Intersection of harmonics and Capelli identities for symmetric pairs

Abstract

We consider a see-saw pair consisting of a Hermitian symmetric pair $(G_R,K_R)$ and a compact symmetric pair $(M_R,H_R)$ , where $(G_R,H_R)$ and $(K_R,M_R)$ form a real reductive dual pair in a large symplectic group. In this setting, we get Capelli identities which explicitly represent certain $K_C$ -invariant elements in $U({\mathfrak{g}}_C)$ in terms of $H_C$ -invariant elements in $U({\mathfrak{m}}_C)$ . The corresponding $H_C$ -invariant elements are called Capelli elements.

We also give a decomposition of the intersection of $O_{2n}$ -harmonics and $S{p}_{2n}$ -harmonics as a module of $G{L}_n=O_{2n}\cap S{p}_{2n}$ , and construct a basis for the $G{L}_n$ highest weight vectors. This intersection is in the kernel of our Capelli elements.