The direct integral of some weighted Bergman spaces

Abstract

Let G be the abelian Lie group R(n) x R(k)/Z(k), acting on the complex space X = R(n+k) x iG. Let F be a strictly convex function on R(n+k). Let H be the Bergman space of holomorphic functions on X which are square-integrable with respect to the weight e(-F). The G-action on X leads to a unitary G-representation on the Hilbert space H. We study the irreducible representations which occur in H by means of their direct integral. This problem is motivated by geometric quantization, which associates unitary representations with invariant Kahler forms. As an application, we construct a model in the sense that every irreducible G-representation occurs exactly once in H.