Cherednik and Hecke algebras of varieties with a finite group action

Details Cherednik and Hecke algebras of varieties with a finite group action Cherednik and Hecke algebras of varieties with a finite group action

2004-06-24

arxiv.org/abs/math/0406499v3

Mathematics

Quantum Algebra

21 pages. latex; in the second version small errors were corrected and exposition was improved; a new example was added at the

Scientific paper

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 H_{t,c}(G,V), 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 several 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. 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 \pi_2(X)=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.

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