Generalized Whittaker functions and
representation theory of SU(2, 2)
2. Definition of the generalized Whittaker functions
2.1. The Siegel parabolic subgroup. Let G denote the special
unitary group SU(2, 2) and G has two realization G\, G2 for example.
G! = ^ e S X ( 4 , C )
G2 = lgeSL(4,C)
Here we denote by I2 the unit matrix of size two. We fix the isomor-
phism ^i2 : G\ 3 g 1— •
€ G2 hereafter. Here,
and also put (^21 = V7^1-
We use realization G2 as G = SU(2, 2) in this section and take the
Siegel parabolic subgroup Ps of G, i.e. maximal parabolic subgroup
with abelian unipotent radical.
We write LstxNs the Levi decomposition of Ps and these subgroups
are realized as follows.
Ls = l(g) =
Ns = n(B) =
lB = Be M2(C)
2.2. Characters of TVs- We define a unitary character £ of iVs
associated with a Hermitian matrix i/^ =
. For n = n(£?) e