Abstract
For a real reductive dual pair the Capelli identities define a homomorphism C \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 ρ \rho -shift. We view a dual pair as a Lie supergroup and offer a construction of the homomorphism C \mathcal {C} based solely on the Harish-Chandra’s radial component maps. Thus we provide a geometric interpretation of the ρ \rho -shift.
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