Admissible restrictions of irreducible representations of reductive Lie groups: symplectic geometry and discrete decomposability

Let $G$ be a real reductive Lie group, $L$ a compact subgroup, and $\pi$ an irreducible admissible representation of $G$. In this article we prove a necessary and sufficient condition for the finiteness of the multiplicities of $L$-types occurring in $\pi$ based on symplectic techniques. This leads us to a simple proof of the criterion for discrete decomposability of the restriction of unitary representations with respect to noncompact subgroups (the author, Ann. Math. 1998), and also provides a proof of a reverse statement which was announced in [Proc.ICM 2002, Thm.D]. A number of examples are presented in connection with Kostant’s convexity theorem and also with non-Riemannian locally symmetric spaces.