The plancherel formula of l 2(n 0 GIpsi) where g is a p adic group

THE PLANCHEREL FORMULA OF L
2
(N
0
\ G; ψ) WHERE G IS A p-ADIC
GROUP.
TANG U-LIANG
(BSc. (Hons), NUS)
A THESIS SUBMITTED
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN
MATHEMATICS
DEPARTMENT OF MATHEMATICS
NATIONAL UNIVERSITY OF SINGAPORE
2011
Acknowledgments
The author wishes to thank Professor Gordon Savin, Professor Gan Wee Teck and
Associate Professor Loke Hung Yean for initiating this project, suggestions, comments
and guidance.
i

Contents
Acknowledgments i
Summary v
Chapter 1. 1
1.1. Introduction and statement of main results 2
1.2. The Cartan and Iwasawa decompositions of G 4
1.3. Parabolic subgroups and Schwartz spaces 7
Chapter 2. 13
2.1. Whittaker functions 14
2.2. The Harish-Chandra transform 17
Chapter 3. 21

3.1. The Plancherel measure of L
2
(G) 22
3.2. The discrete spectrum of L
2
(N
0
\ G; ψ) 24
3.3. The proof of Lemma 3.2.1.2 25
Chapter 4. 31
4.1. The Whittaker transform 32
4.2. Proof of Theorem 4.1.3.2 36
4.3. The Plancherel formula for L
2
(N
0
\ G; ψ) 40
Bibliography 45
iii

Summary
We study the right regular representation on the space L
2
(N
0
\ G; ψ) where G is
a quasi-split p-adic group and ψ a non-degenerate unitary character of the unipotent
subgroup N
0
of a minimal parabolic subgroup of G. We obtain the direct integral

decomposition of this space into its constituent representations. In particular, we deduce
that the discrete spectrum of L
2
(N
0
\ G; ψ) consists precisely of ψ generic discrete series
representations and derive the Plancherel formula for L
2
(N
0
\ G; ψ).
v

CHAPTER 1
1.1. INTRODUCTION AND STATEMENT OF MAIN RESULTS
1.1. Introduction and statement of main results
1.1.1. Let ψ be a nondegenerate unitary character of the unipotent radical N
0
of a
minimal standard parabolic subgroup of a connected quasi-split p-adic group, G. Define
L
2
(N
0
\ G; ψ) as the space of functions on G which transform according to ψ, i.e.
f(ng) = ψ(n)f(g) and are square integrable modulo N
0
. This space becomes a unitary
representation of G via right translation.
The purpose of this work is to obtain the Plancherel formula for this unitary

representation. Dinakar Ramakrishnan first studied the case for GL(2) in [Ram2]
obtaining a Plancherel formula for the archimedean and non-archimedean group. Nolan
Wallach then proved this result for arbitrary real reductive groups (see [Wa, Chapter
14]). Indeed, we have found out that much Wallach’s arguments can be adapted for the
p-adic case.
There are two crucial steps in proving the Plancherel formula for L
2
(N
0
\ G; ψ).
One of them is to prove the surjectivity of a certain map from the space of Schwartz
functions on G to the space of Schwartz functions on N
0
\ G. This is the author’s
original contribution.
Secondly we must define a Whittaker transform which transforms certain smooth
functions on orbits of discrete series representations to Schwartz functions on N
0
\ G
analogous to the Harish-Chandra wave packet map. We refer the reader to Section 4.1.3
for the precise definition of this transform. In order to define this map, we require the a
certain Jacquet integral (this integral is defined later in the paper) extend to a
holomorphic function. This required fact is a consequence of the results of Casselman in
[C-S] (see also [Jac] and [Shah]).
1.1.2. Now to state our main result. Let P = MN be a standard parabolic
subgroup of G with M and N its Levi and unipotent subgroup respectively Let ψ
M
denote the restriction of ψ to M ∩N
0
. We take a ψ

M
generic discrete series
representation (σ, H
σ
) and consider the unitarily induced representation I
G
P
(σ ⊗ ν)
where ν ∈ Im(X
ur
(M)) = ia
M
/L runs over all unramified unitary characters of M. We
refer the reader to section 1.3.4 for the definitions of the relevant notations.
Now let W h
ψ
M
(H
σ
) denote the (one dimensional) space of Whittaker functionals
on σ. Let
H
σ,ν
= I
G
P
(σ, ν) ⊗ W h
ψ
M
(H

σ
)
and consider the direct integral
I
σ,M
=
⊕
ia
M
/L
H
σ,ν
˜µ(σ, ν) dν
where ˜µ(σ, ν) is a certain normalization of the Plancherel measure on ia/L.
Let
W (G|M) := {w ∈ W
G
| w.M = M}/W
M
where W
G
and W
M
denote the Weyl group of G and its Levi M respectively. If
σ ∈ E
2
ψ
M
, then w.σ is defined and denote E
2

ψ
M
(M)/W (G|M) to be the set of
isomorphism classes of square integrable representations of M which are ψ
M
generic
modulo the action of W (G|M).
We prove that
2
1.1. INTRODUCTION AND STATEMENT OF MAIN RESULTS
Theorem 1.1.2.1. There exists a unitary linear isomorphism from
M⊂G
σ∈E
2
ψ
M
(M)/W (G|M)
I
σ,M
onto L
2
(N
0
\ G; ψ) where each of the I
σ,M
is a G-module.
This is a refinement of the Plancherel formula for a particular symmetric space
(i.e. N
0
\ G) of “polynomial growth” studied in [B].

We also compute the explicit normalization of ˜µ(σ, ν). If µ(σ, ν) dν denotes the
Plancherel measure on ia
M
/L, then
˜µ(σ, ν) dν =
1
|W (G|M)|γ(G|M)c
2
(G|M)
µ(σ, ν) dν.
It is worth noting that the Plancherel formula for L
2
(G) is used in the proof of
the Plancherel formula for L
2
(N
0
\ G; ψ). This explains the formal similarity with the
two formulas.
1.1.3. This paper is organized as follows. From Section 1.2.1 until the end of
Chapter 1 we give a simplified exposition of Bruhat-Tits theory adequate for our
purposes and prove Lemma 1.2.2.1. This is a key lemma required to prove the crucial
Lemma 3.2.1.2. We also describe the notations and conventions for parabolic subgroups
and tori needed to describe unitary parabolic induction. Finally, we end by giving a
description of the Schwartz spaces on G and N
0
\ G needed later.
In Chapter 2 we discuss the general theory of Whittaker functions for discrete
series representations and tempered representations, the Harish-Chandra transform for
functions in C

∗
(N
0
\ G; ψ) and conclude with an application of this theory to a result of
Savin, Khare and Larsen.
Chapter 3 is where we discuss aspects of the Placherel measure on G, the
multiplicity one property for L
2
(N
0
\ G; ψ) and prove Lemma 3.2.1.2.
Finally, we set the stage for deriving the full Plancherel formula in Chapter 4.
Our main result is Theorem 4.1.3.2. Theorem 1.1.2.1 stated in this introduction is
precisely Corollary 4.3.2.1.
1.1.4. While this paper was being written, it was brought to the author’s attention
that Erez Lapid and Mao Zhengyu had obtained an explicit form of the Whittaker
function and its asymptotics on a split group G in [L-Z]. Theorem 2.1.3.3 is a direct
corollary of their results.
They conjectured the following: Let W (π) denote the Whittaker model of a
generic representation and suppose that
Z
G
N
0
\G
|W (g)|
2
dg is finite for all
W (g) ∈ W (π), then π is , square integrable. By Theorem 2.2.1.5 and Theorem 3.2.1.3
we conclude that this conjecture is true. Sakellaridis and Venkatesh has also announced

a proof of this conjecture when G is a split group.
Patrick Delorme has obtained the results of this work independantly in [D1] and
[D2]. However, our approach differs slightly from his treatment. We also thank
Professor Delorme for pointing out a gap in the previous version of Proposition 2.2.1.3.
3
1.2. THE CARTAN AND IWASAWA DECOMPOSITIONS OF G
1.2. The Cartan and Iwasawa decompositions of G
1.2.1. We begin by fixing a p-adic field k with ring of integers o and normalized
absolute value |.| = q
−val(.)
. Let G be a connected quasi split reductive group defined
over k. Let G = G(k) be its k-rational points. Let A
0
the maximal k split torus and M
0
its (abelian) centralizer with M
0,0
the maximal compact subgroup of M
0
.
With W denoting the Weyl group of G with respect to A
0
, let W be the affine
Weyl group extending the Weyl group W . We may identify W as the semidirect product
of W and D = A
0
/A
1
0
where A

1
0
is the maximal open compact subgroup of A
0
. Let Σ
aff
and
nd
Σ denote the affine root system and set of nondivisible roots of G respectively.
Let K be the special maximal open compact subgroup fixing a special point x
0
in the
apartment stabilized by A
0
. Denote by Σ
0
the roots of Σ
aff
vanishing on x
0
. If B is the
subgroup of G fixing pointwise a chamber in the apartment with vertex x
0
, then B is an
Iwahori subgroup. One knows that there is a bijection λ :
nd
Σ → Σ
0
since every root
α ∈ Σ

0
is a positive multiple of a unique root in
nd
Σ. We write λ(α) as λ
α
α.
The Iwahori decomposition for G and K is
G =
w∈W
BwB
and
K =
w∈W
BwB
respectively with both these unions disjoint.
We will write the Iwasawa decomposition of G as N
0
A
0
K where N
0
is the
unipotent subgroup of a choice of minimal parabolic P
0
of G containing M
0
.
Define
nd
Σ

+
to be the system of positive roots determined by P
0
and let ∆
denote the set of simple roots. For  > 0, define
A
+
(
−1
) := {a ∈ A
0
| |α(a)| ≤ 
−1
∀α ∈ ∆}.
Writing A
+
as A
+
(1), we have the Cartan decomposition G = KA
+
K.
Let {H
m
}
m≥1
denote a system of ‘good’ open compact subgroups which generates
the topology on G (see [Sil1]). By ‘good’ we simply require that these collection of
subgroups satisfy the triangle decomposition and are normal in K. Without loss of
generality we assume that if w ∈ W , then w /∈ H
i

for any i ≥ 1.
1.2.2. Our objective in this subsection is to prove a crucial lemma which gives the
interplay between the Cartan and Iawasawa decompositions on a group. To do this, we
will need to describe the root subgroups of N
0
and their corresponding filtrations. A
simplified version adequate for quasi split groups can be found in [Ca2] and it is to this
paper which we will refer to for definitions and notation. So let N
0
(α) be a certain
subset of N
0
indexed by each α ∈ Σ
aff
. Then for α ∈
nd
Σ
+
and m ∈ Z define
N
α,m
:= N
0
(λ(α) + m).
We have that N
α,m+1
 N
α,m
and if N
α

:=
m∈Z
N
α,m
, then
N
0
=
α∈
nd
Σ
+
N
α
4
1.2. THE CARTAN AND IWASAWA DECOMPOSITIONS OF G
in any order.
Write N
−
0,m
=
α∈
nd
Σ
+
N
−α,m
, N
0,m
=

α∈
nd
Σ
+
N
α,m
. Then B = N
−
0,1
M
0,0
N
0,0
. If
m ≥ 0, H
m
= N
−
0,m
(M
0
∩ H
m
)N
0,m
.
Set q
α
= [N
0

(α − 1) : N
0
(α)] and q
α/2
= q
α+1
/q
α
. We remark that it is possible
that q
α/2
= 1. For convenience, q
m/2
α/2
q
m
α
= [N(α + 1) : N (α + m + 1)] is shortened to
α
q
m
.
Fix an open compact subgroup H = H
m
as defined above of K. We determine a
parametrization of G/H cosets in a single N
0
\ G/H coset. Let {g
i
} be a set of coset

representatives of N
0
\ G/H. By the Iwasawa decomposition we may assume g
i
= ak
where a ∈ A and k ∈ K where k comes from a set of coset representatives of K/H.
Under the left action of N
0
, the stabilizer of g
i
H is N
0
∩ g
i
Hg
−1
i
. Let
[N
0
/N
0
∩ g
i
Hg
−1
i
] be a set of N
0
∩ g

i
Hg
−1
i
-coset representatives in N
0
. Then each G/H
coset in N
0
g
i
H is parametrized the set [N
0
/N
0
∩ g
i
Hg
−1
i
].
Since H is normal in K, N
0
∩ g
i
Hg
−1
i
= N
0

∩ aHa
−1
= aN
0,m
a
−1
. Written
component wise
aN
α,m
a
−1
= N
0
(λ(α) + m + λ
α
val(α(a)))
where we recall that λ
α
is the positive multiple arising from the bijection between
nd
Σ
and Σ
0
.
Now chose an integer n so that N
0,n
 N
0,m
. Also, choose  > 0 small enough so

that if a satisfies
val(α(a)) ≥
1
λ
α
(−m + n)
for all simple roots α ∈ ∆ then a ∈ A
+
(
−1
). Then a ∈ A
+
(
−1
) contains the set of all
a ∈ A such that N
0
∩ aHa
−1
⊂ N
0,n
.
Lemma 1.2.2.1. For a fixed a
1
∈ A
+
(
−1
) and fixed k
1

∈ K, the coset
H
m
na
1
k
1
H
m
is not equals to any coset of the form
H
m
wa
0
w
−1
kH
m
with a
0
∈ A
+
and w ∈ W as we run over all n ∈ N
0
− ((N
0
∩ aH
m
a
−1

)N
0,m
),
Proof. Firstly for any n ∈ N
0
− ((N
0
∩ aH
m
a
−1
)N
0,m
), we rewrite the element
g = na
1
k
1
according to the KA
+
K decomposition.
Assume that a
1
∈ A
+
(
−1
) is such that N
0
∩ a

1
H
m
a
−1
1
 N
0,m
. There are three
cases.
(1) If n ∈ N
0,0
− ((N
0
∩ aH
m
a
−1
)N
0,m
), then
g = na
1
k
1
= nw
2
a
2
w

−1
2
k
1
where w
−1
2
a
1
w
2
= a
2
∈ A
+
. The expression in the right hand side of the
equation has been decomposed according to the Cartan decomposition as nw
2
and w
−1
2
k
1
are contained in K.
5
1.2. THE CARTAN AND IWASAWA DECOMPOSITIONS OF G
(2) If n /∈ N
0,0
but a
−1

1
na
1
∈ K, then na
1
k
1
= a
1
(a
−1
1
na
1
)k
1
= w
2
a
2
w
−1
2
n
0
k
1
where
n
0

= a
−1
na ∈ K and a
2
∈ A
+
defined as above. As w
−1
2
n
0
k
1
∈ K, the KA
+
K
decomposition of na
1
k
1
is
w
2
a
2
w
−1
2
n
0

k
1
.
(3) If n ∈ N
0
does not satisfy the properties of the previous two cases, then
decompose na
1
= b
1
w
1
a
2
b
2
according to the Iwahori decomposition with
b
1
, b
2
∈ B, w
1
∈ W and a
2
∈ D. Since na
1
and a
−1
1

na
1
/∈ K, a
2
and w
1
are both
not the identity element. Then
na
1
k
1
= b
1
w
1
w
2
a
3
w
−1
2
b
2
k
1
where w
−1
2

a
2
w
2
= a
3
∈ A
+
is the Cartan decomposition for g as b
1
w
1
w
2
and
w
−1
2
b
2
k
1
are elements of K.
For case (1), let us assume that there exists w
0
∈ W and a
0
∈ A
+
such that

nw
2
a
2
w
−1
2
k
1
∈ H
m
w
0
a
0
w
−1
0
k
1
H
m
.
Since the Cartan decomposition gives a disjoint union over all KˇaK where ˇa ∈ A
+
/A
1
0
,
we may assume that a

2
= a
0
so that nw
2
∈ H
m
w
0
and w
−1
2
k
1
∈ w
−1
0
k
1
H
m
. But this
means that w
2
= w
0
implying that n ∈ H
m
which contradicts the hypothesis. The
lemma is proven in this case.

For case (3) we argue in a similar fashion. So suppose that
b
1
w
1
w
2
a
3
w
−1
2
b
2
k
1
∈ H
m
w
0
a
0
w
−1
0
k
1
H
m
.

As before, a
3
= a
0
so that b
1
w
1
w
2
∈ H
m
w
0
. We may as well assume b
1
∈ H
m
and that
w
1
w
2
= w
0
(for otherwise the resulting contradiction proves the lemma already). Now
on the other hand
w
−1
2

b
2
k
1
∈ w
−1
0
k
1
H
m
implying that
w
−1
1
b
2
∈ w
−1
0
H
m
since H
m
is normal. Once again, we may assume b
2
∈ H
m
so that w
0

= w
1
. This forces
w
2
to be the identity element which leads to a contradiction.
Finally in order to apply the argument of the previous cases above to case (2), we
must exclude the possibility that n
0
is in H
m
.
Indeed, if n
0
∈ H
m
, we will derive a contradiction. Write the component of an
arbitrary n ∈ N
0
in N
α
as n
α
. Since a
−1
1
na
1
∈ K, n
α

∈ N
α
is contained in at most
N
α,λ
α
val(α(a))
. However by our assumption on n
0
, n
α
∈ N
α,λ
α
val(α(a))
satisfies
a
−1
1
n
α
a
1
∈ N
α,m
for all roots α ∈
nd
Σ
+
. This implies that n

α
∈ N
α,λ
α
val(α(a))+m
i.e.
n ∈ N
0
∩ a
1
Ha
−1
1
. This contradicts the hypothesis and proves the lemma.
If N
0,m
⊃ N
0
∩ a
1
Ha
−1
1
, then a
1
∈ A
+
so that it is clear that for any n /∈ N
0,m
,

H
m
na
1
k
1
H
m
is never of the form of as H
m
wa
0
w
−1
k
1
H
m
. 
6
1.3. PARABOLIC SUBGROUPS AND SCHWARTZ SPACES
1.3. Parabolic subgroups and Schwartz spaces
1.3.1. As per Harish-Chandra’s philosophy of descent we study the Plancherel
decomposition of L
2
(N
0
\ G; ψ) by reducing a dense space of ‘test’ functions to its
correspondingly defined subspaces on Levi subgroups of ‘standard’ parabolic subgroups
on G. In this section, we describe both these parabolic subgroups, their unramified

characters and the space of ‘test’ functions which we will study in detail.
We introduce the notion of standard parabolic subgroups and define a dense
subspace of L
2
(N
0
\ G; ψ) which we will study in this paper.
Fix a minimal parabolic subgroup P
0
as in the previous section. A standard
parabolic pair is a pair (P, A) consisting of a parabolic subgroup G ⊃ P ⊃ P
0
and
A
0
⊃ A ⊃ Z
G
where Z
G
denotes the (split component of the) center of G.
It is known that all such pairs are in one-to-one correspondence with subsets of
∆. By abuse of notation we identify (P, A) with P and write this correspondence as
θ → P
θ
where we agree that P
∅
= P
0
and P
∆

= G. It is well known that any standard parabolic
corresponds to a subset θ ⊂ ∆. Conversely, to each subset θ, one can associate a
standard parabolic P
θ
, M
θ
, its Levi and If P
θ
is a standard parabolic, we write its
Langlands decomposition as P
θ
= M
θ
N
θ
. Where there is no cause for confusion, we will
drop the θ from notation. Moreover where the context is clear, N
0
always denotes N
∅
.
A smooth unitary character ψ of N
0
is said to be nondegenerate if and only if for
any α ∈ ∆, ψ restricted to N
α
is non-trivial.
1.3.2. Let δ
P
denote the modular character of the parabolic subgroup P. By the

Iwasawa decomposition G = N
θ
M
θ
K, we write g ∈ G as g = n
P
θ
(g)m
P
θ
(g)k(g).
We define the constant
γ(P
θ
) =
¯
N
θ
δ
¯
P
θ
(m
P
θ
(¯n))d¯n
where
¯
N
θ

denotes the unipotent subgroup of the opposite parabolic of P
θ
. (c.f. pg. 240
[Wal].) If we normalize measures on N
θ
so that meas(N
θ
∩K) = 1 then it is known that
γ(P
θ
) depends only on the Levi subgroup of P
θ
, M
θ
. Thus we may write
γ(P
θ
) = γ(G|M
θ
). If α ∈
nd
Σ(P
θ
, A
θ
), let A
α
denote the (identity component) of the
kernel of α and M
α

the centralizer of A
α
in G. Then M
α
⊃ M
θ
for every root
α ∈
nd
Σ(P
θ
, A
θ
). Define
c(G|M
θ
) = γ(G|M
θ
)
−1
α∈
nd
Σ(P
θ
,A
θ
)
γ(M
α
|M

θ
).
If (π, V ) is a unitary representation, let , 
π
denote the Hermitian inner product on V .
Let us also agree to fix the measure on K to be 1.
Lemma 1.3.2.1. If (σ, H
σ
) is any unitary representation then I
G
¯
P
θ
σ = I(σ) is a
unitarily induced representation from
¯
P
θ
to G. Furthermore for any u, v ∈ I(σ)
K
u(k), v(k)
σ
dk = γ(G|M
θ
)
−1
N
θ
u(n), v(n)
σ

dn.
7
1.3. PARABOLIC SUBGROUPS AND SCHWARTZ SPACES
Proof. For the purposes of this proof, Ind
G
P
σ will denote unnormalized induction
instead of the usual normalized induction. Recall the integral decomposition formula
(c.f. [Wal, pg. 240]).
G
f(g) dg = γ(G|M)
−1
N×M×
¯
N
f(nm¯n)δ
−1
P
(m) d¯n dm dn
for any parabolic subgroup P and f ∈ C
∞
c
(G).
For any f ∈ C
∞
c
(G) let P
δ
¯
P

θ
(f) denote the projection of f into Ind
G
¯
P
θ
δ
¯
P
θ
. This is
given by
¯
P
θ
δ
−1
¯
P
θ
(p)f(p) dp.
Let
I
N
θ
(w) :=
N
θ
w(n) dn
for any w ∈ Ind

G
¯
P
θ
δ
¯
P
θ
provided this integral converges.
Consider
I
N
θ
P
δ
¯
P
θ
(f) =
N
θ
¯
P
θ
δ
−1
¯
P
θ
(p)f(pn) dp dn

=
¯
N
θ
M
θ
N
θ
δ
−1
¯
P
θ
(m)f(¯nmn)dn dm d¯n
= γ(G|M
θ
)
¯
N
θ
M
θ
K
δ
−1
¯
P
θ
(m)f(¯nmk) dk dm dn
= γ(G|M

θ
)
K
P
δ
¯
P
θ
(f)(k)dk.
We may choose f such that P
δ
¯
P
θ
(f)(k) = u(k), v(k)
σ
proving the lemma. 
1.3.3. Let Σ(P
0
, A
0
) denote the set of all positive roots of G with respect to P
0
. The
resulting roots by restricting to A
θ
is denoted Σ(P
θ
, A
θ

).
Define a
0,R
:= (Hom
k
(A
0
, k
×
) ⊗
Z
R)
∗
where ∗ denotes the real dual of the vector
space. Letting A
θ
denote the center of M
θ
we may define in an analogous fashion, the
vector space a
θ,R
. Then there is a canonical decomposition
a
0,R
= a
θ,R
⊕ a
θ
R
realizing a

θ,R
as a subspace of a
0,R
. The complexified vector spaces are denoted a
θ,C
.
Let
a
+
0,R
:= {H ∈ a
0,R
| ∀α ∈ Σ(P
0
, A
0
), α, H > 0}.
We call this the (open) positive chamber of a
0,R
with respect to P
0
. Similarly we define
a
+
θ,R
:= {H ∈ a
θ,R
| ∀α ∈ Σ(P
θ
, A

θ
), α, H > 0}.
Define the ‘log’ map H
M
θ
: M
θ
→ a
θ,R
where H
M
θ
(m) is the element of a
θ,R
such
that
q
−ν,H
M
θ
(m)
= |ν(m)|
for all ν ∈ Hom
k
(M
θ
, k
×
). Let M
+

θ
:= H
−1
M
θ
(a
+
θ,R
) and A
+
θ
:= M
+
θ
∩ A
θ
.
8
1.3. PARABOLIC SUBGROUPS AND SCHWARTZ SPACES
We describe a partition of A
+
0
. For any 0 <  ≤ 1, define
A
+
0
(θ, ) = {a ∈ A
+
0
| |α(a)| ≤  ∀α ∈ ∆ − θ

 <|α(a)| ≤ 1 ∀α ∈ θ}.
Then we have a disjoint union
A
+
0
=
θ
A
+
0
(θ, ).
1.3.4. Using the Lie algebra a
∗
θ,C
, we may construct the unramified unitary
characters of M
θ
. Eventually, we will see that these characters will constitute the
‘continuous’ portion of the Plancherel decomposition in a manner made precise in the
later sections of this paper.
Let Hom(G, C
×
) denote the group of continuous homomorphisms from G to C
×
.
If χ ∈ Hom
k
(G, k
×
) then |χ|

k
is defined by |χ|
k
(g) = |χ(g)|
k
. Let G
1
:=
χ
Ker|χ|
k
and
let X
ur
(G) := Hom(G/G
1
, C
×
). We call this set of characters the set of unramified
characters of G. These definitions apply to any Levi subgroup M
θ
of any standard
proper parabolic subgroups of G by replacing G with M
θ
.
There is a surjection from a
∗
θ,C
onto X
ur

(M
θ
) defined by χ ⊗s → (g → |χ(g)|
s
). If
ν ∈ a
∗
θ,C
, the corresponding character in X
ur
(M
θ
) is denoted χ
ν
. The kernel of this map
is a lattice of the form (2π
√
−1/ log q)R where R is a lattice of Hom
k
(M
θ
, k
×
) ⊗
Z
Q.
This endows X
ur
(M
θ

) with the structure of a complex algebraic variety isomorphic to
(C
×
)
d
where d is the dimension of a
θ,R
.
Given χ ∈ X
ur
(M
θ
), suppose that λ ∈ a
∗
θ,C
projects to χ. We define χ = λ.
This is well defined as λ is independent of the choice of λ. If χ ∈ Hom(A
θ
, C
×
), then
the character |χ| extends uniquely to an element of X
ur
(M
θ
) taking positive real values.
Set χ = |χ|.
Consider the group ImX(A
θ
) consisting of characters of χ ∈ Hom(A

θ
, C
×
)
satisfying χ = 0. Then there is a surjection with finite kernel of ImX
ur
(M
θ
) onto
ImX(A
θ
). The latter group is compact and we give it the Haar measure normalized so
that its total mass is 1. We pull back the measure on ImX(A
θ
) to a measure on
ImX
ur
(M
θ
) denoting this as dν.
1.3.5. To end this chapter, we describe the space of ‘test’ functions which is
amenable to our analysis. Our goal is to define a space closely related to
Harish-Chandra’s space of Schwartz functions on G. To do this, we will need to define a
suitable radial function on the homogeneous space N
0
\ G. We will use this function to
regulate the growth of our Schwartz functions as it approaches ‘infinity’ .
We start by considering a k-rational representation ρ : G → GL(V ) with compact
kernel. We denote the (i, j)-th entry of the matrix ρ(g) as ρ(g)
ij

. By pulling back to G,
define ||g|| := sup
ij
sup{|ρ(g)
ij
|, |ρ(g
−1
)
ij
|} and σ(g) := log
q
||g|| and
σ
∗
(g) := inf
z∈Z
G
σ(gz) where Z
G
denotes the center of G.
Adopting the same notation in [Wal], we let Ξ(g) denote the zonal spherical
function of G and Ξ
M
for the same for each standard Levi subgroups M of G. So then
we define a space of functions, C(H
j
\ G/H
j
; χ) as the space of H
j

bi-invariant smooth
9
1.3. PARABOLIC SUBGROUPS AND SCHWARTZ SPACES
complex valued functions on G such that Z
G
acts by a unitary character χ. For any
f ∈ C(H
j
\ G/H
j
) define
q
1,r
(f) := sup
g∈G
|f(g)|Ξ
−1
(g)(1 + σ
∗
(g))
r
where r > 0 . Define
C
∗
(G; χ) :=
j
{f ∈ C(H
j
\ G/H
j

; χ) | q
1,n
(f) < ∞ ∀n ∈ N}.
Then {q
1,n
}
n∈N
is a family of continuous seminorms making C
∗
(G; χ) into a Frechet
space. It is well known that C
∗
(G; χ) is dense (with respect to the L
2
norm) in L
2
(G; χ).
Now let ψ be a nondegenerate character of N
0
and Z
G
acts by the same character
χ as above. However, in the interest of reducing notational clutter, we agree to suppress
this from subsequent notation. Define C(N
0
\ G/H
j
; ψ) to be the space of right H
j
invariant complex valued functions of G such that f(ng) = ψ(n)f(g) for all n ∈ N

0
and
g ∈ G.
Consider any f ∈ C(N
∅
\ G/H
k
). By the Iwasawa decomposition G = N
∅
M
∅
K
write g = n
∅
(g)m
∅
(g)k(g) with m(g) ∈ M
∅
, n
∅
(g) ∈ N
∅
and k(g) ∈ K. Although m
∅
(g) is
only defined up to M
∅
∩K, δ
−
1

2
P
0
(m) = 1 for any m ∈ M
∅
∩K implying that for any r > 0,
q
2,r
(f) := sup
g∈G
|f(g)|δ
−
1
2
P
0
(m
∅
(g))(1 + inf
n∈N
∅
σ
∗
(nm
∅
(g)))
r
is well defined.
Now define the space
C

∗
(N
∅
\ G; ψ) :=
j
{f ∈ C(N
∅
\ G/H
j
; ψ) | q
2,n
(f) < ∞ ∀n ∈ N}.
As before the set {q
2,n
}
n∈N
for a family of continuous seminorms on C
∗
(N
∅
\ G; ψ)
making it into a Frechet space. It is dense in L
2
(N
∅
\ G; ψ).
Let P
θ
be a proper standard parabolic subgroup of G with M
θ

its Levi subgroup
and writing M
θ
∩ N
∅
as N
∗
. Write σ
M
θ
∗
(m) := inf
a∈A
θ
σ(ma). This generalizes σ
∗
defined
earlier. To be precise, σ
∗
= σ
G
∗
.
We define
q
M
θ
1,r
(f) := sup
m∈M

θ
|f(m)|Ξ
M
θ
(m)
−1
(1 + σ
M
θ
∗
(m))
r
,
q
M
θ
2,r
(f) := sup
m∈M
θ
|f(m)|δ
−
1
2
P
0
∩M
θ
(m
∅

(m))(1 + inf
n∈N
∅
∩M
θ
σ
M
θ
∗
(nm
∅
(m)))
r
.
and
q
G;M
θ
2,r
(f) := sup
m∈M
θ
|f(m)|δ
−
1
2
P
0
∩M
θ

(m
∅
(m))(1 + inf
n∈N
∅
∩M
θ
σ
∗
(nm
∅
(m)))
r
.
Define the following Frechet spaces
C
∗
(M
θ
) :=
j
{f ∈ C(H
j
∩ M
θ
\ M
θ
/M
θ
∩ H

j
) | q
M
θ
1,n
(f) < ∞ ∀n ∈ N},
C
∗
(N
∗
\ M
θ
; ψ) :=
j
{f ∈ C(N
∗
\ M
θ
/M
θ
∩ H
j
; ψ
θ
) | q
M
θ
2,n
(f) < ∞ ∀n ∈ N},
10

1.3. PARABOLIC SUBGROUPS AND SCHWARTZ SPACES
and
C
∗
(G; N
∗
\ M
θ
; ψ) :=
j
{f ∈ C(N
∗
\ M
θ
/M
θ
∩ H
j
; ψ
θ
) | q
G;M
θ
2,n
(f) < ∞ ∀n ∈ N}
where ψ
θ
=: Res
N
∅

N
∗
(ψ).
Then we have an inclusion
C
∗
(G; N
∗
\ M
θ
; ψ
θ
) ⊂ C
∗
(N
∗
\ M
θ
; ψ
θ
).
We will refer to these spaces generically as Schwartz spaces.
Note that we may choose (ρ, V ) so that ρ(a) is diagonal for all a ∈ M
∅
and ρ(n)
is upper triangular for all n ∈ N
0
. Then 1 + inf
n∈N
0

σ
∗
(nm
∅
(g)) is equivalent to
1 + σ
∗
(m
∅
(g)). For convenience, we will write this as 1 + σ
∗
(a(g)).
11

CHAPTER 2
2.1. WHITTAKER FUNCTIONS
2.1. Whittaker functions
2.1.1. The purpose of this chapter is to give a brief theory on the asymptotic
behavior of Whittaker functions for ψ-generic discrete series representations. Our main
results will be Theorem 2.1.3.3 and Proposition 2.1.3.4. In the latter part of the chapter,
we will see that the image of any embedding of a smooth generic representation into
L
2
(N
0
\ G; ψ) is lie in the space of Schwartz functions.
2.1.2. Let (π, V ) be a smooth finitely generated representation of G and V

denote
the algebraic complex linear dual of V . Define (π


(g)λ)(v) = λ(π(g
−1
)v) and let V
∨
denote the smooth points of π

. Let π
∨
be the representation obtained by restricting π

to V
∨
. Then (π
∨
, V
∨
) is a smooth representation of G.
Now consider a fixed λ ∈ V

and write
W (v, λ)(g) := λ(π(g)v)
with v ∈ V . Since V is smooth, W (v, λ) is K
j
invariant on the right for some open
compact subgroup. Note that if ˇv ∈ V
∨
, then W (v, ˇv)(g) is a matrix coefficient of V .
Fix any (not necessarily nondegenerate) character ψ of N
0

. Fix a standard
parabolic subgroup P
θ
= P = MN of G corresponding to θ ⊂ ∆ (P possibly equals to
G) for the discussion throughout this subsection. Let N
∗
and ψ
θ
be the unipotent
subgroup and character defined as in the previous section. Then define
V (N
∗
, ψ
θ
) := span{π(n)v − ψ
θ
(n)v | v ∈ V, n ∈ N
∗
}
and
V (N) := span{π(n)v − v | v ∈ V, n ∈ N}.
Define vector spaces
V
N
∗
,ψ
θ
= V/V (N
∗
, ψ

θ
)
and
V
N
= V/V (N).
Then
r
G
P
(V ) = V
N
⊗ δ
−1/2
P
is the Jacquet restriction functor sending smooth, finitely generated admissible
representations of G to smooth, finitely generated admissible representations of M.
Now we require ψ to be a nondegenerate character. Define
W h
ψ
(V ) := Hom
N
0
(π, C
ψ
).
A representation π is said to be ψ-generic if W h
ψ
(V ) is nontrivial. Now assume
that this is the case for π and consider any nonzero λ ∈ W h

ψ
(V ) and nonzero v ∈ V .
Then W (v, λ)(g) is not identically zero as a function on G and satisfies
W (v, λ)(ng) = ψ(n)W (v, λ)(g)
for any g ∈ G and n ∈ N
0
.
We say that W(v, λ) is a Whittaker function and W (v, λ)(1) = λ(v) a Whittaker
functional. All these can be generalized to Levi subgroups of G as well, by substituting
G for M, N
0
for N
∗
and ψ for ψ
θ
.
14
2.1. WHITTAKER FUNCTIONS
Let Φ
θ
be the canonical map from (V
N
θ
)
N
∗
,ψ
θ
to V
N

0
,ψ
introduced in [C-S] (see
also [Ca1]) . If v ∈ V , we write ˜v for its image in (V
N
θ
)
N
∗
,ψ
θ
. Recall the following lemma
of Casselman. (See [C-S, Proposition 6.3 and 6.4]).
Lemma 2.1.2.1. Fix a standard parabolic subgroup P
θ
of G. Let λ ∈ W h
ψ
(V ) and
v ∈ V . Then there exists  > 0 such that
W (v, λ)(a) = W (˜v, λ ◦ Φ
θ
)(a)
for any a ∈ A
0
satisfying |α(a)| <  ∀α ∈ ∆ − θ.
Notice that when θ = ∅, this is Proposition 6.3 in [C-S].
2.1.3. For a fixed but arbitrary θ ⊂ ∆, consider an A
θ
-finite complex valued smooth
function f on A

θ
. Then it is well known that
f(a) =
ν
ν(a)P (H
M
θ
(a))
where P (x) is a polynomial on the Lie algebra a
θ,R
of A
θ
and ν a smooth character of
A
θ
. The characters occurring in the decomposition above are known as the exponents of
f. Let E(A
θ
, f) denote the set of exponents of f.
Lemma 2.1.3.1. [Ca1, Proposition 4.4.4.] Let θ ⊂ ∆,  ∈ (0, 1] and p > 0. Let
f : A
+
0
(θ, ) → C be a complex valued function such that
(1) f is the restriction to A
+
0
(θ, ) of an A
θ
-finite function;

(2) the center of G, Z
G
acts by a unitary character on f and
(3) f is invariant under right translation by some open subgroup A
K
i
of A
1
0
.
Then |f|
p
is integrable on A
+
0
(θ, )/A
K
i
Z
G
if and only if |χ(a)| < 1 for all a ∈ A
+
θ
and
each χ ∈ E(A
θ
, f).

It is well known that if a representation of G, (π, V ) is smooth and admissible,
then so is V

N
. Thus the center of M, A, acts locally finitely on V
N
so that
V
N
=
ν
(V
N
)
ν
as generalized eigenspaces where ν are smooth characters of A. The set of all characters
appearing in this decomposition are called exponents of V with respect to P denoted
E(P, V ). If V is finitely generated then this set is finite.
An irreducible smooth representation is said to be a discrete series (resp.
tempered) representation of G if the center acts by a unitary character and its matrix
coefficients (modulo the center) are in L
2
(G) (resp. L
2+
(G) for any  > 0). The
following result is well known (see [Wal, Proposition III.1.1 and Proposition III.2.2]).
Proposition 2.1.3.2. Suppose (π, V ) is a discrete series representation (resp.
tempered representation) of G. Then it is necessary and sufficient that for every
standard parabolic subgroup P and every ν ∈ E(P, V ), |δ
−
1
2
P

(a)ν(a)| < 1 (resp.
|δ
−
1
2
P
(a)ν(a)| ≤ 1)for all a ∈ A
+
.

15
2.1. WHITTAKER FUNCTIONS
Theorem 2.1.3.3. Let ψ be an nondegenerate additive unitary character of N
0
and let (π, V ) be an irreducible ψ-generic discrete series representation, then V embeds
into L
2
(N
0
\ G; ψ).
Proof. Any Whittaker function is of the form W(v, λ)(g) for λ ∈ (V
N
0
,ψ
)

. Now
choose K
i
as the largest open compact subgroup of K such that

K
i
π(k)v dk = v

does not vanish. Let [K/K
i
] denote a (finite) set of coset representatives of K/K
i
. Then
for each k
j
∈ [K/K
i
], set v

j
= k
j
v

.
Then,
N
0
\G/Z
G
|W (g)|
2
d¯g =
A

0
/A
1
0
Z
G
K
δ
−1
0
(a)|W (ak)|
2
dadk
=
A
0
/A
1
0
Z
G
k
j
∈[K/K
i
]
K
i
δ
−1

0
(a)|W (ak
j
k
i
)|
2
dadk
i
=
A
0
/A
1
0
Z
G
j
δ
−1
0
(a)|W (v

j
, λ)(a)|
2
da.
Thus, the finiteness of that integral depends upon the square integrability of
W (v


j
, λ)(a) on A
0
/A
1
0
Z
G
. Since ψ is nondegenerate, we observe that W (v

j
, λ)(a) is
supported inside a translate of A
+
0
([C-S, Proposition 6.1]). By replacing v

j
with π(a
j
)v

j
for some suitable a
j
∈ A
0
, we may even assume that this support is contained in A
+
0

.
Choose  = min{
θ
}
θ
where 
θ
is obtained by applying Lemma 2.1.2.1 to P
θ
(with
θ ⊂ ∆). Recall that A
+
0
is partitioned into
θ
A
+
0
(θ, ). We let
f(a) := δ
−1/2
0
(a).W (v

, λ)(a) and restrict f(a) to each partition so that we may then
apply Lemma 2.1.3.1 with p = 2.
We must check all three conditions of this lemma. Condition (2) is clear and (3)
is satisfied by A
K
i

= A
1
0
. As π is admissible, so is V
N
and thus A
θ
acts locally finitely on
(V
N
θ
)
N
∗
,ψ
θ
. Therefore W(˜v

, λ ◦ Φ
θ
)(a) satisfies condition (1).
Clearly the exponents of W (˜v

, λ ◦Φ
θ
)(a) are in E(P
θ
, V ). By Proposition 2.1.3.2,
this will imply the square integrability of W (v


, λ) in A
+
0
. This proves the theorem. 
We introduce the notion of moderate growth for functions on N
0
\ G. Define
A(N
0
\ G; ψ) to be the union over all open compact subgroups K
i
of G of subspaces of
functions in C(N
0
\ G/K
i
; ψ) defined by the following growth condition: There exists a
constant C > 0 and r > 0 such that
|f(g)| ≤ Cδ
1
2
0
(m
0
(g))(1 + σ
∗
(a(g)))
r
.
Proposition 2.1.3.4. Assume that (π, V ) is a tempered and ψ-generic irreducible

representation of G. Then for any v ∈ V and λ ∈ W h
ψ
(V ), W (v, λ) ∈ A(N
0
\ G; ψ).
Proof. By [Wal, Proposition III.2.2] we know that if π is tempered, then for each
standard parabolic subgroup P
θ
, an exponent ν ∈ E(θ, V ) satisfies |δ
−
1
2
P
(a)ν(a)| ≤ 1. If
this inequality was strict for all parabolic subgroups and all exponents, then π is a
16
2.2. THE HARISH-CHANDRA TRANSFORM
discrete series and hence by (the proof of) Theorem 2.1.3.3, W (v, λ) is in A(N
0
\ G; ψ).
Thus we assume that for some parabolic subgroup, P
θ
, there exists an exponent such
that |δ
−
1
2
P
(a)ν(a)| = 1. In this case, using Lemma 2.1.2.1 and arguing as in Theorem
2.2.1.5 we obtain the result. 

2.2. The Harish-Chandra transform
2.2.1. If P is a parabolic subgroup of G, then
¯
P will denote its opposite parabolic
subgroup. Given f ∈ C
∗
(N
0
\ G; ψ) define the Harish-Chandra transform
f
P
(m) = δ
1/2
P
(m)
¯
N
f(¯nm)d¯n = δ
1/2
¯
P
(m)
¯
N
f(m¯n)d¯n.(2.2.1.1)
Lemma 2.2.1.1. Let P
θ
be a standard parabolic subgroup and N
θ
its unipotent

subgroup. For any f ∈ C
∗
(G), there exists a continuous seminorm q(f) such that
N
0
×
¯
N
θ
|f(n
0
¯n)|dn
0
d¯n < q(f).
Proof. Using the Iwasawa decomposition, write ¯n = n
0
(¯n)a
0
(¯n)k(¯n). Then the
integral can be written as
N
0
×
¯
N
θ
|f(n
0
a
0

(¯n)k(¯n)|dn
0
d¯n.
By [Wal, Proposition II.4.5], we know that for any integer d > 0, one can find and
integer r > 0 (and seminorm q
1,r
(f)) such that this integral is majorized by
¯
N
θ
δ
0
(a
0
(¯n))
1
2
(1 + σ
∗
(a
0
(¯n)))
−d
d¯n.
But we know by [Wal, Lemme II.4.2] that this integral converges for sufficiently large
d. 
Before we prove the next Proposition, we require the following Lemma
Lemma 2.2.1.2. Fix an open compact subgroup H. Then for any
f ∈ C
∗

(N
0
\ G; ψ)
H
,
|f(g)| ≤ C
H
q
2,r
(f)δ
−
1
2
0
(a
0
(g))(1 + σ
∗
(g))
−r
where q
2,r
(f) is the seminorm on the Schwartz space C
∗
(N
0
\ G; ψ) and C
H
a constant
which depends only on the open compact subgroup H and possibly r.

Proof. In the paragraph before the statement of Lemma 3.3.1.1 in Section 3.3.1, we
see that f is essentially supported in A
+
(
−1
H
) where 
H
> 0 depends only on the open
compact subgroup H. We define the following radial function on A
+
(
−1
H
): For any x, y
in the lattice D ⊂ a
0
we let d(x, y) be the usual Euclidean distance between two
elements. Using a large scale equivalence, m between X = N
0
\ G and a
0
(see [B,
Sections 4.1 and 4.6]), we may define a radial function r
X
(γ) for any γ ∈ X by
r
X
(γ) = d(m(γ), m(γ
0

)) where γ
0
is the element in X which satisfies
|α(H
−1
M
0
(m(γ
0
)))| = 
−1
H
for all α ∈ ∆. One sees easily that r
X
(•) is essentially equivalent
to σ
∗
(•) on G because of the Cartan decomposition on G.
17
2.2. THE HARISH-CHANDRA TRANSFORM
Let p : G → X be the usual projection of G onto X given by
g → a(g)k(g).
Then since A
+
(
−1
H
) is a cone containing A
+
, it is clear that there exists a constants C

H,1
and C
H,2
so that
C
H,1
(1 + σ
∗
(a(g))) ≤ (1 + r
X
(p(g)) ≤ C
H,2
(1 + σ
∗
(a(g)))
for all g such that a(g) ∈ A
+
(
−1
H
).
This proves the lemma. 
Proposition 2.2.1.3. The integral in (2.2.1.1) converges absolutely and
uniformly over compact sets in M, f
P
∈ C
∗
(G; N
0
∩M \M; ψ) and f → f

P
is continuous
in the topology induced by seminorms on C
∗
(N
0
\ G; ψ) and C
∗
(G; N
0
∩ M \ M; ψ).
Proof. Clearly it suffices for us to prove that f
P
lies in C
∗
(G; N
0
∩ M \ M; ψ).
Fix a suitably small open compact subgroup H so that f ∈ C(N
0
\G; ψ)
H
. Write
¯nm = n(¯nm)a(¯nm)k(¯nm) using the Iwasawa decomposition. If we write m ∈ M as
m = n
1
m
1
k
1

where n
1
∈ N
0
∩ M, m
1
∈ A
0
∩ M and k
1
∈ K ∩M, then
¯nm = ¯nn
1
m
1
k
1
= n
1
m
1
.((n
1
m
1
)
−1
¯n(n
1
m

1
))k
1
.
This implies that
a(¯nm) = m
1
a((n
1
m
1
)
−1
¯n(n
1
m
1
))
and
n(¯nm) = n
1
m
1
n((n
1
m
1
)
−1
¯n(n

1
m
1
))m
−1
1
.
Let
¯
n = (n
1
m
1
)
−1
¯n(n
1
m
1
) so that
n(¯nm)a(¯nm) = n
1
m
1
n(
¯
n)a(
¯
n).
Note that n

1
m
1
∈ M and
¯
n ∈
¯
N.
By Lemma 2.2.1.2, a function f ∈ C
∗
(N
0
\ G; ψ)
H
satisfies
|f(¯nm)| ≤ C
H
q
2,d
(f)δ
1
2
0
(a(¯nm))(1 + σ
∗
(¯nm))
−d
.
We note that σ
∗

(n(¯nm)a(¯nm)) = σ
∗
(n
1
m
1
n(
¯
n)a(
¯
n)k(
¯
n)) = σ
∗
(n
1
m
1
¯
n) as σ
∗
is
bi-K-invariant. Also, from the proof of [Wal, Lemme II.3.1], we have that
2σ
∗
(n
1
m
1
¯

n) ≥ σ
∗
(
¯
n)
σ
∗
(n
1
m
1
¯
n) ≥ σ
∗
(n
1
m
1
)
so that
(1 + σ(n
1
m
1
¯
n))
2
= 1 + 2σ(n
1
m

1
¯
n) + (σ(n
1
m
1
¯
n))
2
≥ 1 + σ
∗
(
¯
n)/2 + σ
∗
(n
1
m
1
) + σ
∗
(n
1
m
1
)σ
∗
(
¯
n)/2

≥
1
2
(1 + σ
∗
(
¯
n))(1 + σ
∗
(n
1
m
1
))
18