AN APPLICATION OF THE LITTLEWOOD RESTRICTION FORMULA TO THE KOSTANT-RALLIS THEOREM

Abstract. Consider a symmetric pair (G,K) of linear algebraic groups with g ∼ = k ⊕ p, where k and p are defined as the +1 and-1 eigenspaces of the involution defining K. We view the ring of polynomial functions on p as a rep-resentation of K. Moreover, set P(p) = ⊕∞d=0 Pd(p), where Pd(p) is the space of homogeneous polynomial functions on p of degree d. This decomposition provides a graded K-module structure on P(p). A decomposition of Pd(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 Pd(p) are within the hypoth-esis 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 Pd(p) is interpreted as a q–analog of the Kostant-Rallis theorem