Langlands group

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 cuspidal automorphic representations of GLn(AF), where AF denotes the adeles of F.[1]

References

  1. ↑ Kottwitz 1984, §12
  • Arthur, James (2002), "A note on the automorphic Langlands group", Canad. Math. Bull., 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, doi:10.1215/S0012-7094-84-05129-9, MR 0757954
  • 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
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