A rank-three condition for invariant $(1,2)$-symplectic almost Hermitian structures on flag manifolds

This paper considers invariant $\left( 1,2\right) $-symplectic almost Hermitian structures on the maximal flag manifod associated to a complex semi-simple Lie group $G$. The concept of cone-free invariant almost complex structure is introduced. It involves the rank-three subgroups of $G$, and generalizes the cone-free property for tournaments related to $\mathrm{Sl} \left( n,\Bbb{C}\right) $ case. It is proved that the cone-free property is necessary for an invariant almost-complex structure to take part in an invariant $\left( 1,2\right) $-symplectic almost Hermitian structure. It is also sufficient if the Lie group is not $B_{l}$, $l\geq 3$, $G_{2}$ or $F_{4} $. For $B_{l}$ and $F_{4}$ a close condition turns out to be sufficient.