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? 



Springer corresponding for nullcones other than the standard nilpotent cone
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.


