Langlands group

 ( Langlands Program ) 

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 adele ring of F.