Reversibility and real adjoint orbits of linear maps

Abstract

We extend the classical results on the classification of the reversible elements of the group $\mathrm{GL}(n,\mathbb{C})$ (and $\mathrm{GL}(n,\mathbb{R})$) to $\mathrm{GL}(n,\mathbb{H})$ using an infinitesimal version of the classical reversibility, namely adjoint reality in the Lie algebra set-up. We also provide a new proof of such a classification for the general linear groups over $\mathbb{R}$ and $\mathbb{C}$. Further, we classify the real adjoint orbits in the Lie algebra $\mathfrak{gl}(n,\mathbb{D})$ for $\mathbb{D}=\mathbb{R},\mathbb{C}$ or $\mathbb{H}$.