Langlands group
Encyclopedia
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 a group over a field
k, 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-function
s, particularly the automorphic L-functions.
The L-group is used heavily in the Langlands conjectures of Robert Langlands
. It is used to make precise statements from ideas that automorphic form
s 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 form
s 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 extension
s have related automorphic representations.
(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 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).
of the connected component with the Galois group.
There are some variations of the definition of the L-group, as follows:
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.
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
, 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 a group over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
k, LG is an extension of the absolute Galois group
Absolute Galois group
In mathematics, the absolute Galois group GK of a field K is the Galois group of Ksep over K, where Ksep is a separable closure of K. Alternatively it is the group of all automorphisms of the algebraic closure of K that fix K. The absolute Galois group is unique up to isomorphism...
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
Weil group
In mathematics, a Weil group, introduced by , is a modification of the absolute Galois group of a local or global field, used in class field theory. For such a field F, its Weil group is a profinite group generally denoted WF...
. 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-function
L-function
The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...
s, particularly the automorphic L-functions.
The L-group is used heavily in the Langlands conjectures of Robert Langlands
Robert Langlands
Robert Phelan Langlands is a mathematician, best known as the founder of the Langlands program. He is an emeritus professor at the Institute for Advanced Study...
. It is used to make precise statements from ideas that automorphic form
Automorphic form
In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms...
s are in a sense functorial in the group G, when k is a global field
Global field
In mathematics, the term global field refers to either of the following:*an algebraic number field, i.e., a finite extension of Q, or*a global function field, i.e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of Fq, the field of rational functions...
. It is not exactly G with respect to which automorphic form
Automorphic form
In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms...
s 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 extension
Field extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...
s 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 datumRoot datum
In mathematics, the root datum of a connected split reductive algebraic group over a field is a generalization of a root system that determines the group up to isomorphism...
(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 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).
Definition for groups over more general fields
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- LG = LGo×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 (or even just quasisplit). 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 groupWeil groupIn mathematics, a Weil group, introduced by , is a modification of the absolute Galois group of a local or global field, used in class field theory. For such a field F, its Weil group is a profinite group generally denoted WF...
, 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.)
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 (or Weil 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
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
, so that 'functoriality' has meaning. The definition on the complex Lie groups is as expected, but L-homomorphisms must be 'over' the Weil group.