Stability of branching laws for spherical varieties and highest weight modules

Masatoshi Kitagawa

Research output: Contribution to journalArticle

2 Citations (Scopus)

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 (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/m(x0)M with respect to some reductive subgroup L ⊂ 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 ∈-family.

Original languageEnglish
Pages (from-to)144-149
Number of pages6
JournalProceedings of the Japan Academy Series A: Mathematical Sciences
Volume89
Issue number10
DOIs
Publication statusPublished - 2013 Dec 1

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Keywords

  • Branching rule
  • Highest weight module
  • Multiplicity-free representation
  • Semisimple lie group
  • Spherical variety
  • Symmetric pair

ASJC Scopus subject areas

  • Mathematics(all)

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