In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Langlands classification is a classification of irreducible representations of a reductive Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

 G, suggested by 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...

 (1973). More precisely, it classifies the irreducible admissible (g,K)-modules,
for g a Lie algebra of a reductive Lie group G, with maximal compact subgroup

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....

 K, in terms of tempered representation

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 Lp spacefor any ε > 0.-Formulation:...

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

Anthony Knapp

Anthony W. Knapp is a mathematician at the US State University of New York, Stony Brook working on representation theory who classified the tempered representations of a semisimple Lie group.He won the Leroy P...

 and Gregg Zuckerman

Gregg Zuckerman

Gregg Jay Zuckerman is a mathematician at Yale University who discovered Zuckerman functors and translation functors, and with Anthony Knapp classified the irreducible tempered representations ofsemisimple Lie groups.-Publications:...

.

Notation

• g is the Lie algebra of a real reductive Lie group G in the Harish-Chandra class
  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 subgroups...
  
  .
• K is a maximal compact subgroup of G, with Lie algebra k.
• ω is a Cartan involution 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
  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...
  
   of a in g.
• Δ is a set of simple root
  Simple root
  in mathematics the term simple root can refer to one of two unrelated notions:*A simple root of a polynomial is a root of multiplicity one*A simple root in a root system is a member of a subset determined by a choice of positive roots...
  
  s of Σ.

Classification

The Langlands classification states that the irreducible admissible representation

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....

s of (g,K) are parameterized by triples
where

• F is a subset of Δ
• Q is the standard parabolic subgroup of F, with Langlands decomposition
  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 :...
  
   Q = MAN
• σ is an irreducible tempered representation of the semisimple Lie algebra 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)

Representation theory of SL2(R)

In mathematics, the main results concerning irreducible unitary representations of the Lie group SL are due to Gelfand and Naimark , V...

.

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 insetaed of tempered representations. The problem with doing this is that it is tricky to decide when two irreducible representations are the same.