In mathematics, the Langlands group is a conjectural
group
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Groups of people
* Cultural group, a group whose members share the same cultural identity
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''L''
''F'' attached to each local or global field ''F'', that satisfies properties similar to those of the
Weil group
In mathematics, a Weil group, introduced by , is a modification of the absolute Galois group of a local or global field, used in class field theory. For such a field ''F'', its Weil group is generally denoted ''WF''. There also exists "finite lev ...
. 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 local case, the cuspidal automorphic representations of GL
''n''(A
''F''), where A
''F'' denotes the
adele
Adele Laurie Blue Adkins (, ; born 5 May 1988), professionally known by the mononym Adele, is an English singer and songwriter. After graduating in arts from the BRIT School in 2006, Adele signed a reco ...
s of ''F''.
Notes
References
*
*
*{{citation, mr=0546619
, last=Langlands, first= R. P.
, chapter=Automorphic representations, Shimura varieties, and motives. Ein Märchen, title= Automorphic forms, representations and L-functions , series=Proc. Sympos. Pure Math., pages= 205–246, volume=33, isbn=978-0-8218-1437-6 , date=1979-06-30
Langlands program