Reversibility of Hermitian Isometries

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 $g^{-1}=hg{h}^{-1}$. 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 $ $\mathrm{S}\mathrm{p}$(n) \ltimes $\mathbb{H}$^n$,$$\mathrm{U}$(n) \ltimes $\mathbb{C}$^n$and$$\mathrm{S}\mathrm{U}$(n) \ltimes $\mathbb{C}$^n$. We also give a new proof of the classification of strongly reversible elements in $\mathrm{S}\mathrm{p}(n)$.