Journal of the Mathematical Society of Japan

Intersection of harmonics and Capelli identities for symmetric pairs

Soo Teck LEE, Kyo NISHIYAMA, and Akihito WACHI

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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 a real reductive dual pair in a large symplectic group. In this setting, we get Capelli identities which explicitly represent certain K C -invariant elements in U( g C ) in terms of H C -invariant elements in U( 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 S p 2n -harmonics as a module of G L n = O 2n S p 2n , and construct a basis for the G L n highest weight vectors. This intersection is in the kernel of our Capelli elements.

Article information

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

First available in Project Euclid: 5 November 2008

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Zentralblatt MATH identifier

Primary: 17B35: Universal enveloping (super)algebras [See also 16S30]
Secondary: 22E46: Semisimple Lie groups and their representations 16S32: Rings of differential operators [See also 13N10, 32C38] 15A15: Determinants, permanents, other special matrix functions [See also 19B10, 19B14]

harmonics Capelli identity Weil representation invariant theory


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.

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