Weil Group - 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, L_F is the Weil–Deligne group of F, but when F is global, the existence of L_F is still conjectural. The Langlands correspondence for F is a "natural" bijection 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.