Lusztig sheaves and integrable highest weight modules in symmetrizable case

Abstract: The present paper continues the work of [2]. 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. Its Grothendieck group gives a realization of the integrable highest weight $\mathbf{U}-$module $\Lambda_{\lambda}$, and modulo the traceless ones Lusztig sheaves provide the (signed) canonical basis of $\Lambda_{\lambda}$. As an application, the symmetrizable crystal structures on Nakajima's quiver varieties and Lusztig's nilpotent varieties of preprojective algebras are deduced.