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Homogeneous compatible almost complex structures on symplectic manifolds arestudied, focusing on those which are special, meaning that their Chern-Ricciform is a multiple of the symplectic form. Non Chern-Ricci flat ones are provento be covered by co-adjoint orbits. Conversely, compact isotropy co-adjointorbits of semi-simple Lie groups are shown to admit special compatible almostcomplex structures whenever they satisfy a necessary topological condition.Some classes of examples including twistor spaces of hyperbolic manifolds anddiscrete quotients of Griffiths period domains of weight two are discussed.

Special homogeneous almost complex structures on symplectic manifolds