Let G be a real reductive Lie group, L a compact subgroup, and π an irreducible admissible representation of G. This paper proves a necessary and sufficient condition for the finiteness of the multiplicities of L-types occurring in π based on symplectic techniques. This leads us to a new proof of the criterion for the discrete decomposability of the restriction of unitary representations with respect to noncompact subgroups (the author, Ann. Math. 1998). A number of examples are presented in connection with Kostant's convexity theorem and also with non-Riemannian locally symmetric spaces.
© Toshiyuki Kobayashi