelization of GJ In this chapter we use the following notations. • F denotes a number field, • O is its ring of integers, • {p} is the set of all places of F , • Fp is the completion of F at p, • Op is the closure of O in Fp , • A is the ring of adeles of F . The character ψ of A is defined to be the product of the local additive standard characters we used before, as explained in Section 2.2. It is a character of F \A, i.e. ψ( + x) = ψ(x)

for all x ∈ A,  ∈ F.

For an adele m ∈ A the symbol ψ m denotes the character x → ψ(mx) of A. The characters of F \A are then exactly the ψ m with m ∈ F . The adelization of the Jacobi group is defined as   GJ (A) = GJ (Fp ) : GJ (Op ) , p

the restricted direct product of the local Jacobi groups GJ (Fp ) with respect to the open compact subgroups GJ (Op ) at the finite places. The group GJ (F ) is embedded diagonally in GJ (A) as a discrete subgroup. This follows from the analogous statements for the groups G = SL(2) and H. We will be concerned with the homogeneous space GJ (F )\GJ (A). One checks that the following integration formula for suitable functions Φ on GJ (F )\GJ (A) holds:    Φ(g) dg = Φ(hM ) dh dM. GJ (F )\GJ (A)

G(F )\G(A) H(F )\H(A)

(G = SL(2) as before). Since the measure of G(F )\G(A) is finite (well known) and that of H(F )\H(A) is too (being compact), it follows from this formula that GJ (F )\GJ (A) has finite measure.

(7.1)

7.1. Adelization of GJ

175

For G(A) we have the well known strong approximation theorem, whereafter G(F )G∞ is dense in G(A), the ∞ always denoting all the infinite components. Strong approximation does also hold for the Heisenberg group, because F A∞ is dense in A. Thus we conclude that the Jacobi group fulfills strong approximation, too: GJ (F )GJ∞ is dense in GJ (A).

(7.2)

In particular, with  K0J := GJ (Op ) p∞

it follows that GJ (A) = GJ (F )GJ∞ K0J .

(7.3)

We look at the special case F = Q. By (7.3), the injection GJ (R) → GJ (A) yields a bijection ΓJ \GJ (R)/K∞ Z(R)  GJ (Q)\GJ (A)/K∞ K0 Z(A),

(7.4)

which is easily seen to be a homeomorphism. Here we remind the reader that we put K∞ = SO(2, R), and Z is the center of GJ . As already remarked in Section 1.4, the real Jacobi group GJ (R) acts on H × C in the following way:     aτ + b z + λτ + μ ab (λ, μ, κ).(τ, z) = , . cd cτ + d cτ + d In particular, hM (i, 0) = (τ, λτ + μ),

where τ = M (i), h = (λ, μ, κ).

The stabilizer of the special point (i, 0) ∈ H × C is the group K∞ Z(R). Hence there is a homeomorphism GJ (R)/K∞ Z(R)  H × C.

(7.5)

Taking (7.4) and (7.5) together, we see that there is a homeomorphism ΓJ \H × C  GJ (Q)\GJ (A)/K∞ K0 Z(A).

(7.6)

Classical Jacobi forms may therefore be lifted to functions on the homogeneous space GJ (Q)\GJ (A). This will be carried out in Section 7.4. But before that, we discuss the Schr¨odinger-Weil representation, which is fundamental for the representation theory of GJ in the global case also.

176

7. Global Considerations

7.2 The global Schr¨ odinger-Weil representation First we introduce the global Schr¨ odinger representation πSm of the adelized Heisenberg group H(A) on the Hilbert space L2 (A). It is the unique unitary representation which acts on the dense subspace S(A) of L2 (A) as   πSm (λ, μ, κ)f (x) = ψ m (κ + (2x + λ)μ)f (x + λ) (λ, μ, κ ∈ A). It is easily seen that the global Schr¨ odinger representation is the tensor product of local ones. Hence from the local Stone–von Neumann theorems one deduces: 7.2.1 Theorem. (The global Stone–von Neumann theorem) i) πSm is the unique irreducible unitary representation of H(A) with central character ψ m . ii) Every smooth unitary representation of H(A) with central character ψ m is isomorphic to a direct sum of Schr¨ odinger representations πSm . Just as in the local case, this theorem allows us to construct a global Weil representation, which will be a projective representation of the adelized group G(A), or alternatively, a representation of a certain two-fold cover of G(A), namely the global metaplectic group Mp(A). We collect some facts about this group. To begin with, there is a short exact sequence of topological groups 1 −→ {±1} −→ Mp(A) −→ G(A) −→ 1.

(7.7)

This sequence does not split, but is given by a nontrivial cocycle λ ∈ H 2 (G(A), {±1}). Hence Mp(A) may be realized as the set G(A) × {±1}, endowed with the multiplication (M, ε)(M  , ε ) = (M M  , λ(M, M  )εε ) for all M, M  ∈ G(A), ε, ε ∈ {±1}. The cocycle λ is the product of local cocycles λp for all places p of F :  λ(M, M  ) = λp (Mp , Mp ) for M = (Mp ), M  = (Mp ). p

This is well defined because the λp were designed in such a way that for p  2 the cocycle λp is trivial on G(Op ) × G(Op ). Unfortunately, a product formula does not hold for the λp in the sense that λ(M, M  ) = 1 for all M, M  ∈ G(F ) would hold. However, there is something which is almost as good (cf. [Ge2] 2.2 or [Sch1] 5.2):

7.2. The global Schr¨ odinger-Weil representation

177

7.2.2 Lemma. There is a function η : G(A) → {±1} such that λ(M, M  ) = η(M )η(M  )η(M M  )

for all M, M  ∈ G(F ).

Therefore the sequence (7.7) splits over G(F ), and we have an injection G(F ) −→ Mp(A),

−→ (M, η(M )).

M

The image of this map is denoted by Mp(F ). This is something like the F rational points of Mp, although the metaplectic group is not algebraic and Mp(A) is not quite the restricted direct product of the local groups Mp(Fp ). m We turn back to the global Weil representation πW of G(A) resp. Mp(A) on 2 L (A) and give some explicit formulas for it. These look slightly simpler than in the local cases, due to the fact that there is a product formula for the Weil constant γm . For every Φ ∈ S(A), b, x ∈ A, and a ∈ I (the idel group of F ) the following holds:





  1b πW Φ (x) 01     m a 0 πW Φ (x) 0 a−1 m

m (πW (w)Φ)(x)

=

ψ m (bx2 )Φ(x),

=

|a|1/2 Φ(ax),  ˆ Φ(x) = Φ(y)ψ m (2xy) dy.

=

A



 0 1 of G(F ) ⊂ G(A), and | | is −1 0 the product over all places of the absolute values of Fp , normalized such that the product formula holds. As in the local cases, the Schr¨ odinger and the Weil representation can be put together to obtain the global Schr¨ odinger-Weil m J (A)). representation πSW of GJ (A) (resp. of a two-fold cover G Here w denotes as usual the element

For the next section it is important to have an automorphic version of the Schr¨ odinger representation. From now on the element m ∈ A is chosen from F . To every function f ∈ S(A) we associate a theta function ϑf : H(A) → C by ϑf (h) =



(πSm (h)f )(ξ),

h ∈ H(A).

ξ∈F

It is not hard to see that this series converges and defines a continuous function on H(A) which is right invariant by H(F ). Furthermore it satisfies the relation

178

7. Global Considerations

ϑ(hz) = ψ m (z)ϑ(h) for all h ∈ H(A) and z ∈ Z(A). We compute its L2 -norm:  ϑf 22 = |ϑf (h)|2 dh H(F )\H(A)/Z(A)



 2   (πSm (h)f )(ξ) dh 

= H(F )\H(A)/Z(A)



ξ

  2   ψ m ((2ξ + λ)μ)f (ξ + λ) dμ dλ 

=

F \A F \A





=

ξ



ψ m ((2ξ1 + λ)μ − (2ξ2 + λ)μ)f (ξ1 + λ)f (ξ2 + λ) dμ dλ

F \A F \A ξ1 ,ξ2

 

=

F \A



|f (ξ + λ)|2 dλ

ξ

|f (λ)|2 dλ = f 22 .

=

(7.8)

A

The map f → ϑf may therefore be extended to a norm-preserving map of Hilbert spaces ϑ : L2 (A) −→ L2 (H(F )\H(A))m , where the space on the right consists of all measurable functions Φ : H(A) → C which fulfill Φ(hz) = ψ m (z)Φ(h)

for all  ∈ H(F ), h ∈ H(A), z ∈ Z(A), (7.9)

and

 |Φ(h)|2 dh < ∞. H(F )\H(A)/Z(A)

If we associate to such a Φ the function fΦ : A

−→ C,  λ −→ Φ((0, μ, 0)(λ, 0, 0)) dμ, F \A

then, using some standard Fourier analysis on F \A, it is not hard to see that Φ → fΦ gives an inverse map to ϑ. In particular, for f ∈ S(A),  f (λ) = ϑf ((0, μ, 0)(λ, 0, 0)) dμ for all λ ∈ A. (7.10) F \A

179

7.3. Automorphic representations

Thus ϑ is an isomorphism of Hilbert spaces which intertwines the H(A)-action on both sides. In particular, the right regular representation on the space L2 (H(F )\H(A))m is irreducible, which may be interpreted as a multiplicityone result for automorphic representations of the Heisenberg group. If we consider H(A) as a subgroup of GJ (A) and conjugate H(F ) by an element of G(A), we obtain the following generalization. 7.2.3 Proposition. For any M ∈ G(A), denote by H(F )M the conjugate group M −1 H(F )M . Then the right regular representation of H(A) on the space L2 (H(F )M \H(A))m is isomorphic to πSm . In particular, it is irreducible. It is also possible to lift the Schr¨ odinger-Weil representation from L2 (A) to functions living on the Jacobi group. Just assign to f ∈ S(A) the theta function  m ϑf (g) = (πSW (g)f )(ξ), g ∈ G(A). ξ∈F

This is an isometrical map if the norm on the image is given by integration over the Heisenberg group only, just as in (7.9). If the image of L2 (A) is denoted by Rm SW , then we can state the following. 7.2.4 Proposition. The representation of GJ (A) on Rm by right translation is SW m isomorphic to πSW . For every element ϑ ∈ Rm the following holds: SW for all  ∈ GJ (F ), g ∈ GJ (A).

ϑ(g) = η()λ(, g)ϑ(g)

Here the functions η and λ are carried over trivially from G(A) to GJ (A). We do not give the proof of this invariance property, which rests on the Poisson summation formula (cf. [We]).

7.3 Automorphic representations Let L2 (GJ (F )\GJ (A))m be the Hilbert space of measurable functions Φ : GJ (A) → C which satisfy Φ(gz) = ψ m (z)Φ(g) and

for all g ∈ GJ (A),  ∈ GJ (F ), z ∈ Z(A)

 |Φ(g)|2 dg < ∞. GJ (F )\GJ (A)/Z(A)

Similarly, let L2 (Mp(F )\Mp(A))− be the Hilbert space of measurable functions ϕ : Mp(A) → C which satisfy ϕ((g, ε)) = εϕ(g, 1)

for all  ∈ Mp(F ), g ∈ G(A), ε ∈ {±1}

180

7. Global Considerations

and



 |ϕ(g)|2 dg =

Mp(F )\Mp(A)/S

|ϕ(M, 1)|2 dM < ∞, G(F )\G(A)

where we have denoted by S the two-element subgroup (1, ±1) ∈ Mp(A). 7.3.1 Definition. An irreducible, unitary representation of GJ (A) (resp. of Mp(A)) is called automorphic if it appears as a subrepresentation of the right regular representation of GJ (A) (resp. Mp(A)) on L2 (GJ (F )\GJ (A))m (resp. L2 (Mp(F )\Mp(A))− ). The following lemma is decisive. 7.3.2 Lemma. Let U ⊂ L2 (GJ (F )\GJ (A))m be a subspace which is invariant under right translation by elements of H(A), and such that the representation of H(A) thus defined is equivalent to πSm . Then there exists a function ϕ in L2 (Mp(F )\Mp(A))− such that every Φ ∈ U is of the form for all M ∈ G(A), h ∈ H(A),

Φ(M h) = ϕ(M, 1)ϑf (M h) for some f ∈ L2 (A).

Proof: For M ∈ G(A) fixed, we compare the map U Φ

∼

−→ L2 (H(F )M \H(A)),  

−→ h → Φ(M h) ,

which intertwines the H(A)-action and is an isomorphism by Proposition 7.2.3, to the composition of H(A)-intertwining maps ∼

∼

∼

2 M U −→ L2 (H(F )\H(A))m −→ Rm \H(A))m , SW −→ L (H(F )

(7.11) where the first map is restriction, the inverse of the second is also restriction, and the third one maps ϑ to the function h → ϑ(M h) (the space Rm is SW defined before Proposition 7.2.4). By Schur’s lemma, the two isomorphisms U → L2 (H(F )M \H(A)) thus obtained differ by a scalar, which we denote by ϕ(M ). This defines the function ϕ on G(A). Going through the definitions, we see that Φ(M h) = ϕ(M )ϑf (M h)

for all M ∈ G(A), h ∈ H(A),

where ϑf is the image of Φ under the first map in (7.11). Now, using (7.8), it is easy to see that ϕ satisfies the correct L2 -condition, which allows us to lift  it to a function in L2 (Mp(F ), Mp(A))− .

181

7.3. Automorphic representations

Now we can state the main result of this section, which makes the abstract m isomorphism π  π ˜ ⊗ πSW from Section 2.6 concrete in the global case. Similar statements can be found in the real case in Takase [Ta] (Section 11) and in the adelic case in Pyatetski-Shapiro [PS2] (equation 6.8). 7.3.3 Theorem. There is a natural isomorphism of Hilbert spaces L2 (Mp(F )\Mp(A))− ⊗ L2 (A) ϕ⊗f

∼

−→ L2 (GJ (F )\GJ (A))m ,  

−→ M h → ϕ(M, 1)ϑf (M h) .

This isomorphism is an intertwining map for the representation of GJ (A) on both sides, where the action of GJ (A) on L2 (A) is the Schr¨ odinger-Weil reprem sentation πSW , and right translation on the other parts. Restriction to cuspidal functions yields another isomorphism ∼

L20 (Mp(F )\Mp(A))− ⊗ L2 (A) −→ L20 (GJ (F )\GJ (A))m . Proof: Denoting by Φ the image of ϕ ⊗ f , it is easy to see by Proposition 7.2.4 that Φ has the correct transformation property. We use the calculation (7.8) to see that Φ is indeed an L2 -function and the map is norm-preserving:    2 |Φ(g)| dg = |Φ(hM )|2 dh dM GJ (F )\GJ (A)/Z(A)

G(F )\G(A) H(F )\H(A)/Z(A)



m |ϕ(M, 1)|2 πW (M )f 2 dm

= G(F )\G(A)



|ϕ(M, 1)|2 f 2 dm = ϕ2 f 2.

= G(F )\G(A)

By the preceding lemma and the global Stone–von Neumann theorem, our map is surjective, whence the first isomorphism. Now suppose U is a subspace like in Lemma 7.3.2 which lies in the cuspidal subspace L20 (GJ (F )\GJ (A))m . If ϕ is the function appearing in this lemma, the element f in S(A) is arbitrary, and Φ = ϕ ⊗ ϑf ∈ U , then for almost all g = M h ∈ GJ (A)  0 = Φ(ng) dn N J (F )\N J (A)



=

ϕ F \A F \A

(7.10)









=

ϕ F \A

     1x 1x M, 1 ϑf (0, μ, 0)M h dx dμ 01 01

      1x 1x M, 1 πSW M h f (0) dx 01 01

182

7. Global Considerations 

 =

ϕ

    1x M, 1 dx · πSW (M h)f (0). 01

F \A

It is easy to see that h can be chosen such that (πSW (M h)f )(0) = 0. It follows that     1x ϕ M, 1 dx = 0 for almost all M ∈ G(A), 01 F \A



i.e., ϕ is cuspidal.

7.3.4 Corollary. Every automorphic representation π of GJ factorizes as a restricted tensor product 5 π= πp p

with irreducible, unitary representations of the local groups GJ (Fp ). Proof: The corresponding statement for automorphic representations π ˜ of the metaplectic group is known to be true, cf. [F]. For suitable π ˜ we therefore have  5   5  5   m m m ππ ˜ ⊗ πSW  π ˜p ⊗ πSW,p  π ˜p ⊗ πSW,p .  p

p

p

7.3.5 Corollary. The map m π ˜ −→ π := π ˜ ⊗ πSW

gives a 1-1 correspondence between (genuine) automorphic representations of Mp(A) and automorphic representations of GJ (A) with central character ψ m . The cuspidal representations correspond to the cuspidal ones. Further, this correspondence is compatible with local data: If 5 5 π ˜= π ˜p , π= πp p

p

are the decompositions of π ˜ and π in local components, then m πp = π ˜p ⊗ πSW,p .

7.3.6 Corollary. The space L20 (GJ (F )\GJ (A))m decomposes in a discrete direct sum of irreducible representations, each occurring with finite multiplicity. Proof: This is true for the metaplectic group. The proof runs along the lines sketched in [Ge1], §5, for GL(2). 

183

7.4. Lifting of Jacobi forms

7.4 Lifting of Jacobi forms Before discussing the lifting of Jacobi forms to the adele group, we make some general remarks on Fourier expansions. Let Φ be a function on GJ (F )\GJ (A), which for simplicity we assume to be continuous, although the following considerations also make sense, with slight modifications, for more general types of functions, e.g. measurable ones. We assume Φ(gz) = ψ m (z)Φ(g)

for all g ∈ GJ (A), z ∈ Z(A).

(7.12)

For n, r ∈ F we define the character ψ n,r of N J (A) as    1x n,r ψ (0, μ, 0) = ψ(nx + rμ). 01 If (n, r) runs through F 2 , then ψ n,r obviously runs through all characters of N J (F )\N J (A)  (F \A)2 . 7.4.1 Definition. For n, r ∈ F , the (n, r)-Whittaker-Fourier coefficient of Φ is defined as the continuous function  WΦn,r (g) = Φ(ug)ψ n,r (u) du, N J (F )\N J (A)

if this integral makes sense. 7.4.2 Lemma. Let n, r ∈ F and Φ as above. i) We have WΦn,r (ug) = ψ n,r (u)WΦn,r (g)

for all g ∈ GJ (A), u ∈ N J (A).

ii) If a ∈ F ∗ , λ ∈ F , then  WΦn,r

    a 0 (λ, 0, 0)g = WΦn ,r (g) 0 a−1

for all g ∈ GJ (A),

where n = a2 (n + mλ2 + rλ), r = a(r + 2mλ). Proof: The first item is obvious, while the second one follows easily from the left GJ (F )-invariance of Φ and (7.12).  7.4.3 Remark. As the discriminant of the pair (n, r) ∈ F 2 we denote the number N = 4mn − r2 .

184

7. Global Considerations

The discriminant of the pair (n , r ) with n , r as in the lemma is then N  = 4mn − r = a2 N. 2

Conversely, given a pair (n , r ) ∈ F 2 such that this latter equation holds for some a ∈ F ∗ , we can define λ = (r a−1 − r)/(2m), and the above equations between (n, r) and (n , r ) hold. We can thus state the following: If we know   WΦn,r , then we know WΦn ,r for all pairs (n , r ) ∈ F 2 whose discriminant differs from N = 4mn − r2 by a square in F ∗ . The pairs (n, r) ∈ F 2 (and thus the Whittaker-Fourier coefficients of Φ) are partitioned into orbits, indexed by N ∈ F/F ∗2 = {0} ∪ F ∗ /F ∗2 . In order to reconstruct Φ, we have to know at least one Whittaker-Fourier coefficient from each orbit. This is in contrast to the situation for GL(2), where it is well known that an automorphic function is completely determined by its first Fourier coefficient. In our case, the pairs with discriminant 0 make up one orbit. 7.4.4 Definition. The function Φ as above is called cuspidal if WΦ0,0 = 0, i.e.  Φ(ng) dn = 0 for all g ∈ GJ (A). N J (F )\N J (A)

The above remarks show that if Φ is cuspidal, then not only the (0, 0)-Whittaker-Fourier coefficient vanishes, but all Whittaker-Fourier coefficients with discriminant 0 do also. This is in contrast with the situation in the real case, where there are finitely many conditions to check (cf. the discussion in Section 4.2). Now we come to what is announced in the title of this section. The decomposition (7.3) allows us to lift classical Jacobi forms, as defined in 4.1.1, to the adele group. The process is familiar from the theory of elliptic modular forms, which are usually lifted to GL(2, A). The number field is from now on Q. To a Jacobi form f ∈ Jk,m the lift Φf is the function GJ (A) → C defined in the following way. According to (7.2), an element g ∈ GJ (A) may be decomposed as with γ ∈ GJ (Q), g∞ ∈ GJ (R), k0 ∈ K0J .

g = γg∞ k0 Then

   Φf (g) := f 

k,m

 g∞ (i, 0) = jk,m (g∞ , (i, 0))f (g∞ (i, 0)).

This is well defined since ΓJ = GJ (Q) ∩ K0J . 7.4.5 Proposition. Under the map f → Φf the space Jk,m is isomorphic to the space Am,k of functions Φ : GJ (A) → C having the following properties:

7.4. Lifting of Jacobi forms

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i) Φ(γg) = Φ(g) for all g ∈ GJ (A), γ ∈ GJ (Q). ii) Φ(gz) = ψ m (z)Φ(g) for all g ∈ GJ (A), z ∈ Z(A). iii) Φ(gk∞ ) = jk,m (k∞ , (i, 0))Φ(g) for all g ∈ GJ (A), k∞ ∈ K∞ . iv) Φ(gk0 ) = Φ(g) for all g ∈ GJ (A), k0 ∈ K0J . v) Φ is C ∞ as a function on GJ (R), and as such LX− Φ = LY− Φ = 0. vi) Φ is slowly increasing. The condition v) may be substituted by v’) Φ is C ∞ as a function on GJ (R), and as such  5 LC Φ = k 2 − 3k + Φ. 4 cusp The restriction to Jk,m yields an isomorphism onto the space Am,k of functions Φ which fulfill i)–vi) and are moreover cuspidal, i.e.  Φ(ng) dn = 0 for all g ∈ GJ (A).

N J (F )\N J (A)

Proof: These are routine arguments, except perhaps for the assertion that v) may be substituted by v’). This follows from Corollary 4.3.5, since our lifting may be carried out in two steps: First we lift to GJ (R), obtaining the space described in Corollary 4.3.5, and after that lift these functions to GJ (A) by strong approximation. We check the cusp condition. Let  f (τ, z) = c(n, r)e(nτ + rz) 4mn−r 2 ≥0

be the Fourier expansion of our classical Jacobi form f . Then by straightforward calculations we have for the Whittaker-Fourier coefficients of the lift Φ    2 2 a 0 WΦn,r (λ, 0, 0) = c(n, r)ak e−2π(na +raλ+mλ ) (a ∈ R>0 , λ ∈ R), −1 0a if n, r ∈ Z with 4mn − r2 ≥ 0, while WΦn,r = 0 otherwise. By the remark made after Definition 7.4.4 we see that Φ is cuspidal in the sense of this definition if and only if f is a cusp form. 

186

7. Global Considerations

2 J J 7.4.6 Proposition. Acusp m,k is a subspace of L0 (G (Q)\G (A))m , and f → Φf is cusp cusp an isomorphism of Hilbert spaces Jk,m → Am,k .

Proof: This follows from (7.6) in Section 7.1 and the fact that the classical Petersson scalar product in equation (13) on page 27 of [EZ] is finite for cusp forms.  Let Φ : GJ (A) → C be the lift of a Jacobi form f with weight k and index m. If f is not a cusp form, then it need not be true that Φ ∈ L2 (GJ (Q)\GJ (A))m . Nevertheless, a slight modification of Lemma 7.3.2 (leaving out the L2 -conditions) shows that Φ can be written in the form  Φ(M h) = ϕi (M, 1)ϑfi (M h) (7.13) i

with genuine functions ϕi : Mp(Q)\Mp(A) → C and Schwartz functions fi in S(A). We can derive information on the fi from this equation, letting the Heisenberg group act on both sides, which does not affect the ϕi . Let us begin with the infinite place. The holomorphy of f implies Y− Φ = 0. Assuming the functions ϕi linearly independent, it follows that Y− ϑfi = 0, or equivalently, Y− fi = 0, for all i. But the real Schr¨ odinger representation contains (up to scalars) only one vector annihilated by Y− , namely the function 2

F∞ (x) = e−2πmx ∈ S(R) (remember in the context of Jacobi forms we assume m > 0). Hence each fi may be assumed to be of the form fi = F∞ ⊗ Fi

with Fi ∈ S(A0 ).

Now we continue to get information about the Fi by letting the finite parts of ˆ and the Heisenberg group act on (7.13). The left side is invariant under H(Z), again the linear independence of the ϕi implies that each ϑfi , and therefore ˆ Now from Lemma 6.3.2 one can read each Fi , is right invariant under H(Z). off a basis for the H(Zp )-invariant vectors in S(Qp ) of the local Schr¨ odinger m representation πSW,p ; it is given by  ν  fp,ν = char Zp + , ν ∈ Zp /2mZp . 2m ˆ The H(Z)-invariant vectors in S(A0 ) therefore have a basis consisting of the 2m elements  5 ν  fν = char Zp + , ν ∈ Z/2mZ. 2m p