|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.