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A Capelli Harish-Chandra homomorphism

Author(s): Tomasz Przebinda
Journal: Trans. Amer. Math. Soc. 356 (2004), 1121-1154.
MSC (2000): Primary 22E46, 17B35
Posted: August 26, 2003
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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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Additional Information:

Tomasz Przebinda
Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
Email: tprzebin@crystal.math.ou.edu

DOI: 10.1090/S0002-9947-03-03316-6
PII: S 0002-9947(03)03316-6
Received by editor(s): September 4, 2002
Posted: August 26, 2003
Additional Notes: This research was partially supported by NSF grant DMS 0200724. Part of the work was done while the author was visiting the Institute for Mathematical Sciences, National University of Singapore, in 2001
Copyright of article: Copyright 2003, American Mathematical Society

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