Invariant almost Hermitian structures on flag manifolds

Let $G$ be a complex semi-simple Lie group and form its maximal flag manifold $\Bbb{F}=G/P=U/T$ where $P$ is a minimal parabolic subgroup, $U$ a compact real form and $T=U\cap P$ a maximal torus of $U$. We study $U$ -invariant almost Hermitian structures on $\Bbb{F}$. The $\left( 1,2\right) $ -symplectic (or quasi-K\"{a}hler) 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 $\left( 1,2\right)$-symplectic structures a classification of the whole set of invariant structures is provided, showing, in particular, that near K\"{a}hler invariant structures are K\"{a}hler, except in the $A_{2}$ case.