Lusztig sheaves and integrable highest weight modules in symmetrizable cases

TL;DR

This paper extends Lusztig's geometric approach to realize integrable highest weight modules and their tensor products for symmetrizable quantum groups using sheaves on quivers with automorphisms.

Contribution

It constructs a category of Lusztig sheaves for symmetrizable cases and relates it to modules and bases, advancing geometric representation theory.

Findings

Realization of highest weight modules via Lusztig sheaves

Construction of canonical bases for modules and tensor products

Deductions of crystal structures on quiver varieties

Abstract

The present paper continues the work of [10] and [6]. For any symmetrizable generalized Cartan Matrix $C$ and the corresponding quantum group $\mathbf{U}$, we consider the associated quiver $Q$ with an admissible automorphism $a$. We construct the category $\widetilde{\mathcal{Q}/\mathcal{N}}$ of the localization of Lusztig sheaves for the quiver with the automorphism of corresponding framed quiver and 2-framed quiver. Their Grothendieck groups give realizations of integrable highest weight module $L(\lambda)$ and the tensor product of integrable highest weights $\mathbf{U}-$module $L(\lambda_1)\otimes L(\lambda_2)$, and modulo the traceless ones Lusztig sheaves provide the (signed) canonical basis of $L(\lambda)$ and $L(\lambda_1)\otimes L(\lambda_2)$. As an application, the symmetrizable crystal structures on Nakajima's quiver/tensor product varieties and Lusztig's nilpotent varieties of preprojective algebras are deduced.