I understand the ordinary Springer correspondence gives a bijection between orbits in the nilpotent cone for the adjoint representation and irreducible representations of the Weyl group, through action of the Weyl group on the top intersection cohomology. (I'm still learning about this, I know a very vague picture). My question is: does the same method give a correspondence (not injective nor surjective) between orbits in the nullcone of any representation (not the adjoint representation), and representations (not necessarily irreducible) of the Weyl group? Or is the construction so specific that it breaks down entirely when we replace the nilpotent cone by more generally, a nullcone, and we cannot get any Weyl group representations out of it?
Let me make another guess as to what you are seeing: The Lie/algebraic group acts on the coordinate ring for the closure of the nilpotent orbit (since it acts on the nilpotent orbit). Take the weight 0 weight space of this coordinate ring; the Weyl group (realized as N(T)/T) acts on it since T acts trivially on this weight space. In the case of SL_n, this gives the total cohomology ring of the Springer fiber associated to the conjugate partition (if I remember correctly, and I might not, this result is due to DeConcini and Procesi back in the early 80s). I seem to recall there is a way to recover the grading on the cohomology ring, which allows you to recover the top cohomology (and hence the original Springer correspondence), but cannot remember what it is. This isomorphism between the weight 0 subring and the cohomology ring is specific to SL_n. What I just wrote does make sense for any nullcone, though a priori there is no guarantee that you get a finite dimensional representation of the Weyl group. 


Certainly, there's no reason to believe there will be. The Springer correspondence depends very strongly on the existence and structure on the Springer resolution. 


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. 

