Positive Systems of Kostant Roots

TL;DR

This paper investigates the structure of Kostant root systems in simple Lie algebras, establishing criteria for the existence of certain parabolic subalgebras in classical types and highlighting differences in exceptional types.

Contribution

It provides a characterization of parabolic subalgebras containing specific root spaces for classical Lie algebras, and shows these results do not extend straightforwardly to exceptional types.

Findings

Necessary and sufficient condition for classical Lie algebras.

Failure of the condition for exceptional Lie algebras.

Discussion of non-saturated subsets.

Abstract

Let $\mathfrak{g}$ be a simple complex Lie algebra and let $\mathfrak{t} \subset \mathfrak{g}$ be a toral subalgebra of $\mathfrak{g}$. As a $\mathfrak{t}$-module $\mathfrak{g}$ decomposes as \[\mathfrak{g} = \mathfrak{s} \oplus \big(\oplus_{\nu \in \mathcal{R}} \mathfrak{g}^\nu\big)\] where $\mathfrak{s} \subset \mathfrak{g}$ is the reductive part of a parabolic subalgebra of $\mathfrak{g}$ and $\mathcal{R}$ is the Kostant root system associated to $\mathfrak{t}$. When $\mathfrak{t}$ is a Cartan subalgebra of $\mathfrak{g}$ the decomposition above is nothing but the root decomposition of $\mathfrak{g}$ with respect to $\mathfrak{t}$; in general the properties of $\mathcal{R}$ resemble the properties of usual root systems. In this note we study the following problem: "Given a subset $\mathcal{S} \subset \mathcal{R}$, is there a parabolic subalgebra $\mathfrak{p}$ of $\mathfrak{g}$ containing $\mathcal{M} = \oplus_{\nu \in \mathcal{S}} \mathfrak{g}^\nu$ and whose reductive part equals $\mathfrak{s}$?". Our main results is that, for a classical simple Lie algebra $\mathfrak{g}$ and a saturated $\mathcal{S} \subset \mathcal{R}$, the condition $(\operatorname{Sym}^\cdot(\mathcal{M}))^{\mathfrak{s}} = \mathbf{C}$ is necessary and sufficient for the existence of such a $\mathfrak{p}$. In contrast, we show that this statement is no longer true for the exceptional Lie algebras $\mathrm{F}_4, \mathrm{E}_6, \mathrm{E}_7$, and $\mathrm{E}_8$. Finally, we discuss the problem in the case when $\mathcal{S}$ is not saturated.