Real adjoint orbits of special linear groups

Abstract

Let $G$ be a 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 the strongly ${\mathrm{Ad}}_G$-real orbits in the special linear Lie algebra $\mathfrak{sl}(n,\mathbb{F})$ for $\mathbb{F}=\mathbb{C}$ or $\mathbb{H}$.

Final version, 17 pages, To appear in Illinois Journal of Mathematics