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~ - COHOMOLOGY
SPACES A S S O C I A T E D W I T H A CERTAIN GROUP E X T E N S I O N Ichiro
Satake
The purpose of this paper is to give a new aspect on unitary representations type.
of a semi-direct product
of Lie groups of a certain
We shall be concerned with unitary r e p r e s e n t a t i o n s
on the (square-integrable)
~ - cohomology
spaces
obtained
attached to a pair
(G,K) of a Lie group G and a compact subgroup K such that the homospace G/K has a structure
geneous
precisely,
of a h e r m i t i a n manifold.
we assume the conditions
a summary of basic facts concerning
(AI-2)
and
K = K 2 x KI,
(Gi,Ki)
(i : 1,2)
ditions
We will obtain,
additional
conditions
5.)
((CI-4)
in sections
As a typical example,
pair satisfying the conditions sociated w i t h a symplectic : a non-degenerate
tion
where
(AI-2),
space
7 and 9), a r e l a t i o n s h i p
a symplectic r e p r e s e n t a t i o n
ry of theta-functions Kuga's
fibre varieties.
form on V × V) (see sec-
product
G = G2G 1
of this type appears
In this
(GI,K I) is any
(V : a real vector space,
p: G 1 ÷ Sp(V,A)
(see [13]),
are satis-
G 2 is a nilpotent group as-
a l t e r n a t i n g bilinear
A group extension
sat-
under certain
all these conditions
(V,A)
3), and where the semi-direct
(B2)'
product
we assume the con-
fied, we shall consider in section i0 the case where
A
In
spaces for (G,K) and for (Gi,K I) and
between the ~ - eohomology (G2,K2).
spaces.
being two pairs
(More precisely,
in section
Part I is
G is a semi-direct
isfying the above conditions. (BI-2)
i.)
such ~- cohomology
Part II, we shall consider the case where
G = G2G 1
in section
(More
is defined by
satisfying
a condition
in the classical
theo-
and also in a recent theory of special
case, our result yields
a
139
complete description of the irreducible unitary representations of discrete series in terms of the corresponding irreducible jective) unitary representations
I.
of G
(pro-
of G I.
~ - cohomology spaces a t t a c h e d to a homogeneous h e r m i t i a n m a n i f o l d i.
Let G be a connected unimodular Lie group and K a compact
subgroup of G. respectively.
We denote by
~
and ~
the Lie algebras of G and K,
Then there exists a subspace 144 of
~
such that one
has (i)
~
=
~ + +~
(direct sum),
We fix such a subspace ~ of
9,
@,
144-. . . .
ad(k)~
c~4~
once and for all.
will be denoted by
~¢
for all
k CK.
The complexifications ,
P~
, ~
....
We
assume that there is given a complex structure J on ~4~ satisfying the following conditions. (AI)
J is K-invariant
(i.e., J commutes with all ad(k)l~4~ ,
and the (±i)-eigenspace algebra of [~g
~c
'
,~±] C44g±
There exists a
of J in
~e
is a complex sub-
(From the first condition, ,
subalgebra of ~ ¢ (A