Sheaves of Twisted Cherednik Algebras as Universal Filtered Formal Deformations

TL;DR

This paper extends Etingof's result by proving that sheaves of twisted Cherednik algebras serve as universal filtered formal deformations of differential operator sheaves on non-affine varieties with group actions.

Contribution

It generalizes the universality of Cherednik algebra sheaves from affine to non-affine varieties, using Hochschild cohomology and algebraic extension techniques.

Findings

Established quasi-isomorphisms between Hochschild complexes and differential forms.

Proved the sheaves form universal filtered formal deformations.

Connected deformation classes with parameter spaces of Cherednik algebras.

Abstract

According to a statement by Pavel Etingof, in the special case of an affine variety $X$ with a faithful action by a finite group $G$, the sheaf of (twisted) Cherednik algebras $\mathcal{H}_{1, c, \psi, X, G}$ with formal parameters $c, \psi$ is a universal formal deformation of $\mathcal{D}_X\rtimes G$ where $\mathcal{D}_X$ is the sheaf of differential operators on $X$. In the current note, we generalize Etingof's result to the non-affine case. We prove that for a generic smooth analytic or algebraic variety $X$, the sheaf $\mathcal{H}_{1, c, \psi, X, G}$ with formal $c$ and $\psi$ is a universal filtered formal deformation of $\mathcal{D}_X\rtimes G$. To that aim, we first construct quasi-isomorphisms between the Hochschild (co)chain complex of $\mathcal{D}_X\rtimes G$ and the $G$-invariant part of the direct sum over all elements $g$ in $G$ of sheaves of holomorphic differential forms on the cotangent bundles of the $g$-fixed point submanifolds in $X$. Finally, we combine these quasi-isomorphisms with results from the theory of algebraic extensions for sheaves of filtered associative algebras to establish a bijective correspondence between the space of isomorphism classes of filtered infinitesimal deformations of $\mathcal{D}_X\rtimes G$ and the parameter space of $\mathcal{H}_{1, c, \psi, X, G}$.