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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?
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What context are you working in? you mention that you have uncountably many nilpotent orbits - sounds rather unfamiliar, and so the usual questions people ask about the nilpotent cone might not be relevant.. – David Ben-Zvi Nov 30 '09 at 18:45
yes uncountably many nilpotent orbits would be a bit strange in the strict sense of the word "nilpotent orbit" as orbits on the adjoint representations but i'm working over nullcones for representations other than the adjoint representation (direct sums of the standard module for the reductive group and the adjoint representation, for instance), and in that case there are uncountably many orbits in the nullcone. But one of the things I've done to make it "finite" is group together the orbits into sheets/strata (in the sense of Hesselink/Mumford), making the set of sheets finite. – Vinoth Nov 30 '09 at 23:19
great, thanks! are nullcones defined in terms of rings of invariant polynomials? – David Ben-Zvi Nov 30 '09 at 23:27
the nilcone is the subvariety where the positive degree invariant polynomials on the vector space vanish. – Ben Webster Dec 1 '09 at 0:38
up vote 1 down vote accepted

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..)

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I'll just note: as a general rule, it's a very bad idea to take cohomology of anything in the Zariski topology.

Also, nilcones are always contractible (in both the Zariski and classical topology), so you won't get anywhere with any theory that's homotopy invariant (intersection cohomology isn't).

I'm not sure what else to tell you; I'd say that intersection cohomology is by far the most obvious answer to your question, so I'm not sure why you're so eager to rule it out (unless it's already known).

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The stalks of intersection cohomology sheaves for unipotent orbit closures with coefficients in any equivariant simple local system have been computed (at least in the sense that there is an algorithm for their computation) by Lusztig towards the end of the character sheaves papers. – Kevin McGerty Dec 1 '09 at 3:40
But you're still talking about the adjoint representation, right? The question was about the nilcone of the an arbitrary representation; is that buried somewhere in Lusztig's work? – Ben Webster Dec 1 '09 at 4:10
ahh yes sorry I wasn't ruling out intersection cohomology, that's the first one I'm going to compute I was just asking for other cohomologies that might be interesting to do, in addition. As I understand, nothing much has been done of the specific nullcone I'm doing (according to some experts at my uni), so intersection cohomology is still interesting to do in my case. – Vinoth Dec 1 '09 at 4:28

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