Sheaves of Twisted Cherednik Algebras as Universal Filtered Formal Deformations

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 ${\mathcal{D}}_X⋊G$ 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 ${\mathcal{D}}_X⋊G$. To that aim, we first construct quasi-isomorphisms between the Hochschild (co)chain complex of ${\mathcal{D}}_X⋊G$ 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 ${\mathcal{D}}_X⋊G$ and the parameter space of ${\mathcal{H}}_{1,c,\psi ,X,G}$.

26 pages; The title was shortened. The abstract was slightly altered. Sections 3.2.1 and 3.2.2 were newly added. Section 3.2.3 and 3.2.4 were slightly improved and expanded. Technical errors and typos were corrected

References in corpus (2)

• Calabi-Yau algebras
• Chiral differential operators via Batalin-Vilkovisky quantization