Classification of symmetric pairs with discretely decomposable restrictions of (g,K)-modules

We give a complete classification of reductive symmetric pairs (g, h) with the following property: there exists at least one infinite-dimensional irreducible (g,K)-module X that is discretely decomposable as an (h,H \cap K)-module. We investigate further if such X can be taken to be a minimal representation, a Zuckerman derived functor module A_q(\lambda), or some other unitarizable (g,K)-module. The tensor product $\pi_1 \otimes \pi_2$ of two infinite-dimensional irreducible (g,K)-modules arises as a very special case of our setting. In this case, we prove that $\pi_1 \otimes \pi_2$ is discretely decomposable if and only if they are simultaneously highest weight modules.Comment: To appear in Crelles J. (19 pages

Localization of cohomological induction

We give a geometric realization of cohomologically induced (g,K)-modules. Let (h,L) be a subpair of (g,K). The cohomological induction is an algebraic construction of (g,K)-modules from a (h,L)-module V. For a real semisimple Lie group, the duality theorem by Hecht, Milicic, Schmid, and Wolf relates (g,K)-modules cohomologically induced from a Borel subalgebra with D-modules on the flag variety of g. In this article we extend the theorem for more general pairs (g,K) and (h,L). We consider the tensor product of a D-module and a certain module associated with V, and prove that its sheaf cohomology groups are isomorphic to cohomologically induced modules.Comment: 24 pages, typos correcte

On orbits in double flag varieties for symmetric pairs

Let $G$ be a connected, simply connected semisimple algebraic group over the complex number field, and let $K$ be the fixed point subgroup of an involutive automorphism of $G$ so that $(G, K)$ is a symmetric pair. We take parabolic subgroups $P$ of $G$ and $Q$ of $K$ respectively and consider the product of partial flag varieties $G/P$ and $K/Q$ with diagonal $K$-action, which we call a \emph{double flag variety for symmetric pair}. It is said to be \emph{of finite type} if there are only finitely many $K$-orbits on it. In this paper, we give a parametrization of $K$-orbits on $G/P \times K/Q$ in terms of quotient spaces of unipotent groups without assuming the finiteness of orbits. If one of $P \subset G$ or $Q \subset K$ is a Borel subgroup, the finiteness of orbits is closely related to spherical actions. In such cases, we give a complete classification of double flag varieties of finite type, namely, we obtain classifications of $K$-spherical flag varieties $G/P$ and $G$-spherical homogeneous spaces $G/Q$.Comment: 47 pages, 3 tables; add all the details of the classificatio