Trace Densities and Algebraic Index Theorems for Sheaves of Formal Cherednik Algebras

TL;DR

This paper develops a novel sheaf construction for Cherednik algebras on orbifolds, introduces trace density maps, and proves an algebraic index theorem linking Hochschild homology to Chen-Ruan cohomology.

Contribution

It introduces a new formal geometric construction of Cherednik sheaves, defines trace density maps for Hochschild complexes, and establishes an algebraic index theorem relating algebraic and topological invariants.

Findings

Trace density maps generalize existing constructions for differential operators.

Hypercohomology of Hochschild complexes matches Chen-Ruan orbifold cohomology.

Euler characteristic of Hochschild chain complex relates to orbifold's topological Euler characteristic.

Abstract

We show how a novel construction of the sheaf of Cherednik algebras on a quotient orbifold Y=X/G by virtue of formal geometry in author's prior work leads to results for the sheaf of Cherednik algebra which until recently were viewed as intractable. First, for every orbit type stratum in $X$, we define a trace density map for the Hochschild chain complex of the sheaf of Cherednik algebras, which generalises the standard Engeli-Felder's trace density construction for the sheaf of differential operators. Second, by means of the newly obtained trace density maps, we prove an isomorphism in the derived category of complexes of $\mathbb C_Y[[\hbar]]$-modules which computes the hypercohomology of the Hochschild chain complex $\mathcal{C}_{\bullet}$ of the sheaf of formal Cherednik algebras. We show that this hypercohomology is isomorphic to the Chen-Ruan cohomology of the orbifold $X/G$ with values in the ring of formal power series $\mathbb C[[\hbar]]$. We infer that the Hochschild chain complex of the sheaf of skew group algebras $\mathcal{D}_X\rtimes G$ has a well-defined Euler characteristic which is proportional to the topological Euler characteristic of $X/G$. Finally, we prove an algebraic index theorem.