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

Abstract

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 ${\mathbf{M}}_{\mathbf{\theta }}$ on $\mathcal{H},$ where $\{{\theta }_i{\}}_{i=1}^n$ is a homogeneous system of parameters associated to $G$ and $\mathbf{\theta }=({\theta }_1,\dots ,{\theta }_n)$ is a polynomial map of ${\mathbb{C}}^n$. We show that it admits a family $\{{\mathbb{P}}_{\varrho }\mathcal{H}:\varrho \in \hat{G}\}$ of non-trivial joint reducing subspaces, where $\hat{G}$ 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 $\varrho \in \hat{G},$ the multiplication operator ${\mathbf{M}}_{\mathbf{\theta }}$ 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 $\mathbf{\theta }(\Omega )$. 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 $G\subseteq \mathrm{Aut}(\Omega ).$

Extensively revised with new results and applications that includes version 1 as one of the sections