Moscow Mathematical Journal
Volume 17, Issue 4, October–December 2017 pp. 635–666.
Cherednik and Hecke Algebras of Varieties with a Finite Group Action
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. 2010 Math. Subj. Class. 20C08, 33D80.
Authors:
Pavel Etingof (1)
Author institution:(1) Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Summary:
Keywords: Cherednik algebra, reflection hypersurface, Hecke algebra, variety with a finite group action.
Contents
Full-Text PDF