On $(chi,b)$-factors of cuspidal automorphic representations of unitary groups II

We derive a precise relation of poles of Eisenstein series associated to the cuspidal datum $chiotimessigma$ and lowest occurrence of theta lifts of a cuspidal automorphic representation $sigma$ of a unitary group, where $chi$ is conjugate self-dual character. We also give a refined result on non-vanishing of periods of Eisenstein series and first occurrence of theta lifts. This gives constraints on existence of $(chi,b)$-factors in the global $A$-parameter of $sigma$.