  We compute the space of Poisson traces on symmetric powers of affine
symplectic varieties. In the case of symplectic vector spaces, we also consider
the quotient by the diagonal translation action, which includes the quotient
singularities C^{2n-2}/S_n associated to the type A Weyl group S_n and its
reflection representation C^{n-1}. We also compute the full structure of the
natural D-module, previously defined by the authors, whose solution space over
algebraic distributions identifies with the space of Poisson traces. As a
consequence, we deduce bounds on the numbers of finite-dimensional irreducible
representations and prime ideals of quantizations of these varieties. Finally,
motivated by these results, we pose conjectures on symplectic resolutions, and
give related examples of the natural D-module. In an appendix, the second
author computes the Poisson traces and associated D-module for the quotients
C^{2n}/D_n associated to type D Weyl groups. In a second appendix, the same
author provides a direct proof of one of the main theorems.
