G1 structures on flag manifolds

Abstract

Let $U/K_{\Theta }$ be a generalized flag manifold, where $K_{\Theta }$ is the centralizer of a torus in $U$. We study $U$-invariant almost Hermitian structures on $U/K_{\Theta }$. The classification of these structures are naturally related with the system $R_t$ of t-roots associated to $U/K_{\Theta }$. We introduced the notion of connectedness by triples zero sum in a general set of linear functional and proved that t-roots are connected by triples zero sum. Using this property, the invariant G1 structures on $U/K_{\Theta }$ are completely classified. We also study the Kähler form and classified the invariant quasi Kähler structures on $U/K_{\Theta }$, in terms of t-roots.