Intersection of harmonics and Capelli identities for symmetric pairs

2005/10/03 by Lee, Soo Teck, Nishiyama, Kyo, Wachi, Akihito
#17B35 #22E46 16S32 15A15 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.math/0510033

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 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 ∩ Sp_2n, and construct a basis for the GL_n highest weight vectors. This intersection is in the kernel of our Capelli elements.

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