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.