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

Kavli Affiliate: Toshiyuki Kobayashi

| First 5 Authors: Toshiyuki Kobayashi, , , ,

| Summary:

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.

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