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

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.