Intersection of harmonics and Capelli identities for symmetric pairs

Abstract

We consider a see-saw pair consisting of a Hermitian symmetric pair $(G_{\bm{R}}, K_{\bm{R}})$ and a compact symmetric pair $(M_{\bm{R}}, H_{\bm{R}})$ , where $(G_{\bm{R}}, H_{\bm{R}})$ and $(K_{\bm{R}}, M_{\bm{R}})$ form a real reductive dual pair in a large symplectic group. In this setting, we get Capelli identities which explicitly represent certain $K_{\bm{C}}$ -invariant elements in $U(\mathfrak{g}_{\bm{C}})$ in terms of $H_{\bm{C}}$ -invariant elements in $U(\mathfrak{m}_{\bm{C}})$ . The corresponding $H_{\bm{C}}$ -invariant elements are called Capelli elements.

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