Fourier Coefficients of Automorphic Forms and ArthurClassification

A DISSERTATION

SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL

OF THE UNIVERSITY OF MINNESOTA

BY

Baiying Liu

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

Doctor of Philosophy

Prof. Dr. Dihua Jiang

May, 2013

c© Baiying Liu 2013

ALL RIGHTS RESERVED

Acknowledgements

There are many people that have earned my gratitude for their contribution to my time

in graduate school.

First and foremost, I would like to take this opportunity to express my deepest grat-

itude to my advisor Prof. Dihua Jiang, for introducing me to the topics of Fourier coef-

ficients of automorphic forms, automorphic descent, constructions of square-integrable

automorphic representations, and representations of p-adic groups, for sharing with me

his wonderful ideas and insights to various problems of mathematics, for his constant

encouragement and support.

I would like to thank Prof. James Arthur, Prof. David Ginzburg, Prof. David

Soudry for helpful conversations and communications. And I would like to thank Prof.

Freydoon Shahidi for helpful comments on the paper of Fourier coefficients of automor-

phic forms of GLn.

I also would like to thank my committee members, Prof. Paul Garrett, Prof. Kai-

Wen Lan and Prof. Richard McGehee, for reviewing my thesis and serving on the

committee of my thesis defence. I really appreciate their help.

Finally, I would like to thank my academic elder brothers Dr. Lei Zhang and Dr.

Xin Shen, for their helpful discussion and comments.

i

Abstract

Fourier coefficients play important roles in the study of both classical modular

forms and automorphic forms. For example, it is a well-known theorem of Shalika

and Piatetski-Shapiro that cuspidal automorphic forms of GLn(A) are globally generic,

that is, have non-degenerate Whittaker-Fourier coefficients, which is proved by taking

Fourier expansion. For general connected reductive groups, there is a framework of

attaching Fourier coefficients to nilpotent orbits. For general linear groups and classical

groups, nilpotent orbits are parametrized by partitions. Given any automorphic repre-

sentation π of general linear groups or classical groups, characterizing the set nm(π) of

maximal partitions with corresponding nilpotent orbits providing non-vanishing Fourier

coefficients is an interesting question, and has applications in automorphic descent and

construction of endoscopic lifting.

In this thesis, first, we extend the Fourier expansion of cuspidal automorphic forms

of GLn(A) to any automorphic form occurring in the discrete spectrum of GLn(A).

Then, we determine the set nm(π) for any residual representation ∆(τ,m) ofGL2mn(A)

(with τ an irreducible unitary cuspidal automorphic representation of GL2n(A)) and cer-

tain residual representation E∆(τ,m) of Sp4mn(A) constructed from ∆(τ,m) by Jiang, Liu

and Zhang.

Next, we consider certain set of irreducible cuspidal automorphic representations of

Sp4mn(A) which are nearly equivalent to the residual representation E∆(τ,m). We show

that this set decomposes naturally into two disjoint sets, corresponding to certain sets

of irreducible cuspidal automorphic representations of Sp4mn−2n(A) and Sp4mn+2n(A),

respectively. This extends the Ginzburg-Jiang-Soudry correspondences between certain

automorphic forms on Sp4n(A) and Sp2n(A).

At last, we recall Arthur’s classification of the discrete spectrum and a conjecture of

Jiang towards understanding Fourier coefficients of automorphic forms in automorphic

L2-packets, and briefly discuss the relation between them and the above results in this

thesis.

ii

Contents

Acknowledgements i

Abstract ii

1 Introduction 1

1.1 On Fourier Coefficients of Automorphic Forms of GLn . . . . . . . . . . 2

1.2 On Extension of Ginzburg-Jiang-Soudry Correspondences to Certain Au-

tomorphic Forms on Sp4mn(A) and Sp4mn±2n(A) . . . . . . . . . . . . . 4

1.3 On Arthur Classification and Jiang’s Conjecture . . . . . . . . . . . . . 7

2 On Fourier Coefficients of Automorphic Forms of GLn 9

2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Discrete Spectrum of GLn . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Structure of discrete spectrum . . . . . . . . . . . . . . . . . . . 11

2.2.2 Fourier expansion for cuspidal automorphic forms . . . . . . . . 12

2.3 Fourier Expansion for the Discrete Spectrum . . . . . . . . . . . . . . . 13

2.3.1 Families of Fourier coefficients . . . . . . . . . . . . . . . . . . . 14

2.3.2 Fourier expansion: step one . . . . . . . . . . . . . . . . . . . . . 15

2.3.3 Fourier expansion: step two . . . . . . . . . . . . . . . . . . . . . 18

2.4 Proof of Lemma 2.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Fourier Coefficients for GLn . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5.1 Fourier coefficients for GLn . . . . . . . . . . . . . . . . . . . . . 27

2.5.2 Fourier coefficients for the discrete spectrum of GLn . . . . . . . 33

2.6 Proof of Theorem 2.5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

iii

3 On Fourier Coefficients of Automorphic Forms of Symplectic Groups 51

3.1 Fourier Coefficients Automorphic Forms of Symplectic Groups Attached

to Nilpotent Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2 Fourier-Jacobi Coefficients and Automorphic Descent . . . . . . . . . . . 61

4 On Extension of Ginzburg-Jiang-Soudry Correspondences to Certain

Automorphic Forms on Sp4mn(A) and Sp4mn±2n(A) 63

4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2 Proof of Theorem 4.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3 Proof of Lemma 4.2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.3.1 ω0-term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3.2 ω1-term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.4 Proof of Part (1) of Theorem 4.1.2 . . . . . . . . . . . . . . . . . . . . . 86

4.5 Proof of Theorem 4.4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.6 Proof of Part (2) of Theorem 4.1.2 . . . . . . . . . . . . . . . . . . . . . 110

4.7 Irreducibility of Certain Descent Representations . . . . . . . . . . . . . 110

5 On Arthur Classification and Jiang’s Conjecture 115

References 117

iv

Chapter 1

Introduction

In the study of classical modular forms, it is known that the Fourier coefficients carry

significant arithmetic information. For example, the Sato–Tate Conjecture says that, for

certain holomorphic cusp forms corresponding to (non-CM) elliptic curves over rational

field, the angles corresponding to their Fourier coefficients distribute in a certain way,

this is related to the number of points of the reductions modulo primes of elliptic

curves (this conjecture has been proved by Barnet-Lamb, Geraghty, Harris and Taylor

[BLGHT11]).

Fourier coefficients also play an important role in the study of automorphic forms.

For example, a basic and fundamental result in the theory of automorphic forms for

GLn(A) is that cuspidal automorphic forms are globally generic, that is, have non-

vanishing Whittaker-Fourier coefficients, due to Shalika [S74] and Piatetski-Shapiro

[PS79] independently. This has been used to prove the strong multiplicity one theo-

rem for cuspidal automorphic representations of GLn(A).

As another example, for classical groups, Ginzburg, Rallis and Soudry ([GRS11])

developed the theory of automorphic descent by studying certain Fourier coefficients

of special type residual representations, which produces the inverse of the Langlands

functorial transfers from classical groups to the general linear groups.

1

2

1.1 On Fourier Coefficients of Automorphic Forms of GLn

It is a well-known theorem of Shalika and Piatetski-Shapiro that cuspidal automorphic

forms ϕ of GLn(A) are globally generic, that is, have non-vanishing Whittaker-Fourier

coefficients. This theorem is proved using the Fourier expansion of ϕ along the standard

maximal unipotent subgroup Un of GLn. Since it is not abelian, Shalika and Piatetski-

Shapiro’s idea is to consider abelian subgroups of Un consisting of only column elements

and take the Fourier expansion column-by-column.

Theorem 1.1.1 (Shalika [S74], Piatetski-Shapiro [PS79]). Assume that ϕ is a cuspidal

automorphic form of GLn(A). Then,

ϕ(g) =∑

γ∈Un−1(F )\GLn−1(F )

Wψ(ϕ, ιn(γ)g), (1.1)

where ιn(γ) =

(γ 0

0 1

), and

Wψ(ϕ, g) :=

∫Un(k)\Un(A)

ϕ(ug)ψ−1Un

(u)du,

with

ψUn(u) := ψ(u1,2 + u2,3 + · · ·+ un−1,n). (1.2)

Moreover, the Fourier expansion in (1.1) is absolutely convergent and uniformly con-

verges on any compact set in g.

Wψ(ϕ, g) is called a non-degenerate Whittaker Fourier coefficient of ϕ.

In [JL12], joint with Jiang, we extend Theorem 1.1.1 to any automorphic forms of

GLn(A) occurring in the discrete spectrum.

The explicit construction of the discrete spectrum of GLn(A) was conjectured by

Jacquet ([J84]) and then proved by Moeglin and Waldspurger ([MW89]). It turns out

that an irreducible automorphic representation π of GLn(A) occurring in the discrete

spectrum is parametrized by a pair (τ, b) with τ an irreducible unitary cuspidal auto-

morphic representation of GLa(A), for some pair a, b of integers such that n = ab. In

particular, if π is also cuspidal, then b = 1. The irreducible automorphic representation

parametrized by (τ, b) can be denoted by E(τ,b) (or ∆(τ, b) sometimes), which is called

a Speh residual representation if b > 1. For more discussion, see Section 2.2.1.

3

Theorem 1.1.2 (Jiang and Liu [JL12]). Assume that ϕ ∈ E(τ,b), n = ab. Then,

ϕ(g) =∑γ

Wψ(τ,b)(ϕ, γg), (1.3)

where γ is a certain rational matrix which will be specified in Theorem 2.3.3, and

Wψ(τ,b)(ϕ, g) :=

∫Un(k)\Un(A)

ϕ(ug)ψUn−1

(u)du,

with

ψUn(u) := ψ(n−1∑i=1

ui,i+1 −b−1∑j=1

uja,ja+1). (1.4)

Moreover, the Fourier expansion in (1.3) is absolutely convergent and uniformly con-

verges on any compact set in g.

Wψ(τ,b)(ϕ, g) is called a degenerate Whittaker Fourier coefficient of ϕ if b > 1.

Note that after comparing the characters in (1.2) and (1.4), we can see that the

character occurring in the Fourier expansion of ϕ ∈ E(τ,b) only involves “simple roots

inside the cuspidal support”. Also note that the Fourier coefficients in both (1.1) and

(1.3) have the same integration domain. When b = 1, Theorem 1.1.2 reduces to Theorem

1.1.1.

If one takes an arbitrary automorphic form of GLn(A), one can still apply the idea

of Shalika and Piatetski-Shapiro to do the Fourier expansion, but the resulting formula

might be complicated. Such a general formula can be found in [[Y93], Proposition

2.1.3]. However, Theorem 1.1.2 gives the explicit summation domain for γ in terms of

the cuspidal support of the residual representation E(τ,b).

For classical groups, on one hand, it is very hard to get a Fourier expansion as

above, due to the complicate group structures. On the other hand, there exist non-

generic cuspidal automorphic forms in general. Therefore, the idea of “taking off simple

roots outside of cuspidal support” doesn’t work, since the degenerate Whittaker Fourier

coefficients like Wψ(τ,b)(ϕ, g) will all be killed by cuspidality.

For general linear groups and classical groups, there is a general framework of at-

taching Fourier coefficients to nilpotent orbits which are classified by partitions, gener-

alizing the Whittaker-Fourier coefficients. See Section 2.5.1 (for GLn) and Section 3.1

(for Sp2n), for explicit discussions. Also, see [[J12], Section 4] for more discussions and

related applications.

4

Given an automorphic representation π, let n(π) be the set of all partitions such

that there is a corresponding nilpotent orbit providing non-vanishing Fourier coefficient

to π. Let nm(π) be the subset of n(π) consisting of only maximal elements under the

natural ordering of partitions. Characterizing the set nm(π) is an interesting question,

and has applications to automorphic descent and endoscopic constructions.

In Sections 2.5 and 2.6, we will figure out the set nm(E(τ,b)) (n = ab). Note that

Ginzburg in [G06] gives a sketch of a proof of this result (Proposition 5.3, [G06]) with

an argument combining local and global methods. We give here a global proof with full

details.

Theorem 1.1.3 (Ginzburg [G06], Jiang and Liu [JL12]).

nm(E(τ,b)) = {[ab]}.

Now, for E(τ,b), we have two kinds of Fourier coefficients, one is Wψ(τ,b)(ϕ, g), obtained

from the Fourier expansion, the other one is attached to the nilpotent orbit correspond-

ing to the partition [ab]. One may ask that what is the relation between them? It turns

out that they share the same non-vanishing property.

1.2 On Extension of Ginzburg-Jiang-Soudry Correspon-

dences to Certain Automorphic Forms on Sp4mn(A) and

Sp4mn±2n(A)

Let τ be an irreducible unitary cuspidal automorphic representation of GL2n(A), with

the properties that L(s, τ,∧2) has a simple pole at s = 1, and L(12 , τ) 6= 0.

Let Pr = MrNr be the maximal parabolic subgroup of Sp2l with Levi subgroup Mr

isomorphic to GLr × Sp2l−2r. Using the normalization in [Sh10], the group XSp2lMr

of all

continues homomorphisms from Mr(A) to C×, which is trivial on Mr(A)1 (see [MW95]),

will be identified with C by s→ λs.

Let ∆(τ,m) be a Speh residual representation in the discrete spectrum of GL2mn(A).

For any φ ∈ A(N2mn(A)M2mn(F )\Sp4mn(A))∆(τ,m), following [L76] and [MW95], an

residual Eisenstein series can be defined by

E(φ, s)(g) =∑

γ∈P2mn(F )\Sp4mn(F )

λsφ(γg).

5

It converges absolutely for real part of s large and has meromorphic continuation to

the whole complex plane C. By [JLZ12], this Eisenstein series has a simple pole at m2 ,

which is the right-most pole. Denote the representation generated by these residues at

s = m2 by E∆(τ,m). This residual representation is square-integrable.

In Sections 4.2 and 4.3, we will prove the following theorem.

Theorem 1.2.1. Assume that F is any number field.

(1) For any irreducible automorphic representation π of Sp4mn(A) which is nearly

equivalent to E∆(τ,m), [(2n)2m] is a maximal possible partition providing non-vanishing

Fourier coefficients for π.

(2)

nm(E∆(τ,m)) = [(2n)2m].

In [GRS03], for any irreducible cuspidal automorphic representation π of symplectic

groups or their double covers, Ginzburg, Rallis and Soudry found a maximal partition

p(π) which has all even pieces, providing non-vanishing Fourier coefficients for π.

We assume that F is not totally imaginary, and consider N Sp4mn , the set of irre-

ducible cuspidal automorphic representations π which are nearly equivalent to E∆(τ,m)

and

p(π) = [(2n)2m−1(2n1)s1(2n2)s2 · · · (2nk)sk ],

with 2n ≥ 2n1 > 2n2 > · · · > 2nk, k ≥ 1, which is less or equal than [(2n)2m].

N Sp4mn can be naturally decomposed into a disjoint union of two sets NSp4mn ∪N ′Sp4mn

, where NSp4mn consists of elements having a nonzero Fourier coefficient FJψ−1n−1

(for definition, see Section 3.2), while N ′Sp4mnconsists of elements having no nonzero

Fourier coefficients FJψ−1n−1

.

Let N ′Sp4(m−1)n+2n

(τ, ψ) be a set of irreducible genuine cuspidal automorphic rep-

resentations σ4(m−1)n+2n of Sp4(m−1)n+2n(A), defined in Section 4.1. Then there is a

correspondence between NSp4mn(τ, ψ) and N ′

Sp4(m−1)n+2n

(τ, ψ), as follows.

Theorem 1.2.2. Assume that F is a number field which is not totally imaginary.

(1) There is a surjective map

Ψ : NSp4mn(τ, ψ)→ N ′

Sp4(m−1)n+2n(τ, ψ)

6

σ4mn 7→ D4mn2n,ψ−1(σ4mn).

(2) Under certain assumption, Ψ is also injective.

Let NSp4mn+2n

(τ, ψ) be a set of irreducible genuine cuspidal automorphic represen-

tations σ4mn+2n of Sp4mn+2n(A), also defined in Section 4.1. Then there is a similar

correspondence between NSp4mn+2n

(τ, ψ) and N ′Sp4n(τ, ψ), as follows.

Theorem 1.2.3. Assume that F is a number field which is not totally imaginary.

(1) There is a surjective map

Ψ : NSp4mn+2n

(τ, ψ)→ N ′Sp4n(τ, ψ)

σ4mn+2n 7→ D4mn+2n2n,ψ1 (σ4mn+2n).

(2) Under certain assumption, Ψ is also injective.

The assumptions made in Part (2) of the above two theorems will be made clear in

Section 4.1. The descent functors D4mn2n,ψ−1 and D4mn+2n

2n,ψ1 will be defined in Section 3.2.

Theorem 1.2.2 and Theorem 1.2.3 together give us the following diagram about cor-

respondences between various sets of irreducible cuspidal automorphic representations:

...

Sp4mn−2n(A)

Sp4mn(A)

Sp4mn+2n(A)

...

...

↓ D4mn−2n2n,ψ1

N4mn−2n⋃N ′4mn−2n

↓ D4mn2n,ψ−1

N4mn⋃

N ′4mn

↓ D4mn+2n2n,ψ1

N4mn+2n⋃N ′4mn+2n

↓ D4mn+4n2n,ψ−1

...

7

In the above diagram, for short, we write that N4mn := NSp4mn , N ′4mn := N ′Sp4mn,

N4mn±2n := NSp4mn±2n

, and N ′4mn±2n := N ′Sp4mn±2n

.

Note that the case of m = 1 of Theorem 1.2.2 is proved by Ginzburg, Jiang and

Soudry in [GJS12]. We use the same idea here to extend the result to higher rank cases.

Also note that, they actually proved more, since they knew that E∆(τ,1) is irreducible

(Theorem 2.5 of [GJS12]), so they were able to include it in the domain of the map Ψ.

Here, we avoid many difficult issues by letting the domain of the map Ψ consist of only

irreducible cuspidal automorphic representations.

In Theorem 1.2.2 and Theorem 1.2.3, we assume that F is a number field which is

not totally imaginary, the reason is that when F is a totally imaginary number field,

our construction will stop at some point, and can not go to higher levels. The explicit

explanation of this phenomenon will appear elsewhere.

From Theorem 1.2.1, for the residual representation E∆(τ,m), nm(E∆(τ,m)) = [(2n)2m].

From its proof, and by Lemma 2.6 [GRS03] or Lemma 3.1 [JL13b], we can see that it

has a nonzero Fourier coefficient attached to the partition [(2n)14mn−2n] with respect

to the character ψ[(2n)14mn−2n],−1. In Section 4.7, when F is any number field, we show

that if E∆(τ,m) is irreducible, then D4mn2n,ψ−1(E∆(τ,m)) is irreducible. The result can be

stated as follows.

Theorem 1.2.4. Assume that F is any number field.

(1) D4mn2n,ψ−1(E∆(τ,m)) is square-integrable and is in the discrete spectrum.

(2) Assume that E∆(τ,m) is irreducible, then D4mn2n,ψ−1(E∆(τ,m)) is also irreducible.

Note that in general, it is difficult to prove the irreducibility of certain descent

representations. The case of m = 1 of Theorem 1.2.4 was proved in Theorem 4.1 of

[GJS12], noting that by Theorem 2.5 of [GJS12], E∆(τ,1) is irreducible. Also note that,

the irreducibility of D4n2n,ψ−1(E∆(τ,1)) actually has already been proved by Jiang and

Soudry in [JS03], using different methods. Irreducibility of E∆(τ,m) and some other

residual representations will be considered in the future.

1.3 On Arthur Classification and Jiang’s Conjecture

Recently, Arthur ([Ar12]) classified the discrete spectrum of symplectic groups and

special orthogonal groups up to automorphic L2-packets Πψ(εψ) which are parametrized

8

by Arthur parameters ψ. Towards understanding Fourier coefficients of automorphic

representations in automorphic L2-packets, Jiang ([J12]) made a conjecture (Conjecture

5.0.2), saying that for any π ∈ Πψ(εψ), the set nm(π) of maximal partitions providing

non-vanishing Fourier coefficients for π has a natural upper bound, given by information

of ψ, and there is at least one element π ∈ Πψ(εψ) such that nm(π) achieves the upper

bound.

In Chapter 5, we will briefly discuss the relation between the results in Chapter 2

and 4 and Arthur’s Classification and Jiang’s Conjecture.

Now, we discuss the contents by chapters. In Chapter 2, we extend the result

of Shalika and Piatetski-Shapiro on Fourier expansion of cuspidal automorphic forms

to the whole discrete spectrum of GLn(A) (Sections 2.3 and 2.4), then determine the

Fourier coefficients of irreducible non-cuspidal (residual) automorphic representations of

GLn(A) in terms of nilpotent orbits (Sections 2.5 and 2.6). In Chapter 3, we recall the

definition of Fourier coefficients of automorphic forms of symplectic groups attached

to their nilpotent orbits and the basic properties they satisfy (Section 3.1), and the

notion of Fourier-Jacobi coefficients and automorphic descent (Section 3.2). In Chapter

4, we prove Theorem 1.2.1 (Sections 4.2 and 4.3), Theorems 1.2.2 and 1.2.3 (Sections

4.4, 4.5 and 4.6), Theorem 1.2.4 (Section 4.7). In Chapter 5, we briefly recall Arthur’s

classification of the discrete spectrum of Sp2n(A) and Jiang’s conjecture, and briefly

discuss the relation with the results in Chapters 2 and 4.

In this thesis, F is any number field, except in Sections 4.4, 4.5 and 4.6, F is a

number field which is not totally imaginary. A = AF is the adele ring of F . ψ is a

non-trivial character of F\A, except in Chapter 5, ψ denotes Arthur parameters. This

should not cause any confusion.

Chapter 2

On Fourier Coefficients of

Automorphic Forms of GLn

In this Chapter, we extend this Fourier expansion of cuspidal automorphic forms of

GLn(A) to the whole discrete spectrum of the space of all square-integrable automor-

phic forms of GLn(A) and determine the Fourier coefficients of irreducible non-cuspidal

(residual) automorphic representations of GLn(A) in terms of nilpotent orbits. The

main results of this chapter have been published in [JL12] joint with Prof. Dihua Jiang.

2.1 Overview

For the general linear group GLn, it is a well-known theorem that any nonzero cuspidal

automorphic form ϕ on GLn(A) is globally generic, i.e. has a nonzero Whittaker-Fourier

coefficient (which will be defined in Section 2.2), which was proved by Shalika ([S74]) and

Piatetski-Shapiro ([PS79]), independently, using the Fourier expansion of the cuspidal

automorphic form ϕ in terms of its Whittaker-Fourier coefficients. This important fact

for cuspidal automorphic forms on GLn(A) distinguishes the theory of GLn from that

of other reductive algebraic groups, since over a general reductive algebraic group, there

do exist non-generic cuspidal automorphic forms.

In general, Fourier coefficients of automorphic forms over G(A), where G is a re-

ductive algebraic group defined over F , may be defined in terms of nilpotent orbits of

G(F ). However, the Fourier coefficients in a Fourier expansion of automorphic forms

9

10

over G(A) may be different from the notion of Fourier coefficients attached to nilpotent

orbits, although when G = GLn and the automorphic forms are cuspidal, they coincide.

Hence it is important to study the relations between the two different notions of Fourier

coefficients. In this chapter, we do this for the non-cuspidal discrete series automorphic

forms on GLn(A), with the hope that some of the ideas and methods may be extendable

to the discrete spectrum of classical groups.

First, we extend the Fourier expansion to automorphic forms on GLn(A), which oc-

cur in the discrete spectrum of square-integrable automorphic forms on GLn(A). This is

done in Section 2.3 (Theorem 2.3.3). A technical, but very useful lemma (Lemma 2.3.2)

is proved in Section 2.4. Based on the Fourier expansion in Section 2.3, we determine

the degenerate Whittaker-Fourier coefficients along the standard maximal unipotent

subgroup Un of GLn with degenerate characters for all non-cuspidal discrete series au-

tomorphic representations of GLn(A), following the terminology used by Zelevinsky in

[Z80, Section 8.3]. Note that the notion of degenerate Whittaker-Fourier coefficients is

easy to use when the group is GLn, as one can see in the Fourier expansion in Section 2.3.

However, for other reductive groups, there are cuspidal automorphic forms, which have

no nonzero Whittaker-Fourier coefficients, and hence such degenerate Whittaker-Fourier

coefficients are all zero. Therefore, it is natural to introduce the notion of Fourier coef-

ficients attached to nilpotent orbits for automorphic forms on general reductive groups

([MW87], [GRS03], and [G06]).

In Section 2.5, we define the notion of Fourier coefficients attached to nilpotent orbits

for automorphic forms on GLn(A), and determine the relation between the degenerate

Whittaker-Fourier coefficients from the Fourier expansion and the Fourier coefficients

attached to nilpotent orbits for the residual spectrum of GLn(A) (Theorem 2.5.4). We

remark that Ginzburg in [G06] gives a sketch of a proof of this result (Proposition 5.3,

[G06]) with an argument combining local and global methods. We give here a global

proof with full details. In Section 2.6, we show that the Fourier coefficient for any

non-cuspidal, discrete series automorphic form of GLn(A) obtained from Theorem 2.5.4

is the biggest Fourier coefficient according to the partial ordering of nilpotent orbits

(Theorem 2.5.5). Finally, Theorem 2.5.6, which is the combination of Theorems 2.5.4

and 2.5.5 and Corollary 2.3.4, extends the results of Shalika and of Piatetski-Shapiro

on cuspidal automorphic forms of GLn(A) to the whole discrete spectrum of GLn(A).

11

In other words, we prove the following main result (Theorem 2.5.6) of this paper.

Theorem 2.1.1 (Fourier Coefficients for Discrete Spectrum of GLn). Let τ be an ir-

reducible unitary cuspidal automorphic representation of GLa(A) and let n = ab with

b ≥ 1. Define the residual representation E(τ,b) of GLn(A) as in Section 2.1. Let

p = [p1p2 · · · pr] be a partition of n with p1 ≥ p2 ≥ · · · ≥ pr > 0 and denote by [ab] the

partition of all parts equal to a. Then the following hold.

(1) The residual representation E(τ,b) has a nonzero ψ[ab]-Fourier coefficient, whose

definition is given in Section 5.

(2) For any partition p = [p1p2 · · · pr] of n, if p1 > a, then the residual representation

E(τ,b) has no nonzero ψp-Fourier coefficients.

2.2 Discrete Spectrum of GLn

We first recall from [MW89] the structure of the discrete spectrum of GLn and from

[S74] and [PS79] the Fourier expansion of any cuspidal automorphic form on GLn.

2.2.1 Structure of discrete spectrum

Take n = ab with a, b ≥ 1 integers. It was a conjecture of Jacquet ([J84]) and then a

theorem of Moeglin and Waldspurger ([MW89]) that an irreducible automorphic rep-

resentation π of GLn(A) occurring in the discrete spectrum of the space of all square-

integrable automorphic forms on GLn(A) is parametrized by a pair (τ, b) with τ an

irreducible unitary cuspidal automorphic representation of GLa(A), for some pair a, b

of integers such that n = ab. In particular, if π is also cuspidal, then b = 1.

More precisely, we take the Borel subgroup Bn = TnUn to be the subgroup of all

upper triangular matrices in GLn, where Tn consists of all diagonal matrices in GLn.

The triple (GLn, Bn, Tn) determines the structure of the root system of GLn. For n = ab

with b > 1, take the standard parabolic subgroup Pab = MabNab of GLab, with the Levi

part Mab isomorphic to GL×ba = GLa×· · ·×GLa(b−times). Then (Pab , τ⊗b) is a cuspidal

datum of GLab(A). Following the theory of Langlands ([L76] and [MW95]), there is an

Eisenstein series E(φτ⊗b , s, g) attached to (Pab , τ⊗b), where s = (s1, · · · , sb) ∈ Cb. This

Eisenstein series converges absolutely for the real part of s belonging to a certain cone

12

and has meromorphic continuation to the whole complex space Cb. Moreover, it has an

iterated residue at

s0 = Λb := (b− 1

2,b− 3

2, · · · , 1− b

2),

given by

E−1(φτ⊗b , g) = lims→Λb

b−1∏i=1

(si − si+1 − 1)E(φτ⊗b , s, g) (2.1)

which is square-integrable, and hence belongs to the discrete spectrum of the space of

all square-integrable automorphic forms of GLab(A). Denote by E(τ,b) the automorphic

representation generated by all the residues E−1(φτ⊗b , g). It is a theorem of Moeglin

and Waldspurger ([MW89]) that E(τ,b) is irreducible, and any irreducible non-cuspidal

automorphic representation occurring in the discrete spectrum of GLn(A) is of this form

for some a ≥ 1 and b > 1 such that n = ab, and has multiplicity one.

2.2.2 Fourier expansion for cuspidal automorphic forms

Recall Bn = TnUn is the Borel subgroup fixed in Section 2.1. We write elements of Un

to be u = (ui,j), which is upper triangular. Let ψ be a non-trivial character of A, which

is trivial on k. We define a non-degenerate character of Un(A) by

ψUn(u) := ψ(u1,2 + u2,3 + · · ·+ un−1,n). (2.2)

It is clear that ψUn is trivial on Un(F ). For an automorphic form ϕ on GLn(A), the

(non-degenerate) Whittaker-Fourier coefficient of ϕ is given by

Wψ(ϕ, g) :=

∫Un(k)\Un(A)

ϕ(ug)ψ−1Un

(u)du. (2.3)

When ϕ is cuspidal, the following well-known Fourier expansion of ϕ is proved indepen-

dently in [S74] and [PS79]:

ϕ(g) =∑

γ∈Un−1(F )\GLn−1(F )

Wψ(ϕ, ιn(γ)g), (2.4)

where ιn(γ) =

(γ 0

0 1

). As a consequence, one deduces easily from this Fourier expan-

sion that any nonzero cuspidal automorphic form ϕ has a nonzero Whittaker-Fourier

coefficient, and hence is generic.

13

Now let us consider the residual representations of GLn(A). Take n = ab. The

residual representation E(τ,b) can also be generated by the residues at the point (s1, s2) =

( b−12 ,−1

2) of the Eisenstein series with support

(GLa ×GLa(b−1), τ ⊗ E(τ,b−1)).

From the calculation of Shahidi (Chapter 7 of [Sh10]), it is clear to see that the residual

representation E(τ,b) has a nonzero Whittaker-Fourier coefficient only if the residual

representation E(τ,b−1) has a nonzero Whittaker-Fourier coefficient. By the induction

argument, it is enough to show that E(τ,2) is not generic. This follows from Theorem

7.1.2 of [Sh10]. We summarize the discussion as

Proposition 2.2.1. Any irreducible, non-cuspidal, automorphic representation occur-

ring in the discrete spectrum of GLn(A) is non-generic, i.e. has no nonzero Whittaker-

Fourier coefficients.

We note that this global result can also be proved by using Zelevinsky classification

theory of irreducible smooth representations of GLn over a p-adic local field ([Z80]).

2.3 Fourier Expansion for the Discrete Spectrum

Fourier expansion for automorphic forms on GLn(A) is an important tool to study

properties of automorphic forms. A general expansion is given in Proposition 2.1.3 of

[Y93]. However, it is not easy to use such a general expansion in order to establish an

analogue of the Fourier expansion (2.4) for the automorphic forms in the non-cuspidal

discrete spectrum of GLn(A). In this section, we write the Eisenstein series in a more

explicit form and study the vanishing and non-vanishing of certain Fourier coefficients

of the residual representations in order to obtain the exact extension of (2.4) to the

whole discrete spectrum of GLn(A).

For n = ab with b > 1, take the standard parabolic subgroup P = Pab = MN of

GLab with Levi part M = GL×ba . Consider the normalized induced representation

I(τ, s, b) = IndG(A)P (A)(τ | · |

s1 ⊗ · · · ⊗ τ | · |sb),

where τ is an irreducible unitary cuspidal automorphic representation of GLa(A) and

s = {s1, . . . , sb} ∈ Cb,

14

For any section φ ∈ I(τ, s, b), let iφ be a complex function over N(A)M(F )\G(A)

defined by

iφ(g) = φ(g)(Iba). (2.5)

where Iba is the identity in the group M . Then, when the real part of s belongs to a

certain cone, the Eisenstein series can be expressed as

E(φ, s, g) =∑

γ∈P (F )\GLab(F )

iφ(γg). (2.6)

For any parabolic subgroup Q = LV of GLn, the constant term of the Eisenstein

series along Q is defined by

E(φ, s, g)Q =

∫V (F )\V (A)

E(φ, s, vg)dv. (2.7)

Then the constant term of the residue E−1(φ, g) along Q is given by

E−1(φ, g)Q = lims→Λb

b−1∏i=1

(si − si+1 − 1)E(φ, s, g)Q. (2.8)

It follows from Proposition 2.1.7 in [MW95] that the constant term E−1(φ, g)Q is always

zero unless P ⊆ Q.

2.3.1 Families of Fourier coefficients

In order to study the Fourier expansion for the discrete spectrum of GLn, we introduce

two families of Fourier coefficients. Let α, β, γ and δ be four non-negative integers, such

that

α+ β + γ · δ = n.

Consider the standard parabolic subgroup

Qα,1(n−α) = Lα,1(n−α)Vα,1(n−α)

of GLn with the Levi part Lα,1(n−α) = GLα ×GL×(n−α)1 . We define Q0

α,1(n−α) to be the

subgroup of Qα,1(n−α) = (qi,j) with qi,i = 1 for all i > α. Note that

V1,1(n−1) = V0,1n = Un

15

is the standard maximal unipotent subgroup of GLn.

For non-negative integers α, β, γ and δ as given above, we define two types of (de-

generate) characters ψnβ+;γ·δ and ψnβ;γ·δ of Vα,1(n−α) as follows:

ψnβ+;γ·δ(v) := ψ(vα,α+1 + vα+1,α+2 + · · ·+ vα+β−1,α+β)

·ψ(vα+β+1,α+β+2 + · · ·+ vα+β+δ−1,α+β+δ)

·ψ(vα+β+δ+1,α+β+δ+2 + · · ·+ vα+β+2δ−1,α+β+2δ)

· · · (2.9)

·ψ(vα+β+(γ−1)·δ+1,α+β+(γ−1)·δ+2 + · · ·+ vn−1,n);

and

ψnβ;γ·δ(v) := ψ(vα+1,α+2 + · · ·+ vα+β−1,α+β)

·ψ(vα+β+1,α+β+2 + · · ·+ vα+β+δ−1,α+β+δ)

·ψ(vα+β+δ+1,α+β+δ+2 + · · ·+ vα+β+2δ−1,α+β+2δ)

· · · (2.10)

·ψ(vα+β+(γ−1)·δ+1,α+β+(γ−1)·δ+2 + · · ·+ vn−1,n).

Note that ψnβ+;γ·δ(v) = ψ(vα,α+1) · ψnβ;γ·δ(v). The corresponding Fourier coefficients of

the residue E−1(φ, g) are given by:

Eψnβ+;γ·δ−1 (φ, g) :=

∫[Vα,1(n−α) ]

E−1(φ, vg)ψnβ+;γ·δ(v)−1dv, (2.11)

Eψnβ;γ·δ−1 (φ, g) :=

∫[Vα,1(n−α) ]

E−1(φ, vg)ψnβ;γ·δ(v)−1dv, (2.12)

where [Vα,1(n−α) ] := Vα,1(n−α)(F )\Vα,1(n−α)(A). For simplicity of notation, if there is no

confusion, we also use ψβ+;γ·δ for ψnβ+;γ·δ, and ψβ;γ·δ for ψnβ;γ·δ.

2.3.2 Fourier expansion: step one

We consider first a preliminary version of the Fourier expansion for the residue E−1(φ, g),

following the idea of the Fourier expansion for cuspidal automorphic forms on GLab(A)

(n = ab from now on) given in [S74] and [PS79].

16

Consider the standard parabolic subgroup Qab−1,1 = Lab−1,1Vab−1,1 of GLab. The

unipotent radical Vab−1,1 is abelian and is isomorphic to F⊕(ab−1). Hence we have the

following Fourier expansion for the residue E−1(φ, g) along Vab−1,1(F )\Vab−1,1(A):

E−1(φ, g) = E−1(φ, g)Qab−1,1+

∑γab∈Q0

ab−2,1(2)(F )\Q0

ab−1,1(F )

Eψ1+;0·0−1 (φ, γabg),

with α = ab− 1, β = 1 and γ = δ = 0.

If a = 1, then b = n and the residual representation Eτ,n is one-dimensional. Hence

it has no non-trivial Fourier coefficients. That is, Eψ1+;0·0−1 (φ, g) is identically zero (this

will be generalized in Lemma 2.3.2 below). which implies that

E−1(φ, g) = E−1(φ, g)Qab−1,1= E

ψ1;0·0−1 (φ, g) = E

ψ0;1·1−1 (φ, g). (2.13)

If a > 1, then Qab−1,1 does not contain P . By the cuspidal support of the residue,

the constant term E−1(φ, g)Qab−1,1is always zero. Hence we obtain

E−1(φ, g) =∑

γab∈Q0

ab−2,1(2)(F )\Q0

ab−1,1(F )

Eψ1+;0·0−1 (φ, γabg). (2.14)

Note that Eψ1+;0·0−1 (φ, g) is left ιab−1,ab(Q

0ab−2,1(F ))-invariant, where the subgroupQ0

ab−2,1

is the GLab−1-analogue of subgroup Q0ab−1,1 of GLab and

ιab−1,ab(h) :=

(h 0

0 1

)(2.15)

for h ∈ GLab−1. Next consider the unipotent radical Vab−2,1 of Q0ab−2,1, which is abelian

and isomorphic to F⊕(ab−2). Hence we have the Fourier expansion of Eψ1+;0·0−1 (φ, g) along

Vab−2,1(F )\Vab−2,1(A):

Eψ1+;0·0−1 (φ, g) = E

ψ1+;0·0−1 (φ, g)ιab−1,ab(Qab−2,1)

+∑

γab−1∈Q0

ab−3,1(2)(F )\Q0

ab−2,1(F )

Eψ2+;0·0−1 (φ, γab−1g).

It is easy to see from the definition that the constant term

Eψ1+;0·0−1 (φ, g)ιab−1,ab(Qab−2,1) = E

ψ0;1·2−1 (φ, g).

17

If a = 2, we show that the Fourier coefficient Eψ2+;0·0−1 (φ, γab−1g) is zero (Lemma

2.3.2 below). Hence, when a = 2, from (2.14), we obtain

E−1(φ, g) =∑

γab∈Q0

2(b−1),1(2)(F )\Q0

2b−1,1(F )

Eψ0;1·2−1 (φ, γabg). (2.16)

If a > 2, then the term Eψ0;1·2−1 (φ, g) contains as an inner integration the con-

stant term E−1(φ, g)Pab−2,2of E−1(φ, g) along the standard maximal parabolic subgroup

Pab−2,2 of GLab with Levi part isomorphic to GLab−2 × GL2. Since Pab−2,2 does not

contain P when a > 2, the constant term E−1(φ, g)Pab−2,2must be zero. Hence, when

a > 2, we have

Eψ1+;0·0−1 (φ, g) =

∑γab−1∈Q0

ab−3,1(2)(F )\Q0

ab−2,1(F )

Eψ2+;0·0−1 (φ, γab−1g). (2.17)

With (2.14), we obtain, when a > 2, that the residue E−1(φ, g) is equal to∑γab

∑γab−1∈Q0

ab−3,1(2)(F )\Q0

ab−2,1(F )

Eψ2+;0·0−1 (φ, ιab−1,ab(γab−1)γabg).

where γab runs over Q0ab−2,1(2)(F )\Q0

ab−1,1(F ). Note that

Q0ab−2,1(2) = ιab−1,ab(Q

0ab−2,1)Vab−1,1

and ιab−1,ab(Q0ab−2,1) normalizes Vab−1,1. Note also that

ιab−2,ab(Q0ab−3,1(2))Vab−1,1 = Q0

ab−3,1(3) ,

where ιab−2,ab := ιab−2,ab−1 ◦ ιab−1,ab. We obtain, when a > 2, the following Fourier

expansion for the residue E−1(φ, g):

E−1(φ, g) =∑

γab∈Q0

ab−3,1(3)(F )\Q0

ab−1,1(F )

Eψ2+;0·0−1 (φ, γabg). (2.18)

We continue with the expansion (2.18) and repeat the above argument, and finally

we obtain the following expansion:

E−1(φ, g) =∑

γab∈Q0

a(b−1),1(a)(F )\Q0

ab−1,1(F )

Eψ0;1·a−1 (φ, γabg), (2.19)

which uses Lemma 2.3.2 below. Note that this expansion generalizes (2.13) and (2.16),

from a = 1, 2 to general a. We record the above discussion as follows.

18

Proposition 2.3.1. Let E−1(φ, g) be the residue of the Eisenstein series E(φ, s, g) as

defined in (2.1).

(1) It has the following Fourier expansion:

E−1(φ, g) =∑

γ∈Q0

a(b−1),1(a)(F )\Q0

ab−1,1(F )

Eψab0;1·a−1 (φ, γg).

(2) The above Fourier expansion is absolutely convergent and uniformly converges on

any compact set in g.

Note that Part (2) is clear since in each step, the Fourier expansion in the argument

is absolutely convergent and uniformly converges on any compact set in g. Also, if we

assume that b = 1, then Eτ,1 = τ , and the Fourier expansion in Part (1) recovers the

Fourier expansion in (2.4) for any cuspidal automorphic form ϕτ in the space of τ , given

in [S74] and [PS79].

Now we state the technical lemma, which will be proved in §4.

Lemma 2.3.2. Let E−1(φ, g) be the residue of the Eisenstein series E(φ, s, g) as defined

in (2.1). The Fourier coefficient Eψa+;γ·a−1 (φ, g) is identically zero, for α = a(b− γ − 1),

β = δ = a, and γ = 0, 1, 2, · · · , b− 2.

2.3.3 Fourier expansion: step two

As we remarked after Proposition 2.3.1, if b = 1, then Eτ,1 = τ , and we finish the

Fourier expansion. If b > 1, we are able to do further Fourier expansion from the

Fourier expansion in Part (1) of Proposition 2.3.1.

Consider the subgroup Q0a(b−1)−1,1 of GLa(b−1), which is the GLa(b−1)-analogue of

Q0ab−1,1. In this case, Va(b−1)−1,1 is the unipotent radical of Q0

a(b−1)−1,1, which is isomor-

phic to F⊕(a(b−1)−1). It is clear that the Fourier coefficient Eψ0;1·a−1 (φ, g) in (2.19) is left

ιa(b−1),ab(Va(b−1)−1,1(F ))-invariant, where for h ∈ GLa(b−1), ιa(b−1),ab(h) is the block-

diagonal matrix diag(h, Ia) in GLab. Hence we have the following Fourier expansion for

Eψ′a(b−1),1(a)

−1 (φ, g):

Eψ0;1·a−1 (φ, g) = E

ψ0;1·a−1 (φ, g)ιa(b−1),ab(Q

0a(b−1)−1,1

)

+∑γ

Eψab0;1·a;ψ

a(b−1)

1+;0·0−1 (φ, ιa(b−1),ab(γ)g)

19

where γ runs over Q0a(b−1)−2,1(2)(F )\Q0

a(b−1)−1,1(F ) and

Eψab0;1·a;ψ

a(b−1)

1+;0·0−1 (φ, g) :=

∫vEψab0;1·a−1 (φ, vg)ψ

a(b−1)1+;0·0 (v)−1dv. (2.20)

Here the integration dv is over

ιa(b−1),ab(Va(b−1)−1,1(F ))\ιa(b−1),ab(Va(b−1)−1,1(A)).

As long as a > 1, which we always assume from now on, the constant term

Eψ0;1·a−1 (φ, g)ιa(b−1),ab(Q

0a(b−1)−1,1

)

contains as an inner integration the constant term of the residue E−1(φ, g) along the

maximal parabolic subgroup Pa(b−1)−1,a+1 of GLab, which is identically zero since this

maximal parabolic subgroup does not contain P . Hence we obtain (a > 1)

Eψ0;1·a−1 (φ, g) =

∑γ

Eψab0;1·a;ψ

a(b−1)

1+;0·0−1 (φ, ιa(b−1),ab(γ)g) (2.21)

where γ runs over Q0a(b−1)−2,1(2)(F )\Q0

a(b−1)−1,1(F ). As in §3.1, we can continue and

obtain the following Fourier expansion

Eψ0;1·a−1 (φ, g) =

∑γ

Eψab0;1·a;ψ

a(b−1)0;1·a

−1 (φ, ιa(b−1),ab(γ)g) (2.22)

where γ runs over Q0a(b−2),1(a)(k)\Q0

a(b−1)−1,1(k), and

Eψab0;1·a;ψ

a(b−1)0;1·a

−1 (φ, g) =

∫vEψab0;1·a−1 (φ, vg)ψ

a(b−1)0;1·a (v)−1dv.

Here the integration dv is over

ιa(b−1),ab(Va(b−2),1(a)(F ))\ιa(b−1),ab(Va(b−2),1(a)(A)).

Note that the proof here uses the technical lemma (Lemma 2.3.2 again).

Since ιa(b−1),ab(Va(b−2),1(a)Va(b−1),1(a)) = Va(b−2),1(2a) , we see that

Eψab0;1·a;ψ

a(b−1)0;1·a

−1 (φ, g) =

∫vE−1(φ, vg)ψab0;2·a(v)−1dv

= Eψab0;2·a−1 (φ, g) (2.23)

20

where the integration dv is over Va(b−2),1(2a)(F )\Va(b−2),1(2a)(A). Note here that α =

a(b− 2), β = 0, γ = 2, and δ = a.

From the above discussion, we obtain

E−1(φ, g) =∑γ1

∑γ2

Eψ0;2·a−1 (φ, γ2γ1g), (2.24)

where γ1 runs over Q0a(b−1),1(a)(F )\Q0

ab−1,1(F ) and γ2 runs over

ιa(b−1),ab(Q0a(b−2),1(a)(F ))\ιa(b−1),ab(Q

0a(b−1)−1,1(F )).

For general b > 1, such that n = ab, we repeat the above argument and obtain, by

means of the inductive argument and Lemma 2.3.2 for each step, the following Fourier

expansion for the residue E−1(φ, g).

E−1(φ, g) =∑γ1

· · ·∑γb

Eψ0;b·a−1 (φ, γb · · · γ1g), (2.25)

where γi runs over

ιa(b−i+1),ab(Q0a(b−i),1(a)(F ))\ιa(b−i+1),ab(Q

0a(b−i+1)−1,1(F ))

for i = 1, 2, · · · , b. Note here that α = β = 0, γ = b and δ = a. We state this as follows.

Theorem 2.3.3 (Fourier Expansion). Let E−1(φ, g) be the residue of the Eisenstein

series E(φ, s, g) as defined in (2.1).

(1) It has the following Fourier expansion:

E−1(φ, g) =∑γ1

· · ·∑γb

Eψab0;b·a−1 (φ, γb · · · γ1g),

where γi runs over

ιa(b−i+1),ab(Q0a(b−i),1(a)(F ))\ιa(b−i+1),ab(Q

0a(b−i+1)−1,1(F ))

for i = 1, 2, · · · , b.

(2) The above Fourier expansion is absolutely convergent and uniformly converges on

any compact set in g.

21

As a consequence, we obtain

Corollary 2.3.4. Let E−1(φ, g) be the residue of the Eisenstein series E(φ, s, g) as

defined in (2.1) with cuspidal datum (Pab , τ⊗b). Then the degenerate Whittaker-Fourier

coefficient Eψab0;b·a−1 (φ, g) is nonzero for some choice of φ.

More generally, there is a notion of Fourier coefficients of an automorphic form

on GLn(A) parametrized by partitions of n or nilpotent orbits, which works better

for general reductive groups and will be introduced for GLn in Section 5. By using the

ordering of partitions, we will show that the Fourier coefficient Eψab0;b·a−1 (φ, g) is essentially

the one attached to the biggest partition among all partitions to which the residue

E−1(φ, g) can have non-zero Fourier coefficients attached.

2.4 Proof of Lemma 2.3.2

In order to prove Lemma 2.3.2, we need Lemma 4.1, which will also be used in the proofs

of Theorem 2.5.4 and Lemma 6.1. In the following, we take the standard section for the

Eisenstein series E(φ, s, g), via the natural isomorphism of vector spaces: I(τ, 0, b) ∼=I(τ, s, b), which takes φ to φ(s), for any section φ ∈ I(τ, 0, b), canonically. In this way,

the Eisenstein series is given by, when the real part of s belongs to a certain cone,

E(φ, s, g) =∑

γ∈P (F )\GLab(F )

iφ(s)(γg).

Lemma 2.4.1. Let Qi := Qai,a(b−i) = LiVi be the standard maximal parabolic subgroup

of GLab with Levi part Li ∼= GLai × GLa(b−i), where 1 ≤ i ≤ b − 1. Then there is a

section

f ∈ IndGLab(A)Qi(A) (| · |−

b−i2 E(τ,i) ⊗ | · |

i2E(τ,b−i)),

such that

E−1(φ, g)Qi = f(g)(Iai × Ia(b−i)).

Proof. We first calculate the constant term of the Eisenstein series E(φ, s, g) along the

parabolic subgroup Qi. This Eisenstein series has cuspidal support on the standard

parabolic subgroup Pab = MabNab . In this proof, we use P = MN to simplify the

notation.

22

To do this we introduce the set MWcLi

, which consists of elements ω−1 ∈ W (GLab)

(the Weyl group of GLab), with the properties that ω(α) > 0 for any α ∈ Φ+M , the

positive roots in M , ω−1(β) > 0 for any β ∈ Φ+Li

, the positive roots in Li, and ωMω−1 ⊆Li. By Proposition II.1.7(ii) of [MW95], we have

E(φ, s, g)Qi =∑

ω−1∈MW cLi

∑γ∈(ωPω−1∩Li)(k)\Li(k)

iM(ω, s)φ(s)(γg)

=∑

ω−1∈MW cLi

EQi(M(ω, s)φ(s), ωs, g),

where M(ω, s) is the intertwining operator corresponding to ω.

Note that MWcLi

has total Cib elements which are of following forms: for any i num-

bers {l1, . . . , li} in {1, . . . , b} with increasing order, and the complement b− i numbers

{m1, . . . ,mb−i} = {1, . . . , b} \ {l1, . . . , li} with increasing order, the corresponding ele-

ment ω−1 ∈ MWcLi

is defined as

ω−1 : j 7→ lj and i+ f 7→ mf (2.26)

for j ∈ {1, 2, · · · , i} and f ∈ {1, 2, · · · , b − i}. Note here that by e 7→ n we mean that

ω−1 takes the e-th block to the n-th block (with the block size a× a).

To compute E−1(φ, g)Qi , we use the fact that the multi-residue operator

lims→Λb

b−1∏i=1

(si − si+1 − 1)

(where Λb is defined in (2.1)) and the constant term operator are interchangeable. Using

the same argument as in the proof of Lemma 2.4 of [OS08], we deduce that after applying

the multi-residue operator, the only term left is the one corresponding to ω−1i , where

ωi =

(0 Iai

Ia(b−i) 0

).

Indeed, given an element ω−1 as in (2.26), let

∆1(ω) = {1 ≤ j ≤ b− 1|ω(j) > ω(j + 1)},

∆2(ω) = {1 ≤ j ≤ b− 1|ω(j + 1)− ω(j) = 1} \ {ω−1(i)}.

Then the normalized intertwining operator

N(ω, s) :=∏

j∈∆1(ω)

(sj − sj+1 − 1)M(ω, s)

23

is holomorphic at Λb, and the normalized Eisenstein series∏j∈∆2(ω)

(sj − sj+1 − 1)EQi(N(ω, s)φ(s), ωs, g)

is holomorphic at Λb. Therefore, the term corresponding to ω−1 survives after applying

the multi-residue operator if and only if ω−1 has the property that

∆1(ω) ∪∆2(ω) = {1, . . . , b− 1},

which is equivalent to that ω 6= Iab, and there is no 1 ≤ j ≤ b− 1, such that ω(j + 1)−ω(j) > 1. Note that if ω = Iab, then

∆1(ω) ∪∆2(ω) = {1, . . . , b− 1} \ {i} 6= {1, . . . , b− 1}.

Since the property that there is no 1 ≤ j ≤ b− 1, such that ω(j+ 1)−ω(j) > 1, implies

that

ω : j 7→ i+ j and b− i+ f 7→ f (2.27)

for j ∈ {1, 2, · · · , b − i} and f ∈ {1, 2, · · · , i}. This means that ω =

(0 Iai

Ia(b−i) 0

),

or ω = Iab. After applying the multi-residue operator, the only term left is the one

corresponding to ω−1i , where ωi =

(0 Iai

Ia(b−i) 0

). Therefore, we prove the following

identity

E−1(φ, g)Qi = EQi−1(M−1(ωi)φ, µai,a(b−i), g),

where µai,a(b−i) = (− b−i2 ; i2) ∈ C2. We embed C2 to Cb by

(s1, s2) ↪→ (s1, . . . , s1; s2, . . . , s2)

with i-copies of s1 and (b − i)-copies of s2, and identify C2 with the image. Note that

(Λb = ( b−12 , b−3

2 . . . , 1−b2 ))

ωiΛb =(2i− b− 1

2,2i− b− 3

2, . . . ,

1− b2

;b− 1

2,b− 3

2, . . . ,

2i− b+ 1

2)

=(i− 1

2, . . . ,

1− i2

;(b− i)− 1

2, . . . ,

1− (b− i)2

) + (−b− i2

;i

2)

=Λai,a(b−i) + µai,a(b−i),

24

where Λai,a(b−i) = ( i−12 , i−3

2 . . . , 1−i2 ; (b−i)−1

2 , (b−i)−32 , . . . , 1−(b−i)

2 ).

As discussed on Page 10 of [OS08] (or in the proof of the Proposition 2.3 of [OS08]),

EQi−1(M−1(ωi)φ, µai,a(b−i), g) defines a surjective intertwining operator from I(τ,Λb, b)

onto IndGLab(A)Qi(A) (| · |−

b−i2 E(τ,i) ⊗ | · |

i2E(τ,b−i)). Hence there exists a section

f ∈ IndGLab(A)Qi(A) (| · |−

b−i2 E(τ,i) ⊗ | · |

i2E(τ,b−i))

such that

E−1(φ, g)Qi = EQi−1(M−1(ωi)φ, µQai,a(b−i) , g) = f(g)(Iai × Ia(b−i)).

This finishes the proof.

Now, we are ready to prove the technical Lemma 2.3.2. This is to show that for

the residue E−1(φ, g) of the Eisenstein series E(φ, s, g) as defined in (2.1), the Fourier

coefficient Eψaba+;γ·a−1 (φ, g) is identically zero, for α = a(b − γ − 1), β = δ = a, and

γ = 0, 1, 2, · · · , b− 2. Note that when γ = b− 1,

ψaba+;(b−1)·a = ψaba;(b−1)·a = ψab0;b·a.

By using the cuspidal support of the residue E−1(φ, g), we have the following Fourier

expansion for the Fourier coefficient Eψaba+;γ·a−1 (φ, g)

Eψaba+;γ·a−1 (φ, g) (2.28)

=∑

ε∈Q0a(b−γ−2),1∗ (F )\Q0

a(b−γ−1)−1,1∗ (F )

Eψab2a;γ·a−1 (φ, εg)

+∑

ε+∈Q0a(b−γ−2)−1,1∗ (F )\Q0

a(b−γ−1)−1,1∗ (F )

Eψab

2a+;γ·a−1 (φ, ε+g).

Here we use Q0m,1∗ := Q0

m,1ab−mto simplify the notation.

We show that the Fourier coefficient Eψab2a;γ·a−1 (φ, g) is identically zero. Based on

this, the vanishing of Eψaba+;γ·a−1 (φ, g) is equivalent to the vanishing of E

ψab2a+;γ·a−1 (φ, ε+g).

By using the inductive argument on la for l = 1, 2, · · · , b − γ, which is based on the

vanishing of the Fourier coefficient Eψabla;γ·a−1 (φ, g) for each l = 2, 3, · · · , b − γ, it follows

25

that the vanishing of Eψaba+;γ·a−1 (φ, g) is equivalent to the vanishing of E

ψab(b−γ)a+;γ·a−1 (φ, ε+g),

which is the same as Eψab

(b−γ)a;γ·a−1 (φ, ε+g).

Hence, in order to prove Lemma 2.3.2, it is enough to prove the following lemma.

Lemma 2.4.2. For γ = 0, 1, 2, · · · , b− 2 and l = 2, 3, · · · , b− γ, the Fourier coefficient

Eψabla;γ·a−1 (φ, g) is identically zero.

Proof. First, when γ = 0 and l = b − γ = b, the Fourier coefficient Eψabba;0·a−1 (φ, g) is

exactly the Whittaker-Fourier coefficient of the residue E−1(φ, g), which is identically

zero by Proposition 2.2.1.

In the following, we use Lemma 4.1 twice to reduce the general case to the above

special case with lower rank. Hence those Fourier coefficients must all be zero identically.

We assume that l < b− γ, and we show, by using Lemma 4.1, that this will reduce

to the case l = b− γ, which will be treated next.

Recall the parabolic subgroup Qb−γ−l = Lb−γ−lVb−γ−l of GLab from Lemma 4.1,

with Lb−γ−l = GLa(b−γ−l) × GLa(γ+l). By the definition of the Fourier coefficient

Eψabla;γ·a−1 (φ, g), the constant term of the residue E−1(φ, g) along Qb−γ−l is an inner inte-

gration of Eψabla;γ·a−1 (φ, g). More precisely, we have

Eψabla;γ·a−1 (φ, g) = [E−1(φ, g)Qb−γ−l ]

ψa(γ+l)la;γ·a . (2.29)

Note here that the ψa(γ+l)la;γ·a -Fourier coefficient is taken from the subgroup GLγ+l, which

is the second factor in the Levi subgroup Lb−γ−l.

By Lemma 4.1, there exists a section f belonging to

IndGLab(A)Qb−γ−l(A)(| · |

− γ+l2 E(τ,b−γ−l) ⊗ | · |

b−γ−l2 E(τ,γ+l)),

such that

E−1(φ, g)Qb−γ−l = f(g)(Ia(b−γ−l) × Ia(γ+l)).

Since the ψa(γ+l)la;γ·a -Fourier coefficient of the constant term E−1(φ, g)Qb−γ−l is taken from

the subgroup GLγ+l, it suffices to show that the residual representation E(τ,γ+l) of

GLa(γ+l)(A) has no non-zero ψa(γ+l)la;γ·a -Fourier coefficients. This reduces the problem

from GLab to GLa(γ+l). Note that this reduces the general case l < b − γ to the case

l = b− γ for b = γ + l.

26

Now take E−1(φτ⊗(γ+l) , g) from the space of the residual representation E(τ,γ+l) of

GLa(γ+l)(A). Consider the standard maximal parabolic subgroup Ql,γ = Ll,γVl,γ of

GLa(γ+l) with Levi part Ll,γ = GLla × GLγa. By the definition of the ψa(γ+l)la;γ·a -Fourier

coefficient of E−1(φτ⊗(γ+l) , g), the constant term of the residue E−1(φτ⊗(γ+l) , g) along

Ql,γ occurs as an inner integration in the ψa(γ+l)la;γ·a -Fourier coefficient of E−1(φτ⊗(γ+l) , g).

As before, we write it more precisely as follows:

[E−1(φτ⊗(γ+l) , g)]ψa(γ+l)la;γ·a = [E−1(φτ⊗(γ+l) , g)Ql,γ ]ψ

lala;0·a;ψγa0;γ·a .

After taking the constant term along Ql,γ , E−1(φτ⊗(γ+l) , g)Ql,γ is an automorphic func-

tion over GLla(A)×GLγa(A). Note here that the ψlala;0·a-Fourier coefficient is taking on

GLla(A) and the ψγa0;γ·a-Fourier coefficient is taken on GLγa(A).

By Lemma 4.1 again (applied to GLa(γ+l)), it is enough to show that the residual

representation E(τ,l) has no non-zero ψla2a;0·a-Fourier coefficients or the residual represen-

tation E(τ,γ) has no non-zero ψγa0;γ·a-Fourier coefficients. It is clear that the character

ψlala;0·a is exactly the Whittaker character of GLla(A). By Proposition 2.2.1, the residual

representation E(τ,l) is not generic, and hence it has no non-zero ψlala;0·a-Fourier coeffi-

cients.

This completes the proof of Lemma 2.3.2.

2.5 Fourier Coefficients for GLn

In Theorem 2.3.3 and Corollary 2.3.4, we show that the residue E−1(φ, g) of the Eisen-

stein series E(φ, s, g), with cuspidal datum (Pab , τ⊗b), has a nonzero degenerate Whit-

taker Fourier coefficient Eψab0;b·a−1 (φ, g). In this section, we give the definition of Fourier

coefficients of automorphic forms attached to nilpotent orbits or partitions of n, and

show that this degenerate Whittaker-Fourier coefficient for the residue E−1(φ, g) is

analogous to the Whittaker-Fourier coefficient for the cuspidal automorphic forms on

GLn(A), as remarked at the end of Section 3. In other words, we show that according to

the partial ordering of partitions or nilpotent orbits, the Fourier coefficient Eψab0;b·a−1 (φ, g)

is equivalent (for the non-vanishing property) to the biggest possible Fourier coefficient

that the residue E−1(φ, g) can possibly have.

27

2.5.1 Fourier coefficients for GLn

We consider the Fourier coefficients of automorphic forms of GLn(A) attached to nilpo-

tent F -orbits under the GLn(F )-adjoint action, following the idea of [GRS03] and [G06]

for the global theory and of [MW87] for the local theory.

When G = GLn, each nilpotent F -orbit O of GLn(F ) has an element in the standard

Jordan form, which is unique up to permutation (conjugation by a certain Weyl group

element), and hence is characterized by a standard partition of n: p = [p1p2 · · · pr] with

p1 ≥ p2 ≥ · · · ≥ pr > 0 and n =∑r

i=1 pi. We denote by Jp the nilpotent lower triangular

Jordan matrix in the nilpotent k-orbit Op determined by the partition p. Since Jp is of

standard Jordan form, there is an one-dimensional toric subgroup Hp of GLn(F ):

Hp(t) := diag(H[p1](t),H[p2](t), · · · ,H[pr](t)) (2.30)

with H[pi](t) := diag(tpi−1, tpi−3, · · · , t3−pi , t1−pi) for i = 1, 2, · · · , r, and t ∈ F×, such

that

∀ t ∈ F×,Ad(Hp(t))(Jp) = t−2Jp.

Take J −p to be the opposite to Jp. It is clear that

{Jp, log(Hp),J −p }

generates the F -sl2 attached to the F -orbit Op. Under the adjoint action, the Lie

algebra gln(F ) of GLn(F ) decomposes into a direct sum of Ad(Hp)-eigenspaces:

gln(F ) = g−m ⊕ · · · ⊕ g−2 ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ g2 ⊕ · · · ⊕ gm (2.31)

for some m, where gl := {X ∈ gln(F ) | Ad(Hp(t))(X) = tl ·X}.Let Vp,j(F ) (with j = 1, 2, · · · ,m) denote the unipotent subgroup of GLn(k) whose

Lie algebra is ⊕ml=jgl. Let Lp(F ) be the algebraic subgroup of GLn(F ) such that its

Lie algebra is g0. It is easy to check that Jp belongs to g−2. Under the adjoint ac-

tion, the set Ad(Lp(F ))(Jp) is Zariski open dense in the dual space of the affine space

Vp,2(F )/Vp,3(F ). Hence one may use the representative Jp of the F -orbit Op to define

a (generic) character ψp of Vp,2(F ). Let Qp be the standard parabolic subgroup of GLn

corresponding to the partition p. The Levi subgroup Mp is GLp1×GLp2×· · ·×GLpr . It

is clear that the intersection Mp∩Vp,2 is Up1 ×Up2 ×· · ·×Upr where Upi is the standard

28

maximal unipotent subgroup (the radical of the standard Borel subgroup) of GLpi . We

define a character of Vp,2 as follows: for any v ∈ Vp,2,

ψp(v) :=ψ(tr(Jp log(v)))

=ψ(v1,2 + · · ·+ vp1−1,p1)

· ψ(vp1+1,p1+2 + · · ·+ vp1+p2−1,p1+p2)

· · · (2.32)

· ψ(vp1+···+pr−2+1,p1+···+pr−2+2 + · · ·+ vp1+···+pr−1−1,p1+···+pr−1)

· ψ(vp1+···+pr−1+1,p1+···+pr−1+2 + · · ·+ vn).

Note that ψp also defines a non-degenerate (Whittaker) character of Mp.

Let φ be an automorphic form on GLn(A). We define the ψp-Fourier coefficient of

φ attached to the partition p or the nilpotent orbit Op by the following integral:

φψp(g) :=

∫Vp,2(F )\Vp,2(A)

φ(vg)ψ−1p (v)dv. (2.33)

Note that the definition of the ψp-Fourier coefficient of φ depends on the choice of the

representative Jp (and the semisimple element Hp).According to the k-rational version of the Jacobson-Morozov Theorem ([MW87]),

the Fourier coefficient of φ can be defined by means of any choice of representatives in

the nilpotent k-orbit Op. Since φ is automorphic, the vanishing or non-vanishing of the

ψp-Fourier coefficient of φ only depends on the k-orbit Op.Let π be an irreducible automorphic representation of GLn(A) occurring as a sub-

space of the discrete spectrum of square-integrable automorphic functions on GLn(A).

We say that π has a ψp-Fourier coefficient if there is a function φ ∈ π such that φψp(g)

is non-zero. As discussed above, the property that π has a ψp-Fourier coefficient only

depends on the k-orbit Op.For gi, as defined in (2.31), let G+

i (G−i , respectively) be the union of all one-

parameter subgroups Xα(x) whose Lie algebra is in gi, with positive (negative, respec-

tively) roots α, in the root system determined by (GLn, Bn, Tn). It is easy to see that

both G+1 and G−1 have group structures and are abelian. In the following, by saying

that one entry in GLn is in G+i or G−i , we mean that the corresponding element in the

associated one-parameter subgroup is in G+i or G−i .

29

Recall that Vp,1(k) is the unipotent subgroup of GLn(k) whose Lie algebra is g1 ⊕g2 ⊕ · · · ⊕ gm. Following [MW87], we define

(Jp)] = {X ∈ g | tr(Jp[X,X ′]) = 0, ∀X ′ ∈ g}.

Define V ′p,2 = exp(g1 ∩ (Jp)])Vp,2, which is a normal subgroup of Vp,1(k). From the

definition of (Jp)], it is easy to see that the character ψp on Vp,2 can be trivially ex-

tended to V ′p,2, which we still denote by ψp. It turns out that Vp,1/ kerV ′p,2(ψp) has

a Heisenberg structure W ⊕ Z (see [MW87], Section I.7), where W ∼= Vp,1/V′p,2, and

Z ∼= V ′p,2/ kerV ′p,2(ψp). Note that the symplectic form on W is the one inherited from

the Lie algebra bracket, i.e., for w1, w2 ∈ W (here, we identify w ∈ W with any of it’s

representatives in Vp,1 such that log(w) ∈ g1),

〈w1, w2〉 = tr(Jp log([w1, w2]))

= tr((Jp[log(w1), log(w2)]).

The non-degeneracy of this symplectic form can be checked easily as following: for fixed

w1 ∈ W , if 〈w1, w2〉 ≡ 0, for any w2 ∈ W , i.e., tr(Jp[log(w1), log(w2)]) ≡ 0, for any

w2 ∈W , i.e.,

tr(Jp[log(w1), X ′]) ≡ 0,

for any X ′ ∈ g1, which implies that tr(Jp[log(w1), X ′]) ≡ 0, for any X ′ ∈ g, i.e.,

log(w1) ∈ (Jp)], that is, w1 = 0 ∈ Vp,1/V ′p,2.

Lemma 2.5.1. V ′p,2 = Vp,2.

Proof. As discussed at the beginning of this subsection, the partition p = [p1p2 · · · pr]gives rise to the sl2-triple

{Jp, log(Hp),J −p }.

If, under the adjoint action of Hp on the Lie algebra gln(k), the space g1 as defined in

(2.31) is zero, there is nothing to prove. In the following we assume that g1 is not zero.

To prove V ′p,2 = Vp,2, it suffices to prove that g1 ∩ (Jp)] = {0}.

30

First let’s describe Vp,2. Elements in Vp,2 have the following form:

v =

. . .

ni qi,j

. . .

0 nj. . .

. . .

Ipi 0. . .

pi,j Ipj. . .

,

where ni ∈ Upi (nj ∈ Upj , respectively), the standard maximal unipotent subgroup

of GLpi (GLpj , respectively), qi,j ∈ Mpi×pj , and pi,j ∈ Mpj×pi satisfy some conditions.

Since in this lemma, we only need to care about (i, j) such that pi and pj are of different

parity, we describe the conditions for qi,j and pi,j only for the case that pi and pj are

of different parity: qi,j ∈Mpi×pj with qi,jl,m = 0, for l ≥ m+pi−pj−1

2 , pi,j ∈Mpj×pi with

pi,jl,m = 0, for m ≤ l +pi−pj−1

2 + 1.

According to the structure of the space g1, we define abelian groups Y and X, which

are given by

Y =∏

1≤i<j≤r, pi and pj are of different parity

Y i,j , (2.34)

Y i,j =

pj∏l=1

Xαi,jl

(yi,jl ),

where αi,jl = e∑i−1m=1 pm+

pi−pj−1

2+l− e∑j−1

m=1 pm+l; and

X =∏

1≤i<j≤r, pi and pj are of different parity

Xi,j , (2.35)

Xi,j =

pj∏l=1

Xβi,jl

(xi,jl ),

where βi,jl = e∑j−1m=1 pm+l

− e∑i−1m=1 pm+

pi−pj−1

2+l+1

. Then, we can see that g1 = log(X)⊕

log(Y ). Therefore, to show that g1 ∩ (Jp)] = {0}, we only need to show that (log(X)⊕log(Y ))∩ (Jp)] = {0}. It suffices to show that for 1 ≤ i < j ≤ r, such that pi and pj are

of different parity, and for any l = 1, . . . , pj , both log(Xαi,jl

(yi,jl )) and log(Xβi,jl

(xi,jl ))

are not in (Jp)], where 0 6= yi,jl , xi,jl ∈ k

×. This is true, since by direct calculation, when

31

yi,jl , xi,jl 6= 0,

tr(Jp[log(Xβi,jl

(xi,jl )), log(Xαi,jl

(yi,jl ))]) = −xi,jl yi,jl 6= 0.

This completes the proof of the lemma.

Therefore, by Lemma 2.5.1 and the discussion above, Vp,1/ kerVp,2(ψp) has a Heisen-

berg structure W ⊕ Z, where Z ∼= Vp,2/ kerVp,2(ψp), and X ⊕ Y is a polarization of W ,

where X,Y are defined in (2.35), (2.34).

In more explicit calculations of Fourier coefficients of automorphic forms, there is a

very useful lemma, which has been used in many occasions and is now formulated in a

general term in Corollary 7.1 of [GRS11]. In order to fit it better for our use in this

paper, we reformulate it in a slightly different way and use a slightly different argument

to prove the GLn-analogue of the useful lemma.

Let C be an F -subgroup of a maximal unipotent subgroup of GLn, and let ψC be

a non-trivial character of [C] = C(F )\C(A). X, Y are two unipotent F -subgroups,

satisfying the following conditions:

(1) X and Y normalize C;

(2) X ∩C and Y ∩C are normal in X and Y , respectively, (X ∩C)\X and (Y ∩C)\Yare abelian;

(3) X(A) and Y (A) preserve ψC ;

(4) ψC is trivial on (X ∩ C)(A) and (Y ∩ C)(A);

(5) [X, Y ] ⊂ C;

(6) there is a non-degenerate pairing (X ∩ C)(A) × (Y ∩ C)(A) → C∗, given by

(x, y) 7→ ψC([x, y]), which is multiplicative in each coordinate, and identifies

(Y ∩C)(F )\Y (F ) with the dual of X(F )(X∩C)(A)\X(A), and (X∩C)(F )\X(F )

with the dual of Y (F )(Y ∩ C)(A)\Y (A).

Let B = CY and D = CX, and extend ψC trivially to characters of [B] =

B(F )\B(A) and [D] = D(F )\D(A), which will be denoted by ψB and ψD respectively.

Here is the reformulation of the useful lemma, the proof of which is valid for the general

group H(A) as in [GRS11].

32

Lemma 2.5.2. Assume the quadruple (C,ψC , X, Y ) satisfies the above conditions. Let

f be an automorphic form on GLn(A). Then∫[C]f(cg)ψ−1

C (c)dc ≡ 0, ∀g ∈ GLn(A),

if and only if ∫[D]f(ug)ψ−1

D (u)du ≡ 0, ∀g ∈ GLn(A),

if and only if ∫[B]f(vg)ψ−1

B (v)dv ≡ 0,∀g ∈ GLn(A).

Proof. By symmetry, we only need to show that∫[C]f(cg)ψ−1

C (c)dc ≡ 0, ∀g ∈ GLn(A),

if and only if ∫[D]f(ug)ψ−1

D (u)du ≡ 0, ∀g ∈ GLn(A).

Since ∫[D]f(ug)ψ−1

D (u)du =

∫X(F )(X∩C)(A)\X(A)

∫[C]f(cxg)ψ−1

C (c)dcdy,

we know that if ∫[C]f(cg)ψ−1

C (c)dc ≡ 0, ∀g ∈ GLn(A),

then ∫[D]f(ug)ψ−1

D (u)du ≡ 0, ∀g ∈ GLn(A).

On the other hand, by Formula (7.5) of [GRS11],∫[C]f(cg)ψ−1

C (c)dc =∑

y′∈(Y ∩C)(F )\Y (F )

∫[D]f(uy′g)ψ−1

D (u)du,

which implies that if ∫[D]f(ug)ψ−1

D (u)du ≡ 0, ∀g ∈ GLn(A),

then ∫[C]f(cg)ψ−1

C (c)dc ≡ 0, ∀g ∈ GLn(A).

This completes the proof of the lemma.

33

Note that when we apply this lemma in the remaining of the paper, we always denote

ψB and ψD by ψC for convenience.

The following corollary gives an important property of the ψp-Fourier coefficients for

automorphic forms on GLn(A). The corresponding case for symplectic group is given

in Lemma 1.1, [GRS03].

Corollary 2.5.3. Let p = [p1p2 · · · pr] be a standard partition of n, i.e. p1 ≥ p2 ≥· · · ≥ pr > 0 and n =

∑ri=1 pi. Let φ be an automorphic form on GLn(A). Then φψp,

the ψp-Fourier coefficient of φ is non-vanishing if and only if the following integral is

non-vanishing: ∫[Y ]

∫[Vp,2]

φ(vyg)ψ−1p (v)dvdy

where the subgroup Y is defined in (2.34).

Proof. This is a consequence of Lemma 2.5.1 and Lemma 2.5.2. In fact, by Lemma 2.5.1

and the discussion before it, we know that Vp,1/ kerVp,2(ψp) has a Heisenberg structure

W ⊕ Z, where Z ∼= Vp,2/ kerVp,2(ψp), and X ⊕ Y is a polarization of W , where X,Y

are defined in (2.35), (2.34). This implies directly that the quadruple (Vp,2, ψp, X, Y )

satisfies all the conditions for Lemma 2.5.2.

2.5.2 Fourier coefficients for the discrete spectrum of GLn

Recall from [S74] and [PS79] that any nonzero irreducible cuspidal automorphic rep-

resentation π of GLn(A) is generic, i.e. has a non-zero Whittaker-Fourier coefficient.

From the definition, the Whittaker-Fourier coefficient of π is the one attached to the

partition p = [n].

In the following, we assume that n = ab with b > 1, and consider π = E(τ,b), the

residual representation of GLab(A) with cuspidal support (Pab , τ⊗b).

Theorem 2.5.4. For any residue E−1(φ, ·) in the residual representation E(τ,b) of

GLab(A) with cuspidal support (Pab , τ⊗b), the ψab0;b·a-Fourier coefficient of E−1(φ, ·),

denoted by Eψab0;b·a−1 (φ, g), is non-vanishing for some choice of data if and only if the

ψ[ab]-Fourier coefficient of E−1(φ, ·), denoted by Eψ

[ab]

−1 (φ, g), is non-vanishing for some

choice of data.

34

Proof. If a = 1, then b = n and the residual representation Eτ,b is just χ ◦ det, a

character of GLn(A), which of course has only the trivial Fourier coefficient attached to

the partition [1n] of n. The theorem holds for this case.

When n = a, Eψn0;1·n−1 (φ, g) = E

ψ[n]

−1 (φ, g). We are done for this case, since in this case

the parabolic subgroup is trivial, i.e. the whole group GLn, and hence the automorphic

form considered is cuspidal.

We only need to consider the case 1 < a < n. In order to use the induction argument,

we assume that for n = a, 2a, . . . , a(b− 1), the equivalence of the non-vanishing of both

Fourier coefficients holds. We are going to prove that it will also be true for n = ab.

We start from Eψ

[ab]

−1 (φ, g), the ψ[ab]-Fourier coefficient of the residue E−1(φ, ·). In

order to apply Lemma 2.5.2 to the following integral

Eψ

[ab]

−1 (φ, g) =

∫V

[ab],2(F )\V

[ab],2(A)

E−1(φ, vg)ψ−1[ab]

(v)dv, (2.36)

we conjugate it by a Weyl element ω of GLn which conjugates the one-parameter toric

subgroup H[ab] in (2.30) corresponding to the partition [ab] to the following toric sub-

group:

diag(H[a](t); ta−1Ib−1, t

a−3Ib−1, · · · , t3−aIb−1, t1−aIb−1)

where H[a](t) = diag(ta−1, ta−3, · · · , t3−a, t1−a). Note that ω is of the form diag(Ia, ω′),

where ω′ permutes the toric subgroup H[ab−1] in (2.30) corresponding to the partition

[ab−1] to the toric subgroup in GLa(b−1):

diag(ta−1Ib−1, ta−3Ib−1, · · · , t3−aIb−1, t

1−aIb−1).

Let U[ab],2 = ωV[ab],2ω−1. Then any element of U[ab],2 has the following form:

u =

(n1 q

0 n2

)(Ia 0

p Ia(b−1)

),

where n1 ∈ Ua, the standard maximal unipotent subgroup of GLa; n2 ∈ U[ab−1],2 :=

ω′V[ab−1],2ω′−1; q ∈ Ma×a(b−1) with ql,m = 0, for m ≤ l(b − 1); and p ∈ Ma(b−1)×a with

pl,m = 0, for l > (m− 1)(b− 1). We define

ψU[ab−1],2

(u) := ψV[ab−1],2

(ω′−1uω′).

35

Therefore, after conjugating by ω, the integral (2.36) becomes∫∗E−1(φ,

(n1 q

0 n2

)(Ia 0

p Ia(b−1)

)ωg)ψ−1

[a] (n1)ψ−1U

[ab−1],2(n2)d∗, (2.37)

where∫∗ =

∫p

∫q

∫n2

∫n1

, d∗ = dn1dn2dqdp, ψ[a] is a non-degenerate character of GLa.

We are going to apply Lemma 2.5.2 consecutively in order to replace the integration

on the variable p by corresponding integration on the variable q. To do so, we define a

sequence of unipotent subgroups of GLab (R for ‘row’, C for ‘column’). For 1 ≤ s ≤ a−1,

define

Rs =

{(Ia q

0 Ia(b−1)

)}where q ∈Ma×a(b−1) with the property that ql,m = 0 if l 6= s or l = s,m > l(b− 1). For

2 ≤ s ≤ a, define

Cs =

{(Ia 0

p Ia(b−1)

)}where p ∈ Ma(b−1)×a with the property that pl,m = 0 if m 6= s or m = s, l > (m −1)(b− 1). It is easy to see that for 1 ≤ s ≤ a− 1, Rs(k) ∼= Cs+1(k) ∼= ks(b−1), as abelian

groups.

Write U[ab],2 = U[ab],2

∏as=2Cs, where U[ab],2 consists of elements in U[ab],2 with p-part

(as indicated in the subgroup Cs) being zero. For 1 ≤ s ≤ a − 1, write Rs =∏si=1R

is,

where Ris ⊂ G+−2(i−1). For 2 ≤ s ≤ a, write Cs =

∏s−1i=1 C

is, where Cis ⊂ G−2i.

The key point here is to apply Lemma 2.5.2 to the integration on the variables in∏as=2Cs. To do this, we will deal with the subgroups Cs for s = 2, 3, · · · , a, one by one.

First we apply Lemma 2.5.2 to the integration on C2-part. To do so, consider the

quadruple

(U[ab],2

a∏s=3

Cs, ψ[a]ψU[ab−1],2, R1

1, C12 ).

Note that both R11 and C1

2 normalize U[ab],2 and preserve ψ[a]ψU[ab−1],2, R1

1 ⊂ G+0 , the

conjugation by R11 will change some entries in G+

i with i ≥ 2, but not attached to any

character, and C12 ⊂ G

−2 , the conjugation by C1

2 only changes some entries in G+i or G−i

with i ≥ 4. It is easy to see that the quadruple (U[ab],2

∏as=3Cs, ψ[a]ψU[ab−1],2

, R11, C

12 )

satisfies all the other conditions for Lemma 2.5.2. By Lemma 2.5.2, the integral (2.37)

36

is non-vanishing if and only if the following integral is non-vanishing∫∗E−1(φ,

(n1 q

0 n2

)(Ia 0

p Ia(b−1)

)g)ψ−1

[a] (n1)ψ−1U

[ab−1],2(n2)d∗, (2.38)

where∫∗ =

∫p

∫q

∫n2

∫n1

, d∗ = dn1dn2dqdp,

(Ia 0

p Ia(b−1)

)∈∏as=3Cs, and

(n1 q

0 n2

)∈

U[ab],2R1.

The next step is to apply Lemma 2.5.2 to the integration on variables in the subgroup

C3. Since C3 = C13 · C2

3 , we have to consider the quadruples

(U[ab],2R1

a∏s=4

CsC23 , ψ[a]ψU[ab−1],2

, R12, C

13 )

and

(U[ab],2R1R12

a∏s=4

Cs, ψ[a]ψU[ab−1],2, R2

2, C23 ).

After finishing this step, we come to consider C4 = C14 · C2

4 · C34 . This time we consider

consecutively three quadruples

(U[ab],2R1R2

a∏s=5

CsC24C

34 , ψ[a]ψU[ab−1],2

, R13, C

14 ),

(U[ab],2R1R2R13

a∏s=5

CsC34 , ψ[a]ψU[ab−1],2

, R23, C

24 ),

and

(U[ab],2R1R2R13R

23

a∏s=5

Cs, ψ[a]ψU[ab−1],2, R3

3, C34 ).

By repeating the same procedure, we end up considering the subgroup Ca =∏a−1i=1 C

ia.

To finish this step, we apply Lemma 2.5.2 to the following (a− 1) quadruples

(U[ab],2R1 · · ·Ra−2

i−1∏l=1

Rla−1

a−1∏j=i+1

Cia, ψ[a]ψU[ab−1],2, Ria−1, C

ia)

with i = 1, 2, · · · , a− 1. After finishing all the steps, we obtain that the integral (2.38)

is non-vanishing if and only if the following integral is non-vanishing∫∗E−1(φ,

(n1 q

0 n2

)g)ψ−1

[a] (n1)ψ−1U

[ab−1],2(n2)d∗, (2.39)

37

where∫∗ =

∫q

∫n2

∫n1

, d∗ = dn1dn2dq, and(n1 q

0 n2

)∈ U[ab],2R1R2 · · ·Ra−1.

Rewrite the integral (2.39) as follows:∫n2

∫n1,q

E−1(φ,

(n1 q

0 Ia(b−1)

)(Ia 0

0 n2

)g)ψ−1

[a] (n1)ψ−1U

[ab−1],2(n2)d∗, (2.40)

by changing of variable q · n−12 7→ q. Note that q · n−1

2 has the same structure as q.

Now consider the inner integral∫n1,q

E−1(φ,

(n1 q

0 Ia(b−1)

)g)ψ−1

[a] (n1)dn1dq, (2.41)

which is exactly EψV

1a−1,ab−a+1

−1 (φ, g), using the notation in Lemma 2.6.2, since the n1

is integrated over [Ua], the maximal unipotent subgroup of GLa, q is integrated over

[M ′a×a(b−1)], where

M ′a×a(b−1) = {q = (qi,j) ∈Ma×a(b−1) | qa,j = 0, ∀1 ≤ j ≤ a(b− 1)},

and ψ[a](n1) is the Whittaker character of Ua. Then by Lemma 2.6.2, the integral (2.41)

is actually equal to EψV1a,ab−a−1 (φ, g) (for notation, see Lemma 2.6.2), i.e., the following

integral ∫n1,q

E−1(φ,

(n1 q

0 Ia(b−1)

)g)ψ−1

[a] (n1)dn1dq,

where any entry in any row of q is integrated over F\A. Therefore, the integral (2.40)

becomes∫n2

∫n1,q

E−1(φ,

(n1 q

0 Ia(b−1)

)(Ia 0

0 n2

)g)ψ−1

[a] (n1)ψ−1U

[ab−1],2(n2)d∗, (2.42)

where any entry in any row of q is integrated over F\A.

Note that the inner integral (2.42)∫n1,q

E−1(φ,

(n1 q

0 Ia(b−1)

)g)ψ−1

[a] (n1)dn1dq

=

∫n1

E−1(φ,

(n1 0

0 Ia(b−1)

)g)Qa,a(b−1)

ψ−1[a] (n1)dn1,

38

whereQa,a(b−1) is the parabolic subgroup ofGLab with its Levi subgroupGLa×GLa(b−1).

By Lemma 4.1, there is a section

f ∈ IndGLab(A)Qa,a(b−1)(A)(| · |

− b−12 τ ⊗ | · |

12E(τ,b−1)),

such that

E−1(φ,

(n1 0

0 Ia(b−1)

)g)Qa,a(b−1)

= f(

(n1 0

0 Ia(b−1)

)g)(Ia × Ia(b−1)).

Recall the standard Iwasawa decomposition

GLab(A) = Qa,a(b−1)(A) ·K,

with K =∏vKv being the standard maximal compact subgroup of GLab(A). Then for

any g ∈ GLab(A), we write g = h1(g)h2(g)v(g)k(g), where h1(g) ∈ GLa(A) is identified

with diag(h1(g), Ia(b−1)), h2(g) ∈ GLa(b−1)(A) is identified with diag(Ia, h2(g)), v(g) ∈Va,a(b−1)(A) and k(g) ∈ K.

Now, for n1 ∈ Ua, write(n1 0

0 Ia(b−1)

)g = n1h1(g)h2(g)v(g)k(g).

Then we have

E−1(φ,

(n1 0

0 Ia(b−1)

)g)Qa,a(b−1)

= f(n1h1(g)h2(g)k(g))(Ia × Ia(b−1)).

By definition, we have

f(n1h1(g)h2(g)k(g)) =|h1(g)|−b−1

2 |h2(g)|12

· (τ(n1h1(g))⊗ E(τ,b−1)(h2(g)))(f(k(g))).

When k(g) ∈ K, f(k(g)) is a vector in the space of τ ⊗ E(τ,b−1). Since the sections

defining the Eisenstein series are of K-finite, we may assume that

f(k(g)) =

nk(g)∑j=1

fk(g)j ⊗ φk(g)

j ,

39

where fk(g)j ∈ τ , and φ

k(g)j ∈ E(τ,b−1). Hence we have

(τ(n1h1(g))⊗ E(τ,b−1)(h2(g)))(f(k(g)))

=nk(g)∑j=1

τ(n1h1(g))(fk(g)j )⊗ E(τ,b−1)(h2(g))(φ

k(g)j ).

By definition, we have

τ(n1h1(g))(fk(g)j )(Ia) = f

k(g)j (n1h1(g))

and

E(τ,b−1)(h2(g))(φk(g)j )(Ia(b−1)) = φ

k(g)j (h2(g)).

It follows that

E−1(φ,

(n1 0

0 Ia(b−1)

)g)Qa,a(b−1)

=nk(g)∑j=1

|h1(g)|−b−1

2 |h2(g)|12 f

k(g)j (n1h1(g))φ

k(g)j (h2(g)).

Therefore, we have∫n1

E−1(φ,

(n1 0

0 Ia(b−1)

)g)Qa,a(b−1)

ψ−1[a] (n1)dn1

=

∫n1

nk(g)∑j=1

|h1(g)|−b−1

2 |h2(g)|12 f

k(g)j (n1h1(g))φ

k(g)j (h2(g))ψ−1

[a] (n1)dn1

=nk(g)∑j=1

∫n1

|h1(g)|−b−1

2 fk(g)j (n1h1(g))ψ−1

[a] (n1)dn1|h2(g)|12φ

k(g)j (h2(g)).

Hence the integral (2.42) becomes

nk(g)∑j=1

|h1(g)|−b−1

2

∫n1

fk(g)j (n1h1(g))ψ−1

[a] (n1)dn1

·|h2(g)|12

∫n2

φk(g)j (n2h2(g))ψ−1

U[ab−1],2

(n2)dn2.

(2.43)

40

By the induction assumption, the integral∫n2

φk(g)j (n2h2(g))ψ−1

U[ab−1],2

(n2)dn2

is non-vanishing if and only if the following integral is non-vanishing:∫[Ua(b−1)]

φk(g)j (n2h2(g))ψ

a(b−1)0;(b−1)·a(n2)−1dn2.

Therefore, the integral (2.43), hence the integral (2.42) is non-vanishing if and only if

the following integral is non-vanishing:∫[Ua(b−1)]

∫n1,q

E−1(φ,

(n1 q

0 Ia(b−1)

)(Ia 0

0 n2

)g)

ψ−1[a] (n1)ψ

a(b−1)0;(b−1)·a(n2)−1dn1dqdn2

=

∫[Uab]

E−1(φ, vg)ψab0;b·a(u)−1du.

This finishes the proof of the theorem.

Furthermore, we will prove the following theorem in the next section.

Theorem 2.5.5. Let p = [p1p2 · · · pr] be a standard partition of n, i.e. p1 ≥ p2 ≥· · · ≥ pr > 0. If p1 > a, then the residual representation E(τ,b) of GLab(A) with cuspidal

support (Pab , τ⊗b) has no nonzero ψp-Fourier coefficients, that is, for any E−1(φ, g) ∈

E(τ,b), the ψp-Fourier coefficient Eψp−1(φ, g) is identically zero.

Combining Theorems 2.5.4 and 2.5.5 with Corollary 2.3.4, we obtain the following

extension to the residual spectrum of GLn(A) of the theorem of Shalika ([S74]) and

of Piatetski-Shapiro ([PS79]), independently, that all nonzero irreducible cuspidal au-

tomorphic representations of GLn(A) are generic, i.e. have nonzero Whittaker-Fourier

coefficients.

Theorem 2.5.6. Let p = [p1p2 · · · pr] be a partition of n with p1 ≥ p2 ≥ · · · ≥ pr > 0

and denote by [ab] the partition of all parts equal to a. For the residual representation

E(τ,b) with cuspidal support (Pab , τ⊗b), belonging to the discrete spectrum of GLn(A), the

following hold.

(1) The residual representation E(τ,b) has a nonzero ψ[ab]-Fourier coefficient.

41

(2) For any partition p = [p1p2 · · · pr] of n, if p1 > a, then the residual representation

E(τ,b) has no nonzero ψp-Fourier coefficients.

It is clear that Part (1) follows from Theorem 2.5.4 and Corollary 2.3.4, and Part

(2) is Theorem 2.5.5. Note that if we use the notation of [G06], Theorem 2.5.6 implies

that O(E(τ,b)) = {O[ab]}.

2.6 Proof of Theorem 2.5.5

In this section, we will prove the vanishing property of Fourier coefficients of the residue

E−1(φ, g) attached to the partitions either bigger than or not related to the partition

[ab]. To this end, we need to prove the following two key lemmas.

Lemma 2.6.1. Let p = [p1p2 · · · pr] be a standard partition of n, i.e. p1 ≥ p2 ≥ · · · ≥pr > 0. If p1 > a, then the ψ

ε1,··· ,εr−1p -Fourier coefficient

Eψε1,··· ,εr−1p

−1 (φ, g) :=

∫[Uab]

E−1(φ, ug)ψε1,··· ,εr−1p (u)−1du ≡ 0, (2.44)

where

ψε1,··· ,εr−1p (u) :=

ψ(u1,2 + · · ·+ up1−1,p1 + ε1up1,p1+1)

· ψ(up1+1,p1+2 + · · ·+ up1+p2−1,p1+p2 + ε2up1+p2,p1+p2+1)

· · ·

· ψ(up1+···+pr−2+1,p1+···+pr−2+2 + · · ·+ up1+···+pr−1−1,p1+···+pr−1

+ εr−1up1+···+pr−1,p1+···+pr−1+1)

· ψ(up1+···+pr−1+1,p1+···+pr−1+2 + · · ·+ uab),

and εi ∈ {0, 1}, i = 1, · · · , r − 1.

Proof. We separate the proof into two steps: (I) ε1 = 0; and (II) ε1 = 1.

Step (I). ε1 = 0. Since p1 > a, there are two cases to be considered: (1) p1 6= as

for all 1 < s ≤ b; and (2) p1 = as for some 1 < s ≤ b.Case (1). Let Qp1,ab−p1 be the parabolic subgroup of GLab with Levi isomorphic to

GLp1×GLab−p1 . By the definition of the ψε1,··· ,εr−1p -Fourier coefficient, E

ψε1,··· ,εr−1p

−1 (φ, g)

42

has the constant term of the residue E−1(φ, g) along Qp1,ab−p1 as an inner integral. More

precisely,

Eψε1,··· ,εr−1p

−1 (φ, g) = [E−1(φ, g)Qp1,ab−p1 ]ψ[p1];ψ

ε2,··· ,εr−1[p2···pr ] .

Since p1 6= s ·a for all 1 ≤ s ≤ b, P * Qab−pr,pr , which implies that E−1(φ, g)Qp1,ab−p1 =

0. Therefore, Eψε1,··· ,εr−1p

−1 (φ, g) = 0.

Case (2). If p1 = ab, then Eψε1,··· ,εr−1p

−1 (φ, g) = 0, since E(τ,b) is not generic, and

Eψε1,··· ,εr−1p

−1 (φ, g) is a Whittaker-Fourier coefficient. From now on, we assume that p1 =

as, with 1 < s < b.

Recall from Lemma 4.1 that Qas,a(b−s) is the parabolic subgroup of GLab with Levi

isomorphic to GLas × GLa(b−s). By the definition of the ψε1,··· ,εr−1p -Fourier coefficient,

Eψε1,··· ,εr−1p

−1 (φ, g) has the constant term of the residue E−1(φ, g) along Qas,a(b−s) as an

inner integral. As before, we have

Eψε1,··· ,εr−1p

−1 (φ, g) = [E−1(φ, g)Qp1,ab−p1 ]ψ[p1];ψ

ε2,··· ,εr−1[p2···pr ] .

After taking the constant term along Qas,a(b−s), E−1(φ, g)Qas,a(b−s) is an automorphic

function over GLsa(A) × GLa(b−s)(A). Note here that the ψ[p1]-Fourier coefficient is

taken on GLsa(A) and the ψε2,··· ,εr−1

[p2···pr] -Fourier coefficient is taken on GLa(b−s)(A).

By Lemma 4.1, it is enough to show that the residual representation E(τ,s) has no

non-zero ψ[p1]-Fourier coefficients or the residual representation E(τ,b−s) has no non-zero

ψε2,··· ,εr−1

[p2···pr] -Fourier coefficients. It is clear that the character ψ[p1] is exactly the Whittaker

character of GLsa(A). By Proposition 2.2.1, the residual representation E(τ,s) is not

generic, and hence it has no non-zero ψ[p1]-Fourier coefficients.

Hence, if p1 > a and ε1 = 0, then Eψε1,··· ,εr−1p

−1 (φ, g) = 0.

Step (II). We assume that ε1 = 1. If εi = 1, for all 1 ≤ i ≤ r − 1, then ψε1,··· ,εr−1p

is a non-degenerate character of GLab, and hence Eψε1,··· ,εr−1p

−1 (φ, g) = 0.

So, we may assume i < r − 1 to be the first number such that εi = 0. By applying

the proof of Step (I) to the partition [(∑i

j=1 pj)pi+1 · · · pr], which is still either bigger

than or not related to the partition [ab], we deduce that Eψε1,··· ,εr−1p

−1 (φ, g) = 0.

This completes the proof of the lemma.

43

Lemma 2.6.2. Let V1m−1,ab−m+1 be the unipotent radical of the parabolic subgroup

Q1m−1,ab−m+1 with Levi part GL×(m−1)1 ×GLab−m+1. Let

ψV1m−1,ab−m+1(v) = ψ(v1,2 + · · ·+ vm−1,m),

and

ψV1m−1,ab−m+1(v) = ψ(v1,2 + · · ·+ vm−2,m−1)

be two characters of V1m−1,ab−m+1. Define EψV

1m−1,ab−m+1

−1 (φ, g) by∫[V1m−1,ab−m+1]

E−1(φ, vg)ψ−1V1m−1,ab−m+1

(v)dv. (2.45)

Then, if m > a, EψV

1m−1,ab−m+1

−1 (φ, g) ≡ 0; and if m = a,

EψV

1m−1,ab−m+1

−1 (φ, g) = EψV1m,ab−m−1 (φ, g).

Proof. For i = 1, . . . , ab − 1, let Ri be the subgroup of Uab such that any element

u = (uj,l) ∈ Ri, uj,l = 0, unless j = i.

Since EψV

1m−1,ab−m+1

−1 (φ, g) is left Rm(F )-invariant, we take Fourier expansion of

EψV

1m−1,ab−m+1

−1 (φ, g) along [Rm] = Rm(F )\Rm(A):

EψV

1m−1,ab−m+1

−1 (φ, g)

=EψV1m,ab−m−1 (φ, g)

+∑

γ∈Q01,ab−m−1(k)\GLab−m(k)

EψV1m,ab−m−1 (φ,diag(Im, γ)g).

(2.46)

Since both EψV1m,ab−m−1 (φ, g) and E

ψV1m,ab−m−1 (φ, g) are left Rm+1(F )-invariant, we can

take the Fourier expansion of them along [Rm+1] = Rm+1(F )\Rm+1(A). We repeat this

process for each term in the Fourier expansion of EψV1m,ab−m−1 (φ, g) or E

ψV1m,ab−m−1 (φ, g)

along the following sequence [Rm+2], . . . , [Rab−1]. After plugging back all these Fourier

expansion to (2.46), we can see that EψV

1m−1,ab−m+1

−1 (φ, g) can be written as a summation,

each term of which is of the form (2.44), and is identically zero, if m > a, by Lemma

2.6.1.

44

If m = a, then from (2.46), we can see that

EψV

1m−1,ab−m+1

−1 (φ, g) = EψV1m,ab−m−1 (φ, g),

since EψV1m,ab−m−1 (φ, g) ≡ 0 from the above discussion.

This finishes the proof of the lemma.

Before proving the general case of Theorem 2.5.5, we prove the vanishing of Fourier

coefficients of E−1(φ, ·) corresponding to the orbits [p11ab−p1 ] with p1 > a. The idea

of the proof for this special case is applicable to the general case. Note that in the

proof of the Proposition 5.3 of [G06], the vanishing of Fourier coefficients of E−1(φ, ·)corresponding to the general bigger than or not related orbits is sketched by reducing to

the proof of that corresponding to the special partition [(a + 1)1ab−a−1], which is then

proved by using local argument. We prove it below using global argument.

Proposition 2.6.3. The Fourier coefficient of E−1(φ, ·) corresponding to the partition

p = [p11ab−p1 ], p1 > a is identically zero.

Proof. We separate the proof into two cases: (1) p1 is odd; and (2) p1 is even.

Case (1). We assume that p1 is odd. From the definition, any element in V[p11ab−p1 ],2

has the following form:

u =

(n1 q

0 Iab−p1

)(Ip1 0

p Iab−p1

),

where n1 ∈ Up1 , the standard maximal unipotent subgroup of GLp1 , q ∈ Mp1×ab−p1

with ql,m = 0, for l ≥ p1−12 + 1, and p ∈Mab−p1×p1 with pl,m = 0, for m ≤ p1−1

2 + 1.

The ψ[p11ab−p1 ]-Fourier coefficient of E−1(φ, ·), Eψ

[p11ab−p1 ]

−1 (φ, g), can be rewritten as

∫p

∫q

∫n1

E−1(φ,

(n1 q

0 Iab−p1

)(Ip1 0

p Iab−p1

)g)ψ−1

[p1](n1)dn1dqdp, (2.47)

where ψ[p1](n1) is a non-degenerate character of GLp1 .

For p1−12 + 1 ≤ s ≤ p1 − 1, define the following unipotent subgroup of GLab:

Rs =

{(Ip1 q

0 Iab−p1

): q ∈Mp1×ab−p1 , ql,m = 0, l 6= s

}.

45

For p1−12 + 2 ≤ s ≤ p1, define the following unipotent subgroup of GLab:

Cs =

{(Ip1 0

p Iab−p1

): p ∈Mab−p1×p1 , ql,m = 0,m 6= s

}.

Then we can see that Rs(F )\Rs(A) ∼= Cs(F )\Cs(A) ∼= (F\A)ab−p1 . Note that Rs ⊂G+

−2(s− p1−12−1)

and Cs ⊂ G−2(s− p1−1

2−1)

.

Write V[p11ab−p1 ],2 = V[p11ab−p1 ],2

∏p1

s=p1−1

2+2Cs, where V[p11ab−p1 ],2 consists of ele-

ments in V[p11ab−p1 ],2 with p-part zero.

Now we are ready to apply Lemma 2.5.2 to the integral in (2.47) first with the

quadruple

(V[p11ab−p1 ],2

p1∏s=

p1−12

+3

Cs, ψ[p1], R p1−12

+1, C p1−1

2+2

),

then with the quadruple

(V[p11ab−p1 ],2R p1−12

+1

p1∏s=

p1−12

+4

Cs, ψ[p1], R p1−12

+2, C p1−1

2+3

),

and keep doing the same thing until the final step with the quadruple

(V[p11ab−p1 ],2R p1−12

+1· · ·Rp1−2, ψ[p1], Rp1−1, Cp1).

This calculation shows that the integral (2.47) is identically zero if and only if the

following integral is identically zero∫q

∫n1

E−1(φ,

(n1 q

0 Iab−p1

)g)ψ−1

[p1](n1)dn1dqdp, (2.48)

where

(n1 q

0 Iab−p1

)∈ V[p11ab−p1 ],2R p1−1

2+1· · ·Rp1−1, that is, all first (p1 − 1)-rows of q

are integrated over k\A, and the last row of q is zero.

Note that for each step, we can easily check the conditions for Lemma 2.5.2. For

the first quadruple

(V[p11ab−p1 ],2

p1∏s=

p1−12

+3

Cs, ψ[p1], R p1−12

+1, C p1−1

2+2

),

46

the conjugation by R p1−12

+1will change some entries in G+

i with i ≥ 2, but the changing

of variables does not change the character, the conjugation by C p1−12

+2will change some

entries in G+i or G−i with i ≥ 4. For 1 ≤ j ≤ p1−1

2 − 2, when we consider the quadruple

(V[p11ab−p1 ],2

j+1∏l=1

R p1−12

+l

p1∏s=

p1−12

+j+4

Cs, ψ[p1], R p1−12

+1+j+1, C p1−1

2+1+j+2

),

the conjugation by R p1−12

+1+j+1will change some entries in

G+i ∩ V[p11ab−p1 ],2

j+1∏l=1

R p1−12

+l

p1∏s=

p1−12

+j+4

Cs

with i ≥ −2j, but the changing of variables does not change the character, the conju-

gation by C p1−12

+1+j+2will change some entries in G+

i with i ≥ 4.

Note that the integral in (2.48) is actually EψV

1p1−1,ab−p1+1

−1 (φ, g), which is identically

zero by Lemma 2.6.2. This finishes the proof of the case of p1 odd.

Case (2). Assume that p1 is even. From the definition, any element in V[p11ab−p1 ],2

has the following form:

u =

(n1 q

0 Iab−p1

)(Ip1 0

p Iab−p1

),

where n1 ∈ Up1 , the standard maximal unipotent subgroup of GLp1 , q ∈ Mp1×ab−p1

with ql,m = 0, for l ≥ p1

2 , and p ∈Mab−p1×p1 with pl,m = 0, for m ≤ p1

2 + 1.

The ψ[p11ab−p1 ]-Fourier coefficient of E−1(φ, ·), Eψ

[p11ab−p1 ]

−1 (φ, g), can also be rewrit-

ten as ∫p

∫q

∫n1

E−1(φ,

(n1 q

0 Iab−p1

)(Ip1 0

p Iab−p1

)g)ψ−1

[p1](n1)dn1dqdp, (2.49)

where ψ[p1](n1) is a non-degenerate character of GLp1 .

By Corollary 2.5.3, we only have to show that the following integral is identically

zero: ∫y,p,q,n1

E−1(φ,

(n1 q

0 Iab−p1

)(Ip1 0

p Iab−p1

)yg)ψ−1

[p1](n1)dn1dqdpdy, (2.50)

47

where y ∈ [Y ].

For p1

2 + 1 ≤ s ≤ p1 − 1, define the following unipotent subgroup of GLab:

Rs =

{(Ip1 q

0 Iab−p1

): q ∈Mp1×ab−p1 , ql,m = 0, l 6= s

}.

For p1

2 + 2 ≤ s ≤ p1, define the following unipotent subgroup of GLab:

Cs =

{(Ip1 0

p Iab−p1

): p ∈Mab−p1×p1 , ql,m = 0,m 6= s

}.

Then we can see that Rs(F )\Rs(A) ∼= Cs(F )\Cs(A) ∼= (F\A)ab−p1 . Note that Rs ⊂G+−2(s− p1

2)+1

and Cs ⊂ G−2(s− p12

)+1.

Write Y V[p11ab−p1 ],2 = Y V[p11ab−p1 ],2

∏p1

s=p12

+2Cs, where V[p11ab−p1 ],2 consists of ele-

ments in V[p11ab−p1 ],2 with p-part zero.

Now we apply Lemma 2.5.2 to the integral (2.50) first with the quadruple

(Y V[p11ab−p1 ],2

p1∏s=

p12

+3

Cs, ψ[p1], R p12

+1, C p12

+2),

and then with the following quadruple

(Y V[p11ab−p1 ],2R p12

+1

p1∏s=

p12

+4

Cs, ψ[p1], R p12

+2, C p12

+3),

and keep doing the same thing until the last step with the quadruple

(Y V[p11ab−p1 ],2R p12

+1 · · ·Rp1−2, ψ[p1], Rp1−1, Cp1).

This calculation shows that the integral (2.50) is identically zero if and only if the

following integral is identically zero∫y

∫q

∫n1

E−1(φ,

(n1 q

0 Iab−p1

)yg)ψ−1

[p1](n1)dn1dqdy, (2.51)

where

(n1 q

0 Iab−p1

)∈ Y V[p11ab−p1 ],2R p1

2· · ·Rp1−1, i.e., all first (p1 − 1)-rows of q are

integrated over k\A, the last row of q is zero. For each step, the conditions for Lemma

2.5.2 can be easily checked as the case of p1 odd.

48

Note that the integral in (2.51) is actually EψV

1p1−1,ab−p1+1

−1 (φ, g), which is identically

zero by Lemma 2.6.2.

This completes the proof of the proposition.

Now, we prove the general case of Theorem 2.5.5. As we mentioned, the idea will

be similar to the case of special orbits [p11ab−p1 ], p1 > a, in Proposition 2.6.3.

First, by Corollary 2.5.3, we only have to show that the following integral is identi-

cally zero: ∫[Y ]

∫[Vp,2]

E−1(φ, vyg)ψ−1Vp,2

(v)dvdy. (2.52)

Note that we will use the notation introduced in Section 5 accordingly.

First we conjugate the integration variables in the integral (2.52) by a Weyl element

ω of GLn (n = ab) which conjugates the toric subgroup Hp of GLab in (2.30) attached

to the partition p to the toric subgroup:

diag(H[p1](t); tp2−1, · · · , t1−p2),

where after the first block of size p1, the exponents of t are of non-increasing order.

Note that ω is of the form diag(Ia, ω′), where ω′ is a Weyl element of GLn−p1 , which

conjugates the toric subgroupH[p2···pr] ofGLn−p1 in (2.30) corresponding to the partition

[p2 · · · pr] to the toric subgroup of GLn−p1 :

diag(tp2−1, · · · , t1−p2),

where the exponents of t are of non-increasing order. For example, for the partition

[p2 · · · pr] = [(32)2], ω′ is the Weyl element of GL8, which conjugates the toric subgroup

diag(t2, 1, t−2; t2, 1, t−2; t, t−1) to the toric subgroup: diag(t2, t2, t, 1, 1, t−1, t−2, t−2).

Let Up,2 = ωY Vp,2ω−1. Then any element of Up,2 has the following form:

u =

(n1 q

0 n2

)(Ip1 0

p Iab−p1

),

where n1 ∈ Ua, the standard maximal unipotent subgroup of GLa, and

n2 ∈ U[p2···pr],2 := ω′Y[p2···pr],2V[p2···pr],2ω′−1

49

with Y[p2···pr],2 being the corresponding Y for the partition [p2 · · · pr]. Denote ψUp,2(u) :=

ψVp,2(ω−1uω). Hence integral (2.52) equals

∫[Up,2]

E−1(φ,

(n1 q

0 n2

)(Ip1 0

p Iab−p1

)ωg)ψ−1

Up,2(u)du. (2.53)

Consider the group Up,2 ∩ Up1,ab−p1 , where Up1,ab−p1 is the unipotent subgroup of

the parabolic subgroup of G with Levi GLp1 ×GLab−p1 . Let i be the index of the first

row of Up,2 ∩ Up1,ab−p1 with zero entries. Let Rj be the subgroup of Up1,ab−p1 with

zeros everywhere except the complement of Up,2 ∩ Up1,ab−p1 in the j-th row, i ≤ j ≤p1 − 1. Similarly, let Cj be the subgroup of U−p1,ab−p1

with zeros everywhere except the

complement of Up,2 ∩ U−p1,ab−p1in the j-th column, i + 1 ≤ j ≤ p1. Then, we can see

that Rj ∼= Cj+1.

Write Up,2 = Up,2 ·∏p1s=i+1Cs, where Up,2 consists of elements in Up,2 with U−p1,ab−p1

-

part zero. For i ≤ j ≤ p1 − 1, write Rj =∏mjl=1R

lj , where Rlj consists of all the entries

in G+klj

, with klj decreasing. For i+ 1 ≤ j ≤ p1, write Cj =∏mj−1

l=1 C lj , where C lj consists

of all the entries in G−klj

, with klj increasing. Note that Rlj∼= C lj+1.

Now we are ready to apply Lemma 2.5.2 to the integral (2.53) with a sequence of

quadruples: (Up,2∏p1s=i+2Cs

∏mil=2C

li+1, ψUp,2 , R

1i , C

1i+1), and then

(Up,2R1i

p1∏s=i+2

Cs

mi∏l=3

C li+1, ψUp,2 , R2i , C

2i+1),

and keep going until (Up,2R1i · · ·R

mi−1i

∏p1s=i+2Cs, ψUp,2 , R

mii , Cmii+1). This finishes the

first step. Then we go with a next sequence of quadruples

(Up,2Ri

p1∏s=i+3

Cs

mi+1∏l=2

C li+2, ψUp,2 , R1i+1, C

1i+2),

· · · ,

(Up,2RiR1i+1 · · ·R

mi+1−1i+1

p1∏s=i+3

Cs, ψUp,2 , Rmi+1

i+1 , Cmi+1

i+2 );

50

and keep doing this until the last step with a sequence of quadruples

(Up,2Ri · · ·Rp1−2

mp1−1∏l=2

C lp1, ψUp,2 , R

1p1−1, C

1p1

),

· · · ,

(Up,2Ri · · ·Rp1−2R1p1−1 · · ·R

mp1−1−1p1−1 , ψUp,2 , R

mp1−1

p1−1 , Cmp1−1p1 ).

Note that here for convenience, we denote all the characters in all the above quadruples

by ψUp,2 . The above calculation shows that the integral (2.53) is identically zero if and

only if the following integral is identically zero∫∗

∫q

∫n1

E−1(φ,

(n1 q

0 u[p2...pr]

)g)ψ−1

[p1](n1)ψ−1[p2...pr]

(u[p2...pr])dn1dqdu∗, (2.54)

where q ∈ Up1,ab−p1(F )\Up1,ab−p1(A) with only the last row being zero,∫∗ means∫

U[p2...pr ],2and du∗ is du[p2...pr]. For each step, the conditions for Lemma 2.5.2 can

be checked easily as in proof of Proposition 2.6.3.

Note that the integral (2.54) contains the following integral as an inner integral∫q

∫n1

E−1(φ,

(n1 q

0 Iab−p1

)g)ψ−1

[p1](n1)dn1dq,

which is actually EψV

1p1−1,ab−p1+1

−1 (φ, g). By Lemma 2.6.2, EψV

1p1−1,ab−p1+1

−1 (φ, g) is iden-

tically zero. This completes the proof of Theorem 2.5.5.

Chapter 3

On Fourier Coefficients of

Automorphic Forms of

Symplectic Groups

In this chapter, we first recall the definition of Fourier coefficients of automorphic forms

of symplectic groups attached to their nilpotent orbits and the basic properties they

satisfy. Then we recall the notion of Fourier-Jacobi coefficients and automorphic descent.

We mainly follow [CM93], [GRS03], [G06] and [GRS11].

3.1 Fourier Coefficients Automorphic Forms of Symplectic

Groups Attached to Nilpotent Orbits

We recall the definition of Fourier coefficients attached to unipotent orbits of symplectic

groups. Most results in this section will be published in [JL13b]. Let Gn = Sp2n be the

symplectic group with a Borel subgroup B = TU , where the maximal torus T consists

of all diagonal matrices of form: diag(t1, · · · , tn; t−1n , · · · , t−1

1 ), and the unipotent radical

of B consists all upper unipotent matrices in Sp2n.

Let F be a number filed and F be the algebraic closure of F . It is well known that

the set of all unipotent adjoint orbits of Gn(F ) is parameterized by the set of partitions

of 2n whose odd parts occur with even multiplicity (see [CM93], [N11] and [W01], for

51

52

instance). We may call them symplectic partitions of 2n.

If we consider Gn = Sp2n over F , the symplectic partitions of 2n parameterize

the F -stable unipotent orbits of Gn(F ). In order to define the Fourier coefficients of

automorphic forms of Gn(A) attached to an F -unipotent orbit, where A is the ring of

adeles of F , we introduce certain explicit elements in the F -unipotent orbit. We do this

following [CM93], and see also [N11] and [W01].

Given a symplectic partition p = [p1, . . . , pt], the Recipe 5.2.2 for Type Cn ([CM93,

Page 77]) gives a way to attach to p a standard sl2-triple. It is easy to see that

this method also works over F by using [N11] and [W01]. One may break the se-

ries {p1, . . . , pt} into chunks of the following two types: {2k + 1, 2k + 1} or {2k}, and

rewrite the partition as p = [pe11 · · · perr ], with p1 ≥ p2 ≥ · · · ≥ pr, where ei = 1 if pi is

even, ei = 2, if pi is odd. From now on, general symplectic partitions will always be

written in this form.

We assign a block of consecutive indices for each pi inductively. For p1, choose the

block {1, . . . , q1}, where q1 = p1

2 if e1 = 1, and q1 = p1 if e1 = 2. Assume that for pi−1,

i ≥ 2, the block of consecutive indices has been chosen to be {∑i−2

j=1 qj+1, . . . ,∑i−1

j=1 qj}.Then the block for pi is {

∑i−1j=1 qj + 1, . . . ,

∑ij=1 qj}, where qi = pi

2 if ei = 1, and qi = pi

if ei = 2.

Next, to each pi, we assign a set of negative roots Xi as follows: for 1 ≤ i ≤ r, define

Xi = {αi1 = eN+2 − eN+1, . . . , αiqi−1 = eN+qi − eN+qi−1, α

iqi = −2eN+qi}

if ei = 1, and define

Xi = {αi1 = eN+2 − eN+1, . . . , αiqi−1 = eN+qi − eN+qi−1}

if ei = 2, where N =∑i−1

j=1 qj . Then we define

Xp,a =

r∑i=1

qi−1∑j=1

xαij(1

2) +

∑ei=1

xαiqi(ai),

where ai ∈ F ∗/(F ∗)2 if ei = 1, which defines a = {ai | for ei = 1}, and for each root

α, xα(x) is defined to be the a root vector in the corresponding root subspace. Then,

we can find Yp,a a sum of positive root vectors and a semisimple element Hp,a, such

that {Hp,a, Yp,a, Xp,a} is a standard sl2-triple. Since we will only use Xp,a to define the

53

Fourier coefficients attached to the partition p = [pe11 · · · perr ], we will not make Yp,a and

Hp,a more explicit here.

To this sl2-triple {Hp,a, Yp,a, Xp,a}, we define a one-dimensional toric subgroup Hpof Gn(F ): for t ∈ F ∗,

Hp(t) := diag(T1(t), . . . Tr(t), T∗r (t), . . . , T ∗1 (t)), (3.1)

where

Ti(t) =

diag(tqi−1, . . . , t) if ei = 1;

diag(tqi−1, . . . , t1−qi) if ei = 2.

It is easy to see that under the adjoint action,

Ad(Hp(t))(Xp,a) = t−2Xp,a,∀t ∈ F ∗.

Let g be the Lie algebra of Gn. Under the adjoint action of Hp, g has the following

direct sum decomposition into Hp-eigenspaces:

g = g−m ⊕ · · · ⊕ g−2 ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ g2 ⊕ · · · ⊕ gm, (3.2)

for some positive integer m, where gl := {X ∈ g|Ad(Hp(t))(X) = tlX}.Let Vp,j (j = 1, . . . ,m) be the unipotent subgroup of Gn(F ) with Lie algebra ⊕ml=jgl.

Let Lp be the algebraic subgroup of Gn(F ) with Lie algebra g0. Under the adjoint

action, the set Ad(Lp(F ))(Xp,a) is Zariski open dense in dual space of Vp,2(F )/Vp,3(F ).

Note that when passing to F from F , this Zariski open dense orbit decomposes as a

union of F -rational orbits, which form the corresponding F -stable orbit defined over

F . By Proposition 5 of [N11] (see also [W01]), when the ai’s in a run through all

the square classes in F ∗/(F ∗)2, Ad(Lp(F ))(Xp,a) gives all the F -rational orbits in the

corresponding F -stable orbit defined over F .

Each F -rational Ad(Lp(F ))(Xp,a) defines a character of Vp,2 as follows: for v ∈Vp,2(A) and for a nontrivial character ψ of F\A,

ψp,a(v) :=ψ(tr(Xp,a log(v)))

=

r∏i=1

ψ(v∑i−1j=1 qj+1,

∑i−1j=1 qj+2 + · · ·+ v∑i

j=1 qj−1,∑ij=1 qj

)

· ψ(∑ei=1

aiv∑ij=1 qj ,

∑ij=1 qj

).

(3.3)

54

From now on, we may take Gn(A) = Sp2n(A) or Sp2n(A), unless specified otherwise.

Since Gn(A) = Sp2n(A) splits over any unipotent subgroup, the above discussion holds

for Gn(A) = Sp2n(A).

For an arbitrary automorphic form ϕ on Gn(A), the ψp,a-Fourier coefficient of ϕ is

defined by

ϕψp,a(g) :=

∫[Vp,2]

ϕ(vg)ψp,a(v)−1dv, (3.4)

where for a group G, we denote the quotient G(F )\G(A) simply by [G]. Since ϕ is

automorphic, the non-vanishing of the ψp,a-Fourier coefficient of ϕ depends only on the

F -rational orbit Ad(Lp(F ))(Xp,a). When an irreducible automorphic representation π

of Gn(A) is generated by automorphic forms ϕ, we say that π has a nonzero ψp,a-Fourier

coefficient or a nonzero Fourier coefficient attached to p if there exists an automorphic

form ϕ in the space of π with a nonzero ψp,a-Fourier coefficient ϕψp,a(g), for some choice

of a.

For any irreducible automorphic representation π of Gn(A), as in [J12], we define

nm(π) to be the set of all symplectic partitions p which have the properties that π has

a nonzero ψp,a-Fourier coefficient for some choice of a, and for any p′ > p (with the

natural ordering of partitions), π has no nonzero Fourier coefficients attached to p′.

Recall that Vp,1(F ) is the unipotent subgroup of Gn(F ) whose Lie algebra is g1 ⊕g2 ⊕ · · · ⊕ gm. Following [MW87], we define

(Xp,a)] = {X ∈ g | tr(Xp,a[X,X

′]) = 0, ∀X ′ ∈ g}.

Define V ′p,2 = exp(g1 ∩ (Xp,a)])Vp,2, which is a normal subgroup of Vp,1(F ). From the

definition of (Xp,a)], it is easy to see that the character ψp,a on Vp,2 can be trivially

extended to V ′p,2, which we still denote by ψp,a. It turns out that Vp,1/ kerV ′p,2(ψp,a) has

a Heisenberg structure W ⊕ Z (see [MW87], Section I.7), where W ∼= Vp,1/V′p,2, and

Z ∼= V ′p,2/ kerV ′p,2(ψp,a). Note that the symplectic form on W is the one inherited from

the Lie algebra bracket, i.e., for w1, w2 ∈ W (here, we identify w ∈ W with any of it’s

representatives in Vp,1 such that log(w) ∈ g1),

〈w1, w2〉 := tr(Xp,a log([w1, w2])) = tr(Xp,a[log(w1), log(w2)]).

The non-degeneracy of this symplectic form can be checked easily as following: for fixed

55

w1 ∈W , if 〈w1, w2〉 ≡ 0 for any w2 ∈W , that is,

tr(Xp,a[log(w1), log(w2)]) ≡ 0

for any w2 ∈W , then, tr(Xp,a[log(w1), X ′]) ≡ 0, for any X ′ ∈ g1. Therefore,

tr(Xp,a[log(w1), X ′]) ≡ 0,

for any X ′ ∈ g. Hence, log(w1) ∈ (Xp,a)], that is, w1 = 0 ∈ Vp,1/V ′p,2.

As in the case of GLn (see Lemma 2.5.1), we can prove that actually V ′p,2 = Vp,2,

i.e., g1 ∩ (Xp,a)] = {0}. This is expected always true if the nilpotent orbit is attached

to the sl2-triple.

Lemma 3.1.1. The two subgroups V ′p,2 and Vp,2 are equal. In particular, the quotient

group Vp,1/ kerVp,2(ψp,a) has a Heisenberg structure W ⊕ Z, where W ∼= Vp,1/Vp,2 is

symplectic, and Z ∼= Vp,2/ kerVp,2(ψp,a) is the center.

Proof. If g1 is zero, there is nothing to prove. We assume that g1 6= {0}. To prove

V ′p,2 = Vp,2, it suffices to prove that g1 ∩ (Xp,a)] = {0}.

First, we are going to describe structure of elements in Vp,2. By definition, elements

v in Vp,2 can be written as a product of

. . .

ni qi,j1 qi,j2 qi,i

. . .

0 nj qj,j (qi,j2 )∗

. . .

0 0 (nj)∗ (qi,j1 )∗

. . .

0 0 0 (ni)∗

. . .

56

and

. . .

Iqi 0 0 0. . .

pi,j1 Iqj 0 0. . .

pi,j2 pj,j Iqj 0. . .

pi,i (pi,j2 )∗ (pi,j1 )∗ Iqi. . .

,

where ni ∈ Nqi (nj ∈ Nqj ), the maximal upper-triangular unipotent subgroup of GLqi

(GLqj , respectively), qi,j1 , qi,j2 ∈ Mqi×qj , qi,i ∈ Mqi×qi , (qi,j1 )∗, (qi,j2 )∗ ∈ Mqj×qi , q

j,j ∈Mqj×qj , p

i,j1 , pi,j2 ∈ Mqj×qi , p

i,i ∈ Mqi×qi , (pi,j1 )∗, (pi,j2 )∗ ∈ Mqi×qj , and pj,j ∈ Mqj×qj ,

satisfying certain conditions. By definition, g1 6= 0 if there are parts pi and pj having

different parity. The above “certain conditions” are needed for the case that the parts

pi and pj are of different parity. We only need to describe them for qi,j1 , qi,j2 and pi,j1 , pi,j2 .

There are two sub-cases to be considered: (1) pi is even and pj is odd and (2) pi is odd

and pj is even, assuming that i < j.

For Case (1) where pi is even and pj is odd, we have that qi = pi2 and qj = pj . Then

qi,jk ∈ Mqi×qj with k = 1, 2 and qi,jk (l,m) = 0 for l ≥ m +pi−pj−1

2 ; and pi,jk ∈ Mqj×qi

with k = 1, 2 and pi,jk (l,m) = 0 for m ≤ l +pi−pj−1

2 + 1.

For Case (2) where pi is odd and pj is even, we have that qi = pi and qj =pj2 . Then

qi,j1 ∈Mqi×qj with qi,j1 (l,m) = 0 for l ≥ m+pi−pj−1

2 ; and qi,j2 ∈Mqi×qj with qi,j2 (l,m) = 0

for l ≥ m+pj2 +

pi−pj−12 . Also pi,j1 ∈Mqj×qi with pi,j1 (l,m) = 0 for m ≤ l+ pi−pj−1

2 + 1;

and pi,j2 ∈Mqj×qi with pi,j2 (l,m) = 0 for m ≤ l +pj2 +

pi−pj−12 + 1.

For Case (1), we define abelian groups Y i,j1 and Xi,j

1 as follows. First we define

Y i,j1 =

pj+1

2∏l=1

Xαi,jl

(yi,jl )

pj−1

2∏m=1

Xβi,jm

(yi,jm ),

where αi,jl = e∑i−1k=1 qk+

pi−pj−1

2+l

+ en−∑rk=j+1 qk−l+1 for 1 ≤ l ≤ pj+1

2 , and βi,jm =

e∑i−1k=1 qk+

pi−pj−1

2+m− e∑j−1

k=1 qk+mfor 1 ≤ m ≤ pj−1

2 . Here to each given root α, Xα(x)

57

is the corresponding one-dimensional root subgroup. Then we define

Xi,j1 =

pj+1

2∏l=1

Xγi,jl

(xi,jl )

pj−1

2∏m=1

Xδi,jm

(xi,jm ),

where γi,jl = e∑j−1k=1 qk+l

− e∑i−1k=1 qk+

pi−pj−1

2+l+1

for 1 ≤ l ≤ pj−12 ;

γi,jpj+1

2

= e∑i−1k=1 qk+

pi−pj−1

2+pj+1

2

− e∑j−1k=1 qk+

pj+1

2

;

and δi,jm = −en−∑rk=j+1 qk−m+1 − e∑i−1

k=1 qk+pi−pj−1

2+m+1

for 1 ≤ m ≤ pj−12 .

For Case (2), we define abelian groups Y i,j2 and Xi,j

2 as follows. First, we define

Y i,j2 =

pj2∏l=1

Xαi,jl

(yi,jl )

pj2∏

m=1

Xβi,jm

(yi,jm ),

where αi,jl = e∑i−1k=1 qk+

pi−pj−1

2+l− e∑j−1

k=1 qk+lfor 1 ≤ l ≤ pj

2 , and

βi,jm = e∑i−1k=1 qk+

pi−pj−1

2+pj2

+m+ en−

∑rk=j+1 qk−m+1

for 1 ≤ m ≤ pj2 . Then we define

Xi,j2 =

pj2∏l=1

Xγi,jl

(yi,jl )

pj2∏

m=1

Xδi,jm

(yi,jm ),

where γi,jl = e∑j−1k=1 qk+l

− e∑i−1k=1 qk+

pi−pj−1

2+l+1

for 1 ≤ l ≤ pj2 , and

δi,jm = −en−∑rk=j+1 qk−m+1 − e∑i−1

k=1 qk+pi−pj−1

2+pj2

+m+1

for 1 ≤ m ≤ pj2 .

Define Y i,j = Y i,jk and Xi,j = Xi,j

k for k = 1, 2, depending on the case of the pair

(pi, pj). Then define

Y =∏

1≤i<j≤r, pi and pj are of different parity

Y i,j , (3.5)

and

X =∏

1≤i<j≤r, pi and pj are of different parity

Xi,j . (3.6)

58

It follows that g1 = log(X) ⊕ log(Y ). To show that g1 ∩ (Xp,a)] = {0}, it is enough to

show that

(log(X)⊕ log(Y )) ∩ (Xp,a)] = {0}.

In other words, it suffices to show that for 1 ≤ i < j ≤ r such that pi and pj are of

different parity,

(log(Xi,j)⊕ log(Y i,j)) ∩ (Xp,a)] = {0}.

If (pi, pj) is in Case (1), i.e., pi is even, pj is odd, then by direct calculation, we

obtain that when 1 ≤ l ≤ pj−12 ,

tr(Xp,a[log(Xδi,jl

(xi,jl )), log(Xαi,jl

(yi,jl ))]) = −xi,jl yi,jl ;

tr(Xp,a[log(Xγi,jl

(xi,jl )), log(Xβi,jl

(yi,jl ))]) = −xi,jl yi,jl .

and when l =pj+1

2 ,

tr(Xp,a[log(Xαi,jl

(yi,jl )), log(Xγi,jl

(xi,jl ))]) = −aixi,jl yi,jl .

This implies that (log(Xi,j1 )⊕ log(Y i,j

1 )) ∩ (J −p )] = {0}.If (pi, pj) is in Case (2), i.e., pi is odd, pj is even, then by direct calculation, we get

that for 1 ≤ l ≤ pj2 ,

tr(Xp,a[log(Xγi,jl

(xi,jl )), log(Xαi,jl

(yi,jl ))]) = −xi,jl yi,jl ;

tr(Xp,a[log(Xδi,jl

(xi,jl )), log(Xβi,jl

(yi,jl ))]) = −xi,jl yi,jl .

Therefore, we also have that (log(Xi,j2 )⊕ log(Y i,j

2 ))∩ (Xp,a)] = {0}. This completes the

proof of this lemma.

From the proof of Lemma 3.1.1, we have

Corollary 3.1.2. The subspaces X and Y as defined in (3.6), (3.5) give a polarization

X ⊕ Y of W .

Note that the proof in Lemma 2.5.2 works for the general group H(A) as in [GRS11],

in particular, works for Gn(A) = Sp2n(A), Sp2n(A). We recall this lemma below.

Let C be an F -subgroup of a maximal unipotent subgroup of Gn, and let ψC be

a non-trivial character of [C] = C(F )\C(A). X, Y are two unipotent F -subgroups,

satisfying the following conditions:

59

(1) X and Y normalize C;

(2) X ∩C and Y ∩C are normal in X and Y , respectively, (X ∩C)\X and (Y ∩C)\Yare abelian;

(3) X(A) and Y (A) preserve ψC ;

(4) ψC is trivial on (X ∩ C)(A) and (Y ∩ C)(A);

(5) [X, Y ] ⊂ C;

(6) there is a non-degenerate pairing (X ∩ C)(A) × (Y ∩ C)(A) → C∗, given by

(x, y) 7→ ψC([x, y]), which is multiplicative in each coordinate, and identifies

(Y ∩C)(F )\Y (F ) with the dual of X(F )(X∩C)(A)\X(A), and (X∩C)(F )\X(F )

with the dual of Y (F )(Y ∩ C)(A)\Y (A).

Let B = CY and D = CX, and extend ψC trivially to characters of [B] =

B(F )\B(A) and [D] = D(F )\D(A), which will be denoted by ψB and ψD respectively.

Lemma 3.1.3. Assume the quadruple (C,ψC , X, Y ) satisfies the above conditions. Let

f be an automorphic form on Gn(A). Then∫[C]f(cg)ψ−1

C (c)dc ≡ 0, ∀g ∈ Gn(A),

if and only if ∫[D]f(ug)ψ−1

D (u)du ≡ 0, ∀g ∈ Gn(A),

if and only if ∫[B]f(vg)ψ−1

B (v)dv ≡ 0, ∀g ∈ Gn(A).

For simplicity, we will use ψC to denote its extensions ψB and ψD in the remaining

of the paper when we use Lemma 3.1.3. Applying Lemma 3.1.3 to the ψp,a-Fourier

coefficients for automorphic forms on Gn(A), we obtain a different proof of a useful

result [GRS03, Lemma 1.1]. For completeness and convenience, we include it here.

Corollary 3.1.4. Let p = [pe11 pe22 · · · perr ] be a standard symplectic partition of n with

p1 ≥ p2 ≥ · · · ≥ pr > 0 and n =∑r

i=1 eipi. Then the ψp,a-Fourier coefficient ϕψp,a of

60

an automorphic form ϕ on Gn(A) is non-vanishing if and only if the following integral

is non-vanishing ∫[Y ]

∫[Vp,2]

ϕ(vyg)ψp,a(v)−1dvdy

is non-vanishing; and if and only if the following integral∫[X]

∫[Vp,2]

ϕ(vxg)ψp,a(v)−1dvdx

is non-vanishing. Here the subgroups X and Y are defined in (3.6) and (3.5), respec-

tively.

Proof. By Lemma 3.1.1 and Corollary 3.1.2, Vp,1/ kerVp,2(ψp,a) has a Heisenberg struc-

ture W⊕Z, where Z ∼= Vp,2/ kerVp,2(ψp,a), and X⊕Y is a polarization of W , where X,Y

are defined in (3.6), (3.5). This implies directly that the quadruple (Vp,2, ψp,a, X, Y ) sat-

isfies all the conditions for Lemma 3.1.3.

The following lemma follows from Corollary 7.2 of [GRS11], and is proved in the

proof of Theorem 2.1 of [GJS12] (see formula (2.26) and its proof therein), the argument

also appears in the proof of [GRS99, Lemma 1, Page 895]. This lemma will not be used

in this paper, but for convenience of future reference, we include it here.

Lemma 3.1.5 ([GJS12] (2.26), [GRS99] Lemma 1, Page 895). Assume the quadruple

(C,ψC , X, Y ) as in Lemma 3.1.3. π is an irreducible automorphic representation of

Gn(A). Let f ∈ π. Then there is an f ′ ∈ π, such that∫[B]f(vg)ψ−1

B (v)dv =

∫[D]f ′(ug)ψ−1

D (u)du,∀g ∈ Gn(A).

Proof. We sketch the proof as follows. Without loss of generality (via replacing f by

its right g translation), we only have to prove∫[B]f(v)ψ−1

B (v)dv =

∫[D]f ′(u)ψ−1

D (u)du.

By Lemma 7.1 of [GRS11],∫[B]f(v)ψ−1

B (v)dv =

∫(Y ∩C)(A)\Y (A)

∫[D]f(uy)ψ−1

D (u)dudy.

61

By Corollary 7.2 of [GRS11], there exit f1, . . . , fr ∈ π, and Schwartz functions φ1, . . . , φr ∈S((Y ∩ C)(A)\Y (A)), such that for all y ∈ (Y ∩ C)(A)\Y (A),∫

[D]f(uy)ψ−1

D (u)du =r∑i=1

φi(y)

∫[D]fi(uy)ψ−1

D (u)du.

Therefore,∫[B]f(v)ψ−1

B (v)dv =

∫(Y ∩C)(A)\Y (A)

∫[D]f(uy)ψ−1

D (u)dudy

=

∫(Y ∩C)(A)\Y (A)

r∑i=1

φi(y)

∫[D]fi(uy)ψ−1

D (u)dudy

=

∫[D]

r∑i=1

∫(Y ∩C)(A)\Y (A)

φi(y)fi(uy)dyψ−1D du.

If we take

f ′(u) =r∑i=1

φi ∗ fi(u) =r∑i=1

∫(Y ∩C)(A)\Y (A)

φi(y)fi(uy)dy,

then f ′ =∑r

i=1 φi ∗ fi ∈ π and we obtain∫[B]f(v)ψ−1

B (v)dv =

∫[D]f ′(u)ψ−1

D du.

3.2 Fourier-Jacobi Coefficients and Automorphic Descent

Let Gn(A) = Sp2n(A), Sp2n(A). Fourier-Jacobi coefficients are slight invariants of the

Fourier coefficients attached to partition [(2k)12n−2k]. They play an important role in

the theory of automorphic descent. For reference, see [GRS11].

Let Nk be the unipotent radical of the parabolic subgroup Pk of Sp2n with Levi iso-

morphic to GLk1×Sp2n−2k. For n ∈ Nk, let j(n) = (nk,k+1, nk,k+2, . . . , nk,2n−k+1). Then

j(Nk) is a Heisenberg group, denoted by H2n−2k. Let W+ be the subgroup of H2n−2k

consists of elements (nk,k+1, nk,k+2, . . . , nk,n, 0, . . . , 0). For any a ∈ F ∗/(F ∗)2, let θφψ−a

be the theta series on H2n−2k(A) o Sp2n−2k(A), defined using the Weil representation

corresponding the central character ψ−a (ψ−a(x) = ψ(−ax)), φ ∈ S(W+(A)) the set of

Schwartz-Bruhat functions on W+(A).

62

Let Gn−k(A) = Sp2n−2k(A), if Gn(A) = Sp2n(A), and = Sp2n−2k(A), if Gn(A) =

Sp2n(A). Assume that π is an automorphic representation of Gn(A). Then for any

ξπ ∈ π, the Fourier-Jacobi coefficient of ξπ is defined as follows:

FJφψak−1(ξπ)(g) :=

∫Nk(F )\Nk(A)

ξπ(ng′)ψ−1k−1(n)θφ

ψ−a(j(n)g′′)dn, (3.7)

where g ∈ Gn−k(A), if Gn(A) = Sp2n(A), then g = g′′ projects to g′, if Gn(A) =

Sp2n(A), then g′ = g′′ project to g; φ ∈ S(W+(A)); and ψk−1(n) = ψ(∑k−1

i=1 ni,i+1).

Sometimes we drop φ in the notion of FJφψak−1, and only call it a Fourier coefficient for

convenience.

Let D2n2k,ψa(π) be the representation of Gn−k(A) generated by FJφψak−1

(ξπ), as ξπ runs

through π, and φ runs through S(W+(A)). D2n2k,ψa(π) is called the descent of π with

respect to the character ψa.

For discussion on properties and applications of automorphic descent, see [GRS11].

For discussion on the history of the idea of automorphic descent and generalization to

the construction of endoscopic lifting, see [J12].

Chapter 4

On Extension of

Ginzburg-Jiang-Soudry

Correspondences to Certain

Automorphic Forms on Sp4mn(A)and Sp4mn±2n(A)

In this chapter, first, we compute the set nm(E∆(τ,m)) of maximal partitions providing

non-vanishing Fourier coefficients for certain residual representation E∆(τ,m) of Sp4mn(A)

constructed by Jiang, Liu and Zhang [JLZ12], where τ is an irreducible cuspidal auto-

morphic representation of GL2n(A) of symplectic type with L(12 , τ) 6= 0, and F is any

number field. We show that nm(E∆(τ,m)) = [(2n)2m].

Then, we assume that F is not totally imaginary, and consider the set of irre-

ducible cuspidal automorphic representations of Sp4mn(A) which are nearly equivalent

to E∆(τ,m), and with following as a maximal partition providing non-vanishing Fourier

coefficients:

[(2n)2m−1(2n1)s1(2n2)s2 · · · (2nk)sk ],

with 2n ≥ 2n1 > 2n2 > · · · > 2nk, k ≥ 1. We show that this set decomposes naturally

as a disjoint union of two subsets, which correspond to certain cuspidal representations

63

64

of Sp4mn±2n(A), respectively. This generalizes the result of Ginzburg, Jiang and Soudry

[GJS12], in which they proved that there is a correspondence between certain automor-

phic representations of Sp4n(A) and Sp2n(A). We also consider the irreducibility of

certain descent representations. Most results in this chapter will be published in [L13].

4.1 Overview

Fourier coefficients play important roles in the study of automorphic forms. For sym-

plectic and special orthogonal groups G, on one hand, there is a general framework

of attaching Fourier coefficients to nilpotent orbits, which are classified by partitions

(for symplectic groups, see [GRS03] and Chapter 3). On the other hand, the discrete

spectrum of G(A) was classified by Arthur up to automorphic L2-packets ([Ar12]).

Therefore, given an irreducible automorphic representation π in the discrete spectrum

of G(A), it is an interesting question to characterize the set nm(π) of maximal partitions

corresponding to nilpotent orbits which provide non-vanishing Fourier coefficients.

In [JLZ12], Jiang, Liu and Zhang explicitly constructed certain families of residual

representations which are in the discrete spectrum for symplectic and special orthogo-

nal groups. These explicitly constructed residual representations are expected to play

important roles in descent and endoscopic constructions in the future. In this chapter,

first, we compute the set nm(E∆(τ,m)) of maximal partitions providing non-vanishing

Fourier coefficients for these residual representations E∆(τ,m).

Let τ be an irreducible unitary cuspidal automorphic representation of GL2n(A),

with the properties that L(s, τ,∧2) has a simple pole at s = 1, and L(12 , τ) 6= 0.

By Proposition 3.2 of [GJS12], there is an irreducible representation π of Sp2n(A),

which is ψ1-generic, lifts weakly to τ with respect to ψ.

Let Pr = MrNr be the maximal parabolic subgroup of Sp2l with Levi subgroup Mr

isomorphic to GLr × Sp2l−2r. Using the normalization in [Sh10], the group XSp2lMr

of all

continues homomorphisms from Mr(A) to C×, which is trivial on Mr(A)1 (see [MW95]),

will be identified with C by s→ λs. Let Pr be the pre-image of Pr in Sp2l.

Let ∆(τ,m) be a Speh representation in the discrete spectrum of GL2mn(A). For any

φ ∈ A(N2mn(A)M2mn(F )\Sp4mn(A))∆(τ,m), following [L76] and [MW95], an residual

65

Eisenstein series can be defined by

E(φ, s)(g) =∑

γ∈P2mn(F )\Sp4mn(F )

λsφ(γg).

It converges absolutely for real part of s large and has meromorphic continuation to

the whole complex plane C. By [JLZ12], this Eisenstein series has a simple pole at m2 ,

which is the right-most pole. Denote the representation generated by these residues at

s = m2 by E∆(τ,m). This residual representation is square-integrable.

Our first main result can be stated as follows.

Theorem 4.1.1. Assume that F is any number field.

(1) For any irreducible automorphic representation π of Sp4mn(A) which is nearly

equivalent to E∆(τ,m), [(2n)2m] is a maximal possible partition providing non-vanishing

Fourier coefficients for π.

(2)

nm(E∆(τ,m)) = [(2n)2m].

Part (1) of Theorem 4.1.1 can be easily deduced from the main results in [JL13a].

Part (2) of Theorem 4.1.1 was discussed by Ginzburg in [G08] with a quite sketchy

argument. A fully detailed proof will be given in Section 4.2.

It is expected that for any irreducible automorphic representation π of classical

groups, nm(π) is a singleton, see [Ka87], [MW87], [M98] and [GRS03]. If we assume

this, then Theorem 4.1.1 implies that [(2n)2m] is an upper bound for nm(π), for any

irreducible automorphic representation π of Sp4mn(A) which is nearly equivalent to

E∆(τ,m).

In [GRS03], for any irreducible cuspidal automorphic representation π of symplectic

groups or their double covers, Ginzburg, Rallis and Soudry found a maximal partition

p(π) which has all even pieces, providing non-vanishing Fourier coefficients for π.

We assume that F is not totally imaginary, and consider N Sp4mn , the set of irre-

ducible cuspidal automorphic representations π which are nearly equivalent to E∆(τ,m),

and

p(π) = [(2n)2m−1(2n1)s1(2n2)s2 · · · (2nk)sk ],

with 2n ≥ 2n1 > 2n2 > · · · > 2nk, k ≥ 1, which is less or equal than [(2n)2m].

66

N Sp4mn can be naturally decomposed into a disjoint union of two sets NSp4mn ∪N ′Sp4mn

, where NSp4mn consists of elements having a nonzero Fourier coefficient FJψ−1n−1

(for definition, see Section 3.2, or [GRS11] Section 3.2), whileN ′Sp4mnconsists of elements

having no nonzero Fourier coefficients FJψ−1n−1

.

For any φ ∈ A(N2kn(A)M2kn(F )\Sp4kn+2n(A))µψ∆(τ,k)⊗π, following [L76] and [MW95],

an residual Eisenstein series can be defined by

E(φ, s)(g) =∑

γ∈P2kn(F )\Sp4kn+2n(F )

λsφ(γg).

It converges absolutely for real part of s large and has meromorphic continuation to the

whole complex plane C. By similar argument as in [JLZ12], this Eisenstein series has a

simple pole at k+12 , which is the right-most pole. Denote the representation generated by

these residues at s = k+12 by E∆(τ,k)⊗π. This residual representation is square-integrable.

Let N ′Sp4(m−1)n+2n

(τ, ψ) be the set of irreducible genuine cuspidal automorphic repre-

sentations σ4(m−1)n+2n of Sp4(m−1)n+2n(A), which are nearly equivalent to the residual

representation E∆(τ,m−1)⊗π, have no nonzero Fourier coefficients FJψ1n−1

, and

p(σ4(m−1)n+2n) = [(2n)2(m−1)(2n1)s1(2n2)s2 · · · (2nk)sk ],

with 2n ≥ 2n1 > 2n2 > · · · > 2nk, k ≥ 1.

Let NSp4mn+2n

(τ, ψ) be the set of irreducible genuine cuspidal automorphic repre-

sentations σ4mn+2n of Sp4mn+2n(A), which are nearly equivalent to the residual repre-

sentation E∆(τ,m)⊗π, have a nonzero Fourier coefficient FJψ1n−1

, and

p(σ4mn+2n) = [(2n)2m(2n1)s1(2n2)s2 · · · (2nk)sk ],

with 2n ≥ 2n1 > 2n2 > · · · > 2nk, k ≥ 1.

For any σ4(m−1)n+2n ∈ N ′Sp4(m−1)n+2n

(τ, ψ), and for any

φ ∈ A(N2mn(A)M2mn(F )\Sp4mn+2n(A))µψτ⊗σ4(m−1)n+2n,

it is easy to see that the corresponding Eisenstein series has a simple pole at s =

m (see Section 5 of [GJS12]). Let Eτ,σ4(m−1)n+2nbe the residual representation of

Sp4mn+2n(A) generated by the corresponding residues. This residual representation

is square-integrable.

67

For any σ4mn ∈ N ′Sp4mn(τ, ψ), and for any

φ ∈ A(N2mn(A)M2mn(F )\Sp4mn+2n(A))τ⊗σ4mn ,

it is easy to see that the corresponding Eisenstein series has a simple pole at s = 2m+12

(see Section 5 of [GJS12]). Let Eτ,σ4mn be the residual representation of Sp4(m+1)n(A)

generated by the corresponding residues. This residual representation is square-integrable.

For any σ4mn ∈ NSp4mn(τ, ψ), let D4mn

2n,ψ−1(σ4mn) be the ψ−1-descent of σ4mn from

Sp4mn(A) to Sp4(m−1)n+2n(A) (for definition, see Section 3.2, or [GRS11] Section 3.2).

Note that by tower property (see Theorem 7.10 of [GRS11]), D4mn2n,ψ−1(σ4mn) is cuspidal.

For any σ4mn+2n ∈ NSp4mn+2n(τ, ψ), let D4mn+2n

2n,ψ1 (σ4mn+2n) be the ψ1-descent of

σ4mn+2n from Sp4mn+2n(A) to Sp4mn(A). Note that by tower property (see Theorem

7.10 of [GRS11]), D4mn+2n2n,ψ1 (σ4mn+2n) is also cuspidal.

Our second main result is that there are correspondences between NSp4mn(τ, ψ) and

N ′Sp4(m−1)n+2n

(τ, ψ), and between NSp4mn+2n

(τ, ψ) and N ′Sp4n(τ, ψ), as follows.

Theorem 4.1.2. Assume that F is a number field which is not totally imaginary.

(1) There is a surjective map

Ψ : NSp4mn(τ, ψ)→ N ′

Sp4(m−1)n+2n(τ, ψ)

σ4mn 7→ D4mn2n,ψ−1(σ4mn).

(2) If for any σ4(m−1)n+2n ∈ N ′Sp4(m−1)n+2n

(τ, ψ), Eτ,σ4(m−1)n+2nis irreducible, then

Ψ is also injective.

Note that the case of m = 1 is proved by Ginzburg, Jiang and Soudry in [GJS12].

We use the same idea here. Also note that, they actually proved more, since they knew

that E∆(τ,1) is irreducible (Theorem 2.5 of [GJS12]), so they were able to include it in

the domain of the map Ψ. Here, we avoid many difficult issues by letting the domain

of the map Ψ consist of only irreducible cuspidal automorphic representations.

Theorem 4.1.3. Assume that F is a number field which is not totally imaginary.

(1) There is a surjective map

Ψ : NSp4mn+2n

(τ, ψ)→ N ′Sp4n(τ, ψ)

68

σ4mn+2n 7→ D4mn+2n2n,ψ1 (σ4mn+2n).

(2) If for any σ4mn ∈ N ′Sp4mn(τ, ψ), Eτ,σ4mn is irreducible, then Ψ is also injective.

Due to the similarity of the proofs of Theorem 4.1.2 and Theorem 4.1.3, we only

give the proof for Theorem 4.1.2.

Theorem 4.1.2 and Theorem 4.1.3 together give us the following diagram about cor-

respondences between various sets of irreducible cuspidal automorphic representations:

...

Sp4mn−2n(A)

Sp4mn(A)

Sp4mn+2n(A)

...

...

↓ D4mn−2n2n,ψ1

N4mn−2n⋃N ′4mn−2n

↓ D4mn2n,ψ−1

N4mn⋃

N ′4mn

↓ D4mn+2n2n,ψ1

N4mn+2n⋃N ′4mn+2n

↓ D4mn+4n2n,ψ−1

...

In the above diagram, for short, we write that N4mn := NSp4mn , N ′4mn := N ′Sp4mn,

N4mn±2n := NSp4mn±2n

, and N ′4mn±2n := N ′Sp4mn±2n

.

Remark 4.1.4. In Theorem 4.1.2 and Theorem 4.1.3, we assume that F is a number

field which is not totally imaginary, the reason is that when F is a totally imaginary

number field, our construction will stop at some point, and can not go to higher levels.

The explicit explanation of this phenomenon will appear elsewhere.

From Theorem 4.1.1, for the residual representation E∆(τ,m), nm(E∆(τ,m)) = [(2n)2m].

From its proof, and by Lemma 2.6 [GRS03] or Lemma 3.1 [JL13b], we can see that it

has a nonzero Fourier coefficient attached to the partition [(2n)14mn−2n] with respect

69

to the character ψ[(2n)14mn−2n],−1. In Section 4.7, when F is any number field, we show

that if E∆(τ,m) is irreducible, then D4mn2n,ψ−1(E∆(τ,m)) is irreducible. The result can be

stated as follows.

Theorem 4.1.5. Assume that F is any number field.

(1) D4mn2n,ψ−1(E∆(τ,m)) is square-integrable and is in the discrete spectrum.

(2) Assume that E∆(τ,m) is irreducible, then D4mn2n,ψ−1(E∆(τ,m)) is also irreducible.

Note that in general, it is difficult to prove the irreducibility of certain descent

representations. The case of m = 1 of Theorem 4.1.5 was proved in Theorem 4.1 of

[GJS12], noting that by Theorem 2.5 of [GJS12], E∆(τ,1) is irreducible. Also note that,

the irreducibility of D4n2n,ψ−1(E∆(τ,1)) actually has already been proved by Jiang and

Soudry in [JS03], using different methods.

At the end of this overview, we discuss contents by sections. In Section 4.2, we will

show Theorem 4.1.1, whose proof is reduced to that of Lemma 4.2.4, which will be given

in Section 4.3. The Section 4.4, we will prove Part (1) of Theorem 4.1.2, whose proof is

reduced to that of Theorem 4.4.6, which will be given in Section 4.5. In Section 4.6, we

completes the proof of Theorem 4.1.2, by proving its Part (2). In Section 4.7, we prove

Theorem 4.1.5. We assume that F is not totally imaginary only in Sections 4.4–4.6.

4.2 Proof of Theorem 4.1.1

In this section, we prove Theorem 4.1.1, which is done by proving the following two

theorems. The first one can be easily deduced from the main results in [JL13a]. The

second one actually has already been proved by Ginzburg in [G08] with a quite sketchy

argument. To be complete, we give a fully detailed proof here. F is any number field

in this section, Sections 4.3 and 4.7.

Theorem 4.2.1. For any irreducible automorphic representation π of Sp4mn(A) which

is nearly equivalent to E∆(τ,m), [(2n)2m] is a maximal possible partition providing non-

vanishing Fourier coefficients for π.

Proof. This theorem can be easily deduced from the main results in [JL13a], by looking

at its unramified components. Details are omitted here.

70

Theorem 4.2.2 (Ginzburg, Theorem 1 [G08]).

nm(E∆(τ,m)) = [(2n)2m].

Proof. By Theorem 4.2.1, we only have to show that E∆(τ,m) has a nonzero Fourier

coefficient attached to [(2n)2m].

We will prove this by induction on m. For m = 1, this is proved in the book

[GRS11]. Note that when m = 1, E∆(τ,1) has a nonzero Fourier coefficient attached

to the partition [(2n)12n], and the descent to Sp2n(A) is generic (see Theorem 3.1 of

[GRS11]). Therefore, E∆(τ,1) has a nonzero Fourier coefficient attached to the composite

partition [(2n)12n] ◦ [(2n)], which implies that E∆(τ,1) has a nonzero Fourier coefficient

attached to the partition [(2n)2] by Lemma 2.6 of [GRS03], or Lemma 3.1 of [JL13b].

For definition of composite partitions, we refer to Section 1 of [GRS03].

Now we assume that the theorem is true for the case of m−1, and consider the case

of m ≥ 2.

Take any ϕ ∈ E∆(τ,m), its Fourier coefficients attached to p = [(2n)2m] are of the

following forms

ϕψp,a(g) =

∫[Vp,2]

ϕ(vg)ψ−1p,a(v)dv, (4.1)

where a = {a1, a2, . . . , a2m} ⊂ (F ∗/(F ∗)2)2m. For definitions of the unipotent group

Vp,2 and its character ψp,a, see Section 3.1 or Section 2 of [JL13b].

Assume that T is the maximal split torus in Sp4mn, consists of elements

t = diag(t1, t2, . . . , t2mn, t−12mn, . . . , t

−12 , t−1

1 ).

Let ω1 be the Weyl element of Sp4mn, sending elements t ∈ T to the following torus

elements:

t′ = diag(t(0), t(1), t(2), . . . , t(n), t(n),∗, . . . , t(2),∗, t(1),∗, t(0),∗), (4.2)

where

t(0) =diag(t1, tn+1, t2, tn+2, . . . , ti, tn+i, . . . , tn, t2n)

t(j) =diag(t2n+j , t3n+j , . . . , tin+j , . . . , t(2m−1)n+j),

for 1 ≤ j ≤ n.

71

Note that Sp4(m−1)n can be embedded into Sp4mn via g 7→ diag(I2n, g, I2n). Iden-

tify Sp4(m−1)n with its image under this embedding. Denote the restriction of ω1 to

Sp4(m−1)n by ω′1.

Conjugating cross by ω1, the Fourier coefficient ϕψp,a becomes:∫[Up,2]

ϕ(uω1g)ψω1p,a(u)−1du, (4.3)

where Up,2 = ω1Vp,2ω−11 , and ψω1

p,a(u) = ψp,a(ω−11 uω1).

Now, we describe the structure of elements in Up,2. Any element in Up,2 has the

following form:

u =

z2n q1 q2

0 u′ q∗1

0 0 z∗2n

I2n 0 0

p1 I(4m−4)n 0

0 p∗1 I2n

, (4.4)

where z2n ∈ V2n , the unipotent radical of the parabolic Q2n of GL2n with Levi iso-

morphic to GL×n2 ; u′ ∈ U[(2n)2m−2],2 := ω′1V[(2n)2m−2],2ω′−11 ; q1 ∈ M2n×(4m−4)n, p1 ∈

M(4m−4)n×2n, satisfy certain conditions, which we do not specify at this moment;

q2 ∈ M(2n)×(2n), such that qt2v2n − v2nq2 = 0, where v2n is a matrix only with ones

on the second diagonal. Note that

ψω1p,a(

z2n q1 q2

0 I(4m−4)n q∗1

0 0 z∗2n

)

=ψ(z2n(1, 3) + · · ·+ z2n(i, i+ 2) + · · ·+ z2n(2n− 2, 2n))

·ψ(a1q2(2n− 1, 2) + a2q2(2n, 1)),

where a1, a2 come from the a = {a1, a2, . . . , a2m} occurred in the Fourier coefficient

ϕψp,a .

Since to show that E∆(τ,m) has a nonzero Fourier coefficient attached to the partition

p = [(2n)2m], we only need to show that it has a nonzero Fourier coefficient ϕψp,a for

some a, we consider the following special type of a:

a = {1,−1, a3, . . . , a2m},

where a3, . . . , a2m are arbitrary elements in F ∗/(F ∗)2.

72

Let A =

(1 −1

1 1

), and ε = diag(A, . . . , A; I(4m−4)n;A∗, . . . , A∗), as in (2.31) of

[GJS12]. Conjugating cross the integral in (4.3) by ε, it becomes:∫[Uεp,2]

ϕ(uεω1g)ψω1,εp,a (u)−1du, (4.5)

where U εp,2 = εUp,2ε−1 whose elements have the same structure as Up,2 (see (4.4)), and

ψω1,εp,a (u) = ψω1

p,a(ε−1uε).

Note that now

ψω1,εp,a (

z2n q1 q2

0 I(4m−4)n q∗1

0 0 z∗2n

)

=ψ(z2n(1, 3) + · · ·+ z2n(i, i+ 2) + · · ·+ z2n(2n− 2, 2n))

·ψ(q2(2n− 1, 1)).

Note that the a in (2.18) and (2.35) of [GJS12] is −1 here.

Let ν be the following Weyl element of Sp4n which is defined on Page 14 of [GJS12],

also in (4.9) of [GRS99]:

νi,2i−1 = 1, i = 1, . . . , 2n,

ν2n+i,2i = −1, i = 1, . . . , n,

ν2n+i,2i = 1, i = n+ 1, . . . , 2n,

νi,j = 0, otherwise.

(4.6)

Write ν as

(ν1 ν2

ν3 ν4

), where νi’s are of size 2n× 2n.

Let ω2 =

ν1 ν2

I(4m−4)n

ν3 ν4

, a Weyl element of Sp4mn. Conjugating cross the

integral in (4.5) by ω2, it becomes:∫[Uε,ω2p,2 ]

ϕ(uω2εω1g)ψω1,ε,ω2p,a (u)−1du, (4.7)

where U ε,ω2p,2 = ω2U

εp,2ω

−12 , and ψω1,ε,ω2

p,a (u) = ψω1,εp,a (ω−1

2 uω2).

73

Now let’s describe the structure of elements in U ε,ω2p,2 . Any element in U ε,ω2

p,2 has the

following form:

u =

z2n q1 q2

0 u′ q∗1

0 0 z∗2n

I2n 0 0

p1 I(4m−4)n 0

p2 p∗1 I2n

, (4.8)

where z2n ∈ V2n, the standard maximal unipotent subgroup of GL2n as before; u′ ∈U[(2n)2m−2],2 := ω′1V[(2n)2m−2],2ω

′−11 ; q1 ∈ M2n×(4m−4)n, with q1(i, j) = 0, for j ≤ (2m −

2)i; q2, p2 ∈ M(2n)×(2n), with q2(i, j) = p2(i, j) = 0, for j ≤ i; p1 ∈ M(4m−4)n×2n, with

p1(i, j) = 0, for i ≥ (2m− 2)(j − 1).

Note that now

ψω1,ε,ω2p,a (

z2n q1 q2

0 I(4m−4)n q∗1

0 0 z∗2n

)

=ψ(z2n(1, 2) + · · ·+ z2n(n, n+ 1)− z2n(n+ 1, n+ 2)− · · · − z2n(2n− 1, 2n)).

(4.9)

To proceed, we need to define some unipotent subgroups. Let R1i =

∏ij=1R

1i,j , for

1 ≤ i ≤ n, where R1i,j =

∏2m−2s=1 Xαij,s

, and αij,s = ei − e2n+(2m−2)(i−j+1)−s+1. Let

R1i =

∏ij=1R

1i,j , for n + 1 ≤ i ≤ 2n − 1, where R1

i,j =∏2m−2s=1 Xαij,s

, and αij,s =

ei − e2n+(2m−2)(i−j+1)−s+1, if j ≥ i − n + 1; R1i,j =

∏2m−2s=1 Xαij,s

, and αij,s = ei +

e2mn−(2m−2)(i−n)+(2m−1)(j−1)+s, if j ≤ i− n.

Let C1i =

∏ij=1C

1i,j , for 1 ≤ i ≤ n, where C1

i,j =∏2m−2s=1 Xβij,s

, and βij,s =

e2n+(2m−2)(i−j+1)−s+1 − ei+1. Let C1i =

∏ij=1C

1i,j , for n + 1 ≤ i ≤ 2n − 1, where

C1i,j =

∏2m−2s=1 Xβij,s

, and βij,s = e2n+(2m−2)(i−j+1)−s+1 − ei+1, if j ≥ i − n + 1; C1i,j =∏2m−2

s=1 Xβij,s, and βij,s = −e2mn−(2m−2)(i−n)+(2m−1)(j−1)+s − ei+1, if j ≤ i− n.

Let R2i =

∏ij=1Xαij

, for 1 ≤ i ≤ n, where αij = ei + e2n−i+j , and R2i =

∏2n−ij=1 Xαij

,

for n + 1 ≤ i ≤ 2n − 1, where αij = ei + ei+j . Let C2i =

∏ij=1Xβij

, for 1 ≤ i ≤ n,

where βij = −e2n−i+j − ei+1, and C2i =

∏2n−ij=1 Xβij

, for n + 1 ≤ i ≤ 2n − 1, where

βij = −ei+j − ei+1.

Now, we are ready to apply Lemma 3.1.3 repeatedly to the integration over

2n−1∏i=1

C2i C

1i .

74

We will consider the following pairs of groups:

(R21R

11, C

21C

11 ), . . . , (R2

nR1n, C

2nC

1n);

(R2n+1, C

2n+1), (R1

n+1, C1n+1); . . . ; (R2

2n−1, C22n−1), (R1

2n−1, C12n−1).

Let U ε,ω2p,2 be the subgroup of U ε,ω2

p,2 with p1 and p2-parts zero. Then, U ε,ω2p,2 =

U ε,ω2p,2

∏2n−1i=1 C2

i C1i . Let W = U ε,ω2

p,2 W = U ε,ω2p,2 , and ψW = ψω1,ε,ω2

p,a .

First, we apply Lemma 3.1.3 to (R21R

11, C

21C

11 ). For this, we need to consider the

quadruple (W∏2n−1i=2 C2

i C1i , ψW , Xα1

1R1

1,1, Xβ11C1

1,1). It is easy to see that this quadruple

satisfies all the conditions for this lemma. By this lemma, the integral in (4.7) is non-

vanishing if and only if the following integral is non-vanishing:∫[R2

1R11W

∏2n−1i=2 C2

i C1i ]ϕ(rwcω2εω1g)ψW (w)−1dcdwdr. (4.10)

Note that R21 = Xα1

1, and R1

1 = R11,1.

Then we apply Lemma 3.1.3 to (R22R

12, C

22C

12 ). For this, we need to consider the

following sequence of quadruples:

(R21R

11W

2n−1∏i=3

C2i C

1iXβ2

2R1

2,2, ψW , Xα21R1

2,1, Xβ21C1

2,1),

(R21R

11Xα2

1R1

2,1W2n−1∏i=3

C2i C

1i , ψW , Xα2

2R1

2,2, Xβ22C1

2,2).

Applying this lemma twice, we get that the integral in (4.10) is non-vanishing if and

only if the following integral is non-vanishing:∫[R2

1R22R

11R

12W

∏2n−1i=3 C2

i C1i ]ϕ(rwcω2εω1g)ψW (w)−1dcdwdr. (4.11)

Then we continue the above procedure, applying Lemma 3.1.3 to pairs

(R23R

13, C

23C

13 ), . . . , (R2

nR1n, C

2nC

1n).

For the pair (R2nR

1n, C

2nC

1n), we need to consider the following sequence of quadruples:

(

n−1∏i=1

R2iR

1i W

2n−1∏i=n+1

C2i C

1i

n∏s=2

Xβns R1n,s, ψW , Xαn1

R1s,1, Xβn1

C1n,1),

· · · ,

(n−1∏i=1

R2iR

1i

n−1∏s=1

XαlsR1n,sW

2n−1∏i=n+1

C2i C

1i , ψW , XαnnR

1n,n, XβnnC

1n,n).

75

Applying Lemma 3.1.3 repeatedly, we get that the integral in (4.11) is non-vanishing if

and only if the following integral is non-vanishing:∫[∏ni=1R

2iR

1i W

∏2n−1i=n+1 C

2i C

1i ]ϕ(rwcω2εω1g)ψW (w)−1dcdwdr. (4.12)

Before applying Lemma 3.1.3 to pairs (R2s, C

2s ), (R1

s, C1s ), s = n+1, n+2, . . . , 2n−1,

we need to take Fourier expansion along the one-dimensional root subgroup Xes+es .

And then we need to consider the pair (R2s, C

2s ) first, then (R1

s, C1s ).

For example, for s = n + 1, we first take the Fourier expansion of the integral in

(4.12) along the one-dimensional root subgroup Xes+es . Under the action of GL1, we get

two kinds of Fourier coefficients corresponding to the two orbits of the dual of [Xes+es ]:

the trivial one and the non-trivial one. For the Fourier coefficients attached to the

non-trivial orbit, we can see that there is an inner integral ϕ[(2n+2)14mn−2n−2],{a}, which

is identically zero by Theorem 4.2.1. Therefore only the Fourier coefficient attached to

the trivial orbit, which actually equals to the integral in (4.12), survives.

Then, we can apply Lemma 3.1.3 to pairs (R2n+1, C

2n+1), (R1

n+1, C1n+1). We need to

consider the following sequence of quadruples:

(n∏i=1

R2iR

1i WXen+1+en+1

2n−1∏i=n+2

C2i

2n−1∏t=n+1

C1t

n−1∏s=2

Xβns , ψW , Xαn+11

, Xβn+11

),

· · · ,

(n∏i=1

R2iR

1i

n−2∏s=1

Xαns WXen+1+en+1

2n−1∏i=n+2

C2i

2n−1∏t=n+1

C1t , ψW , Xαn+1

n−1, Xβn+1

n−1),

(n∏i=1

R2iR

1iR

2n+1WXen+1+en+1

2n−1∏i=n+2

C2i

2n−1∏t=n+2

C1t

n+1∏s=2

C1n+1,s, ψW , R

1n+1,1, C

1n+1,1),

· · · ,

(n∏i=1

R2iR

1iR

2n+1

n∏s=1

R1n+1,sWXen+1+en+1

2n−1∏i=n+2

C2i

2n−1∏t=n+2

C1t , ψW , R

1n+1,n+1, C

1n+1,n+1),

Applying Lemma 3.1.3 2n times, we get that the integral in (4.12) is non-vanishing if

and only if the following integral is non-vanishing:∫[∏n+1i=1 R2

iR1i WXen+1+en+1

∏2n−1i=n+2 C

2i C

1i ]ϕ(rwcω2εω1g)ψW (w)−1dcdwdr. (4.13)

76

After repeating the above procedure to the pairs (R2s, C

2s ), (R1

s, C1s ), s = n + 1, n +

2, . . . , 2n−1, we get that the integral in (4.13) is non-vanishing if and only if the following

integral is non-vanishing:∫[∏2n−1i=1 R2

iR1i W

∏2n−1j=n+1 Xej+ej ]

ϕ(uω2εω1g)ψW (w)−1du. (4.14)

Now, let’s see the structure of elements in∏2n−1i=1 R2

iR1i W

∏2n−1j=n+1Xej+ej . Any element

in∏2n−1i=1 R2

iR1i W

∏2n−1j=n+1Xej+ej has the following form:

w =

z2n q1 q2

0 u′ q∗1

0 0 z∗2n

,

where z2n ∈ V2n, the standard standard maximal unipotent subgroup of GL2n, u′ ∈U[(2n)2m−2],2 := ω′1V[(2n)2m−2],2ω

′−11 , q1 ∈ M(2n)×(4m−4)l, with the last row zero, q2 ∈

M(2n)×(2n), with q2(2n, 1) = 0. And by (4.9), the restriction of ψW to the z2n-part gives

a Whittaker character of GL2n. Note that ψW (

z2n 0 0

0 I4mn−4n 0

0 0 z∗2n

) = ψ(z2n(1, 2) +

· · ·+ z2n(n, n+ 1)− z2n(n+ 1, n+ 2)− · · · − z2n(2n− 1, 2n)).

It is clear that the integral in 4.14 is non-vanishing if and only if the following integral

is non-vanishing: ∫[∏2n−1i=1 R2

iR1i W

∏2n−1j=n+1Xej+ej ]

ϕ(uω2εω1g)ψ′W (w)−1du, (4.15)

where ψ′W (

z2n 0 0

0 I4mn−4n 0

0 0 z∗2n

) = ψ(∑2n−1

i=1 z2n(i, i+ 1)).

Then it is easy to see that the integral in (4.15) has an inner integral which is exactly

ϕψN

12n−1 , using notation in Lemma 4.2.6. On the other hand, we know that by Lemma

4.2.6, ϕψN

12n−1 = ϕψN

12n . Therefore, the integral in (4.14) becomes∫[U ]ϕ(uω2εω1g)ψ

U(u)−1du, (4.16)

77

where any element in U has the following form:

u = u(z2l, u′, q1, q2) =

z2n q1 q2

0 u′ q∗1

0 0 z∗2n

,

where z2n ∈ V2n, the standard standard maximal unipotent subgroup of GL2n, u′ ∈U[(2n)2m−2],2 := ω′1V[(2n)2m−2],2ω

′−11 , q1 ∈ M(2n)×(4m−4)n, q2 ∈ M(2n)×(2n), such that

qt2v2n − v2nq2 = 0, where v2n is a matrix only with ones on the second diagonal.

Hence, the integral in (4.16) can be written as∫u(z2n,u′,0,0)

ϕ(uω2εω1g)P2nψU (u)−1du, (4.17)

where ϕP2n is the constant term of ϕ along the parabolic subgroup P2n = M2nU2n of

Sp4mn with Levi isomorphic to GL2n × Sp(4m−4)n.

By Lemma 4.2.4, there is an automorphic function

f ∈ A(N2n(A)M2n(F )\Sp4mn(A))τ |·|−

2m−12 ⊗E∆(τ,m−1)

,

such that

ϕ(g)P2n = f(g), ∀g ∈ Sp4mn(A).

Therefore, ϕ(uω2εω1g)P2n is an automorphic form in τ |·|−2m−1

2 ⊗ E∆(τ,m−1). Note that

the restriction of ψU

to the z2n-part gives us a Whittaker coefficient, and the restriction

to the u′-part gives a Fourier coefficient of E∆(τ,m−1) attached to the partition [(2n)2m−2]

up to a conjugation of the Weyl element ω′1. On the other hand, τ is generic, and by

induction assumption, E∆(τ,m−1) has a nonzero Fourier coefficient attached to the parti-

tion [(2n)2m−2]. Therefore, we can make the conclusion that E∆(τ,m) has a nonzero ψp,a-

Fourier coefficient attached to the partition [(2n)2m], for some a = {1,−1, a3, . . . , a2m},ai ∈ F ∗/(F ∗)2, for 3 ≤ i ≤ 2m. Hence, O(E∆(τ,m)) = [(2n)2m].

Since the proof of Lemma 4.2.4 will be given in the next section, we have proved the

theorem, up to Lemma 4.2.4.

Remark 4.2.3. By similar argument, we can also prove that for the residual represen-

tation E∆(τ,k)⊗π of Sp4kn+2n(A),

nm(E∆(τ,k)⊗π) = [(2n)2k+1].

We will not give the details here.

78

The following lemma is an Sp-analogue of Lemma 2.4.1. It gives formulas for certain

constant terms of automorphic forms in E∆(τ,m). The proof of this lemma will be given

in the next section.

Lemma 4.2.4. Let P2ni = M2niN2ni, 1 ≤ i ≤ m be the parabolic subgroup of Sp4mn

with Levi part M2ni∼= GL2ni × Sp4n(m−i). Let ϕ be an arbitrary automorphic form in

E∆(τ,m). Denote by ϕ(g)P2ni the constant term of ϕ along P2ni. Then for 1 ≤ i ≤ m−1,

there is an automorphic function

f ∈ A(N2ni(A)M2ni(F )\Sp4mn(A))∆(τ,i)|·|−

2m−i2 ⊗E∆(τ,m−i)

,

such that

ϕ(g)P2ni = f(g),∀g ∈ Sp4mn(A).

For i = m, there is an automorphic function

f ∈ A(N2mn(A)M2mn(F )\Sp4mn(A))∆(τ,m)|·|−

m2,

such that

ϕ(g)P2mn = f(g), ∀g ∈ Sp4mn(A).

Next, we prove two important lemmas, which can be viewed as an Sp-analogue of

Lemmas 2.6.1 and 2.6.2.

Lemma 4.2.5. Let N12mn be the maximal unipotent subgroup of Sp4mn consists of upper

triangular matrices. For p ≥ 2n, define a character of N12mn as follows:

ψεp(n) :=ψ(n1,2 + · · ·np−1,p + np,p+1)

·ψ(ε1np+1,p+2 + · · ·+ ε2mn−pn2mn,2mn+1),(4.18)

where n ∈ N12mn, εi ∈ {0, 1}, for 1 ≤ i ≤ 2mn − p − 1, ε2mn−p ∈ F/(F ∗)2, ε =

{ε1, . . . , ε2mn−p}. Then for any automorphic form ϕ ∈ E∆(τ,m), the following ψεp-Fourier

coefficient is identically zero:

ϕψεp(g) :=

∫[N12mn ]

ϕ(ng)ψεp(n)−1dn ≡ 0. (4.19)

79

Proof. If εi 6= 0, ∀1 ≤ i ≤ 2mn− p, then ψεp is a generic character of Gn. Since E∆(τ,m)

is not generic, it has no nonzero ψεp-Fourier coefficients, i.e., ϕψ

εp ≡ 0, ∀ϕ ∈ E∆(τ,m).

Assume that 1 ≤ i ≤ 2mn− p is the first number such that εi = 0. Then ϕψεp has an

inner integral ϕPp+i , which is the constant term of ϕ along Pp+i, the parabolic subgroup

of Sp4mn with Levi isomorphic to GLp+i × Sp4mn−2(p+i). Note that p+ i > 2n.

By the cuspidal support of E∆(τ,m), we can see that ϕPp+i = 0 unless p + i = 2ns

with 2 ≤ s ≤ m. By Lemma 4.2.4, there is an automorphic function

f ∈ A(N2ns(A)M2ns(F )\Sp4mn(A))∆(τ,s)|·|−

2m−s2 ⊗E∆(τ,b−s)

,

such that

ϕ(g)P2ns = f(g),∀g ∈ Sp4mn(A),

Therefore, after taking the constant term, ϕ(g)P2ns is an automorphic function over

GL2ns(A) × Sp4n(m−s)(A). Note that the character ψ(n1,2 + · · · + n2ns−1,2ns) gives a

Whittaker character of GL2ns. Since by Proposition 2.2.1, ∆(τ, s) is not generic for

s > 1, i.e., it has no nonzero Whittaker Fourier coefficients, we conclude that ϕψεp ≡ 0.

This completes the proof of the lemma.

Lemma 4.2.6. Let N1p be the unipotent radical of the parabolic subgroup P1p of Sp4mn

with Levi part isomorphic to GL×p1 × Sp4mn−2p. Let

ψN1p(n) := ψ(n1,2 + · · ·+ np,p+1),

and

ψN1p(n) := ψ(n1,2 + · · ·+ np−1,p),

be two characters of N1p. For any automorphic form ϕ ∈ E∆(τ,m), define ψN1pand

ψN1p-Fourier coefficients as follows:

ϕψN1p (g) :=

∫[N1p ]

ϕ(ng)ψN1p(n)−1dn, (4.20)

ϕψN1p (g) :=

∫[N1p ]

ϕ(ng)ψN1p(n)−1du. (4.21)

Then ϕψN1p ≡ 0,∀p ≥ 2n; and ϕψN

12n−1 = ϕψN

12n .

80

Proof. First we assume that p ≥ n. Let Xep+1+ep+1 be the root subgroup corresponding

to the root ep+1 + ep+1. Since the conjugating action of Xep+1+ep+1 normalizes N1p ,

and preserves the character ψN1p, we can take the Fourier expansion of ϕψN1p along

Xep+1+ep+1 .

Under the action of GL1, we will get Fourier coefficients corresponding the two

orbits of the dual of Xep+1+ep+1 : the trivial one and the non-trivial one. Note that the

non-trivial orbit gives us Fourier coefficients which are exactly the Fourier coefficients

attached to [(2(p + 1))14mn−2q−2], with p + 1 > n. On the other hand, by Theorem

4.2.1, all the Fourier coefficients attached to the non-trivial orbit vanish, and the Fourier

coefficient attached to the trivial orbit, i.e., the Fourier coefficient with respect to the

trivial character, survives. Hence, ϕψN1p becomes:∫[Xep+1+ep+1 ]

∫[N1p ]

ϕ(nxg)ψN1p(n)−1dndx. (4.22)

Let Ri, p+ 1 ≤ i ≤ 2mn− 1 be the following subgroup of N12mn :

Ri :={n ∈ N12mn |n(j, l) = 0,∀(j, l) 6= (p+ 1, w),

p+ 2 ≤ w ≤ 4mn− p− 1}.

Since Rp+1 normalizes the group Xep+1+ep+1N1p , and preserves the character ψN1p,

we can take the Fourier expansion of the integral in (4.22) along Rp+1. Since the

subgroup Ip+1 × Sp4mn−2p−2 (image of Sp4mn−2p−2 in Sp4mn under the embedding

g 7→ diag(Ip+1, g, Ip+1)) of Sp4mn acts on the dual space of [Rp+1] with two orbits: the

trivial one and the non-trivial one, the integral in (4.22) becomes:∑γ

ϕψN

1p+1 (γg) + ϕψN1p (g), (4.23)

where γ is in some quotient space which we will not specify here.

Then we continue the above process of Fourier expansion for the two kinds of Fourier

coefficients ϕψN

1p+1 and ϕψN

1p+1 along the pair of groups (Xep+2+ep+2 , Rp+2). We will

get four kinds of Fourier coefficients:∫[N1p+2 ]

ϕ(ng)ψεN1p+2

(n)−1dn,

81

where

ψεN1p+2

(n) :=ψ(n1,2 + · · ·np−1,p + np,p+1)

·ψ(ε1np+1,p+2 + ε2np+2,p+3),

and ε = {ε1, ε2}, ε1, ε2 ∈ {0, 1}. Then we can continue the above process of Fourier

expansion for each of these four kinds of Fourier coefficients along the pair of groups

(Xep+3+ep+3 , Rp+3). We will get the following kinds of Fourier coefficients:∫[N1p+3 ]

ϕ(ng)ψεN1p+3

(n)−1dn,

where

ψεN1p+3

(n) :=ψ(n1,2 + · · ·np−1,p + np,p+1)

·ψ(ε1np+1,p+2 + ε2np+2,p+3 + ε2np+3,p+4),

and ε = {ε1, ε2, ε3}, ε1, ε2, ε3 ∈ {0, 1}.Keep repeating the above procedure, we will get Fourier coefficients of the following

form: ϕψεp , with ε = {ε1, . . . , ε2mn−p}, εi ∈ {0, 1}, 1 ≤ i ≤ 2mn − p − 1, and ε2mn−p ∈

F/(F ∗)2.

By Lemma 4.2.5, all Fourier coefficients of this kind are zero, if p ≥ 2n. Therefore,

ϕψN1p ≡ 0, if p ≥ 2n.

For p = 2n− 1, by (4.23), we can see that ϕψN

12n−1 = ϕψN

12n , since ϕψN

12n ≡ 0 by

the above discussion.

This completes the proof of the lemma.

4.3 Proof of Lemma 4.2.4

In this section, we will prove Lemma 4.2.4. Before we start, we recall the definition of

the Eisenstein series in Section 1:

E(φ, s)(g) =∑

γ∈P2mn(F )\Sp4mn(F )

λsφ(γg),

where φ ∈ A(N2mn(A)M2mn(F )\Sp4mn(A))∆(τ,m). Assume that ϕ = Ress=m2E(φ, s).

82

To compute ϕP2ni = (Ress=m2E(φ, s))P2ni , we use the fact that the residue operator

and the constant term operator are interchangeable. So, first, we are going to calculate

the constant term of E(φ, s) along P2ni. We follow the calculation in Section 2 of

[JLZ12].

EP2ni(φ, s)(g)

=

∫N2ni(F )\N2ni(A)

E(φ, s)(ug)du

=∑

ω−1∈P2mn\Sp4mn/P2ni

∑γ∈Mω

2ni(F )\M2ni(F )

∫[Nω

2ni]

∫N2ni,ω(A)

λsφ(ω−1γu′u′′g)du′du′′

(4.24)

where we define Mω2ni := ωP2mnω

−1∩M2ni and Nω2ni := ωP2mnω

−1∩N2ni and [Nω2ni] :=

Nω2ni(F )\Nω

2ni(A). Note that the unipotent radical N2ni can be decomposed as a prod-

uct N2ni,ωNω2ni, where N2ni,ω satisfies N2ni,ω ∩ Nω

2ni = {1} and N2ni = N2ni,ωNω2ni =

Nω2niN2ni,ω.

Using the explicit calculation (see Chapter 4 of [GRS11]) about the generalized

Bruhat decomposition P2mn\Sp4mn/P2ni, we can see that all the double cosets are killed

by the cuspidal support of the Eisenstein series except two, which have the following

representatives: ω0 = Id, and

ω1 =

0 0 I2ni 0

I2n(m−i) 0 0 0

0 0 0 I2n(m−i)

0 −I2ni 0 0

.

Then

EP2ni(φ, s)(g) = EP2ni(φ, s)ω0(g) + EP2ni(φ, s)ω1(g), (4.25)

where

EP2ni(φ, s)ωj (g)

=∑

γ∈Mωj2ni(F )\M2ni(F )

∫[N

ωj2ni]

∫N2ni,ωj

(A)λsφ(ωj

−1γu′u′′g)du′du′′,

j = 0, 1.

We will consider these two terms separately in the next two subsections.

83

4.3.1 ω0-term

Write elements in N2ni as follows:

u(X,Z,W ) =

I2ni X Z W

I2n(m−i) Z′

I2n(m−i) X′

I2ni

.

Note that P2mn∩M2ni\M2ni∼= P2n(m−i)\Sp4n(m−i), where P2n(m−i) is the parabolic

subgroup of Sp4n(m−i) with Levi isomorphic to GL2n(m−i) × 1Sp0 . Then the ω0-term of

the constant term can be written as

EP2ni(φ, s)ω0(g) =∑

γ∈P2n(m−i)(F )\Sp4n(m−i)(F )

∫[N2ni]

λsφ(γug)du. (4.26)

The integral can be calculated as follows:

∫[N2ni]

λsφ(γug)du

=

∫[N2ni]

λsφ(uγg)du

=

∫[M2ni×2n(m−i)]

∫[N2mn∩U2ni]

λsφ(u′u(X)γg)du′dX

=

∫[M2ni×2n(m−i)]

λsφ(u(X)γg)dX,

where u(X) = u(X, 0, 0) with X ∈M2ni×2n(m−i).

As in Section 2.2 of [JLZ12], the integral∫

[M2ni×2n(m−i)]λsφ(u(X)γg)dX can be

viewed as the constant term of the automorphic function in ∆(τ,m): x 7→ φ(diag(x, x∗)g),

along the maximal parabolic subgroup Q2ni,2n(m−i) of GL2mn with Levi isomorphic to

GL2ni ×GL2n(m−i). We will denote it by φQ2ni,2n(m−i) .

Let P2ni,2n(m−i) = M2ni,2n(m−i)N2ni,2n(m−i) be a standard parabolic subgroup of

Sp4mn, whose Levi M2ni,2n(m−i) ∼= GL2ni ×GL2n(m−i) × 1Sp0 .

The following lemma is parallel to Lemma 2.1 of [JLZ12].

Lemma 4.3.1. The constant term λsφQ2ni,2n(m−i) belongs to the following space

A(N2ni,2n(m−i)(A)M2ni,2n(m−i)(F )\Sp4mn(A))∆(τ,i)|·|s−

m−i2 ⊗∆(τ,m−i)|·|s+

i2⊗1Sp0

.

84

Proof. The proof is omitted here, since it is almost the same as that of Lemma 2.1 of

[JLZ12], word-by-word.

Therefore, by (4.26) and Lemma 4.3.1, we can see that EP2ni(φ, s)ω0 belongs to the

following space

A(N2ni(A)M2ni(F )\Sp4mn(A))∆(τ,i)|·|s−

m−i2 ⊗(∆(τ,m−i)|·|s+

i2 o1Sp0 )

.

Hence, the residue operator will kill EP2ni(φ, s)ω0 , since s = m2 is not a pole of the

Eisenstein series on Sp4n(m−i) with inducing data ∆(τ,m− i)| · |s+i2 ⊗ 1Sp0 .

4.3.2 ω1-term

Note that Nω12ni = {u(Z) = u(0, Z, 0) | Z ∈ M2ni×2n(m−i)}. and Mω1

2ni(F )\M2ni(F ) is

isomorphic to P2n(m−i)(F )\Sp4n(m−i)(F ).

Therefore, we have

EP2ni(φ, s)ω1(g)

=∑

γ∈P2n(m−i)(F )\Sp4n(m−i)

∫N2ni,ω1

(A)

∫[M2ni×2n(m−i)]

λsφ(ω−11 γu(Z)ug)dZdu

=∑

γ∈P2n(m−i)(F )\Sp4n(m−i)

∫N2ni,ω1

(A)

∫[M2n(m−i)×2ni]

λsφ(u′(X)ω−11 uγg)dXdu,

(4.27)

where

u′(X) =

I2n(m−i) X

I2ni

I2ni X ′

I2n(m−i)

for X ∈M2n(m−i)×2ni.

As in case of ω0, the integral∫

[M2n(m−i)×2ni]φ(u′(X)g)dX can be viewed as the con-

stant term of the automorphic function in ∆(τ,m): x 7→ φ(diag(x, x∗)g), along the max-

imal parabolic subgroup Q2n(m−i),2ni of GL2mn with Levi isomorphic to GL2n(m−i) ×GL2ni. We will denote it by φQ2n(m−i),2ni .

Let P2n(m−i),2ni = M2n(m−i),2niU2n(m−i),2ni be a standard parabolic subgroup of

Sp4mn, whose Levi M2n(m−i),2ni ∼= GL2n(m−i)×GL2ni× 1Sp0 . Then, by Lemma 4.3.1, the

constant term λsφQ2n(m−i),2ni belongs to the following space

A(N2n(m−i),2ni(A)M2n(m−i),2ni(F )\Sp4mn(A))∆(τ,m−i)|·|s−

i2⊗∆(τ,i)|·|s+

m−i2 ⊗1Sp0

.

85

Note that the outer integral in (4.27) is the intertwining operator corresponding to

ω1, which maps

A(N2n(m−i),2ni(A)M2n(m−i),2ni(F )\Sp4mn(A))∆(τ,m−i)|·|s−

i2⊗∆(τ,i)|·|s+

m−i2 ⊗1Sp0

to

A(N2ni,2n(m−i)(A)Mai,a(b−i)(F )\Sp4mn(A))∆(τ,i)|·|−(s+m−i

2 )⊗∆(τ,m−i)|·|s−i2⊗1Sp0

.

Note that τ is self-dual.

Therefore, by (4.27), and the the above discussion, we can see that EP2ni(φ, s)ω1

belongs to the following space

A(N2ni(A)M2ni(F )\Sp4mn(A))∆(τ,i)|·|−(s+m−i

2 )⊗(∆(τ,m−i)|·|s−i2 o1Sp0 )

.

And for 1 ≤ i ≤ m− 1, after taking the residue operator, Ress=m2EP2ni(φ, s)ω1 belongs

to the following space

A(N2ni(A)M2ni(F )\Sp4mn(A))∆(τ,i)|·|−

2m−i2 ⊗E∆(τ,m−i)

,

since s = m2 is the rightmost simple pole of the (normalized) Eisenstein series with

inducing data ∆(τ,m− i)| · |s−i2 ⊗ 1Sp0 , and it is not a pole of the intertwining operator

corresponding to ω1.

Therefore, for 1 ≤ i ≤ m− 1,

ϕP2ni

=(Ress=m2E(φ, s))P2ni

=Ress=m2

(EP2ni(φ, s))

=Ress=m2

(EP2ni(φ, s)ω1),

belongs to the following space

A(N2ni(A)M2ni(F )\Sp4mn(A))∆(τ,i)|·|−

2m−i2 ⊗E∆(τ,m−i)

.

For i = m, ϕP2mn = Ress=m2

(EP2mn(φ, s)ω1) belongs to the following space

A(N2mn(A)M2mn(F )\Sp4mn(A))∆(τ,m)|·|−

m2 ⊗1Sp0

,

86

since EP2mn(φ, s)ω1 belongs to the following space

A(N2mn(A)M2mn(F )\Sp4mn(A))∆(τ,m)|·|−s⊗1Sp0,

and s = m2 is a simple pole of the intertwining operator corresponding to ω1.

This completes the proof of Lemma 4.2.4.

4.4 Proof of Part (1) of Theorem 4.1.2

In this section, we will prove that Ψ is surjective, which comes from the computation

of composition of two descents

D4mn2n,ψ−1 ◦ D4mn+2n

2n,ψ1 (Eτ,σ4(m−1)n+2n),

where σ4(m−1)n+2n ∈ N ′Sp4(m−1)n+2n

(τ, ψ), and

D4mn+2n2n,ψ1 (Eτ,σ4(m−1)n+2n

) ⊂ NSp4mn(τ, ψ).

It turns out that there is a similar identity:

D4mn2n,ψ−1 ◦ D4mn+2n

2n,ψ1 (Eτ,σ4(m−1)n+2n) = σ4(m−1)n+2n,

as in Proposition 5.2 of [GJS12]. We will prove in the next section that Ψ is well-defined,

i.e., D4mn2n,ψ−1(σ4mn) is irreducible (see Theorem 4.4.6).

Note that, from this section to Section 6, we assume that F is a number field which

is not totally imaginary, unless specified.

To start, we begin with the following theorem which generalizes Theorem 6.3 of

[GRS11] and is true for any number field.

Theorem 4.4.1. Assume that F is any number field.

(1) Let µi, 1 ≤ i ≤ r, be characters of F ∗v , a ∈ F ∗v , then

FJψak−1(IndSp2n

Pm1,...,mkνα1χ1(detGLm1

)⊗ · · · ⊗ ναkχk(detGLmk ))

∼=IndSp2n−2k

Pm1−1,...,mk−1µψ−aν

α1χ1(detGLm1−1)⊗ · · · ⊗ ναkχk(detGLmk−1).(4.28)

87

(2) Let µi, 1 ≤ i ≤ r, be characters of F ∗v , a, b ∈ F ∗v , then

FJψbk−1(Ind

Sp2nPm1,...,mk

µψaνα1χ1(detGLm1

)⊗ · · · ⊗ ναkχk(detGLmk ))

∼=IndSp2n−2k

Pm1−1,...,mk−1χ baνα1χ1(detGLm1−1)⊗ · · · ⊗ ναkχk(detGLmk−1),

(4.29)

where χ ba

is a quadratic character of F ∗v defined by Hilbert symbol as follows: χ ba(x) =

( ba , x).

Proof. The proof is the same as Theorem 6.3 of [GRS11]. The key calculation is reduced

to that in Proposition 6.6 of [GRS11]. Explicitly,

γψaγψ−b

=γψaγψ−aχ ba

=χ ba.

Next, we prove the equality mentioned at the beginning of this section, which will

imply later that Ψ is surjective.

Theorem 4.4.2. (1)

D4mn2n,ψ−1 ◦ D4mn+2n

2n,ψ1 (Eτ,σ4(m−1)n+2n) 6= 0.

(2)

D4mn2n,ψ−1 ◦ D4mn+2n

2n,ψ1 (Eτ,σ4(m−1)n+2n) = σ4(m−1)n+2n.

Proof. The proof of Part (1) is similar to that of Theorem 4.2.2 and that of Theorem

2.1 of [GJS12]. The proof of Part (2) is similar to those of Theorem 5.1 and Proposition

5.2 of [GJS12].

Proof of Part (1). Note that descents D4mn+2n2n,ψ1 and D4mn

2n,ψ−1 are defined in Section

3.2 or Section 3.2 of [GRS11].

By Corollary 3.1.4 or Lemma 1.1 [GRS03], and the discussion at the end of Section

1 of [GRS03], D4mn2n,ψ−1 ◦D4mn+2n

2n,ψ1 (Eτ,σ4(m−1)n+2n) 6= 0 if and only if the following integral

is non-vanishing: ∫[Y2V[(2n)14mn−2n],2]

∫[Y1V[(2n)14mn],2]

ϕ(vv1g)

ψ−1[(2n)14mn],1

(v)ψ−1[(2n)14mn−2n],−1

(v1)dvdv1,

(4.30)

88

where ϕ ∈ Eτ,σ4(m−1)n+2n, g ∈ Sp4mn−2n(A), embedded into Sp4mn+2n(A) via the map

g 7→ diag(I2n, g, I2n), Y1, Y2 are the groups defined in (3.5) corresponding to the parti-

tions [(2n)14mn] and [(2n)14mn−2n] respectively.

Explicitly, let N1n be the unipotent radical of the parabolic P1n of Sp4mn+2n with

the Levi subgroup isomorphic to GLn1 × Sp4mn, then Y1V[(2n)14mn],2 is a subgroup of

N1n consists of elements v with vn,j = 0, for n + 1 ≤ j ≤ 2mn + n. ψ[(2n)14mn],1(v) =

ψ(∑n−1

i=1 vi,i+1 + vn,4mn+n+1).

Identify Sp4mn with its embedding into Sp4mn+2n via the map g 7→ diag(In, g, In).

Let N1n be the unipotent radical of the parabolic P1n of Sp4mn with the Levi subgroup

isomorphic to GLn1 × Sp4mn−2n, then Y1V[(2n)14mn−2n],2 is a subgroup of N1n consists of

elements v with vn,j = 0, for n + 1 ≤ j ≤ 2mn. ψ[(2n)14mn−2n],−1(v) = ψ(∑n−1

i=1 vi,i+1 −vn,4mn−n+1).

Let ω be the Weyl element of GL2n defined in (4.31) of [GRS99]:

ω2i,i = 1, i = 1, . . . , n,

ω2i−1,i+n = 1, i = 1, . . . , n,

ωi,j = 0, otherwise.

Let

ω1 =

ω

I4mn−2n

ω∗

∈ Sp4mn+2n(F ). (4.31)

We identify Sp4mn+2n(F ) with the subgroup Sp4mn+2n(F ) × 1 of Sp4mn+2n(A), as in

[GJS12].

Conjugating cross the integral in 4.30 by ω1, it becomes:∫[W1]

ϕ(wω1g)ψ−1W1

(w)dw, (4.32)

where

W1 = ω1Y2V[(2n)14mn−2n],2Y1V[(2n)14mn],2ω−11 ,

if ω−11 wω1 = v1v, v1 ∈ Y2V[(2n)14mn−2n],2, v ∈ Y1V[(2n)14mn],2, then

ψW1(w) = ψ[(2n)14mn],1(v)ψ[(2n)14mn−2n],−1(v1).

89

Note that the metaplectic cover splits over W1(A), and W1(A)× 1 is a subgroup of

Sp4mn+2n(A). We identify W1 with W1 × 1.

Elements in W1 are of the following form

w =

Z q1 q2

I4mn−2n q∗1

Z∗

, (4.33)

where q1(i, j) = 0, for i = 2n−1, 2n, 1 ≤ j ≤ 2mn−n, Z ∈ GL2n has the form (4.34) of

[GRS99]. Write Z as an n× n matrix of 2× 2 block matrices Z = ([Z]i,j), 1 ≤ i, j ≤ n,

then [Z]n,1 = · · · = [Z]n,n−1 = 0, [Z]n,n = I2; [Z]i,i is lower unipotent, for i < n; [Z]i,j

is lower triangular, for i < j; [Z]i,j is lower nilpotent, for j < i < n. And

ψW1(w) = ψ(

n−1∑i=1

tr([Z]i,i+1) + (q2(2n, 1)− q2(2n− 1, 2))). (4.34)

Let R1i =

∏ij=1R

1i,j , for 1 ≤ i ≤ n − 1, where R1

i,j = Xαi,j , the root subgroup

corresponding to the root αi,j = e2i − e2(j−1)+1. Let R1 =∏n−1i=1 R

1i . Actually R1 is

the subgroup of W1 consists of lower unipotent matrices. Write W1 = R1W1, with

R1 ∩ W1 = {1}.Let C1

i =∏ij=1C

1i,j , for 1 ≤ i ≤ n − 1, where C1

i,j = Xβi,j , the root subgroup

corresponding to the root βi,j = e2(j−1)+1 − e2(i+1). Let C1 =∏n−1i=1 C

1i .

We consider the quadruple

(W1

n−2∏i=1

R1i

n−1∏j=2

R1n−1,j , ψW1 , R

1n−1,1, C

1n−1,1). (4.35)

It is easy to see that this quadruple satisfies all the conditions for Lemma 3.1.3. Hence,

by Lemma 3.1.3, the integral in (4.32) is non-vanishing if and only if the following

integral is non-vanishing:∫[C1n−1,1W1

∏n−2i=1 R1

i

∏n−1j=2 R

1n−1,j ]

ϕ(cwrω1g)ψ−1W1

(w)drdwdc. (4.36)

90

We continue to consider the following sequence of quadruples:

(C1n−1,1W1

n−1∏i=1

R1i

n−1∏j=3

R1n−1,j , ψW1 , R

1n−1,2, C

1n−1,2),

(

2∏k=1

C1n−1,kW1

n−2∏i=1

R1i

n−1∏j=4

R1n−1,j , ψW1 , R

1n−1,3, C

1n−1,3),

· · · ,

(n−2∏k=1

C1n−1,kW1

n−2∏i=1

R1i , ψW1 , R

1n−1,n−1, C

1n−1,n−1).

(4.37)

Applying Lemma 3.1.3 (n− 1) times, we get that the integral in (4.36) is non-vanishing

if and only if the following integral is non-vanishing:∫[C1n−1W1

∏n−2i=1 R1

i ]ϕ(cwrω1g)ψ−1

W1(w)drdwdc. (4.38)

Then, we repeat the above procedure for the pairs

(R1n−2, C

1n−2), . . . , (R1

1, C11 ).

For example, after repeating the above procedure for the pairs

(R1n−2, C

1n−2), . . . , (R1

s+1, C1s+1),

we need the following sequence of quadruples for the pair (R1s, C

1s ):

(n−1∏i=s+1

C1i W1

s−1∏i=1

R1i

s∏j=2

R1s,j , ψW1 , R

1s,1, C

1s,1),

(

n−1∏i=s+1

C1i C

1s,1W1

s−1∏i=1

R1i

s∏j=3

R1s,j , ψW1 , R

1s,2, C

1s,2),

· · · ,

(n−1∏i=s+1

C1i

s−1∏k=1

C1s,kW1

s−1∏i=1

R1i , ψW1 , R

1s,s, C

1s,s).

(4.39)

After applying Lemma 3.1.3 s times, the integral in (4.38) is non-vanishing if and only

if the following integral is non-vanishing:∫[∏n−1i=s C

1i W1

∏s−1j=1 R

1j ]ϕ(cwrω1g)ψ−1

W1(w)drdwdc. (4.40)

91

After repeating the above procedure for all the pairs

(R1n−2, C

1n−2), . . . , (R1

1, C11 ),

we will see that the integral in (4.38) is non-vanishing if and only if the following integral

is non-vanishing: ∫[C1W1]

ϕ(cwω1g)ψ−1W1

(w)dwdc. (4.41)

Note that C1W1 = ω1V[(2n)214mn−2n],2ω−11 , and

ψC1W1

(cw) = ψW1(w) = ψ[(2n)214mn−2n],{−1,1}(ω−11 cwω1).

Let A =

(1 −1

1 1

), and ε = diag(A, . . . , A; I(4m−4)n;A∗, . . . , A∗), as in (2.31) of

[GJS12]. Conjugating cross the integral in (4.41) by ε, it becomes:∫[W2]

ϕ(wεω1g)ψ−1W2

(w)dw, (4.42)

where W2 = εC1W1ε−1, ψW2(w) = ψ

C1W1(ε−1wε).

Elements in W2 are of the following form

w =

Z q1 q2

I4mn−2n q∗1

Z∗

, (4.43)

where q1(i, j) = 0, for i = 2n− 1, 2n, 1 ≤ j ≤ 2mn− n, Z is in the unipotent radical of

the parabolic subgroup of GL2n with the Levi subgroup isomorphic to GLn2 . And

ψW2(w) = ψ(

2n−2∑i=1

Zi,i+2 + q2(2n− 1, 1)). (4.44)

Let ν =

(ν1 ν2

ν3 ν4

)be the Weyl element in (4.6). Let

ω2 =

ν1 ν2

I4mn−2n

ν3 ν4

, (4.45)

92

a Weyl element of Sp4mn+2n.

Conjugating cross the integral in (4.42) by ω2, it becomes:∫[W3]

ϕ(wω2εω1g)ψ−1W3

(w)dw, (4.46)

where W3 = ω2W2ω−12 , ψW3(w) = ψW2(ω−1

2 wω2).

Elements in W3 have the following form:

w =

Z q1 q2

0 I4mn−2n q∗1

0 0 Z∗

I2n 0 0

p1 I4mn−2n 0

p2 p∗1 I2n

, (4.47)

where Z is in the standard maximal unipotent subgroup of GL2n; q1(i, j) = 0, for

n+1 ≤ i ≤ 2n, and for i = n, 1 ≤ j ≤ 2mn−n; q2(i, j) = p2(i, j) = 0, for 1 ≤ j ≤ i ≤ 2n;

p1(i, j) = 0, for 1 ≤ j ≤ n, and for j = n+ 1, n+ 1 ≤ i ≤ 2n.

Let C be the unipotent subgroup consisting of elements of the following form:I2n 0 0

p1 I4mn−2n 0

0 p∗1 I2n

,

where p1(i, j) = 0 for any i, j, except j = n + 1, 1 ≤ i ≤ 2mn − n. Let R be the

unipotent subgroup consisting of elements of the following form:I2n q1 0

0 I4mn−2n q∗1

0 0 I2n

,

where q1(i, j) = 0 for any i, j, except i = n, 1 ≤ j ≤ 2mn− n.

Write W3 = CW3, with C ∩ W3 = {1}. Consider the quadruple (W3, ψW3 , R, C). It

is easy to see that this quadruple satisfies all the conditions of Lemma 3.1.3. Hence, by

Lemma 3.1.3, the integral (4.46) is non-vanishing if and only if the following integral is

non-vanishing: ∫[W4]

ϕ(wω2εω1g)ψ−1W4

(w)dw, (4.48)

where W4 = RW3, for w = rw′, r ∈ R, w ∈ W3, ψW4(rw′) = ψW3(w′).

93

Let R2i =

∏ij=1Xαij

, for 1 ≤ i ≤ n, where αij = ei + e2n−i+j , and R2i =

∏2n−ij=1 Xαij

,

for n + 1 ≤ i ≤ 2n − 1, where αij = ei + ei+j . Let C2i =

∏ij=1Xβij

, for 1 ≤ i ≤ n,

where βij = −e2n−i+j − ei+1, and C2i =

∏2n−ij=1 Xβij

, for n + 1 ≤ i ≤ 2n − 1, where

βij = −ei+j − ei+1.

For n+ 1 ≤ i ≤ 2n− 1, let R3i be the unipotent subgroup consisting of elements of

the following form: I2n q1 0

0 I4mn−2n q∗1

0 0 I2n

,

where q1(k, j) = 0 for any k, j, except k = i, 1 ≤ j ≤ 4mn− 2n. For n+ 1 ≤ i ≤ 2n− 1,

let C3i be the unipotent subgroup consisting of elements of the following form:

I2n 0 0

p1 I4mn−2n 0

0 p∗1 I2n

,

where p1(k, j) = 0 for any k, j, except j = i + 1, 1 ≤ k ≤ 4mn − 2n. Write W4 =∏2n−1i=1 C2

i

∏2n−1j=n+1C

3j W4, with

∏2n−1i=1 C2

i

∏2n−1j=n+1C

3j ∩ W4 = {1}.

Then, we apply Lemma 3.1.3 to the pairs (R21, C

21 ), (R2

2, C22 ), . . . , (R2

n, C2n). For ex-

ample, for (R2s, C

2s ), 1 ≤ s ≤ n, we need to consider the following sequence of quadruples:

(

s−1∏i=1

R2i

2n−1∏j=n+1

C3j W4

2n−1∏t=s+1

C2t

s∏l=2

Xβsl, ψW4 , Xβs1

, Xαs1),

(s−1∏i=1

R2iXαs1

2n−1∏j=n+1

C3j W4

2n−1∏t=s+1

C2t

s∏l=3

Xβsl, ψW4 , Xβs2

, Xαs2),

· · · ,

(

s−1∏i=1

R2i

s−1∏k=1

Xαsk

2n−1∏j=n+1

C3j W4

2n−1∏t=s+1

C2t , ψW4 , Xβss , Xαss).

After applying Lemma 3.1.3 to all the pairs (R21, C

21 ), (R2

2, C22 ), . . . , (R2

n, C2n), we get

that the integral (4.48) is non-vanishing if and only if the following integral is non-

vanishing: ∫[∏ni=1R

2i W4

∏2n−1t=n+1 C

2t C

3t ]ϕ(rwcω2εω1g)ψ−1

W4(w)dcdwdr, (4.49)

94

Next, we apply Lemma 3.1.3 to the pairs

(R2n+1, C

2n+1), (R3

n+1, C3n+1); · · · ; (R2

2n−1, C22n−1), (R3

2n−1, C32n−1).

Note that before applying Lemma 3.1.3 to each pair (R2s, C

2s ), n + 1 ≤ s ≤ 2n − 1, we

need to take the Fourier expansion along the one-dimensional root subgroup Xes+es , as

in the proof of Theorem 4.2.2.

For example, for s = n + 1, we first take the Fourier expansion of the integral

in (4.49) along the one-dimensional root subgroup Xes+es . Under the action of GL1,

we get two kinds of Fourier coefficients corresponding to the two orbits of the dual of

[Xes+es ]: the trivial one and the non-trivial one. For any Fourier coefficient attached to

the non-trivial orbit, we can see that there is an inner integral ϕ[(2n+2)14mn−2],{a}, which

is identically zero by (1) in the proof of Theorem 4.4.4. Therefore only the Fourier

coefficient attached to the trivial orbit, which actually equals to the integral in (4.49),

survives.

After applying Lemma 3.1.3 to all the pairs

(R2n+1, C

2n+1), (R3

n+1, C3n+1); · · · ; (R2

2n−1, C22n−1), (R3

2n−1, C32n−1),

the integral (4.49) is non-vanishing if and only if the following integral is non-vanishing:

∫[∏2n−1i=1 R2

i

∏2n−1t=n+1R

3tXet+etW4]

ϕ(rxwω2εω1g)ψ−1W4

(w)dwdxdr, (4.50)

Note that elements in∏2n−1i=1 R2

i

∏2n−1t=n+1R

3tXet+etW4 have the following form:

w =

Z q1 q2

0 I4mn−2n q∗1

0 0 Z∗

, (4.51)

where Z is in the standard maximal unipotent subgroup of GL2n; the last row of q1 is

zero. And ψW4(

Z 0 0

0 I4mn−2n 0

0 0 Z∗

) = ψ(∑n−1

i=1 Zi,i+1 −∑2n−1

j=n Zi,i+1).

Clearly the integral (4.50) is non-vanishing if and only if the following integral is

non-vanishing:∫[∏2n−1i=1 R2

i

∏2n−1t=n+1R

3tXet+etW4]

ϕ(rxwω2εω1g)ψ′−1W4

(w)dwdxdr, (4.52)

95

where ψ′W4(

Z 0 0

0 I4mn−2n 0

0 0 Z∗

) = ψ(∑2n−1

i=1 Zi,i+1). Note that the integral (4.52) is

exactly ϕψN

12n−1 , using notation in Lemma 4.2.6. On the other hand, we know that by

Lemma 4.2.6, ϕψN

12n−1 = ϕψN

12n . Note that Lemma 4.2.6 also applies to metaplectic

groups.

Therefore, the integral in (4.52) becomes∫[U ]ϕ(uω2εω1g)ψ

U(u)−1du, (4.53)

where any element in U has the following form:

u = u(Z, q1, q2) =

Z q1 q2

0 I4mn−2n q∗1

0 0 Z∗

,

where Z is in the standard standard maximal unipotent subgroup of GL2n, q1 ∈M(2n)×(4m−4)n, q2 ∈ M(2n)×(2n), such that qt2v2n − v2nq2 = 0, where v2n is a matrix

only with ones on the second diagonal. ψU

(u) = ψ(∑2n−1

i=1 ui,i+1).

Hence, the integral in (4.53) can be written as∫u(Z,0,0)

ϕ(uω2εω1g)P2n

ψU

(u)−1du, (4.54)

where ϕP2n is the constant term of ϕ along the pre-image of the parabolic subgroup

P2n = M2nU2n of Sp4mn+2n with Levi isomorphic to GL2n × Sp(4m−2)n.

By the similar calculation as in the proof of Lemma 4.2.4, or the calculation at the

end of Theorem 2.1 of [GJS12], there is an automorphic function

f ∈ A(N2n(A)M2n(F )\Sp4mn+2n(A))τ |·|−m⊗σ4(m−1)n+2n,

such that

ϕ(g)P2n

= f(g),∀g ∈ Sp4mn+2n(A).

Therefore, the integral (4.54) is the Whittaker Fourier coefficient of an element in

τ , hence not identically zero. This completes the proof of Part (1).

96

Proof of Part (2). By definition of Fourier-Jacobi coefficients ((3.7) or (3.14)

[GRS11]), for φ1 ∈ S(A2mn−n), φ2 = φ21⊗φ22, φ21 ∈ S(An), φ22 ∈ S(A2mn−n), we need

to compute the composition of two Fourier-Jacobi coefficients FJφ1

ψ−1n−1

and FJφ2

ψ1n−1

:

FJφ1

ψ−1n−1

◦ FJφ2

ψ1n−1

(ξ)(g)

=

∫[V[(2n)14mn−2n],1]

∫[V[(2n)14mn],1]

ϕ(uvg)θ2mn,φ2

ψ−1 (l2(u)vg)ψ−1[(2n)14mn],1

(u)du

θ2mn−n,φ1

ψ1 (l1(v)g)ψ−1[(2n)14mn−2n],−1

(v)dv,

(4.55)

where ϕ ∈ Eτ,σ4(m−1)n+2n, g ∈ Sp4mn−2n(A), the theta series are defined in Section 1.2

[GRS11]. V[(2n)14mn−2n],1 and V[(2n)14mn],1, ψ[(2n)14mn],1 and ψ[(2n)14mn−2n],−1 are defined

in Chapter 3. V[(2n)14mn−2n],1 and V[(2n)14mn],1 are as Nn in (3.7) or (3.14) [GRS11].

Explicitly, V[(2n)14mn],1 is the unipotent radical of the parabolic subgroup P 4mn+2n1n of

Sp4mn+2n with Levi subgroup isomorphic to GLn1 ×Sp4mn. V[(2n)14mn−2n],1 is the unipo-

tent radical of the parabolic subgroup P 4mn1n of Sp4mn, with Levi subgroup isomor-

phic to GLn1 × Sp4mn−2n. Note that Sp4mn is embedded into Sp4mn+2n via the map

g 7→ diag(In, g, In), and we identify it with the image. Then for u ∈ V[(2n)14mn],1,

ψ[(2n)14mn],1(u) = ψ(∑n−1

i=1 ui,i+1), l2(u) =∏2mni=1 Xαi(un,n+i), with αi = en − en+i. And

for v ∈ V[(2n)14mn−2n],1, ψ[(2n)14mn−2n],−1(v) = ψ(∑n−1

i=1 vn+i,n+i+1), with βi = e2n− e2n+i

for 1 ≤ i ≤ 2mn− n, l1(v) =∏2mn−ni=1 Xβi(v2n,2n+i).

First, we want to unfold the theta series θ2mn,φ2

ψ−1 (l2(u)vg). Write l2(u) as l2(u) =

(q1, q2, q3; z), where q1, q3 ∈ An, q2 ∈ A4mn−2n, z ∈ A. Then

θ2mn,φ2

ψ−1 (l2(u)vg)

=∑

ξ∈F 2mn

ω2mnψ−1 (l2(u)vg)φ2(ξ)

=∑

ξ1∈Fn,ξ2∈F 2mn−n

ω2mnψ−1 ((q1, q2, q3; z)vg)φ2(ξ1, ξ2)

=∑

ξ1∈Fn,ξ2∈F 2mn−n

ω2mnψ−1 ((ξ1, 0, 0; 0)(q1, q2, q3; z)vg)φ2(0, ξ2)

=∑

ξ1∈Fn,ξ2∈F 2mn−n

ω2mnψ−1 ((q1 + ξ1, q2, q3; z + ξ1νnq

′3)vg)φ2(0, ξ2)

(4.56)

97

=∑

ξ1∈Fn,ξ2∈F 2mn−n

ω2mnψ−1 ((0, q2, q3; z + ξ1)(q1 + ξ1, 0, 0; 0)vg)φ2(0, ξ2)

=∑

ξ1∈Fn,ξ2∈F 2mn−n

ω2mnψ−1 ((0, q2, q3; z + ξ1)vg(q1 + ξ1, 0, 0; 0))φ2(0, ξ2)

where νn is the matrix only with 1’s on the second diagonal, ξ1 = 2ξ1νnq′3 + q3νnq

′1,

q2 = q2 + (q1 + ξ1)(n)v, (q1 + ξ1)(n) is the n-th coordinate of the vector q1 + ξ1. Note

that (q1 + ξ1, 0, 0; 0)) commutes with g.

Plugging (4.56) into the integral in (4.55), collapsing the summation over ξ1 with

the integration over q1, changing variables for q2 and z, we will have

FJφ1

ψ−1n−1

◦ FJφ2

ψ1n−1

(ξ)(g)

=

∫[V[(2n)14mn−2n],1]

∫[V ′

[(2n)14mn],1]

∫Anϕ(uvgq1)

∑ξ2∈F 2mn−n

ω2mnψ−1 (l2(u)vg(q1, 0, 0; 0))φ2(0, ξ2)ψ−1

[(2n)14mn],1(u)du

θ2mn−n,φ1

ψ1 (l1(v)g)ψ−1[(2n)14mn−2n],−1

(v)dq1dv,

(4.57)

where q1 =∏ni=1Xαi(q1(n, n + i)), with αi = en − en+i; V

′[(2n)14mn],1 is a subgroup of

V[(2n)14mn],1 consisting of elements u with un,n+i = 0, for 1 ≤ i ≤ n; l2(u) = (0, q2, q3; z).

Note that, V ′[(2n)14mn],1 is actually Y V[(2n)14mn],2, where Y is defined in (3.5) correspond-

ing to the partition [(2n)14mn], and V[(2n)14mn],2 is defined Chapter 3. For short, we will

write q1 for (q1, 0, 0; 0).

By Formula (1.4) [GRS11],∑ξ2∈F 2mn−n

ω2mnψ−1 ((0, q2, q3; z)vg(q1, 0, 0; 0))φ2(0, ξ2)

=∑

ξ2∈F 2mn−n

ω2mnψ−1 ((0, q2, 0; z)vg(q1, 0, 0; 0))φ2(0, ξ2)

=∑

ξ2∈F 2mn−n

ω2mnψ−1 ((0, q2, 0; z)vg(q1, 0, 0; 0))φ2(q1, ξ2)

=θ2mn−n,φ22

ψ−1 ((q2, z)vg)φ21(q1),

(4.58)

since we assumed that φ2 = φ21 ⊗ φ22. Let l3(u) = (q2, z), for u ∈ V ′[(2n)14mn],1. Then,

98

the integral in (4.57) becomes∫[V[(2n)14mn−2n],1]

∫[V ′

[(2n)14mn],1]

∫Anϕ(uvgq1)φ21(q1)dq1

θ2mn−n,φ22

ψ−1 (l3(u)vg)ψ−1[(2n)14mn],1

(u)duθ2mn−n,φ1

ψ1 (l1(v)g)ψ−1[(2n)14mn−2n],−1

(v)dv

=

∫[V[(2n)14mn−2n],1]

∫[V ′

[(2n)14mn],1]ϕ′(uvg)θ2mn−n,φ22

ψ−1 (l3(u)vg)ψ−1[(2n)14mn],1

(u)du

θ2mn−n,φ1

ψ1 (l1(v)g)ψ−1[(2n)14mn−2n],−1

(v)dv,

(4.59)

where ϕ′(uvg) =∫An ϕ(uvgq1)φ21(q1)dq1. Note that we still have that ϕ′ ∈ Eτ,σ4(m−1)n+2n

.

Let ω1 be the Weyl element of Sp4mn+2n in (4.31). Conjugating cross by ω1, the

integral in (4.59) becomes∫[W ′1]

ϕ′(wω1g)θ2mn−n,φ22

ψ−1 (l4(w)g)θ2mn−n,φ1

ψ1 (l5(w)g)ψ−1W ′1

(w)dw, (4.60)

where W ′1 = ω1V[(2n)14mn−2n],1V′

[(2n)14mn],1ω−11 , its elements have the form as in (4.33),

except that there is no requirement that q1(i, j) = 0, for i = 2n−1, 2n, 1 ≤ j ≤ 2mn−n.

For w =

Z q1 q2

I4mn−2n q∗1

Z∗

as in (4.33), l4(w) = (q1(2n − 1), q2(2n − 1, 2)), l5(w) =

(q1(2n), q2(2n, 1)), q1(2n−1), q1(2n) are the (2n−1)-th, (2n)-th rows of q1, respectively.

And ψW ′1(w) = ψ(∑n−1

i=1 tr([Z]i,i+1), with notation as in (4.34).

Next, we repeat the steps from (4.35) to (4.41), and use Lemma 3.1.5 whenever

Lemma 3.1.3 is used. We will get that the integral in (4.60) becomes∫[W ′′1 ]

ϕ′′(wω1g)θ2mn−n,φ22

ψ−1 (l4(w)g)θ2mn−n,φ1

ψ1 (l5(w)g)ψ−1W ′′1

(w)dw, (4.61)

where W ′′1 is unipotent radical of the parabolic subgroup P 4mn+2n2n of Sp4mn+2n with

Levi subgroup isomorphic to GLn2 × Sp4mn−2n. ϕ′′ ∈ Eτ,σ4(m−1)n+2n. And for w ∈ W ′′1 ,

ψW ′′1 (w) = ψ(∑2n−2

i=1 wi,i+2).

Next, we want to unfold the two theta series as before. To do this, we need to use

certain property of theta series as in (5.9) [GJS12]:

θ2mn−n,φ22

ψ−1 ((x1, y1; z1)g)θ2mn−n,φ1

ψ1 ((x2, y2; z1)g)

=θ4mn−2n,φ22⊗φ1

ψ ((x1, x2, y2,−y1; z2 − z1)g),(4.62)

99

where for w ∈ W ′1, we write l4(w) = (x1, y1; z1), l5(w) = (x2, y2; z2), x1, y1, x2, y2 ∈A2mn−n, and

g =

A −B

A B

C D

−C D

,

if we write g =

((A B

C D

), ε

). Let φ3 = φ22 ⊗ φ1.

Let

γ =

I2mn−n 0 1

2I2mn−n 0

I2mn−n 0 −12I2mn−n 0

0 I2mn−n 0 12I2mn−n

0 −I2mn−n 0 12I2mn−n

∈ Sp8mn−4n.

Then,

(x1, x2, y2,−y1)γ = (x1 + x2, y1 + y2,1

2(x1 − x2),−1

2(y1 − y2)),

g := γ−1gγ =

((g

g∗

), 1

).

(4.63)

Therefore, by (4.63), the right hand side of (4.62) becomes:

θ4mn−2n,φ3

ψ ((x1, x2, y2,−y1; z2 − z1)g)

=θ4mn−2n,φ′3ψ ((x1 + x2, y1 + y2,

1

2(x1 − x2),−1

2(y1 − y2); z2 − z1)g),

(4.64)

where φ′3 = ω4mn−2nψ (γ−1)φ3.

Let A =

(1 −1

1 1

), and ε = diag(A, . . . , A; I(4m−4)n;A∗, . . . , A∗), as in (2.31) of

[GJS12]. Conjugating cross the integral in (4.61) by ε, it becomes:∫[W ′2]

ϕ′′(wεω1g)θ4mn−2n,φ′3ψ (l6(w)g)ψ−1

W ′2(w)dw

=

∫[W ′2]

ϕ′′(wgεω1)θ4mn−2n,φ′3ψ (l6(w)g)ψ−1

W ′2(w)dw

=

∫[W ′2]

ϕ′′′(wg)θ4mn−2n,φ′3ψ (l6(w)g)ψ−1

W ′2(w)dw,

(4.65)

100

where W ′2 = εW ′′1 ε−1, and elements in W ′2 are as in (4.43), except that there is no re-

quirement that q1(i, j) = 0, for i = 2n−1, 2n, 1 ≤ j ≤ 2mn−n, still ϕ′′′ ∈ Eτ,σ4(m−1)n+2n.

Note that g commutes with εω1. And ψW ′2(w) = ψ(∑2n−2

i=1 wi,i+2 + w2n−1,4mn+1). For

w =

Z q1 q2

I4mn−2n q∗1

Z∗

as in (4.43), for i = 2n − 1, 2n, write the i-th row of q1 as

q1(i) = (xi, yi), then

l6(w) = (x2n, y2n,1

2x2n−1,−

1

2y2n−1,−q2(2n− 1, 1)).

Then, we unfold the theta series θ4mn−2n,φ′3ψ (l6(w)g) as in (4.56), the integral in

(4.65) becomes: ∫[W ′3]

∫A4mn−2n

ϕ′′′(wgξ)φ3(ξ)dξψ−1W ′2

(w)dw

=

∫[W ′3]

ϕ(4)(wg)ψ−1W ′3

(w)dw

(4.66)

where ξ =∏2mn−ni=1 Xαi(ξ(i))

∏2mn−nj=1 Xβj (ξ(2mn−n+ j)), with αi = e2n− e2n+i, βj =

e2n + e2mn+n−j+1, and ϕ(4)(wg) =∫A4mn−2n ϕ

′′′(wgξ)φ3(ξ), still ϕ(4) ∈ Eτ,σ4(m−1)n+2n.

And W ′3 is the subgroup of W ′2 consisting of elements w =

Z q1 q2

I4mn−2n q∗1

Z∗

, with

q1(2n) = 0.

Conjugate cross the integral in (4.66) by the Weyl elements ω2 in (4.45), it becomes:∫[W4]

ϕ(4)(wω2g)ψ−1W4

(w)dw

=

∫[W4]

ϕ(4)(wgω2)ψ−1W4

(w)dw

=

∫[W4]

ϕ(5)(wg)ψ−1W4

(w)dw,

(4.67)

where W4, ψW4 are exactly as in (4.48).

Now, we repeat the steps from (4.48) to (4.50), and use Lemma 3.1.5 whenever

Lemma 3.1.3 is used. Then, we get that the integral in (4.67) becomes:∫[W5]

ϕ(5)(wg)ψ−1W5

(w)dw, (4.68)

101

where

W5 =

2n−1∏i=1

R2i

2n−1∏t=n+1

R3tXet+etW4

as in (4.50). And given w =

Z q1 q2

0 I4mn−2n q∗1

0 0 Z∗

as in (4.51), ψW5(w) = ψ(∑n−1

i=1 Zi,i+1−

∑2n−1j=n Zi,i+1).

Let t = diag(In−1,−In+1; I4mn−2n;−In+1, In−1) ∈ Sp4mn+2n(F ). Since ϕ(5) is auto-

morphic, the integral in (4.68) becomes:∫[W5]

ϕ(5)(twg)ψ−1W5

(w)dw

=

∫[W5]

ϕ(5)(wg)ψ′,−1W5

(w)dw,

(4.69)

after changing variable w 7→ t−1w, and ψ′W5

(w) = ψ(∑2n−1

i=1 Zi,i+1), for

w =

Z q1 q2

0 I4mn−2n q∗1

0 0 Z∗

∈W5.

Note that the integral on the right hand side of the identity in (4.69) is exactly

ϕψN

12n−1 , using notation in Lemma 4.2.6. On the other hand, we know that by Lemma

4.2.6, ϕψN

12n−1 = ϕψN

12n . Note that Lemma 4.2.6 also applies to metaplectic groups.

Therefore, the integral in (4.69) becomes∫[U ]ϕ(5)(ug)ψ

U(u)−1du, (4.70)

where U and ψU

are exactly as in (4.53).

Now, it follows easily from the end of the proof of Part (1) that as a function of

g ∈ Sp4mn−2n, the integral in (4.70) gives a section in σ4(m−1)n+2n. Since starting

from the integral in (4.55), we always get equalities, FJφ1

ψ−1n−1

◦FJφ2

ψ1n−1

(ξ) ∈ σ4(m−1)n+2n.

Therefore,

D4mn2n,ψ−1 ◦ D4mn+2n

2n,ψ1 (Eτ,σ4(m−1)n+2n) ⊂ σ4(m−1)n+2n.

On the other hand, by Part (1),

D4mn2n,ψ−1 ◦ D4mn+2n

2n,ψ1 (Eτ,σ4(m−1)n+2n) 6= 0.

102

Since σ4(m−1)n+2n is irreducible, we have that

D4mn2n,ψ−1 ◦ D4mn+2n

2n,ψ1 (Eτ,σ4(m−1)n+2n) = σ4(m−1)n+2n.

This finishes the proof of Part (2), and completes the proof of the theorem.

Remark 4.4.3. Note that in the proof of Part (2), we could easily get a similar identity

as in Theorem 5.1 [GJS12], but for simplicity, we did not write it down explicitly.

Theorem 4.4.2 easily implies the following result.

Theorem 4.4.4. p = [(2n)2m(2n1)s1(2n2)s2 · · · (2nk)sk ] is a maximal partition provid-

ing non-vanishing Fourier coefficients for Eτ,σ4(m−1)n+2n.

Proof. By Theorem 4.4.2,

[(2n)14mn] ◦ [(2n)14mn−2n] ◦ [(2n)2m−2(2n1)s1(2n2)s2 · · · (2nk)sk ]

is a composite partition providing non-vanishing Fourier coefficients for Eτ,σ4(m−1)n+2n.

By Lemma 2.6 of [GRS03] or Lemma 3.1 of [JL13b], [(2n)2m(2n1)s1(2n2)s2 · · · (2nk)sk ]

is a partition providing non-vanishing Fourier coefficients for Eτ,σ4(m−1)n+2n.

Since by Theorem 4.4.2, D4mn2n,ψ−1 ◦ D4mn+2n

2n,ψ1 (Eτ,σ4(m−1)n+2n) = σ4(m−1)n+2n, and

p(σ4(m−1)n+2n) = [(2n)2(m−1)(2n1)s1(2n2)s2 · · · (2nk)sk ],

to show the maximality of p, we just have to show that

(1) at the step of taking D4mn+2n2n,ψ1 , Eτ,σ4(m−1)n+2n

has no nonzero Fourier coefficients

attached to the symplectic partitions [(2l)14mn+2n−2l] for any l ≥ n + 1, or [(2l +

1)214mn+2n−4l−2], for any l ≥ n;

(2) at the step of taking D4mn2n,ψ−1 , D4mn+2n

2n,ψ1 (Eτ,σ4(m−1)n+2n) has no nonzero Fourier

coefficients attached to the symplectic partitions [(2l)14mn−2l] for any l ≥ n + 1, or

[(2l + 1)214mn−4l−2], for any l ≥ n.

We will show (1) and (2) using calculations of unramified local components. Let v

be a finite place such that Eτ,σ4(m−1)n+2n,v is unramified. This means that both τv and

σ4(m−1)n+2n,v are also unramified. Assume that τv = ×ni=1ναiχi ××ni=1ν

−αiχ−1i , where

ναi(·) = |det(·)|αi , 0 ≤ αi <12 , and χi’s are unitary unramified characters of F ∗v , for

1 ≤ i ≤ n. Since π lifts weakly to τ with respect to ψ, πv = µψ ×ni=1 ναiχi o 1

Sp0.

103

Since by the definition of the set N ′Sp4(m−1)n+2n

(τ, ψ), σ4(m−1)n+2n is an irreducible

cuspidal automorphic representation of Sp4(m−1)n+2n(A), which is nearly equivalent to

the residual representation E∆(τ,m−1)⊗π, similarly as Lemma 3.1 of [GRS05], it is easy

to see that Eτ,σ4(m−1)n+2n,v is the unique unramified component of the following induced

representation

IndSp4mn+2n(F ∗v )

P(2m+1)n (F ∗v )µψ ⊗ni=1 ν

αiχi(detGL2m+1)⊗ 1Sp0

, (4.71)

where P(2m+1)n is the parabolic subgroup of Sp4mn+2n with Levi isomorphic toGLn2m+1×1Sp0 , and P(2m+1)n is its full pre-image in Sp4mn+2n.

By Lemma 3.2 of [JL13a], we can easily see that (1) holds. By (4.29) of Theorem

4.4.1,

FJψ1n−1

(IndSp4mn+2n(F ∗v )

P(2m+1)n (F ∗v )µψ ⊗ni=1 ν

αiχi(detGL2m+1)⊗ 1Sp0

)

∼=IndSp4mn(F ∗v )P(2m)n (F ∗v ) ⊗

ni=1 ν

αiχi(detGL2m)⊗ 1Sp0,

(4.72)

which is actually irreducible, by results in [Jan96], and is an unramified local component

of D4mn+2n2n,ψ1 (Eτ,σ4(m−1)n+2n

). Again, by Lemma 3.1 of [JL13a], we can easily see that (2)

also holds. Therefore, we have shown that p = [(2n)2m(2n1)s1(2n2)s2 · · · (2nk)sk ] is a

maximal partition providing non-vanishing Fourier coefficients for Eτ,σ4(m−1)n+2n, which

completes the proof of this theorem.

To continue, we prove that D4mn+2n2n,ψ1 (Eτ,σ4(m−1)n+2n

) is a cuspidal representation,

every component of which is inside NSp4mn(τ, ψ).

Theorem 4.4.5. D4mn+2n2n,ψ1 (Eτ,σ4(m−1)n+2n

) ⊂ NSp4mn(τ, ψ).

Proof. We will prove that

(1) D4mn+2n2n,ψ1 (Eτ,σ4(m−1)n+2n

) is a cuspidal representation;

(2) every irreducible component of D4mn+2n2n,ψ1 (Eτ,σ4(m−1)n+2n

) is inside NSp4mn(τ, ψ).

To prove (1), we will show that for any element in D4mn+2n2n,ψ1 (Eτ,σ4(m−1)n+2n

), the

constant terms along all maximal parabolic subgroups of Sp4mn are all zero.

Recall that P 4mnr = M4mn

r N4mnr (with 1 ≤ r ≤ 2mn) is the standard parabolic

subgroup of Sp4mn with Levi part M4mnr isomorphic to GLr × Sp4mn−2r, N

4mnr is the

104

unipotent radical. Take any ξ ∈ Eτ,σ4(m−1)n+2n, we will calculate the constant term of

FJφψ1n−1

(ξ) along P 4mnr , which is denoted by CN4mn

r(FJφ

ψ1n−1

(ξ)).

By Theorem 7.8 of [GRS11],

CN4mnr

(FJφψ1n−1

(ξ))

=r∑

k=0

∑γ∈P 1

r−k,1k(F )\GLr(F )

∫L(A)

φ1(i(λ))FJφ2

ψ1n−1+k

(CN4mn+2nr−k

(ξ))(γλβ)dλ,(4.73)

where N4mn+2nr−k is the unipotent radical of the parabolic subgroup P 4mn+2n

r−k of Sp4mn+2n

with Levi isomorphic to GLr−k×Sp4mn+2n−2r+2k, and it is identified with it’s image in

Sp4mn+2n; P 1r−k,1k is a subgroup of GLr consisting of matrices of the form

(g x

0 z

), with

z ∈ Uk, the standard maximal unipotent subgroup of GLk; for a ∈ GLj , j ≤ 2mn+ n,

a = diag(a, I4mn+2n−2j , a∗); L is a unipotent subgroup, consisting of matrices of the

form λ =

(Ir 0

x In

)∧, and i(λ) is the last row of x; β =

(0 Ir

In 0

)∧; φ = φ1 ⊗ φ2,

with φ1 ∈ S(Ar), φ2 ∈ S(A2mn−r); CN4mn+2nr−k

(ξ) is restricted to Sp4mn+2n−2r+2k(A),

then we apply the Fourier-Jacobi coefficient FJφ2

ψ1n−1+k

, taking automorphic forms on

Sp4mn+2n−2r+2k(A) to Sp4mn−2r(A).

By the cuspidal support of ξ, CN4mn+2nr−k

(ξ) is identically zero, unless r = k or r−k =

2n. When r = k, the corresponding term is zero, because FJφ2

ψ1n−1+r

(ξ) is zero, by

Theorem 4.4.4. When r − k = 2n, the restriction of CN4mn+2n2n

(ξ) to Sp4mn−2n(A) is

actually a vector in σ4(m−1)n+2n. Hence, FJφ2

ψ1n−1+k

(CN4mn+2nr−k

(ξ)) is identically zero,

for 0 ≤ k ≤ r, because σ4(m−1)n+2n has no nonzero Fourier coefficients FJψ1n−1

, and

p(σ4(m−1)n+2n) = [(2n)2(m−1)(2n1)s1(2n2)s2 · · · (2nk)sk ]. So, when r − k = 2n, the

corresponding term is also zero.

Therefore, we have shown that CN4mnr

(FJφψ1n−1

(ξ)) is identically zero for any 1 ≤ r ≤

2mn, and for any ξ ∈ Eτ,σ4(m−1)n+2n, which implies that D4mn+2n

2n,ψ1 (Eτ,σ4(m−1)n+2n) is a

cuspidal representation. This completes the proof of (1).

To prove (2), for every irreducible component π of D4mn+2n2n,ψ1 (Eτ,σ4(m−1)n+2n

), we need

to show that the follows hold:

(2-1) p(π) = [(2n)2m−1(2n1)s1(2n2)s2 · · · (2nk)sk ];

(2-2) π is nearly equivalent to the residual representation E∆(τ,m);

105

(2-3) π has a nonzero Fourier coefficient FJψ−1n−1

.

(2-1) is obvious by Theorem 4.4.4, and by Lemma 2.6 of [GRS05] or Lemma 3.1

of [JL13b]. (2-2) follows easily from (4.72), because the induced representation on the

right hand side of (4.72) is also an unramified component of E∆(τ,m).

To show (2-3), as in the proof of Proposition 3.4 of [GJS12], we need to consider the

following integral

〈ϕπ, FJφψ1n−1

(ξ)〉 =

∫[Sp4mn]

ϕπ(h)FJφψ1n−1

(ξ)(h)dh, (4.74)

which is nonzero for some data ϕπ ∈ π, ξ ∈ Eτ,σ4(m−1)n+2n, since π is an irreducible

component of D4mn+2n2n,ψ1 (Eτ,σ4(m−1)n+2n

).

Assume that ξ = Ress=m+12E(φs, ·), then from (4.74), we know that the following

integral is also nonzero for some choice of data:

〈ϕπ, FJφψ1n−1

(E(φs, ·))〉 =

∫[Sp4mn]

ϕπ(h)FJφψ1n−1

(E(φs, ·))(h)dh. (4.75)

Then, by the unfolding in Theorem 3.3 of [GJRS11] (take m = 2n, r = n there), the

non-vanishing of the integral in (4.74) implies the non-vanishing of FJψ−1n−1

(π).

This finishes the proof of (2), and completes the proof of the theorem.

The next theorem implies that Ψ is well-defined. We will prove it in the next section.

Theorem 4.4.6. D4mn2n,ψ−1(σ4mn) is irreducible, and

D4mn2n,ψ−1(σ4mn) ∈ N ′

Sp4(m−1)n+2n(τ, ψ),

for any σ4mn ∈ NSp4mn(τ, ψ).

Now by Theorems 4.4.2, 4.4.5, 4.4.6, we are able to conclude the Part (1) of Theorem

4.1.2.

Theorem 4.4.7 (Part (1) of Theorem 4.1.2). There is a surjective map

Ψ : NSp4mn(τ, ψ)→ N ′

Sp4(m−1)n+2n(τ, ψ)

σ4mn 7→ D4mn2n,ψ−1(σ4mn).

106

Proof. Theorem 4.4.6 implies that Ψ is well-defined. Theorem 4.4.2 and Theorem 4.4.5

imply that for any σ4(m−1)n+2n ∈ N ′Sp4(m−1)n+2n

(τ, ψ), take any irreducible component

π of D4mn+2n2n,ψ1 (Eτ,σ4(m−1)n+2n

), which is inside NSp4mn(τ, ψ), then

Ψ(π)

=D4mn2n,ψ−1(π)

⊆D4mn2n,ψ−1 ◦ D4mn+2n

2n,ψ1 (Eτ,σ4(m−1)n+2n)

=σ4(m−1)n+2n.

Theorem 4.4.5 also implies that D4mn2n,ψ−1(π) 6= 0. Since σ4(m−1)n+2n is irreducible, actu-

ally we have D4mn2n,ψ−1(π) = σ4(m−1)n+2n.

Hence Ψ is surjective.

4.5 Proof of Theorem 4.4.6

For any σ4mn ∈ NSp4mn(τ, ψ), we know that the Eisenstein series corresponding to

IndSp4(m+1)n(A)

P2n(A) τ |·|s ⊗ σ4mn

has a simple pole at s = m+12 . Let Eτ,σ4mn be the residual representation of Sp4(m+1)n(A)

generated by the corresponding residues.

First, we have a similar result as that in Theorem 4.4.2.

Theorem 4.5.1. (1)

D4mn+2n2n,ψ1 ◦ D4(m+1)n

2n,ψ−1 (Eτ,σ4mn) 6= 0.

(2)

D4mn+2n2n,ψ1 ◦ D4(m+1)n

2n,ψ−1 (Eτ,σ4mn) = σ4mn.

Proof. The proof is very similar to that of Theorem 4.4.2. We omit it here.

Next, we have a similar result as at in Theorem 5.8 of [GJS12].

Theorem 4.5.2. For any σ4mn ∈ NSp4mn(τ, ψ), the representation D4(m+1)n

2n,ψ−1 (Eτ,σ4mn)

is square-integrable. Moreover, there is an irreducible representation σ4(m−1)+2n, which

is a component of

D4mn2n,ψ−1(σ4mn) ⊂ N ′

Sp4(m−1)n+2n(τ, ψ),

107

such that the representation space of D4(m+1)n2n,ψ−1 (Eτ,σ4mn) has a nontrivial intersection

with the representation space of Eτ,σ4(m−1)n+2n.

Proof. We follow the constant term calculation in the proof of the Theorem 4.4.5.

Recall that P 4mn+2nr = M4mn+2n

r N4mn+2nr (with 1 ≤ r ≤ 2mn + n) is the stan-

dard parabolic subgroup of Sp4mn+2n with Levi part M4mn+2nr isomorphic to GLr ×

Sp4mn+2n−2r, N4mn+2nr is the unipotent radical, and P 4mn+2n

r is the pre-image of

P 4mn+2nr in Sp4mn+2n. Take any ξ ∈ Eτ,σ4mn , we will calculate the constant term of

FJφψ−1n−1

(ξ) along P 4mn+2nr , which is denoted by CN4mn+2n

r(FJφ

ψ−1n−1

(ξ)).

By Theorem 7.8 of [GRS11],

CN4mn+2nr

(FJφψ−1n−1

(ξ))

=r∑

k=0

∑γ∈P 1

r−k,1k(F )\GLr(F )

∫L(A)

φ1(i(λ))FJφ2

ψ−1n−1+k

(CN4mn+4nr−k

(ξ))(γλβ)dλ,(4.76)

where N4mn+4nr−k is the unipotent radical of the parabolic subgroup P 4mn+4n

r−k of Sp4mn+4n

with Levi isomorphic to GLr−k × Sp4mn+4n−2r+2k; P1r−k,1k is a subgroup of GLr con-

sisting of matrices of the form

(g x

0 z

), with z ∈ Uk, the standard maximal unipotent

subgroup of GLk; for a ∈ GLj , j ≤ 2mn + 2n, a = diag(a, I4mn+4n−2j , a∗); L is a

unipotent subgroup, consisting of matrices of the form λ =

(Ir 0

x In

)∧, and i(λ) is the

last row of x; β =

(0 Ir

In 0

)∧; φ = φ1 ⊗ φ2, with φ1 ∈ S(Ar), φ2 ∈ S(A2mn+n−r);

CN4mn+4nr−k

(ξ) is restricted to Sp4mn+4n−2r+2k(A), then we apply the Fourier-Jacobi coef-

ficient FJφ2

ψ−1n−1+k

, taking automorphic forms on Sp4mn+4n−2r+2k(A) to Sp4mn+2n−2r(A).

By the cuspidal support of ξ, CN4mn+4nr−k

(ξ) is identically zero, unless r = k or r−k =

2n. When r = k, the corresponding term is zero, because FJφ2

ψ−1n−1+r

(ξ) is zero, by

Theorem 4.4.4. When r−k = 2n, the restriction of CN4mn+4n2n

(ξ) to Sp4mn(A) is actually

a vector inside σ4mn. Hence, FJφ2

ψ−1n−1+k

(CN4mn+4nr−k

(ξ)) is not zero for k = 0, and is

identically zero for 1 ≤ k ≤ r, because σ4mn has a nonzero Fourier coefficient FJψ−1n−1

,

and p(σ4mn) = [(2n)2m−1(2n1)s1(2n2)s2 · · · (2nk)sk ].

108

Therefore,

CN4mn+2n2n

(FJφψ−1n−1

(ξ))

=

∫L(A)

φ1(i(λ))FJφ2

ψ−1n−1

(CN4mn+4n2n

(ξ))(λβ)dλ.(4.77)

By similar calculation as in the proof of Lemma 4.2.4, when restricting to GL2n(A)×Sp4mn(A),

CN4mn+4n2n

(ξ) ∈ δ12

P 4mn+4n2n

|det|−2m+1

2 τ ⊗ σ4mn.

As in the proof of Theorem 2.5 [GJS12], we need to calculate the automorphic

exponents attached to this nontrivial constant term (for definition see I.3.3 [MW95]).

We consider the action of

g = diag(g, I4mn−2n, g∗) ∈ GL2n(A)× Sp4mn−2n(A).

Since r = 2n, β =

(0 I2n

In 0

)∧. βdiag(In, g, In)β−1 = diag(g, I4mn, g

∗) =: g. Then

changing variables in (4.77) λ 7→ gλg−1 will give a Jacobian |det(g)|−n. On the other

hand, by Formula (1.4) [GJS12], the action of g on φ1 gives γψ(det(g))|det(g)|12 . There-

fore, the g acts by τ(g) with character

δ12

P 4mn+4n2n

|det(g)|−2m+1

2 |det(g)|−nγψ(det(g))|det(g)|12

=γψ(det(g))δ12

P 4mn+2n2n

|det(g)|−m.

Hence, by Langlands square-integrability criterion (Lemma I.4.11 [MW95]), the auto-

morphic representation D4(m+1)n2n,ψ−1 (Eτ,σ4mn) is square-integrable. And as a representation

of GL2n(A)× Sp4mn−2n(A),

CN4mn+4n2n

(D4(m+1)n2n,ψ−1 (Eτ,σ4mn)) = γψδ

12

P 4mn+2n2n

|det|−mτ ⊗D4mn2n,ψ−1(σ4mn). (4.78)

By (4.78), it is easy to see that any non-cuspidal irreducible subrepresentation of

D4(m+1)n2n,ψ−1 (Eτ,σ4mn) must be an irreducible subrepresentation of Eτ,σ4(m−1)n+2n

, for some

irreducible subrepresentation σ4(m−1)+2n of D4mn2n,ψ−1(σ4mn).

To prove D4mn2n,ψ−1(σ4mn) ⊂ N ′

Sp4(m−1)n+2n

(τ, ψ), we need to show that for every irre-

ducible component σ of D4mn2n,ψ−1(σ4mn),

109

(1) σ is cuspidal;

(2) p(σ) = [(2n)2m−2(2n1)s1(2n2)s2 · · · (2nk)sk ];

(3) σ is nearly equivalent to the residual representation E∆(τ,m−1)⊗π;

(4) σ has no nonzero Fourier coefficient FJψ1n−1

.

(1) follows easily from the tower property (Theorem 7.10 [GRS11]). (2) is implied

by Lemma 2.6 [GRS03] or Lemma 3.1 [JL13b]. (3) can be read out easily from the right

hand side of (4.72) by (4.28) of Theorem 4.4.1. Note that the right hand side of (4.72) is

an unramified component of E∆(τ,m), hence unramified component of σ4mn. By Theorem

5.2 [JL13b], as a cuspidal representation, σ4mn has no nonzero Fourier coefficient with

respect to character ψ[(2n)2m−1(2n1)s1 (2n2)s2 ···(2nk)sk ],a, where a = {−1, 1} ∪ a′. Now (4)

follows easily from Lemma 3.1 [JL13b].

Therefore, the representation space of D4(m+1)n2n,ψ−1 (Eτ,σ4mn) has a nontrivial intersection

with the representation space of Eτ,σ4(m−1)n+2n, for some component σ4(m−1)n+2n of

D4mn2n,ψ−1(σ4mn) ⊂ N ′

Sp4(m−1)n+2n(τ, ψ).

This completes the proof of the theorem.

Proof of Theorem 4.4.6.

For any σ4mn ∈ NSp4mn(τ, ψ), by Theorem 4.5.1,

D4mn+2n2n,ψ1 ◦ D4(m+1)n

2n,ψ−1 (Eτ,σ4mn) = σ4mn. (4.79)

By Theorem 4.5.2, there is an irreducible representation σ4(m−1)+2n, which is a com-

ponent of D4mn2n,ψ−1(σ4mn) ⊂ N ′

Sp4(m−1)n+2n

(τ, ψ), such that the representation space of

D4(m+1)n2n,ψ−1 (Eτ,σ4mn) contains an irreducible subrepresentation π of Eτ,σ4(m−1)n+2n

. Since

σ4mn is irreducible, by (4.79),

σ4mn = D4mn+2n2n,ψ1 (π) ⊂ D4mn+2n

2n,ψ1 (Eτ,σ4(m−1)n+2n). (4.80)

Therefore,

D4mn2n,ψ−1(σ4mn) ⊂ D4mn

2n,ψ−1 ◦ D4mn+2n2n,ψ+1 (Eτ,σ4(m−1)n+2n

)

= σ4(m−1)n+2n,

by Theorem 4.4.2. Hence, D4mn2n,ψ−1(σ4mn) = σ4(m−1)n+2n, irreducible as an element in

N ′Sp4(m−1)n+2n

(τ, ψ).

This completes the proof of Theorem 4.4.6, showing that Ψ is well-defined. �

110

4.6 Proof of Part (2) of Theorem 4.1.2

In this section, we will prove that Ψ is injective. For this, we need to assume that for

any σ4(m−1)n+2n ∈ N ′Sp4(m−1)n+2n

(τ, ψ), Eτ,σ4(m−1)n+2nis irreducible.

For any σ4mn ∈ NSp4mn(τ, ψ), by Theorem 4.4.6, D4mn

2n,ψ−1(σ4mn) = σ4(m−1)n+2n ∈N ′Sp4(m−1)n+2n

(τ, ψ), which is irreducible. To show Ψ is injective, we only need to show

that σ4mn is uniquely determined by σ4(m−1)n+2n.

By (4.80), σ4mn = D4mn+2n2n,ψ1 (π) ⊂ D4mn+2n

2n,ψ1 (Eτ,σ4(m−1)n+2n), where π is an irreducible

subrepresentation of Eτ,σ4(m−1)n+2n. Since we assume that Eτ,σ4(m−1)n+2n

is irreducible,

we have that π = Eτ,σ4(m−1)n+2n. Hence σ4mn = D4mn+2n

2n,ψ1 (Eτ,σ4(m−1)n+2n), which means

that σ4mn is uniquely determined by σ4(m−1)n+2n.

This completes the proof of Part (2) of Theorem 4.1.2.

4.7 Irreducibility of Certain Descent Representations

In Theorem 4.2.2, for the residual representation E∆(τ,m), we proved that nm(E∆(τ,m)) =

[(2n)2m]. From the proof, and by Lemma 2.6 [GRS03] or Lemma 3.1 [JL13b], we can

see that it has a nonzero Fourier coefficient attached to the partition [(2n)14mn−2n] with

respect to the character ψ[(2n)14mn−2n],−1. In this section, F is any number field. We

show that if E∆(τ,m) is irreducible, then D4mn2n,ψ−1(E∆(τ,m)) is irreducible. The result can

be stated as follows.

Theorem 4.7.1. Assume that F is any number field.

(1) D4mn2n,ψ−1(E∆(τ,m)) is square-integrable and is in the discrete spectrum.

(2) Assume that E∆(τ,m) is irreducible, then D4mn2n,ψ−1(E∆(τ,m)) is also irreducible.

Proof. Proof of Part (1). As in Theorem 4.5.2, we follow the constant term calculation

in the proof of the Theorem 4.4.5.

Recall that P 4mn−2nr = M4mn−2n

r N4mn−2nr (with 1 ≤ r ≤ 2mn − n) is the stan-

dard parabolic subgroup of Sp4mn−2n with Levi part M4mn−2nr isomorphic to GLr ×

Sp4mn−2n−2r, N4mn−2nr is the unipotent radical, and P 4mn−2n

r is the pre-image of

P 4mn−2nr in Sp4mn−2n. Take any ξ ∈ E∆(τ,m), we will calculate the constant term

of FJφψ−1n−1

(ξ) along P 4mn−2nr , which is denoted by CN4mn−2n

r(FJφ

ψ−1n−1

(ξ)).

111

By Theorem 7.8 of [GRS11],

CN4mn−2nr

(FJφψ−1n−1

(ξ))

=

r∑k=0

∑γ∈P 1

r−k,1k(F )\GLr(F )

∫L(A)

φ1(i(λ))FJφ2

ψ−1n−1+k

(CN4mnr−k

(ξ))(γλβ)dλ,(4.81)

where N4mnr−k is the unipotent radical of the parabolic subgroup P 4mn

r−k of Sp4mn with Levi

isomorphic to GLr−k×Sp4mn−2r+2k; P1r−k,1k is a subgroup of GLr consisting of matrices

of the form

(g x

0 z

), with z ∈ Uk, the standard maximal unipotent subgroup of GLk;

for a ∈ GLj , j ≤ 2mn, a = diag(a, I4mn−2j , a∗); L is a unipotent subgroup, consisting of

matrices of the form λ =

(Ir 0

x In

)∧, and i(λ) is the last row of x; β =

(0 Ir

In 0

)∧; φ =

φ1⊗φ2, with φ1 ∈ S(Ar), φ2 ∈ S(A2mn−n−r); CN4mnr−k

(ξ) is restricted to Sp4mn−2r+2k(A),

then we apply the Fourier-Jacobi coefficient FJφ2

ψ−1n−1+k

, taking automorphic forms on

Sp4mn−2r+2k(A) to Sp4mn−2n−2r(A).

By the cuspidal support of ξ, CN4mnr−k

(ξ) is identically zero, unless r = k or r−k = 2ln,

1 ≤ l ≤ m − 1. When r = k, the corresponding term is zero, because FJφ2

ψ−1n−1+r

(ξ) is

zero, by Theorem 4.4.4. When r−k = 2ln, 1 ≤ l ≤ m−1, FJφ2

ψ−1n−1+k

(CN4mn+4nr−k

(ξ)) is not

zero for k = 0, and is identically zero for 1 ≤ k ≤ r, because nm(E∆(τ,m)) = [(2n)2m].

Therefore, CN4mn+2nr

(FJφψ−1n−1

(ξ)) 6= 0, only for r = 2ln, 1 ≤ l ≤ m − 1. And for

1 ≤ l ≤ m− 1,

CN4mn−2n2ln

(FJφψ−1n−1

(ξ))

=

∫L(A)

φ1(i(λ))FJφ2

ψ−1n−1

(CN4mn2ln

(ξ))(λβ)dλ.(4.82)

To prove square-integrability of D4mn2n,ψ−1(E∆(τ,m)), it turns out we only need to con-

sider r = 2(m− 1)n, which will be clear from the following discussion.

For r = 2(m− 1)n,

CN4mn−2n2(m−1)n

(FJφψ−1n−1

(ξ))

=

∫L(A)

φ1(i(λ))FJφ2

ψ−1n−1

(CN4mn2(m−1)n

(ξ))(λβ)dλ.(4.83)

112

By Lemma 4.2.4, when restricting to GL2(m−1)n × Sp4n,

CN4mn2(m−1)n

(ξ) ∈ δ12

P 4mn2(m−1)n

|det|−m+1

2 ∆(τ,m− 1)⊗ E∆(τ,1).

As in the proof of Theorem 2.5 [GJS12], to calculate the automorphic exponent

attached to this nontrivial constant term (for definition see I.3.3 [MW95]). we need to

consider the action of

g = diag(g, I4n, g∗) ∈ GL2(m−1)n(A)× Sp4n(A).

Since r = 2(m− 1)n, β =

(0 I2(m−1)n

In 0

)∧. βdiag(In, g, In)β−1 = diag(g, I6n, g

∗) =: g.

Then changing variables in (4.83) λ 7→ gλg−1 will give a Jacobian |det(g)|−n. On the

other hand, by Formula (1.4) [GJS12], the action of g on φ1 gives γψ(det(g))|det(g)|12 .

Therefore, the g acts by τ(g) with character

δ12

P 4mn2(m−1)n

|det(g)|−m+1

2 |det(g)|−nγψ(det(g))|det(g)|12

=γψ(det(g))δ12

P 4mn−2n2(m−1)n

|det(g)|−m2 .

Therefore,

CN4mn−2n2(m−1)n

(FJφψ−1n−1

(ξ))

∈γψδ12

P 4mn−2n2(m−1)n

|det|−m2 ∆(τ,m− 1)⊗D4n

2n,ψ−1(E∆(τ,1)).(4.84)

By Theorem 2.3 [GJS12], we know that D4n2n,ψ−1(E∆(τ,1)) is an irreducible, genuine, ψ-

generic, cuspidal automorphic representation of Sp2n(A), which lifts to τ with respect

to ψ. Hence, as a representation of GL2(m−1)n(A)× Sp2n(A),

CN4mn−2n2(m−1)n

(D4mn2n,ψ−1(E∆(τ,m)))

=γψδ12

P 4mn−2n2(m−1)n

|det|−m2 ∆(τ,m− 1)⊗D4n

2n,ψ−1(E∆(τ,1)).(4.85)

Since, the cuspidal exponent of ∆(τ,m − 1) is {(2−m2 , 4−m

2 , . . . , m−22 )}, the cuspi-

dal exponent of CN4mn−2n2(m−1)n

(FJφψ−1n−1

(ξ)) is {(2−2m2 , 4−2m

2 , . . . ,−1)}. Hence, by Langlands

square-integrability criterion (Lemma I.4.11 [MW95]), the automorphic representation

D4mn2n,ψ−1(E∆(τ,m)) is square-integrable. And it is in the discrete spectrum.

113

This completes the proof of Part (1).

Proof of Part (2). The proof is similar as that in Theorem 4.4.6. We just sketch

all the steps needed.

Recall that P 4mn+4n2n = M4mn+4n

2n N4mn+4n2n is the parabolic subgroup of Sp4mn+4n

with Levi subgroup M4mn+4n2n isomorphic to GL2n × Sp4mn. For any

φ ∈ A(N4mn+4n2n (BA)M4mn+4n

2n (F )\Sp4mn+4n(A)τ⊗E∆(τ,m),

the corresponding Eisenstein series defined as follows has a pole at s = m+12 :

E(φ, s)(g) =∑

γ∈P 4mn+4n2n (F )\Sp4mn+4n(F )

λsφ(γg).

The resulting residual representation generated by all the residues is actually E∆(τ,m+1).

Then, by similar argument as in the proof of Theorem 4.4.2, we get that

D4mn+2n2n,ψ1 ◦ D4(m+1)n

2n,ψ−1 (E∆(τ,m+1)) 6= 0,

D4mn+2n2n,ψ1 ◦ D4(m+1)n

2n,ψ−1 (E∆(τ,m+1)) = E∆(τ,m).(4.86)

Note that, as indicated at the end of the proof of Theorem 4.4.2, the irreducibility

of E∆(τ,m) plays an essential role in proving the equality in (4.86).

From Part (1), we see that D4mn2n,ψ1(E∆(τ,m)) is square-integrable and is in the dis-

crete spectrum. For any irreducible component π of D4mn2n,ψ1(E∆(τ,m)), for any φ ∈

A(N4mn+2n2n (BA)M4mn+2n

2n (F )\Sp4mn+2n(A))µψτ⊗π, the corresponding Eisenstein series

defined as follows has a pole at s = m. Denote the residual representation generated by

all the residues by Eτ,π.

Since π is irreducible, also by similar argument as in the proof of Theorem 4.4.2, we

get that

D4mn2n,ψ−1 ◦ D4mn+2n

2n,ψ1 (Eτ,π) 6= 0,

D4mn2n,ψ−1 ◦ D4mn+2n

2n,ψ1 (Eτ,π) = π.(4.87)

Then, using similar argument as in the proof of Theorem 4.5.2, there is an irreducible

component π ofD4mn2n,ψ1(E∆(τ,m)), such that the representation space ofD4mn+4n

2n,ψ1 (E∆(τ,m+1))

has a nontrivial intersection with the representation space of Eτ,π.

114

Since we assume that E∆(τ,m) is irreducible, by the identity in (4.86), we have

E∆(τ,m) = D4mn+2n2n,ψ1 (π) ⊆ D4mn+2n

2n,ψ1 (Eτ,π). Therefore,

D4mn2n,ψ−1(E∆(τ,m)) ⊆ D4mn

2n,ψ−1 ◦ D4mn+2n2n,ψ1 (Eτ,π) = π,

by (4.87). Hence, D4mn2n,ψ−1(E∆(τ,m)) = π, irreducible.

This completes the proof of the theorem.

Remark 4.7.2. Write π = D4n2n,ψ−1(E∆(τ,1)). For

φ ∈ A(N4mn−2n2(m−1)n(A)M4mn−2n

2(m−1)n\Sp4mn−2n(A))µψ∆(τ,m−1)⊗π,

it is easy to see that the corresponding Eisenstein series has a simple pole at m2 . Denote

the residual representation by E∆(τ,m−1),π.

From the proof of Part (1) of Theorem 4.7.1, it is easy to see that if the residual repre-

sentation E∆(τ,m−1),π is irreducible, then actually, we have proved that D4mn2n,ψ1(E∆(τ,m)) =

E∆(τ,m−1),π. And, with the assumption that E∆(τ,m−1),π is irreducible, using similar ar-

gument as in Theorem 4.7.1, we can also prove that D4mn−2n2n,ψ−1 (E∆(τ,m−1),π) is irreducible,

square-integrable and is in the discrete spectrum. And, if we assume in addition that

E∆(τ,m−1) is irreducible, then we have D4mn−2n2n,ψ−1 (E∆(τ,m−1),π) = E∆(τ,m−1).

Chapter 5

On Arthur Classification and

Jiang’s Conjecture

Recently, Arthur gives the classification of the discrete spectrum of symplectic groups

and special orthogonal groups. We recall the case of symplectic groups as follows.

Theorem 5.0.1 (Arthur [Ar12]).

L2disc(Sp2n(F )\Sp2n(A)) = ⊕

ψ∈Ψ2(Sp2n)Πψ(εψ),

where Ψ2(Sp2n) is a set of (global) Arthur parameters: ψ = ψ1 � · · · � ψr, ψi’s are

all different. Each ψi = (τi, bi) is called a simple Arthur parameter, where τi is an

irreducible unitary cuspidal automorphic representation of GLai(A), bi ∈ Z≥1, 2n+ 1 =∑ri=1 aibi, and with certain condition on central characters of τi ([Ar12], Section 1.4).

And, each simple Arthur parameter is of orthogonal type, that is, if τi is of symplectic

type (i.e., L(s, τi,∧2) has a pole at s = 1), then bi is even; if τi is of orthogonal type

(i.e., L(s, τi,Sym2) has a pole at s = 1), then bi is odd. εψ is a linear character defined

explicitly in terms of symplectic ε-factors ([Ar12], Section 1.5).

For each ψ ∈ Ψ2(Sp2n), Πψ(εψ) is called the automorphic L2-packet corresponding

to ψ. Note that Arthur also proved that each π ∈ Πψ(εψ) occurs in the discrete spec-

trum with multiplicity one. Towards understanding Fourier coefficients of automorphic

representations in automorphic L2-packets, Jiang made a conjecture in [J12] as follows:

115

116

Conjecture 5.0.2 (Jiang [J12]). Assume that G is a general linear group or a classical

group. (1) Given an Arthur parameter ψ = �ri=1(τi, bi) ∈ Ψ2(G) for G, let p(ψ) =

[(b1)a1(b2)a2 · · · (br)ar ]. Then, for any π ∈ Πψ(εψ), nm(π) is bounded above by the

partition ηg∨,g(p(ψ)), where ηg∨,g is the Barbasch-Vogan duality map (see Definition A1

[BV85], or Section 3.5 [Ac03]).

(2) There is at lease one π ∈ Πψ(εψ), such that nm(π) = ηg∨,g(p(ψ)).

In the format of Arthur classification, the discrete spectrum of GLn(A) is a disjoint

union of automorphic L2-packets Πψ(εψ) = {E(τ,b)}, corresponding to Arthur parameters

ψ = (τ, b). Based on previous discussion for GLn, nm(E(τ,b)) = {[ab]}. Note that for

the case of GLn, the Barbasch-Vogan duality map ηg∨,g is just transpose, and [ab] =

[ba]t = ηg∨,g([ba]). Therefore, we can summarize the work of Shalika, Piatetski-Shapiro,

Ginzburg, Jiang and Liu as following:

Theorem 5.0.3 (Shalika, Piatetski-Shapiro, Ginzburg, Jiang and Liu). The GLn-case

of Conjecture 5.0.2 is true.

Note that by Section 6.2 of [JLZ12], we know that the Arthur parameter of E∆(τ,m) is

ψ = (τ, 2m)�(1GL1 , 1). Therefore, formally, we can interpret Theorem 4.1.1 as consider-

ing a particular case of Conjecture 5.0.2, and Theorems 4.1.2 and 4.1.3 as understanding

Fourier coefficients of certain irreducible cuspidal automorphic representations in the au-

tomorphic L2-packet corresponding to the Arthur parameter ψ = (τ, 2m) � (1GL1 , 1).

More cases of Conjecture 5.0.2 are considered in [JL13a].

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