On Lusztig Sheaves for Tame Quivers

Let $Q$ be a connected quiver of tame type, and let $U_v^{-}$ be the positive part of the corresponding quantized affine algebra. In Lusztig’s construction of the canonical basis of $U_v^{-}$, the simple perverse sheaves occurring, up to shift, as direct summands of the flag sheaf complexes are given by intersection cohomology complexes $\mathrm{IC}(X(c,m),{\mathcal{L}}_{\chi })$. We study the restrictions of these intersection cohomology complexes to the locally closed subsets $Y(c,m)$ obtained by fixing the preprojective, non-homogeneous regular, and preinjective summands in the category $\mathrm{rep}Q$. We prove pointwise purity on the loci where the homogeneous regular part is a direct sum of pairwise non-isomorphic quasi-simple modules, and establish a semisimple decomposition of the restrictions to the corresponding larger strata. The graded multiplicities of the local systems associated with irreducible characters of symmetric groups determine the transition coefficients from the canonical basis to the PBW basis constructed via quiver representations. This gives a geometric explanation of the positivity of these coefficients and the unitriangularity of the transition matrix.