Approximate Converse Theorem
Min Lee
Submitted in partial fulfillment of the
requirements for the degree of
Doctorate of Philosophy
in the Graduate School of Arts and Sciences
COLUMBIA UNIVERSITY
2011
c
2011
Min Lee
All Rights Reserved
ABSTRACT
Approximate Converse Theorem
Min Lee
The theme of this thesis is an “approximate converse theorem” for globally unramified
cuspidal representations of P GLpn, Aq, n
¥
2, which is inspired by [19] and [3]. For
a given set of Langlands parameters for some places of Q, we can compute ¡ 0 such
that there exists a genuine globally unramified cuspidal representation, whose Langlands
parameters are within of the given ones for finitely many places.
C ONTENTS
Chapter 1. Introduction
1
1.1
Cuspidal representations and Maass forms . . . . . . . . . . . . . . . . . .
1
1.2
Approximate converse theorem . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3
Format of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
Chapter 2. Automorphic functions for SLpn, ZqzGLpn, Rq{ pOpn, Rq R q
9
2.1
Parabolic subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Coordinates for GLpn, Rq . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3
Invariant differential operators . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4
Maass forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5
Hecke operators and Hecke-Maass forms . . . . . . . . . . . . . . . . . . 29
2.6
Eisenstein series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Chapter 3. Automorphic cuspidal representations for A zGLpn, Aq
9
39
3.1
Local representations for GLpn, Qv q . . . . . . . . . . . . . . . . . . . . . 39
3.2
Adelic automorphic forms and automorphic representations . . . . . . . . . 43
3.3
Principal series for GLpn, Qv q . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4
Spherical generic unitary representations of GLpn, Qv q . . . . . . . . . . . 49
3.5
Quasi-Automorphic parameter and Quasi-Maass form . . . . . . . . . . . . 51
Chapter 4. Annihilating operator 6np
58
4.1
Harmonic Analysis for GLpn, Rq{pR Opn, Rqq . . . . . . . . . . . . . . 58
4.2
Annihilating operator 6np
4.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Example for Hδ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
i
Chapter 5. Approximate converse theorem
79
5.1
Approximate converse theorem . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2
Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.3
Proof of Theorem1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Acknowledgments
It is pleasure to thank my advisor Dorian Goldfeld for introducing me to the topic
of this dissertation and for his invaluable advice during the preparation of this document.
I would also like to thank everyone from whom I learned so much during my graduate
studies, especially Andrew Booker, Youngju Choie, Herve Jacquet, Sug Woo Shin, Andreas
Strömbergsson, Akshay Venkatesh and Shouwu Zhang. Finally, I would like to thank my
parents, my grand mother and Jingu for all their support.
ii
1
Chapter 1
I NTRODUCTION
1.1
Cuspidal representations and Maass forms
Let A be the ring of adeles over Q. Let n ¥ 2 be an integer and π be a cuspidal automorphic
representation for A zGLpn, Aq. By the tensor product theorem ([11], [17], [8]), there
exists an irreducible admissible generic unitary local representation πv of Q
v zGLpn, Qv q
¤ 8 of Q, such that π b1v πv .
for each place v
Here the local representation πv is
spherical except at finitely many places. Define
aC pnq :
#
pα1, . . . , αnq P
n
¸
αj
Cn j 1
0
+
.
¤ 8. For any α pα1, . . . , αnq P aCpnq, there exists an unramified character
χα (for the minimal parabolic subgroup of GLpn, Qv q) defined by
Fix a place v
x1
χα
..
.
xn
For each place v
:
n
¹
| xj | v
1
2
pn2j 1q
αj
,
x1
for
..
.
xn
j 1
P GLpn, Qv q
.
¤ 8, if πv is spherical, then there exists σv P aCpnq such that πv πv pσv q,
where πv pσv q is the irreducible spherical principal series representation (or the irreducible
spherical subquotient of the reducible principal series representation), associated to the
character χσv . We call σv the Langlands parameter associated to πv . For a cuspidal automorphic representation π
bv πv define
"
σ : σv
*
πv is spherical and
aC n .
πv πv σv
P pq
p q
Then σ is called the automorphic parameter for π. By the multiplicity one theorem (first
proved by Casselman [5] for GLp2q in 1975, the strong version for GLp2q proved in [17]
2
by Jacquet and Langlands in 1970, and generalized separately by Shalika [25], PiatetskiShapiro [23] and by Gelfand and Kazhdan [10]), it follows that the automorphic parameter
σ is uniquely determined by π.
At the Conference on Analytic number theory in higher rank groups, P. Sarnak suggested the following problem:
Given a positive number X, a set S of places and a representation πv of GLpn, Qv q
P S), give an algorithm to determine whether or not there is a global automorphic
representation π with cpπ q X and σv within of πv for v P S (in whatever reasonable
sense). Here cpπ q is the analytic conductor of π.
(for v
In this thesis, this problem is solved for the globally unramified case.
bv πv be a globally unramified cuspidal representation of AzGLpn, Aq. Then
πv is spherical for every v ¤ 8, and σ tσv P aC pnq | πv πv pσv q, @v ¤ 8u is the
automorphic parameter of π such that π b1v πv pσv q : π pσ q. There exists a unique (up to
constant) Hecke-Maass form fσ (associated to π pσ q) on SLpn, Zq for the generalized upper
half plane Hn R zGLpn, Rq{Opn, Rq. The Whittaker-Fourier coefficients of fσ are deLet π
termined by the automorphic parameter σ. Moreover, there is a one-to-one correspondence
between unramified cuspidal representation of A zGLpn, Aq and Hecke-Maass forms on
SLpn, Zq.
The existence of Maass forms on SLp2, Zq was first proved by Selberg [24] in 1956.
He used the trace formula as a tool to obtain Weyl’s law, which gives an asymptotic count
3
for the number of Maass forms with Laplacian eigenvalue |λ|
¤ X as X Ñ 8. In 2001,
Selberg’s method was extended by Miller [21] to obtain Weyl’s law for Maass forms on
SLp3, Zq. In 2004, Müller [22] further extended Selberg’s method to obtain Weyl’s law for
Maass forms on SLpn, Zq, n ¥ 2.
More recently, in 2007, Lindenstrauss and Venkatesh obtained Weyl’s law for Maass
forms on GpZqzGpRq{K8 [19] where G is a split semisimple group over Q and K8
€G
is the maximal compact subgroup. In the Appendix [19], they explain a constructive proof
of the existence of Maass forms. Our work is inspired by this proof; for a given set of
Langlands parameters, we give an explicit bound, which ensures that there exists a genuine
unramified cuspidal representation within the boundary of the finite subset of the given parameters.
1.2
Approximate converse theorem
The converse theorem of Cogdell and Piatetski-Shapiro ([6], [7]) proves that an L-function
is the Mellin transform of an automorphic form on GLpnq if it satisfies a certain infinite
class of twisted functional equations. In this thesis we introduce the approximate converse
theorem whose main aim is to prove that an L-function is the Mellin transform of a function
which is very close to an actual automorphic form provided a finite set of conditions are
satisfied. We now explicitly describe these conditions and quantify the notion of closeness
in this context.
4
Let M be a set of places of Q including 8. Let n ¥ 2 be an integer. Define
`M :
"
`v P aC pnq πv p`v q is an irreducible unitary spherical
generic representation for Q
v zGLpn, Qv q, pv
*
P Mq
.
Then `M is called the quasi-automorphic parameter for M . For example, the automorphic
parameter for a cuspidal automorphic representation is a quasi-automorphic parameter.
We use the usual quasi-mode construction for a given quasi-automorphic parameter
P `M , for
each v P M . By summing these constructed coefficients, we define a function F` pz q on
the upper half plane Hn R zGLpn, Rq{Opn, Rq which is essentially a finite Whittaker`M . The Whittaker-Fourier coefficient can be constructed from the parameters `v
M
Fourier expansion. The function F`M is called a quasi-Maass form associated to `M . In
general the quasi-Maass form is not automorphic; but it is an eigenform of the Casimir
pj q
operators ∆n (for j
1, 2, . . . , n 1), such that
∆pnj q F`M pz q λp8j q p`8 q F`M pz q
pj q
with eigenvalues λ8 p`8 q
P C.
Also, for each positive integer N
¥ 1, the quasi-Maass
form is an eigenfunction of the Hecke operator TN , such that
TN F`M pz q A`M pN q F`M pz q
with eigenvalues A`M pN q
P C. For each j 1, . . . , n 1, and a prime q P M , define the
Hecke operators
j¸
1
p
jq
Tq p1qk Tqk 1 Tqpjk1q,
k 0
p1q T r , for any integer r ¥ 0 .
q
Tqr
Then
Tqpj q F`M pz q λpqj q p`q q F`M pz q
5
pj q
for λq p`q q P C.
8 and let `M and σM 1 be quasiautomorphic parameters for M and M 1 , respectively. Let S € M X M 1 be a finite subset
including 8. Let ¡ 0. The quasi-automorphic parameters `M and σM 1 are -close for S if
Let M and M 1 be sets of places of Q including
λpj q p`8 q λpj q pσ8 q2
8
8
j 1
n¸1
n
Fix the fundamental domain F
n
t u
2
¸¸
pj q
λ `q
q
q PS, j 1
p q λpqjqpσq q2 .
finite
SLpn, ZqzHn (described in Proposition 2.8; and
based on [14]). Define an automorphic lifting
Fr`M pz q F`M pγz q,
P Hn and γ P SLpn, Zq, which is uniquely determined by γz P Fn. Then Fr` is
automorphic for SLpn, Zq, and square-integrable. But it is neither smooth nor cuspidal in
for z
M
general. We can get the distance between the given quasi-automorphic parameter `M and
a genuine automorphic parameter, by determining the distance between the quasi-Maass
form F`M and its automorphic lifting.
For δ
¡
0 and a finite set S of places of Q including
8, let B npδ; S q be a region
bounded by δ and finite primes in S, around the neighborhood of the boundary of the
fundamental domain Fn , as described in (5.1). Let Hδ be a smooth compactly supported,
bi-pR Opn, Rqq-invariant function on R zGLpn, Rq, which is given in §4.3.
¥ 2 be an integer, M be a set
t`v P aCpnq, v P M u be a quasi-automorphic
Theorem 1.1. (Approximate converse theorem) Let n
of places of Q including
8, and let `M
parameter. Let F`M be a quasi-Maass form associated to `M . Define
n
6 p`8, `pq :
pn
p
t u
2
¹
¹
¹
¤ jk ¤n 1¤i1 ik ¤n
k 1 1 j1
1 pp`8,i1 `8,ik qp`p,j1 `p,jk
q
6
p δ p`8 q 0 where H
pδ
and assume that p6np p`8 , `p q 0 for some prime p P M . Assume that H
is the spherical transform of Hδ . Let S be a finite subset of M including 8.
Then there exists a genuine unramified cuspidal automorphic representation π pσ q for
A zGLpn, Aq with an automorphic parameter σ
tσv P aCpnq, v ¤ 8u such that `M
and σ are -close for S where
: p
Hδ p`8 q
sup Fr`M
p q
B n δ;S
2 pn
p `
2
Cp n, δ; S
M
F`
p
2 ³
³
q
6 p 8, `pq T8 T8 |WJ py; `8q|2
d y
for some
0 δ
where Cp pn, δ; S q
T
max
¤ 21 ln j 1,2,...,n
1
pj q
(
λ `8 max
8
$
'
'
& »
p q 0¤t¤1 '
'
%
Hn ,
u z t
p q
, 1
/
/
.
1d z /
/
-
1
¡ 0 is a constant and WJ p; `8q is the Whittaker function on Hn. Here
¡ 1 is a positive constant determined by δ, the prime p and n.
In this theorem, we see that the closeness for the given quasi-automorphic parameter
and a geuniue automorphic parameter mainly depends on the difference between the quasiMaass form of the given quasi-automorphic parameter and its automorphic lifting on the
neighborhood of the boundary of the fundamental domain Fn . This theorem does not give
uniqueness. However, by Remark 8 in [4], if the difference between Fr`M and F`M is small
enough when S is sufficiently large, then the cuspidal representation should be uniquely determined. The neighborhood for the boundary of the fundamental domain becomes much
larger as the primes in S becomes bigger. The formula for Cp pn, δ; S q is given in (5.5).
A more general result is described in Theorem 5.1, where we take arbitrary δ
¡ 0 and an
arbitrary compactly supported bi-pR Opn, Rqq-invariant smooth function Hδ , such that
7
xδ p`8 q
H
0 for the given Langlnads parameter `8 at 8.
It is an interesting problem to
choose Hδ so that the in Theorem 1.1 (or in Theorem 5.1) is as small as possible.
The constant p6np p`8 , `p q turns out to be an eigenvalue of the annihilating operator
which maps L2 pSLpn, ZqzHn q to cuspidal functions, i.e., 6np Fr`M
pidal automorphic function. The annihilating operator
Hδ
6np,
is a smooth cus-
6np plays an important role in the
proof of the approximate converse theorem. It is constructed by following Lindenstrauss
and Venkatesh [19]. They observe that there are strong relations between the spectrum of
the Eisenstein series at different places. From this observation, they construct the convolution operator ℵ, whose image is purely cuspidal. They use this operator ℵ to get Weyl’s
law for cusp forms in [19]. For example, for automorphic functions on SLp2, Zq, for any
prime p,
ℵ Tp p
?
1
4
?
∆ p 1
4
∆
and it also has a rigorous interpretation in terms of convolution operators. More detailed
explanation and explicit description of 6np are given in chapter 4.
In the 1970’s a number of authors considered the problem of computing Maass forms
on P SLp2, Zq numerically. The first notable algorithms for computing Maass forms on
P SLp2, ZqzH2 are due to Stark in [26] and Hejhal in [15]. In [27], Hejhal’s algorithm was
used by Then to compute large Laplace eigenvalues on P SLp2, ZqzH2 .
In [3], Booker, Strömbergsson and Venkatesh compute the Laplace and Hecke eigenvalues for Maass forms, to over 1000 decimal places, for the first few Maass forms on
P SLp2, ZqzH2 . Their paper is another inspiration and source for the approximate converse
theorem. In particular, we followed the method for verification of their computation in
8
Proposition 2, [3]. The in the approximate converse theorem may recover (38) in [3]
weakly, with good choices for δ and Hδ , for the case n 2 and S
δ
8. In [3] they choose
¤ 4?1 λ for a given Laplacian eigenvalue λ, and Hδ as
Hδ pz q where z
#
3 1 2δ 2
0,
1
2
y
x2
y
1
y
2
1
,
2
if 21 y xy
otherwise,
1
y
1 ¤ δ2
2
;
p y0 x1 q P H2. See [3] for more details.
Recently, Booker and his student Bian computed the Laplace and Hecke eigenvalues for
Maass forms on P SLp3, ZqzH3 [2], [1]. Moreover, Mezhericher presented an algorithm for
evaluating a (quasi-)Maass form for SLp3, Zq in his thesis [20]. We expect that we might
use the approximate converse theorem to certify Bian’s computations.
1.3
Format of Thesis
The main theorem is stated and proved in chapter 5. In chapter 2, we review the theory of
automorphic forms for SLpn, ZqzHn and introduce notations. The main reference for this
chapter is [12]. In chapter 3, we review the theory of automorphic cuspidal representation
for A zGLpn, Aq. The main reference for this chapter is [13]. In §3.5, we define the
quasi-automorphic parameter and the quasi-Maass form of the given quasi-automorphic
parameter. The annihilating operator
6np is defined in chapter 4.
annihilating operator are proved in §4.2.
Several properties of the
9
Chapter 2
AUTOMORPHIC FUNCTIONS FOR
SLpn, ZqzGLpn, Rq{ pOpn, Rq Rq
2.1
Parabolic subgroups
Let n ¥ 1 be an integer. For an integer 1 ¤ r
¤ n, define an ordered partition of n to be a
set of integers pn1 , . . . , nr q where 1 ¤ n1 , . . . , nr ¤ n and n1 nr n.
Definition 2.1. (Parabolic subgroups) Fix an integer n
¥
1 and let R be a commu-
tative ring with identity 1. A subgroup P of GLpn, Rq is said to be parabolic if there
exists an ordered partition pn1 , . . . , nr q of n and an element g
P
GLpn, Rq such that
gPn ,...,n pRqg1 where Pn ,...,n pRq is the standard parabolic of GLpn, Rq associated to the partition pn1 , . . . , nr q defined by
,
$
A
/
'
1
.
&
.
..
Pn ,...,n pRq : P GLpn, Rq Ai P GLpni , Rq, 1 ¤ i ¤ r . (2.1)
/
'
P
1
1
r
r
r
1
%
-
Ar
The integer r is termed the rank of the parabolic subgroup Pn1 ,...,nr pRq. Define
Mn1 ,...,nr pRq :
$
'
& A1
'
%
..
Ai
.
Ar ,
/
.
P GLpni, Rq, 1 ¤ i ¤ r/
-
(2.2)
to be the standard Levi subgroup of Pn1 ,...,nr pRq and call gMn1 ,...,nr pRqg 1 a Levi factor of
P . Define
Un1 ,...,nr pRq :
$
'
& In1
'
%
..
.
Inr
P
p
GL n, R
where Ik is the k k identity matrix for an integer k
,
/
.
q/ ,
-
(2.3)
¥ 1, to be the unipotent radical of
Pn1 ,...,nr pRq. Call gUn1 ,...,nr pRqg 1 the unipotent radical of P .
10
Two standard parabolic subgroups Pn1 ,...,nr pRq and Pn11 ,...,n1r pRq of GLpn, Rq corre-
n11 n1r are said to be associated if
tn1, . . . , nr u tn11, . . . , n1r u. We write Pn ,...,n pRq Pn1 ,...,n1 pRq, if Pn ,...,n and Pn1 ,...,n1
sponding to the partitions n
n1 nr
r
1
r
1
r
1
r
1
are associated.
Let n
¥ 1 be an integer and fix an ordered partition pn1, . . . , nr q of n.
For each j
1, . . . , r define a map
mnj : Pn1 ,...,nr pRq Ñ GLpnj , Rq,
mn1 pg q
g
...
mn2 pg q . . .
..
.
such that
..
.
(2.4)
Pn1 ,...,nr R
P
p q
pmn pgq P GLpnj , Rqq.
j
mnr pg q
The standard parabolic subgroup associated to the partition n n1
nr (denoted by
Pn1 ,...,nr pRq) is defined to be the group of all matrices of the form
mn1 pg q
g
mn pg q
2
where mni pg q P GLpni , Rq for i 1, . . . , r.
...
...
..
.
..
.
mnr pg q
GL n, R
P
p
q
(2.5)
11
Let n ¥ 1 be an integer. For r
n, let
N pn, Rq : U1,1,...,1 pRq
Apn, Rq :
$
1
x
x
.
.
.
x
'
1,2
1,3
1,n
'
'
'
1 x2,3 . . . x2,n '
&
.. x
..
i,j
.
. '
'
'
1 x1n,n '
'
%
1
$
'
& a1
.
..
M1,1,...,1 R
0
'
%
p q
an
(2.6)
,
/
/
/
/
/
.
P R for 1 ¤ i j ¤ n/ € GLpn, Rq,
/
/
/
/
-
,
/
.
aj P R for j 1, . . . , n/ € GLpn, Rq,
-
and
P pn, Rq P1,1,...,1 pRq N pn, Rq Apn, Rq.
Here P pn, Rq is called the minimal parabolic subgroup of GLpn, Rq.
2.2
Coordinates for GLpn, Rq
¥ 2 be an integer. Define the
generalized upper half plane Hn to be a set of matrices z P GLpn, Rq and z xy such
Definition 2.2. (Generalized Upper half plane) Let n
that
1 x1,2 x1,3 . . . x1,n
1 x2,3 . . . x2,n .
.
.
.
x
P N pn, Rq
.
.
1 xn1,n 1
and
y1 yn1
y
(2.7)
..
.
y1
A n, R ,
P p
1
q
py1, . . . , yn1 ¡ 0q.
(2.8)
12
By the Iwasawa Decomposition,
GLpn, Rq N pn, RqApn, RqOpn, Rq,
(2.9)
GLpn, Rq{ pR Opn, Rqq, i.e., for any g P GLpn, Rq there exist unique x P
N pn, Rq as in (2.7), and y P Apn, Rq as in (2.8), some k P Opn, Rq and a positive real
so Hn
number d, such that
g
d xy k d z k,
pz xy P Hnq.
(2.10)
Remark 2.3. Let Wn denote the Weyl group of GLpn, Rq, consisting of all n n matrices
in SLpn, ZqX Opn, Rq which have exactly one 1 in each row and column. The Weyl group
Wn acts on the diagonal matrices as a permutation group. For any w
P Wn there exists a
unique permutation σw on n symbols such that
a1
w. ..
:
.
a1
w
..
an
1
w
.
aσw p1q
..
for any diagonal matrix
..
with ai
.
an
(2.11)
aσw pnq
an
a1
.
P R (or ai P C).
By the Cartan decomposition,
GLpn, Rq Opn, RqApn, RqOpn, Rq.
So for any g
(2.12)
P GLpn, Rq there exist k1, k2 P Opn, Rq and a unique Apgq P A1pn, R q (up
to the conjugation by the Weyl group Wn ) such that
g
| det g| k1 Apgq k2,
1
n
(2.13)
where
A1 pn, R
q
a P Apn, R
q | det a 1
(
.
(2.14)
13
For an integer n ¥ 1, define the set
apnq : tα pα1 , . . . , αn q P Rn
| α1 αn
0u .
(2.15)
¥ 2 be an integer. For g P GLpn, Rq define ln : GLpn, Rq Ñ apnq
Definition 2.4. Let n
such that
lnpApg qq : pln a1 , . . . , ln an q P apnq
a1
(2.16)
¡ 0 as in (2.13). This lnpApgqq is uniquely
determined up to the Weyl group action, i.e., permutation for Apg q. Moreover, ln a1 ln an 0 since det Apg q 1.
where Apg q
..
with a1 , . . . , an
.
an
Conversely, define exp : apnq Ñ A1 pn, R
q such that
eh1
expphq : q
(2.17)
|| ln Apgq|| : pln a1q2 pln anq2
(2.18)
..
P p
1
A n, R
.
hn
e
for any h ph1 , . . . , hn q P apnq.
For any g
P GLpn, Rq define
a
for Apg q a1
..
with a1 , . . . , an
.
an
¡ 0 and a1 an 1 as in (2.13).
Lemma 2.5. (Relations between the coordinates for generalized upper half plane and
Cartan decomposition) Let n
¥ 2 be an integer. For any g P GLpn, Rq, by the Iwasawa
decomposition and Cartan decomposition, we have
g
d xy kIwa p| det g|q
e α1
1
n
k1 ..
k2 ,
.
e
αn
pkIwa, k1, k2 P Opn, Rqq,
14
where d ¡ 0 with dn det y
1
xi,j
x
..
.
| det g|,
P p
y1 yn1
q
N n, R ,
y
..
P p
A n, R
.
1
q,
1
and lnpApg qq pα1 , . . . , αn q P apnq. Then
e2|| ln Apgq||
¤ y1 ¤ e2|| ln Apgq||,
e4|| ln Apgq|| ¤ yj ¤ e4|| ln Apgq|| ,
Proof. Let z : xy
(2.19)
p for j 2, . . . , n 1q.
P Hn. Then for g P GLpn, Rq,
z
xy pdet yq
1
n
eα1
..
k1 1
k2 kIwa ,
.
e αn
and
z tz
xy2 tx pdet yq
e2α1
2
n
..
k
t
k.
.
e2αn
So
e2α1
xy 2 pdet y q n t x k 2
..
t
k,
.
e2αn
for k
k1 P Opn, Rq. Compare the diagonal parts. For the left hand side, for j 1, . . . , n,
we have
yj1
on the diagonal, where yj1
hand side, for j
yj1 1 x2j,j
1
yn1 x2j,n ,
pxn,n 1q
pdet yq py1 ynj q2 and yn1 pdet yq
2
n
1, . . . , n, we have
2 2α1
kj,1
e
2
kj,n
e2αn
2
n
. For the right
15
on the diagonal, where
k1,1 . . . k1,n
k2,1 . . . k2,n k .
P Opn, Rq.
.
.. ..
...
kn,1 . . . kn,n
So for any j
1, . . . , n,
yj1
2 2α
e
¤ yj1 yj1 1x2j,j 1 yn1 x2j,n kj,1
2
2
¤ pkj,1
kj,n
qe2|| ln Apgq|| e2|| ln Apgq||.
1
2
kj,n
e2αn
Since || ln Apg q|| || ln Apg 1 q||, we also have
yj11
¤ e2|| ln Apgq||,
p for j 1, . . . , nq
so
e2|| ln Apgq||
For j
¤ yj1 ¤ e2|| ln Apgq||,
p for j 1, . . . , nq.
1, . . . , n 1, we have
|| ln Apgq|| ¤ n1 lnpdet yq
ln y1
ln ynj
¤ || ln Apgq||
and
|| ln Apgq|| ¤ n1 lnpdet yq ¤ || ln Apgq||.
Therefore, we have
2|| ln Apgq|| ¤ ln y1 ¤ 2|| ln Apgq|||
and
4|| ln Apgq|| ¤ ln yj ¤ 4|| ln Apgq||,
p for j 1, . . . , n 1q.
16
Definition 2.6. (Siegel Sets) Fix a, b ¥ 0. We define the Siegel set Σa,b
of all matrices of the form
€ Hn to be the set
1 x1,2 x1,3 . . . x1,n
y1 y2 yn1
1 x2,3 . . . x2,n y1 y2 yn2
.. .
.
.
.
.
.
.
y1 1 xn1,n
1
1
with |xi,j | ¤ b for 1 ¤ i j
P Hn,
¤ n and yi ¡ a for 1 ¤ i ¤ n 1.
Definition 2.7. (Fundamental Domain) For n
¥ 2, we define Fn to be the subset of the
Siegel set Σ ?3 , 1 , satisfying:
2
2
• for any z
P Hn, there exists γ P SLpn, Zq such that γz P Fn,
• for any z
P Fn, γz R Fn for any γ P SLpn, Zq (with γ In).
Then Fn becomes a fundamental domain for SLpn, Zq and
Fn
SLpn, ZqzHn.
We introduce the partial Iwasawa decomposition for Hn to describe the fundamental
domain explicitly. Let n ¥ 2 be an integer. For any z
P Hn, we may write
y1 yn1
y1 yn2
1 x1,2 . . . x1,n1 x1,n
1 x2,3
...
x2,n
.. .
.
..
..
z
. 1
xn1,n 1
where
x1,n
.. In1
. y1 z 1
xn1,n 0 ... 0
1
0 ... 0
..
.
y1
1
0
.. .
(2.20)
0
1
1 . . . x1,n1
y2 yn1
.
1
n1
.
..
..
.. z . P H .
1
1
17
The following proposition is an interpretation of §2 in [14].
Proposition 2.8. (Explicit Description of the Fundamental domain) Let n
¥ 2 be an
integer and Fn be the closure of the fundamental domain Fn .
(1) for n 2, the closure of the fundamental domain F2 is the set of z
for x, y
p 10 x1 q
y 0
0 1
P R and y ¡ 0 satisfying
x2
y2
¥ 1 and |x| ¤ 12 .
(2) for n ¡ 2, the closure of the fundamental domain Fn is the set of
x1
.. . y1 z 1
z In1
xn1 0 ... 0
1
0 ... 0
for x1 , . . . , xn1
(i) z 1
b1
..
.
(ii) for any
bn1
a
c1 ... cn1
c1 x 1
1
i.e., if z
pa
0
1
P R and y1 ¡ 0 satisfying the following conditions:
P Fn1;
pa
0
.. .
P GLpn, Zq{tInu, we have
cn1 xn1 q
x1,2 ... x1,n1
1 ... x2,n1
..
..
.
.
1
x1
x2
..
.
xn
1
2
y12
pc1
y1 yn1
. . . cn1
y1 yn2
qz 1 t z 1
..
.
c1
..
.
cn1
¥ 1;
P Hn, then
1
cn1xn1,nq2
y12 c21 py2 yn1 q2 pc1 x1,2 c2 q2 py2 yn2 q2 pc1x1,j cj xj1,j cj q2py1 ynj q2 pc1x1,n1 cn2xn2,n1 cn1q2 ¥ 1;
c1 x1,n
P Hn
18
(iii) |xj | ¤
1
2
for j
1, . . . , n 1.
Proof. The proof is again an interpretation of §2 in [14]. Let SP n denote the space of
quadratic forms of determinant 1, which is identified by
Opn, RqzGLpn, Rq
Opn, Rq g
t
SP n
g g : Z.
P GLpn, Zq{ tInu acts on SP n discontinuously by Z ÞÑ Z rγ s :
Then γ
Z
Ñ
ÞÑ
P SP n can be represented as
t
γZγ. Every
1 xn1 . . . x1
y 1 0
...
0
0
0
Z .
.
1
..
In1
y n1 Z 1 ..
0
0
P SP n1 and x1, . . . , xn1 P R by the partial Iwasawa decomposition
(2.20). By repeating this, we can get the Iwasawa decompositon for Z P SP n , namely
with y
¡ 0,
Z1
1
Z
y 1 y12
py1y2q2
..
.
py1 yn1q
2
1
..
.
xi,j
..
.
..
.
1
¡ 0, xi,j P R, (for 1 ¤ i j ¤ n). In [14], by using the partial Iwasawa decomposition,
the fundamentaldomain
n SP n { pGLpn, Zq{tIn uq is described
1 x F
... x y 0
...
0
0
0
as the set of all Z P SP n satisfying:
..
..
1
.
I .
y
Z
with y1 , . . . , yn1 , y
1
n 1
1
n 1
(i) Z 1
(ii) for any
a tb
c D
n 1
0
0
P Fn ;
1
P GLpn, Zq{tInu, a P Z, b, c 1, Zq, we must have
pa
xn1 cn1
cn1 ..
.
c1
P Zn1 and D P M atpn x1c1q2
y n 1
n
cn1
cn1
1 . . . . c1 Z .. c1
¥ 1;
19
(iii) for j
1, 2, . . . , n 2, we have
0 ¤ xn1
Let
wn
1
..
.
|xj | ¤ 21 .
12 ,
1
wn1
1
wn1
1
.
(2.21)
We may identify SP n by Hn via
Hn Ý

Ñ
ÞÑ
z
Then for any z
Z
wn
xn1,n
0 ... 0
1
1
n
2
n
where
So for γ
x1,n
..
.
I n 1
wn pdet zq
pdet zq
Z1
pdet zq z
z
1
n
t
n
SP
0
.
y1 z 1 ..
0
0 ... 0 1
pdet zq
1
n
t
pdet zq n1 z wn : Z.
P Hn and z1 P Hn1,
z wn
1 xn1 . . . x1
0
.
2
P SP n ,
1
.
n
p
n
1
q
pdet zq Z
.
In1
0
wn1 pdet z1q
z1
1
n 1
t
pdet z1q
z 1 wn1
1
n 1
P SP n1.
P SLpn, Zq,
γz
and wn t γwn
ÞÑ wn pdet zq
P SLpn, Zq.
1
n
γz
t
pdet zq
1
n
γz wn
Z rwn tγwns,
Therefore by using the fundamental domain Fn for SP n , we
can get the explicit description for the fundamental domain Fn
SLpn, ZqzHn.
Consider GLpn, Rq or GLpn, Rq{R , with Haar measure dg, which is normalized as
»
p
q
O n,R
dk
»
O n,R R
p
q{
dk
1.
20
By [12], the left invariant GLpn, Rq-measure d z on Hn can be given explicitly by the
formula
d z
dx d y,
where d x 2.3
n
1
¹
kpnkq1 dy .
dxi,j , d y yk
k
1¤i j ¤n
k 1
(2.22)
¹
Invariant differential operators
¥ 2 be an integer and glpn, Rq be the Lie algebra of GLpn, Rq with the Lie bracket
r, s given by rα, β s αβ βα for α, β P glpn, Rq. The universal enveloping algebra
of glpn, Rq can be realized as an algebra of differential operators Dα acting on smooth
functions f : GLpn, Rq Ñ C. The action is given by
Let n
B
B
Dα f pg q : f pg expptαqq f pg
Bt
Bt
t0
tgα q
(2.23)
t 0
P glpn, Rq. For any α, β P glpn, Rq, Dα β Dα Dβ and Drα,βs rDα, Dβ s Dα Dβ Dβ Dα . Here is the composition of differential operators. The differential
operators Dα with α P glpn, Rq generate an associative algebra Dn defined over R.
for α
For 1
¤
i, j
¤
elsewhere. Let Di,j
P glpn, Rq be the matrix with 1 at the i, jth entry and 0
for 1 ¤ i, j ¤ n.
n, let Ei,j
DE
i,j
Definition 2.9. (Casimir operators) Let n
pj q
¥ 2 be an integer.
For j
1, . . . , n 1, we
define Casimir operators ∆n given by
1
∆pj q n
p1q
j
n
¸
1 i 1
1
n
¸
ij
Let ∆n : ∆n be the Laplace operator.
1
Di1 ,i2
Di ,i Di
2 3
j 1 ,i1
.
(2.24)
21
For n ¥ 2, define Z pDn q to be the center of the algebra of differential operators Dn . It
p1q
pn1q
P Z pDnq. Moreover every dif-
is well known that the Casimir operators ∆n , . . . , ∆n
ferential operator which lies in Z pDn q can be expressed as a polynomial (with coefficients
p1q
pn1q , i.e.,
in R) in the Casimir operators ∆n , . . . , ∆n
Z pDn q R ∆pn1q , . . . , ∆pnn1q
(see [12]).
There is a standard procedure to construct simultaneous eigenfunctions of all differential operators of D
P Z pDnq.
Let n
¥ 2 be an integer and ν pν1, . . . , νn1q P Cn1.
Define
Iν pg q :
n
¹1 n¹1
b
yi i,j
νj
(2.25)
i 1 j 1
1
x1,2
1
x1,n
x2,n
.
..
. ..
1
y1 yn1
.
| det g|
y
yi ¡ 0, with 1 ¤ i j ¤ n and k P Opn, Rq. Here
where g
1
n
"
bi,j
For j
pijn iqpn j q
..
k
P
j
j
¤n
¥n
1
GLpn, Rq for xi,j , yi
P
R,
1
if i
if i
.
1, . . . , n 1, let
Bj pν q : j n¸j
k νk
pn j q k 1
Then
Iν ..
.
y1
1
Clearly, Bj pν
µq Bj pν q
k νnk .
k 1
y1 yn1
j¸1
Bj pµq for any ν, µ P Cn1 .
n
¹1
j 1
p q.
B ν
yj j
(2.26)
22
pν1, . . . , νn1q P Cn1, the function Iν is an eigenfunction of Z pDnq. Define
λν : Z pDn q Ñ C to be the character such that
For ν
DIν
for any D
λν pDqIν
(2.27)
P Z pDnq. For any D1, D2 P Z pDnq, we have
λν pD1 D2 q λν pD1 q λν pD2 q,
λν pD1
D2 q λν pD1 q
λν pD2 q.
The Weyl group Wn (defined in Remark 2.3) acts on ν
way. For any w
pν1, . . . , νn1q in the following
P Wn ,
w.ν : µ pµ1 , . . . , µn1 q P Cn1 if and only if Iν 1 py q Iµ 1 pwy q,
for y
y1 yn1
n
..
.
1
n
(2.28)
¡ 0. Here ν n1 pν1 n1 , . . . , νn1 n1 q. Then
λw.ν for any ν P Cn1, i.e., Iν and Iw.ν have the same
, y1 , . . . , yn1
P Wn, we have λν
eigenvalues for Z pDn q.
for any w
Let a pnq apnq as in (2.15) and
aC pnq : a pnq bR C
(2.29)
t` p`1, . . . , `nq P Cn | `1 Definition 2.10. Let n ¥ 2 and ν
`n
0u .
pν1, . . . , νn1q P Cn1. The Langlands parameter for
ν is defined to be
`8 pν q p`8,1 pν q, . . . , `8,n pν qq P aC pnq,
where
$
&
n2 1
Bn1 pν q,
n 2j 1
`8,j pν q :
Bnj pν q Bnj
% n1 2
B1pν q,
2
1
pν q
for j 1,
for 1 j n,
for j n.
(2.30)
23
Here Bj (for j
1, . . . , n 1) are defined in (2.26). Then `8,1pν q Remark 2.11.
`8,n pν q 0.
(i) The Langlands parameter is the same parameter defined in p.314-315,
[12].
(ii) For any ν
pν1, . . . , νn1q P Cn1,
n
¹1
py1 ynj q`8
,j
pν q n2j2
1
p
n
°n j
¹
`8,k pν q
y k 1
j
n 2k 1
2
q
j 1
j 1
n
¹1
p q.
B ν
yj j
j 1
(iii) Let f : Hn
Ñ C be an eigenfunction of Z pDnq of type ν. Define
`8 pf q : `8 pν q.
(2.31)
(iv) The Weyl group Wn acts on the Langlands parameter as a permutation group. For
any w
P Wn there exists a permutation σw on n symbols such that
`8 pw.ν q σw p`8 pν qq p`8,σw p1q pν q, . . . , `8,σw pnq pν qq.
(2.32)
p`8,1, . . . , `8,nq P aCpnq with
`8,n 0, we can get ν P Cn1, satisfying (2.30). For j 1, . . . , n 1,
(v) Since (2.26) and (2.30) are linear, from the given `8
`8,1
let
νj p`8 q :
1
p`8,j
n
`8,j
1
Then `8 pν q ν p`8 pν qq. For j
1q ,
ν p`8 q : pν1 p`8 q, . . . , νn1 p`8 qq.
(2.33)
1, . . . , n 1, define
λp8j q p`8 q : λν p`8 q p∆pnj q q.
The following Lemma is given in [12].
(2.34)
24
¥
Lemma 2.12. (Eigenvalues for Di,j ) Let n
1 ¤ i, j
2 and ν
pν1, . . . , νn1q P
Cn1 . For
¤ n and k 1, 2, . . . , we have
k
Di,j
Iν
"
n 2j 1
2
0,
`8,j pν q
k
Iν ,
if i j,
otherwise.
(2.35)
Here `8 pν q p`8,1 pν q, . . . , `8,n pν qq P aC pnq is defined in (2.30).
Proposition 2.13. (The Laplace eigenvalue) Let n
¥
2 and ν
pν1, . . . , νn1q.
The
Laplace eigenvalue is
λν p∆n q 1
1
n3 n `8,1 pν q2
24
2
`8,n pν q2 ,
where ∆n is the Laplace operator in Definition 2.9. Here `8 pν q
(2.36)
p`8,1pν q, . . . , `8,npν qq
is the Langlands parameter for ν, defined in (2.30).
Proof. For any y
y1 yn1
..
.
1
P Apn, R q with y1, . . . , yn1 ¡ 0, consider ∆nIν pyq.
Then
∆n Iν py q 1¸¸
Di,j
2 i1 j 1
n
12
#
n
n
¸
Dj,j
Dj,iIν pyq
Dj,j Iν pyq
j 1
¸
¤ ¤
pDi,j Dj,i
+
Dj,i Di,j q Iν py q .
1 i j n
For 1 ¤ i, j, i1 , j 1 ¤ n, we have rEi,j , Ei1 ,j 1 s δi1 ,j Ei,j 1 δi,j 1 Ei1 ,j where δi,j
So
rDi,j , Di1,j1 s DrE
1 1s
i,j ,Ei ,j
For 1 ¤ i j
δi1,j Di,j1 δi,j1 Di1,j .
¤ n, we have
Di,j
Dj,i
Dj,i Di,j
2Di,j Dj,i
Dj,j
Di,i,
"
1,
0,
if i j,
.
otherwise
25
and
Dj,iIν py
q
Di,j
BBt BBt Iν 1
2
For 1 ¤ i j
pI2
y1 yn1
..
t1
.
..
.
1
t2 Ej,i q
1
t2
1
..
. 1 ..
.
t1 t2 0
1
1
1
¤ n, we have
t1 Ei,j qpI2
.
1 . . . t1
.
..
. ..
1
..
.
..
.
pt22
1q 2 . . .
1
..
.
t2
pt22 1q 12
..
.
pt22
1q 2
1
..
.
1
1
1
..
.
pt22
1q 2 . . .
1
..
t1 pt22
t2
pt22 1q 12
.
pt22
..
.
..
1q
1q 2
1
1
2
.
mod Opn, Rq,
1
so Di,j
Dj,iIν pyq 0 and
1
∆n Iν py q 2
#
¤ ¤
pDj,j Di,iq 1 i j n
2
Dj,j
Iν py q
j 1
Here
¸
n
¸
n
¸
j 2
¤ ¤
Dj,j
pn 2j
j 1
.
n¸1
i 1
j 2
1qDj,j .
n
¸
Di,i pj 1qDj,j i 1
n
¸
pDj,j Di,iqIν pyq
+
1 i j n
j¸1
¸
pn iqDi,i
26
Therefore,
1
∆n Iν py q 2
#
n
¸
2
Dj,j
pn 2j
1qDj,j Iν py q
j 1
n
¸
21
j 1
#
n 2j
2
pn 2j
2
`8,j pν q
1
1q
"
n 2j
2
¸
pn 2j
21
`8,j pν q2 4
j 1
n
2.4
#
+
n
1¸
1q2
*
`8,j pν q
1
*
Iν py q
Iν py q
+
1 3
`8,j pν q2 Iν py q.
p
n nq 24
2 j 1
Maass forms
Definition 2.14. (Automorphic Function) For an integer n ¥ 2, an automorphic function
for SLpn, Zq is a function f : Hn
Ñ C such that
f pγz q f pz q
for any γ
P SLpn, Zq and z P Hn.
Consider L2 pSLpn, ZqzH2 q to be the space of automorphic functions f : Hn
satisfying
||f || :
2
2
For f1 , f2
»
p qz
SL n,Z Hn
|f pzq|2
d z
Ñ
C
8.
P L2 pSLpn, ZqzHnq, define the inner product
hf1 , f2 i :
»
p qz
SL n,Z Hn
f1 pz qf2 pz q d z.
(2.37)
27
Definition 2.15. (Cuspidal function) Let n
¥ 2 be an integer and let f : Hn Ñ C be an
automorphic function for SLpn, Zq. The function f is cuspidal if
»
f puz q du 0
pRq
pZqqzU
pSLpn,ZqXU
nr n and r ¥ 2. Here Un ,...,n
n1 ,...,nr
n1 ,...,nr
for any partition n1
(2.38)
1
r
is the unipotent radical
defined in (2.3).
Let L2cusp pSLpn, ZqzHn q denote the space of automorphic cuspidal functions. Let
8 pSLpn, ZqzHn q) denote the space of smooth automorphic
C 8 pSLpn, ZqzHn q (resp. Ccusp
functions (resp. smooth automorphic cuspidal functions) and C 8
(resp. C 8
X L2 pSLpn, ZqzHnq
X L2cusp pSLpn, ZqzHnq) denote the space of smooth automorphic functions
(resp. smooth automorphic cuspidal functions), which are square integrable.
Let n
¥ 2 be an integer and f P L2cusp pSLpn, ZqzHnq. By Theorem 5.3.2 [12], f has
the following Fourier expansion:
f pz q
¸
8̧
P p qz p q 8̧
¸
γ N n 1,Z SL n 1,Z m1 1
P p qz p q
γ N n 1,Z SL n 1,Z m1 1
8̧
¸
Wf
γ
1
mn2 1 mn1 0
8̧
¸
mn2 1 mn1 0
1
(2.39)
z; pm1 , . . . , mn1 q
Wf py γ ; pm1 , . . . , mn1 qq
e2πipm x γ
1 n 1,n
where p γ
mn2 xγ2,3 mn1 xγ1,2
q
q z xγ yγ P Hn for z xy P Hn with x, y, xγ , yγ given as in (2.7) and (2.8).
28
Here the sum is independent of the choice of coset representative γ. Further
Wf pz; pm1 , . . . , mn1 qq
:
1
(2.40)
»
f puz qe2πipm1 un1,n pN pn,RqX
SLpn,ZqqzN pn,Rq
»
f puzqe2πipm1un1,n ZzR
ZzR
»
u1,2 ... u1,n
..
.. ..
.
.
.
1 un1,n
1
mn2 u2,3 mn1 u1,2
q d u
q d u
P N pn, Rq and du ±1¤i k¤n
N pn, Rq is the minimal unipotent radical defined in (2.6).
with u
mn2 u2,3 mn1 u1,2
dui,j as in (2.22). Here
¥ 2 be an integer and ν pν1, . . . , νn1q P Cn1.
A smooth automorphic function f : Hn Ñ C is a Maass form of type ν if:
Definition 2.16. (Maass forms) Let n
(i) f is an eigenform of Z pDn q i.e., for j
∆pnj q f
1, . . . , n 1,
λν p∆pnjqqf,
where λν is the Harish-Chandra character of type ν defined in (2.27);
(ii) f is square-integrable, i.e.,
||f || 2
2
»
p qz
SL n,Z
Hn
|f pzq|2
d z
8;
(iii) f is cuspidal.
¥ 2 be an integer. For each ν P
Cn1 and 1, we define a function WJ p ; ν, q : Hn Ñ C such that
1 u1,2 . . . u1,n
p
1q
»
1 . . . u2,n 1
WJ pz; ν, q Iν .. z (2.41)
..
..
Definition 2.17. (Jacquet’s Whittaker function) Let n
t
p
N n,R
q
n
u
2
.
1
exp p2πi pu1,2 u2,3 un1,nqq du,
.
.
1
29
for
³
p
q d u
N n,R
³8
³8
±
8 8 1¤i j ¤ndui,j . The function WJ pz; ν, q is called Jacquet’s
Whittaker function or Whittaker function of type ν.
Remark 2.18. The Whittaker function of type ν is an eigenfunction of Z pDn q of type ν,
i.e., for any D
P Z pDnq,
DWJ pz; ν, q λν pDq WJ pz; ν, q.
Let f be a Maass form of type ν
pν1, . . . , νn1q P Cn1. Then by (9.1.2) [12], f has
a Fourier-Whittaker expansion of the form
f pz q 8̧
¸
P p qz p q
γ N n 1,Z SL n 1,Z m1 1
m1 |mn1 |
WJ ..
8̧
8̧
Af pm1 , . . . , mn1 q
±n1
kpnkq{2
k1 |mk |
mn2 1 mn1 0
mn1
γ
z; ν,
1
mn1
.
m1
|
|
(2.42)
1
where Af pm1 , . . . , mn1 q
P C.
Here Af pm1 , . . . , mn1 q is called the pm1 , . . . , mn1 qth
Fourier coefficient of f for each 1 ¤ m1 , . . . , mn2
2.5
P Z and nonzero mn1 P Z.
Hecke operators and Hecke-Maass forms
Recall the general definition of Hecke operators from [12]. Let X be a topological space.
Consider a group G that acts continuously on X. Let Γ be a discrete subgroup of G, and
set
CG pΓq : g
P G rΓ : pg1Γgq X Γs 8 and rg1Γg : pg1Γgq X Γs 8
to be the commensurator group of Γ in G. For any g
P CGpΓq, we have a decomposition of
a double coset into disjoint right cosets of the form
ΓgΓ ¤
i
(
Γαi .
30
For each such g
P CGpΓq, the Hecke operator Tg : L2pΓzX q Ñ L2pΓzX q is defined by
Tg f pxq ¸
f pαi xq,
i
where f
P L2pΓzX q, x P X. Fix a semiring ∆ where Γ € ∆ € CGpΓq. The Hecke ring
consists of all formal sums
for ck
P Z and gk P ∆.
¸
ck Tgk
k
Since two double cosets are either identical or totally disjoint, it
follows that unions of double cosets are associated to elements in the Hecke ring. If there
exists an antiautomorphism g
for every g
ÞÑ g satisfying pghq hg, Γ Γ and pΓgΓq ΓgΓ
P ∆, the Hecke ring is commutative.
Definition 2.19. (Hecke Operator TN ) Let f : Hn
N
¥ 1, we define a Hecke operator
TN f pz q :
Ñ C be a function. For each integer
c1 c1,2 . . . c1,n
¸
c2 . . . c2,n 1
f . z .
n 1
..
±n c N,
. .. N 2
j
j 1
0¤ci,j cj p1¤i j ¤nq
cn
(2.43)
Clearly T1 is the identity operator.
For n 2, the Hecke operators are self-adjoint with respect to the inner product (2.37),
P L2 pSLp2, ZqzH2q and any integer N ¥ 1, we have hTN f1, f2i hf1 , TN f2 i. For n ¥ 3, the Hecke operator is no longer self-adjoint, but the adjoint operator
i.e., for any f1 , f2
is again a Hecke operator and the Hecke operator commutes with its adjoint, so it is a
normal operator. The following Theorem is proved in Theorem 9.3.6, [12].
Theorem 2.20. Let n ¥ 2 be an integer. Consider the Hecke operators TN for any integer
N
¥ 1, defined in (2.43). Let TN be the adjoint operator which satisfies
hTN f, gi hf, TN gi
(2.44)
31
for all f, g
P L2 pSLpn, ZqzHnq. Then TN is another Hecke operator which commutes with
TN so that TN is a normal operator. Explicitly,
N
T f pz q N
¸
1
N
n 1
2
±n
p ¤ ¤ q
c
N,
j 1 j
0 ci,j cj 1 i j n
¤
f 1
c1 c1,2 . . . c1,n
..
c2 . . . c2,n .
.. .
.
. . N
N
cn
z .
(2.45)
Definition 2.21. (Hecke-Maass form) Let n ¥ 2. A Maass form f is called a Hecke-Maass
form if it is an eigenfunction of all Hecke operators TN for N
¥ 1.
Assume that f is a Hecke-Maass form then f has the Fourier-Whittaker expansion as
in (2.42). Let Af pm1 , . . . , mn1 q
0 m1 , . . . , mn1
P
C be the pm1 , . . . , mn1 qth Fourier coefficients for
P Z. Since f is a Hecke-Maass form, Af p1, 1, . . . , 1q 0. Assume that
Af p1, . . . , 1q 1. Then we have the following (multiplicative) relations (see [12]):
• TN f
Af pN, 1, . . . , 1qf for any integer N ¥ 1;
• for pm1 , . . . , mn1 q P Zn1 , we have
Af pmn1 , . . . , m1 q Af pm1 , . . . , mn1 q;
• for pm1 , . . . , mn1 q P Zn1 , and a nonzero integer m, we have
Af pm, 1, . . . , 1qAf pm1 , . . . , mn1 q
¸
±n
| Af
c
m,
j 1 j
c1 m1 ,...,cn 1 mn 1
|
m1 cn m2 c1
mn1 cn2
,
,...,
c1
c2
cn1
(2.46)
.
1, 2, . . . ,
Let p be a prime. Then for any k
Af ppk , 1, . . . , 1qAf p1,
. . . , 1, p, 1, . . . , 1q
loooomoooon
r
Af pploooooomoooooon
, 1, . . . , 1, p, 1, . . . , 1q
k
r
(2.47)
k1
Af pploooooooomoooooooon
, 1, . . . , 1, p, 1, . . . , 1q,
r 1
32
for r
1, . . . , n 2, and
Af ppk , 1, . . . , 1qAf p1, . . . , 1, pq Af ppk , 1, . . . , 1, pq
Definition 2.22. Let n ¥ 2 and fix a prime p. For j
Af ppk1 , 1, . . . , 1q.
1, . . . , n 1, define
j¸
1
p
jq
Tp p1qk Tpk 1 Tppjk1q
k0
p1q
p0q
and Tpr Tpr for any integer r ¥ 0 and Tp is an identity operator.
prq
Lemma 2.23. (Eigenvalues of Hecke operators Tp ) Let n
¥
(2.48)
2 be an integer and
f be a Maass form for SLpn, Zq. Then f has the Fourier-Whittaker expansion as in
(2.42) and let Af pm1 , . . . , mn1 q
P
C be the pm1 , . . . , mn1 qth Fourier coefficients for
m1, . . . , mn1 P Z. Assume that f is an eigenfunction for Tp
Af p1, . . . , 1q 1. Then for r 1, . . . , n 1,
0
Tpprq f
j
for j
0, . . . , n and
Af p1,
. . . , 1, p, 1, . . . , 1qf
loooomoooon
r
for any prime p.
prq
1, . . . , n 1) and the multiplicative relations
prq
in (2.47), we get the eigenvalue of Tp (for r 1, . . . , n 1).
Proof. By using the definition of Tp (for r
Definition 2.24. Let n
pj q
¥
eigenfunction for Tp for j
pj q
2 be an integer and fix a prime p. Let f : Hn
Ñ
C be an
1, . . . , n 1 as in Definition 2.22, i.e., for j 1, . . . , n 1
there exists λp pf q P C such that
Tppj q f pz q λppj q pf q f pz q,
pz P Hnq
Define the parameters
`p pf q : p`p,1 pf q, . . . , `p,n pf qq P aC pnq
(2.49)
33
such that
n¸1
1
p1qj λppjqpf qxj p1qnxn n
¹
1 p`p,j pf q x .
j 1
j 1
n
Here a pnq € C is the complex vector space defined in (2.29). So, for j
C
¸
λppj q pf q ¤ kj ¤n
: λpjqp` pf qq.
pp`p,k1 pf q 1, . . . , n 1,
p qq
`p,kj f
(2.50)
1 k1
p
Remark 2.25.
p
(i) Let f be a Hecke-Maass form with pm1 , . . . , mn1 qth Fourier coeffi-
cients Af pm1 , . . . , mn1 q
P C as in (2.42). Assume that Af p1, . . . , 1q 1. Then we
have
Af p1,
. . . , 1, p, 1, . . . , 1q loooomoooon
¤ ¤
pp`p,j1 pf q p qq λpj q p` pf qq
p
p
`p,jr f
1 j1 ... jr n
r
for any 1 ¤ r
¸
¤ n 1.
(ii) The parameter is given by the equation
1 Af pp, 1, . . . , 1qx
Af p1, p, 1, . . . , 1q
p1qr Af p1,
. . . , 1, p, 1, . . . , 1qxr
loooomoooon
p1qnxn 0
(2.51)
r
and it has solutions p`p,1 pf q , . . . , p`p,n pf q . This equation comes from the pth factor
of the L-function of the Hecke-Maass form f . For s P C with <psq ¡ 1, let
8̧
Lp pf ; sq :
Af ppk , 1, . . . , 1qpks
(2.52)
k 0
p1 Af pp, 1 . . . , 1qps p1qr Af p1,
. . . , 1, p, 1, . . . , 1qprs loooomoooon
r
p1q pnsq1
n
n
¹
j 1
1 p`p,j pf qs
1
34
(see [12]). Conversely, if the parameters `p pf q
P aCpnq is given then we can determine Af p1,
. . . , 1, p, 1, . . . , 1q for each r 1, . . . , n 1.
loooomoooon
r
Recall the definition of dual Maass forms and the properties from Proposition 9.2.1,
[12].
Proposition 2.26. (Dual Maass forms) Let φpz q be a Maass form of type ν
Cn1 . Then
φrpz q : φ w t
p z 1 q w
t
,
w
p1q
1
..
.
n
u
2
pν1, . . . , νn1q P
1
pνn1, . . . , ν1q for SLpn, Zq. The Maass form φr is called
the dual Maass form. If Apm1 , . . . , mn1 q is the pm1 , . . . , mn1 qth Fourier coefficient of
r If φ φ,
r then the
φ then Apmn1 , . . . , m1 q is the corresponding Fourier coefficient of φ.
is a Maass form of type νr
Maass form φ is called the self-dual Maass form.
Remark 2.27. Let f be a Hecke-Maass form of type ν. Then the dual Maass form fr of type
νr has the following Langlands parameters.
(i) v
8: Since Bj pνrq Bnj pν q for j 1, . . . , n 1, we have
`8 pfrq `8 pνrq `8,nj
1
pν q `8,nj 1pf q,
p for j 1, . . . , nq
so
`8 pfrq `8 pf q.
(ii) v
p, prime: Since
Afrp1,
. . . , 1, p, 1, . . . , 1q Af p1,
. . . , 1, p, 1, . . . , 1q,
loooomoooon
loooomoooon
j
n j
we have
`p pfrq `p pf q.
p for j 1, . . . , n 1q,
35
2.6
Eisenstein series
We defined parabolic subgroups, their Levi parts and unipotent radicals in Definition 2.1.
Then for each partition n n1
nr with rank 1 r
¤ n, we have the factorization
Pn1 ,...,nr pRq Un1 ,...,nr pRq Mn1 ,...,nr pRq.
P Pn ,...,n pRq, we have
mn pg q
0
m
n pg q
g P Un ,...,n pRq It follows that for any g
r
1
1
2
1
r
...
...
..
.
0
0
..
.
mnr pg q
where mni pg q P GLpni , Rq for i 1, . . . , r.
Let n ¥ 2 be an integer and fix a partition n n1
nr with 1 ¤ n1 , . . . , nr
n.
For each i 1, . . . , r, let φi be either a Maass form for SLpni , ZqzHn of type µi pµi,1, . . . , µi,n 1q P Cn 1 or a constant with µi p0, . . . , 0q. For t pt1, . . . , tr q P Cr
with n1 t1 nr tr 0, define a function
i
i
i
IPn1 ,...,nr p; t; φ1 , . . . , φr q : Pn1 ,...,nr pRq Ñ C
by the formula
IPn1 ,...,nr pg; t; φ1 , . . . , φr q :
r
¹
φi pmni pg qq |detpmni pg qq|ti
for g
P Pn ,...,n pRq.
1
r
i 1
(2.53)
1, . . . , r, let φipmn pgkqq φipmn pgqq and |detpmn pgkqq| |detpmn pgqq|
pg; t; φ1, . . . , φr q IP
pz; t; φ1, . . . , φr q for g dz k
where k P Opn, Rq. So IP
with z P Hn , d P R and k P Opn, Rq. Let η1 0 and ηi n1 ni1 for i 2, . . . , r.
There exists a unique ν pν1 , . . . , νn1 q P Cn1 (up to the action of the Weyl group Wn )
For each i
i
n1 ,...,nr
i
i
n1 ,...,nr
i
36
such that IPn1 ,...,nr p; t; φ1 , . . . , φr q is an eigenfunction for Z pDn q of type ν. Furthermore,
Iν py q n
¹1
k 1
r
¹
pq
B ν
yk k
(2.54)
y1 ynηj nj
nj tj
j 1
for any y
..
.
P Apn, R q (see Proposition 10.9.1, [12]).
Then for
1
1 ¤ i ¤ n 1,
Bi pν q p q p q
B µ
n k t
yn k ηj j nj jk j
k 1
y1 yn1
¹
nj 1
$
Bipnn1 q µ1
'
'
'
'
'
'
&
p q pn iqt1,
if n n1 1 ¤ i ¤ n 1
nj1tj1 Bipnη n qpµj q pn ηj iqtj
if 2 ¤ j ¤ r and n ηj nj 1 ¤ i ¤ n ηj 1
nj tj ,
if i n ηj nj .
n1 t1
j
'
'
'
'
n1 t1
'
'
%
j
(2.55)
Therefore, by (2.30), for 1 ¤ j
¤ r and ηj
`8,i pν q n
nj
2
1 ¤ i ¤ ηj
tj
ηj
nj , we have
`8,iηj pµj q.
(2.56)
¥ 2 be an integer and fix an ordered partition
n n1 nr with 1 ¤ n1 , . . . , nr n. For each i 1, . . . , r, let φi be either a
Maass form for SLpni , ZqzHn of type µi pµi,1 , . . . , µi,n 1 q P Cn 1 or a constant with
µi p0, . . . , 0q. Let t pt1 , . . . , tr q P Cr with n1 t1 nr tr 0. Define the Eisenstein
Definition 2.28. (Eisenstein series) Let n
i
i
i
series by the infinite series
EPn1 ,...,nr pz; t; φ1 , . . . , φr q
¸
:
γ
for z
P Hn .
PpPn1 ,...,nr pZqXSLpn,ZqqzSLpn,Zq
(2.57)
IPn1 ,...,nr pγz; t; φ1 , . . . , φr q
37
Remark 2.29.
(i) Since IPn1 ,...,nr p; t; φ1 , . . . , φr q is actually a function on Hn , the Eisen-
stein series (2.57) are well-defined on Hn , but they are not square-integrable.
(ii) Eisenstein series are automorphic, i.e., for any γ
P SLpn, Zq, we have
EPn1 ,...,nr pγz; t; φ1 , . . . , φr q EPn1 ,...,nr pz; t; φ1 , . . . , φr q,
pz P Hnq.
(iii) Each Eisenstein series is an eigenfunction of type ν of Z pDn q where ν is given by the
formula (2.54).
The Fourier coefficients for Eisenstein series are given in Proposition 10.9.3 [12].
¤ n1 , . . . , n r n. For each i 1, . . . , r, let φi be either a Hecke-Maass form for SLpni , ZqzHn or a
constant. Let t pt1 , . . . , tr q P Cr with n1 t1 nr tr 0. Then the Eisenstein series
EP
pz; t; φ1, . . . , φr q is an eigenfunction of the Hecke operators TN (for any N ¥ 1)
Proposition 2.30. Let n
¥ 2 and fix a partition n n1 nr with 1
i
n1 ,...,nr
with eigenvalues
At;φ1 ,...,φr
N
pN q N n2 1
n 1
2
¸
r
¹
¤ P
j 1
C1 Cr N,
1 Cj Z
¸
Aφj pCj qCj
nj 1
2
Aφ1 pC1 q Aφr pCr q C1
n1 1
2
¤ P
t1
tj ηj
(2.58)
n2 1
2
C2
t2 η2
Cr
nr 1
2
tr ηr
,
C1 Cr N,
1 Cj Z
where η1
0 and ηj n1 nj 1 (for j
2, . . . , r).
Here Aφj pCj q is the Hecke
eigenvalue of TCj for φj .
Remark 2.31. If φj is a constant, then
TCj φj
Cj
n 1
j
¹
¸
nj 1
2
d1 dn Cj
and Aφj pCj q Cj
nj 1
2
dkk1 φj
k 1
¸
d1 dn Cj
n 1
j
¹
k 1
dkk1 .
(2.59)
38
Now, we can extend the parameters defined in Definition 2.24. Let n
¤ n1, . . . , nr n. For each i 1, . . . , r, let φi
be either a Maass form for SLpni , ZqzHn or a constant. Let t pt1 , . . . , tr q P Cr with
n1 t1 nr tr 0. Let E pz q : EP
pz; t; φ1, . . . , φr q then
partition n
¥ 2 and fix a
n1
nr with 1
i
n1 ,...,nr
AE ppk q p
p q
k n 1
2
¸
r
¹
¤ P
j 1
C1 Cr pk ,
1 Cj Z
for a prime p and k
Aφj pCj qCj
nj 1
2
tj ηj
¥ 0. By (2.52), define
`p pE q : p`p,1 pE q, . . . , `p,n pE qq P aC pnq
(2.60)
in the following way. For <psq ¡ 0, s P C, we have
8̧
AE ppk qpks
r
¹
j 1
k 0
8̧ Aφj ppkj qp
kj
n n
j
2
kj s tj ηj
kj 0
r ¹
¹
nj
p`p,k pφj q
nj n
2
tj ηj
ps
1
j 1k 1
n
¹
p`p,k pE q ps
1
.
k 1
Then for i 1, . . . , r and ηi
1¤j
¤ ηi
ni , it follows that
`p,j pE q `p,j ηi pφi q n
i
n
2
Lemma 2.32. Let n ¥ 2 and fix a partition n n1
ti
ηi .
(2.61)
nr with 1 ¤ n1 , . . . , nr
n. For
1, . . . , r, let φi be either a Hecke-Maass form for SLpni, ZqzHn or a constant.
Let t pt1 , . . . , tr q P Cr with n1 t1 nr tr 0. Let E : EP
pz; t; φ1, . . . , φr q.
By (2.56) and (2.61), for i 1, . . . , r and ηi 1 ¤ j ¤ ηi ni , we have
n n
i
` pE q p1q
t η
`
pφ q
each i
i
n1 ,...,nr
v,j
where "
0,
1,
if v
if v
8;
8,
2
and ηi
n1 i
i
v,j ηi
ni1 and η1
0.
i
39
Chapter 3
AUTOMORPHIC CUSPIDAL REPRESENTATIONS FOR
AzGLpn, Aq
3.1
Local representations for GLpn, Qv q
Let G be a group and let V be a complex vector space. A representation of G on V is a pair
of pπ, V q where
π : G Ñ EndpV q t set of all linear maps: V
ÑVu
is a homomorphism. We let π pg q.v denote the action of π pg q on v and π pg 1 g 2 q π pg 1 q.π pg 2 q
for all g 1 , g 2
P G. The vector space V is called the space of the representation pπ, V q. If the
group G and the vector space V are equipped with topologies, then we shall also require
the map G V
Ñ V given by pg, vq Ñ πpgq.v to be continuous. A representation pπ, V q is
said to be irreducible if V 0 and V has no closed π-invariant subspace other than 0 and V .
Let V be a space of functions f : G Ñ C and π R be the action given by right translation,
pπR phqf qpgq f pghq,
p@g, h P Gq.
Then pπ R , V q is a representation of G.
In this section we review the properties of local representations of GLpn, Qv q. The
main reference is [13].
8. Let V be a
complex vector space equipped with a positive definite Hermitian form p , q : V V Ñ C.
Let n
¥
1 be an integer. Consider the archimedean case with v
40
A unitary representation of GLpn, Rq consists of V and a homomorphism π : GLpn, Rq Ñ
GLpV q such that the function pg, v q ÞÑ π pg q.v is a continuous function GLpn, RqV
ÑV,
and
pπpgq.v, wq pv, πpg1q.wq,
p for all v, w P V, g P GLpn, Rqq.
The representation pπ, V q has the trivial central character if π
a P R , v
a
..
PV.
As in §2.3, for an integer n
.v
.
a
v for any
¥ 1, the Lie algebra glpn, Rq of GLpn, Rq consists of the
additive vector space of n n matrices with coefficients in R with Lie bracket given by
rα, β s αβ βα for any α, β P glpn, Rq. The universal enveloping algebra of glpn, Rq
is an associative algebra which contains glpn, Rq. The Lie bracket and the associative
product on U pglpn, Rqq are compatible, in the sense that rα, β s α β β α for
all α, β P U pglpn, Rqq. The universal enveloping algebra U pglpn, Rqq can be realized as
an algebra of differential operators acting on smooth functions F : GLpn, Rq Ñ C as in
?
(2.23). Set i 1. For any α, β P glpn, Rq we define a differential operator Dα iβ
acting on F by the rule
Dα
The differential operators Dα
iβ
iβ
: Dα
iDβ .
generate an algebra of differential operators which is iso-
morphic to the universal enveloping algebra U pgq where g glpn, Cq.
Fix an integer n ¥ 1. Let K8
Opn, Rq. We define a pg, K8q-module to be a complex
vector space V with actions
πg : U pgq Ñ EndpV q,
such that for each v
P
Ñ GLpV q,
(3.1)
V the subspace of V spanned by tπK8 pk q.v
| k P K8 u is finite
π K 8 : K8
41
dimensional, and the actions πg and πK8 satisfy the relations
πg pDα qπK8 pk q πK8 pk qπg pDk1 αk q
for all α P g and k
P K8. Further, we require that
πg pDα q.v
1
pπK8 pexpptαqq.v vq
lim
tÑ0 t
for all α P glpn, Rq such that exppαq P Opn, Rq. We denote this pg, K8 q-module as pπ, V q
where π
pπg, πK8 q.
Let pπ, V q be the pg, K8 q-module. For each v
€ V to be
the span of tπK8 pk q.v | k P K8 u and define a homomorphism ρv : K8 Ñ GLpWv q given
by ρv pk q.w πK8 pk q.w for all k P K8 and w P Wv . Then pπ, V q is admissible, if for each
finite dimensional representation pρ, W q of K8 , the span of tv P V | pρv , Wv q pρ, W qu
is finite dimensional. Let pπ, V q be a pg, K8 )-module. Then it is said to be unramified or
spherical if there exists a nonzero vector v P V such that
πK8 pk q.v v
PV
define a vector space Wv
p for all k P K8q.
Otherwise, it is said to be ramified.
The pg, K8 q-module pπ, V q is said to be unitary if there exists a positive definite Hermitian form p ,
q : V V Ñ C which is invariant in the sense that
pπK8 pkq.v, wq for all v, w
v, πK8 pk 1 q.w , pπg pDα q.v, wq pv, πg pDα q.wq ,
P V , k P K8 and α P glpn, Rq.
Theorem 3.1. Fix an integer n ¥ 1.
42
(i) If pπ, V q is a unitary representation of GLpn, Rq then there is a dense subspace
€ V such that ppπg, πK8 q, Vpg,K8qq is a unitary pg, K8q-module called the
underlying pg, K8 q-module of pπ, V q.
Vpg,K8 q
(ii) If ppπg , πK8 q, V q is a unitary pg, K8 q-module, then there exists a unitary representation pπ, VGLpn,Rq q of GLpn, Rq such that ppπg , πK8 q, V q is isomorphic to the underlying pg, K8 q-module of pπ, VGLpn,Rq q.
(iii) A unitary representation of GLpn, Rq is irreducible if and only if its underlying
pg, K8q-module is irreducible. Moreover, two irreducible unitary representations of
GLpn, Rq are isomorphic if and only if their underlying pg, K8 q-modules are isomorphic.
Proof. Theorem 14.8.11 in [13].
Let n
¥
1 be an integer. Consider a prime v
p 8, and G GLpn, Qp q. A
representation of GLpn, Qp q is a pair of pπ, V q where V is a complex vector space and
π : GLpn, Qp q
Ñ GLpV q is a homomorphism. Such a representation is smooth if for any
vector ξ P V there exists an open subgroup Uξ € GLpn, Qp q such that π pg qξ ξ for any
g P Uξ . It is admissible if for any r ¥ 1, the space
tξ P V | πpkqξ ξ, for all k P Kr u
is finite dimensional where
Kr
tk P GLpn, Zpq | k In P pr M atpn, Zpqu .
If pπ, V q is an irreducible smooth representation of GLpn, Qp q then there exists a unique
multiplicative character ωπ : Q
p
a
P
Q
p and ξ
P
Ñ
C such that π
a
..
ξ
.
a
ωπ paqξ for any
V . This character ωπ is called the central character associated to the
43
representation pπ, V q. A smooth representation pπ, V q of GLpn, Qp q is said to be unitary
if V is equipped with a positive definite Hermitian form p, q : V
pπpgqv, πpgqwq pv, wq ,
V Ñ C and
p@g P GLpn, Qpqq.
A representation pπ, V q of GLpn, Qp q is termed unramified or spherical if there exists a
nonzero GLpn, Zp q fixed vector ξ 3.2
P V . Otherwise it is said to be ramified.
Adelic automorphic forms and automorphic representations
Fix an integer n ¥ 1. Let A be the ring of adeles over Q and
K pn, Aq : Opn, Rq
¹
GLpn, Zp q
p
be the standard maximal compact subgroup of GLpn, Aq. In this section we review adelic
automorphic forms and automorphic representations for pA GLpn, QqqzGLpn, Aq. As in
the previous section, the main reference is [13].
Definition 3.2. Let n ¥ 1 be an integer and φ : GLpn, Aq Ñ C be a function.
(i) Smoothness: A function φ is said to be smooth if for every fixed g0
P
GLpn, Aq,
€ GLpn, Aq, containing g0 and a smooth function φU8 :
GLpn, Rq Ñ C such that φpxq φU8 px8 q for all x tx8 , x2 , . . . , xp , . . .u P U .
there exists an open set U
(ii) Moderate growth: For each place v of Q define a norm function ||
||v on GLpn, Qv q
by ||g ||v : max pt|gi,j |v , 1 ¤ i, j ¤ nu Y t| det g |v uq. Define a norm function || ||
±
on GLpn, Aq by ||g || : v ||gv ||v . Then we say a function φ is of moderate growth
if there exist constants C, B ¡ 0 such that |φpg q| C ||g ||B for all g P GLpn, Aq.
(iii) K pn, Aq-finiteness: A function φ is said to be right K pn, Aq-finite if the set
tφpgkq | k P K pn, Aqu, of all right translates of φpgq generates a finite dimensional
vector space.
44
(iv) Z pU pgqq-finiteness: Let Z pU pgqq denote the center of the universal enveloping al-
glpn, Cq. Then we say a function φ is Z pU pgqq-finite if the set
tDφpgq | D P Z pU pgqqu generates a finite dimensional vector space.
gebra of g
Definition 3.3. (Adelic automorphic form on GLpn, Aq with trivial central character)
Let n ¥ 1 be an integer. An automorphic form for GLpn, Aq with trivial central character
is a smooth function φ : GLpn, Aq Ñ C which satisfies the following five properties:
(i) φpγg q φpg q, @g
P GLpn, Aq, γ P GLpn, Qq;
(ii) φpzg q φpg q, @g
P GLpn, Aq, z P A;
(iii) φ is right K pn, Aq-finite;
(iv) φ is Z pU pgqq-finite;
(v) φ is of moderate growth.
An adelic automorphic form φ is a said to be a cusp form (or cuspidal) if
ϕP pg q »
p qz p q
U Q U A
ϕpug q du 0
for any proper parabolic subgroups P pAq of GLpn, Aq and for all g
P GLpn, Aq. Here U
is the unipotent radical of the parabolic subgroup P defined in Definition 2.1.
Let ApA zGLpn, Aqq denote the C-vector space of all adelic automorphic forms for
GLpn, Aq with the trivial central character. Let Acusp pA zGLpn, Aqq denote the C-vector
space of all adelic cuspidal forms for GLpn, Aq with the central character.
¥ 1, let GLpn, Afiniteq denote the
multiplicative subgroup of all afinite P GLpn, Aq of the form afinite tIn , a2 , a3 , . . . , ap , . . .u
Let Afinite denote the finite adeles. For an integer n
45
where ap
P GLpn, Qpq for all finite primes p and ap P GLpn, Zpq for all but finitely many
primes p. We define the action
πfinite : GLpn, Afinite q Ñ GLpApA zGLpn, Aqqq
as follows. For φ P ApA zGLpn, Aqq let
πfinite pafinite q.φpg q : φpgafinite q,
for all g
P GLpn, Aq, afinite P GLpn, Afiniteq.
Definition 3.4. Let n ¥ 1 be an integer. Let g glpn, Cq and K8
pg, K8q GLpn, Afiniteq-module to be a complex vector space V
πg : U pgq Ñ EndpV q,
πK8 : K8
Ñ GLpV q,
Opn, Rq. We define a
with actions
πfinite : GLpn, Afinite q Ñ GLpV q,
such that pπg , πK8 q and V form a pg, K8 q-module, and the actions pπg , πK8 q and πfinite
commute. The ordered pair pppπg , πK8 q, πfinite q , V q is said to be a pg, K8 q GLpn, Afinite qmodule.
(i) The representation pppπg , πK8 q, πfinite q , V q is smooth if every vector v
P V is fixed by
some open compact subgroup of GLpn, Afinite q under the action πfinite .
pppπg, πK8 q, πfiniteq , V q is admissible if it is smooth and for any
fixed open compact subgroup K 1 € GLpn, Afinite q, and any fixed finite-dimensional
representation ρ of SOpn, Rq, the set of vectors in V fixed by K 1 and generate a
subrepresentation under the action of SOpn, Rq (which is isomorphic to ρ) spans a
(ii) The representation
finite dimensional space.
(iii) The representation
pppπg, πK8 q, πfiniteq , V q is irreducible if it is nonzero and has no
proper nonzero subspace preserved by the actions π.
46
Definition 3.5. (Automorphic representation) Let n
¥ 1 be an integer. An automorphic
(resp. cuspidal) representation with the trivial central character is an irreducible smooth
pg, K8q GLpn, Afiniteq-module which is isomorphic to a subquotient of ApAzGLpn, Aqq
(resp. Acusp pA zGLpn, Aqq).
3.3
Principal series for GLpn, Qv q
Again the main reference for this section is [13].
¤ 8. The
modular quasi-character of the minimal (standard) parabolic subgroup P pn, Qv q is defined
Definition 3.6. (Modular quasi-character) Let n
¥
2 and fix a prime v
as
a1
δv ..
.
an
a1
for any
..
.
an
:
n
¹
|aj |vn2j
1
(3.2)
j 1
P P pn, Qv q. Here P pn, Qv q is the minimal parabolic subgroup defined
in (2.6).
Let χ : Apn, Qv q
Ñ C be a character. Then we can extend the character χ to the
minimal parabolic P pn, Qv q as
a1
a1
..
..
χ χ .
.
an
a1
where
..
.
an
an
P P pn, Qv q.
Definition 3.7. (Principal series) Let n ¥ 2 be an integer and fix a prime v
¤ 8. Let χ be
47
a character of Apn, Qv q. Denote
p
q
IndP pn,Qv qv pχq
GL n,Q
:
(3.3)
#
f : GLpn, Qv q Ñ
+
1
2
f
is
locally
constant,
f
umg
δ
m
χ
m
f
g
,
v
C .
for all u N n, Qv , m P n, Qv , g GL n, Qv
P p
q
p
q p q p q p q
P p q P p q
q pχq where π R phqf pg q Define a homomorphism π : GLpn, Qv q Ñ GL
p q
GLpn,Qv q
GLpn,Qv q
R
f pghq for any g, h P GLpn, Qv q and f P IndP pn,Qv q . Then π , IndP pn,Qv q pχq is called
R
p
GL n,Q
IndP n,Qv v
the principal series representation of GLpn, Qv q associated to χ.
For each v
¤ 8 and n ¥ 1 define
Kv pnq :
"
Opn, Rq,
GLpn, Zp q,
Let χ : Apn, Qv q{pKv pnq X Apn, Qv qq
if v
if v
8
p, finite prime.
Ñ C be a character, i.e., a spherical character of
Apn, Qv q. There exists
`v pχq p`v,1 pχq, . . . , `v,n pχqq P Cn
such that
χ a1
..
.
an
a
n
¹
|aj |v`
v,j
pχq .
(3.4)
j 1
1 for any a P Qv , then `v,1pχq `v,n pχq 0 so `v pχq P aC pnq. If χ is unitary, then `v,j pχq P iR for j 1, . . . , n.
If χ is trivial on the center, i.e., χ
..
.
a
Let `
P
Cn and χv p`q be the spherical character of Apn, Qv q which is associated to
the parameter ` as in (3.4). When the representation π
R
p
p
q
q pχv p`qq is not ir-
GL n,Q
, IndP n,Qv v
reducible, there exists a unique spherical subconstituent. Denote πv p`q as the spherical
48
q
subconstituent of π
p q pχv p`qq . It is called the spherical representation assoGLpn,Q q
ciated to ` (or χv p`q). We abuse notation and denote IndP pn,Qv qv pχv p`qq as the vector space
R
p
GL n,Q
, IndP n,Qv v
of the representation πv p`q.
For each ` p`1 , . . . , `n q P Cn , define the function ϕ` with parameter ` P Cn by
a1
..
ϕ` .
n
¹
k :
for any
..
.
an
a1
|aj |`v δv j
n
¹
..
.
(3.5)
an
|aj |v`
j
n 2j 1
2
j 1
P P pn, Qv q and k P Kv pnq.
Then ϕ`
` P Cn , and it is unique.
Let n
2
j 1
an
a1
1
pn,Q q pχ p`qq for any
P IndPGLpn,Q
q v
v
v
¥ 2 be an integer and v 8. For ν pν1, . . . , νn1q P Cn1 we have already
defined the eigenfunction Iν of Casimir operators of type ν in (2.25) and defined the Langlands parameter `8 pν q
P Cn in Definition 2.10. Then Iν pzq ϕ`8pνqpzq. Conversely, for
each ` p`1 , . . . , `n q P aC pnq, as in (2.33), and for j 1, . . . , n 1, we have
νj p`q 1
p`j
n
`j 1 1q,
and
°n
1
ϕ` pz q pdetpz qq n p k1 `k
n
¹1
°nj
p`
yj k 1 k
n 2k 1
2
n 2k 1
2
q
n1
q ¹ py y q`j
1
nj
j 1
n 2j 1
2
Iνp`qpzq.
j 1
¥ 2 be an integer and ` P aCpnq. Fix a place v ¤ 8. Then πv p`q
is an irreducible spherical representation of GLpn, Qv q associated to ` with trivial central
Definition 3.8. Let n
character. Define:
49
• v
8, for j 1, . . . , n 1,
λp8j q p`q : λν p`q p∆pnj q q
(3.6)
where ν p`q P Cn1 as in (2.33) and λν p`q is the Harish-Chandra character defined in
(2.27);
• v
p 8, for j 1, . . . , n 1
¸
λppj q p`q :
¤ kj ¤n
pp`k1 `p,kj
q.
(3.7)
1 k1
Spherical generic unitary representations of GLpn, Qv q
3.4
Definition 3.9. (Additive character) Fix a prime v
defined by
ev pxq :
"
e2πitxu
e2πix
8 or v 8. Let ev : Qv Ñ C be
if v
if v
8,
8,
where
tx u if v
" ° 1
j
0,
j
k aj p , if x °8
j
k aj p
j
P Qp with k ¡ 0, 0 ¤ aj ¤ p 1,
otherwise,
p 8.
Definition 3.10. (Whittaker model for a representation of GLpn, Qv q) Fix an integer
¥ 1 and v p a finite prime or v 8. Let ev : Qv Ñ C be the additive character in
Definition 3.9. Fix a character ψv : N pn, Qv q Ñ C of the form
n
1 u1,2 . . .
u1,n
1 u2,3
.
.
.
.
.
.
ψv : ev pa1 u1,2
.
.
. 1 un1,n 1
for ui,j
P Qv , p1 ¤ i j ¤ nq with ai P Qv , pi 1, . . . , n 1q.
an1 un1,n q
(3.8)
50
p, let pπ, V q be a complex representation of GLpn, Qpq. A Whittaker model
for pπ, V q relative to ψp is the representation pπ 1 , W q pπ, V q where W is a space of
Whittaker functions relative to ψ, i.e., of locally constant functions W : GLpn, Qp q Ñ
(i) For v
C satisfying
W pug q ψp puqW pg q
for all u P N pn, Qp q, g
P GLpn, Qpq and π1 is given by the right translation.
8, let pπ, V q be a pg, K8q-module where g glpn, Cq and K8 Opn, Rq.
Following Theorem 3.1, we refer to pπ, V q as a representation of GLpn, Rq. A Whittaker model for pπ, V q relative to ψ8 is the representation pπ 1 , W q pπ, V q where W
(ii) For v
is a space of Whittaker functions relative to ψ8 , i.e., of smooth functions of moderate
growth satisfying
W pug q ψ8 puqW pg q
for all u P N pn, Rq, g
P GLpn, Rq and π1 is given as in (3.1).
1
ui,j
..
exppu1,2 un1,nq. Then
8, let ψ8
.
1
Jacquet’s Whittaker WJ p ; ν, 1q for some ν P Cn1 defined in (2.41) is the Whittaker func-
Remark 3.11. For v
tion relative to ψ8 . Moreover, for every automorphic cuspidal smooth function f , the
Fourier
coefficient
Wf p ; pm1 , . . . , mn1 qq defined in (2.40) is also a Whittaker function relative to an additive
character ψ8
1
for m1 , . . . , mn1
ui,j
..
.
P Z.
1
expp2πipm1un1,n mj unj,nj
1
Definition 3.12. (Generic representation of GLpn, Qv q) Fix an integer n
mn1 u1,2 qq
¥ 1, let v be a
8, and let ψv be an additive character as in (3.8). A representation
pπ, V q of GLpn, Qv q is said to be generic relative to ψv if it has a Whittaker model relative
finite prime or v
to ψv as in Definition 3.10.
51
Definition 3.13. (Spherical generic character) Let n
¥ 2 be an integer and v ¤ 8 be a
prime. If there exist
• an integer 0 ¤ r
n2 and t1, . . . , tr P R,
• real numbers α1 , . . . , αr
P p0, 12 q,
such that
`v
p`v,1, . . . , `v,nq P aCpnq
pα1 it1, α1 it1, . . . , αr
(3.9)
itr , αr
itr , itr 1 , . . . , itnr q,
then the character χv p`v q : P pn, Qv q Ñ C is called a spherical generic character.
Theorem 3.14. (Classification of irreducible spherical unitary generic representations)
Let n ¥ 2 be an integer and v
¤ 8 be a place of Q. Let π be an irreducible spherical uni
tary generic representaiton of Q
v zGLpn, Qv q. Then there exists ` P aC pnq which satisfies
the condition in Definition 3.13 such that π πv p`q.
3.5
Quasi-Automorphic parameter and Quasi-Maass form
Let A be the ring of adeles over Q. Let n
¥ 2 be an integer. Let π be a cuspidal automor-
phic representation of GLpn, Aq. The representation π is unramified or spherical if there
P Vπ (the complex vector space of π), such that πpkqv v for any
±
k P K pn, Aq Opn, Rq p GLpn, Zp q.
exists a vector v Let π be an unramified cuspidal automorphic representation of A zGLpn, Aq. Then by
the tensor product theorem ([11], [17], [8]), there exist local generic spherical unitary representations πv of Q
v zGLpn, Qv q for v
¤ 8 such that π v¤8
b 1πv . Since πv ’s are generic,
52
tσv P aCpnq, v ¤ 8u
where σv satisfies conditions in Definition 3.13 for any v ¤ 8, such that πv πv pσv q. So,
π b1v πv pσv q and we may denote
unitary, and spherical, there exist an automorphic parameter σ
π pσ q : b1v πv pσv q π.
(3.10)
¥ 2 be an integer and let M
t`v P aCpnq, v P M u satisfy the conditions in
Definition 3.15. (Quasi-Automorphic Parameters) Let n
be a set of primes including
8.
Let `M
Definition 3.13. Then `M is called a quasi-automorphic parameter for M .
By the tensor product theorem combined with the multiplicity one theorem, for any unramified cuspidal automorphic representation π for A zGLpn, Aq, there is an automorphic
parameter σ for
t8, 2, 3, . . . , u such that πpσq π as in (3.10) and σ is also a quasi-
automorphic parameter. There exists a unique Hecke-Maass form Fσ of type ν pσ8 q such
that
• ∆n Fσ
pj q
λp8jqpσ8qFσ ,
pj q
λppjqpσpqFσ ,
• Tp Fσ
( for j
(for j
1, . . . , n 1),
1, . . . , n 2),
See [13] for more explanation. So, `v pFσ q
for any finite prime p.
`v pσv q for any v ¤ 8.
Conversely, let F
P Cn1. Then there exists a unique unramified cuspidal
automorphic representation π pσF q for A zGLpn, Aq such that
be a Hecke-Maass form of type ν
• ν pσF,8 q ν, so `8 pν q `8 pF q `8 pσF,8 q;
• σF,p
`ppF q and for each r 1, . . . , n 1, we have
AF p1,
. . . , 1, p, 1, . . . , 1q λpprq pσF,p q,
loooomoooon
r
where AF p1, . . . , 1, p, 1, . . . , 1q is the p1, . . . , 1, p, 1 . . . , 1qth Fourier coefficient as in
Lemma 2.23.
53
¥ 2 be an integer and M be a set of primes
t`v P aCpnq, v P M u be a quasi-automorphic parameter for M .
Definition 3.16. (Quasi-Maass Form) Let n
including
8.
Let `M
Let
L¥
±
P
qn,
(if M is a finite set)
q M,
finite prime
(3.11)
L 8,
P Hn ,
and define for z
F`M pz q (if M is an infinite set),
¸
L
¸
P p qz p q
γ N n 1,Z SL n 1,Z m1 1
m1 |mn1 |
WJ ..
L
¸
¸
mn2 1
| |¤
mn 1 0,
mn 1 L
A`M pm1 , . . . , mn1 q
±n1
kpnkq{2
k1 |mk |
mn1
γ
z; ν `8 ,
1
mn1
.
m1
(3.12)
p q|
|
,
1
where A`M p1, . . . , 1q 1 and A`M pm1 , . . . , mn1 q P C satisfy the multiplicative condition
in (2.46), if this series is absolutely convergent. For r
1, . . . , n 1 and any prime q P M ,
A`M p1,
. . . , 1, q , 1, . . . , 1q λpqj q p`q q
loooomoooon
r
¸
¤ jr ¤n
pp`q,j1 `q,jr
q.
1 j1
Then F`M is called a quasi-Maass form of `M of length L.
Remark 3.17.
F`M pz q (i) By Theorem 9.4.7, [12], we can rewrite (3.12) as
¸
L
¸
P p qz p q
γ N n 1,Z SL n 1,Z m1 1
e2πimpa 1
n 1,1 x1,n
L
¸
A`M pm1 , . . . , mn1 q
(3.13)
±n1
kpnkq{2
k 1 | m k |
mn1 0,
|mn1 |¤L
an1,n1 x1 q e2πipm2 xγ2 mn2 1
m1 |mn1 |
WJ ¸
..
.
m1
mn1 xγn1
q
γ
y ; ν `8 , 1
1
p q
54
where
γ
a1,1
a2,1
..
.
a1,n1
a2,n1
..
.
N n
P p 1, ZqzSLpn 1, Zq
an1,n1
and xγ , y γ are defined by p γ 1 q z xγ y γ P Hn by Iwasawa decomposition, for
xγ P N pn, Rq as in (2.7) and y γ P Apn, R q as in (2.8).
an1,1
(ii) For any γ
P Pn1,1pZq X SLpn, Zq, we have
F`M pγz q F`M pz q,
(iii) For j
(3.14)
1, . . . , n 1
∆pnj q F`M
and for any finite prime q
λp8jqp`8qF`
M
,
P M , j 1, . . . , n 1,
Tqpj q F`M
λpqjqp`q qF`
pj q
where λv p`v q is defined in Definition 3.8 for v
v
pz P Hnq.
M
,
P M . Moreover, `v pF` q `v for any
M
P M . For any integer 1 ¤ m ¤ L, we have
Tm F`M
A` pm, 1, . . . , 1qF`
M
M
.
Definition 3.18. (-closeness) Let n ¥ 2 and ¡ 0.
¤ 8, let πv p`v q and πv pσv q be irreducible unramified unitary generic representations of GLpn, Qv q as in Theorem 3.14 with parameters `v , σv P aC pnq satisfying
the condition in Definition 3.13. The representations πv p`v q and πv pσv q are -close if
(i) For v
m
¸
pj q
λ `v
v
p q λpvjqpσv q2 j 1
where m n 1 for v
8 and m t n2 u for v 8.
(3.15)
55
(ii) Let M and M 1 be sets of primes including 8 and let `M and σM 1 be quasi-automorphic
parameters for M and M 1 respectively as in Definition 3.15. Let S
€ M X M 1 be a
finite subset including 8. The quasi-automorphic parameters `M and σM 1 are -close
for S if
λpj q p`8 q λpj q pσ8 q2
8
8
j 1
n
t u
2
¸¸
pj q
λ `q
q
q PS, j 1
n¸1
p q λpqjqpσq q2 .
(3.16)
finite
We obtain a condition for -closeness with a given quasi-automorphic parameter in the
following Lemma. The idea of the lemma and its proof are generalizations of Lemma 1 in
[3], 3.1.
¥ 2 be an integer and M be a set of places of Q including 8. Let
`M be a quasi-automorphic parameter for M as in Definition 3.15. Let S € M be a finite
subset including 8. If there exists a smooth function f P L2 pSLpn, ZqzHn q, which is
Lemma 3.19. Let n
cuspidal, such that
n¸1
||
∆pnj q λp8j q p`8 q f ||22
j 1
for some n
t u
2
¸¸
P
q S,
finite
||
Tqpj q λpqj q p`q q f ||22
||f ||22
(3.17)
j 1
¡ 0, then there exists an unramified cuspidal automorphic representation πpσq
as in (3.10) such that the parameters `M and σ are -close for S.
Proof. By the spectral decomposition, the space L2cusp pSLpn, ZqzHn q is spanned by HeckeMaass forms uj pz q with ||uj ||22
1 for j 1, 2, . . .. For each uj there exists an unramified
cuspidal automorphic representation π pσj q b1v π pσj,v q such that `v puj q `v pσj,v q for any
v ¤ 8.
For any f
P L2cusp pSLpn, ZqzHnq,
f pz q 8̧
j 1
hf, uj i uj pz q.
56
For ¡ 0, let
| σj and `M are -close for S u ,
U p`M q : tuj
and define
¸
Pr pf qpz q :
P p q
hf, uj i uj pz q
P
L2cusp pSLpn, ZqzHn q .
uj U `M
Assume that f is a smooth automorphic function which satisfies (3.17). Then
¸
||Prpf q||22 ||f ||22 ¥ ||f ||
8̧
R p q
|hf, uj i|2
uj U `M
2
2
|hf, uj i|2 j 1
$
n1
1 & ¸ pkq
λ8 σj,8
%
p
t
¸ ¸
P
q S,
finite
k 1
n
u
2
||
,
2 .
λpqj q `q ,
.
p q
finite
∆pnkq λp8kq p`8 q f ||22
n
u
2
¸ ¸
λpkq σj,q
q
q PS, k 1
q λp8kqp`8q2
k 1
$
1
&n¸
||f ||22 1 % ||
¡ 0.
t
Tqpkq λpqkq p`q q f ||22
k 1
p q
-
Therefore, U p`M q H.
Definition 3.20. (Automorphic Lifting of Quasi-Maass forms) Let n
set of primes including
8 and let `M
¥ 2 and M be a
be a quasi-automorphic parameter. Let F`M be a
quasi-Maass form of `M . Define
Fr`M pz q : F`M pγz q,
p for any z P Hn and a unique γ P SLpn, Zq such that γz P Fnq.
(3.18)
We say Fr`M is an automorphic lifting of a quasi-Maass form. Here Fn is the fundamental
domain described in Proposition 2.8.
Remark 3.21.
(i) Let n ¥ 2. Define
€n :
F
¤
P p q
γ PSLpn,Zq
1
p q
γ SL n 1,Z ,
γ
1
Fn
(3.19)
57
where Fn is the fundamental domain described in Proposition 2.8. By (3.14), we have
Fr`M pz q F`M pz q,
(ii) Since F`M is square-integrable, Fr`M
uous and is not cuspidal in general.
pz P F€nq.
(3.20)
P L2 pSLpn, ZqzHnq. However Fr is not contin-
58
Chapter 4
A NNIHILATING OPERATOR 6np
4.1
Harmonic Analysis for GLpn, Rq{pR Opn, Rqq
For vectors v
pv1, . . . , vnq, v1 pv11 , . . . , vn1 q P Cn, the inner product h , i : Cn Cn Ñ
C denotes the usual inner product
hv, v 1 i :
We define the norm ||v || :
n
¸
vj vj1
P C.
j 1
a
hv, vi. For any w
P Wn, define
w.v : vσw p1q , . . . , vσw pnq
(4.1)
where σw is the permutation on n symbols corresponding to w defined by
vσw p1q
Then for any v, v 1
..
.
w
v1
..
vσw pnq
P Cn and w P Wn
1
w .
.
vn
hw.v, w.v 1 i hv, v 1 i .
For n
A1 pn, R
¥ 2, apnq is isomorphic to the Lie algebra of A1pn, R q which is isomorphic to
q Apn, Rq{pR pOpn, Rq X Apn, Rqqq via the exponential map in (2.17).
Let χ : A1 pn, R
q Ñ C be a character. Since A1pn,
R q Apn, Rq{pR Opn, Rqq,
a
n
..
as discussed in §3.3, there exists `8 pχq P C such that χ
±nj1 |aj |8`8 pχq
.
a
with `8,1 pχq `8,n pχq 0, and it is a one-to-one correspondence. Define
1
,j
n
a pnq : t` p`1 , . . . , `n q P Rn
| `1 `n 0 u ,
aC pnq : a pnq ia pnq Hompa8 pnq, C q.
(4.2)
59
Then aC pnq is isomorphic to the group of characters of A1 pn, R
isomorphic to the set of unitary characters of A1 pn, R
structure. For any character χ : A1 pn, R
χpaq e
p q pq,
q and this has the R-vector space
q Ñ C, we may write
p for a h`8 χ ,ln a i
q and iapnq € aCpnq is
a1
..
.
an
P A1pn, R qq.
Let
ρ
n 2j
2
1
n
j 1
P apnq.
(4.3)
Then for any a P A1 pn, R q, we have
δ8paq
ehρ,ln ai
1
2
,
where δ8 is the modular quasi-character defined in (3.6).
By the Iwasawa decomposition, for any g
P GLpn, Rq, we have
HIwa pg q P apnq, npg q P N pn, Rq, k pg q P Opn, Rq
such that g
For each ` P aC pnq and g
|det g|8 npgq exppHIwapgqq kpgq.
1
n
P GLpn, Rq, we defined the function ϕ`pgq in (3.5). Then
ϕ` pg q eh`
For any w
(4.4)
pq
ρ,HIwa g i
Iνp`qpgq.
(4.5)
P Wn, the Weyl group, we have
ϕ`ρ pwg q eh`,HIwa pwgqi
eh`,wH
Iwa
pgqi ehw1 .`,HIwa pgqi ϕ
This explains the definition of the action of the Weyl group Wn on ν
pg q.
w.` ρ
P Cn1 in (2.28).
60
Let H paC pnqqWn be the space of holomorphic functions on aC pnq, which are invariant
under the action of Wn , the Weyl group of GLpn, Rq. Define the spherical transform:
Cc8 Opn, RqzGLpn, Rq{pR Opn, Rqq
Ñ H paCpnqqW
n
ã
Definition 4.1. (Spherical Transform) For any compactly supported, smooth, bi-Opn, Rqinvariant function k
P Cc8 pOpn, RqzGLpn, Rq{pOpn, Rq Rqq, the spherical transform
p q P C is defined as the corresponding eigenvalue of the convolution operator associated
p
k `
to k, i.e.,
ϕ` k pg q »
GLpn,Rq{R
ϕ` pgξ 1 qk pξ qdξ
pkp`q ϕ`pgq,
(4.6)
and
p q
p
k `
»
GL n,R R
p
q{
eh`
p q k pξ 1 qdξ,
ρ,HIwa ξ i
where ϕ` is the eigenfunction of Z pDn q with the parameter `, defined in (4.5).
Definition 4.2. For each ` P aC pnq, define
β` p g q :
»
Opn,Rq{R
ϕ` pξg q dξ
»
Opn,Rq{R
eh`
p q dξ,
ρ,HIwa ξg i
(4.7)
P GLpn, Rq. Then β` is called the spherical function of type `. Moreover, β` is
pR Opn, Rqq-bi-invariant function. i.e., for any ξ1, ξ2 P Opn, Rq, we have
for any g
β` pξ1 g ξ2 q β` pg q.
The spherical function is again an eigenfunction of the convolution operator whose
eigenvalue is the corresponding spherical transform. For any compactly supported, smooth,
bi-Opn, Rq-invariant function k
P Cc8 pOpn, RqzGLpn, Rq{pR Opn, Rqqq, we have
β` k pg q p
k p`qβ` pg q.
61
We recall the following inversion formula for the spherical transform as given in [18].
For any bi-pR Opn, Rqq-invariant, compactly supported smooth function k, we have
k pg q 1
n!
»
ia n
pq
p q pq
p q
p
k α βα g φP lanch α dα,
(4.8)
where
φP lanch pαq ¹
¤ ¤
1 k j n
Γ R αk
Γ α
p αj 1qΓRpk
j
R p k αj qΓR pk
2
j q ,
1q (4.9)
for α pα1 , . . . , αn q P ia pnq, and
ΓR psq π 2 Γ
s
s
2
.
We recall the Paley-Wiener theorem from [19].
Theorem 4.3. (Paley-Wiener)
Cc8 pOpn, RqzGLpn, Rq{pR Opn, Rqq, such that k pg q
P
GLpn, Rq with || ln Apg q|| ¡ δ for some δ ¡ 0. Then the spherical transform p
k P
H paC pnqqW . Moreover, for any integer N , there exists a constant CN ¡ 0 such that
(i) Let k
P
0 for any g
n
p k ` p q ¤ CN p1 ||`||qN eδ||<p`q||
for any ` P aC pnq.
¡ 0) satisfies the following condition. For any
integer N there exists a constant CN ¡ 0 such that
(ii) Assume that Rδ
P H paCpnqqW
n
(for δ
|Rδ p`q| ¤ CN p1 ||`||qN eδ||<p`q||
(4.10)
P aCpnq. Then there exists Hδ P Cc8 pOpn, RqzGLpn, Rq{pR Opn, Rqqq
xδ Rδ and
with Hδ pg q 0 for any g P GLpn, Rq, || ln Apg q|| ¡ δ such that H
for any `
Hδ pg q »
ia n
pq
Rδ p`qβ` pg q dµP lanch p`q.
62
¥ 2 be an integer and f be a smooth function on Hn. For D P Z pDnq and
any k P Cc8 pOpn, RqzGLpn, Rq{pR Opn, Rqqq, since D is invariant under the action of
GLpn, Rq, we have
Let n
D pf k q pz q pDf q k pz q
if the integral is absolutely convergent. Let S be a Hecke operator and f be a function on
Hn . Then
S pf k q pz q pSf q k pz q
when the integral is absolutely convergent. Therefore, the convolution operator associated
P Cc8 pOpn, RqzGLpn, Rq{pR Opn, Rqqq commutes with the Hecke operators S if the integral is absolutely convergent and also commutes with any D P Z pDn q
to the function k
if the function is smooth and the integral is absolutely convergent.
Let D pOpn, RqzGLpn, Rq{ pR Opn, Rqqq be the space of Opn, Rq-bi-invariant compactly supported distributions on GLpn, Rq{R . For any compactly supported distribu-
P D pOpn, RqzGLpn, Rq{ pR Opn, Rqqq, the spherical transform Tpp`q (for any
` P aC pnq) is defined to be the scalar by which T acts on the function ϕ` . Furthermore, by
[16], for any R P H pa8 pnqqW satisfying an inequality
tion T
n
|Rp`q| ¤ C p1 ||`||qN eδ||<p`q||,
p` P aCpnqq
(4.11)
for some positive constants C, N and δ, there exists a distribution bi-pR Opn, Rqq-invariant
distribution T such that its spherical transform Tpp`q Rp`q for any ` P aC pnq.
For any T
P D pOpn, RqzGLpn, Rq{ pR Opn, Rqqq we define the spectral norm
||T ||spec :
sup
P p q
π8 p p qq
` a n ,
C
` χ , unitary
|Tpp`q|,
(if finite).
(4.12)
63
Annihilating operator 6np
4.2
The annihilating operator maps L2 pSLpn, ZqzHn q
Ñ
L2 pSLpn, ZqzHn q, and has the
property that it has a purely cuspidal image.
Lemma 4.4. (Construction of 6np ) Let n ¥ 2 and fix a prime p. For any `1
`2
p`1,1, . . . , `1,nq,
p`2,1, . . . , `2,nq P aCpnq, define
n
6pnp p`1, `2q :
t u
2
¹
¹
¹
¤ jk ¤n 1¤i1 ik ¤n
1 pp`1,i1 `1,ik
qp`2,j1 `2,jk
q . (4.13)
k 1 1 j1
Then there exists an operator denoted 6np , which is a polynomial in convolution operators
(associated to some compactly supported bi-pOpn, Rq R q-distributions), and in Hecke
operators at p, satisfies
6npf pzq 6pnpp`8pf q, `ppf qq f pzq,
pz P Hnq.
Here f is a smooth function on Hn which is also an eigenfunction of Z pDn q and the Hecke
operators at p. The parameter `8 pf q
`8pν q, as in (2.30), since f is an eigenfunction of
type ν P Cn1 , and the parameter `p pf q is defined in (2.49).
Remark 4.5. Before proving this Lemma we give an example of 6np for the cases n 2 and
n 3.
(i) For n 2, we have
62p Tp
2
Tp2 2Tp Lκ
1
(4.14)
pp`q where Lκ is the convolution operator associated to the distribution κ such that κ
p`1
p`2 for any ` p`1 , `2 q P aC p2q. This operator satisfies 62p
ℵ constructed in 2, [19].
ℵ2 for the operator
64
(ii) Let n 3. For j
1, 2, 3, define the compactly supported bi-ROpn, Rq-distributions
κj such that
κp1 p`q p`1
p`2
`
κ
y
1 p`q p 1
p`3 ,
2
y
κp2 p`q κ
1 p`q
3κp1 p`q,
p2 p`q κp1 p`q,
κz
3 p`q κ
and
p`3 ,
p`2
2
y
κ
2 p`q κp1 p`q 3κy
1 p`q,
y
κ
3 p`q κy
2 p`q κy
1 p`q,
for any ` p`1 , `2 , `3 q P aC pnq. Then
63p TpLκ
3
Tp3 TppTpp2qq2Lκ
Tp2 Tpp2q Lκ
pTpp2qq2Lκ pTpp2qq3
Tp2 Lκ2
Tpp2q Lκ3 .
2
1
Proof for Lemma 4.4. For any w1 , w2
(4.15)
1
P Wn (the Weyl group of GLpn, Rq), we have
6pnppw1.`1, w2.`2q 6pnpp`1, `2q,
where 6pnp p`1 , `2 q is holomorphic and satisfies the condition (4.11) for both `1 , `2
For each 1 ¤ k
P aCpnq.
¤ t n2 u, consider the polynomial
¹
¤ jk ¤n
1 xpp`j1 `j,k
q
1 j1
1 B1,k p`qx p1qr Br,k p`qxr p1qd pnqxd pnq
k
k
for any ` p`1 , . . . , `n q P aC pnq, where dk pnq k!pnn!kq! . For each 1 ¤ r
¤ dk pnq 1, the
coefficients
Br,k p`q P H paC pnqqWn
satisfy (4.11). For 1 ¤ k
¹
¤ t n2 u, we have,
¹
¤ jk ¤n 1¤i1 ik ¤n
dr¸
k pnq
1 j1
j 0
aj,k p`1 q bj,k p`2 q
1 pp`1,i1 `1,ik
qp`2,j1 `2,jk
q
65
for `1
p`1,1, . . . , `1,nq, `2 p`2,1, . . . , `2,nq P aCpnq and some positive integer drk pnq. So,
pq
dr¸
k n
aj,k p`1 q bj,k p`2 q
j 0
d pnq
k
¸
¹
¤ ik ¤n
1 i1
¹
`2,ik
r 0
d pnq
k
¸
¤ ik ¤n
Br,k p`2 qprp`1,i1 q
`1,ik
q .
r 0
1 i1
For 1 ¤ j
Br,k p`1 qprp`2,i1 ¤ drk pnq, we have,
aj,k p`1 q, bj,k p`2 q P H paC pnqqWn
satisfies (4.11) because aj,k p`1 q (resp. bj,k p`2 q) is a polynomial in Br,k p`1 q (resp. Br,k p`2 q)
(for 1
¤
r
¤ dk pnq).
So there exist compactly supported bi-pR Opn, Rqq-invariant
pkq whose spherical transform is a p` q. For each 1
j,k 1
distributions κj
1¤j
¤
k
¤
n
t u
2
and
¤ drk pnq, let Lκp q be the convolution operator associated to the distribution κpjkq.
k
j
We also have the p-adic version of Theorem 4.3 as explained in [19] (also see [9]). So
pkq such that
there exist Hecke operators Sj
pk q
Sj f
bj,k p`ppf qq f
where f is an eigenfunction of Hecke operators with parameter `p pf q P aC pnq.
Therefore,
n
6np t u
2
¹
k 1
pq
dr¸
k n
j 0
pk q
Sj Lκpkq j
and
6npf p6npp`8pf q, `ppf qq f
where f is an eigenfunction of Casimir operators and the Hecke operators.
66
Since we use distributions to define
smooth functions. For any δ
6np, the operator is well defined in the space of
¡ 0, let
U pδ q : tg
P GLpn, Rq | || ln Apgq|| ¤ δu .
(4.16)
Let Hδ be a bi-pR Opn, Rqq-invariant compactly supported smooth function with supppHδ q €
U pδ q, i.e., Hδ pg q 0 for any g
R U pδq. We define the operator Hδ 6np to be
Hδ 6np F
6nppF Hδ q
(4.17)
Ñ C which makes the integral convergent. By the Paley-Wiener
Theorem 4.3, the operator Hδ 6np is a polynomial in convolution operators (associated to the
bi-pR Opn, Rqq-invariant, compactly supported smooth functions), and in Hecke operators at the prime p. Then the operator Hδ 6np can be defined for the functions in L2 pHn q
for a function F : Hn
and
n
z
pn
x
H
δ 6p p`1 , `2 q Hδ p`1 q 6p p`1 , `2 q
where `j
(4.18)
p`j,1, . . . , `j,nq P aCpnq for j 1, 2.
Proposition 4.6. Let n
¥
2 and p be a prime. Let E pz q be an Eisenstein series as in
Definition 2.28. Then
6pnpp`8pE q, `ppE qq 0
and
6npE 0
(4.19)
for any prime p. Let φ be a self-dual Hecke-Maass form as in Proposition 2.26. Then
6pnpp`8pφq, `ppφqq 0
Moreover, for any constant C
and
6npφ 0.
(4.20)
P C,
6npC 0.
(4.21)
67
¤ n1, . . . , nr n and r ¥ 2. For each i 1, . . . , r
let φi be either a Hecke-Maass form for SLpni , ZqzHn of type µi pµi,1 , . . . , µi,n 1 q P
Cn 1 or a constant with µi p0, . . . , 0q. Let t pt1 , . . . , tr q P Cr with n1 t1 nr tr 0. Let E pz q : EP
pz; t; φ1, . . . , φr q be an Eisenstein series as in Definition 2.28. Let
η1 0 and ηi n1 ni1 for i 1, . . . , r. By Lemma 2.32, we have
η ¸n
η ¸n
n n
i
t η n ` pE q
` pE q Proof. Let n
n1 nr with 1
i
i
i
n1 ,...,nr
i
i
i
8,j
i
2
j ηi 1
i
i
i
p,j
j ηi 1
for any prime p. Therefore,
1 pp`8,ηi 1 pE q `8,ηi
ni
pE qqp`p,ηi 1 pE q `p,ηi
ni
pE qq 0
and 6pnp p`8 pE q, `p pE qq 0.
Let φ be a self-dual Maass form for SLpn, Zq. Then by Remark 2.27,
`v pφq `v pφq
¤ 8. So either there exists 1 ¤ j ¤ n such that
`v,j pφq 0 or there exist 1 ¤ j j 1 ¤ n such that `v,j pφq `v,j 1 pφq 0. Therefore
6pnpp`8pφq, `ppφqq 0.
up to permutations, for any place v
Let C
P C be a constant. Then
C
for any z
C I0pzq
P Hn and
`8 pC q n 2j2
So 1 p`8,j pC q`p,j pC q
1
n
j 1
and
`p pC q n 2j
2
1
n
.
j 1
0 for any j 1, . . . , n. Therefore, 6pnpp`8pC q, `ppC qq 0.
68
The idea of the following theorem and its proof is in [19].
Theorem 4.7. Let n ¥ 2 be an integer and p be a prime. Let δ
¡ 0 and Hδ 0 be a bi-pR Opn, Rqq-invariant compactly supported smooth function with supppHδ q € U pδ q. Then the
space of the image of Hδ 6np on L2 pSLpn, ZqzHn q is cuspidal and infinite dimensional. So
there are infinitely many non self-dual Hecke-Maass forms.
Proof. The Langlands spectral decomposition states that
L2 pSLpn, ZqzHn q
L2cont pSLpn, ZqzHnq ` L2residue pSLpn, ZqzHnq ` L2cusp pSLpn, ZqzHnq
where L2cusp denote the space of Maass forms, L2residue consists of iterated residues of
Eisenstein series and L2cont is the space spanned by integrals of Eisenstein series. The
Eisenstein series are studied in §2.6. So, for any f
P
L2 pSLpn, ZqzHn q there exists
fcont pz q P L2cont , fresidue pz q P L2residue and fcusp pz q P L2cusp such that
f pz q fcont pz q
fresidue pz q
fcusp pz q.
By Proposition 4.6, for any Eisenstein series E and constant C, we have 6np E
6npC 0.
Since the invariant integral operators and Hecke operators preserve the space of cuspidal
functions,
Hδ 6np f
Hδ 6npfcusp P L2cusp pSLpn, ZqzHnq .
Therefore the image of Hδ 6np on L2 pSLpn, ZqzHn q is cuspidal.
We will show that the image of Hδ 6np on L2 pSLpn, ZqzHn q is infinite dimensional. First
we show that it is non-zero. Take α8
pα8,1, . . . , α8,nq,
αp
pαp,1, . . . , αp,nq P aCpnq
xδ pα8 q 6pn pα8 , αp q 0 and α8 and αp satisfies the condition in Definition 3.13.
such that H
p
69
As in Definition 3.16, we construct a quasi-Maass form F of type ν pα8 q for t8, pu and of
length L 8 such that
F pz q ¸
AF pk1 , . . . , pkn1
¸
P p qz p q
¥
γ N n 1,Z SL n 1,Z k1 ,...,kn1 0
1
p2
pk1 °n1
j 1
p q
kj n j j
kn1
WJ ..
γ
.
1
z; ν pα8 q, 1.
1
where
AF p1,
. . . , 1, p, 1, . . . , 1q loooomoooon
¸
¤ kj ¤n
ppαp,k1 αp,kj
q,
p for j 1, . . . , n 1q
1 k1
j
and AF ppk1 , . . . , pkn1 q satisfies the multiplicative condition (2.46) and (2.47). Then
xδ pα8 q 6pn pα8 , αp q F pz q
Hδ 6np F pz q H
p
for z
P Hn .
Let Fr be the automorphic lifting of F as in Definition 3.20. Then Fr
and Hδ 6np Fr
Hδ 6np Fr
Let
Σ T :
P
C8
P L2 pSLpn, ZqzHnq
X L2 pSLpn, ZqzHnq is cuspidal as we show above.
To show that
0, we need Lemma below.
$
1 x1,2 . . . x1,n
y1
'
'
'
&
1 . . . x2,n .. ..
'
.
. '
'
%
yn1
yi T,
n H
for i 1, . . . , n
1
..
.
y1
1
P
¡
,
/
/
/
.
1/
/
/
-
(4.22)
then
ΣT
€
¤
P
p qXSLpn,Zq
γ Pn1,1 R
γFn
pT ¡ 1q.
70
Lemma 4.8. Let
¡ max
T
then for any z
#
expp4δ q, exp
+
n! ln p
n
2 t 2 u 1 ! n t n2 u !
(4.23)
P ΣT ,
xδ pα8 q6pn pα8 , αp q F pz q.
Hδ 6np Frpz q H
p
By Lemma 4.8, for any z
P ΣT , and for any T as in (4.23), we have
xδ pα8 q 6pn pα8 , αp q F pz q 0.
Hδ 6np Frpz q Hδ 6np F pz q H
p
So Hδ 6np Fr
0. Therefore, the image of Hδ 6np on L2 pSLpn, ZqzHnq is not empty.
Assume that the space of image of 6np on L2 pSLpn, ZqzHn q is finite dimensional. Let
6
n
pU
:
"
uj , a Hecke-Maass form of type µj P C n 1
H.
6npuj 0, *
and ||u||22 1
Then it is the basis of the space of image of Hδ 6np . Since we assume that it is finite dimensional, it follows that
B
6npU is a finite set.
Suppose that the number of elements of
6npU is
8, where B is the positive integer and
6npU tu1, . . . , uB u .
Then there are c1 , . . . , cB
P C such that
Hδ 6np Frpz q B
¸
j 1
cj uj pz q.
(4.24)
71
Compare Fourier coefficients on both sides. For nonnegative integers k1 , . . . , kn1 , the
ppk , . . . , pk qth Whittaker-Fourier coefficient for Hδ 6npFr is
n 1
1
WHδ 6np Fr z; pk1 , . . . , pkn1
1 v1,2 . . . v1,n
»1
»1
1 . . . v2,n Hδ 6npFr z
..
..
.
0
0
. 1
e2πipp
For each j
pk2 vn2,n1
1
y1 yn1
WHδ 6np Fr B
¸
..
dv1,2.
n 1,n
P
cj Aj p , . . . , p
p
1
2
°n1
i 1
kn1
p q
..
WHδ 6np Fr .
pk1 WJ ..
j 1
p
1
2
xδ pα8 q 6pn pα8 , αp q
and H
p
°n1
k1
k
; p , . . . , p n1 i 1
kn1
p q
ki n i i
z; µj , 1
.
pk1 kn1
Hxδ pα8q 6pnppα8, αpq WJ cj Aj p , . . . , p
kn1
1
1
k1
By (4.24) we
ki n i i
P Hn. For z P ΣT , we have
y1 yn1
B
¸
Cn1 , and let
k1
k
; p , . . . , p n1 .
k1
j 1
q dv
n 1
1
for any z
pk1 v1,2
P C be the ppk , . . . , pk qth Fourier coefficient of uj .
1, . . . , B, the Hecke-Maass form uj is of type µj
Aj ppk1 , . . . , pkn1 q
have
k1 v
n 1,n
..
pk1 WJ .
p q
z; ν α8 , 1
1
kn1
..
.
z; µj , 1,
1
0 by our assumption. Fix k1 kn1 0. Since B is
a finite positive integer, it is possible to assume that ν pα8 q µj for j 1, . . . , B. Then
72
there are c11 , . . . , c1B
P C such that for y1, . . . , yn1 ¡ T , and
y1 yn1
WJ ..
; ν, 1
.
B
¸
y1 yn1
c1j WJ ..
.
j 1
1
; µj , 1
1
0 (for j 1, . . . , B). Assume that c11 0. Since WJ is an eigenfuncp iq
tion of ∆n , for y1 , . . . , yn1 ¡ T and for any i 1, . . . , n 2, we have
y1 yn1
..
∆pniq λp8iq pα8 q WJ . ; ν pα8 q, 1 0,
for at least one c1j
1
so
B
¸
c1j
piq pα q
λp8iq p`8 pµj qq λ8
8
y1 yn1
WJ j 1
..
; µj .
0,
1
p iq
pj q
where λ8 pα8 q and λ8 p`8 pµj qq (for j
1, . . . , B and i 1, . . . , n 1) are eigenvalues of
piq
∆n as in (2.34). Since we assume that ν pα8 q µ1 , . . . , µB , there exists i 1, . . . , n 1
such that
λp8iq p`8 pµj qq λp8iq pα8 q 0,
Again, there exist c22 , . . . , c2M
y1 yn1
WJ P C such that
..
.
p for j 1, . . . , B q.
; µ1 , 1
1
B
¸
c2j WJ j 2
y1 yn1
..
.
; µj , 1
1
¡ T and c2j 0 for at least one j 2, . . . , B. So in a similar manner, we
deduce that there exists µ P tµ1 , . . . , µB u such that
y1 yn1
..
WJ . ; µ, 1 0
for y1 , . . . , yn1
1
¡ T . This gives a contradiction. Therefore, 6npU should be an infinite
set. It follows that the image of Hδ 6np on L2 pSLpn, ZqzHn q is infinite dimensional.
for any y1 , . . . , yn1
73
To complete the proof of the Theorem, we give a proof of Lemma 4.8.
Proof for Lemma 4.8. Let κ be a compactly supported function with support in U pδ q. Let
t
¡ expp4δq. For any z P Σt, assume that || ln Apzh1q|| ¤ δ for some h P GLpn, Rq. By
Iwasawa decomposition,
y1 yn1
z
x
..
.
1
for x P N pn, Rq, y1 , . . . , yn1
h
¡ 0 and
| detphq|
±n1
1
n
n j
j 1 vj
P N pn, Rq, v1, . . . , vn1 ¡
1, . . . , n 1, we have
for u
v1 vn1
u
..
k,
.
1
0 and k
expp4δ q ¤
P Opn, Rq.
yj
vj
Then by Lemma 2.5, for j
¤ expp4δq.
So,
vj
¥ yj expp4δq ¥ t expp4δq ¡ 1.
Then Frphq F phq because Σ1
p q
Fr κ z
»
GLpn,Rq{R
»
GL n,R R
p
q{
P R satisfies (4.23).
for any z P ΣT ,
Let T
€ ”γPP n 1,1
n
pRqXSLpn,Zq γF . So for z P Σt , we have
Frphqκpzh1 q dh
F phqκpzh1 q dh F
κpzq κppα8q F pzq.
For a non-negative integer B
¤ exp 2p
t
n! ln p
1 ! n
n
u
2
q p q!
t
n
u
2
, and
TpB Frpz q AF ppB , 1, . . . , 1q F pz q.
The operator Hδ 6np is a polynomial in Hecke operators and convolution operators associated
with compactly supported functions which have support in U pδ q. By combining the above
74
computations, for any z
P ΣT , we obtain
xδ pα8 q 6pn pα8 , αp q F pz q.
Hδ 6np Frpz q H
p
Lemma 4.9. Let n ¥ 2 and p be a prime. Then
n
||6 || ¤
n
2
pf 2
t u
2
¹
k p n2 1 q
p n2 1
p
p q
k n2 1
n2 1
4dk pnq
||f ||22
(4.25)
k 1
P C 8 X L2 pSLpn, ZqzHnq. Here dk pnq k!pnn!kq! for k 1, . . . , t n2 u. Moreover,
for any Hδ P Cc8 pOpn, RqzGLpn, Rq{pR Opn, Rqqq and δ ¡ 0, there exists a positive
real number CH 8 such that
for any f
δ
n
|| 6 || ¤
Hδ np f 22
2
CH
δ
t u
2
¹
k p n2 1 q
p n2 1
p
p q
k n2 1
n2 1
4dk pnq
||f ||22
,
(4.26)
k 1
for any f
P L2 pSLpn, ZqzHnq.
Proof. Since 6np kills the continuous part, we only need to consider cuspidal functions. For
any cuspidal function f
P C 8 X L2 pSLpn, ZqzHnq, we have
f pz q 8̧
hf, uj i uj pz q ,
j 1
where uj pz q’s are Hecke-Maass forms for SLpn, Zq of type µj with ||uj ||22
||6npf ||22 ¤
8̧ 1. So
2
pn
2
p `8 uj , `p uj hf, uj i .
6 p p q p qq |
|
n 1
If there exists a constant A ¡ 0 such that 6pnp p`8 puj q, `p puj qq ¤ A for any uj , then
||6npf ||22 ¤ A2 ||f ||22.
75
By (4.13),
pn
`
,
`
p 1 2 6 p
q
n
t2u
¹
k1
¹
¹
1 pp`1,i1 ¤ jk ¤n 1¤i1 ik ¤n
`1,ik
qp`2,j1 1 j1
`2,jk q (4.27)
¤
n
t u
2
¹
¹
¹
¤ jk ¤n 1¤i1 ik ¤n
1
pp`
1,i1
`1,ik `2,j1
`2,jk
q k 1 1 j1
n
¤
t u
2
¹
¹
¹
¤ jk ¤n 1¤i1 ik ¤n
p<p`1,i1 1
`1,ik `2,j1
`2,jk
q
k 1 1 j1
for any `1 , `2
P aCpnq. Recall the following theorem from [12].
¥ 2. Let f be a Hecke-Maass
form for SLpn, Zq. Then for j 1, . . . , n and any prime v ¤ 8, including 8,
Theorem 4.10. (Luo-Rudnick-Sarnak) Fix an integer n
< p`v,j pf qq ¤
For `1
1
2
n2 1
1
.
(4.28)
`8puq and `2 `ppuq for any Hecke-Maass forms u, the last line of the (4.27)
is less than or equal to
t u 2dk pnq
2
¹
¹ n
1
p
pk q xj
,
j 1
°
2
dk pnq xpkq 0. So for each k 1, . . . , t n u,
kpn 1q
and j2
j
1
n2 1
2
pk q
2¹
dk pnq 2¹
dk pnq pk q
x
x1
2dk pnq
pk q pk q xj
xj
p
q
2
2
1 p
p
1 p
j 1
j 1
2¹
dk pnq
pk q
pk q x
x
j
j
p 2
p 2
j 1
2¹
dk pnq kpn2 1q
k p n2 1 q
2 1
2 1
n
n
¤
p
p
,
j 1
k 1
pk q ¤
where xj
since p
p q
k n2 1
n2 1
¡ 1. Therefore, for any Hecke-Maass form u,
pn
p `8 u , `p u 6 p p q p qq ¤
n
¹
k 1
k p n2 1 q
p n2 1
p
p q
k n2 1
n2 1
2dk pnq
.
76
By Theorem 4.3, for any Hecke-Maass form u, we have
x
Hδ `8 u p p qq ! p1 ||`8puq||q1eδ||<p`8puqq||.
By Theorem 4.10, ||<p`8 puqq|| is bounded. So there exists a real positive constant CHδ
such that
x
Hδ `8 u p p qq ¤ CH
δ
for any Hecke-Maass form u. Since Hδ 6np f is cuspidal for any f
P L2 pSLpn, ZqzHnq it
follows that
|| 6 || ¤
Hδ np f 22
2
CH
δ
n
¹
k p n2 1 q
p n2 1
p
p q
k n2 1
n2 1
4dk pnq
||f ||22
.
k 1
4.3
Example for Hδ
For any g
P GLpn, Rq, by (2.13), we have
g
|det g|
1
n
k1 Apg q k2 ,
( for k1 , k2
P Opn, Rq),
then define
upg q :
where Apg q
e2a1
ea1
e2a 1,
1
1 2a1
trpApg q2 q 1 e
n
n
..
.
such that a1 , . . . , an
ean
e2an
P R and a1 1,
(4.29)
an
0. Then since
n
0 ¤ upg q,
( for g
P GLpn, Rq),
and
upg q
1 ¤ expp2|| ln Apg q||q,
( for any g
P GLpn, Rq).
77
We generalize the function used in [3]. Let φ : r0, 8q
with supppφq € r0, 1s and
»8
φpxq dx 1.
0
For any δ
Ñ r0, 8q be a smooth function
¡ 0, let Y e 11 . Define
2δ
Hδ pg q : φ pY upg qq ,
( for g
P GLpn, Rq).
(4.30)
Then Hδ is a compactly supported smooth bi-pR Opn, Rqq-invariant function. Since
φpY upg qq 0 for upg q ¡
1
Y
, we have
supppHδ q € tg
P GLpn, Rq | || ln Apgq|| ¤ δ u .
For example, for x P r0, 8q, let
#
φpxq :
where c ³1
tp11tq
with a support p0, 1q.
For any g
tr
gn,1
gg
xp11xq
,
for 0 x 1;
otherwise,
P GLpn, Rq, we have
¸
¤ ¤
2
gi,j
: |det g| ||g||2 |det g|
2
n
2
n
So,
upg q 1
||g||2 1,
n
¡ 0 such that
2δ
e
1 »
Hn ,
u z t
p q
( for g
e2δ 1
1 d z
¤
P GLpn, Rq).
¤ 1 and
1,
tr Apg q2 .
1 i,j n
Lemma 4.11. Take δ
(4.31)
dt. Then φ is always non-negative and it is a smooth function
... g1,n
..
.
... gn,n
..
.
t
exp
0,
exp
0
g1,1
1
c
p for 0 ¤ t ¤ 1q,
78
and let Hδ be a function defined in (4.30). Then we have
»
Hn
Hδ pz q d z
¤
1.
e 11 . Then
Proof. Let Y
2δ
»
Hn
Hδ pz q d z »
Hn
Y1
φpY upz qq d z »8
0
φptq
»8 »
0
Hn ,
u z t
p q
»
1 d z dt.
Hn ,
t
u z
Y
p q
Since φptq 0 for t ¡ 1, we have 0 ¤
»
Hn
t
Y
¤ Y1 ¤ 1. So
Hδ pz q d z ¤
»8
0
φptq dt 1.
φpY tq d z dt
79
Chapter 5
A PPROXIMATE CONVERSE THEOREM
5.1
Approximate converse theorem
Let S be a finite set of primes including 8. Let qS : max tv
and qS : 1 for S
!
B n pS; δ q : z
P S | v 8u for S t8u
t8u. For δ ¡ 0, define
)
R F€n || ln Apz1τ q|| ¤ δ for some τ P Fn
)
¤!
€n
z P Fn || ln Apz 1 τ q|| ¤ δ for some τ R F
(5.1)
,
α1
q
/
S
/
/
/
.
/
n
€
.
/
z
F
for
some
.
.
¤
n αn
z F qS
/
'
/
'
'
αn t n2 u /
non-negative integers α1
/
'
/
'
/
'
%
$ α
qS 1
'
&
¤ for some z Fn , for some
..
€n z F
.
non-negative integers α
'
αn
1
%
αn
$
'
'
'
'
'
'
&
R
P
R
P
qS
,
/
.
t n2 u /
-
€n is the extended fundamental domain defined in (3.19).
where F
We state the main theorem.
Theorem 5.1. (Approximate Converse Theorem) Let n ¥ 2 be an integer and M be a set
of primes including 8 and at least one finite prime. Let `M
t`v P aCpnq, v P M u be a
quasi-automorphic parameter for M and F`M be a quasi-Maass form of `M of length L as
in Definition 3.16. Let Fr`M be the automorphic lifting of F`M in (3.18). Assume that there
P M such that 6pnpp`8, `pq 0. Let S € M be a finite subset including 8.
Choose arbitrary δ ¡ 0 and an arbitrary bi-pR Opn, Rqq-invariant compactly supported
exists a prime p
80
p δ p`8 q 0.
smooth function Hδ with supppHδ q € U pδ q, satisfying H
Then there exists an unramified cuspidal automorphic representation π pσ q b1v πv pσv q
with an automorphic parameter σ as in (3.10) such that `M and σ are -close for S where
:
2
Cp n, S, Hδ ; `8
M
F`
sup Fr`M
p q
B n S;δ
p
2
2 pn
p
L F`M
`
p `8 , `p H
δ 8 p q p
q 6p
q
.
q2
(5.2)
Here Cp pn, S, Hδ ; `8 q is a positive constant (which is determined by `8 , the prime p and
Hδ ) given explicitly as
n
Cp pn, S, Hδ ; `8 q :Vol pB pS; δ q X F
n
#
n
kpn2 1q
p n2 1
p
p q
k n2 1
n2 1
k 1
n¸1 »
j 1
q
t u
2
¹
Hn
pj q
∆
n
t
2
CH
δ
λp8jqp`8q Hδ pτ q dτ
n
u
2
¸
¸
q S,
finite prime
j 1
P
q p
j n 1
2
4dk pnq
(5.3)
2
¸
q
¤ kj ¤n
q k1 1 k1
2 ,
/
.
kj ,
/
-
and
LpF`M q2 :
L
¸
A`M m1 , . . . , mn1
±n1
kpnkq{2
mn2 1 0|mn1 |¤L
k1 mk
m1
mn1
»8
» 8 ..
WJ .
T
T
m1 1
L
¸
| |
| |
where dk pnq k!pnn!kq! for k
for Hδ . For r
p
¸
2
q 2
, 1 d y,
p q
y; ν `8
1
1, . . . , t n2 u and CH ¡ 0 is a constant defined in Lemma 4.9
δ
1, . . . , n 1, we have
A`M p1,
. . . , 1, q , 1, . . . , 1q loooomoooon
r
" °
¤ kr ¤n q
1 k1
0,
p`q,k1 `q,kr
q , if q P M,
otherwise,
81
and A`M p1, . . . , 1q
1 while A` pm1, . . . , mn1q is determined by the multiplicative relations in (2.46) for pm1 , . . . , mn1 q P Zn1 . Here T is a constant such that
T
Remark 5.2.
M
¥ max
#
expp4δ q, exp
+
n! ln p
n
2 t 2 u 1 ! n t n2 u !
.
(i) If in (5.2) is sufficiently small, then by Remark 8 [4], π pσ q is uniquely
determined.
(ii) The constant in (5.2) mainly depends on sup Fr`M z
p q
B n δ,S
2
z . It is an inter-
p q F` p q
M
esting problem to choose Hδ so that is as small as possible.
(iii) Taking δ and Hδ is also important to get a good . We give an example for Hδ in §4.3.
(iv) For any finite set S, since the space B n pS; δ q is bounded, there are finitely many
γ1 , . . . , γr
P SLpn, Zq such that
γ1 Fn Y Y γr Fn
 B npδ, S q
and
sup Fr`M z
p q
B n δ,S
2
z p q F` p q M
sup
p qXFn
B n δ,S
|F` pγj zq F` pzq|2 | j 1, . . . , r
M
(v) For an unramified cuspidal representation π
analytic conductor
C pπ q :
n
¹
M
(
.
bv πv pσv q of AzGLpn, Aq, define an
p1 |σ8,j |q
j 1
pσ8,1, . . . , σ8,nq P aCpnq. Fix Q ¥ 2. By [4], for any
unramified cuspidal representation π bv pσv q of A zGLpn, Aq with C pπ q ¤ Q,
there exists a prime p ! log Q such that p6np pσ8 , σp q is sufficiently large.
as in [4], where σ8
82
5.2
Proof of Theorem 5.1
xδ p`8 q 0. By Theorem 4.7, since 6pn p`8 , `p q Proof of Theorem 5.1. Take Hδ such that H
p
0, the automorphic function Hδ 6np Fr`M
6np
If
6np
Hδ
Fr`M
Fr`M
0 and
Hδ Hδ 6npFr` P
M
satisfies (3.17) for L2cusp pSLpn, ZqzHn q .
¡ 0 then, by Lemma 3.19, there exists an unrami-
fied cuspidal representation with an automorphic parameter σ, which is -close to `M . Let
ν : ν p`8 q as in (2.33).
To get the lower bound for ||Hδ 6np Fr`M ||22 , we use the following Lemma.
¥ 2, let f be a square-integrable, cuspidal, automorphic,
smooth function for Hn . For T ¥ 1,
Lemma 5.3. For an integer n
8̧
||f || ¡
2
2
8̧
mn2 1 mn1 0 T
m1 1
y1 yn1
»8
¸
»8
T
|Wf py; m1, . . . , mn2, mn1q|2
d y
±
j pnj q1 dy . Here
¡
0 and d y jn11 yj
j
y
1
Wf pz; m1 , . . . , mn1 q is the Fourier coefficient of f for m1 , . . . , mn1 P Z, defined by
where y
..
, y1 , . . . , yn1
.
1
Wf pz; m1 , . . . , mn1 q
»
z
»
z
Z R
where u Let T
ΣT, 1
2
1
ui,j
..
¥ max
, ui,j
1
expp4δ q, exp
6
n
p
q d u
P R for 1 ¤ i j ¤ n and du ±1¤i j¤n dui,j .
"
€ Fn,
mn2 u2,3 mn1 u1,2
Z R
.
f puz qe2πipm1 un1,n Fr`
M
*
p
2
t
n! ln p
1 ! n
n
u
2
q p q!
t
n
u
2
¡ 1. By Lemma 4.8, for any z P
Hδ pzq Hxδ p`8q6pnpp`8, `pq F` pzq.
M
83
P ΣT,
Then for z
1
2
, and for integers 1 ¤ m1 , . . . , mn2 , |mn1 | ¤ L, the pm1 , . . . , mn1 qth
Fourier coefficient for Hδ 6np Fr`M is
pz; m1, . . . , mn1q
Hxδ p`8q6pnpp`8, `pq A±` npm1 1, . . .k,pnmnkq{21q
k1 |mk |
m1 mn2 |mn1 |
2πipm x .. WJ . y; ν p`8 q, 1e
WHδ 6np Fr`
M
M
m2 xn2,n1
1 n 1,n
mn1 x1,2
q.
1
Therefore, by Lemma 5.3,
L
¸
||Hδ 6npFr` ||22 ¥
M
m1 1
»8
T
L
¸
¸
|mn1 |¤L
mn2 1 0
m1
» 8 WJ T Consider the case when v
8.
x
Hδ `8 pnp `8 , `p
p q6 p
mn2|mn1|
q
2
A`M pm1 , . . . , mn1 q ±n1
kpnkq{2 k 1 | m k |
(5.4)
2
, 1 d y.
..
. y; ν p`8 q
1
1, . . . , n 1, there exist λp8jqp`8q P C as
For j
in Definition 3.8 for the corresponding character associated to the parameter `8 . So for
j
1, . . . , n 1, we have
||
∆pnj q λp8j q p`8 q
n
¤
t u
2
¹
6np
kpn2 1q
p n2 1
p
Fr`M
p q
k n2 1
n2 1
Hδ ||22
4dk pnq
||
∆pnj q λp8j q p`8 q Fr`M
Hδ ||22
,
k 1
pj q
since the operator 6np commutes with the invariant differential operators ∆n . Since
∆pnj q λp8j q p`8 q F`M
for any z
||
Hδ pzq 0
P Hn, it follows that
∆pnj q λp8j q p`8 q Fr`M
Hδ ||22 || ∆pnjq λp8jqp`8q Fr` F` Hδ ||22
r
|| F` F` p∆pnjq λp8jqp`8qqHδ ||22
M
M
M
M
84
and
||
Fr`M
¤
»
M
»
n
Bn
p
q{
pS;δqX
p q
B n S;δ
Fr`
Bn
pS;δq
M
r
F`M
ξ ∆pnj q
F` p q M
2
F`M Vol B n S; δ
p p
2
p∆pnjq λp8jqp`8qqHδ pξ 1z dξ dz
F` pξ q M
»
Fn
sup Fr`M
¤
p∆pnjq λp8jqp`8qqHδ ||22
GL n,R R
F
»
F`
λp8jqp`8qqHδ dξ
»
qXF q
n
GL n,R R
p
q{
pj q
∆
n
p
2
d z
λp8jqp`8qqHδ dξ
2
.
q 8 and q P S. For j 1, . . . , t n2 u there exists λpqjqp`q q P
Consider the case when v
C, as in Definition 3.8, for the corresponding character associated to the parameter `q . Since
Hδ 6np commutes with Hecke operators, it follows that
||
Tqpj q λpqj q p`q q Hδ 6np Fr`M ||22
n
¤
t u
2
¹
kpn2 1q
p n2 1
p
p q
k n2 1
n2 1
4dk pnq
CH2 ||
δ
Tqpj q λpqj q p`q q Fr`M ||22 ,
k 1
1, . . . , t n2 u. Since
for j
pj q λpj q p`
Tq
q
qq
F`M
0 and Fr` pzq F` pzq for z P Fn, we
M
M
have
||
» 2
pj q
p
jq
p
jq
2
pj q r
Tq λq p`q q F`M ||2 Tq λq p`q q Fr`M pz q d z
Fn
Fn
pj q in (2.48) for each j
By the definition of Tq
7
»
pj q such that
pj q r
F`M
Tq
2
F` pzq
M
d z.
1, . . . , t n2 u, there exists a positive integer
Tq
Tqpj q Fr`M
F` pzq M
q
p q
j n 1
2
1
q
p q
j n 1
2
¸
1
¤ ¤¤kj ¤j
cpk1 ,...,kj q Tqk1 Tqkj Fr`M
0 k1
pj q
7p¸
Tq q k 1
Fr`M
F` pCk zq,
M
F ` pz q
M
85
where cpk1 ,...,kj q
P Z and Ck ’s are upper triangular matrices with integer coefficients which
are determined by Hecke operators in the first line. So
»
Fn
pj q r
T
F`M
q
2
M
¤ Vol pB pS; δq X F q n
n
q p
d z
F` pzq
j n 1
2
2
T pj q 1 q
Fn
pj q
7p¸
Tq q 1
p q
j n 1
2
q
sup Fr`M
p q
B n S;δ
r
F`M
2
F`M Ck z d z
p
¤ kj ¤n
q
k1
2
F`M Vol pB pS; δq X F q kj q
k 1
2
¸
q
¤
»
n
sup Fr`M
n
1 k1
p q
B n S;δ
2
F`M .
To complete the proof of the main theorem, we give the proof of Lemma 5.3.
Proof of Lemma 5.3. Let f : SLpn, ZqzHn
Ñ C be a cuspidal automorphic function,
which is smooth and square integrable. For j 1, . . . , n 1, let
unj
1
:
u1,nj
..
.
Inj
unj,nj
0j nj
1
0nj j 1 1
Ij
P p
N n, R
q
P R and 0ab is an a b matrix with 0 for every entry. Here
N pn, Rq € GLpn, Rq is the set of n n unitary upper triangular matrices. We follow the
argument in 5.3, [12]. Let n ¥ 2 be an integer. Fix j 1, . . . , n 1. For m1 , . . . , mj P Z,
where u1,nj 1 , . . . , unj,nj
1
define
fj pz; m1 , . . . , mj q :
»
»
z
e p
z
f pun un1 unj 1 z q
Z R
Z R
2πi m1 un1,n
where
d u
n j 1
n
¹j
k 1
mj unj,nj
duk,nj 1 .
1
q d u
n
dunj
1,
86
Then for m1 , . . . , mn1
P Z,
fn1 pz; m1 , . . . , mn1 q Wf pz; m1 , . . . , mn1 q.
Let f0 pz q : f pz q with z
P Hn.
By following the proof of Theorem 5.3.2, [12], we can
also prove the following.
(i) For j
1, . . . , n 1, we have
fj pz; m1 , . . . , mj q
»
z
»
z
Z R
(ii) Fix j
fj 1 punj 1 z; m1 , . . . , mj 1 qe2πimj unj,nj
1
d unj 1 .
Z R
1, . . . , n 2. For positive m1, . . . , mj1 P Z, we have
fj 1 pz; m1 , . . . , mj 1 q
8̧
¸
P
p qz p q
γnj
fj
z; m1 , . . . , mj 1 , mj .
Ij
mj 1 γnj Pnj 1,1 Z SL n j,Z
(iii) For positive integers m1 , . . . , mn2 , we have
fn2 pz; m1 , . . . , mn2 q Since the Siegel set Σ1, 1
2
||f || 2
2
»
Fn
¸
P
0 mn1 Z
¸
P
0 mn1 Z
fn1 pz; m1 , . . . , mn2 , mn1 q
Wf pz; m1 , . . . , mn2 , mn1 q .
€ Fn,
|f pzq|
2
d z
¡
»8
1
»8»
1
1
2
1
2
»
1
2
1
2
|f pzq|2
d z.
87
Then
»8
1
»8
»8»
1
1
1
2
12
»8»
1
2
1
»
1
2
21
1
2
|f pzq|2
»
8̧
1
2
γn1
f1 1
d z
1
2
¸
P
p qz p q
m1 1 γn1 Pn2,1 Z SL n 1,Z
0
.. .
f pz qe2πim1 pγn1,1 x1,n γn1,n1 xn1,n q
; m1 d z,
0
y1 z 1
0 ... 0 1
pγ where γn1
γn1
n 1,1
pγ n 1,1
... γn1,n1
»
1
2
21
q P Pn2,1pZqzSLpn 1, Zq. For a positive integer m1 and
q P Pn2,1pZqzSLpn 1, Zq, it follows that
... γn1,n1
»
1
2
12
f pz qe
p
2πim1 γn1,1 x1,n
γn1
f1 1 n1
γn1,n1 xn1,n q ¹ dx
k
k1
0
.. . y1 z 1
0
0 ... 0 1
So,
»8»8»
1
1
»8
1
1
2
21
»
»8»
1
1
2
12
1
2
12
|f pzq|2
»
d z
8̧
1
2
12 m1 1
¸
γn1
f1 1
γn1 PPn2,1 pZqzSLpn1,Zq 2
y1 z 1
; m1 0
0 ... 0 1
¹
0
.. .
¤ ¤ dxi,j d y
1 i j n 1
¥
8̧ » 8
m1 1 1
»8»
1
1
2
1
2
»
1
2
1
2
2
0
.. 1
y1 z
.
f1 ; m1 0
0 ... 0 1
¹
¤ ¤ 1 i j n 1
dxi,j d y.
88
Then using
x1,n1
..
In2
.
y2 z 2
z1 xn2,n1 0 ... 0
1
0 ... 0
¡ 0, z2 P Hn2, for 0 m1 P Z, we obtain
with y2
x1,n1
..
.
0
.. . 0
1
0
..
.
0
.. .
0
.. .
2
I
y
y
z
n
2
1
2
f1 ;
m
xn2,n1 0 0 0 1 0 . . . 0
1
00 . . . 0 y1 0
0
...
0
1
0
...
0
1
0
..
.
0
.. .
0
1
γn2
y1 y2 z 2
1 0
f2 ;
m
,
m
0 0 1 2 m2 1 γn2 PPn3,1 zSLpn2,Zq
0 1 0 . . . 0 y1 0
8̧
¸
0
e2πim pγ 2
where γn2
pγ ||f || ¥
2
2
n 2,1
... γn2,n2
8̧
8̧ » 8
m1 1 m2 1 1
...
n 2,1 x1,n 1
γn2,n2 xn2,n1
q,
q. We get again,
»8»
1
1
2
12
»
0
.
y1 y2 z 2 ..
f2 0
0 . . . 0 y1
0
...
0
1
2
12
2
;
m
,
m
0 1 2 0
1
0
.. .
¹
¤ ¤ dxi,j d y.
1 i j n 2
After continuing this process inductively for n 1 steps, we finally obtain
||f || ¡
2
2
8̧
m1 1
where y
y1 yn1
8̧
¸
»8
mn2 1 mn1 0 1
..
, y1 , . . . , yn1
.
y1
1
»8
1
¡ 0.
|Wf py; m1, . . . , mn2, mn1q|2
d y
89
5.3
Proof of Theorem1.1
Let Hδ pz q
δ
φpY upzqq where φ is a function defined in (4.31) and Y e 11 . Choose
2δ
¡ 0 such that
$
'
'
& »
(
1 d z
max λp8j q p`8 q j 1,...,n1 max
'
'
%
Then for j
Hn ,
u z t
p q
1, . . . , n 1,
»
pj q
∆
n
Hn
»
¤
λp8jqp`8q
Hn
pj q
λ `8 8
pq
p q
As in Lemma 4.11, we have
pj q
λ `8 8
p q
»
Hn
pe2δ 1q ¤ 1.
/
/
-
¤¤
0 t 1
Hδ pτ q d τ
pj q
∆ Hδ τ d τ
n
,
/
/
.
»
Hn
φpY upτ qq d τ
φpY upτ qq d τ
¤
»8
λp8j q `8 φt Y
0
p q
pq
»8
0
»
1 d τ dt
Hn ,
t
u τ
Y
p q
φptq dt 1.
So,
»
Hn
pj q
∆
n
λp8jqp`8q Hδ pτ q dτ
¤
»
Hn
pj q
∆ Hδ τ d τ
n
pq
1,
then
n
Cp pn, δ; S q :VolpB pS; δ q X F
n
n
Hn
t
2
CH
δ
n
u
2
pj q
∆ φ Y
n
k p n2 1 q
p n2 1
p
u τ d τ
p p qq
¸
¸
q S,
finite prime
j 1
k 1
n¸1 »
j 1
q
t u
2
¹
P
q p
j n 1
2
p q
k n2 1
n2 1
1
¤ kj ¤n
1 k1
(5.5)
2
¸
q
4dk pnq
2 q k1 kj .
90
By (5.2) and (5.3), we have Theorem 1.1.
91
B IBLIOGRAPHY
[1] Ce Bian. Computing GLp3q automorphic forms. Bull. London Math. Soc. (2010),
42(5):827–842, 2010.
[2] Andrew Booker. Uncovering a new L-function. Notices of the AMS, 55:1088–1094,
2008.
[3] Andrew Booker, Andreas Strömbergsson, and Akshay Venkatesh. Effective Computation of Maass cusp forms. IMRN, 2006.
[4] Farrell Brumley. Effective multiplicity on on GLN and narrow zero-free regions
for Rankin-Selberg L-functions. American Journal of Mathematics, 128:1455–1474,
2006.
[5] W. Casselman. On some results of Atkin and Lehner. Math.Ann., 201:301–314, 1973.
[6] J. W. Cogdell and I.I. Piatetski-Shapiro. Converse theorems for GLn . Inst. Hautes
Études Sci. Publ. Math., 79:157–214, 1994.
[7] J. W. Cogdell and I.I. Piatetski-Shapiro. Converse theorems for GLn . II. J. Reine
Angew. Math., 507:165–188, 1999.
[8] D. Flath. Decomposition of representations into tensor products. In Automorphic
Forms, Representations, and L-functions, volume 33 of Proceedings of Symposia in
Pure Mathematics, pages 170–183. American Mathematical Society, Providence, RI,
1979.
[9] R. Gangolli. On the Plancherel formula and the Paley-Wiener theorem for spherical
functions on semisimple Lie groups. Ann. of Math., 93:150–165, 1971.
[10] I. Gelfand and D. Kazhdan. Representations of the group GLpn, K q whre K is a local
field. Lie groups and their Representations. John Wiley and Sons, New York, 1975.
[11] I. M. Gelfand, M. Graev, and I.I. Piatetski-Shapiro. Representation theory and automorphic functions. Academic Press, Inc., Boston, MA, 1990.
[12] Dorian Goldfeld. Automorphic Forms and L-functions for the group GLpn, Rq. Number 99 in Cambridge studies in advanced mathematics. Cambridge University Press,
Cambridge, 2006.
[13] Dorian Goldfeld and Joseph Hundley. Automorphic representations and L-functions
for the general linear group. Cambridge studies in advanced mathematics. Cambridge
University Press, Cambridge, prepare.
92
[14] Douglas Grenier. On the shape of fundamental domains in GLpn, Rq. Pacific Journal
of Math., 160:53–65, 1993.
[15] Dennis A. Hejhal. On eigenfunctions of the Laplacian for Hecke triangle groups. In
Emerging applications of number theory (Minneapolis, MN, 1996), volume 109 of
IMA Vol. Math. Appl., pages 291–315. Springer, New York, 1999.
[16] Sigurdur Helgasson. Groups and Geometric Analysis: Integral geometry, invariant
differential operators, and spherical functions. Number 83 in Mathematical Surveys
and Monographs. American Mathematical Society, Providence, RI, 2000.
[17] Hervé Jacquet and Robert P. Langlands. Automorphic forms on GLp2q, volume 114
of Lecture Notest in Mathematics. Spring-Verlag, Berlin-New York, 1970.
[18] Erez Lapid and Werner Müller. Spectral asymptotics for arithmetic quotients of
SLpn, Rq{SOpnq. Duke Mathematical Journal, 149:117–155, 2009.
[19] Elon Lindenstrauss and Akshay Venkatesh. Existence and Weyl’s law for Spherical
Cusp forms. GAFA, 17:220–251, 2007.
[20] Boris Mezhericher. Evaluating Jacquet’s Whittaker functions and Maass forms for
SLp3, Zq. PhD thesis, Columbia University, 2008.
[21] Stephen D. Miller. On the existence and temperedness of cusp forms for SL3 pZq. J.
Reine Angew. Math., 533:127–169, 2001.
[22] Werner Müller. Weyl’s law for the cuspidal spectrum of SLn . Ann. of Math., 165:275–
333, 2007.
[23] I.I. Piatetski-Shapiro. Multiplicity One Theorems. In Automorphic Forms, Representations, and L-functions, volume 33 of Proceedings of Symposia in Pure Mathematics, pages 209–212. American Mathematical Society, Providence, RI, 1979.
[24] Atle Selberg. Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. (N.S.),
20:47–87, 1956.
[25] J. Shalika. The Multiplicity One Theorem on GLpnq. Ann. Math., 100:171–193,
1974.
[26] Harold Stark. Fourier coefficients of Maass waveforms. In Modular forms (Durham,
1983), pages 263–269. Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood,
Chichester, 1984.
[27] Holger Then. Maass cusp forms for large eigenvalues. Math. Comp. 74, 249:363–381,
2005.