MODULES OVER CONVOLUTION-ALGEBRAS FROM EQUIVARIANT DERIVED CATEGORIES, I

In this paper, we provide a general (functorial) construction of modul es over convolution algebras (i.e., where the multiplication is provid ed by a convolution operation) starting with an appropriate equivarian t derived category. The construction is sufficiently general to be app licable to different situations. One of the main applications is to th e construction of modules over the graded Hecke algebras associated to complex reductive groups starting with equivariant complexes on the u nipotent variety. It also applies to the affine quantum enveloping alg ebras of type A,. As is already known, in each case the algebra can be realized as a convolution algebra. Our construction turns suitable eq uivariant derived categories into an abundant source of modules over s uch 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 equi variant perverse sheaves and also obtain a general multiplicity formul a for the simple modules in the composition series of the modules cons tructed here. (C) 1998 Academic Press.