CIRM – Research in Pairs — Determination of Reducing Submodules of Hilbert Modules Possessing an Invariant Kernel

Let G be a finite group and Ω be a domain in C n such that G acts on Ω. Assume that H ⊆ Hol(Ω) be a Hilbert space with a G-invariant reproducing kernel K, then {P% : % ∈ Gb} is a family of orthogonal projections which add up to the identity operator on H, where Gb is the unitary dual of G. One of the basic consequences is the description of reducing submodules of H, viewed as a module over the ring of invariants C[z] G, via this family of projections. We are interested to study the questions of minimality of these reducing submodules and mutual inequivalence over the invariant ring C[z] G, where G is a finite pseudo-reflection group. We feel that the Chevalley-Shephard-Todd theorem characterizing the finite pseudo-reflection groups in terms of their invariant ring and our extension of this theorem in the category of holomorphic functions could be a key step towards this study