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.
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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.