Doubling constructions: local and global theory, with an application to global functoriality for non-generic cuspidal representations

Abstract

A fundamental difficulty in the study of automorphic representations, representations of $p$-adic groups and the Langlands program is to handle the non-generic case. In this work we develop a complete local and global theory of tensor product $L$-functions of $G\times GL_k$, where $G$ is a symplectic group, split special orthogonal group or the split general spin group, that includes both generic and non-generic representations of $G$. Our theory is based on a recent collaboration with David Ginzburg, where we presented a new integral representation that applies to all cuspidal automorphic representations. Here we develop the local theory over any field (of characteristic $0$), define the local $\gamma$-factors and provide a complete description of their properties. We then define $L$- and $\epsilon$-factors, and obtain the properties of the completed $L$-function. By combining our results with the Converse Theorem, we obtain a full proof of the global functorial lifting of cuspidal automorphic representations of $G$ to the natural general linear group.Comment: Added Appendix A by Dmitry Gourevitch, and appendices B and C by Eyal Kapla