Strongly real adjoint orbits of complex symplectic Lie group

TL;DR

This paper studies the properties of elements in the complex symplectic Lie algebra that are related to their negatives via symplectic group actions, introducing classifications for strongly real elements and skew-Hamiltonian matrices.

Contribution

It proves the existence of skew-involutions relating elements to their negatives and classifies strongly real elements and related matrices in the symplectic Lie algebra.

Findings

Existence of skew-involutions for all elements in the Lie algebra.

Classification of strongly real elements in the symplectic Lie algebra.

Characterization of skew-Hamiltonian matrices similar to their negatives.

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.