The Direct Integral of Some Weighted Bergman Spaces

Abstract

Let G be the abelian Lie group R n × R k /Z k , acting on the complex space X = R n+k × 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 Kähler forms. As an application, we construct a model in the sense that every irreducible G-representation occurs exactly once in H.