In representation theory, a branch of mathematics, the Langlands dualLG of a reductive algebraic groupG is a group that controls the representation theory ofG. If G is defined over a fieldk, 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. 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 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 Gwith 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 fieldK we can construct its root datum, where X* is the lattice of characters of a maximal torus, X* the dual lattice, Δ the roots, and Δv the coroots. A connected reductive algebraic group over K is uniquely determined 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, we can define a dual root datum 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 fieldK, then its Langlands dual groupLG 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 is GLn.
Definition for groups over more general fields
Now suppose that G is a reductive group over some field k with separable closureK. Over K, G has a root datum, and this comes with an action of the Galois group Gal. The identity componentLGo 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. 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:
Instead of using the full Galois group Gal of the separable closure, one can just use the Galois group of a finite extension over which G is split. The corresponding semidirect product then has only a finite number of components and is a complex Lie group.
Suppose that k is a local, global, or finite field. Instead of using the absolute Galois group of k, one can use the absolute Weil group, which has a natural map to the Galois group and therefore also acts on the root datum. The corresponding semidirect product is called the Weil form of the L-group.
For algebraic groupsG over finite fields, Deligne and Lusztig introduced a different dual group. As before, G gives a root datum with an action of the absolute Galois group of the finite field. The dual groupG* is then the reductive algebraic group over the finite field associated to the dual root datum with the induced action of the Galois group.
Applications
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 into the Langlands dual group of G. A more general formulation of the conjectures is Langlands functoriality, which says that given a 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.
