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Admissible restrictions of irreducible representations of reductive Lie groups: symplectic geometry and discrete decomposability
Pure and Applied Mathematics Quarterly ( IF 0.7 ) Pub Date : 2020-12-01 , DOI: 10.4310/pamq.2020.v16.n5.a3
Shin-ichi Matsumura 1
Affiliation  

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.

中文翻译:

还原李群不可约表示的可容许限制:辛几何和离散可分解性

假设$ G $是一个真实的还原李群,$ L $是一个紧凑子群,$ \ pi $是$ G $的不可约的可容许表示。在本文中,我们基于辛技术证明了$ \ pi $中出现的$ L $类型的多重性的有限性的充要条件。这使我们简单证明了关于非紧致子群的unit表示限制的离散可分解性标准(作者,Ann。Math。1998),并且还提供了在[ Proc。 。ICM 2002,D]。结合Kostant凸定理和非黎曼局部对称空间,给出了许多例子。
更新日期:2020-12-01
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