Reversibility of Hermitian Isometries

Abstract

An element $g$ in a group $G$ is called reversible (or real) if it is conjugate to $g^{-1}$ in $G$, i.e., there exists $h$ in $G$ such that $g^{-1}=hgh^{-1}$. The element $g$ is called strongly reversible if the conjugating element $h$ is an involution (i.e., element of order at most two) in $G$. In this paper, we classify reversible and strongly reversible elements in the isometry groups of $\mathbb{F}$-Hermitian spaces, where $\mathbb{F}=\mathbb{C}$ or $\mathbb{H}$. More precisely, we classify reversible and strongly reversible elements in the groups $ \mathrm{Sp}(n) \ltimes \mathbb{H}^n$, $\mathrm{U}(n) \ltimes \mathbb{C}^n$ and $\mathrm{SU}(n) \ltimes \mathbb{C}^n$. We also give a new proof of the classification of strongly reversible elements in $\mathrm{Sp}(n)$.