Langlands quotient - Knowlpedia

Let $I(P,\sigma,\nu)$ be a standard module of a real reductive group, with $\operatorname{Re}\nu$ in the chosen open positive chamber. The Langlands classification theorem gives this module a unique irreducible quotient. That quotient,

J(P,\sigma,\nu),

is its Langlands quotient. The term applies to the irreducible admissible $(\mathfrak g,K)$ -module and, after choosing a compatible globalization, to the corresponding representation of $G$ . It is not required to be unitary: unitarity is an additional property of particular Langlands parameters.

Classification statement

Every irreducible admissible representation of $G$ is the Langlands quotient of some standard module. With a fixed positive system, the inducing data are unique up to the standard conjugacies and equivalences on the Levi factor. This makes the quotient construction a parametrization theorem, not merely a way to manufacture examples.

Example

For a real rank-one group, a positive-parameter principal series can be reducible at special parameters. Its composition series may contain both a finite-dimensional constituent and an infinite-dimensional constituent, but the chosen chamber and quotient orientation select exactly one as $J(P,\sigma,\nu)$ . Replacing the positive parameter by its negative can exchange which constituent appears as a quotient rather than a submodule.

Conventions and scope

References

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, uniqueness of the irreducible quotient, and the Langlands classification.
Robert P. Langlands, On the Classification of Irreducible Representations of Real Algebraic Groups, Institute for Advanced Study, 1973. Author PDF. Relevant: §3, Lemmas 3.13–3.14 on the irreducible quotient and uniqueness, and §4, Lemma 4.2 on exhaustion.