Rational Cherednik algebras and Hecke algebras attached to complex orbifolds

Let X be a complex manifold, and G a discrete group of holomorphic transformations of X. To this data I will assign a sheaf of algebras on the orbifold X/G, called the sheaf of rational Cherednik algebras. This sheaf is the deformation of the sheaf G*D(X), the smash product of G with the sheaf of differential operators on X. If X is an affine space and G is a complex reflection group, then the algebra of global sections of this sheaf is the usual rational Cheredik algebra. Using this generalization of rational Cherednik algebras, I wil constuct interesting flat deformations of group algebras, which include usual, affine, and double affine Hecke algebras, quantizations of Del Pezzo surfaces, etc. The (non-disjoint) classes of groups to be discussed are: 1) finite real and complex reflection groups; 2) orbifold surface groups; 3) crystallographic groups; 4) hyperbolic reflection groups.