The weighted Bergman spaces and complex reflection groups

We consider a bounded domain $\Omega \subseteq \mathbb C^d$ 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 $\Omega$ is isometrically isomorphic to a weighted Bergman space on the quotient domain $\Omega/G.$ Consequently, formulae involving the weighted Bergman kernels and projections of $\Omega$ 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.