We consider a see-saw pair consisting of a Hermitian symmetric pair and a compact symmetric pair , where and form a real reductive dual pair in a large symplectic group. In this setting, we get Capelli identities which explicitly represent certain -invariant elements in in terms of -invariant elements in . The corresponding -invariant elements are called Capelli elements.
We also give a decomposition of the intersection of -harmonics and -harmonics as a module of , and construct a basis for the highest weight vectors. This intersection is in the kernel of our Capelli elements.
"Intersection of harmonics and Capelli identities for symmetric pairs." J. Math. Soc. Japan 60 (4) 955 - 982, October, 2008. https://doi.org/10.2969/jmsj/06040955