Homogeneous almost Kähler manifolds and the Chern-Einstein equation

Given a non-compact semisimple Lie group G we describe all homogeneous spaces G/L carrying an invariant almost-Kähler structure (ω, J). When L is abelian and G is of classical type, we classify all such spaces which are Chern-Einstein, i.e. which satisfy ρ = λω for some λ ∈ R, where ρ is the Ricci form associated to the Chern connection.