2016/12/08 by Dimitrov, Ivan, Roth, Mike
#FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1612.02851
Let \mathfrakg be a simple complex Lie algebra and let \mathfrakt ⊂ \mathfrakg be a toral subalgebra of \mathfrakg. As a \mathfrakt-module \mathfrakg decomposes as \mathfrakg = \mathfraks ⊕ (⊕ν∈ R \mathfrakgν) where \mathfraks ⊂ \mathfrakg is the reductive part of a parabolic subalgebra of \mathfrakg and R is the Kostant root system associated to \mathfrakt. When \mathfrakt is a Cartan subalgebra of \mathfrakg the decomposition above is nothing but the root decomposition of \mathfrakg with respect to \mathfrakt; 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 \mathfrakp of \mathfrakg containing M = ⊕ν∈ S \mathfrakgν and whose reductive part equals \mathfraks?". Our main results is that, for a classical simple Lie algebra \mathfrakg and a saturated S ⊂ R, the condition (Sym^⋅(M))^\mathfraks = C is necessary and sufficient for the existence of such a \mathfrakp. 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 S is not saturated.