  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.
