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.