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arXiv · 2307.16131

Lusztig sheaves and integrable highest weight modules

Jiepeng Fang·Yixin Lan·Jie Xiao
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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 $\pm 1$.

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BibTeXRIS

Jiepeng Fang, Yixin Lan, Jie Xiao. 2023-07-30. Lusztig sheaves and integrable highest weight modules. https://arxiv.org/abs/2307.16131

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