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 Sp_2n-harmonics as a module of GL_n = O_2n ∩ Sp_2n, and construct a basis for the GL_n highest weight vectors. This intersection is in the kernel of our Capelli elements.