Functoriality for the Classical Groups over Function Fields

Langlands' functoriality for generic representations from the split classical groups to an appropriate GLN is established. The functorial lift or transfer to GLN is obtained with the help of a converse theorem once the analytic properties of L-functions are studied using the Langlands–Shahidi approach. This paper is mostly devoted to understanding L-functions for the classical groups over a global function field, since the Langlands–Shahidi method has only been developed over number fields. To overcome many difficulties, stability of γ-factors under twists by highly ramified characters is used together with multiplicativity. Finally, by analyzing the image of functoriality, a proof of the Ramanujan conjecture for generic representations is obtained.