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December 2013 Stability of branching laws for spherical varieties and highest weight modules
Masatoshi Kitagawa
Proc. Japan Acad. Ser. A Math. Sci. 89(10): 144-149 (December 2013). DOI: 10.3792/pjaa.89.144

Abstract

If a locally finite rational representation $V$ of a connected reductive algebraic group $G$ has uniformly bounded multiplicities, the multiplicities may have good properties such as stability. Let $X$ be a quasi-affine spherical $G$-variety, and $M$ be a $(\mathbf{C}[X],G)$-module. In this paper, we show that the decomposition of $M$ as a $G$-representation can be controlled by the decomposition of the fiber $M/\mathfrak{m}(x_{0})M$ with respect to some reductive subgroup $L \subset G$ for sufficiently large parameters. As an application, we apply this result to branching laws for simple real Lie groups of Hermitian type. We show that the sufficient condition on multiplicity-freeness given by the theory of visible actions is also a necessary condition for holomorphic discrete series representations and symmetric pairs of holomorphic type. We also show that two branching laws of a holomorphic discrete series representation with respect to two symmetric pairs of holomorphic type coincide for sufficiently large parameters if two subgroups are in the same $\epsilon$-family.

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Masatoshi Kitagawa. "Stability of branching laws for spherical varieties and highest weight modules." Proc. Japan Acad. Ser. A Math. Sci. 89 (10) 144 - 149, December 2013. https://doi.org/10.3792/pjaa.89.144

Information

Published: December 2013
First available in Project Euclid: 2 December 2013

zbMATH: 1290.22007
MathSciNet: MR3161536
Digital Object Identifier: 10.3792/pjaa.89.144

Subjects:
Primary: 22E46
Secondary: 20G05 , 32M15 , 57S20

Keywords: branching rule , highest weight module , multiplicity-free representation , semisimple Lie group , spherical variety , symmetric pair

Rights: Copyright © 2013 The Japan Academy

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Vol.89 • No. 10 • December 2013
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