Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Monogamous subvarieties of the nilpotent cone

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

TL;DR

This work extends Kostant-type uniqueness of $\sl_2$-triples to nilpotent cones in small characteristic by introducing and analyzing monogamous subvarieties. It proves the existence of a unique maximal closed $G$-stable monogamous subvariety $\mathcal V \subset \mathcal N$, which is irreducible and equal to an orbit closure, and it provides a characterization of $\mathcal V$ via Serre's $G$-complete reducibility, notably through an $A_1$-$G$-cr criterion. The authors treat both bad and good characteristics, employing case-by-case (and computational) arguments in bad characteristic and cocharacter-based, Kostant-like arguments in good characteristic to establish monogamy and maximality, including exceptional types. Consequently, they obtain a unified statement: $\mathcal V$ is the unique maximal closed $G$-stable monogamous subvariety of the nilpotent cone and is governed by an $A_1$-$G$-cr framework, linking orbit-closure geometry with $G$-complete reducibility. The results pave the way for a deeper understanding of the nilpotent cone's geometry in all types and connect to broader questions about rigidity and reducibility in modular representation theory.

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.

Monogamous subvarieties of the nilpotent cone

TL;DR

This work extends Kostant-type uniqueness of -triples to nilpotent cones in small characteristic by introducing and analyzing monogamous subvarieties. It proves the existence of a unique maximal closed -stable monogamous subvariety , which is irreducible and equal to an orbit closure, and it provides a characterization of via Serre's -complete reducibility, notably through an --cr criterion. The authors treat both bad and good characteristics, employing case-by-case (and computational) arguments in bad characteristic and cocharacter-based, Kostant-like arguments in good characteristic to establish monogamy and maximality, including exceptional types. Consequently, they obtain a unified statement: is the unique maximal closed -stable monogamous subvariety of the nilpotent cone and is governed by an --cr framework, linking orbit-closure geometry with -complete reducibility. The results pave the way for a deeper understanding of the nilpotent cone's geometry in all types and connect to broader questions about rigidity and reducibility in modular representation theory.

Abstract

Let be a reductive algebraic group over an algebraically closed field of prime characteristic not , whose Lie algebra is denoted . We call a subvariety of the nilpotent cone monogamous if for every , the -triples with are conjugate under the centraliser . Building on work by the first two authors, we show there is a unique maximal closed -stable monogamous subvariety and that it is an orbit closure, hence irreducible. We show that can also be characterised in terms of Serre's -complete reducibility.
Paper Structure (8 sections, 15 theorems, 17 equations, 1 figure, 3 tables)

This paper contains 8 sections, 15 theorems, 17 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nilpotent orbits and Hasse diagrams
  4. G-cr subalgebras
  5. Monogamy of V
  6. Bad characteristic
  7. Good characteristic
  8. Proof of Theorems \ref{['thm:mainKos']} and \ref{['thm:mainGcr']}

Key Result

Theorem 1.1

Let $G$ be a simple algebraic group over an algebraically closed field $\Bbbk$ of characteristic $p > 2$. Then $\mathcal{V}$ is the unique maximal $G$-stable closed monogamous subvariety of $\mathcal{N}$. Furthermore, $\mathcal{V}$ is irreducible, being the closure of a single orbit as specified in

Figures (1)

  • Figure 1: Full Hasse diagram for $G_2$ when $p=3$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • Lemma 2.4
  • Proposition 2.5
  • ...and 22 more