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

Abstract

A. Etingof introduces a sheaf of Cherednik algebras ℋ 1,c,X,G , attached to a complex algebraic variety X with an action by a finite group G, by means of generators and relations. In a previous paper the author showed how to realize the sheaf of Cherednik algebras on a general global quotient orbifold X/G by gluing sheaves of flat sections of flat holomorphic vector bundles on orbit type strata in X. In the current note we use this realization of the sheaf of global Cherednik algebras to define trace density maps for ℋ 1,c,X,G attached to the various orbit type strata. The maps are not entirely explicit. By means of the trace density morphisms we obtain a morphism between the hypercohomology of the sheaf of Hochschild chain complexes of ℋ 1,c,X,G for a Coxeter group G and the Chen-Ruan orbifold cohomology of X/G. We subsequently show that for any complex reflection group G, when c is a formal parameter, the hyercohomology of the sheaf of Hochschild chain complexes of ℋ 1,c,X,G is isomorphic to the Chen-Ruan cohomology of X/G with values in the Laurent field. Finally, we present an algebraic index theorem for the sheaf of formal global Cherendik algebras ℋ 1,(( )),X,G localized at . C H,(( )) n−l,l 14 4.2. Construction of the trace density maps 15 5. Algebraic index theorem 20 Acknowledgemnts 24 References 24