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.