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