Homogeneous para-Kähler Einstein manifolds

A para-Kähler manifold can be defined as a pseudo-Riemannian manifold $(M,g)$ with a parallel skew-symmetric para-complex structure $K$, that is, a parallel field of skew-symmetric endomorphisms with $K^2=\operatorname{Id}$ or, equivalently, as a symplectic manifold $(M,\omega)$ with a bi-Lagrangian structure $L^\pm$, that is, two complementary integrable Lagrangian distributions. A homogeneous manifold $M = G/H$ of a semisimple Lie group $G$ admits an invariant para-Kähler structure $(g,K)$ if and only if it is a covering of the adjoint orbit $\operatorname{Ad}_Gh$ of a semisimple element $h$. A description is given of all invariant para-Kähler structures $(g,K)$ on such a homogeneous manifold. With the use of a para-complex analogue of basic formulae of Kähler geometry it is proved that any invariant para-complex structure $K$ on $M=G/H$ defines a unique para-Kähler Einstein structure $(g,K)$ with given non-zero scalar curvature. An explicit formula for the Einstein metric $g$ is given. A survey of recent results on para-complex geometry is included.