Cherednik and Hecke algebras of orbifolds

The rational Cherednik algebra is attached to a finite
group G acting on a vector space V, i.e. to the orbifold V/G.
It is generated by group elements, functions on V, and Dunkl operators. I willexplain how the theory of Cherednik algebras can be extended to an arbitrary orbifold (algebraic or complex analytic), and how to define the KZ functor for such algebras. This leads to a construction of a flat deformation of the group algebra of the orbifold fundamental group of a complex orbifold Y whose universal cover has a vanishing second cohomology with rational coefficients. These deformations include all known Hecke algebras (usual, complex reflection, affine, double affine), as well as many new examples (e.g., Hecke algebras attached to Mostow hyperbolic complex reflection groups).
The talk is based on my paper math.QA/0406499.