Inrepresentation theory, a branch of mathematics, the Langlands dual LG of a reductive algebraic group G (also called the L-groupofG) 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 groupofk 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. Here, the letter L in the name also indicates 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 conjecturesofRobert 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.
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.
IfG is a connected reductive algebraic group over the algebraically closed field K, then its Langlands dual group LG is the complex connected reductive group whose root datum is dual to that of G.
Examples: The Langlands dual group LG has the same Dynkin diagram as G, except that components of type Bn are changed to components of type Cn and vice versa. If G has trivial center then LG is simply connected, and if G is simply connected then LG has trivial center. The Langlands dual of GLn(K) is GLn(C).
Now suppose that G is a reductive group over some field k with separable closure K. Over K, G has a root datum, and this comes with an action of the Galois group Gal(K/k). The identity component LGo of the L-group is the connected complex reductive group of the dual root datum; this has an induced action of the Galois group Gal(K/k). The full L-group LG is the semidirect product
of the connected component with the Galois group.
There are some variations of the definition of the L-group, as follows:
The Langlands conjectures imply, very roughly, that if G is a reductive algebraic group over a local or global field, then there is a correspondence between "good" representations of G and homomorphisms of a Galois group (or Weil group or Langlands group) into the Langlands dual group of G. A more general formulation of the conjectures is Langlands functoriality, which says (roughly) that given a (well behaved) homomorphism between Langlands dual groups, there should be an induced map between "good" representations of the corresponding groups.
To make this theory explicit, there must be defined the concept of L-homomorphism of an L-group into another. That is, L-groups must be made into a category, so that 'functoriality' has meaning. The definition on the complex Lie groups is as expected, but L-homomorphisms must be 'over' the Weil group.