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In representation theory, a branch of mathematics, the Langlands dual ''L''''G'' of a reductive algebraic group ''G'' (also called the ''L''-group of ''G'') is a group that controls the representation theory of ''G''. If ''G'' is defined over a field ''k'', then ''L''''G'' is an extension of the absolute Galois group of ''k'' by a complex Lie group. There is also a variation called the Weil form of the ''L''-group, where the Galois group is replaced by a Weil group. The Langlands dual group is also often referred to as an ''L-group''; here the letter ''L'' indicates also the connection with the theory of L-functions, particularly the ''automorphic'' L-functions. The Langlands dual was introduced by in a letter to A. Weil. The ''L''-group is used heavily in the Langlands conjectures of Robert Langlands. It is used to make precise statements from ideas that automorphic forms are in a sense functorial in the group ''G'', when ''k'' is a global field. It is not exactly ''G'' with respect to which automorphic forms and representations are functorial, but ''L''''G''. This makes sense of numerous phenomena, such as 'lifting' of forms from one group to another larger one, and the general fact that certain groups that become isomorphic after field extensions have related automorphic representations. ==Definition for separably closed fields== From a reductive algebraic group over a separably closed field ''K'' we can construct its root datum (''X'' *, Δ,''X'' *, Δv), where ''X'' * is the lattice of characters of a maximal torus, ''X'' * the dual lattice (given by the 1-parameter subgroups), Δ the roots, and Δv the coroots. A connected reductive algebraic group over ''K'' is uniquely determined (up to isomorphism) by its root datum. A root datum contains slightly more information than the Dynkin diagram, because it also determines the center of the group. For any root datum (''X'' *, Δ,''X'' *, Δv), we can define a dual root datum (''X'' *, Δv,''X'' *, Δ) by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots. If ''G'' is a connected reductive algebraic group over the algebraically closed field ''K'', then its Langlands dual group ''L''''G'' is the complex connected reductive group whose root datum is dual to that of ''G''. Examples: The Langlands dual group ''L''''G'' has the same Dynkin diagram as ''G'', except that components of type ''B''''n'' are changed to components of type ''C''''n'' and vice versa. If ''G'' has trivial center then ''L''''G'' is simply connected, and if ''G'' is simply connected then ''L''''G'' has trivial center. The Langlands dual of ''GL''''n''(''K'') is ''GL''''n''(C). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Langlands dual group」の詳細全文を読む スポンサード リンク
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