2004/06/07 by J. W. Cogdell, James Cogdell, H. H. Kim +4 · 182 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Finite Group Theory Research #Mathematics
paper · pdf · doi:10.1007/s10240-004-0020-z
published in Publications mathématiques de l IHÉS 99, 163-233 (Springer Nature)
crossref issued 2004/06/07 · crossref published 2004/06/07 · crossref published-online 2004/06/07 · openalex publication_date 2004/06/07 · crossref created 2004/06/15 · openalex created_date 2025/10/10 · crossref deposited 2026/02/17 · openalex updated_date 2026/07/23 · crossref indexed 2026/07/28
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 Gn = SO2n+1, SO2n, or Sp2n to an appropriate general linear group GLN, associated to the natural embedding of L-groups, for globally generic cuspidal representations π of Gn(A) over a number field k. We had previously presented functoriality for