Reducing submodules of Hilbert modules and Chevalley-Shephard-Todd theorem

Abstract

Let G be a finite pseudoreflection group, Ω⊆Cn be a bounded domain which is a G-space and H⊆O(Ω) be an analytic Hilbert module possessing a G-invariant reproducing kernel. We study the structure of joint reducing subspaces of the multiplication operator Mθ on H, where {θi}i=1n is a homogeneous system of parameters associated to G and θ=(θ1,…,θn) is a polynomial map of Cn. We show that it admits a family {PϱH:ϱ∈Gˆ} of non-trivial joint reducing subspaces, where Gˆ is the set of all equivalence classes of irreducible representations of G. We prove a generalization of Chevalley-Shephard-Todd theorem for the algebra O(Ω) of holomorphic functions on Ω. As a consequence, we show that for each ϱ∈Gˆ, the multiplication operator Mθ on the reducing subspace PϱH can be realized as multiplication by the coordinate functions on a reproducing kernel Hilbert space of C(deg⁡ϱ)2-valued holomorphic functions on θ(Ω). 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 Ω in Cn which is factored by automorphisms G⊆Aut(Ω).