We begin by recalling a beautiful result of Solomon concerning exterior powers of the reflection representation of a Weyl group W and then explain a conjectural generalization of it to Springer representations. The conjecture is known in some cases (Henderson), but a weak version of it (conjecture of Lehrer-Shoji) is always true and this leads to a decomposition of a (singly- graded) generalized Parking Function Module in terms of Springer representations.

The decomposition yields polynomials in q (for certain integral parameters m) attached to each nilpotent orbit of the corresponding Lie algebra and each local system on the orbit. When the orbit takes a certain form, these polynomials are related to the characteristic polynomials of a hyperplane arrangement attached to the orbit and they turn out to have non-negative coefficients. The polynomials are also q-analogues for numbers that show up in the combinatorics of the non-crossing partition lattice of W (in type A they yield q-analogues of the Catalan numbers, the Narayana numbers, and the Kreweras numbers). This leads to a conjecture describing the orbit structure of a natural cyclic group action on subsets of the non-crossing partition lattice (known as cyclic sieving), which is true in classical types. This is joint work with Vic Reiner.