## Abstract

In this paper, we study the irreducible decomposition of a (ℂ[X];G)-module M for a quasi-affine spherical variety X of a connected reductive algebraic group G over ℂ. We show that for sufficiently large parameters, the decomposition of M with respect to G is reduced to the decomposition of the ‘fiber’ M/m(x_{0})M with respect to some reductive subgroup L of G. In particular, we obtain a method to compute the maximum value of multiplicities in M. Our main result is a generalization of earlier work by F. Satō in [17]. We apply this result to branching laws of holomorphic discrete series representations with respect to symmetric pairs of holomorphic type. We give a necessary and sufficient condition for multiplicity-freeness of the branching laws.

Original language | English |
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Pages (from-to) | 1027-1050 |

Number of pages | 24 |

Journal | Transformation Groups |

Volume | 19 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2014 Nov 18 |

Externally published | Yes |

## ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology