Langlands classification

In mathematics, the Langlands classification is a description of the irreducible representations of a reductive Lie group G, suggested by Robert Langlands (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 of a reductive Lie group G, with maximal compact subgroup 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 of R or C into the Langlands dual group.

The Langlands classification states that the irreducible admissible representations of (g,K) are parameterized by triples

where

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).

There are several minor variations of the Langlands classification. For example:

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