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.

Full Text (PDF format)

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