## Cherednik and Hecke Algebras of Varieties with a Finite Group Action

##### Author(s)

Etingof, Pavel I
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Let G be a finite group of linear transformations of a finite dimensional complex vector space V. To this data one can attach a family of algebras Ht,c(V, G), parametrized by complex numbers t and conjugation invariant functions c on the set of complex reflections in G, which are called rational Cherednik algebras. These algebras have been studied for over 15 years and revealed a rich structure and deep connections with algebraic geometry, representation theory, and combinatorics. In this paper, we define global analogs of Cherednik algebras, attached to any smooth algebraic or analytic variety X with a finite group G of automorphisms of X. We show that many interesting properties of Cherednik algebras (such as the PBW theorem, universal deformation property, relation to Calogero–Moser spaces, action on quasiinvariants) still hold in the global case, and give several interesting examples. Then we define the KZ functor for global Cherednik algebras, and use it to define (in the case π2(X) ⊗ Q = 0) a flat deformation of the orbifold fundamental group of the orbifold X/G, which we call the Hecke algebra of X/G. This includes usual, affine, and double affine Hecke algebras for Weyl groups, Hecke algebras of complex reflection groups, as well as many new examples. Keyword: Cherednik algebra; reflection hypersurface; Hecke algebra; variety with a finite group action

##### Date issued

2017-10##### Department

Massachusetts Institute of Technology. Department of Mathematics##### Journal

Moscow Mathematical Journal

##### Publisher

National Research University, Higher School of Economics (HSE)

##### Citation

Etingof, Pavel. "Cherednik and Hecke Algebras of Varieties with a Finite Group Action." Moscow Mathematical Journal 17, 4 (October 2017): 635-666 © Independent University of Moscow.

Version: Original manuscript

##### ISSN

1609-4514