Langlands dual group
In representation theory, a branch of mathematics, the Langlands dual LG 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 LG 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 Langlands (1967) 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 LG. 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.
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.
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