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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?

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up vote 8 down vote accepted

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.

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Another thing perhaps worth mentioning is that, outside the finite type case, you have to do some real work to find a stratification with respect to which the perverse sheaves constituting the canonical basis are constructible: for affine types, such a stratification is pretty much implicit in Lusztig's Publ IHES paper which you cite, and it uses the classification of representations of tame quivers which he recovers via the McKay correspondence.

In general it is known that the characteristic cycles of the sheaves in the canonical basis lie in a certain Lagrangian variety (this is already established in the paper on quivers and canonical bases). This doesn't give you a stratification however, because they components are conormals to locally closed subvarieties of the moduli space whose union is not the whole space. The same phenomenon happens for character sheaves, though there Lusztig did produce a stratification of the group and show character sheaves have locally constant cohomology on the strata.

You see papers studying this sort of problem in terms of quiver representations at the level of functions on $\mathbb F_q$-points when people try and generalize the "existence of Hall polynomials" outside of finite type quivers and on the quantum group side when people look for "PBW" bases.

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