Monogamous subvarieties of the nilpotent cone

Simon M. Goodwin, Rachel Pengelly, David I. Stewart and Adam R. Thomas

Vol. 336 (2025), No. 1-2, 161–180

DOI: 10.2140/pjm.2025.336.161

Abstract

Let $G$ be a reductive algebraic group over an algebraically closed field of prime characteristic not $2$, whose Lie algebra is denoted $\mathfrak{𝔤}$. We call a subvariety $\mathfrak{𝔛}$ of the nilpotent cone $\mathcal{𝒩}\subset \mathfrak{𝔤}$ monogamous if for every $e\in \mathfrak{𝔛}$, the ${\mathfrak{𝔰}\mathfrak{𝔩}}_2$-triples $(e,h,f)$ with $f\in \mathfrak{𝔛}$ are conjugate under the centraliser $C_G(e)$. Building on work by the first two authors, we show there is a unique maximal closed $G$-stable monogamous subvariety $\mathcal{𝒱}\subset \mathcal{𝒩}$ and that it is an orbit closure, hence irreducible. We show that $\mathcal{𝒱}$ can also be characterised in terms of Serre’s $G$-complete reducibility.

In memory of Gary, who influenced us greatly

Keywords

Lie algebras, positive characteristic, Jacobson–Morozov, complete reducibility, $\mathfrak{sl}_{2}$, nilpotent elements, exceptional groups

Mathematical Subject Classification

Primary: 17B08, 17B45

Milestones

Received: 1 July 2024

Revised: 11 September 2024

Accepted: 13 September 2024

Published: 26 May 2025