Modules over Convolution Algebras from Equivariant Derived Categories, I

In this paper, we provide a general (functorial) construction of modules over convolution algebras (i.e., where the multiplication is provided by a convolution operation) starting with an appropriate equivariant derived category. The construction is sufficiently general to be applicable to different situations. One of the main applications is to the construction of modules over the graded Hecke algebras associated to complex reductive groups starting with equivariant complexes on the unipotent variety. It also applies to the affine quantum enveloping algebras of typeAn. As is already known, in each case the algebra can be realized as a convolution algebra. Our constructionturns suitable equivariant derived categories into an abundant source of modules over such algebras; most of these are new, in that, so far the only modules have been provided by suitable Borel–Moore homology or cohomology with respect to a constant sheaf (or by an appropriate K-theoretic variant.) In a sequel to this paper we will apply these constructions to equivariant perverse sheaves and also obtain a general multiplicity formula for the simple modules in the composition series of the modules constructed here.