Lusztig sheaves and integrable highest weight modules

We consider the localization ${\mathcal{Q}}_{\mathbf{V},\mathbf{W}}\mathrm{/}{\mathcal{N}}_{\mathbf{V}}$ of Lusztig’s sheaves for framed quivers, and define functors $E_i^{(n)},F_i^{(n)},K_i^{\pm },n\in \mathbb{N},i\in I$ between the localizations. With these functors, the Grothendieck group of localizations realizes the irreducible integrable highest weight modules $L(\mathrm{\Lambda })$ of quantum groups. Moreover, the nonzero simple perverse sheaves in localizations form the canonical bases of $L(\mathrm{\Lambda })$. We also compare our realization (at $v\to 1$) 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 $\pm 1$.