Positive Systems of Kostant Roots

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

$$\mathfrak{g} = \mathfrak{s} \oplus \left( \oplus_{\nu \in \mathcal{R}}~ \mathfrak{g}^{\nu}\right)$$

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 $(\text {Sym}^{\cdot }(\mathcal {M}))^{\mathfrak {s}} = \mathbb {C}$ 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 F₄,E₆,E₇, and E₈. Finally, we discuss the problem in the case when $\mathcal {S}$ is not saturated.