ON THE REPRESENTATIONS OF LIE-GROUPS IN T HE DOLBEAULT COHOMOLOGY SPACES

Let G be a connected reductive algebraic group over C with Lie algebra g and let Gamma subset of G be a discrete cocompact subgroup. We prov e that the Dolbeault cohomology G-module H-p,H-q(G/Gamma) is isomorphi c to the tensor product Lambda(p)(g) x H-q(Gamma,C), where the G-actio n on Lambda(p)(g) is induced by the adjoint representation of G and H- q(Gamma, C) is a trivial G-module. As an application, we prove that fo r any compact complex G-homogeneous manifold M the G-action on H-q(M, O-M) is trivial.