Functoriality for the Classical Groups over Function Fields

Abstract

Langlands' functoriality for generic representations from the split classical groups to an appropriate GL N is established. The functorial lift or transfer to GL N 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.