The weighted Bergman spaces and complex reflection groups

Abstract

We consider a bounded domain $\Omega \subseteq C^d$ which is a G-space for a finite complex reflection group G. For each one-dimensional representation of the group G, the relative invariant subspace of the weighted Bergman space on Ω is isometrically isomorphic to a weighted Bergman space on the quotient domain $\Omega /G$. Consequently, formulae involving the weighted Bergman kernels and projections of Ω and $\Omega /G$ are established. As a result, a transformation rule for the weighted Bergman kernels under a proper holomorphic mapping with G as its group of deck transformations is obtained in terms of the character of the sign representation of G. Explicit expressions for the weighted Bergman kernels of several quotient domains (of the form $\Omega /G$) have been deduced to demonstrate the merit of the described formulae.