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
Externally publishedYes

Fingerprint

Spherical Varieties
Branching
Module
Multiplicity
Series Representation
Subgroup
Decompose
Reductive Group
Algebraic Groups
Fiber
Necessary Conditions
Sufficient Conditions

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.

In: Proceedings of the Japan Academy Series A: Mathematical Sciences, Vol. 89, No. 10, 01.12.2013, p. 144-149.

Research output: Contribution to journalArticle

@article{936f926f427440d8b68d3543140e695a,
title = "Stability of branching laws for spherical varieties and highest weight modules",
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.",
keywords = "Branching rule, Highest weight module, Multiplicity-free representation, Semisimple lie group, Spherical variety, Symmetric pair",
author = "Masatoshi Kitagawa",
year = "2013",
month = "12",
day = "1",
doi = "10.3792/pjaa.89.144",
language = "English",
volume = "89",
pages = "144--149",
journal = "Proceedings of the Japan Academy Series A: Mathematical Sciences",
issn = "0386-2194",
publisher = "Japan Academy",
number = "10",

}

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 -