Strongly real adjoint orbits of complex symplectic Lie group

Abstract

We consider the adjoint action of the symplectic Lie group $\mathrm{Sp}(2n,C)$ on its Lie algebra $\mathrm{sp}(2n,C)$. An element $X\in \mathrm{sp}(2n,C)$ is called ${\mathrm{Ad}}_{\mathrm{Sp}(2n,C)}$-real if $-X=\mathrm{Ad}\left(g\right)X$ for some $g\in \mathrm{Sp}(2n,C)$. Moreover, if $-X=\mathrm{Ad}\left(h\right)X$ for some involution $h\in \mathrm{Sp}(2n,C)$, then $X\in \mathrm{sp}(2n,C)$ is called strongly ${\mathrm{Ad}}_{\mathrm{Sp}(2n,C)}$-real. In this paper, we prove that for every element $X\in \mathrm{sp}(2n,C)$, there exists a skew-involution $g\in \mathrm{Sp}(2n,C)$ such that $-X=\mathrm{Ad}\left(g\right)X$. Furthermore, we classify the strongly ${\mathrm{Ad}}_{\mathrm{Sp}(2n,C)}$-real elements in $\mathrm{sp}(2n,C)$. We also classify skew-Hamiltonian matrices that are similar to their negatives via a symplectic involution.