Quantization, Orbifold Cohomology, and Cherednik Algebras

In this note, given an associative algebra A over C, we compute the Hochschild homology of the crossed product C[Sn]⋉A . If A is simple, this homology coincides with the homology of SA. If A satisfies the “Gorenstein” properties of [VB1, VB2], then this computation allows us to compute the Hochschild cohomology of C[Sn] ⋉ A , and of SA for simple A. In particular, we obtain a result conjectured (in a much stronger form) by Ginzburg and Kaledin ([GK], (1.3)): if X is an affine symplectic algebraic variety over C, A+ is a deformation quantization of X , and A = A+[~ ], then the Hochschild cohomology of the algebra SA (which is a quantization of the singular Poisson variety SX = X/Sn) is additively isomorphic to the Chen-Ruan orbifold (=stringy) cohomology of SX with coefficients in C((~)). 1 If X is a surface, this cohomology is isomorphic to the cohomology of the Hilbert scheme Hilbn(X) (the Göttsche formula, [Go]). As a corollary, we get that if H(X,C) = 0 then for n > 1, dimHH(SA) = dimHH(A) if dimX > 2, and dimHH(SA) = dimHH(A) + 1 if dimX = 2 (i.e., X is a surface). This implies that if dimX > 2 then all deformations of SA come from deformations of A; on the other hand, if X is a surface, then there is an additional parameter of deformations of SA, which does not come from deformations of A. In this case, the universal deformation of SA is a very interesting algebra. For example, if X = C, it is the spherical subalgebra of the rational Cherednik algebra attached to the group Sn and its standard representation C (see e.g. [EG]). In general, this deformation exists only over formal series, but we expect that in “good” cases (when X has a compactification to which the Poisson bracket extends by zero at infinity, see [Ko2],[Ar]), this deformation exists “nonperturbatively”. We note that many of the results below are apparently known to experts; we present them with proofs since we could not find an exposition of them in the literature. Acknowledgments. The work of the first author was supported by the NSF grant DMS-9988796. The authors thank M. Artin, Yu. Berest, P. Bressler, V. Dolgushev, V. Ginzburg, L. Hesselholt, D. Kaledin, and M. Kontsevich for useful discussions.