Lusztig sheaves and integrable highest weight modules

Abstract: We consider the localization $\mathcal{Q}{\mathbf{V},\mathbf{W}}/\mathcal{N}{\mathbf{V}}$ of Lusztig's sheaves for framed quivers, and define functors $E^{(n)}{i},F^{(n)}{i},K^{\pm}_{i},n\in \mathbb{N},i \in I$ between the localizations. With these functors, the Grothendieck group of localizations realizes the irreducible integrable highest weight modules $L(\Lambda)$ of quantum groups. Moreover, the nonzero simple perverse sheaves in localizations form the canonical bases of $L(\Lambda)$. We also compare our realization (at $v \rightarrow 1$) with Nakajima's realization via quiver varieties and prove that the transition matrix between canonical bases and fundamental classes is upper triangular with diagonal entries all equal to $\pm 1$.