Reversibility and Real Adjoint Orbits of Linear Maps

Abstract

We extend classical results on the classification of 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} $.