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

We study the right regular representation on the space L2N0\ G; ψ where G isa quasi-split p-adic group and ψ a non-degenerate unitary character of the unipotentsubgroup N0 of a minimal p

THE PLANCHEREL FORMULA OF L2(N0\ G; ψ) WHERE G IS A p-ADIC

GROUP

TANG U-LIANG(BSc (Hons), NUS )

A THESIS SUBMITTEDFOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN

MATHEMATICSDEPARTMENT OF MATHEMATICSNATIONAL UNIVERSITY OF SINGAPORE

2011

The author wishes to thank Professor Gordon Savin, Professor Gan Wee Teck andAssociate Professor Loke Hung Yean for initiating this project, suggestions, commentsand guidance.

i

Acknowledgments i

We study the right regular representation on the space L2(N0\ G; ψ) where G is

a quasi-split p-adic group and ψ a non-degenerate unitary character of the unipotentsubgroup N0 of a minimal parabolic subgroup of G We obtain the direct integral

decomposition of this space into its constituent representations In particular, we deducethat the discrete spectrum of L2(N0\ G; ψ) consists precisely of ψ generic discrete seriesrepresentations and derive the Plancherel formula for L2(N0\ G; ψ)

v

1.1 INTRODUCTION AND STATEMENT OF MAIN RESULTS

1.1 Introduction and statement of main results1.1.1 Let ψ be a nondegenerate unitary character of the unipotent radical N0 of aminimal standard parabolic subgroup of a connected quasi-split p-adic group, G Define

L2(N0\ 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 N0 This space becomes a unitaryrepresentation of G via right translation

The purpose of this work is to obtain the Plancherel formula for this unitaryrepresentation Dinakar Ramakrishnan first studied the case for GL(2) in [Ram2]obtaining a Plancherel formula for the archimedean and non-archimedean group NolanWallach then proved this result for arbitrary real reductive groups (see [Wa, Chapter14]) Indeed, we have found out that much Wallach’s arguments can be adapted for thep-adic case

There are two crucial steps in proving the Plancherel formula for L2(N0\ G; ψ).One of them is to prove the surjectivity of a certain map from the space of Schwartzfunctions on G to the space of Schwartz functions on N0\ G This is the author’s

original contribution

Secondly we must define a Whittaker transform which transforms certain smoothfunctions on orbits of discrete series representations to Schwartz functions on N0\ Ganalogous to the Harish-Chandra wave packet map We refer the reader to Section 4.1.3for the precise definition of this transform In order to define this map, we require the acertain 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 = M N be a standard parabolic

subgroup of G with M and N its Levi and unipotent subgroup respectively Let ψM

denote the restriction of ψ to M ∩ N0 We take a ψM generic discrete series

representation (σ, Hσ) and consider the unitarily induced representation IG

P(σ ⊗ ν)where ν ∈ Im(Xur(M )) = iaM/L runs over all unramified unitary characters of M Werefer the reader to section 1.3.4 for the definitions of the relevant notations

Now let W hψM(Hσ) denote the (one dimensional) space of Whittaker functionals

on σ Let

Hσ,ν = IPG(σ, ν) ⊗ W hψM(Hσ)and consider the direct integral

Iσ,M =

Z ⊕

ia M /LHσ,νµ(σ, ν) dν˜where ˜µ(σ, ν) is a certain normalization of the Plancherel measure on ia/L

We prove that

2

Theorem 1.1.2.1 There exists a unitary linear isomorphism from

X

M ⊂G

X σ∈ E 2

ψM (M )/W (G|M )

Iσ,M

onto L2(N0\ G; ψ) where each of the Iσ,M is a G-module

This is a refinement of the Plancherel formula for a particular symmetric space(i.e N0\ G) of “polynomial growth” studied in [B]

We also compute the explicit normalization of ˜µ(σ, ν) If µ(σ, ν) dν denotes thePlancherel measure on iaM/L, then

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 crucialLemma 3.2.1.2 We also describe the notations and conventions for parabolic subgroupsand tori needed to describe unitary parabolic induction Finally, we end by giving adescription of the Schwartz spaces on G and N0\ G needed later

In Chapter 2 we discuss the general theory of Whittaker functions for discreteseries representations and tempered representations, the Harish-Chandra transform forfunctions in C∗(N0\ G; ψ) and conclude with an application of this theory to a result ofSavin, Khare and Larsen

Chapter 3 is where we discuss aspects of the Placherel measure on G, the

multiplicity one property for L2(N0\ 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 isprecisely Corollary 4.3.2.1

1.1.4 While this paper was being written, it was brought to the author’s attentionthat Erez Lapid and Mao Zhengyu had obtained an explicit form of the Whittakerfunction and its asymptotics on a split group G in [L-Z] Theorem 2.1.3.3 is a directcorollary of their results

They conjectured the following: Let W (π) denote the Whittaker model of ageneric representation and suppose that R

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 G1.2.1 We begin by fixing a p-adic field k with ring of integers o and normalizedabsolute value |.| = q−val(.) Let G be a connected quasi split reductive group definedover k Let G = G(k) be its k-rational points Let A0 the maximal k split torus and M0its (abelian) centralizer with M0,0 the maximal compact subgroup of M0

With W denoting the Weyl group of G with respect to A0, let ÝW be the affine

Weyl group extending the Weyl group W We may identify ÝW as the semidirect product

of W and D = A0/A1

0 where A1

0 is the maximal open compact subgroup of A0 Let Σaffand ndΣ 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 x0 in theapartment stabilized by A0 Denote by Σ0 the roots of Σaff vanishing on x0 If B is thesubgroup of G fixing pointwise a chamber in the apartment with vertex x0, then B is anIwahori subgroup One knows that there is a bijection λ : ndΣ → Σ0 since every root

α ∈ Σ0 is a positive multiple of a unique root in ndΣ We write λ(α) as λαα

The Iwahori decomposition for G and K is

G = a w∈ W Ü

We will write the Iwasawa decomposition of G as N0A0K where N0 is the

unipotent subgroup of a choice of minimal parabolic P0 of G containing M0

Define ndΣ+ to be the system of positive roots determined by P0 and let ∆denote the set of simple roots For  > 0, define

A+(−1) := {a ∈ A0 | |α(a)| ≤ −1∀α ∈ ∆}

Writing A+ as A+(1), we have the Cartan decomposition G = KA+K

Let {Hm}m≥1 denote a system of ‘good’ open compact subgroups which generatesthe topology on G (see [Sil1]) By ‘good’ we simply require that these collection ofsubgroups satisfy the triangle decomposition and are normal in K Without loss ofgenerality we assume that if w ∈ W , then w /∈ Hi for any i ≥ 1

1.2.2 Our objective in this subsection is to prove a crucial lemma which gives theinterplay between the Cartan and Iawasawa decompositions on a group To do this, wewill need to describe the root subgroups of N0 and their corresponding filtrations Asimplified version adequate for quasi split groups can be found in [Ca2] and it is to thispaper which we will refer to for definitions and notation So let N0(α) be a certainsubset of N0 indexed by each α ∈ Σaff Then for α ∈ ndΣ+ and m ∈ Z define

Nα,m:= N0(λ(α) + m)

We have that Nα,m+1 ( Nα,m and if Nα :=S

m∈ZNα,m, then

N0 = Y α∈ nd Σ +

Nα

4

Under the left action of N0, the stabilizer of giH is N0∩ giHgi−1 Let

[N0/N0∩ giHgi−1] be a set of N0∩ giHgi−1-coset representatives in N0 Then each G/Hcoset in N0giH is parametrized the set [N0/N0∩ giHg−1i ]

Since H is normal in K, N0∩ giHg−1i = N0∩ aHa−1 = aN0,ma−1 Written

component wise

aNα,ma−1 = N0(λ(α) + m + λαval(α(a)))where we recall that λα is the positive multiple arising from the bijection between ndΣand Σ0

Now chose an integer n so that N0,n ) N0,m Also, choose  > 0 small enough sothat if a satisfies

val(α(a)) ≥ 1

λα(−m + n)for all simple roots α ∈ ∆ then a ∈ A+(−1) Then a ∈ A+(−1) contains the set of all

a ∈ A such that N0∩ aHa−1 ⊂ N0,n

Lemma 1.2.2.1 For a fixed a1 ∈ A+(−1) and fixed k1 ∈ K, the coset

Hmna1k1Hm

is not equals to any coset of the form

Hmwa0w−1kHmwith a0 ∈ A+ and w ∈ W as we run over all n ∈ N0− ((N0∩ aHma−1)N0,m),

Proof Firstly for any n ∈ N0− ((N0∩ aHma−1)N0,m), we rewrite the element

g = na1k1 according to the KA+K decomposition

Assume that a1 ∈ A+(−1) is such that N0 ∩ a1Hma−11 * N0,m There are threecases

(1) If n ∈ N0,0− ((N0∩ aHma−1)N0,m), then

g = na1k1 = nw2a2w−12 k1

where w2−1a1w2 = a2 ∈ A+ The expression in the right hand side of the

equation has been decomposed according to the Cartan decomposition as nw2

and w−12 k1 are contained in K

5

1.2 THE CARTAN AND IWASAWA DECOMPOSITIONS OF G

(2) If n /∈ N0,0 but a−11 na1 ∈ K, then na1k1 = a1(a−11 na1)k1 = w2a2w−12 n0k1 where

n0 = a−1na ∈ K and a2 ∈ A+ defined as above As w2−1n0k1 ∈ K, the KA+Kdecomposition of na1k1 is

w2a2w−12 n0k1

(3) If n ∈ N0 does not satisfy the properties of the previous two cases, then

decompose na1 = b1w1a2b2 according to the Iwahori decomposition with

b1, b2 ∈ B, w1 ∈ W and a2 ∈ D Since na1 and a−11 na1 ∈ K, a/ 2 and w1 are bothnot the identity element Then

na1k1 = b1w1w2a3w2−1b2k1where w2−1a2w2 = a3 ∈ A+ is the Cartan decomposition for g as b1w1w2 and

w−12 b2k1 are elements of K

For case (1), let us assume that there exists w0 ∈ W and a0 ∈ A+ such that

nw2a2w2−1k1 ∈ Hmw0a0w−10 k1Hm.Since the Cartan decomposition gives a disjoint union over all KˇaK where ˇa ∈ A+/A1

0,

we may assume that a2 = a0 so that nw2 ∈ Hmw0 and w−12 k1 ∈ w−10 k1Hm But thismeans that w2 = w0 implying that n ∈ Hm which contradicts the hypothesis The

lemma is proven in this case

For case (3) we argue in a similar fashion So suppose that

b1w1w2a3w−12 b2k1 ∈ Hmw0a0w0−1k1Hm

As before, a3 = a0 so that b1w1w2 ∈ Hmw0 We may as well assume b1 ∈ Hm and that

w1w2 = w0 (for otherwise the resulting contradiction proves the lemma already) Now

on the other hand

w2−1b2k1 ∈ w0−1k1Hmimplying that

w−11 b2 ∈ w0−1Hmsince Hm is normal Once again, we may assume b2 ∈ Hm so that w0 = w1 This forces

w2 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), wemust exclude the possibility that n0 is in Hm

Indeed, if n0 ∈ Hm, we will derive a contradiction Write the component of anarbitrary n ∈ N0 in Nα as nα Since a−11 na1 ∈ K, nα∈ Nα is contained in at most

Nα,λαval(α(a)) However by our assumption on n0, nα ∈ Nα,λαval(α(a)) satisfies

a−11 nαa1 ∈ Nα,m for all roots α ∈ ndΣ+ This implies that nα ∈ Nα,λαval(α(a))+m i.e

n ∈ N0∩ a1Ha−11 This contradicts the hypothesis and proves the lemma

If N0,m ⊃ N0∩ a1Ha−11 , then a1 ∈ A+ so that it is clear that for any n /∈ N0,m,

Hmna1k1Hm is never of the form of as Hmwa0w−1k1Hm 

6

1.3 Parabolic subgroups and Schwartz spaces1.3.1 As per Harish-Chandra’s philosophy of descent we study the Planchereldecomposition of L2(N0\ G; ψ) by reducing a dense space of ‘test’ functions to itscorrespondingly defined subspaces on Levi subgroups of ‘standard’ parabolic subgroups

on G In this section, we describe both these parabolic subgroups, their unramifiedcharacters and the space of ‘test’ functions which we will study in detail

We introduce the notion of standard parabolic subgroups and define a densesubspace of L2(N0\ G; ψ) which we will study in this paper

Fix a minimal parabolic subgroup P0 as in the previous section A standardparabolic pair is a pair (P, A) consisting of a parabolic subgroup G ⊃ P ⊃ P0 and

A0 ⊃ A ⊃ ZG where ZG 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 7→ Pθwhere we agree that P∅ = P0 and P∆= G It is well known that any standard paraboliccorresponds to a subset θ ⊂ ∆ Conversely, to each subset θ, one can associate a

standard parabolic Pθ, Mθ, its Levi and If Pθ is a standard parabolic, we write its

Langlands decomposition as Pθ = MθNθ Where there is no cause for confusion, we willdrop the θ from notation Moreover where the context is clear, N0 always denotes N∅

A smooth unitary character ψ of N0 is said to be nondegenerate if and only if forany α ∈ ∆, ψ restricted to Nα is non-trivial

1.3.2 Let δP denote the modular character of the parabolic subgroup P By theIwasawa decomposition G = NθMθK, we write g ∈ G as g = nPθ(g)mPθ(g)k(g)

We define the constant

γ(Pθ) = γ(G|Mθ) If α ∈ ndΣ(Pθ, Aθ), let Aα denote the (identity component) of thekernel of α and Mα the centralizer of Aα in G Then Mα ⊃ Mθ for every root

Z K

1.3 PARABOLIC SUBGROUPS AND SCHWARTZ SPACES

Proof For the purposes of this proof, IndGP σ will denote unnormalized inductioninstead of the usual normalized induction Recall the integral decomposition formula(c.f [Wal, pg 240])

Z

Gf (g) dg = γ(G|M )−1

Z

N ×M × ¯ Nf (nm¯n)δ−1P (m) d¯n dm dnfor any parabolic subgroup P and f ∈ Cc∞(G)

For any f ∈ Cc∞(G) let Pδ¯

Pθ(f ) denote the projection of f into IndGP¯θδP¯θ This is

Pδ¯

Pθ(f )(k)dk

We may choose f such that Pδ¯

Pθ(f )(k) = hu(k), v(k)iσ proving the lemma 1.3.3 Let Σ(P0, A0) denote the set of all positive roots of G with respect to P0 Theresulting roots by restricting to Aθ is denoted Σ(Pθ, Aθ)

Define a0,R:= (Homk(A0, k×) ⊗ZR)∗ where ∗ denotes the real dual of the vectorspace Letting Aθ denote the center of Mθ we may define in an analogous fashion, thevector space aθ,R Then there is a canonical decomposition

a0,R = aθ,R⊕ aθRrealizing aθ,R as a subspace of a0,R The complexified vector spaces are denoted aθ,C

We describe a partition of A+0 For any 0 <  ≤ 1, define

1.3.4 Using the Lie algebra a∗θ,C, we may construct the unramified unitary

characters of Mθ Eventually, we will see that these characters will constitute the

‘continuous’ portion of the Plancherel decomposition in a manner made precise in thelater sections of this paper

Let Hom(G, C×) denote the group of continuous homomorphisms from G to C×

If χ ∈ Homk(G, k×) then |χ|k is defined by |χ|k(g) = |χ(g)|k Let G1 :=T

χKer|χ|k andlet Xur(G) := Hom(G/G1, C×) We call this set of characters the set of unramifiedcharacters of G These definitions apply to any Levi subgroup Mθ of any standardproper parabolic subgroups of G by replacing G with Mθ

There is a surjection from a∗θ,C onto Xur(Mθ) defined by χ ⊗ s 7→ (g 7→ |χ(g)|s) If

ν ∈ a∗θ,C, the corresponding character in Xur(Mθ) is denoted χν The kernel of this map

is a lattice of the form (2π√

−1/ log q)R where R is a lattice of Homk(Mθ, k×) ⊗ZQ.This endows Xur(Mθ) with the structure of a complex algebraic variety isomorphic to(C×)d where d is the dimension of aθ,R

Given χ ∈ Xur(Mθ), 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(Aθ, C×), thenthe character |χ| extends uniquely to an element of Xur(Mθ) taking positive real values.Set <χ = <|χ|

Consider the group ImX(Aθ) consisting of characters of χ ∈ Hom(Aθ, C×)

satisfying <χ = 0 Then there is a surjection with finite kernel of ImXur(Mθ) ontoImX(Aθ) The latter group is compact and we give it the Haar measure normalized sothat its total mass is 1 We pull back the measure on ImX(Aθ) to a measure on

ImXur(Mθ) 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 asuitable radial function on the homogeneous space N0\ G We will use this function toregulate the growth of our Schwartz functions as it approaches ‘infinity’

We start by considering a k-rational representation ρ : G → GL(V ) with compactkernel We denote the (i, j)-th entry of the matrix ρ(g) as ρ(g)ij By pulling back to G,define ||g|| := supijsup{|ρ(g)ij|, |ρ(g−1)ij|} and σ(g) := logq||g|| and

σ∗(g) := infz∈ZGσ(gz) where ZG denotes the center of G

Adopting the same notation in [Wal], we let Ξ(g) denote the zonal sphericalfunction of G and ΞM for the same for each standard Levi subgroups M of G So then

we define a space of functions, C(Hj \ G/Hj; χ) as the space of Hj bi-invariant smooth

9

1.3 PARABOLIC SUBGROUPS AND SCHWARTZ SPACES

complex valued functions on G such that ZG acts by a unitary character χ For any

f ∈ C(Hj \ G/Hj) define

q1,r(f ) := sup

g∈G

|f (g)|Ξ−1(g)(1 + σ∗(g))rwhere r > 0 Define

Now let ψ be a nondegenerate character of N0 and ZG acts by the same character

χ as above However, in the interest of reducing notational clutter, we agree to suppressthis from subsequent notation Define C(N0\ G/Hj; ψ) to be the space of right Hjinvariant complex valued functions of G such that f (ng) = ψ(n)f (g) for all n ∈ N0 and

g ∈ G

Consider any f ∈ C(N∅\ G/Hk) By the Iwasawa decomposition G = N∅M∅Kwrite g = n∅(g)m∅(g)k(g) with m(g) ∈ M∅, n∅(g) ∈ N∅ and k(g) ∈ K Although m∅(g) isonly defined up to M∅∩ K, δ−

1 2

P 0 (m) = 1 for any m ∈ M∅∩ K implying that for any r > 0,

As before the set {q2,n}n∈N for a family of continuous seminorms on C∗(N∅\ G; ψ)

making it into a Frechet space It is dense in L2(N∅\ G; ψ)

Let Pθ be a proper standard parabolic subgroup of G with Mθ its Levi subgroupand writing Mθ∩ N∅ as N∗ Write σMθ

∗ (m) := infa∈Aθσ(ma) This generalizes σ∗ definedearlier To be precise, σ∗ = σG∗

P 0 ∩Mθ(m∅(m))(1 + inf

n∈N ∅ ∩MθσMθ

∗ (nm∅(m)))r.and

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 ∈ N0 Then 1 + infn∈N 0σ∗(nm∅(g)) is equivalent to

1 + σ∗(m∅(g)) For convenience, we will write this as 1 + σ∗(a(g))

11

2.1 WHITTAKER FUNCTIONS

2.1 Whittaker functions2.1.1 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 mainresults 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

L2(N0\ 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 V0 denotethe algebraic complex linear dual of V Define (π0(g)λ)(v) = λ(π(g−1)v) and let V∨denote the smooth points of π0 Let π∨ be the representation obtained by restricting π0

to V∨ Then (π∨, V∨) is a smooth representation of G

Now consider a fixed λ ∈ V0 and write

W (v, λ)(g) := λ(π(g)v)with v ∈ V Since V is smooth, W (v, λ) is Kj invariant on the right for some opencompact 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 N0 Fix a standardparabolic subgroup Pθ = P = M N of G corresponding to θ ⊂ ∆ (P possibly equals toG) for the discussion throughout this subsection Let N∗ and ψθ be the unipotent

subgroup and character defined as in the previous section Then define

V (N∗, ψθ) := span{π(n)v − ψθ(n)v | v ∈ V, n ∈ N∗}and

V (N ) := span{π(n)v − v | v ∈ V, n ∈ N }

Define vector spaces

VN∗,ψθ = V /V (N∗, ψθ)and

VN = V /V (N )

Then

rPG(V ) = VN ⊗ δ−1/2P

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 hψ(V ) := HomN0(π, Cψ)

A representation π is said to be ψ-generic if W hψ(V ) is nontrivial Now assumethat this is the case for π and consider any nonzero λ ∈ W hψ(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 ∈ N0

We say that W (v, λ) is a Whittaker function and W (v, λ)(1) = λ(v) a Whittakerfunctional All these can be generalized to Levi subgroups of G as well, by substituting

G for M , N0 for N∗ and ψ for ψθ

14

Let Φθ be the canonical map from (VNθ)N∗,ψθ to VN0,ψ introduced in [C-S] (seealso [Ca1]) If v ∈ V , we write ˜v for its image in (VNθ)N∗,ψθ 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 Pθ of G Let λ ∈ W hψ(V ) and

v ∈ V Then there exists  > 0 such that

W (v, λ)(a) = W (˜v, λ ◦ Φθ)(a)for any a ∈ A0 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 Aθ-finite complex valued smoothfunction f on Aθ Then it is well known that

f (a) =X

ν

ν(a)P (HMθ(a))where P (x) is a polynomial on the Lie algebra aθ,R of Aθ and ν a smooth character of

Aθ The characters occurring in the decomposition above are known as the exponents of

f Let E(Aθ, 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 Aθ-finite function;

(2) the center of G, ZG acts by a unitary character on f and

(3) f is invariant under right translation by some open subgroup AK i of A10

Then |f |p is integrable on A+0(θ, )/AKiZG if and only if |χ(a)| < 1 for all a ∈ A+θ andeach χ ∈ E(Aθ, f )

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 matrixcoefficients (modulo the center) are in L2(G) (resp L2+(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−

2.1 WHITTAKER FUNCTIONS

Theorem 2.1.3.3 Let ψ be an nondegenerate additive unitary character of N0

and let (π, V ) be an irreducible ψ-generic discrete series representation, then V embedsinto L2(N0\ G; ψ)

Proof Any Whittaker function is of the form W (v, λ)(g) for λ ∈ (VN 0 ,ψ)0 Nowchoose Ki as the largest open compact subgroup of K such that

Z

K i

π(k)v dk = v0does not vanish Let [K/Ki] denote a (finite) set of coset representatives of K/Ki Thenfor each kj ∈ [K/Ki], set v0j = kjv0

δ0−1(a)|W (ak)|2dadk

δ0−1(a)|W (v0j, λ)(a)|2da

Thus, the finiteness of that integral depends upon the square integrability of

W (v0j, λ)(a) on A0/A10ZG Since ψ is nondegenerate, we observe that W (vj0, λ)(a) issupported inside a translate of A+0 ([C-S, Proposition 6.1]) By replacing vj0 with π(aj)vj0for some suitable aj ∈ A0, 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 Pθ (with

θ ⊂ ∆) Recall that A+0 is partitioned into S

|f (g)| ≤ Cδ01(m0(g))(1 + σ∗(a(g)))r.Proposition 2.1.3.4 Assume that (π, V ) is a tempered and ψ-generic irreduciblerepresentation of G Then for any v ∈ V and λ ∈ W hψ(V ), W (v, λ) ∈ A(N0\ G; ψ).Proof By [Wal, Proposition III.2.2] we know that if π is tempered, then for eachstandard parabolic subgroup Pθ, an exponent ν ∈ E(θ, V ) satisfies |δ−

1 2

P (a)ν(a)| ≤ 1 Ifthis inequality was strict for all parabolic subgroups and all exponents, then π is a

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discrete series and hence by (the proof of) Theorem 2.1.3.3, W (v, λ) is in A(N0 \ G; ψ).Thus we assume that for some parabolic subgroup, Pθ, there exists an exponent suchthat |δ−

1

P (a)ν(a)| = 1 In this case, using Lemma 2.1.2.1 and arguing as in Theorem

2.2 The Harish-Chandra transform2.2.1 If P is a parabolic subgroup of G, then ¯P will denote its opposite parabolicsubgroup Given f ∈ C∗(N0\ G; ψ) define the Harish-Chandra transform

fP(m) = δ1/2P (m)

Z

¯ N

f (¯nm)d¯n = δ1/2P¯ (m)

Z

¯ N

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∗(N0\ G; ψ)H,

|f (g)| ≤ CHq2,r(f )δ−

1

0 (a0(g))(1 + σ∗(g))−rwhere q2,r(f ) is the seminorm on the Schwartz space C∗(N0\ G; ψ) and CH a constantwhich 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, wesee that f is essentially supported in A+(−1H ) where H > 0 depends only on the opencompact subgroup H We define the following radial function on A+(−1H ): For any x, y

in the lattice D ⊂ a0 we let d(x, y) be the usual Euclidean distance between two

elements Using a large scale equivalence, m between X = N0\ G and a0 (see [B,

Sections 4.1 and 4.6]), we may define a radial function rX(γ) for any γ ∈ X by

rX(γ) = d(m(γ), m(γ0)) where γ0 is the element in X which satisfies

|α(HM−10(m(γ0)))| = −1H for all α ∈ ∆ One sees easily that rX(•) is essentially equivalent

to σ∗(•) on G because of the Cartan decomposition on G

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2.2 THE HARISH-CHANDRA TRANSFORM

Let p : G → X be the usual projection of G onto X given by

g 7→ a(g)k(g)

Then since A+(−1H ) is a cone containing A+, it is clear that there exists a constants CH,1and CH,2 so that

CH,1(1 + σ∗(a(g))) ≤ (1 + rX(p(g)) ≤ CH,2(1 + σ∗(a(g)))for all g such that a(g) ∈ A+(−1H )

Proposition 2.2.1.3 The integral in (2.2.1.1) converges absolutely and

uniformly over compact sets in M , fP ∈ C∗(G; N0∩ M \ M ; ψ) and f 7→ fP is continuous

in the topology induced by seminorms on C∗(N0\ G; ψ) and C∗(G; N0∩ M \ M ; ψ).Proof Clearly it suffices for us to prove that fP lies in C∗(G; N0∩ M \ M ; ψ).Fix a suitably small open compact subgroup H so that f ∈ C(N0\ G; ψ)H Write

¯

nm = n(¯nm)a(¯nm)k(¯nm) using the Iwasawa decomposition If we write m ∈ M as

m = n1m1k1 where n1 ∈ N0∩ M , m1 ∈ A0∩ M and k1 ∈ K ∩ M , then

¯

nm = ¯nn1m1k1 = n1m1.((n1m1)−1n(n¯ 1m1))k1.This implies that

a(¯nm) = m1a((n1m1)−1n(n¯ 1m1))and

n(¯nm) = n1m1n((n1m1)−1n(n¯ 1m1))m−11 Let ¯n = (n1m1)−1n(n¯ 1m1) so that

n(¯nm)a(¯nm) = n1m1n(¯n)a(¯n)

Note that n1m1 ∈ M and ¯n ∈ ¯N

By Lemma 2.2.1.2, a function f ∈ C∗(N0 \ G; ψ)H satisfies

Thus, one has that

P (m1)δ

1 2

0(a(¯n))(1 + σ∗(¯n))−d2 d¯n.The integral is known to converge (c.f [Wal, Lemme II.4.2]) for large enough d Notealso that σ∗(m) ≥ σ∗(m1) implying then that the transformed function lies in the

Define

◦C∗(N0\ G; ψ) = {f ∈ C∗(N0\ G; ψ)|(R(k)f )P ≡ 0

f or all k ∈ K and f or all parabolic subgroups P = Pθ, θ ( ∆}

We refer to this space as the space of discrete functions By the Iwasawa decompositionthis space is stable under right translation by the full group G

Let f ∈ C∗(N0\ G; ψ) and Vf be the space spanned by right translates of G and

VfP the space generated by right translates of M of fP in C∗(N0∩ M \ M ; ψ)

Lemma 2.2.1.4 The representation space Vf P is a quotient of rG

¯

P(Vf)

Proof It is easy to check from definition that (R(m)f )P = δ

1 2

¯

P(m)R(m)fP.Next, let ¯N be the unipotent subgroup opposite of P and consider the space

Vf( ¯N ) If h ∈ Vf( ¯N ), then Z

¯

N 0

R(n)h = 0for some open compact subgroup ¯N0 of ¯N In particular, Vf( ¯N ) is contained in thekernel of f 7→ fP The lemma follows from these two observations 

Theorem 2.2.1.5 Let (π, V ) be a smooth irreducible representation of G whichembeds into L2(N0\ G; ψ) Then the image of V under this embedding is in

◦C∗(N0\ G; ψ)

Proof Let T be the embedding map and define W (v, T )(g) = T (π(g)v)(1) Then

W (v, T ) is a Whittaker function (with (T v)(1) considered a Whittaker functional) Wenote that W (v, T ) is smooth while ψ is nondegenerate and therefore supported in atranslate of A+0 By Lemma 2.1.2.1, the restriction of W (v, T ) on each partition A+0(θ, )coincides with some Aθ-finite function Thus conditions (1) to (3) of Lemma 2.1.3.1 aresatisfied and we may conclude that the exponents (of the Aθ-finite function) must satisfy

|δ0−1(a)ν(a)| < 1 for every a ∈ A+0(θ, ) as W (v, T ) is also square integrable

Since

W (v, T )(a) = ˜W (˜v, T ◦ Φθ)(a) =X

ν

νi(a)Pi(HMθ(a))and we know that |P (HMθ(a))| ∈ O((1 + σ∗(a))n) for some integer n (see [Wal, pg.242]), there exists a constant C > 0 such that

|δ−

1 2

0 (a)W (v, T )(a)| < C(1 + σ∗(a))−r)

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