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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

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Branching theorems for compact symmetric spaces
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by A. W. Knapp PDF
Represent. Theory 5 (2001), 404-436

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.
References
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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
  • Received by editor(s): March 20, 2001
  • Received by editor(s) in revised form: September 10, 2001
  • Published electronically: October 26, 2001
  • © Copyright 2001 Anthony 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
  • MathSciNet review: 1870596