Journal of the Mathematical Society of Japan

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.

Article information

Source
J. Math. Soc. Japan, Volume 60, Number 4 (2008), 955-982.

Dates
First available in Project Euclid: 5 November 2008

https://projecteuclid.org/euclid.jmsj/1225894029

Digital Object Identifier
doi:10.2969/jmsj/06040955

Mathematical Reviews number (MathSciNet)
MR2467866

Zentralblatt MATH identifier
1236.17017

Citation

LEE, Soo Teck; NISHIYAMA, Kyo; WACHI, Akihito. Intersection of harmonics and Capelli identities for symmetric pairs. J. Math. Soc. Japan 60 (2008), no. 4, 955--982. doi:10.2969/jmsj/06040955. https://projecteuclid.org/euclid.jmsj/1225894029

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