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

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Abstract: A compact symmetric space, for purposes of this article, is a quotient , where is a compact connected Lie group and 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 to , where is any of , , or , with . For each of these compact symmetric spaces, one associates another compact symmetric space with the following property: To each irreducible representation of whose space of -fixed vectors is nonzero, there corresponds a canonical irreducible representation of such that the representations and are equivalent. For the situations under study, is equal respectively to , , and , independently of . 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

Email:
aknapp@math.sunysb.edu

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

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