Real adjoint orbits of special linear groups (Q6633897)

Let \(G\) be a linear Lie group with Lie algebra \(\mathfrak{g}\). An element \(X \in \mathfrak{g}\) is called \(\mathrm{Ad}_{G}\)-real if \[-X=g^{-1}Xg\] for some \(g \in G\). Moreover, if there is some involution \(\tau \in G\) such that \(-X=\tau^{-1}X\tau\), then \(X\) is called strongly \(\mathrm{Ad}_{G}\)-real. If \(X \in \mathfrak{g}\) is \(\mathrm{Ad}_{G}\)-real (strongly real), then \(\mathrm{exp}\,X\) is real (strongly real) in \(G\).\N\NIn the paper under review, the authors classify the \(\mathrm{Ad}_{G}\)-real and strongly \(\mathrm{Ad}_{G}\)-real orbits in the special linear Lie algebra \(\mathfrak{sl}(n,k)\) for \(k=\mathbb{C}\) and \(k=\mathbb{H}\).