4
1. INTRODUCTION
Here, D^ is the discrete series representation of SU{1,1) with
Blattner parameter ±p.
2. Moreover, if X ^ for ^e above p, the generalized Whittaker
functions for TT\ with minimal K-type are (up to Laurent poly-
nomials) described by the function
{
e
2n(c1a21-c2al) . . . __ p+^
e
-27r(c1a?-C2ai) ...X =
D
P-
Thus we have
dim
c
G^(^;
X
,O
m o d
=
Here is some historical comments related to our work. In Sp(2,R)
case, which has the same restricted root system of type C2, this prob-
lem has considered by several authors. Niwa [Ni] and Miyazaki [Mi]
obtained explicit integral representation of generalized Whittaker func-
tions. Their results are used for studying non-holomorphic Siegel mod-
ular forms and their Andrianov L-functions (cf. [Ho]). Among others
Miyazaki investigated the large discrete series cases.
We obtain a much simpler integral expression of generalized Whit-
taker functions belonging to the large discrete series representations
than that of Miyazaki. Moreover there is no such representation like
the middle discrete series for Sp(2,IR). It is a new feature peculiar to
SU {2,2).
Acknowledgements. The author would like to express his profound
gratitude to Professor Takayuki Oda and Dr. Takahiro Hayata for their
valuable suggestions and encouragement, and Dr. Takuya Miyazaki for
careful reading of the first version of the manuscript.
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1
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