Functoriality for the classical groups

Abstract

Functoriality is one of the most central questions in the theory of automor-phic forms and representations [1,2,35,36]. Locally and globally, it is a manifesta-tion of Langlands' formulation of a non-abelian class field theory. Now known as the Langlands correspondence, this formulation of class field theory can be viewed as giving an arithmetic parameterization of local or automorphic representations in terms of admissible homomorphisms of (an appropriate analogue) of the Weil-Deligne group into the Langlands dual group or L-group. When this conjectural parameterization is combined with natural homomorphisms of the L-groups it pre-dicts a transfer or lifting of local or automorphic representations of two reductive algebraic groups. As a purely automorphic expression of a global non-abelian class field theory, global functoriality is inherently an arithmetic process. In this paper we establish global functoriality from the split classical groups G n = SO 2n+1 , SO 2n , or Sp 2n to an appropriate general linear group GL N , associated to the natural embedding of L-groups, for globally generic cuspidal representations π of G n (A) over a number field k. We had previously presented functoriality for the case G n = SO 2n+1 in [6], but were limited at that time by a lack of suit-able local tools in the other cases. The present paper is by no means a simple generalization of [6]. There were serious local problems to be overcome in the development of the tools that now allow us to cover all three series of classical groups simultaneously and that will be applicable to other cases of functoriality in the future. In addition, we have completely determined the associated local im-ages of functoriality and as a result are able to present several new applications of functoriality, including both global results concerning the Ramanujan conjecture for the classical groups and various applications to the local representation theory of the classical groups. There are several approaches to the question of functoriality: the trace for-mula, the relative trace formula, and the Converse Theorem. In this work we use the Converse Theorem, which is an L-function method. The Converse Theorem itself states that if one has an irreducible admissible representation Π ⊗ Π v of GL N (A), then Π is in fact automorphic if sufficiently many of its twisted L-functions L(s, Π × τ), with τ cuspidal automorphic representations of smaller GL m (A), are nice [7,9]. As a vehicle for establishing functoriality from cuspidal representations π = ⊗π