Properties of nilpotent orbit complexification

TL;DR

This paper explores the relationship between real and complex nilpotent orbits in semisimple Lie algebras, establishing key properties about their closures, intersections, and real orbit containment within complex orbits.

Contribution

It proves that distinct real nilpotent orbits in the same complex orbit are incomparable, characterizes Lie algebras with intersections across all orbits, and describes real orbit containment in quasi-split cases.

Findings

Distinct real orbits in the same complex orbit are incomparable.

Characterization of Lie algebras intersecting all complex nilpotent orbits.

Description of complex orbits containing real nilpotent orbits in quasi-split cases.

Abstract

We consider aspects of the relationship between nilpotent orbits in a semisimple real Lie algebra $\mathfrak{g}$ and those in its complexification $\mathfrak{g}_{\mathbb{C}}$. In particular, we prove that two distinct real nilpotent orbits lying in the same complex orbit are incomparable in the closure order. Secondly, we characterize those $\mathfrak{g}$ having non-empty intersections with all nilpotent orbits in $\mathfrak{g}_{\mathbb{C}}$. Finally, for $\mathfrak{g}$ quasi-split, we characterize those complex nilpotent orbits containing real ones.