An application of the Littlewood restriction formula to the Kostant-Rallis Theorem

Author:
Jeb F. Willenbring

Journal:
Trans. Amer. Math. Soc. **354** (2002), 4393-4419

MSC (2000):
Primary 22E47, 20G05, 05E05

Published electronically:
June 24, 2002

MathSciNet review:
1926881

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Abstract | References | Similar Articles | Additional Information

Abstract: Consider a symmetric pair of linear algebraic groups with , where and are defined as the +1 and -1 eigenspaces of the involution defining . We view the ring of polynomial functions on as a representation of . Moreover, set , where is the space of homogeneous polynomial functions on of degree . This decomposition provides a graded -module structure on . A decomposition of is provided for some classical families when is within a certain stable range.

The stable range is defined so that the spaces are within the hypothesis of the classical Littlewood restriction formula. The Littlewood restriction formula provides a branching rule from the general linear group to the standard embedding of the symplectic or orthogonal subgroup. Inside the stable range the decomposition of is interpreted as a -analog of the Kostant-Rallis theorem.

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Additional Information

**Jeb F. Willenbring**

Affiliation:
Department of Mathematics, Yale University, 10 Hillhouse Avenue, PO Box 208283, New Haven, Connecticut 06520

Email:
jeb.willenbring@math.yale.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-02-03065-9

Keywords:
Kostant-Rallis theorem,
Littlewood restriction formula,
skew Schur polynomial

Received by editor(s):
October 22, 2001

Published electronically:
June 24, 2002

Additional Notes:
This research was funded by the Yale Gibbs Instructorship as well as the NSF VIGRE postdoctoral fellowship.

Article copyright:
© Copyright 2002
American Mathematical Society