Residues of Rankin-Selberg Zeta integrals and the split non-tempered Gan-Gross-Prasad conjectures
Paul Boisseau
TL;DR
The paper develops a residue-based regularization for Rankin–Selberg periods on GL_n × GL_{n+1} to handle non-tempered (Arthur-type) automorphic representations. It proves global and local non-tempered Gan–Gross–Prasad conjectures, including an Ichino–Ikeda refinement, by expressing global periods as Euler products of regularized local zetas and residues, and by constructing explicit nonzero H-invariant functionals via residues. A key innovation is the residue analysis of Zeta integrals, together with Zydor’s truncation framework, yielding a factorization through Langlands quotients and ensuring non-vanishing criteria tied to L^*-factors. The authors provide both a global-to-local globalization approach and a purely local non-Archimedean proof, and relate their construction to the BZSV formalism, Speh representations, and automorphic-to-local transfer. Collectively, the results settle the split non-tempered GGP conjectures in characteristic zero, with potential extensions to unitary groups and connections to nearby non-tempered periods in the broader Langlands program.
Abstract
We construct a regularization of the Rankin-Selberg period on general linear groups for non-tempered automorphic representations using residues of Zeta integrals. We prove that it satisfies the global non-tempered Gan-Gross-Prasad conjecture and its Ichino-Ikeda refinement. We also build a local version of our regularization and show that it defines a non-zero invariant linear form on non-tempered representations. Combined with previous works of Chan, Chen and Chen, this settles the conjectures over local fields of characteristic zero.