Theta lifting for discrete series representations of real unitary groups

TL;DR

This paper investigates theta lifting for real unitary groups, explicitly determining the lifts of discrete series representations and expressing them as cohomologically induced representations in the weakly fair range.

Contribution

It extends previous results by explicitly describing theta lifts of discrete series representations as cohomologically induced in the weakly fair range.

Findings

Theta lifts of discrete series are expressed as cohomologically induced representations.

Extension of Li's result to less regular infinitesimal characters.

Complete determination of theta lifts for real unitary groups.

Abstract

We study the theta lifting for real unitary groups and completely determine the theta lifts of discrete series representations. In particular, we show that these theta lifts can be expressed as cohomologically induced representations in the weakly fair range. This extends a result of J.-S. Li in the case of discrete series representations with sufficiently regular infinitesimal character, whose theta lifts can be expressed as cohomologically induced representations in the good range.