## 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