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

We give constraints on the existence of $(\chi,b)$-factors in the global $A$-parameter of a genuine cuspidal automorphic representation $\sigma$ of a 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 integrals of residues of Eisenstein series associated to the cuspidal datum $\chi\otimes\sigma$. This complements our previous results for symplectic groups.