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 F-Hermitian spaces, where F = C or H. More precisely, we classify reversible and strongly reversible elements in the groups Sp(n)Hn, U(n)Cn and SU(n)Cn. We also give a new proof of the classification of strongly reversible elements in Sp(n).