Langlands classification for real reductive groups

Statement

Let $G$ be a real reductive Lie group and fix compatible standard parabolic subgroups and positive chambers. The Langlands classification states that every irreducible admissible representation of $G$ , equivalently every irreducible admissible $(\mathfrak g,K)$ -module, is the unique irreducible Langlands quotient

J(P,\sigma,\nu)

of a standard module $I(P,\sigma,\nu)$ . Here $P=MAN$ , $\sigma$ is irreducible tempered data on $M$ , and $\operatorname{Re}\nu$ lies in the chosen open positive chamber. The inducing data are unique up to the prescribed conjugacies and Weyl-group equivalences.

Content of the classification

The theorem has three parts: a standard module has a unique irreducible quotient; every irreducible admissible representation occurs as such a quotient; and ordered inducing data determine that quotient uniquely up to the standard equivalences. Langlands proves the quotient and uniqueness in §3 and exhaustion in §4 Langlands, §3, Lemmas 3.13–3.14, and §4, Lemma 4.2.

The result classifies the admissible dual, not just the unitary dual. A Langlands quotient may fail to be unitarizable.

Example

For $G=\mathrm{SL}(2,\mathbb R)$ , normalized principal series induced from the upper-triangular minimal parabolic are standard modules when the real part of the parameter lies in the positive chamber. At reducibility points, the induced module can have several composition factors, but the theorem selects exactly one irreducible quotient. Other factors are not additional Langlands quotients for the same ordered parameter.

Conventions and scope

Changing from normalized to unnormalized induction shifts the parameter by $\rho_P$ . Reversing the positive chamber can turn the unique-quotient formulation into a unique-subrepresentation formulation. Some sources use relative discrete-series data on a Levi subgroup rather than the equivalent tempered formulation. These are convention changes in the parametrization, not different classifications Knapp, Chapter XIV.

The theorem does not by itself decide which quotients are unitary, nor does it describe their full composition series.

References

Robert P. Langlands, On the Classification of Irreducible Representations of Real Algebraic Groups, Institute for Advanced Study preprint, 1973; reprinted in Representation Theory and Harmonic Analysis on Semisimple Lie Groups, Mathematical Surveys and Monographs 31, American Mathematical Society, 1989. IAS record and author PDF. Relevant: §3, Lemmas 3.13–3.14, and §4, Lemma 4.2.
Anthony W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton Mathematical Series 36, Princeton University Press, 1986. Author-maintained record. Relevant: Chapter XIV on standard modules and the Langlands classification.