# On adjoint orbits in nilpotent ideals of a Borel subalgebra

## TL;DR

This paper investigates the structure of nilpotent elements within a nilpotent ideal of a Borel subalgebra in a complex semisimple Lie algebra, revealing a unique closed orbit under the Borel group action.

## Contribution

It establishes the existence and uniqueness of a closed Borel orbit in the set of ad-nilpotent elements associated with a nilpotent ideal, characterized by minimal roots.

## Key findings

- Unique closed Borel orbit in the set of ad-nilpotent elements.
- The orbit corresponds to a nilpotent element supported on minimal roots.
- Characterization of the orbit via root space decomposition.

## Abstract

Let $\mathfrak{m}$ be a nilpotent ideal in the Borel subalgebra $\mathfrak{b}$ of a complex finite-dimensional semisimple Lie algebra, and $\mathfrak{m}^{\bullet}$ the subset of (ad-)nilpotent elements in $\mathfrak{b}$ such that $\mathfrak{m}$ is the minimal ideal containing them. This set is stable under the adjoint action of the corresponding Borel subgroup $B$. We prove that $\mathfrak{m}^{\bullet}$ contains a unique closed $B$-orbit which is the orbit of a nilpotent element whose support is the set of minimal roots associated to the root space decomposition of $\mathfrak{m}$.