Positive Systems of Kostant Roots

Let 𝔤 be a simple complex Lie algebra and let 𝔱⊂𝔤 be a toral subalgebra of 𝔤 . As a 𝔱 -module 𝔤 decomposes as 𝔤 = 𝔰⊕( ⊕_ν∈ℛ 𝔤^ν) where 𝔰⊂𝔤 is the reductive part of a parabolic subalgebra of 𝔤 and ℛ is the Kostant root system associated to 𝔱 . When 𝔱 is a Cartan subalgebra of 𝔤 the decomposition above is nothing but the root decomposition of 𝔤 with respect to 𝔱 ; in general the properties of ℛ resemble the properties of usual root systems. In this note we study the following problem: “Given a subset 𝒮⊂ℛ , is there a parabolic subalgebra 𝔭 of 𝔤 containing ℳ = ⊕ _ν∈𝒮𝔤^ν and whose reductive part equals 𝔰 ?”. Our main results is that, for a classical simple Lie algebra 𝔤 and a saturated 𝒮⊂ℛ , the condition (Sym^·(ℳ))^𝔰 = ℂ is necessary and sufficient for the existence of such a 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 𝒮 is not saturated.