Invariant almost Hermitian structures on flag manifolds

Abstract

Let G be a complex semi-simple Lie group and form its maximal flag manifold F=G/P=U/T where P is a minimal parabolic (Borel) subgroup, U a compact real form and T= U∩ P a maximal torus of U. We study U-invariant almost Hermitian structures on F . The (1,2)-symplectic (or quasi-Kähler) structures are naturally related to the affine Weyl groups. A special form for them, involving abelian ideals of a Borel subalgebra, is derived. From the (1,2)-symplectic structures a classification of the whole set of invariant structures is provided showing, in particular, that nearly Kähler invariant structures are Kähler, except in the A 2 case.