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.