Real adjoint orbits of the unipotent subgroup

TL;DR

This paper investigates the concept of Ad-reality within the Lie algebra of unipotent upper-triangular matrices, establishing the absence of non-trivial real elements in the standard group and constructing examples in an extended group, with implications for classical reality.

Contribution

It proves the non-existence of non-trivial Ad-real elements in the Lie algebra of unipotent upper-triangular matrices and constructs a class of Ad-real elements in an extended group, advancing understanding of real structures in these groups.

Findings

No non-trivial Ad-real elements in $ _n(K)$ for the standard unipotent group.

Existence of a large class of Ad-real elements in the extended group $ _n^ ext{±}(K)$.

Applications to classical reality in these groups.

Abstract

Let $G$ be a linear Lie group that acts on it's Lie algebra $\mathfrak{g}$ by the adjoint action: $\mathrm{Ad}(g)X=gXg^{-1}$. 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.