Langlands group

Robert Langlands introduced a conjectural group L_F 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, L_F is the Weil group of F, when F is local non-archimedean, L_F is the product of the Weil group of F with SU(2). When F is global, the existence of L_F 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 L_F and, in the local case, the irreducible admissible representations of GL_n(F), in the global case, the cuspidal automorphic representations of GL_n(A_F), where A_F denotes the adeles of F.^[1]

References

• 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