Hesselink strata in small characteristic and Lusztig-Xue pieces

2024/07/31 by Alexander Premet, Premet, Alexander
Mathematics · #14L30 #17B45 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2407.21469

openalex publication_date 2024/07/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a simple algebraic group over an algebraically closed field of characteristic p≥ 0 and \mathfrakg=\rm Lie(G). We show that the nilpotent pieces \rm LX(Δ) introduced by Lusztig coincide with the corresponding Hesselink strata H(Δ) and hence form a partition of the nilpotent cone of \mathfrakg. Similar results are obtained for the unipotent pieces of G. Here Δ runs through the set of all weighted Dynkin diagrams of G. Thanks to the results obtained by Lusztig, Xue and Voggesberger this boils down to establishing the partition property of the pieces \rm LX(Δ) for groups of type \rm E₇ in characteristic 2 and for groups of type \rm E₈ in characteristic 2 and 3.

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