### 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 |
---|---|

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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Stability of branching laws for spherical varieties and highest weight modules.** / Kitagawa, Masatoshi.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Stability of branching laws for spherical varieties and highest weight modules

AU - Kitagawa, Masatoshi

PY - 2013/12/1

Y1 - 2013/12/1

N2 - 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.

AB - 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.

KW - Branching rule

KW - Highest weight module

KW - Multiplicity-free representation

KW - Semisimple lie group

KW - Spherical variety

KW - Symmetric pair

UR - http://www.scopus.com/inward/record.url?scp=84891636313&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84891636313&partnerID=8YFLogxK

U2 - 10.3792/pjaa.89.144

DO - 10.3792/pjaa.89.144

M3 - Article

AN - SCOPUS:84891636313

VL - 89

SP - 144

EP - 149

JO - Proceedings of the Japan Academy Series A: Mathematical Sciences

JF - Proceedings of the Japan Academy Series A: Mathematical Sciences

SN - 0386-2194

IS - 10

ER -