Monogamous subvarieties of the nilpotent cone

TL;DR

This paper characterizes a unique maximal irreducible subvariety within the nilpotent cone of a reductive algebraic group, defined by a conjugacy condition on $ ext{sl}_2$-triples, linking it to $G$-complete reducibility.

Contribution

It establishes the existence and uniqueness of a maximal monogamous subvariety in the nilpotent cone and characterizes it via orbit closures and Serre's $G$-complete reducibility.

Findings

The maximal monogamous subvariety is an orbit closure.

It is unique and irreducible.

Characterization via $G$-complete reducibility.

Abstract

Let $G$ be a reductive algebraic group over an algebraically closed field $k$ of prime characteristic not $2$, whose Lie algebra is denoted $\mathfrak{g}$. We call a subvariety $\mathfrak{X}$ of the nilpotent cone $N \subset \mathfrak{g}$ monogamous if for every $e\in \mathfrak{X}$, the $\mathfrak{sl}_2$-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 $V \subset N$ and that it is an orbit closure, hence irreducible. We show that $V$ can also be characterised in terms of Serre's $G$-complete reducibility.