Robert Langlands introduced a conjectural group LF attached to each local or global field F, coined the Langlands group of F by Robert Kottwitz, that satisfies properties similar to those of the Weil group. In Kottwitz's formulation, the Langlands group should be an extension of the Weil group by a compact group. When F is local archimedean, LF is the Weil group of F, when F is local non-archimedean, LF is the product of the Weil group of F with SU(2). When F is global, the existence of LF is still conjectural, though Arthur (2002) gives a conjectural description of it. The Langlands correspondence for F is a "natural" correspondence between the irreducible n-dimensional complex representations of LF and, in the local case, the irreducible admissible representations of GLn(F), in the global case, the cuspidal automorphic representations of GLn(AF), where AF denotes the adeles of F.
- Kottwitz 1984, §12
- Arthur, James (2002), "A note on the automorphic Langlands group", Canad. Math. Bull., 45 (4): 466–482, MR 1941222, doi:10.4153/CMB-2002-049-1
- Kottwitz, Robert (1984), "Stable trace formula: cuspidal tempered terms", Duke Mathematical Journal, 51 (3): 611–650, MR 0757954, doi:10.1215/S0012-7094-84-05129-9
- Langlands, R. P., "Automorphic representations, Shimura varieties, and motives. Ein Märchen", Automorphic forms, representations and L-functions, Proc. Sympos. Pure Math., 33, pp. 205–246, ISBN 0-8218-1437-0, MR 0546619