Fourier Coefficients of Automorphic Forms and Arthur Classification

126 Pages·2013·0.85 MB·English

Fourier Coefficients of Automorphic Forms and Arthur
Classification
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
(cid:13)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 GL .
n
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 GL (A) are globally generic,
n
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 GL (A) to any automorphic form occurring in the discrete spectrum of GL (A).
n n
Then,wedeterminethesetnm(π)foranyresidualrepresentation∆(τ,m)ofGL (A)
2mn
(withτ anirreducibleunitarycuspidalautomorphicrepresentationofGL (A))andcer-
2n
tainresidualrepresentationE ofSp (A)constructedfrom∆(τ,m)byJiang,Liu
∆(τ,m) 4mn
and Zhang.
Next, we consider certain set of irreducible cuspidal automorphic representations of
Sp (A) which are nearly equivalent to the residual representation E . We show
4mn ∆(τ,m)
that this set decomposes naturally into two disjoint sets, corresponding to certain sets
of irreducible cuspidal automorphic representations of S(cid:102)p (A) and S(cid:102)p (A),
4mn−2n 4mn+2n
respectively. This extends the Ginzburg-Jiang-Soudry correspondences between certain
automorphic forms on Sp4n(A) and S(cid:102)p2n(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 GL . . . . . . . . . . 2
n
1.2 OnExtensionofGinzburg-Jiang-SoudryCorrespondencestoCertainAu-
tomorphic Forms on Sp4mn(A) and S(cid:102)p4mn±2n(A) . . . . . . . . . . . . . 4
1.3 On Arthur Classification and Jiang’s Conjecture . . . . . . . . . . . . . 7
2 On Fourier Coefficients of Automorphic Forms of GL 9
n
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Discrete Spectrum of GL . . . . . . . . . . . . . . . . . . . . . . . . . . 11
n
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 GL . . . . . . . . . . . . . . . . . . . . . . . . . 26
n
2.5.1 Fourier coefficients for GL . . . . . . . . . . . . . . . . . . . . . 27
n
2.5.2 Fourier coefficients for the discrete spectrum of GL . . . . . . . 33
n
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 S(cid:102)p4mn±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 ω -term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
0
4.3.2 ω -term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
1
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
significantarithmeticinformation. Forexample,theSato–TateConjecturesaysthat,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
GL (A) is that cuspidal automorphic forms are globally generic, that is, have non-
n
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 GL (A).
n
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 GL
n
It is a well-known theorem of Shalika and Piatetski-Shapiro that cuspidal automorphic
forms ϕ of GL (A) are globally generic, that is, have non-vanishing Whittaker-Fourier
n
coefficients. ThistheoremisprovedusingtheFourierexpansionofϕalongthestandard
maximal unipotent subgroup U of GL . Since it is not abelian, Shalika and Piatetski-
n n
Shapiro’s idea is to consider abelian subgroups of U consisting of only column elements
n
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 GL (A). Then,
n
(cid:88)
ϕ(g) = Wψ(ϕ,ι (γ)g), (1.1)
n
γ∈Un−1(F)\GLn−1(F)
(cid:32) (cid:33)
γ 0
where ι (γ) = , and
n
0 1
(cid:90)
Wψ(ϕ,g) := ϕ(ug)ψ−1(u)du,
Un(k)\Un(A) Un
with
ψ (u) := ψ(u +u +···+u ). (1.2)
Un 1,2 2,3 n−1,n
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
GL (A) occurring in the discrete spectrum.
n
The explicit construction of the discrete spectrum of GL (A) was conjectured by
n
Jacquet ([J84]) and then proved by Moeglin and Waldspurger ([MW89]). It turns out
that an irreducible automorphic representation π of GL (A) occurring in the discrete
n
spectrum is parametrized by a pair (τ,b) with τ an irreducible unitary cuspidal auto-
morphic representation of GL (A), for some pair a,b of integers such that n = ab. In
a
particular, if π is also cuspidal, then b = 1. The irreducible automorphic representation
parametrized by (τ,b) can be denoted by E (or ∆(τ,b) sometimes), which is called
(τ,b)
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 , n = ab. Then,
(τ,b)
ϕ(g) = (cid:88)Wψ (ϕ,γg), (1.3)
(τ,b)
γ
where γ is a certain rational matrix which will be specified in Theorem 2.3.3, and
(cid:90)
W(ψτ,b)(ϕ,g) := ϕ(ug)ψ(cid:103)Un−1(u)du,
Un(k)\Un(A)
with
n−1 b−1
(cid:88) (cid:88)
ψ(cid:103)Un(u) := ψ( ui,i+1− uja,ja+1). (1.4)
i=1 j=1
Moreover, the Fourier expansion in (1.3) is absolutely convergent and uniformly con-
verges on any compact set in g.
Wψ (ϕ,g) is called a degenerate Whittaker Fourier coefficient of ϕ if b > 1.
(τ,b)
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 only involves “simple roots
(τ,b)
inside the cuspidal support”. Also note that the Fourier coefficients in both (1.1) and
(1.3)havethesameintegrationdomain. Whenb = 1,Theorem1.1.2reducestoTheorem
1.1.1.
If one takes an arbitrary automorphic form of GL (A), one can still apply the idea
n
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 (ϕ,g) will all be killed by cuspidality.
(τ,b)
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 GL ) and Section 3.1
n
(for Sp ), for explicit discussions. Also, see [[J12], Section 4] for more discussions and
2n
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 ) (n = ab). Note that
(τ,b)
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 ) = {[ab]}.
(τ,b)
Now, forE , wehavetwokindsofFouriercoefficients, oneisWψ (ϕ,g), obtained
(τ,b) (τ,b)
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 Sp (A) and
4mn
S(cid:102)p (A)
4mn±2n
Let τ be an irreducible unitary cuspidal automorphic representation of GL (A), with
2n
the properties that L(s,τ,∧2) has a simple pole at s = 1, and L(1,τ) (cid:54)= 0.
2
Let P = M N be the maximal parabolic subgroup of Sp with Levi subgroup M
r r r 2l r
isomorphic to GL ×Sp . Using the normalization in [Sh10], the group XSp2l of all
r 2l−2r Mr
continueshomomorphismsfromM (A)toC×, whichistrivialonM (A)1 (see[MW95]),
r r
will be identified with C by s → λ .
s
Let∆(τ,m)beaSpehresidualrepresentationinthediscretespectrumofGL (A).
2mn
For any φ ∈ A(N (A)M (F)\Sp (A)) , following [L76] and [MW95], an
2mn 2mn 4mn ∆(τ,m)
residual Eisenstein series can be defined by
(cid:88)
E(φ,s)(g) = λ φ(γg).
s
γ∈P2mn(F)\Sp4mn(F)