Branching theorems for compact symmetric spaces

Author:
A. W. Knapp

Journal:
Represent. Theory **5** (2001), 404-436

MSC (2000):
Primary 20G20, 22E45; Secondary 05E15

DOI:
https://doi.org/10.1090/S1088-4165-01-00139-X

Published electronically:
October 26, 2001

MathSciNet review:
1870596

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A compact symmetric space, for purposes of this article, is a quotient $G/K$, where $G$ is a compact connected Lie group and $K$ is the identity component of the subgroup of fixed points of an involution. A branching theorem describes how an irreducible representation decomposes upon restriction to a subgroup. The article deals with branching theorems for the passage from $G$ to $K_{2}\times K_{1}$, where $G/(K_{2}\times K_{1})$ is any of $U(n+m)/(U(n)\times U(m))$, $SO(n+m)/(SO(n)\times SO(m))$, or $Sp(n+m)/(Sp(n)\times Sp(m))$, with $n\leq m$. For each of these compact symmetric spaces, one associates another compact symmetric space $G’/K_{2}$ with the following property: To each irreducible representation $(\sigma ,V)$ of $G$ whose space $V^{K_{1}}$ of $K_{1}$-fixed vectors is nonzero, there corresponds a canonical irreducible representation $(\sigma ’,V’)$ of $G’$ such that the representations $(\sigma |_{K_{2}},V^{K_{1}})$ and $(\sigma ’,V’)$ are equivalent. For the situations under study, $G’/K_{2}$ is equal respectively to $(U(n)\times U(n))/\text {diag}(U(n))$, $U(n)/SO(n)$, and $U(2n)/Sp(n)$, independently of $m$. Hints of the kind of “duality” that is suggested by this result date back to a 1974 paper by S. Gelbart.

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Additional Information

**A. W. Knapp**

Affiliation:
School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540, and Department of Mathematics, State University of New York, Stony Brook, New York 11794

Address at time of publication:
81 Upper Sheep Pasture Road, East Setauket, New York 11733–1729

MR Author ID:
103200

Email:
aknapp@math.sunysb.edu

Keywords:
Branching rule,
branching theorem,
representation

Received by editor(s):
March 20, 2001

Received by editor(s) in revised form:
September 10, 2001

Published electronically:
October 26, 2001

Article copyright:
© Copyright 2001
Anthony W. Knapp