Construction of Sheaves of Cherednik Algebras via Formal Geometry

Abstract

In this note we realize the sheaf of Cherednik algebras $H_{1,c,X,G}$ on a general good complex orbifold $X/G$, originally introduced by Etingof for smooth complex varieties with an action by a finite group, by gluing sheaves of flat sections of flat holomorphic vector bundles on orbit type strata in $X$ which result from a localization procedure. In the case, when $c$ is formal, this construction can be interpreted as a formal deformation of $D_X⋊\mathbb{C}G$ via Gel'fand-Kazhdan formal geometry. Contrary to the original definition of $H_{1,c,X,G}$ the presented construction permits the computation of trace densities, Hochschild homologies and an algebraic index theorem for formal deformations of $D_X⋊\mathbb{C}G$. We also hope that the methods developed here will contribute towards a full proof of Dolgushev-Etingof's conjecture.

54 pages; The title has been made more precise. The old university affiliation has been removed; the email address is updated