In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Langlands classification is a description of the
irreducible representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W, ...
s of a reductive
Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
''G'', suggested by
Robert Langlands
Robert Phelan Langlands, (; born October 6, 1936) is a Canadian mathematician. He is best known as the founder of the Langlands program, a vast web of conjectures and results connecting representation theory and automorphic forms to the study o ...
(1973). There are two slightly different versions of the Langlands classification. One of these describes the irreducible
admissible (''g'', ''K'')-
modules,
for ''g'' a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
of a reductive Lie group ''G'', with
maximal compact subgroup
In mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. T ...
''K'', in terms of
tempered representations of smaller groups. The tempered representations were in turn classified by
Anthony Knapp and
Gregg Zuckerman. The other version of the Langlands classification divides the irreducible representations into
L-packets, and classifies the L-packets in terms of certain homomorphisms of the
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 generally denoted ''WF''. There also exists "finite leve ...
of R or C into the
Langlands dual group
In representation theory, a branch of mathematics, the Langlands dual ''L'G'' 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 f ...
.
Notation
*''g'' is the Lie algebra of a real reductive Lie group ''G'' in the
Harish-Chandra class
In mathematics, Harish-Chandra's class is a class of Lie groups used in representation theory. Harish-Chandra's class contains all semisimple connected linear Lie groups and is closed under natural operations, most importantly, the passage to Levi ...
.
*''K'' is a maximal compact subgroup of ''G'', with Lie algebra ''k''.
*ω is a
Cartan involution
In mathematics, the Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure theory and representation theory. It generalizes the polar decomposition or singular value decom ...
of ''G'', fixing ''K''.
*''p'' is the −1 eigenspace of a Cartan involution of ''g''.
*''a'' is a maximal abelian subspace of ''p''.
*Σ is the
root system
In mathematics, a root system is a configuration of vector space, vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and ...
of ''a'' in ''g''.
*Δ is a set of
simple root
In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer ...
s of Σ.
Classification
The Langlands classification states that the irreducible
admissible representation
In mathematics, admissible representations are a well-behaved class of Group representation, representations used in the representation theory of reductive group, reductive Lie groups and locally compact group, locally compact totally disconnected ...
s of (''g'', ''K'') are parameterized by triples
:(''F'', σ, λ)
where
*''F'' is a subset of Δ
*''Q'' is the standard
parabolic subgroup Parabolic subgroup may refer to:
* a parabolic subgroup of a reflection group
* a subgroup of an algebraic group that contains a Borel subgroup
In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zarisk ...
of ''F'', with
Langlands decomposition
In mathematics, the Langlands decomposition writes a parabolic subgroup ''P'' of a semisimple Lie group as a product P=MAN of a reductive subgroup ''M'', an abelian subgroup ''A'', and a nilpotent subgroup ''N''.
Applications
A key applica ...
''Q'' = ''MAN''
*σ is an irreducible tempered representation of the semisimple Lie group ''M'' (up to isomorphism)
*λ is an element of Hom(''a''
''F'', C) with α(Re(λ)) > 0 for all simple roots α not in ''F''.
More precisely, the irreducible admissible representation given by the data above is the irreducible quotient of a parabolically induced representation.
For an example of the Langlands classification, see the
representation theory of SL2(R)
In mathematics, the main results concerning irreducible unitary representations of the Lie group SL(2, R) are due to Gelfand and Naimark (1946), V. Bargmann (1947), and Harish-Chandra (1952).
Structure of the complexified Lie algebra
We choo ...
.
Variations
There are several minor variations of the Langlands classification. For example:
*Instead of taking an irreducible quotient, one can take an irreducible submodule.
*Since tempered representations are in turn given as certain representations induced from discrete series or limit of discrete series representations, one can do both inductions at once and get a Langlands classification parameterized by discrete series or limit of discrete series representations instead of tempered representations. The problem with doing this is that it is tricky to decide when two irreducible representations are the same.
References
*
*E. P. van den Ban, ''Induced representations and the Langlands classification,'' in (T. Bailey and A. W. Knapp, eds.).
*
Borel, A. and
Wallach, N. ''Continuous cohomology, discrete subgroups, and representations of reductive groups''. Second edition. Mathematical Surveys and Monographs, 67. American Mathematical Society, Providence, RI, 2000. xviii+260 pp.
*
*
*D. Vogan, ''Representations of real reductive Lie groups'', {{isbn, 3-7643-3037-6
Representation theory of Lie groups