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

Abstract

Abstract We show how a novel construction of the sheaf of Cherednik algebras $\mathscr {H}_{1, c, X, G}$ on a quotient orbifold $Y:=X/G$ in author’s prior work leads to results for $\mathscr {H}_{1, c, X, G}$, 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 $\mathscr {H}_{1, c, X, G}$, which generalizes the standard Engeli–Felder’s trace density construction for the sheaf of differential operators $\mathscr {D}_X$. Second, by means of the newly obtained trace density maps, we prove an isomorphism in the derived category of complexes of $\mathbb {C}_{Y}\llbracket \hbar \rrbracket $-modules, which computes the hypercohomology of the Hochschild chain complex of the sheaf of formal Cherednik algebras $\mathscr {H}_{1, \hbar , X, G}$. We show that this hypercohomology is isomorphic to the Chen–Ruan cohomology of the orbifold $Y$ with values in the ring of formal power series $\mathbb {C}\llbracket \hbar \rrbracket $. We infer that the Hochschild chain complex of the sheaf of skew group algebras $\mathscr {H}_{1, 0, X, G}$ has a well-defined Euler characteristic that is equal to the orbifold Euler characteristic of $Y$. Finally, we prove an algebraic index theorem.