Lusztig has defined a category of perverse sheaves on the moduli space of representations of a Dynkin quiver (see [his paper](http://www.jstor.org/pss/2939279)) corresponding to canonical basis vectors. 


I'm interested in the stalks of these perverse sheaves, in particular, the stalk at 0. I believe I've reduced a (seemingly unrelated) conjecture to the question



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> Is the stalk of any one of Lusztig's perverse sheaves at 0 1-dimensional? (A weaker claim, which I think would make me roughly as happy is that the Euler characteristic of a stalk is 1.)
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(The conjecture is that it is 1-dimensional, but I'm fairly agnostic on this point. I wouldn't be surprised either way, though if I had to choose I'd say this conjecture sounds a little unlikely).



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> Is there a proof or counterexample of this claim in the literature? 
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After a quick read-through of some the literature (Lusztig's paper, [Reineke's paper](http://arxiv.org/abs/math.AG/0104284) on resolutions, [Kashiwara and Saito's paper](http://arxiv.org/abs/q-alg/9606009) on crystal structures on components, etc.) I'm not feeling any closer to understanding these stalks, and am having trouble finding anywhere else to look. 




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**EDIT**: This seems like an even longer shot, but what would be even better would be to understand the cohomology of one of these sheaves on a "quiver orbital variety,'' the space of quiver representations (not modulo isomorphism!) which preserve a particular flag. Is there anything about this in the literature?



What's known about the stalks of Lusztig's perverse sheaves on quiver varieties?
The dimensions of the stalks of Lusztig's sheaves give the coefficients when a canonical basis element is expanded in a PBW basis. These stalks satisfy a parity vanishing condition.


For these positive results, see Corollary 10.7 of Lusztig's "Canonical Bases Arising from Quantized Enveloping Algebras".


The first example of a 2-dimensional stalk at zero occurs in type A2, in dimension vector (2,2). If X is the variety of 2x2 matrices of rank at most 1, then it is not hard to see (e.g. using the Deligne construction) that IC(X) is one of Lusztig's sheaves and has a 2-dimensional stalk at zero. Alternatively the canonical basis computation can be carried out, which is implemented in GAP via the package QuaGroup.


