Theta lifting for tempered representations of real unitary groups

TL;DR

This paper investigates theta lifting for real unitary groups, fully characterizing the lifts of tempered representations and extending known results to limits of discrete series in the weakly fair range.

Contribution

It provides a complete description of theta lifts for tempered and limits of discrete series representations of real unitary groups, extending previous results to broader cases.

Findings

Theta lifts of (limits of) discrete series are cohomologically induced in the weakly fair range.

Extension of J.-S. Li's results to less regular infinitesimal characters.

Complete determination of theta lifts for all tempered representations.

Abstract

We study the theta lifting for real unitary groups and completely determine the theta lifts of tempered representations. In particular, we show that the theta lifts of (limits of) discrete series representations 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.