CHAPTER 2
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)
9
*9
-12
12
9 =
-U
h
-h
-12
12
(1)
(2)
Here we denote by I2 the unit matrix of size two. We fix the isomor-
phism ^i2 : G\ 3 g 1—
CgC~l
G2 hereafter. Here,
C =
h
V2\~
1-U
1_2
1-12
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.
GoDPs =
* *
02 *
eG2
We write LstxNs the Levi decomposition of Ps and these subgroups
are realized as follows.
Ls = l(g) =
'r
1
geGL(2,C), detge
Ns = n(B) =
12 B
lB = Be M2(C)
2.2. Characters of TVs- We define a unitary character £ of iVs
associated with a Hermitian matrix i/^ =
ci 7
7
C
2
. For n = n(£?) e
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