Invariant $${\mathcal {G}}_1$$ structures on flag manifolds

Abstract

Let $${\mathbb {F}}_{\Theta }=U/K_\Theta $$ be a partial flag manifold, where $$K_\Theta $$ is the centralizer of a torus in U. We study U-invariant almost Hermitian structures on $${\mathbb {F}}_{\Theta }$$ . The classification of these structures are naturally related with the system $$R_{\mathfrak {t}}$$ of $${\mathfrak {t}}$$ -roots associated to $${\mathbb {F}}_{\Theta }$$ . We introduced the notion of connectedness by triples with zero sum in a general subset of a vector space and proved that the set of $${\mathfrak {t}}$$ -roots satisfies this property. Using this result, the invariant $${\mathcal {G}}_1$$ structures on $${\mathbb {F}}_{\Theta }$$ are completely classified.