Quivers, Perverse Sheaves, and Quantized Enveloping Algebras

Abstract

Let U denote the quantized enveloping algebra associated to asymmetric generalized Cartan matrix (GMC) (a\sb {ij}). Then Uhas a triangular decompositionU=U\sp \otimes U\sp 0\otimes U\sp +. In the case in which(a\sb {ij}) is of type A,D,E, the author has constructed[same journal 3 (1990), no. 2, 447 498; MR 90m:17023] a certainbasis for U\sp (now known as Lusztig's canonical basis) withsome wonderful properties. He even had two different approaches,the elementary method and the geometric method. In this paper heshows how one can extend the geometric method to handle the abovegeneral case.\par Corresponding to the GMC (a\sb {ij}) we havea finite quiver. C. M. Ringel has shown [Invent. Math. 101(1990), no. 3, 583 591; MR\Cite{Ringel90:Hall:583--591}[91i:16024]] (for type A,D,E) howthe algebra U\sp can be constructed from the representationtheory of this quiver. Taking a closer look at the geometry ofsuch representations, the author of the present paper is able toobtain both U\sp and a canonical basis for it. The highlynontrivial generalization from the simply laced to the generalcase is achieved by imitating the definition of character sheaves(cf. the author's fundamental work on this topic, beginning in1985 [Adv. Math. 56 (1985), no. 3, 193 237; MR87b:20055]).\par It is thus fairly sophisticated techniques thatare involved in this geometric method. One advantage over theelementary method is that it gives positivity results for thecanonical basis with respect to comultiplication