  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 real reductive dual pairs in a large symplectic group. In this setting, we
get Capelli identities which explicitly represent certain K_C-invariant
elements in U(Lie(G)_C) in terms of H_C-invariant elements in U(Lie(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} \cap Sp_{2n}, and construct a
basis for the GL_n highest weight vectors. This intersection is in the kernel
of our Capelli elements.
