CHAPTER 1

Introduction

1. Introduction

In this article we study generalized Whittaker functions on G —

SU{2, 2), the special unitary group of signature (2+, 2—), with respect

to the Siegel parabolic subgroup P$. The subgroup Ps is a maximal

parabolic subgroup with abelian unipotent radical Afe. Our concern

is to extend the classical theory of Fourier expansion for holomorphic

modular forms on G to that for non-holomorphic modular forms. The

reason why we investigate this group is because it is of real rank two

and has many new features that never exists for smaller groups.

Let IT be an admissible representation of G and £ be a unitary

character of N§. Then a fundamental problem for the theory of Fourier

expansion of automorphic forms for IT is to investigate the intertwining

space Horned, Ind^£). However this intertwining space is infinite-

dimensional in general. So we introduce a certain larger group R (See

Sect. 2.2 for definition) containing N$.

A generalized Whittaker model for an admissible representation TT

of G is a realization of TT in the induced module from a closed subgroup

which contains Ns- This induced module is a typical example of the

reduced generalized Gelfand-Graev representation [Yal]. In this article

we investigate the intertwining space

GW(TT;V)

= Hom

( 0

^)(7r,C-(i?\G)),

where r\ is an irreducible i?-module such that T]\NS 3 £, K is a maximal

compact subgroup of G and Q is the Lie algebra of G. A function in this

realization is called a generalized Whittaker function for TT. The impor-

tant problem is to determine the dimension of the intertwining space

GW(TT;

rj) and to obtain an explicit formula for generalized Whittaker

functions.

We want to discuss this problem for the discrete series representa-

tions of G. The group G has three types of discrete series:

(1) Holomorphic and anti-holomorphic discrete series representations.

(2) Two discrete series representations which have ordinary Whittaker

1