  We prove a conjecture which expresses the bigraded Poisson-de Rham homology
of the nilpotent cone of a semisimple Lie algebra in terms of the generalized
(one-variable) Kostka polynomials, via a formula suggested by Lusztig. This
allows us to construct a canonical family of filtrations on the flag variety
cohomology, and hence on irreducible representations of the Weyl group, whose
Hilbert series are given by the generalized Kostka polynomials. We deduce
consequences for the cohomology of all Springer fibers. In particular, this
computes the grading on the zeroth Poisson homology of all classical finite
W-algebras, as well as the filtration on the zeroth Hochschild homology of all
quantum finite W-algebras, and we generalize to all homology degrees. As a
consequence, we deduce a conjecture of Proudfoot on symplectic duality,
relating in type A the Poisson homology of Slodowy slices to the intersection
cohomology of nilpotent orbit closures. In the last section, we give an
analogue of our main theorem in the setting of mirabolic D-modules.
