Tame quivers and affine bases I: A Hall algebra approach to the canonical bases

Abstract

In [17], Lusztig gave an explicit construction of the affine canonical basis by simple perverse sheaves. In this paper, we construct a bar-invariant basis by using a PBW basis arising from representations of the corresponding tame quiver. We prove that this bar-invariant basis coincides with Lusztig's canonical basis and obtains a concrete bijection between the elements in theses two bases. The index set of these bases is listed orderly by modules in preprojective, regular non-homogeneous, preinjective components and irreducible characters of symmetric groups. Our results are based on the work of Lin-Xiao-Zhang ([14]) and closely related with the work of Beck-Nakajima ([2]). A crucial method in our construction is a generalization of that in [8].