Computing nilpotent and unipotent canonical forms: a symmetric approach

TL;DR

This paper introduces a symmetric, canonical form for nilpotent and unipotent orbits in classical Lie algebras and groups, using algebraic and geometric methods, applicable across various characteristics and group types.

Contribution

It develops a unified, symmetric approach to compute canonical representatives for nilpotent and unipotent orbits in classical Lie algebras and groups, extending to finite unitary groups.

Findings

Canonical symmetric form for nilpotent orbits with entries in {0,1}

Method to adapt forms for symmetric or skew-symmetric bilinear forms

Complete set of representatives for unipotent classes in finite unitary groups

Abstract

Let $k$ be an algebraically closed field of any characteristic except 2, and let $G = \GL_n(k)$ be the general linear group, regarded as an algebraic group over $k$. Using an algebro-geometric argument and Dynkin-Kostant theory for $G$ we begin by obtaining a canonical form for nilpotent $\Ad(G)$-orbits in $\gl\l_n(k)$ which is symmetric with respect to the non-main diagonal (i.e. it is fixed by the map $f : (x_{i,j})\mapsto (x_{n+1-j,n+1-i})$), with entries in $\{0,1\}$. We then show how to modify this form slightly in order to satisfy a non-degenerate symmetric or skew-symmetric bilinear form, assuming that the orbit does not vanish in the presence of such a form. Replacing $G$ by any simple classical algebraic group we thus obtain a unified approach to computing representatives for nilpotent orbits of all classical Lie algebras. By applying Springer morphisms, this also yields representatives for the corresponding unipotent classes in $G$. As a corollary we obtain a complete set of generic canonical representatives for the unipotent classes in finite general unitary groups $\GU_n(\F_q)$ for all prime powers $q$.