  We compute the Hochschild homology of the crossed product $\Bbb C[S_n]\ltimes
A^{\otimes n}$ in terms of the Hochschild homology of the associative algebra
$A$ (over $\Bbb C$). It allows us to compute the Hochschild (co)homology of
$\Bbb C[W]\ltimes A^{\otimes n}$ where $A$ is the $q$-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 $S^n A$, $A=A_+[\hbar^{-1}]$ is additively
isomorphic to the Chen-Ruan orbifold cohomology of $S^nX$ with coefficients in
$\Bbb C((\hbar))$. We prove that for $X$ satisfying $H^1(X,\Bbb C)=0$ (or $A\in
VB(d)$) the deformation of $S^nX$ ($\Bbb C[S_n]\ltimes A^{\otimes n}$) 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.
