On the non-vanishing of theta liftings of tempered representations of U(p,q)

Abstract

In this paper, we give an explicit determination of the non-vanishing of the theta liftings of tempered representations for unitary dual pairs (U(p,q),U(r,s)) for arbitrary non-negative integers p,q,r,s. For discrete series representations, in terms of Harish-Chandra parameters, we give a complete criterion when the theta lifts are nonzero. For tempered representations, we determine the non-vanishing in terms of the local Langlands correspondence assuming the local Gan–Gross–Prasad conjecture.