Zeroth Poisson homology of symmetric powers of isolated quasihomogeneous surface singularities

Let X be a surface with an isolated singularity at the origin, given by the equation Q(x,y,z)=0, where Q is a weighted-homogeneous polynomial. In particular, this includes the Kleinian surfaces X = C^2/G for G < SL(2,C) finite. Let Y be the n-th symmetric power of X. We compute the zeroth Poisson homology of Y, as a graded vector space with respect to the weight grading. In the Kleinian case, this confirms a conjecture of Alev, that the zeroth Poisson homology of the n-th symmetric power of C^2/G is isomorphic to the zeroth Hochschild homology of the n-th symmetric power of the algebra of G-invariant differential operators on C. That is, the Brylinski spectral sequence degenerates in this case. In the elliptic case, this yields the zeroth Hochschild homology of symmetric powers of the elliptic algebras with three generators modulo their center, for the parameter equal to all but countably many points of the elliptic curve.