Quantization, orbifold cohomology, and Cherednik algebras

We compute the Hochschild homology of the crossed product $\Bbb C[S_n]\ltimes A^{\otimes n}$intermsoftheHochschildhomologyoftheassociativealgebra$A$(over$\Bbb C$)\mathrm{.}ItallowsustocomputetheHochschild(co)homologyof$\Bbb C[W]\ltimes A^{\otimes n}$where$A$isthe$q$-Weyl algebra or any its degeneration and $W$ is the Weyl group of type $A_{n-1}$ or $B_n$. For a deformation quantization $A_{+}$ of an affine symplectic variety $X$ we show that the Hochschild homology of $S^{n}A$, $A=A_{+}[{\mathrm{\hslash }}^{-1}]$ is additively isomorphic to the Chen-Ruan orbifold cohomology of $S^{n}X$ with coefficients in $\mathbb{C}((\mathrm{\hslash }))$. We prove that for $X$ satisfying $H^1(X,\mathbb{C})=0$ (or $A\in VB(d)$)thedeformationof$S^nX$($\Bbb C[S_n]\ltimes A^{\otimes n}$) which does not come from deformations of $X$ ($A$) exists if and only if $\dim X=2$ ($d=2$). In particular if $A$ is $q$-Weyl algebra (its trigonometric or rational degeneration) then the corresponding nontrivial deformations yield the double affine Hecke algebras of type $A_{n-1}$ (its trigonometric or rational versions) introduced by Cherednik.