Doubling constructions: Global functoriality for non-generic cuspidal representations

Abstract

We study the generalized doubling method for pairs of representations of $G\times G{L}_k$ where $G$ is a symplectic group, split special orthogonal group or split general spin group. We analyze the poles of the local integrals, and prove that the global completed $L$-function with a cuspidal representation of $G{L}_k$ twisted by a highly ramified Hecke character is entire. We obtain a new proof of the weak functorial transfer of cuspidal automorphic representations of $G$ to the natural general linear group, which is independent of the trace formula and its prerequisites, by combining our results with the Converse Theorem.

Previous version, V4, was expanded and split into 3 parts: 1. Cai, Friedberg, Gourevitch and Kaplan, The generalized doubling method: (k,c) models, Proc. Amer. Math. Soc. 151 (2023), 2831-2845. 2. Cai, Friedberg and Kaplan, The generalized doubling method: local theory, Geom. Funct. Anal. 32 (2022), 1233-1333 (with an appendix by Gourevitch). 3. This version, to be published in the Ann. of Math

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• Twisted homology for the mirabolic nilradical
• Doubling Constructions and Tensor Product $L$-Functions: coverings of the symplectic group
• Fourier Coefficients for Degenerate Eisenstein Series and the Descending Decomposition
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Cited by in corpus (6)

• Doubling Constructions and Tensor Product $L$-Functions: coverings of the symplectic group
• On the standard $L$-function for $GS{p}_{2n}\times G{L}_1$ and algebraicity of symmetric fourth $L$-values for $G{L}_2$
• Doubling Constructions: the complete L-function for coverings of the symplectic group
• Stability of local gamma factors arising from the doubling method for general spin groups
• Doubling Constructions and Tensor Product $L$-Functions: the linear case
• The special values of the standard $L$-functions for ${\mathrm{GSp}}_{2n}\times {\mathrm{GL}}_1$