A Capelli Harish-Chandra homomorphism

Przebinda, Tomasz

doi:10.1090/S0002-9947-03-03316-6

A Capelli Harish-Chandra homomorphism
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Trans. Amer. Math. Soc. 356 (2004), 1121-1154 Request permission

Abstract:

For a real reductive dual pair the Capelli identities define a homomorphism $\mathcal {C}$ from the center of the universal enveloping algebra of the larger group to the center of the universal enveloping algebra of the smaller group. In terms of the Harish-Chandra isomorphism, this map involves a $\rho$-shift. We view a dual pair as a Lie supergroup and offer a construction of the homomorphism $\mathcal {C}$ based solely on the Harish-Chandra’s radial component maps. Thus we provide a geometric interpretation of the $\rho$-shift.

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