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

Willenbring, Jeb

doi:10.1090/S0002-9947-02-03065-9

An application of the Littlewood restriction formula to the Kostant-Rallis Theorem
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by Jeb F. Willenbring PDF

Trans. Amer. Math. Soc. 354 (2002), 4393-4419 Request permission

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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