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

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