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.

Keywords

reductive group, unitary representation, symmetry breaking, admissible restriction, momentum map, Harish–Chandra module, convexity theorem

2010 Mathematics Subject Classification

Primary 22E46. Secondary 22E45, 43A77, 58-xx.

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Dedicated to Bertram Kostant with admiration to his deep and vast perspectives and with sincere gratitude to his constant encouragement for many years.

This work was partially supported by Grant-in-Aid for Scientific Research (A) (JP18H03669), Japan Society for the Promotion of Science.

Received 30 July 2019

Accepted 13 May 2020

Published 22 December 2021