rt.representation theory - Springer corresponding for nullcones other than the standard nilpotent cone

A couple of things: Springer theory does not give a bijection between orbits of the nilpotent cone and irreducible representations of the Weyl group outside of type A: in general, there is an injective map from the irreducible representations of the Weyl group to the set of equivariant irreducible local systems on nilpotent orbits (but it is not necessarily surjective).

For the action on other representations, there is a paper by Misha Grinberg called "A generalization of Springer theory using nearby cycles" which generalizes Springer theory to a class of polar representations V, which behave sufficiently like case of the adjoint representation. From this point of view, you do not need the resolutions.

In fact in a sense his results show that sometimes there cannot be a resolution: he proves that the Fourier transform of the nearby cycles sheaf is an intersection cohomology sheaf, and this is the analogue of the sheaf you get from the Grothendieck resolution in the ordinary Springer theory. However, in the case of symmetric spaces, he also computes the monodromy action on the local system determining the intersection cohomology sheaf, and shows that it does not have to be semisimple. A consequence of this is that the local system cannot arise from a finite cover, and thus does not come from some sort of resolution. A similar phenomenon was notice by Grojnowski in his thesis on character sheaves on symmetric spaces.