Reducing submodules of Hilbert Modules and Chevalley-Shephard-Todd Theorem

Let $G$ be a finite pseudoreflection group, $\Omega\subseteq \mathbb C^n$ be a bounded domain which is a $G$-space and $\mathcal H\subseteq\mathcal O(\Omega)$ be an analytic Hilbert module possessing a $G$-invariant reproducing kernel. We study the structure of joint reducing subspaces of the multiplication operator $\mathbf M_{\boldsymbol\theta}$ on $\mathcal H,$ where $\{\theta_i\}_{i=1}^n$ is a homogeneous system of parameters associated to $G$ and $\boldsymbol\theta = (\theta_1, \ldots, \theta_n)$ is a polynomial map of $\mathbb C^n$. We show that it admits a family $\{\mathbb P_\varrho\mathcal H:\varrho\in\widehat G\}$ of non-trivial joint reducing subspaces, where $\widehat G$ is the set of all equivalence classes of irreducible representations of $G.$ We prove a generalization of Chevalley-Shephard-Todd theorem for the algebra $\mathcal O(\Omega)$ of holomorphic functions on $\Omega$. As a consequence, we show that for each $\varrho\in \widehat G,$ the multiplication operator $\mathbf M_{\boldsymbol\theta}$ on the reducing subspace $\mathbb P_\varrho \mathcal H$ can be realized as multiplication by the coordinate functions on a reproducing kernel Hilbert space of $\mathbb C^{(\mathrm{deg}\,\varrho)^2}$-valued holomorphic functions on $\boldsymbol\theta(\Omega)$. This, in turn, provides a description of the structure of joint reducing subspaces of the multiplication operator induced by a representative of a proper holomorphic map from a domain $\Omega$ in $\mathbb C^n$ which is factored by automorphisms $G\subseteq {\rm Aut}(\Omega).$