Tempered representation of a p-adic group

Definition

Let $F$ be a nonarchimedean local field and let $G$ be a connected reductive $F$ -group. An irreducible admissible representation $\pi$ of $G(F)$ with unitary central character is tempered if its matrix coefficients belong to

L^{2+\varepsilon}(G(F)/Z_G(F)) \qquad\text{for every }\varepsilon>0.

Here $Z_G$ is the center of $G$ . Equivalently, its unitary realization is weakly contained in the regular representation. For a nonunitary central character, one first makes the standard unitary twist when that formulation is available.

Position in the classification

Every irreducible representation that is square-integrable modulo the center is tempered. Tempered representations are the building blocks in the p-adic Langlands classification: general irreducibles are quotients of inductions of tempered data twisted into a positive chamber.

Parameter-side expectation

The local Langlands correspondence is expected to match tempered representations with bounded L-parameters. For groups where local Langlands is established, this is a theorem subject to the normalization of the correspondence.

References

Harish-Chandra, “Harmonic analysis on reductive $p$ -adic groups,” in Harmonic Analysis on Homogeneous Spaces, Proceedings of Symposia in Pure Mathematics 26, 1973.
William Casselman, “Introduction to the theory of admissible representations of $p$ -adic reductive groups,” unpublished notes, §§4–6. UBC.