# Are the strata of Nakajima quiver varieties simply-connected? Do they have odd cohomology?

Nakajima defined a while back a nice family of varieties, called "quiver varieties" (sometimes with "Nakajima" appended to the front to avoid confusion with other varieties defined in terms of quivers). These are most concisely defined as the moduli of certain representations of certain preprojective algebras.

I'll be interested in the affine (in the sense of "affine variety" not "affine Lie algebra") version of these, which is the moduli space of semi-simple representations of a certain preprojective algebra. This variety is singular, but can be divided into smooth strata which correspond to fixing the size of the automorphism group of the representation (i.e. one stratum for simple representations, one for sums of pairs of non-isomorphic simples, etc.). I would like to know a bit about the geometry of these strata. One basic (and important for me) question is

Are these strata simply connected or equivariantly simply connected for the action of a group?

Actually, I know that they are not simply connected from some very simple examples (such as the nilcone of $\mathfrak{sl}_2$), but in those examples there is an action of a group such that there are no equivariant local systems (if you're willing to think of the quotient as a stack, the quotient is simply connected), so they are "equivariantly simply connected."

Similarly, I'm interested in the cohomology of these strata; I'd like its odd part to vanish. Again, there's no hope of this in the most obvious way. The only way it could happen is equivariantly.

Does the odd (equivariant) cohomology of these strata vanish for any group action?

let me just note in closing: I would be entirely satisfied if these results were only true in the finite type case; I know a lot of geometric statements for quiver varieties go a little sour once you're outside the finite type case.

If anyone knows these or any other results in the literature about the geometry of these strata, I would be very happy to hear them.

As Ben wrote, there are examples of non-simply connected strata.

For a quiver variety corresponding to a fundamental representation, it, a priori, only has a small group action. So it is usually far away from homogeneous.

So I suspect the answer is NO, though I do not know a concrete example.

But, I think the question itself is not right:

It is true that perverse sheaves appearing in the push-forward from natural resolutions' are IC sheaves associated with constant sheaves. But it is not because stratum are simply-connected, or equivariantly simply connected. It is from a very different reason:

This reason cannot be seen if one treat an individual quiver variety separately. One can see it only when one consider various quiver varieties simultaneously, and relate them to a representation. It is basically because there is a component consisting of a single point, and other components are connected' to it in some sense.

(Each component corresponds only to a weight space, and considering several components simultaneously, one get the structure of a representation. The single point corresponds to the highest weight vector.)

In summary, quiver varieties are not homogeneous in a conventional sense, but have substitutive property (I do not know how to call it) when one treat several components simultaneously.

Not really an answer, but I had a similar problem a while ago (I wanted to know the cohomology ring of certain non-free quotients). My supervisor told me to look at Bredon homology, but I found something different to work on. Might be worth a try!