Langlands dual group

In

The L-group is used heavily in the

From a reductive algebraic group over a separably closed field K we can construct its root datum (X^*, Δ,X_*, Δ^v), 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 Δ^v 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_*, Δ^v), we can define a dual root datum (X_*, Δ^v,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 ^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 B_n are changed to components of type C_n 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 GL_n(K) is GL_n(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 ^LG^o 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

^LG = ^LG^o×Gal(K/k)

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(K/k) 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 groups G 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 group G^* is then the reductive algebraic group over the finite field associated to the dual root datum with the induced action of the Galois group. (This dual group is defined over a finite field, while the component of the Langlands dual group is defined over the complex numbers.)

The

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.

• ISBN 0-8218-1437-0
• Langlands, R. (1967), letter to A. Weil
• Mirković, I.; Vilonen, K. (2007), "Geometric Langlands duality and representations of algebraic groups over commutative rings",
  MR 2342692 describes the dual group of G in terms of the geometry of the affine Grassmannian
  of G.