Cherednik and Hecke algebras of varieties with a finite group action

摘要: This paper is an expanded and updated version of the preprint arXiv:math/0406499. It includes a more detailed description basics theory Cherednik Hecke algebras varieties started in arXiv:math/0406499, as well new Section 4, which reviews developments this since 2004 with references to relevant literature. Let $G$ be finite group linear transformations dimensional complex vector space $V$. To data one can attach family $H_{t,c}(V,G)$, parametrized by numbers $t$ conjugation invariant functions $c$ on set reflections $G$, are called rational algebras. These have been studied for over 15 years revealed rich structure deep connections algebraic geometry, representation theory, combinatorics. In paper, we define global analogs algebras, attached any smooth or analytic variety $X$ automorphisms. We show that many interesting properties (such PBW theorem, universal deformation property, relation Calogero-Moser spaces, action quasiinvariants) still hold case, give several examples. Then KZ functor use it (in case $\pi_2(X)\otimes \Bbb Q=0$) flat orbifold fundamental $X/G$, call algebra $X/G$. usual, affine, double affine Weyl groups, reflection