Doubling constructions: Global functoriality for non-generic cuspidal representations

Abstract

We study the generalized doubling method for pairs of representations of $G\times \mathrm {GL}_k$ 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 $\mathrm {GL}_k$ 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.