Lusztig sheaves and integrable highest weight modules

2023/07/30 by Fang, Jiepeng, Lan, Yixin, Xiao, Jie · 2 citations
#16G20 #17B37 #Algebraic Geometry (math.AG) #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2307.16131

Abstract

We consider the localization Q_V,W/N_V of Lusztig's sheaves for framed quivers, and define functors E⁽ⁿ⁾_i,F⁽ⁿ⁾_i,K^±_i,n∈ ℕ,i ∈ 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 → 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 ± 1.

Cited by

• Notes on Chevalley Groups and Root Category II: Compact Lie Groups and Representations
• Positivity properties of canonical bases

Related

• Lusztig sheaves and integrable highest weight modules in the symmetrizable case
• Lusztig sheaves and tensor products of integrable highest weight modules
• Lusztig sheaves, characteristic cycles and the Borel-Moore homology of Nakajima's quiver varieties
• Lusztig sheaves, decomposition rule and restriction rule
• Localization of \fraku-modules. II. Configuration spaces and quantum groups