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Lusztig sheaves and integrable highest weight modules

Jiepeng Fang, Yixin Lan, Jie Xiao

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 $L(Λ)$ of quantum groups. Moreover, the nonzero simple perverse sheaves in localizations form the canonical bases of $L(Λ)$. 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 $\pm 1$.

Lusztig sheaves and integrable highest weight modules

Abstract

We consider the localization of Lusztig's sheaves for framed quivers, and define functors between the localizations. With these functors, the Grothendieck group of localizations realizes the irreducible integrable highest weight modules of quantum groups. Moreover, the nonzero simple perverse sheaves in localizations form the canonical bases of . We also compare our realization (at ) 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 .
Paper Structure (16 sections, 51 theorems, 262 equations)

This paper contains 16 sections, 51 theorems, 262 equations.

Table of Contents

  1. Introduction
  2. Lusztig sheaves and quantum groups
  3. Induction functor and restriction functor
  4. Fourier-Deligne transform
  5. Analysis at sink
  6. Analysis at source
  7. Realization of the integrable highest weight modules
  8. Localizations
  9. The functors of localizations at $i$
  10. Commutative relations in localizations at $i$
  11. The integrable highest weight modules
  12. Compare the functor $E_{i}$ with derivation functors
  13. Compare with Nakajima's realization
  14. Nakajima quiver variety
  15. The left graphs and the isomorphism $\Phi^{\Lambda}$
  16. ...and 1 more sections

Key Result

Theorem 1.1

With the action of linear operators induced by functors $E^{(n)}_{i},F^{(n)}_{i},K^{\pm}_{i}$ for $n\in \mathbb{N},i \in I$, the Grothendieck group $\mathcal{K}_{0}(\Lambda)$ of $\coprod\limits_{\mathbf{V}}\mathcal{Q}_{\mathbf{V},\mathbf{W}}/\mathcal{N}_{\mathbf{V}}$ becomes a $_{\mathcal{A}}\mathbf The morphism $\varsigma^{\Lambda}$ sends the image of constant sheaf $[\overline{\mathbb{Q}}_{l}]$

Theorems & Definitions (103)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Corollary 2.5
  • Corollary 2.6
  • Remark 2.7
  • ...and 93 more