In "Quivers, perverse sheaves and quantized enveloping algebras," Lusztig defines a category of perverse sheaves on the moduli stack of representations of a quiver. These perverse sheaves are defined as summands of the pushforwards of the constant sheaves on stacks of quiver representations along with a choice of invariant flag (and thus, by definition are supported on the nilpotent locus in the moduli stack). They're mostly of interest since they categorify the canonical basis.
My question is: Is there a stratum in this stack where the pull-back of one of these sheaves is not the trivial local system?
Now, in finite type, this is not a concern, since each stratum is the classifying space of a connected algebraic group, and thus simply connected. But I believe in affine or wild type this is no longer true; this was at least my takeaway from the latter sections of "Affine quivers and canonical bases." However, I got a little confused about the relationship between the results of the two papers mentioned above, since they use quite different formalisms, so I hold out some hope that the local systems associated to symmetric group representations aren't relevant to the perverse sheaves for the canonical basis. Am I just hoping in vain?
Best Answer
For affine quivers, except cyclic ones, there are always perverse sheaves attached to nontrivial local systems.
If you just need an example, I recommend you to read McGerty's paper math/0403279, before Lusztig's paper, where the Kronecker quiver case is studied in detail.
I also wrote a survey paper
Crystal, canonical and PBW bases of quantum affine algebras, in Algebraic Groups and Homogeneous Spaces, Ed. V.B.Mehta, Narosa Publ House. 2007, 389–421.