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

Symplectic Geometry;Differential Geometry