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
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