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 F
4,E
6,E
7, and E
8. Finally, we discuss the problem in the case when
\(\mathcal {S}\) is not saturated.
