Strongly real adjoint orbits of complex symplectic Lie group

Abstract

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

9 pages, Final version, To appear in Linear Algebra and its Applications