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?