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

Keywords

weakly symmetric space, pseudo-riemannian manifold, homogeneous manifold, Lorentz manifold, trans-Lorentz manifold

2010 Mathematics Subject Classification

Primary 22E45, 43A80. Secondary 32M15, 53B30, 53B35.

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To the memory of Bert Kostant, a good friend and a mathematical pioneer.

The author’s research was partially supported by a Simons Foundation grant.

Received 25 October 2018

Accepted 24 October 2019

Published 22 December 2021