Unitary Representations, $L^2$ Dolbeault Cohomology, and Weakly Symmetric Pseudo--Riemannian Nilmanifolds

TL;DR

This paper develops a theory connecting weakly symmetric pseudo-Riemannian nilmanifolds of complex type with unitary representations on Dolbeault cohomology spaces, extending classical constructions and providing classifications for key families.

Contribution

It introduces a new framework for constructing and classifying unitary representations on cohomology spaces of pseudo-Riemannian nilmanifolds of complex type, expanding previous work on flag domains.

Findings

Constructed square integrable representations on holomorphic cohomology spaces.

Developed classifications for three main families of weakly symmetric pseudo-Riemannian nilmanifolds.

Extended classical theories to pseudo-Riemannian and nilmanifold settings.

Abstract

We combine recent developments on weakly symmetric pseudo--riemannian nilmanifolds with 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 and the nilmanifold versions 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.