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 \(g^{-1}=hgh^{-1}\). 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 \(\mathrm{Sp}(n)\).