Lusztig sheaves and integrable highest weight modules in the symmetrizable case

This paper continues the work of \cite{fang2023lusztigsheavesintegrablehighest} and \cite{fang2023lusztigsheavestensorproducts}. For a symmetrizable generalized Cartan matrix $C$ and the corresponding quantum group $\mathbf{U}$, we consider an associated quiver $Q$ equipped with an admissible automorphism $a$. We construct a category $\widetilde{\mathcal{Q}/\mathcal{N}}$ obtained from localizations of Lusztig sheaves for the corresponding framed and $2$-framed quivers with automorphism. The Grothendieck groups of these categories realize the integrable highest weight module $L(\lambda)$ and the tensor product $L(\lambda_1)\otimes L(\lambda_2)$ of integrable highest weight $\mathbf{U}$-modules. After quotienting by traceless objects, Lusztig sheaves yield the signed canonical bases of $L(\lambda)$ and $L(\lambda_1)\otimes L(\lambda_2)$. As applications, we recover symmetrizable crystal structures on Nakajima quiver varieties, Nakajima tensor product varieties, and Lusztig nilpotent varieties of preprojective algebras.