(1, 2)-symplectic structures on flag manifolds

Abstract

By using moving frames and directred digraphs, we study invariant (1, 2)-symplectic structures on complex flag manifolds. Let F be a flag manifold with height k-1. We show that there is a k-dimensional family of invariant (1, 2)-symplectic metrics of any parabolic structure on F. We also prove any invariant almost complex structure J on F with height 4 admits an invariant (1, 2)-symplectic metric if and only if J is parabolic or integrable.