  We compute the space of Poisson traces on a classical W-algebra modulo an
arbitrary central character, i.e., linear functionals on such an algebra
invariant under Hamiltonian derivations. This space identifies with the top
cohomology of the corresponding Springer fiber. As a consequence, we deduce
that the zeroth Hochschild homology of the corresponding quantum W-algebra
modulo a central character identifies with the top cohomology of the
corresponding Springer fiber. This implies that the number of irreducible
finite-dimensional representations of this algebra is bounded by the dimension
of this top cohomology, which was established earlier by C. Dodd using
reduction to positive characteristic. Finally, we prove that the entire
cohomology of the Springer fiber identifies with the so-called Poisson-de Rham
homology (defined previously by the authors) of the classical W-algebra modulo
a central character.
