Pure and Applied Mathematics Quarterly
Volume 17 (2021)
Special Issue In Memory of Prof. Bertram Kostant
Guest Editors: Shrawan Kumar, Lizhen Ji, and Kefeng Liu
Admissible restrictions of irreducible representations of reductive Lie groups: symplectic geometry and discrete decomposability
Pages: 1321 – 1343
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.
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.
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