The action of component groups on irreducible components of Springer fibers

2024/09/06 by Duy K. Hoang, Hoang, Do Kien
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2409.04076

openalex publication_date 2024/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

Let G be a simple Lie group. Consider a nilpotent element e∈ \mathfrakg. Let Z_G(e) be the centralizer of e in G, and let A_e:= Z_G(e)/Z_G(e)^o be its component group. Write Irr(B_e) for the set of irreducible components of the Springer fiber B_e. We have an action of A_e on Irr(B_e). When \mathfrakg is exceptional, we give an explicit description of Irr(B_e) as an A_e-set. For \mathfrakg of classical type, we describe the stabilizers for the A_e-action. With this description, we prove a conjecture of Lusztig and Sommers. These results suggest relations (first proposed by Lusztig) between Springer fibers and cells in Weyl groups.

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