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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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: Consider a symmetric pair $(G,K)$ of linear algebraic groups with $\mathfrak{g} \cong \mathfrak{k} \oplus \mathfrak{p}$, where $\mathfrak{k}$ and $\mathfrak{p}$ are defined as the +1 and -1 eigenspaces of the involution defining $K$. We view the ring of polynomial functions on $\mathfrak{p}$ as a representation of $K$. Moreover, set $\mathcal{P}(\mathfrak{p}) = \bigoplus_{d=0}^\infty \mathcal{P}^d(\mathfrak{p})$, where $\mathcal{P}^d(\mathfrak{p})$ is the space of homogeneous polynomial functions on $\mathfrak{p}$ of degree $d$. This decomposition provides a graded $K$-module structure on $\mathcal{P}(\mathfrak{p})$. A decomposition of $\mathcal{P}^d(\mathfrak{p})$is provided for some classical families $(G,K)$ when $d$ is within a certain stable range.

The stable range is defined so that the spaces $\mathcal{P}^d(\mathfrak{p})$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 $\mathcal{P}^d(\mathfrak{p})$ is interpreted as a $q$-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
PII: S 0002-9947(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