Computing the Lusztig--Vogan Bijection

2017/11/01 by David B. Rush, David B Rush, Rush, David B · 2 citations
Mathematics · #05E10 #17B08 #20G05 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:05E10 #msc:17B08 #msc:20G05

paper · pdf · doi:10.48550/arxiv.1711.00148

59 pages

arxiv created 2017/11/01 · openalex publication_date 2017/11/01 · arxiv updated 2017/11/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a connected complex reductive algebraic group with Lie algebra \mathfrakg. The Lusztig--Vogan bijection relates two bases for the bounded derived category of G-equivariant coherent sheaves on the nilpotent cone N of \mathfrakg. One basis is indexed by Λ⁺, the set of dominant weights of G, and the other by Ω, the set of pairs (O, E) consisting of a nilpotent orbit O ⊂ N and an irreducible G-equivariant vector bundle E → O. The existence of the Lusztig--Vogan bijection γ\colon Ω→ Λ⁺ was proven by Bezrukavnikov, and an algorithm computing γ in type A was given by Achar. Herein we present a combinatorial description of γ in type A that subsumes and dramatically simplifies Achar's algorithm.

Citations

• On the equivariant K-theory of the nilpotent cone

Cited by

• Iterated Lusztig-Vogan Bijection and Distinguished Weights
• Asymptotic Hecke algebras and Lusztig-Vogan bijection via affine matrix-ball construction

Related

• Integral exotic sheaves and the modular Lusztig-Vogan bijection
• On the equivariant K-theory of the nilpotent cone
• A Langlands dual realization of coherent sheaves on the nilpotent cone
• Reductive groups, the loop Grassmannian, and the Springer resolution
• A categorification of the Lusztig--Vogan module