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The Theta Correspondence and Periods of Automorphic Forms

by

Patrick Walls

A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Mathematics

University of Toronto

c Copyright 2013 by Patrick Walls

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Abstract

The Theta Correspondence and Periods of Automorphic Forms

Patrick Walls

Doctor of Philosophy

Graduate Department of Mathematics

University of Toronto

2013

The study of periods of automorphic forms using the theta correspondence and the Weil representation

was initiated by Waldspurger and his work relating Fourier coefficients of modular forms of half-integral

weight, periods over tori of modular forms of integral weight and special values of L-functions attached

to these modular forms. In this thesis, we show that there are general relations among periods of auto-

morphic forms on groups related by the theta correspondence. For example, if G is a symplectic group

and H is an orthogonal group over a number field k, these relations are identities equating Fourier

coefficients of cuspidal automorphic forms on G (relative to the Siegel parabolic subgroup) and periods

of cuspidal automorphic forms on H over orthogonal subgroups. These identities are quite formal and

follow from the basic properties of theta functions and the Weil representation; further study is required

to show how they compare to the results of Waldspurger. The second part of this thesis shows that,

under some restrictions, the identities alluded to above are the result of a comparison of nonstandard

relative traces formulas. In this comparison, the relative trace formula for H is standard however the

relative trace formula for G is novel in that it involves the trace of an operator built from theta functions.

The final part of this thesis explores some preliminary results on local height pairings of special cycles

on the p-adic upper half plane following the work of Kudla and Rapoport. These calculations should

appear as the local factors of arithmetic orbital integrals in an arithmetic relative trace formula built

from arithmetic theta functions as in the work of Kudla, Rapoport and Yang. Further study is required

to use this approach to relate Fourier coefficients of modular forms of half-integral weight and arithmetic

degrees of cycles on Shimura curves (which are the analogues in the arithmetic situation of the periods

of automorphic forms over orthogonal subgroups).

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Contents

0 Introduction 1

0.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 The Theta Correspondence 8

1.1 The Weil Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.1 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.2 Translation and Dual Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.1.3 The Heisenberg Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.1.4 Intertwining Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.1.5 The Metaplectic Group and the Weil Representation . . . . . . . . . . . . . . . . . 17

1.2 Dual Reductive Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2.1 Symplectic-Orthogonal Dual Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2.2 Metaplectic-Orthogonal Dual Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.2.3 Unitary-Unitary Dual Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.3 The Theta Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2 Duality Among Periods 28

2.1 Fourier Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2 Orthogonal Periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3 Matching Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4 Example: Matching Functions for PGL2(Qp) . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.5 Duality Among Periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.6 The Main Theorem: Spectral Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.7 Further Directions: Special Values ofL-Functions . . . . . . . . . . . . . . . . . . . . . . . 44

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3 A Comparison of Relative Trace Formulas 48

3.1 Relative Trace of the Kernel for G(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Relative Trace of the Kernel for H(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3 Matching Geometric Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.4 Spectral Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.5 Spectral Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4 Height Pairings of Special Cycles 61

4.1 Special Cycles on the p-adic Upper Half Plane . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2 Height Pairings of Special Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3 The Unramified/Unramified Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.4 The Unramified/Split Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.5 The Split/Unramified Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.6 The Ramified/Ramified Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Bibliography 78

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Chapter 0

Introduction

The study of periods of automorphic forms using the theta correspondence and the Weil representationwas initiated by Waldspurger and his work (see [19], [20], [21] and [22]) relating Fourier coefficients of

modular forms of half-integral weight, periods over tori of modular forms of integral weight and special

values of L-functions attached to these modular forms. In this thesis, we show that there are general

relations among periods of automorphic forms on groups related by the theta correspondence.

This thesis comprises of three separate (but related) results which form Chapter 2, Chapter 3 and

Chapter 4 respectively. The main result is the collection of spectral identities in Theorem 2.6.1, for a

given dual pair (G, H), equating Fourier coefficients of cuspidal automorphic forms on G and periods

of cuspidal automorphic forms on H over orthogonal subgroups. In Theorem 3.5.1, we show, in some

very special cases, that these spectral identities are the result of a comparision of nonstandard relative

trace formulas. Finally, Chapter 4 explores some preliminary results on local height pairings of special

cycles on the p-adic upper half plane. Inspired by the work of the preceding two chapters, these height

pairings should appear as the local factors of arithmetic orbital integrals in a comparison of arithmetic

relative trace formulas. In this introductory section, we give a summary of these results.

0.1 Overview

Let k be a global field, let A be its ring of adeles, let : A/k C be a unitary character and let(G, H) be a dual reductive pair (in the sense of Howe [6]) defined over k. For example, let G = Spn

be the symplectic group of rank n (consisting of symplectic matrices of size 2n) and let H = OV be

the orthogonal group attached to a space V over k of even dimension m equipped with a nondegenerate

symmetric bilinear form ( , ) and corresponding quadratic form Q(v) = 12 (v, v). The results described

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Chapter 0. Introduction 2

below apply to symplectic-orthogonal, metaplectic-orthogonal and unitary-unitary dual reductive pairs

as described in Section 1.2, however, for simplicity, we will restrict ourselves in this introduction to the

symplectic-orthogonal dual pair (Spn, OV). The group G contains a Siegel parabolic subgroup P with

unipotent radical N P

N =n(b) = 1 b0 1

: b Symnwhere Symn denotes symmetric matrices of size n. For a cuspidal automorphic form f

G on G(A) and

symmetric matrix t Symn(k), the t-Fourier coefficient of fG is

Wt(fG) =

N(k)\N(A)

fG(n) t(n) dn

where t(n) = (tr tb) for n = n(b).

Let j = (j1, . . . , jn) Vn be an n-tuple of vectors in V and let T be the stabilizer ofj where H actscomponentwise on Vn. The group T is an orthogonal subgroup of H and for a cuspidal automorphic

form fH on H(A), the T-period offH is

PT(fH) =

T(k)\T(A)

fH() d .

The groups form a dual pair (G, H) for the Weil representation = (relative to ) acting on the

Schwartz space S(VnA ) (where VA = V k A). The elements S(VnA ) define theta functions

(g, h) =

vVn

(g)(h1v)

and the theta correspondence is built by integrating cuspidal automorphic forms on G(A) against theta

functions to produce automorphic forms on H(A) and vice versa. In particular, if is a cuspidal

automorphic representation ofG acting on a subspace V L2(G(k)\G(A)), the theta lift offG V is

fG(h) = G(k)\G(A)

(g, h) fG(g) dg

and the theta lift of is the space of theta lifts fG as and fG vary and is denoted (). Analo-

gously, if is a cuspidal automorphic representation of H acting on a subspace V L2(H(k)\H(A)),the theta lift of fH V is

fH(g) =

H(k)\H(A)

(g, h) fH(h) dh

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and the theta lift of is the space of theta lifts fH as and fH vary and is denoted (). It is very

important to note that theta lifts are not always cuspidal. In the main theorem, we assume that and

are cuspidal representations of G and H respectively such that () = (equivalently, () = ).

To state the main result, we need some extra notation: given a Schwartz function on H(A), R is the

Hecke operator

RfH(h) =

H(A)

(x)fH(hx) dx

and, for v Vn, Q[v] = 12 ((vi, vj )) is the symmetric matrix of inner products of the components of v.The following is a statement of Theorem 2.6.1.

Theorem. Letj1,j2 Vn with Q[j1] = t1 and Q[j2] = t2 such that det t1 = 0 and det t2 = 0, andlet T1 and T2 be the stabilizers in H of j1 andj2 respectively. Let 1, 2 S(VnA ) and let 1 and 2 be

smooth functions on H(A) such that 1 matches 1 relative toj

1 and 2 matches 2 relative toj

2 asin Proposition 2.3.1. Given cuspidal automorphic representations and of G and H respectively such

that () = (equivalently () = ), we have

FGB()

Wt1(12 FG) Wt2(F

G) =

FHB()

PT1(R1R2 FH) PT2(F

H) (1)

FGB()

PT1(2 FG) Wt2(F

G) =

FHB()

PT1(R2 FH) PT2(F

H) (2)

FGB()PT1(1 F

G) PT2(2 FG) =

FHB()PT1(1 2F

H) PT2(FH) (3)

whereB() andB() are orthonormal bases ofV andV respectively.

This result has some special features. First, the relation between the various , , t and T is explicit

and, in low rank cases, computable. Second, it is well-known that the Fourier coefficient Wt(fH) of a

theta lift of fH is expressible in terms of an orthogonal period PT(fH) however Equation 1 allows one,

in principle, to choose the data 1 and 2 to select the Fourier coefficients of any fG in and express

them in terms of a chosen orthogonal basis of , say, involving Hecke eigenforms. Third, the result of

Waldspurger [20] (later generalized by Baruch-Mao [1]) is an equality between twisted central values of

L-functions attached to modular forms of integral weight and squares of Fourier coefficients of modular

forms of half-integral weight; the spectral identity Equation 1 introduces a formula for a pair ofdistinct

Fourier coefficients. Finally, by the Siegel-Weil formula, a special value of an L-function is encoded in

the inner product of theta functions 12 .

Next, we describe the second result of this thesis. When V is anisotropic and n = 1, Equation 1

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can be interpreted as a spectral identity as a result of a comparison of relative trace formulas. First,

let us introduce the kernel for G(A). Given Schwartz functions 1, 2 S(VA), the map which takes acusp form fG on G(A) to its theta lift 2 f

G on H(A) and then to the lift 12 fG on G(A) again is

an operator on the space of cusp forms. In fact, switching the order of integration shows that it is an

integral operator with kernel function

K12 (g1, g2) =

H(k)\H(A)

1(g1, h) 2(g2, h) dh .

Since we have assumed that V is anisotropic, the quotient H(k)\H(A) is compact and this integralis absolutely convergent. Define the trace of the kernel K12 relative the subgroup N N and thecharacters t1 and t2 by

JG(1 2 ; t1, t2) = [NN]

K12 (n1, n2) t1(n1) t2(n2) dn1dn2

where [N N] = (N N)(k)\(N N)(A).The right regular representation ofH(A) on L2(H(k)\H(A)) defines an action of the Schwartz space

S(H(A)) by

RfH(x) =

H(A)

(y) fH(xy) dy

which we unwind to find that it is an integral operator with kernel function

K(x, y) =

H(k)

(x1y) .

Let j1, j2 V and let T1 and T2 be the stabilizers in H ofj1 and j2 respectively. For each S(H(A)),define the trace of the kernel K relative to the subgroup T1 T2 by

JH(; T1, T2) =

[T1T2]

K(1, 2) d1d2

where [T1

T2

] = (T1

T2

)(k)\

(T1

T2

)(A).

Both traces JG(1 2 ; t1, t2) and JH(; T1, T2) have simple geometric expansions whose terms aresums over the double coset T1\H/T2 and we prove in Proposition 3.3.2, stated below, that they are equalby establishing identities term by term.

Proposition. Let j1, j2 V be nonzero with Q(j1) = t1 and Q(j2) = t2, and let T1 and T2 be thestabilizers in H of j1 and j2 respectively. Let 1, 2 S(VA) and let 1 and 2 be smooth functions on

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H(A) such that 1 matches 1 relative to j1 and 2 matches 2 relative to j2 as in Proposition 2.3.1.

Then

JG(1 2 ; t1, t2) = JH(1 2 ; T1, T2)

where 1

2 (h) = H(A) 1(x)2 (x1h) dx is the convolution of 1 and 2 for 2 (x) = 2(x1).Having established that the traces of the kernels for G and H are equal when the input data is

matching, we express each trace spectrally and proceed to produce equalities of traces at the level of

representations. The first step is Corollary 3.4.1, stated below, where we show that the noncuspidal

components of JG(1 2 ; t1, t2) and JH(; T1, T2) are equal and conclude that the cuspidal parts areequal.

Proposition. Let j1, j2 V be nonzero with Q(j1) = t1 and Q(j2) = t2, and let T1 and T2 be thestabilizers in H of j1 and j2 respectively. For al l 1, 2 S(VA) and 1 matching 1 relative to j1 and2 matching 2 relative to j2 as in Proposition 2.3.1, we have

AG0

JG (1 2 ; t1, t2) =

AH0

JH (1 2 ; T1, T2) (4)

whereAG0 andAH

0 are the sets of cuspidal automorphic representations of G and H respectively, the

trace along AG0 is

JG (1 2 ; t1, t2) = FGB()

Wt1(12 FG) Wt2(F

G)

and the trace along AH0 is

JH (1 2 ; T1, T2) =

FHB()

PT1(R1R2 FH) PT2(F

H) .

The final step in the comparison of trace formulas is to use a linear independence argument to produce

equalities of traces at the level of representations which correspond by the theta correspondence. The

following is a statement of Theorem 3.5.1.

Theorem. Letj1, j2 V be nonzero withQ(j1) = t1 andQ(j2) = t2, and letT1 andT2 be the stabilizersin H of j1 and j2 respectively. Let 1, 2 S(VA) and let 1 and 2 be smooth functions on H(A) suchthat 1 matches 1 relative to j1 and 2 matches 2 relative to j2 as in Proposition 2.3.1. Given

cuspidal automorphic representations and of G andH respectively such that() = (equivalently

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() = ), we have

FGB()

Wt1(12 FG) Wt2(F

G) =

FHB()

PT1(R1R2 FH) PT2(F

H) (5)


It should be noted that the method of proving Theorem 2.6.1 was discovered only when the final steps

in the proof of Theorem 3.5.1 were completed. However we have presented the method of Chapter 2

first as the main result since it is more general and to the point.

The final chapter in this thesis explores some preliminary results on height pairings of special cycles

on the p-adic upper half plane following the work of Kudla-Rapoport [11]. Let k = Fp, let W = W(k)

be the Witt ring of k and let B be the quaternion division algebra over Qp with ring of integers OB .

The p-adic upper half plane W is a formal scheme over SpfW and is the moduli space of special formalOB-modules as in the work of Drinfeld [3] (see also [2]). Fix a special formal OB-module X over k. The

ring of endomorphisms EndOB (X) Q is naturally identified with the algebra M2(Qp) of 2 by 2 matricesover Qp and we denote by V the subspace of traceless endomorphisms equipped with the quadratic

form q(j) = det(j). For any j V, there is a special cycle Z(j) which lives on W (see Chapter 4for definitions) and the main result of Kudla-Rapoport [11] is a formula for the local height pairings

(Z(j1), Z(j2)) for j1, j2 V.Since V can be identified with the space of traceless 2 by 2 matrices over Qp, there is a natural action

of PGL2(Qp) on V by conjugation. The results of Chapter 4 are formulas for the local height pairings

(Z(j1), Z(j2)) for fixed j1, j2 V as a function of PGL2(Qp) where j = j 1. The motivation

for studying such height pairings is that they should be the arithmetic analogues of the local integrals

PGL2(Qp)

1(h1j1)2(h1j2) dh

appearing in the geometric expansion of JG(1 2 ; t1, t2) as in Proposition 3.1.1. See the introductionto Chapter 4 for further discussion.

The formula for the pairing (Z(j1), Z(j2)) depends on the arithmetic of the values q(j1) and q(j2).

In Chapter 4, we consider four of the nine possible cases and we outline one such formula as an example

below. In the statement of Proposition 4.4.1 below, is the quadratic residue character, Bj is the set

of points on the Bruhat-Tits tree of PGL2(Qp) fixed under multiplication by j and d(Bj1 ,B

j2) is the

distance from Bj1 to Bj2 . Finally, as expected, these formulas are a synthesis of the formulas given in

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Theorem 6.1 in [11]

Proposition. Let j1, j2 V with q(j1) = p1 and q(j2) = p 2 such that 0 , and areeven, (1) = 1 and (2) = 1. Let GL2(Qp) and let d = d(Bj1 ,Bj2). Then (Z(j1), Z(j2)) = 0if d > /2 + /2, (Z(j

1), Z(j

2)) = 1 if d = /2 + /2, and if d < /2 + /2 then (Z(j

1), Z(j

2)) =

+ + 1 2d

2p(/2+/2d+1)/2 1

p 1 if2 +

2 d is odd and 2 > 2 d,

p(/2+/2d)/2 + 2p(/2+/2d)/2 1

p 1 if2 +

2 d is even and 2 > 2 d,

p/2 + 2p/2 1

p 1 if2 2 d.

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Chapter 1

The Theta Correspondence

The global theta correspondence is a correspondence between automorphic representations of groups

forming a dual reductive pair in the sense of Howe [6], and the main tool in the construction is the

Weil representation [23]. In Section 1.1, we introduce the Heisenberg group, the Weil representation and

the metaplectic group by focusing on the Fourier transform. There are many introductions to the Weil

representation (for example [9], [15], [18] and the original paper by Weil [23]) therefore our discussion

is informal as we merely outline in Summary 1.1.1 and Summary 1.1.2 the elementary properties of

the Heisenberg group, the Weil representation and the metaplectic group. In Section 1.2, we introduce

the dual pairs and the explicit formulas for the Weil representation used in subsequent chapters in this

thesis. In Section 1.3, we introduce the theta functions and the theta correspondence for these dual

pairs. The results in Chapter 2 and Chapter 3 concern cuspidal automorphic representations which

correspond under the theta correspondence and we conclude with a discussion of the results of Moeglin

[14] and Jiang-Soudry [8] on the irreducibility of cuspidal theta lifts. In general, the theta lift of a

cuspidal representation to another randomly chosen group is almost never cuspidal and so it is worth

noting that the situation where cuspidal representations correspond is quite special. For example, the

results of Rallis [16] show that given a cuspidal representation of an orthogonal group OV(A) (for a

rational space V of even dimension), there is an integer n such that the theta lift of to Spn(A) is

cuspidal and the theta lift to Spr(A) is zero for all r < n. Furthermore, the results of Moeglin [14] show

that the theta lift of to Spr(A) is not cuspidal for r > n.

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Chapter 1. The Theta Correspondence 9

1.1 The Weil Representation

Metaplectic groups and the Weil representation form the basis of the theta correspondence and are

defined by twisting the representations of the Heisenberg groups. There are several introductions to the

Weil representation (see [9], [15], [18] and [23] for example) and so our goal in this section is to introduce

these objects in an informal way while referring to the literature when necessary. Our perspective is

elementary and we take the familiar Fourier transform as our starting point. The abstract Fourier

transform for self-dual locally compact abelian groups and its interaction with translation leads to the

Heisenberg group and we enumerate its properties in Summary 1.1.1. Twisting the natural representation

(, S) of the Heisenberg group by symplectic automorphisms produces the unitary operators that define

the Weil representation and we describe them in Summary 1.1.2.

1.1.1 The Fourier Transform

The Fourier transform of an integrable function f L1(R) is the continuous function defined by theintegral f(y) =

R

f(x) e2ixy dx .

The restriction of the Fourier transform to the Schwartz space S(R) of infinitely differentiable, rapidly

decreasing functions defines an automorphism

F : S(R) S(R) : f fsuch that

f(x) = f(x).The definition of the Fourier transform can be exteneded to any locally compact abelian group. In

particular, let G be a locally compact abelian group, let G = Homcont(G,R/Z) be the (continuous)

Pontryagin dual ofG and denote the natural pairing between G and G by x, x for x G and x G.Since G is locally compact, there is a Haar measure dx on G unique up to scalars. In particular, dx

is a Borel measure invariant by translation. The abstract Fourier tranform of an integrable function

f L1(G) is the continuous function on G given by the integral

f(x) = G

f(x) e2ix,x dx .

Just as in the case of the classical Fourier transform, there are Schwartz spaces S(G) L1(G) and

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S(G) L1(G) such that the Fourier transform

F : S(G) S(G) : f fis an isomorphism. The explicit definition of S(G) depends on G however its most important property

is the isomorphism defined by the Fourier transform. Furthermore, there are unique measures on G and

G such that the Fourier inversion formulaf(x) = f(x) holds.

The case G = R is special since R is self-dual in the sense that we have the isomorphism

R R = Homcont(R,R/Z) : y (x xy)

in contrast to the case G = R/Z where (R/Z) = Z. Therefore the Fourier transform is an automorphism

of S(R) depending on the identification ofR with its dual. There are many other examples of self-dual

groups such as finite-dimensional vector spaces over finite fields or locals fields and the adelic points of

a finite-dimensional vector space over a global field. We make the following definition so that we may

introduce the Heisenberg group and the Weil representation for each of these cases simultaneously.

Definition 1.1.1. LetF and V be one of the following pairs:

1. F is a finite field and V is a finite dimensional F-vector space

2. F is a local field and V is a finite dimensional F-vector space

3. F is the adele ring Ak of a global field k and V is a free F-module of finite rank obtained by

extending scalars to F from a finite dimensional k-vector space

Let( , ) : V V F be a nondegenerate symmetric bilinear form on V which we write as a dot product(v, w) = v w, and choose a nonzero unitary additive character

: F C .

The Fourier transform of a Schwartz function S(V) is

(w) = V

(v) (v w) dv

and the map defines an automorphism (depending on the choice ) of the Schwartz space S(V).The Haar measure dv is chosen to be self-dual relative to so that (v) = (v).

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We give some examples of the spaces defined above. In each case, we define a homomorphism from

F to R/Z which we exponentiate to produce a unitary character .

If V is a finite dimensional vector space over a finite field F of characteristic p, then S(V) is the

space of all complex-valued functions on V and we have the map

Ftr Fp R/Z : x

tr xp

where tr is the trace map from F to Fp and the tilde indicates any lift a Z ofa Fp (ie. a a mod p).IfV is a finite dimensional vector space over F = R, then S(V) is the space of infinitely differentiable,

rapidly decreasing functions on V and we have the quotient map R R/Z. And if F = C, we havethe map C R/Z : z 2 Re z given by the trace from C to R.

IfV is a finite dimensional vector space over a p-adic field F (necessarily of finite degree overQp), then

S(V) is the space of locally constant, compactly supported functions on V and we have the compositions

of natural maps

Ftr Qp Qp/Zp Q/Z R/Z

where tr is the trace map from F to Qp.

If V is a free module of finite rank over the adele ring F = Ak of a number field k, then V = vVvis isomorphic to a restricted tensor product of finite dimensional vector spaces Vv over the local fields

kv for all the places v of k. The Schwartz space S(V) is the span of the space of factorizable functions

vv where each v is a Schwartz function on S(Vv) (and, for almost all v, v is the characteristicfunction of a fixed distinguished lattice Lv) and we have the map given by the product of local maps

kv R/Z : v v for all places v

Ak R/Z : (v) v|

v

v finite

v .

The choice to take the difference between the infinite places and the finite places ensures that the

resulting map on Ak is trivial on k.

1.1.2 Translation and Dual Translation

Let (V,F, ( , ), , dv) be as in Definition 1.1.1. The space V acts on itself by translation therefore it acts

on functions

v(x) = (x + v) .

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Translation interacts with the Fourier transform by the formula

v(y) = V

(x + v) (x y) dx

= V (x) ((x v) y) dx= (v y)

V

(x) (x y) dx

= (v y)(y)therefore we define the dual translation v by v V on functions by

v (x) = (x v) (x) .

By construction, the Fourier transform intertwines the actions v and v of V

F v = v F .

Translation and dual translation do not commute as we compute

vw(x) = ((x + v) w) (x + v) = (v w) (x w) (x + v)

and

wv(x) = (x w) (x + v)

and arrive at the fundamental commutation relation

vw = (v w) wv .

Let W = V V and for each element of W define an operator (v, w) = wv on S(V). This is nota representation of W as we compute using the relation v

w = (v w) wv

(v1, w1)(v2, w2) = w1

v1w2

v2

= (v1 w2)w1w2v1v2= (v1 w2) (v1 + v2, w1 + w2) .

This relation suggests that the natural actions of translation and dual translation are a representation

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of a central extension of W.

1.1.3 The Heisenberg Group

Define the bilinear map on W = V

V

c : W W F : (v1, w1; v2, w2) v1 w2 .

Bilinearity implies that c is a 2-cocycle on W and therefore defines a central extension

0 F H(W) W 0 .

Define (v, w, t) = (t)wv and compute using the relation vw = (v w) wv

(v1, w1, t1)(v2, w2, t2) = (t1 + t2)

w1v1

w2v2

= (t1 + t2 + v1 w2)w1+w2v1+v2= (w1 + w2, v1 + v2, t1 + t2 + v1 w2) .

Therefore is a representation of H(W) acting on S(V).

However, our ultimate goal is to have certain automorphisms of W acting on H(W). The automor-

phisms of W which preserve the cocycle define automorphisms of H(W). Therefore, we will adjust the

cocycle by a coboundary so that it is recognizable as a symplectic form on W allowing the symplectic

automorphisms of W to act on this extension.

We must now make the further assumption that the characteristic of F is not 2. Write the bilinear

form c as a sum of a symmetric form and a symplectic form

c(v1, w1; v2, w2) =1

2(v1 w2 + v2 w1) + 1

2(v1 w2 v2 w1) .

Define Q(v, w) = 12 v

w and compute the coboundary map

dQ(v1, w1; v2, w2) = Q(v1 + v2, w1 + w2) Q(v1, w1) Q(v2, w2)

=1

2(v1 w2 + v2 w1) .

Therefore

c = dQ + c

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where we have defined the symplectic form on W

c(v1, w1; v2, w2) =1

2(v1 w2 v2 w1) .

The cocycle c defines a central extension H(W) called the Heisenberg group

0 F H(W) W 0 .

This extension is isomorphic to the original extension H(W) since the two cocycles are cohomologous

however now the symplectic automorphisms

Sp(W) = {A GLF(W) : c(A(v1, w1); A(v2, w2)) = c(v1, w1; v2, w2)}

define automorphisms of H(W) by the formula A(v, w, t) = (A(v, w), t).

Summary 1.1.1. Let(V,F, ( , ), , dv) be as in Definition 1.1.1 and assume 2 = 0 in F.

1. The Heisenberg group H(W) attached to V is the central extension of W = V V defined by thesymplectic form

c(v1, w1; v2, w2) =1

2(v1 w2 v2 w1) .

In particular, H(W) = V V F with the group operation

(v1, w1, t1)(v2, w2, t2) = (v1 + v2, w1 + w2, t1 + t2 +1

2(v1 w2 v2 w1)) .

2. The symplectic automorphisms of W

Sp(W) = {A GLF(W) : c(A(v1, w1); A(v2, w2)) = c(v1, w1; v2, w2)}

define automorphisms of H(W) by the formula A(v, w, t) = (A(v, w), t).

3. The Heisenberg group is equipped with the representation

: H(W) U(S) : (v, w, t)

t +v w

2

wv

into the group U(S) of unitary automorphisms of the Schwartz space S(V) arising from the inter-

action of the Fourier transform with translation.

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1.1.4 Intertwining Operators

The symplectic automorphisms Sp(W) of W act on the Heisenberg group H(W) and therefore on the

functions on H(W). To obtain a left action on functions, we view the symplectic automorphisms ofW

as matrices acting on W by right multiplication

(v, w)

a bc d

= (va + wc,vb + wd)where a,b,c,d EndF(V).

Each g Sp(W) defines a new representation (g, S) where g(v, w, t) = ((v, w)g, t) (and we write = since the character is fixed). Since the central character of the representation (

g, S) is , the

Stone-Von-Neumann Theorem implies that (g, S) = (, S) and so, for all g Sp(W), there is a unitaryoperator r(g) : S(V) S(V) such that the following diagram commutes

S(V)

r(g)

g(v,w,t) // S(V)

r(g)

S(V)

(v,w,t) // S(V)

for all (v, w, t) H(W). The operators r(g) are best described via an intermediate space of functions

on the Heisenberg group.

Let X = V 0 W and Y = 0 V W so that W = X + Y is a complete polarization of W.Let Ind

H(W)H(Y) denote the space of functions on the Heisenberg group obtained as the image of the map

(v, w, t)(0) = (t + 12 v w)(v) on Schwartz functions S(V). In other words,

IndH(W)H(Y)

=

(v, w, t) =

t +v w

2

(v) : S(V)

.

Then, by design, (, S) is isomorphic to the right regular representation of H(W) acting on IndH(W)H(Y)

and the inverse map is given by restriction to V 0 0 H(W), (v, w, t)|V00 = (v). Thenotation indicates that Ind

H(W)H(Y) is the (smooth) induction of the character of the abelian subgroup

H(Y) = {(0, w , t) : w V, t F} H(W) where (0, w , t) = (t). The essential property of thesefunctions is the left invariance by Y

((0, w, 0)(v, w, t)) =

v, w + w, t v w

2

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=

t v w

2+

v (w + w)2

(v)

=

t +v w

2

(v)

= (v, w, t) .

For g Sp(W), we also have the map from the representation (g, S) to the right regular representationof H(W)

g(v, w, t) = g(v, w, t)(0)

but now these functions are left invariant by Y g1. As above, we denote by IndH(W)H(Y g1)

the space of

functions g(v, w, t) = g(v, w, t)(0). We integrate to get invariance by Y

g(v, w, t) = (Y g1Y)\Y g((0, w, 0)(v, w, t)) dwwhere dw is some measure on the quotient (Y g1 Y)\Y. Finally, restricting to V 0 0 H(W)maps the function g to S(V). Since each step was H(W)-equivariant, the result is a H(W)-intertwiningmap r(g) from (g, S) to (, S)

N

(g, S)

r(g) // (, S) g|V00

gy >>Ind

H(W)

H(Y g1) // IndH(W)

H(Y)

OO

gQ

VV

Furthermore, it can be shown (see [18] and [23]) that the measure on (Y g1 Y)\Y can be chosen so

that the operator r(g) is unitary. Finally, we can explicitly write the operator r(g) for g =

a bc d

r(g)(v) = g(v, 0, 0)= (Y g1Y)\Y g((0, w, 0)(v, 0, 0)) dw=

ker(c)\Y

((wc,wd, 0)(va,vb, 0)) dw

=

ker(c)\Y

(va + wc,vb + wd,1

2(wc vb va wd)) dw

=

ker(c)\Y

(1

2(wc vb va wd) + 1

2(va + wc) (vb + wd)) (va + wc) dw

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=

ker(c)\Y

wc vb + va vb

2+

wc wd2

(va + wc) dw

where in the third line we use that fact that since (0, w, 0) g = (wc,wd, 0) then Y Y g = ker(c) g soY g1 Y = ker(c). In particular, we have the familiar formulas

r

1 b0 1

(v) = v vb2

(v) (1.1)

r

a 00 ta1

(v) = |det(a)|1/2F (va) (1.2)r

0 1

1 0

(v) =

V

(w) (v w) dw (1.3)

where | |F is the modulus character of F and dw is the measure on V that is self-dual relative to .

1.1.5 The Metaplectic Group and the Weil Representation

In the previous section, we found a map from the symplectic group to the group of unitary automorphisms

of S(V)

r : Sp(W) U(S) : g r(g)

however this is not a representation of Sp(W). The cohomology class [C] H2

(Sp(W),C

1 ) (whereC1 is the group of complex numbers of norm 1) defined by the 2-cocyle

r(g1)r(g2) = C(g1, g2)r(g1g2)

defines a central extension

1 C1 Mp(W) Sp(W) 1

called the metaplectic group Mp(W) attached to Sp(W). The end result is a representation of the

metaplectic group

: Mp(W) U(S) : (g, z) z r(g)

called the Weil representation.

Summary 1.1.2. Let(V,F, ( , ), , dv) be as in Definition 1.1.1 (assume 2 = 0 in F), let H(W) be theHeisenberg group attached to V and let(, S) be the representation ofH(W) defined in Summary 1.1.1.

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1. The action of the symplectic automorphisms Sp(W) on H(W) defines representations (g, S) of

H(W) where

g(v, w, t) = ((v, w)g, t) .

2. Each representation(g, S) is isomorphic to (, S) and the intertwining operator

r(g) : (g, S) (, S)

is given by the unitary operator

r(g)(v) =

ker(c)\Y

wc vb + va vb

2+

wc wd2

(va + wc) dcw

where g = a bc d

and dcw is the unique Haar measure on ker(c)\Y such that r(g) is unitary.3. The operators r(g) define the 2-cocyle C on Sp(W) with values inC

1 = {z C : |z| = 1} by

r(g1)r(g2) = C(g1, g2)r(g1g2)

and gives the central extension

1 C1 Mp(W) Sp(W) 1

where Mp(W) is the metaplectic group. In particular, Mp(W) = Sp(W) C1 with the operation(g1, z1)(g2, z2) = (g1g2, z1z2C(g1, g2)) and is equipped with the representation

: Mp(W) U(S) : (g, z) z r(g)

called the Weil representation.

The explicit form of the metaplectic group depends on the choice of the 2-cocycle representing the

cohomology class in H2(Sp(W),C1 ) corresponding to the extension Mp(W) of Sp(W). In particular,

the cocycle C can be altered by a coboundary to produce an equivalent form of the metaplectic group.

In the next section, we introduce the dual pairs and the explicit formulas for the Weil representation

(based on the cocycle introduced by Rao [18] which is not the cocycle in Summary 1.1.2) used in the

remainder of this thesis.

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1.2 Dual Reductive Pairs

Throughout this section, let k be a global field with A = Ak its ring of adeles and fix once and for all an

additive unitary class character : Ak/k C. We will introduce the dual pairs and explicit formulas

for the Weil representation (based on the cocycle introduced by Rao [18] which is not the cocycle in

Summary 1.1.2) used in the remainder of this thesis.

1.2.1 Symplectic-Orthogonal Dual Pairs

Let V be a k-vector space ofeven dimension m equipped with a nondegenerate symmetric bilinear form

( , ). Let Q(v) = 12 (v, v) be the corresponding quadratic form and let O V be the k-algebraic group of

automorphisms of V that preserve the symmetric form

OV = { h GL(V) : (hv,hw) = (v, w) for all v, w V } .

Let W be the standard k-vector space of dimension 2n equipped with the nondegenerate skew-

symmetric bilinear form , defined by the matrix

J =

0 1n1n 0

where 1n is the identity matrix of size n. Let Spn = Sp(W) be the k-algebraic group of automorphisms

ofW that preserve the symplectic form. In particular, we view W as the space of row vectors of size 2n

and the automorphisms of W as matrices acting by right multiplication therefore

Spn =

g GL2n : g Jtg = J

.

The Siegel parabolic subgroup P of Spn has a decomposition P = MN where the Levi component M is

M = m(a) = a 00 ta1 : a GLnand the unipotent radical N is

N =

n(b) =1n b

0 1n

: b Symn

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where Symn denotes symmetric matrices of size n.

Let W = V k W and let , be the symplectic form on W defined by

v

1 w

1, v

2 w

2=

1

2(v

1, v

2)w

1, w

2.

Let Sp(W) be the k-algebraic group of automorphisms ofW that preserve the symplectic form. Define

right actions of h OV and g Spn on W by

(v w) h = h1v w and (v w) g = v wg .

It is clear that the algebraic groups OV and Spn are subgroups of Sp(W) such that each is the others

centralizer. The standard polarization W = kn

kn defines a polarization W = Vn

Vn where

Vn = {v = (v1, . . . , vn) : v1, . . . , vn V}

and so v h = h1v whereh1v = (h1v1, . . . , h

1vn) .

By construction, the symplectic form is

v1 + w1,v2 + w2 =1

2 {(v1,w2) (v2,w1)}

where v1 + w1,v2 + w2 Vn Vn. Extending scalars to A, we have the self-dual group VnA equippedwith the nondegenerate symmetric bilinear form (v,w) =

i(vi, wi) therefore we proceed as in the

previous section to define the Heisenberg group H(WA), the metaplectic group Mp(WA) and the Weil

representation.

We will take the form of metaplectic group described by the cocyle defined by Rao [ 18]. In particular,

the metaplectic group Mp(WA) is a central extension of the group of adelic points Sp(WA) of the

symplectic group Sp(W) and fits into the exact sequence

1 C1 Mp(WA) Sp(WA) 1

defined by the cocycle described in [18]. The assumption that m is even implies that the cocycle

is cohomologically trivial when restricted to Spn(A) therefore the extension splits over Spn(A). The

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extension is also split over OV(A) therefore there is an injective homomorphism

Spn(A) OV(A) Mp(WA)

(normalized to produced the formulas below) and so we view Spn(A) and OV(A) as commuting subgroups

of Mp(WA) and we call (Spn(A), OV(A)) a symplectic-orthogonal dual pair.

Let = be the Weil representation of Mp(WA) (relative to ) acting on the space S(VnA ) of

Schwartz functions on VnA . Restricted to the symplectic-orthogonal dual pair by the chosen embedding

above, the Weil representation satisfies

(h)(v) = (h1v) h OV(A)(m(a))(v) = V(det a) |det a|

m2

A (va) m(a) M(A)

(n(b))(v) = (tr bQ[v]) (v) n(b) N(A)

(1.4)

where

V(x) =

x, (1)m(m1)2 det VA

is the character of the quadratic space (V, Q) and

Q[v] =1

2((vi, vj ))

where ((vi, vj )) is the symmetric matrix of inner products of the components ofv = (v1, . . . , vn).

1.2.2 Metaplectic-Orthogonal Dual Pairs

Let V be a k-vector space of odd dimension m equipped with a nondegenerate symmetric bilinear form

( , ). Let Q(v) = 12 (v, v) be the corresponding quadratic form and let O V be the k-algebraic group of

automorphisms of V that preserve the symmetric form.

Let W be the standard symplectic space over k of dimension 2n and let W = Vk W be the symplecticspace over k defined analogous to the construction in the previous section. Note again that in this case

OV and Spn are subgroups of Sp(W) such that each is the others centralizer. The metaplectic group

Mp(WA) is a central extension


defined by the cocycle described in [18] however the assumption that m is odd implies that the cocycle

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is not cohomologically trivial when restricted to Spn(A) and it defines the central extension Mpn(A) of

Spn(A)

1 C1 Mpn(A) Spn(A) 1 .

This extension splits over the Siegel parabolic subgroup P(A) as well as the group of rational pointsSpn(k) and we view each as a subgroup of Mpn(A) by fixed embeddings.

The extension Mp(WA) of Sp(WA) is still split over OV(A) therefore there is an injective homomor-

phism

Mpn(A) OV(A) Mp(WA)

(normalized to produced the formulas below) and so we view Mpn(A) and OV(A) as commuting sub-

groups of Mp(WA) and we call (Mpn(A), OV(A)) a metaplectic-orthogonal dual pair.

Let = be the Weil representation of Mp(WA) (relative to ) acting on the space S(VnA ) of

Schwartz functions on VnA . Restricted to the metaplectic-orthogonal dual pair by the chosen embedding

above, the Weil representation satisfies

(h)(v) = (h1v) h OV(A)(m(a))(v) = V(det a) |det a|

m2

A (va) m(a) M(A)(n(b))(v) = (tr bQ[v]) (v) n(b) N(A)

(z)(v) = z (v) z C1 Mpn(A)

(1.5)

where V(x) = V(x) (x, )1 such that (x, ) = (x)/() is the Weil index (see [23] or [9]).

1.2.3 Unitary-Unitary Dual Pairs

Let K be a quadratic extension of k, let AK be the ring of adeles of K and continue to let A denote

the ring of adeles of k. Let V be a K-vector space of dimension m equipped with a nondegenerate

hermitian form ( , ). In particular, we have (v, w) = (w, u) and (u,v) = (u, v) for ,

K

and the nontrivial element Gal(K/k). Let Q(v) = 12 (v, v) and let UV be the k-algebraic group ofK-linear automorphisms of V that preserve the hermitian form

UV = { h GLK(V) : (hv,hw) = (v, w) for all v, w V } .

Let W be the standard K-vector space of dimension 2n equipped with the nondegenerate skew-

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hermitian form , defined by the matrix

J =

0 1n

1n 0

where 1n is the identity matrix of size n. Let Un be the k-algebraic group of K-linear automorphisms

of W that preserve the skew-hermitian form. In particular, we view W as the space of row vectors of

size 2n and the automorphisms of W as matrices acting by right multiplication therefore

Un =

g GL2n/K : g Jtg = J

.

The Siegel parabolic subgroup P of Un has a decomposition P = MN where the Levi component M is

M =m(a) =

a 00 (ta1)

: a GLn/Kand the unipotent radical N is

N =

n(b) =1n b

0 1n

: b Hermn

where Hermn denotes hermitian matrices of size n.

Define W = VKW as a k-vector space and let , be the symplectic k-bilinear form on W definedby

v1 w1, v2 w2 = 12

trK/k {(v1, v2) w1, w2} .

Let Sp(W) be the k-algebraic group of k-linear automorphisms ofW that preserve the symplectic form.

Define right actions of h UV and g Un on W by

(v w) h = h1v w and (v w) g = v wg .

It is clear that the algebraic groups UV and Un are subgroups of Sp(W) such that each is the others

centralizer. The standard polarization W = Kn Kn defines a polarization W = Vn Vn where

Vn = {v = (v1, . . . , vn) : v1, . . . , vn V}

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and so v h = h1v whereh1v = (h1v1, . . . , h

1vn) .

The metaplectic group Mp(WA) is a central extension of the group of adelic points Sp(WA) of the

symplectic group Sp(W) and fits into the exact sequence


defined by the cocycle described in [18]. Let : AK/K C be a character such that |A = mK/k

where K/k is the character attached to the extension K/k by class field theory. Then defines a

splitting of the metaplectic extension over Un(A) UV(A) (see [4] for the local construction) thereforethere is an injective homomorphism

Un(A) UV(A) Mp(WA)

(depending on ) and so we view Un(A) and UV(A) as commuting subgroups of Mp(WA) and we call

(Un(A), UV(A)) a unitary-unitary dual pair.

Let = , be the Weil representation of Mp(WA) (relative to and , see [4]) acting on the space

S(VnA ) of Schwartz functions1 on VnA . Restricted to the unitary-unitary dual pair, the Weil representation

satisfies

(h)(v) = (h1v) h UV(A)(m(a))(v) = (det a) |det a|

m2

AK(va) m(a) M(A)

(n(b))(v) = (tr bQ[v]) (v) n(b) N(A)(1.6)

and recall the groups Un and UV are defined over k by restriction of scalars and so m(a) M(A) isgiven by a GLn(AK) and n(b) N(A) is given by b Hermn(AK).

1Note that V is defined over K and VA = Vk A = (VK K)k A = VK AK = VAK .

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1.3 The Theta Correspondence

Let (G(A), H(A)) be a dual reductive pair as in the previous section with the Weil representation

acting on S(VnA ). In particular, we have

(G(A), H(A)) =

(Spn(A), OV(A)) symplectic-orthogonal

(Mpn(A), OV(A)) metaplectic-orthogonal

(Un(A), UV(A)) unitary-unitary

and we adopt the convention that G(k) = Spn(k) in the metaplectic case G(A) = Mpn(A). Each

Schwartz function S(VnA ) defines a theta function

(g, h) = vVn (g)(h1v) (1.7)which is an automorphic form on G(A) H(A). In particular, it is left-invariant by G(k) H(k) and isof moderate growth. The map

A(G H) :

is a G(A) H(A)-equivariant map from the Weil representation to the space of automorphic forms onG(A) H(A). The theta functions act as kernel functions which provide a method for transferringautomorphic forms on one group to forms on the other.

Let be an irreducible cuspidal automorphic representation of G(A) acting on a subspace V of

L2(G(k)\G(A)). The theta lift of fG V is the function

fG(h) =

G(k)\G(A)

(g, h) fG(g) dg (1.8)

and the space of functions obtained as and fG vary is the theta lift of

() ={

fG :

S(VnA ) and f

G

}

. (1.9)

The measure dg is chosen to be the Tamagawa measure of G(A) when G is either symplectic or unitary

and, in the metaplectic case, dg is the product of the unit measure on C1 and Tamagawa measure on

Spn(A). The cuspidality of ensures the absolute convergence of the integrals. Furthermore, the map

() : fG fG

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is a G(A) H(A)-equivariant map where the action of G(A) on is diagonal and the action ofG(A) on () is trivial.

Let be an irreducible cuspidal automorphic representation of H(A) acting on a subspace V of

L2(H(k)\H(A)). The theta lift of fH V is the function

fH(g) =

H(k)\H(A)

(g, h) fH(h) dh (1.10)

and the space of functions obtained as and fH vary is the theta lift of

() = {fH : S(VnA ) and fH } . (1.11)

The measure dh is chosen to be the Tamagawa measure of H(A). Again, the map

() : fH fH

is a G(A)H(A)-equivariant map where the action ofH(A) on is diagonal and the action ofH(A)on () is trivial.

Basic questions concerning theta lifts are as follows:

1. When is a theta lift nonzero?

2. When is a theta lift cuspidal?

These questions are quite deep. Solutions to the first question in some cases involve the nonvanishing of

special values of L-functions as in the work of Waldspurger [19] and the Rallis inner product formula as

in the work of Rallis [17]. The answer to the second question appears in the work of Moeglin [14] and

Jiang-Soudry [8] and relies heavily on the regularized Siegel-Weil formula as proved by Rallis-Kudla [10]

and Ichino [7]. We summarize the results on irreduciblity in the symplectic-orthogonal and metaplectic-

orthogonal cases.

Theorem 1.3.1 (Moeglin [14]). Let(Spn(A), OV(A)) be a symplectic-orthogonal dual pair.

1. Let be an irreducible cuspidal automorphic representation of Spn(A) and suppose () con-

tains cuspidal automorphic forms of OV(A). Then () is an irreducible cuspidal automorphic

representation of OV(A) and (()) = .

2. Let be an irreducible cuspidal automorphic representation of OV(A) and suppose () con-

tains cuspidal automorphic forms of Spn(A). Then() is an irreducible cuspidal automorphic

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representation of Spn(A) and (()) = .

Theorem 1.3.2 (Jiang-Soudry [8]). Let(Mpn(A), OV(A)) be a metaplectic-orthogonal dual pair.

1. Let be an irreducible genuine cuspidal automorphic representation of Mpn(A) and suppose ()

contains cuspidal automorphic forms ofOV(A). Then() is an irreducible cuspidal automorphic

representation of OV(A) and (()) = .

2. Let be an irreducible cuspidal automorphic representation of OV(A) and suppose () contains

cuspidal automorphic forms ofMpn(A). Then() is an irreducible genuine cuspidal automorphic

representation of Spn(A) and (()) = .

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Chapter 2

Duality Among Periods

The main result of this thesis is the collection of spectral identities in Theorem 2.6.1 relating Fourier

coefficients and orthogonal periods of cuspidal automorphic forms for a given dual pair (G, H). The

essential calculation is contained in the main duality Proposition 2.5.1 where we show that the Fourier

coefficient of a theta lift Wt(fH) is equal to the period of a Hecke translate PT(RfH) when the

functions and are matching as in Proposition 2.3.1. This calculation is inspired by the work of Wald-

spurger [19] for the dual pair (SL2(A), PGL2(A)). Regarding both operations Wt(fH) and PT(RfH)as linear functionals on a given cuspidal automorphic representation of H(A), we show in Proposi-

tion 2.1.1 and Proposition 2.2.1 that these functionals are given by integrating against certain elements

of . The equality of these functions yield the spectral identities in Theorem 2.6.1 by computing their

various periods.

The method for proving Theorem 2.6.1 contained in this chapter was discovered while attempting the

comparison of relative trace formulas appearing in Chapter 3. In fact, it was only when the comparison

was complete that the more direct (and more general) argument below was formulated. The comparison

of relative trace formulas is included in the next chapter because it is an interesting calculation that

further illuminates the resulting spectral identities.

The spectral identities Theorem 2.6.1 are quite formal and follow from the simplest properties of the

Weil representation and the theta correspondence. The real potential of these identities lies in applying

them in conjunction with the Siegel-Weil formula to produce formulas equating Fourier coefficients,

orthogonal periods and special values of L-functions. In Proposition 2.4.2, we take the first steps in this

direction for PGL2(Qp) (which is the local special orthogonal group, for almost every prime p, when the

rational quadratic space V has dimension 3) by explicitly computing the local functions p matching

28

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Chapter 2. Duality Among Periods 29

the characteristic function of the standard lattice Lp relative to various j. In Section 2.7, we roughly

sketch a method for producing formulas for special values of L-functions in terms of orthogonal periods

using the Siegel-Weil formula following a result of Waldspurger appearing in [19] and [22].

Let k be a global field and let A be the ring of adeles of k. Let (G(A), H(A)) be a dual pair of the

type described in Section 1.2. In particular, we have

(G(A), H(A)) =

(Spn(A), OV(A)) symplectic-orthogonal

(Mpn(A), OV(A)) metaplectic-orthogonal

(Un(A), UV(A)) unitary-unitary

and we adopt the convention that G(k) = Spn(k) in the metaplectic case G(A) = Mpn(A). In each case

there is the Siegel parabolic subgroup P(A) = M(A)N(A) G(A). In our attempt to treat each case

simultaneously, we let

Symn =

Symn symplectic-orthogonal

Symn metaplectic-orthogonal

Hermn unitary-unitary

and in each case we have Symn= N via b n(b).

In the orthogonal case, the quadratic space V defining OV is defined over k and we have the space

of Schwartz functions S(VnA ). In the unitary case, the hermitian space V defining UV is defined over

a quadratic extension K over k. Since V K AK = V k Ak, we again write the space of Schwartzfunctions as S(VnA ) where as always A is the adele ring of k.

The set of irreducible cuspidal automorphic representations of G(A) (resp. H(A)) is denoted AG0

(resp. AH0 ). Given AG0 (resp. AH0 ), let V (resp. V) denote the subspace of L2(G(k)\G(A))(resp. L2(H(k)\H(A))) on which the representation acts. The inner product of square-integrable func-tions on G(k)\G(A) (resp. H(k)\H(A)) is denoted by fG1 , fG2 G (resp. fH1 , fH2 H).

2.1 Fourier Coefficients

For t Symn(k), the t-Fourier coefficient of an automorphic form fG on G(A) is

Wt(fG) =

N(k)\N(A)

fG(n) t(n) dn (2.1)

where t(n) = (tr tb) for n = n(b). The goal of this chapter is to relate the Fourier coefficients of theta

lifts to orthogonal periods as in Proposition 2.5.1. Our first step is to compute the spectral expansion

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of the operation fH Wt(fH).

Proposition 2.1.1. Let and be cuspidal automorphic representations of G and H respectively such

that () = (equivalently () = ). For all t Symn(k) and S(VnA ), the linear functional onV

L2(H(k)

\H(A)) defined by fH

Wt(f

H) is given by the inner product with the function

,,t(h) =

FGB()

FG(h) Wt(FG) (2.2)

whereB() is an orthonormal basis ofV.

Proof. Using the basis B() we write

fH(g) =

FGB()

fH, FG

G

FG(g) .

We find the adjoint property

fH, FG

G=

fH, FG

Hby computing

f

H, FG

G=

G(k)\G(A)

H(k)\H(A)

(g, h) fH(h) dh

FG(g) dg

=

H(k)\H(A)

fH(h)

G(k)\G(A)

(g, h) FG(g) dg

dh .

Therefore, we compute the t-Fourier coefficient

Wt(fH) =

FGB()

f

H, FG

GWt(F

G)

=

FGB()

fH, F

G

HWt(F

G)

=

fH,

FGB()

FG Wt(FG)

H

.

2.2 Orthogonal Periods

Let j Vn and let T be the stabilizer ofj in H where H acts componentwise on Vn. The period overT of a cuspidal automorphic form fH on H(A) is

PT(fH) =

T(k)\T(A)

fH() d . (2.3)

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We call these integrals orthogonal periods because the subgroup T is the orthogonal/unitary group

attached to the subspace {v V : v ji for i = 1, . . . , n} V which is orthogonal to each of thecomponents ofj = (j1, . . . , jn).

Given a Schwartz function

S(H(A)), the Hecke operator R is the convolution operator

RfH(h) =

H(A)

(x) fH(hx) dx .

Looking ahead to Proposition 2.5.1, the orthogonal periods PT(RfH) appear when the integrals defining

Wt(fH) are unfolded.

Proposition 2.2.1. Let AH0 . For all S(H(A)), the linear functional onV L2(H(k)\H(A))defined by fH

PT(Rf

H) is given by the inner product with the function

R,,T(h) =

FHB()

RFH(h) PT(FH) (2.4)

whereB() is an orthonormal basis ofV and (h) = (h1).

Proof. Using the basis B() we write

RfH(h) = FHB() RfH, FHH FH(h) .

We find the adjoint property

RfH, FH

H=

fH, RFH

Hby computing

Rf

H, FH

H=

H(k)\H(A)

H(A)

(x) fH(hx) dx

FH(h) dh

=

H(A)

(x)

H(k)\H(A)

fH(hx) FH(h) dh

dx

= H(k)\H(A) fH(h)H(A) (x1) FH(hx) dxdh .

Therefore, we compute the period over T

PT(RfH) =

FHB()

Rf

H, FH

HPT(F

H)

=

FHB()

fH, RF

H

HPT(F

H)

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=

fH,

FHB()

RFH PT(FH)

H

.

2.3 Matching Functions

The next proposition introduces a notion of matching between the Schwartz functions S(VnA ) definingthe theta functions and the Schwartz functions S(H(A)) defining the Hecke operators R. Thisis the main input into Proposition 2.5.1 which shows that Wt(f

H) and PT(RfH) are equal exactly

when the functions and are matching.

Proposition 2.3.1. Letj Vn with Q[j] = t such that det t = 0, and let T be the stabilizer of j in H.For all S(VnA ) there is a smooth function on H(A) such that

(h1j) =

T(A)

( h) d (2.5)

andv = |H(kv) is compactly supported for all finite places v and rapidly decreasing for all infinite placesv. In this case, we say matches relative to j.

Proof. We may assume that = v v is factorizable therefore we will prove the analogous equality foreach place v. The group H(A) can be written as the restricted product

v H(kv) with respect to the

compact open subgroups Kov = Aut(Lv) where Lv = L OOv for a fixed global O-lattice L V. HereO is the ring of integers of k and Ov is the ring of integers of the competion kv. We will show that v is

the characteristic function of Kov for almost every place v.

Suppose v is a finite place. If Vv is anisotropic, the group H(kv) is compact and so

v(h) =1

vol T(kv)v(h

1j)

is a smooth function of compact support which matches v relative to j.

Suppose Vv is isotropic. Let v be a locally constant compactly supported function and let Kv be

a compact open subgroup of H(kv) such that v(kv) = v(v) for all k Kv and v Vnv . Use the

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notation CS to denote the characteristic function of a set S and write

v(h1j) =

H(kv)/Kv

1j supp v

v(1j) CKv(h)

=

T(kv)\H(kv)/Kv

1j supp v

v(1j)

(T(kv)Kv1)\T(kv)

CKv( h) .

We claim that the outer sum is finite. Since det t = 0, the group H(kv) acts transitively on the sett = {v Vnv : Q[v] = t} and therefore the map 1j is a homeomorphism between T(kv)\H(kv)and t. Since v has compact support, its restriction to the closed subset t has compact support and

therefore the support of v(h1j) in T(kv)\H(kv) is also compact. Finally, since T(kv)\H(kv)/Kv is

discrete, the set { T(kv)\H(kv)/Kv : 1j supp v} is finite.We make the observation

T(kv)

CKv( h) d = vol(T(kv) Kv1)

(T(kv)Kv1)\T(kv)

CKv( h)

for all , h H(kv). Therefore

v(h) =

T(kv)\H(kv)/Kv1j supp v

v(1j)

vol(T(kv)

Kv1)

CKv (h)

is a locally constant compactly supported function which matches v relative to j.

We claim that, for almost all v, the sum above has a single term and v(h) = CK0v (h). For almost

all v, we are in the following situation:

1. v = Lv Lv is n copies of the characteristic function Lv of the lattice Lv = L OOv

2. j1, . . . , jn

Lv

3. det Q[j] Ov

4. Q is Ov-valued on Lv

The symmetric matrix Q[j] represents the quadratic form Q restricted to j = spanOv{j1, . . . , jn}relative to the basis j1, . . . , jn. Since det Q[j] Ov , the lattice j is regular therefore Lv = j is

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an orthogonal direct sum ofOv-lattices where

= {v Lv : (v, w) = 0 for all w j} .

Suppose H(kv) such that ji Lv for each i = 1, . . . , n. Then j is a regular subspace of Lvtherefore we have the orthogonal direct sum Lv = j ofOv-lattices where

= {v Lv : (v,w) = 0 for all w j} .

Since j and j are isometric, we must have that and are isometric by the Wittcancellation property for local rings. In other words, there is some T(kv) (note that T(kv) is theorthogonal group of spankv{j1, . . . , jn}) such that = . Finally, Lv = Lv and so Kov .

Therefore, in this most unramified case, the sum above has a single term and v(h) = CKov (h) is the

characteristic function of Kov . Here we have used the fact that vol(T(kv) Kov ) = 1 for almost all v.

Suppose v is an infinite place. If Vv is anisotropic, the group H(kv) is compact and so

v(h) =1

vol T(kv)v(h

1j)

is a smooth compactly supported function which matches v relative to j.

Suppose Vv is isotropic. We follow the construction in [5, Lemma 1.10, p.92]. Let us simplify the

notation and let H = H(kv) which we view as a real Lie group and let T = T(kv) which we view a Lie

subgroup of H. Define a map f f on the space Cc(H) of continuous compactly supported functionsby

f(h) =

T

f( h) d .

We will proceed as in [5] to show that the map f f is a linear map of Cc(H) onto Cc(T\H).

The map h h1j defines a diffeomorphism from T\H to t = {v Vnv : Q[v] = t}. First supposethat = v is a smooth function of compact support on Vnv which we view by restriction as a smooth

function of compact support on T\H with support in a compact set C T\H. Let C be a compactsubset of H such that (C) = C for the projection map : H T\H. Let CT be a compact subset ofT of positive measure and put C = CT C. Then (C) = C. Select f Cc(G) such that f 0 on G

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and f > 0 on C. Then f > 0 on C and the function

(h) =

f(h)(h1j)

f(h)if (h) C

0 if (h) C

has compact support and = .

If S(Vnv ) then let {i} be a sequence of smooth compactly supported functions such that i uniformly. Then we can choose a sequence {fi} of smooth compactly supported functions and define thesequence of functions {i} as in the construction above so that i = i. There is much flexibility in thechoice of the functions {fi} and so we can choose them such that the sequence {i} converges uniformlyto a smooth, rapidly decreasing function on H which matches relative to j.

2.4 Example: Matching Functions for PGL2(Qp)

One direction of future work would be to use the spectral identities in Theorem 2.6.1 to compute, in

special cases, formulas relating Fourier coefficients and orthogonal periods. Such work would depend, in

part, on explicitly calculating matching functions therefore, in this section, we compute local matching

functions when dim V = 3.

Let Vp be the space of traceless 2 by 2 matrices over Qp equipped with the quadratic form Q(x) =

det(x). In this case, we have O(Vp) = SO(Vp) 1 and SO(Vp) = PGL2(Qp) acting on Vp by conju-gation. Let Lp be the lattice of integral matrices and let Kp SO(Vp) denote the proper automorphismsof Lp. Let j Vp and let Q(j) = p with Z and Zp . There are three possibilities for j:

(unramified) if is even and is a nonsquare, then there is some h O(Vp) such that

h(j) = p/2

0

1 0

(ramified) if is odd, then there is some h O(Vp) such that

h(j) = p(1)/2

0 p1 0

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(split) if is even and is a square, then there is some h O(Vp) such that

h(j) = p/2

1 0

0 1

To compute matching functions we need to have representatives for the double coset space Tp\SO(Vp)/Kpwhere Tp is the stabilizer of j.

Proposition 2.4.1. Letj Vp and let Tp be the stabilizer of j in SO(Vp).

(i) A set of coset representatives of SO(Vp)/Kp is given by

pd u

0 1

: d 0 and 0 u < pd

1 0

u pd

: d > 0, 0 u < pd and p | u

.

(ii) If j =

0 1 0

where Zp is a nonsquare, then Tp\SO(Vp)/Kp = d0

Tp

pd 00 1

Kp.

(iii) If j =

0 p1 0

where Zp , then Tp\SO(Vp)/Kp = d1

Tp

pd 00 1

Kp.

(iv) If j = 1 0

0 1, then Tp\SO(Vp)/Kp = d0 Tppd 1

0 1Kp.Proof. The Iwasawa decomposition states that SO(Vp) = BKp where B are the upper triangular matrices

and so we need only consider elements of B. Let g B and choose a matrix representative that hasintegral entries with at least one entry a unit. There are three possibilities:

Case 1: Write g =

pd1 b0 2

for some d 0, 1, 2 Zp and b Zp. There is a unique u Zand x Zp such that 0 u < pd and b12 = u +pdx. Modify the element g by multiplying on the rightby elements of Kp

pd1 b0 2

11 0

0 12

1 x

0 1

=pd b12

0 1

1 x

0 1

=pd u

0 1

.

Case 2: Write g =

1 b0 pd2

for some d 0, 1, 2 Zp and b Zp. Modify the element g by

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multiplying on the right by elements of Kp

1 b

0 pd2

11 0

0 12

1 b120 1

=

1 b12

0 pd

1 b120 1

=

1 0

0 pd

.

Case 3: Write g =

p1 u0 p2

for some , 0 and 1, 2, u Zp . There is a unique u Z andx Zp such that 0 u < p+ and p u12 = u +p+x. Modify the element g by multiplying on theright by elements of Kp

p1 u

0 p2

11 0

0 12

1 0

pu12 1

0 u12

u12 0

1 0

x 1

=

p u120 p

1 0

pu12 1

0 u12

u12 0

1 0

x 1

=

0 u12

p+u12 p

0 u12

u12 0

1 0

x 1

=

1 0p u12 p

+

1 0

x 1

=

1 0u p+

.

If p u then u Zp andp+ u1

u 0

Kp. There is a unique u Z and x Zp such that0 u < p+ and u1 = u +p+ x. Compute

1 0u p+

p+ u1

u 0

1 x

0 1

=p+ u1

0 1

1 x

0 1

=p+ u

0 1

.

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Therefore, we have the coset representatives in (i).

Given the element j in (ii), we have

Tp =

a b

b a

SO(Vp)

.

Modify the coset representatives obtained above by multiplying on the left by an element of Tp and on

the right by an element of Kp

1 uu1 1

pd u

0 1

1 0

pdu1(1 u21)1 (1 u21)1

=

pd 0

pdu1 1 u21

1 0

pdu1(1 u21)1 (1 u21)1

=

pd 00 1

We also compute 0

1 0

1 0

u pd

0 1

1 0

=pd u

0 1

which can be modified further as above to produce pd 00 1

. Therefore we obtain the representativesin (ii). The other two statements are proved analogously.

Proposition 2.4.2. Letp be the characteristic function of Lp, let j Vp, let Tp be the stabilizer of jin SO(Vp) and let Q(j) = p

with Z and Zp . We write CS for the characteristic function of aset S.

(i) If < 0, then p(h1j) = 0 for all h SO(Vp).

(ii) If j = p/20 1 0

with 0 even and Zp a nonsquare, then

p(h) =1

vol Tp

CKp(h) + /2d=1

(pd +pd1) C[pd]Kp(h)

, [pd] =pd 0

0 1

, (2.6)matches p relative to j.

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(iii) If j = p(1)/2

0 p1 0

with > 0 odd, then

p(h) =1

vol Tp

(+1)/2

d=1

2pd1

C[pd]Kp(h) , [pd

] = pd 00 1

, (2.7)matches p relative to j.

(iv) If j = p/2

1 00 1

with 0 even and Zp a square, then

p(h) =1

vol(Tp

Kp)

CKp(h) +

/2

d=1C[pd,1]Kp(h)

, [pd, 1] =

pd 1

0 1

, (2.8)

matches p relative to j.

Proof. We have ordp Q(v) 0 for all v Lp and Q(h1j) = Q(j) for all h SO(Vp) therefore ifordp Q(j) < 0 then h

1j Lp for all h SO(Vp) and so p(h1j) = 0.In the proof of Proposition 2.3.1, we saw that the function

p(h) =

Tp\SO(Vp)/Kp1j Lp

1

vol(Tp Kp1) CKp(h) (2.9)

is a smooth function of compact support that matches p relative to j.

In the first case, if j = p/2

0 1 0

then:

Tp =

a bb a

PGL2(Qp)

and Tp\SO(Vp)/Kp =

d0

Tp[pd]Kp where [pd] =

pd 00 1

For d > 0, 1

vol(Tp [pd]Kp[pd]) =Tp : Tp [pd]Kp[pd]

vol Tp= p

d +pd1

vol Tp

[pd] j = p/2d 0

p2d 0

, therefore [pd] j Lp if d /2To prove the second point, note that Tp Kp fixes the base point Kp SO(Vp)/Kp in the Bruhat-Titstree of PGL2(Qp) and acts transitively on the p

d +pd1 points at a distance d from the base point while

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the subgroup Tp [pd]Kp[pd] is the stabilizer in Tp of a point at a distance d from the base point.Therefore

p(h) =1

vol Tp

CKp(h) + /2d=1

(pd +pd1) C[pd]Kp(h)

.

If j = p(1)/20 p

1 0

, then

Tp =

a bp

b a

PGL2(Qp) and Tp\SO(Vp)/Kp =

d1

Tp[pd]Kp where [p

d] =

pd 00 1

For d > 0, 1vol(Tp [pd]Kp[pd]) =

Tp : Tp [pd]Kp[pd]

vol Tp

=2pd1

vol Tp

[pd] j = p(+1)/2d 0

p2d1 0

, therefore [pd] j Lp if d ( + 1)/2To prove the second point, note that Tp fixes the midpoint between Kp and [p]Kp in the tree SO(Vp)/Kp

and acts transitively on the 2pd1 points at a distance d 1/2 from that midpoint while the subgroupTp[pd]Kp[pd] is the stabilizer in Tp of the point [pd]Kp which is at a distance d1/2 from the midpointbetween Kp and [p]Kp. Therefore

p(h) =1

vol Tp

(+1)/2

d=1

2pd1

C[pd]Kp(h) .

Finally, if j = p/2

1 00 1

, then

Tp =

a 0

0 b

PGL2(Qp) and Tp\SO(Vp)/Kp = d0 Tp[pd, 1]Kp where [pd, 1] =

pd 10 1

.

For d 0,1

vol(Tp [pd, 1]Kp[pd, 1]1) =1

vol(Tp Kp)

[pd, 1]1 j = p/2d

pd 20 pd

therefore [pd, 1]1 j Lp if d /2To prove the second point, note that Tp acts on the set of vertices {[pd]Kp : d Z} in the tree SO(Vp)/Kpand [pd, 1]Kp[p

d, 1]1 is the stabilizer in SO(Vp) of the point [pd, 1]Kp. Therefore, if Tp stabilizes

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[pd, 1]Kp it must fix Kp also therefore Tp [pd, 1]Kp[pd, 1]1 = Tp Kp. Therefore

p(h) =1

vol(Tp Kp)

CKp(h) +

/2d=1

C[pd,1]Kp(h)

, [pd, 1] =

pd 1

0 1

.

Notice that when d = 0, we have

1 10 1

Kp = Kp.

2.5 Duality Among Periods

The next proposition1 gives the main duality between Fourier coefficients and orthogonal periods. It

is a simple calculation using the notion of matching described in Proposition 2.3.1. An immediate

consequence, stated in the following corollary, is that the functions ,,t(h) and R,,T(h) found in

Proposition 2.1.1 and Proposition 2.2.1 respectively are equal when the functions and are matching

relative to j. Looking ahead, it is the equality of these functions which yield the spectral identities in

Theorem 2.6.1 in the next section.

Proposition 2.5.1. Let j Vn with Q[j] = t such that det t = 0, and let T be the stabilizer of jin H. Let S(VnA ) and let be a smooth function on H(A) which matches relative to j as inProposition 2.3.1. Then, for all fH L2(H(k)\H(A)),

Wt(fH) = PT(Rf

H) . (2.10)

Proof. We simply compute the left hand side by changing the order of integration

Wt(fH) =

N(k)\N(A)

H(k)\H(A)

vVn

(n(b))(h1v) fH(h) t(b) db

= H(k)\H(A) N(k)\N(A) vVn (h1v) (b(Q[v] t)) db fH(h) dh=

H(k)\H(A)

v Vn

Q[v] = t

(h1v) fH(h) dh .

1During the preparation of this thesis, we found a version of this proposition for the dual pair (fSL2, PB) (where fSL2is the metaplectic cover of SL2 and B is a quaternion algebra over k) appearing in a paper by Mao [13].

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Since det t = 0, the sum is a single H(k)-orbit therefore we continue

Wt(fH) =

H(k)\H(A)

T(k)\H(k)

(h11j) fH(h) dh

= T(k)\H(A) (h1j) fH(h) dh=

T(A)\H(A)

(h1j)

T(k)\T(A)

fH( h) d dh .

The function matches relative to j as in Proposition 2.3.1 therefore

Wt(fH) =

T(A)\H(A)

T(A)

(h) d

T(k)\T(A)

fH( h) d dh

=

H(A)

T(k)\T(A)

(h)fH( h) d dh

= T(k)\T(A)

H(A)

(h)fH( h) dhd

and we conclude Wt(fH) = PT(RfH).

Corollary 2.5.1. Let j Vn with Q[j] = t such that det t = 0, and let T be the stabilizer of jin H. Let S(VnA ) and let be a smooth function on H(A) which matches relative to j as inProposition 2.3.1. Given cuspidal automorphic representations and of G and H respectively such

that () = (equivalently () = ), we have

FGB()

FG(h) Wt(FG) =

FHB()

RFH(h) PT(FH) (2.11)


Proof. Proposition 2.5.1 says that the linear functionals fH Wt(fH) and fH PT(RfH) areequal therefore the functions in Proposition 2.1.1 and Proposition 2.2.1 (which are clearly elements of

V) are equal.

2.6 The Main Theorem: Spectral Identities

Equipped with Corollary 2.5.1, we are now prepared to prove our main result: a collection of spectral

identites relating Fourier coefficients and orthogonal periods of cuspidal automorphic forms for a given

dual pair (G, H). As the analysis in the previous sections show, these period relations are manifestations

of the main duality Wt(fH) = PT(Rf

H) in Proposition 2.5.1. These identities are quite formal and, in

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the next section, we explore the possibility of proving formulas involving Fourier coefficients, orthogonal

periods and, via the Siegel-Weil formula, special values of L-functions.

Theorem 2.6.1. Letj1

,j2

Vn with Q[j1

] = t1

and Q[j2

] = t2

such that det t1

= 0 and det t2

= 0,

and let T1 and T2 be the stabilizers in H of j1 and j2 respectively. Let 1, 2 S(VnA ) and let 1 and2 be smooth functions on H(A) such that 1 matches1 relative to j1 and 2 matches2 relative to j2

as in Proposition 2.3.1. Given cuspidal automorphic representations of G and H respectively such that

() = (equivalently () = ), we have

FGB()

Wt1(12 FG) Wt2(F

G) =

FHB()

PT1(R1R2 FH) PT2(F

H) (2.12)

FGB() PT1(2 FG) Wt2(FG) = FHB() PT1(R2 FH) PT2(FH) (2.13)FGB()

PT1(1 FG) PT2(2 F

G) =

FHB()

PT1(1 2FH) PT2(F

H) (2.14)


Proof. Write Equation 2.11 with the data j2, t2, T2, 2 and 2

FGB()

2 FG

(h) Wt2(FG

) = FHB()

R

2 FH

(h) PT2(FH

)

and then apply the linear functional fH Wt1(1fH) to each side

FGB()

Wt1(12 FG) Wt2(F

G) =

FHB()

Wt1(1R2 FH) PT2(F

H) .

Using the equality Wt1(1R2 FH) = PT1(R1R2 F

H) as in Proposition 2.5.1, we arrive at Equation 2.12

FGB()

Wt1(12 FG

) Wt2(FG) = FHB()

PT1(R1R2 FH

) PT2(FH) .

For the second equation, write Equation 2.11 with the data j2, t2, T2, 2 and 2 and then apply the

linear functional fH PT1(fH) to each side to obtain Equation 2.13

FGB()

PT1(2 FG) Wt2(F

G) =

FHB()

PT1(R2 FH) PT2(F

H) .

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For the third equation, write f V in terms of the basis B()

f(g) =

FGB()

f, FGG FG(g)

therefore

PT2(2 f) =

f,

FGB()

PT2(2 FG) FG

G

.

Alternatively, write 2 f in terms of the basis B()

2 f(h) =

FHB()

2 f, FHH FH(h)

and compute PT2(2 f)

PT2(2 f) =

FHB()

2 f, F

H

HPT2(F

H) =

f,

FHB()

PT2(FH) 2F

H

H

therefore FGB()

PT2(2 FG) FG(g) =

FHB()

PT2(FH) 2F

H(g) .

Apply the linear functional fG PT1(1 fG) and we arrive at Equation 2.14

FGB()

PT1(1 FG

) PT2(2 FG) = FHB()

PT1(1 2FH

) PT2(FH) .

2.7 Further Directions: Special Values ofL-Functions

We conclude this chapter by roughly sketching a method for producing formulas for special values of

L-functions in terms of orthogonal periods using the Siegel-Weil formula. Consider the second spectral

identity in Theorem 2.6.1

FGB()

PT1(2 FG) Wt2(F

G) =

FHB()

PT1(R2 FH) PT2(F

H)

applied to the dual pair (SL2(A), OV(A)). Here SL2(A) is the metaplectic cover of SL2(A) and OVis the orthogonal group of the vector space V of traceless elements of a quaternion division algebra

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B over k equipped with the norm form. During the course of Waldspurgers deep study of the theta

correspondence for such dual pairs in [19], [20], [21] and [22], Waldspurger proves the following theorem

which links the orthogonal periods of theta lifts to the special values of L-functions.

Theorem 2.7.1. (Waldspurger, [22, Lemma 45 p.293]) Let be a cuspidal automorphic representation

of SL2(A) and let be an automorphic representation of OV(A) such that () = (equivalently() = ). Let Wald() be the Jacquet-Langlands transfer of to PGL2(A). Let j V be nonzerowith Nm(j) = t, let T be the stabilizer of j in OV, let S(VA) and let f V. There is a finite set Sof places v such that

PT(f) =LS(Wald(), 1/2)

LS(t, 1)

vS

Zv(f,,j, 1/2)

where t

is the quadratic character attached to the extension k(

t),

Zv(f, , j, s) =

N(kv)\SL2(kv)

Wfv (g) (g)v(j)|a(g)|s1/2v dg

are local zeta functions and [N]

f(ng) t(n) dn =

v

Wfv (g)

is the decomposition of the t-Whittaker function of f into local t,v-Whittaker functions.

Proof. Consider the orthogonal decomposition V = V V where V = spanj and V = (spanj).Note that O(V) = T. Suppose (aj + w) = (aj)(w) for all aj + w V V with S(V(A))and S(V(A)). Now we compute

PT(f) =

[T]

[SL2]

vV

(g)(1v) f(g) dgd

=

[SL2]

ajV

(g)(aj)

[T]

wV

(g)(1w) d f(g) dg

= [SL2]

ajV

(g)(aj) E(g, 1/2, ) f(g) dg .

The last line uses the Siegel-Weil formula for the dual reductive pair (SL2(A), T(A)) where the Eisenstein

series is given by the analytic continuation of

E(g,s, ) =

P(k)\SL2(k)

(g)(0) |a(g)|s1/2 , Re s 0 .

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To unfold this integral further, introduce the integral for s C with Re s 0

PT(f; s) =

[SL2]

ajV

(g)(aj) E(g,s, ) f(g) dg

= [SL2] ajV (g)(aj) P(k)\SL2(k) (g)(0) |a(g)|s1/2 f(g) dg=

P(k)\SL2(A)

ajV

(g)(aj) (g)(0) |a(g)|s1/2 f(g) dg

=

P(k)\SL2(A)

ajV

(g)(aj) |a(g)|s1/2 f(g) dg .

Since (g)(aj) = (m(a)g)(j) for all a k, we continue

PT(f; s) =

P(k)\SL2(A) m(a)M(k)(m(a)g)(j) |a(g)|s1/2 f(g) dg

+

P(k)\SL2(A)

(g)(0) |a(g)|s1/2 f(g) dg

where the second term vanishes since f is cuspidal. Finally,

PT(f; s) =

P(k)\SL2(A)

m(a)M(k)

(m(a)g)(j) |a(g)|s1/2 f(g) dg

=

N(k)\SL2(A)

(g)(j) |a(g)|s1/2 f(g) dg

= N(A)\SL2(A) (g)(j) |a(g)|s1/2 [N] f(ng) t(n) dndg.The t-Whittaker function of f can written as a product of local Whittaker functions

[N]

f(ng) t(n) dn =

v

Wfv (g)

therefore the value PT(f) is the value at = 1/2 of the analytic continuation of

PT(f; s) = v N(kv)\SL2(kv) Wfv (g) (g)(j) |a(g)|s1/2 dg .The set S is given by the set of places v such that for all v S the functions v and Wfv are unramifiedand t is an integral unit. Finally, Waldspurger does the calculation for v S

N(kv)\SL2(kv)

Wfv (g) (g)(j) |a(g)|s1/2 dg =Lv(Wald(), 1/2)

Lv(t, 1).

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Equipped with this result of Waldspurger, we would like to make the following conjecture.

Conjecture 2.7.1. Let and be cuspidal automorphic representations of

SL2(A) and OV(A) respec-

tively such that () = (equivalently () = ). There exist automorphic forms fG V andfH V and data j, t , T, , such that Equation 2.13 reduces to

L(Wald(), 1/2)

L(t, 1)

Wt(fG)

fG, fGG?

=1

t

PT(fH)2fH, fHH . (2.15)

The factor 1/

t appearing on the right is due to the operator R acting on fH based on the

calculations in Proposition 2.4.2.

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Chapter 3

A Comparison of Relative Trace

Formulas

The goal of this chapter is to show, in some very special cases, that the spectral identity Equation 2.12

appearing in Theorem 2.6.1 is the result of a comparison of nonstandard relative trace formulas. One

trace formula is standard and given by the trace of the operator R1R2 for H relative to T1 T2. Theother trace formula is novel and is given by the trace of the operator 12 for G relative to N Nand the characters t1 and t2 (see Theorem 2.6.1 for notation). The comparison requires the following

strong assumptions:

n = 1, therefore G(A) is either Sp1(A) = SL2(A), its metaplectic cover Mp1(A) or U1(A).

V is anisotropic.

The first assumption implies that the Siegel parabolic subgroup is the only proper parabolic subgroup

of G(A) and this is essential in the computation of the noncuspidal part of 12 in Proposition 3.4.1.

It is likely that the techniques in the work of Rallis [ 16] can be used to compute the noncuspidal part of

12 in the general situation by calculating the constant terms along all parabolic subgroups.

In general, given a dual reductive pair (G(A), H(A)), the theta lift of a cuspidal automorphic form on

G(A) is not always (in fact, almost never) a cuspidal automorphic form on H(A) as shown in the results

of Rallis [16], Moeglin [14] and Jiang-Soudry [8]. The second assumption ensures that the theta lift of a

cuspidal representation of G(A) is a cuspidal representation of H(A) therefore the operator 12 on

cusp forms is well-defined.

48

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Chapter 3. A Comparison of Relative Trace Formulas 49

The second assumption also impies that G(k)\G(A) is compact therefore the inner product of thetafunctions 12 is absolutely convergent and defines an integral operator with kernel K (where =

1 2 ) on the space of cusp forms L2cusp(G(k)\G(A)). The action of Hecke operators R1R2 unfoldsin the standard way to show that it is an integral operator with kernel K (where = 1 2 ). Thecomparison then follows the standard procedure:

1. Compute the geometric expansions of the traces of K and K (Proposition 3.1.1 and Proposi-

tion 3.2.1).

2. Find a natural bijection between the terms, the orbi

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