The berezin form and harmonic analysis on a pair of hyperboloids

Abstract

We study the canonical and boundary representa� tions on a sphere Ω with an action of the generalized Lorentz group, following the extended interpretation suggested by Molchanov in [10, 11]. In this case, the sphere is not a homogeneous space, the action of the group G is not transitive, and the representations are not unitary. The manifold Ω is the flag space of the overgroup G*. The fundamental role in the theory is played by the Berezin transform (which is the basic object in Berezin quantization), that is, the operator intertwining canonical representations with “conju� gate” parameters. In this paper, we present an decomposition of the Berezin forms and construct harmonic analysis on a pair hyperboloids. The constructions and methods of the paper are of independent interest.