Unitary representations, $L^2$ Dolbeault cohomology, and weakly symmetric pseudo-riemannian nilmanifolds

We combine recent developments on weakly symmetric pseudo--riemannian nilmanifolds with geometric methods for construction of unitary representations square integrable Dolbeault cohomology spaces. This runs parallel to discrete series spaces harmonic forms values in holomorphic vector bundles over flag domains. Some special cases had been described by Satake 1971 and the author 1975. Here we develop a theory complex type nilmanifold versions construct associated (modulo center) those domains note that there are enough such Plancherel Fourier Inversion Formulae there. Finally, most interesting pseudo-riemannian nilmanifolds, so discuss give classifications three basic families type.