Almost Kähler geometry of adjoint orbits of semisimple Lie groups

We study the almost Kähler geometry of adjoint orbits of non-compact real semisimple Lie groups endowed with the Kirillov–Kostant–Souriau symplectic form and a canonically defined almost-complex structure. We give explicit formulas for the Chern–Ricci form, the Hermitian scalar curvature and the Nijenhuis tensor in terms of root data. We also discuss when the Chern–Ricci form is a multiple of the symplectic form, and when compact quotients of these orbits are of Kähler type.