mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, 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,W ...

s of a reductive

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...

''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'')-

module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...

s, for ''g'' a

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...

of a reductive Lie group ''G'', with

maximal compact subgroup In mathematics, a maximal compact subgroup ''K'' of a topological group ''G'' is a subgroup ''K'' that is a compact space, in the subspace topology, and maximal amongst such subgroups. Maximal compact subgroups play an important role in the classi ...

''K'', in terms of

tempered representation In mathematics, a tempered representation of a linear semisimple Lie group is a representation that has a basis whose matrix coefficients lie in the L''p'' space :''L''2+ε(''G'') for any ε > 0. Formulation This condition, as just g ...

s of smaller groups. The tempered representations were in turn classified by

Anthony Knapp Anthony W. Knapp (born 2 December 1941, Morristown, New Jersey) is an American mathematician at the State University of New York, Stony Brook working on representation theory, who classified the tempered representations of a semisimple Lie group ...

and

Gregg Zuckerman Gregg Jay Zuckerman (born 1949) is a mathematician at Yale University who discovered Zuckerman functors and translation functors, and with Anthony W. Knapp classified the irreducible tempered representations of semisimple Lie groups. He received ...

. 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 lev ...

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 fie ...

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 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 representati ...

of ''a'' in ''g''. *Δ is a set of

simple root Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...

s of Σ.

Classification

The Langlands classification states that the irreducible

admissible representation In mathematics, admissible representations are a well-behaved class of representations used in the representation theory of reductive Lie groups and locally compact totally disconnected groups. They were introduced by Harish-Chandra. Real or comp ...

s of (''g'',''K'') are parameterized by triples :(''F'', σ,λ) where *''F'' is a subset of Δ *''Q'' is the standard

parabolic subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ...

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 applicat ...

''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