Complex extensions of semisimple symmetric spaces

Let G/H be a pseudo-Riemannian semisimple symmetric space. The tangent bundle T (G/H) contains a maximal G-invariant neighbourhood Omega of the zero section where the adapted-complex structure exists. Such Omega is endowed with a canonical G-invariant pseudo-Kahler metric of the same signature as the metric on G/H. We use the polar map phi : Omega -> G(C)/H-C to define a G-invariant pseudo-Kahler metric on distinguished G-invariant domains in G(C)/H-C or on coverings of principal orbit strata in G(C)/H-C. In the rank-one case, we show that the polar map is globally injective and the domain phi(Omega) subset of G(C)/H-C is an increasing union of q-complete domains.