Rikard Bøgvad: Using rings of differential operators to construct simple modules for complex reflection groups

Abstract:
Let G be a finite subgroup of the linear group of a finite-dimensional complex vector V, B=S(V) be the symmetric algebra, D the ring of G-invariant differential operators on B, and D^- its subring of negative degree operators. We prove that M -> M^{ann}=Ann_{D^-}(M) defines an isomorphism between the category of D-submodules of B and a category of modules formed as lowest weight spaces. This is applied to give a sufficient criterion for when B^{ann} is a so-called Gelfand model,
i.e., contains a unique copy of each G-representation. When G is a generalized symmetric group, it is then very easy to show that B^{ann} is a Gelfand model. This elucidates i.a. work by Araujo and Oesterle on these models, and shows how it fits into a framework of direct images of D-modules.