Stability of branching laws for spherical varieties and highest weight modules

Proceedings of the Japan Academy, Series A. Mathematical Sciences

article

Masatoshi Kitagawa¹ 

[1] Graduate School of Mathematical Science, The University of Tokyo

关键词: Spherical variety;    multiplicity-free representation;    branching rule;    symmetric pair;    highest weight module;    semisimple Lie group.;   

DOI  :  10.3792/pjaa.89.144

学科分类：数学（综合）

来源: Japan Academy

【 摘 要 】

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.

【 授权许可】

Unknown