Bergman spaces under maps of monomial type

For appropriate domains $\Omega_{1}, \Omega_{2}$ we consider mappings $\Phi_{\mathbf A}:\Omega_{1}\to\Omega_{2}$ of monomial type. We obtain an orthogonal decomposition of the Bergman space $\mathcal A^{2}(\Omega_{1})$ into finitely many closed subspaces indexed by characters of a finite Abelian group associated to the mapping $\Phi_{\mathbf A}$. We then show that each subspace is isomorphic to a weighted Bergman space on $\Omega_{2}$. This leads to a formula for the Bergman kernel on $\Omega_{1}$ as a sum of weighted Bergman kernels on $\Omega_{2}$