Functoriality for the quasisplit classical groups

Abstract

Functoriality is one of the most central questions in the theory of automorphic forms and representations [3, 6, 31, 32]. Locally and globally, it is a manifestation 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 parametrization of local or automorphic representations in terms of admissible homomorphisms of (an appropriate analogue) of the WeilDeligne group into the L-group. When this conjectural parametrization is combined with natural homomorphisms of the L-groups it predicts 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. Global functoriality from a quasisplit classical group G to GLN associated to a natural map on the L-groups has been established in many cases. We recall the main cases: (i) For G a split classical group with the natural embedding of the L-groups, this was established in [10] and [11]. (ii) For G a quasisplit unitary group with the L-homomorphism associated to stable base change on the L-groups, this was established in [29],[26], and [27]. (iii) For G a split general spin group, this was established in [5]. In this paper we consider simultaneously the cases of quasisplit classical groups G. This includes all the cases mentioned in (i) and (ii) above as well as the new case of the quasisplit even special orthogonal groups. Similar methods should work for the quasisplit GSpin groups, and this will be pursued by Asgari and Shahidi as a sequel to [5]. As with the previous results above, our method combines the Converse Theorem for GLN with the Langlands-Shahidi method for controlling the L-functions of the quasisplit classical groups. One of the crucial ingredients in this method is the use of the “stability of local γ–factors” to finesse the lack of the Local Langlands Conjecture at the ramified non-archimedean places. The advance that lets us now

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