Langlands classification for p-adic groups

Statement

Let $F$ be a nonarchimedean local field and let $G$ be a connected reductive $F$ -group. The Langlands classification says that every irreducible admissible representation $\pi$ of $G(F)$ is the unique irreducible quotient of a normalized induced representation

I_P^G(\tau\otimes\chi),

where $P=MN$ is a parabolic subgroup with Levi subgroup $M$ , $\tau$ is an irreducible tempered representation of $M(F)$ , and the real unramified character $\chi$ lies in the chosen open positive Weyl chamber. The induced representation is the standard module, and its distinguished quotient is the Langlands quotient.

The triple $(M,\tau,\chi)$ is unique up to $G(F)$ -conjugacy once the positive chamber convention is fixed.

Boundary cases

If $\chi=1$ , the quotient is tempered and the data lie on the boundary of the positive chamber. Starting instead from an essentially square-integrable representation yields an equivalent refinement of the classification.

The result here is for reductive groups over nonarchimedean local fields. The real-group classification uses analogous standard modules but different categories and analytic input.

Relation to parameters

The extra $\operatorname{SL}_2$ in a nonarchimedean L-parameter records monodromy, while the unbounded real part of the Weil-group image parallels the positive unramified twist $\chi$ . This relation guides the extension of tempered local Langlands to all irreducible admissible representations.

References

Robert P. Langlands, “On the classification of irreducible representations of real algebraic groups,” 1973; the nonarchimedean analogue follows the same standard-module formalism.
Allan J. Silberger, Introduction to Harmonic Analysis on Reductive $p$ -adic Groups, Mathematical Notes 23, Princeton University Press, 1979, Chapter 5.