Computing nilpotent and unipotent canonical forms: a symmetric approach

Let $k$ be an algebraically closed field of any characteristic except 2, and let $G={\text{GL}}_n(k)$ 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 $\text{Ad}(G)$-orbits in $\text{gl}{\text{l}}_n(k)$ 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 ${\text{GU}}_n({\text{F}}_q)$ for all prime powers $q$.