Real adjoint orbits of special linear groups

Abstract

Let G be a linear Lie group with Lie algebra $\mathfrak{g}$. An element $X\in \mathfrak{g}$ is called ${\mathrm{Ad}}_G$-real if $-X=gX{g}^{-1}$ for some $g\in G$. Moreover, if $-X=gX{g}^{-1}$ holds for some involution $g\in G$, then X is called strongly ${\mathrm{Ad}}_G$-real. We have classified the ${\mathrm{Ad}}_G$-real and strongly ${\mathrm{Ad}}_G$-real orbits in the special linear Lie algebra $\mathfrak{s}\mathfrak{l}(n,\mathbb{F})$ for $\mathbb{F}=\mathbb{C}$ or $\mathbb{H}$.