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 (nondisjoint) 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. 