Discrete decomposability of the restriction ofA q(?) with respect to reductive subgroups and its applications

Abstract

LetG′⊂G be real reductive Lie groups and q a θ-stable parabolic subalgebra of Lie (G) ⊗ ℂ. This paper offers a sufficient condition on (G, G′, q) that the irreducible unitary representation $$\mathop {A_q }\limits^--- $$ ofG with non-zero continuous cohomology splits into a discrete sum of irreducible unitary representations of a subgroupG′, each of finite multiplicity. As an application to purely analytic problems, new results on discrete series are also obtained for some pseudo-Riemannian (non-symmetric) spherical homogeneous spaces, which fit nicely into this framework. Some explicit examples of a decomposition formula are also found in the cases whereA q is not necessarily a highest weight module.