As I understand, when we have a nilpotent cone, or a nullcone of a Lie group representation, what seems to be done in a lot of the literature (e.g. Achar&Henderson-"Orbit closures in the enhanced nilpotent cone") is to compute the intersection cohomology sheaves and find polynomials that determine the dimensions of various stalks. 


But what other cohomology theories (that are different to intersection cohomology, I understand sometimes two different cohomology/homology can coincide under special circumstances), would be interesting in nilpotent cones? 


Here's a bit about the problem I'm working on, and some theories that I hope (?) might be interesting, can anyone tell me some more that might be interesting? I am very far from being knowledgeable about cohomology, so if some-one could tell me if the following questions are stupid/trivial/ill-defined or not, please tell me. 


* I have the orbits, which themselves are usually quasi-affine or quasi-projective varieties, which I could compute cohomology of? (perhaps Cech cohomology?)
* The set of orbits inherits a Zariski topology structure from the Zariski topology structure (that coincides with that inherited from the classical topology), perhaps I can compute some (co)homology of this topological space? In my case the set of orbits is uncountably infinite, but I am not completely sure if it has a triangulation - any theory that doesn't involve triangulations?
* As standard, one computes the orbit closures, and instead of doing intersection cohomology of these singular varieties, compute perhaps some of the lower K-groups?
* perhaps Hochschild cohomology of the coordinate rings of some of the affine coordinate rings of these varieties could be interesting?



What cohomology theories would be interesting for nilpotent cones/nullcones?
I think most questions of interest regarding the nilpotent cone have to do with categories of equivariant sheaves on it - either equivariant perverse sheaves, like IC complexes of orbits, or equivariant coherent sheaves, like structure sheaves of orbit closures. So cohomologies that help elucidate the structure of these categories would be great. For example, equivariant IC of orbit closures fits into this, as does ordinary cohomology of orbit closures. Hochschild homology/cohomology of structure sheaves or K-theory likewise control aspects of the category of coherent sheaves, and you could ask for equivariant analogs. 


But I would say history suggests it's best to emphasize two things:


-the full structure of the category (ie what are simples, standards, relations
between them)


-if you have an analog of the Springer resolution for these nullcones, its cohomologies might be even more interesting (or if you'd like, the pushforwards of standard sheaves from there to the nullcone..)


