Real adjoint orbits of the unipotent subgroup

Abstract

Let $G$ be a linear Lie group that acts on it's Lie algebra $\mathfrak{g}$ by the adjoint action: $\mathrm{Ad}(g)X=gX{g}^{-1}$. An element $X\in \mathfrak{g}$ is called ${\mathrm{Ad}}_G$-real if $-X=\mathrm{Ad}(g)X$ for some $g\in G$. An ${\mathrm{Ad}}_G$-real element $X$ is called strongly ${\mathrm{Ad}}_G$-real if $-X=\mathrm{Ad}(\tau )X$ for some involution $\tau \in G$. Let $K=\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$. Let ${\mathrm{U}}_n(K)$ be the group of unipotent upper-triangular matrices over $K$. Let ${\mathfrak{u}}_n(K)$ be the Lie algebra of ${\mathrm{U}}_n(K)$ that consists of $n\times n$ upper triangular matrices with $0$ in all the diagonal entries. In this paper, we consider the $\mathrm{Ad}$-reality of the Lie algebra ${\mathfrak{u}}_n(K)$ that comes from the adjoint action of the Lie group ${\mathrm{U}}_n(K)$ on ${\mathfrak{u}}_n(K)$. We prove that there is no non-trivial ${\mathrm{Ad}}_{{\mathrm{U}}_n(K)}$-real element in ${\mathfrak{u}}_n(K)$. We also consider the adjoint action of the extended group ${\mathrm{U}}_n^{\pm }(K)$ that consists of all upper triangular matrices over $K$ having diagonal elements as $1$ or $-1$, and construct a large class of ${\mathrm{Ad}}_{{\mathrm{U}}_n^{\pm }(K)}$-real elements. As applications of these results, we recover related results concerning classical reality in these groups.