Lusztig sheaves and integrable highest weight modules

TL;DR

This paper constructs a geometric realization of integrable highest weight modules of quantum groups using Lusztig sheaves on framed quivers, connecting canonical bases with perverse sheaves and comparing with Nakajima's approach.

Contribution

It introduces a new localization framework for Lusztig's sheaves, defining functors that realize quantum group modules and their canonical bases, and relates this to Nakajima's quiver variety realization.

Findings

Grothendieck group of localizations realizes irreducible integrable modules

Nonzero simple perverse sheaves form canonical bases

Transition matrix between bases is upper triangular with ±1 diagonal entries

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 $\pm 1$.