Positive Systems of Kostant Roots

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.