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

Abstract

In this paper, we introduce a new family of period integrals attached to irreducible cuspidal automorphic representations $\sigma$ of symplectic groups ${\mathrm{Sp}}_{2n}(\mathbb{A})$, which detects the right-most pole of the $L$-function $L(s,\sigma \times \chi )$ for some character $\chi$ of $F^{\times }\backslash {\mathbb{A}}^{\times }$ of order at most $2$, and hence the occurrence of a simple global Arthur parameter $(\chi ,b)$ in the global Arthur parameter $\psi$ attached to $\sigma$. We also give a characterisation of first occurrences of theta correspondence by (regularised) period integrals of residues of certain Eisenstein series.

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