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

Abstract

This paper considers invariant (1, 2)-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 𝕊l (n,ℂ) case. It is proved that the cone-free property is necessary for an invariant almost-complex structure to take part in an invariant (1, 2)-symplectic almost Hermitian structure. It is also sufficient if the Lie group is not Bl , l ≥ 3, G2 or F4. For Bl and F4 a close condition turns out to be sufficient.