Lusztig sheaves and integrable highest weight modules in symmetrizable cases

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 $L(λ)$ and the tensor product of integrable highest weights $\mathbf{U}-$module $L(λ_1)\otimes L(λ_2)$, and modulo the traceless ones Lusztig sheaves provide the (signed) canonical basis of $L(λ)$ and $L(λ_1)\otimes L(λ_2)$. As an application, the symmetrizable crystal structures on Nakajima's quiver/tensor product varieties and Lusztig's nilpotent varieties of preprojective algebras are deduced.