In this paper, we introduce a new family of period integrals attached to irreducible cuspidal automorphic representations σ of symplectic groups Sp2n(A), which detects the right-most pole of the L-function L(s, σ × χ) for some character χ of F×A × of order at most 2, and hence the occurrence of a simple global Arthur parameter (χ, b) in the global Arthur parameter ψ attached to σ. We also give a characterisation of first occurrences of theta correspondence by (regularised) period integrals of residues of certain Eisenstein series.
Bibliographical noteFunding Information:
∗The research of the first-named author is supported in part by the NSF Grants DMS–1301567 and DMS–1600685. ∗∗ The research of the second-named author is supported in part by National Natu-ral Science Foundation of China (#11601087) and by Program of Shanghai Aca-demic/Technology Research Leader (#16XD1400400). Received December 13, 2016 and in revised form February 8, 2017
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