Langlands group

Not to be confused with Langlands dual group.

In mathematics, the Langlands group is a conjectural group L_F attached to each local or global field F, that satisfies properties similar to those of the Weil group. It was given that name by Robert Kottwitz. 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 James Arthur 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 global case, the cuspidal automorphic representations of GL_n(A_F), where A_F denotes the adeles of F.

References

• Arthur, James (2002), "A note on the automorphic Langlands group" (PDF), Canadian Mathematical Bulletin, 45 (4): 466–482, doi:10.4153/CMB-2002-049-1, MR 1941222
• Kottwitz, Robert (1984), "Stable trace formula: cuspidal tempered terms", Duke Mathematical Journal, 51 (3): 611–650, CiteSeerX 10.1.1.463.719, doi:10.1215/S0012-7094-84-05129-9, MR 0757954
• Langlands, R. P. (1979-06-30), "Automorphic representations, Shimura varieties, and motives. Ein Märchen", Automorphic forms, representations and L-functions, Proc. Sympos. Pure Math., vol. 33, pp. 205–246, ISBN 978-0-8218-1437-6, MR 0546619