Hesselink strata in small characteristic and Lusztig–Xue pieces

Alexander Premet

Vol. 336 (2025), No. 1-2, 415–432

DOI: 10.2140/pjm.2025.336.415

Abstract

Let $G$ be a connected reductive algebraic group over an algebraically closed field of characteristic $p\ge 0$ and $\mathfrak{𝔤}=\mathrm{Lie}<!--FUNCTION\; APPLICATION-->(G)$. We show that the nilpotent pieces $LX(\mathrm{\Delta })$ introduced by Lusztig form a partition of the nilpotent cone of $\mathfrak{𝔤}$ and hence coincide with the Hesselink strata $\mathcal{\mathscr{H}}(\mathrm{\Delta })$ where $\mathrm{\Delta }$ runs through the set of all weighted Dynkin diagrams of $G$. Thanks to earlier results obtained by Lusztig, Xue and Voggesberger this boils down to describing the pieces $LX(\mathrm{\Delta })$ for groups of type $E_7$ in characteristic $2$ and for groups of type $E_8$ in characteristic $2$ and $3$. Our arguments are computer-free, but rely very heavily on the results of Liebeck and Seitz (2012).

In memory of Gary Seitz

Keywords

simple algebraic groups, nilpotent orbits, Hesselink strata

Mathematical Subject Classification

Primary: 14L30, 17B45

Secondary: 14L15, 20G41

Milestones

Received: 30 July 2024

Revised: 30 November 2024

Accepted: 30 November 2024

Published: 26 May 2025