# Construction of Sheaves of Cherednik Algebras via Formal Geometry

## TL;DR

This paper constructs sheaves of Cherednik algebras on complex orbifolds using formal geometry, enabling new computations and contributing to the proof of a significant conjecture in the field.

## Contribution

It introduces a sheaf construction of Cherednik algebras on orbifolds via formal geometry, allowing for explicit calculations and advancing the understanding of their deformation theory.

## Key findings

- Construction of Cherednik sheaves on orbifolds via formal geometry.
- Ability to compute trace densities, Hochschild homologies, and algebraic index theorems.
- Potential progress towards proving Dolgushev-Etingof's conjecture.

## 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 $D_X\rtimes\mathbb CG$ 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 $D_X\rtimes\mathbb CG$. We also hope that the methods developed here will contribute towards a full proof of Dolgushev-Etingof's conjecture.