Open Access
October, 2008 Intersection of harmonics and Capelli identities for symmetric pairs
Soo Teck LEE, Kyo NISHIYAMA, Akihito WACHI
J. Math. Soc. Japan 60(4): 955-982 (October, 2008). DOI: 10.2969/jmsj/06040955


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.


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Soo Teck LEE. Kyo NISHIYAMA. Akihito WACHI. "Intersection of harmonics and Capelli identities for symmetric pairs." J. Math. Soc. Japan 60 (4) 955 - 982, October, 2008.


Published: October, 2008
First available in Project Euclid: 5 November 2008

zbMATH: 1236.17017
MathSciNet: MR2467866
Digital Object Identifier: 10.2969/jmsj/06040955

Primary: 17B35
Secondary: 15A15 , 16S32 , 22E46

Keywords: Capelli identity , harmonics , invariant theory , Weil representation

Rights: Copyright © 2008 Mathematical Society of Japan

Vol.60 • No. 4 • October, 2008
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