Cherednik and Hecke Algebras of Varieties with a Finite Group Action

Let h be a finite dimensional complex vector space, and G be a finite subgroup of GL(h). To this data one can attach a family of algebras Ht,c(h, G), called the rational Cherednik algebras (see [EG]); for t = 1 it provides the universal deformation of G ⋉ D(h) (where D(h) is the algebra of differential operators on h). These algebras are generated by G, h, h with certain commutation relations, and are parametrized by pairs (t, c), where t is a complex number, and c is a conjugation invariant function on the set of complex reflections in G. They have a rich representation theory and deep connections with combinatorics (Macdonald theory, n! conjecture) and algebraic geometry (Hilbert schemes, resolutions of symplectic quotient singularities). The purpose of this paper is to introduce “global” analogues of rational Cherednik algebras, attached to any smooth complex algebraic variety X with an action of a finite group G; the usual rational Cherednik algebras are recovered in the case when X is a vector space and G acts linearly. More specifically, let G be a finite group of automorphisms of X , and let S be the set of pairs (Y, g), where g ∈ G, and Y is a connected component of the set X of gfixed points in X which has codimension 1 in X . Suppose that X is affine. Then we define (in Section 1 of the paper) a family of algebrasHt,c,ω(X,G), where t ∈ C, c is a G-invariant function on S, and ω is a G-invariant closed 2-form on X . This family for t = 1 provides a universal deformation of the algebra H1,0,0(X,G) = G⋉D(X), where D(X) is the algebra of differential operators on X (assuming that ω runs through a space of forms bijectively representing H(X,C)). IfX is not affine, then we define a sheaf of algebrasHt,c,ω,X,G rather than a single algebra. In this case the parameters are the same, except that ω runs over a space representing classes of G-equivariant twisted differential operator (tdo) algebras on X (see [BB], Section 2). We find that much of the theory of rational Cherednik algebras survives in the global case. In particular, one can define the spherical subalgebra, which is both commutative and isomorphic to the center of the Cherednik algebra in the case t = 0. The spectrum of this algebra is “the Calogero-Moser space” of X , which is a global version of the similar space defined in [EG]. This includes, in particular, Calogero-Moser spaces attached to symmetric powers of algebraic curves. One can also define the global analog of the theory of quasiinvariants which was worked out in [FV, EG, BEG] and references therein. These results can be generalized to the case when X is a complex analytic variety. It thus appears that the global Cherednik algebras introduced in this paper deserve further study. One interesting problem is to develop the theory of modules