Standard module of a real reductive group - Imported Knowlpedia Content

Definition

Let $G$ be a real reductive Lie group, let $P=MAN$ be a Langlands decomposition of a standard parabolic subgroup, and choose positive roots for $(P,A)$ . If $\sigma$ is an irreducible tempered representation of $M$ and $\nu\in\mathfrak a_{\mathbb C}^{*}$ has real part in the open positive chamber, the standard module

I(P,\sigma,\nu)=\operatorname{Ind}_{P}^{G} \bigl(\sigma\otimes e^\nu\otimes 1_N\bigr)

is formed using normalized parabolic induction. The same name is used for its smooth globalization and for its underlying admissible $(\mathfrak g,K)$ -module when that choice is clear.

Role in the Langlands classification

The positivity of $\operatorname{Re}\nu$ orders the inducing data and is what distinguishes a standard module from an arbitrary parabolically induced representation. The Langlands classification theorem says that a standard module has a unique irreducible quotient and that every irreducible admissible representation occurs in this way, with its data unique up to the customary conjugacies Langlands, §3, Lemmas 3.13–3.14, and §4, Lemma 4.2. That quotient is the Langlands quotient.

Example

For $G=\mathrm{SL}(2,\mathbb R)$ , take its upper-triangular minimal parabolic. A character of the split diagonal factor with positive real parameter produces a standard principal-series module. At reducibility parameters the induced module can have several irreducible subquotients, but the positive-chamber ordering still singles out one irreducible quotient.

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 the Langlands classification and standard induced modules.
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 canonical irreducible quotient and uniqueness, and §4, Lemma 4.2 on exhaustion.