Periods and $(chi,b)$-factors of Cuspidal Automorphic Forms of Metaplectic Groups

We give constraints on existence of $(chi,b)$-factors in the global $A$-parameter of a genuine cuspidal automorphic representation $sigma$ of the metaplectic group in terms of the invariant, lowest occurrence index, of theta lifts to odd orthogonal groups. We also give a refined result that relates the invariant, first occurrence index, to non-vanishing of period integral of residue of Eisenstein series associated to the cuspidal datum $chiotimessigma$. This complements our previous results for symplectic groups.