### 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(x_{0})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 language | English |
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Pages (from-to) | 144-149 |

Number of pages | 6 |

Journal | Proceedings of the Japan Academy Series A: Mathematical Sciences |

Volume | 89 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2013 Dec 1 |

Externally published | Yes |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)