Representatives for unipotent classes and nilpotent orbits

TL;DR

This paper provides explicit representatives for unipotent classes and nilpotent orbits in simple algebraic groups, facilitating their classification and recognition, especially in exceptional types, using Chevalley bases.

Contribution

It introduces explicit representatives for eminent unipotent and nilpotent classes in simple algebraic groups, aiding classification and recognition tasks.

Findings

Explicit representatives for eminent classes are given.

Recognition theorems for exceptional groups are proved.

Generation of representatives for any class is straightforward.

Abstract

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. The classification of the conjugacy classes of unipotent elements of $G(k)$ and nilpotent orbits of $G$ on $\operatorname{Lie}(G)$ is well-established. One knows there are representatives of every unipotent class as a product of root group elements and every nilpotent orbit as a sum of root elements. We give explicit representatives in terms of a Chevalley basis for the eminent classes. A unipotent (resp. nilpotent) element is said to be eminent if it is not contained in any subsystem subgroup (resp. subalgebra), or a natural generalisation if $G$ is of type $D_n$. From these representatives, it is straightforward to generate representatives for any given class. Along the way we also prove recognition theorems for identifying both the unipotent classes and nilpotent orbits of exceptional algebraic groups.