Tempered unitary representation - Imported Knowlpedia Content

Definition

Let $G$ be a locally compact group and $\pi$ a strongly continuous unitary representation of $G$ . The representation $\pi$ is tempered if it is weakly contained in the left regular representation $\lambda_G$ :

\pi\prec\lambda_G.

Equivalently, the integrated form of $\pi$ factors through the reduced group $C^*$ -algebra $C_r^*(G)$ . Some authors reserve “tempered” for irreducible representations; under that convention, $\pi$ is additionally assumed irreducible. Temperedness depends only on the unitary-equivalence class of $\pi$ , not on a chosen realization of its Hilbert space.

Almost-square-integrable coefficients

For a connected semisimple Lie group with finite center, an irreducible unitary representation is tempered exactly when all its $K$ -finite matrix coefficients lie in $L^{2+\varepsilon}(G)$ for every $\varepsilon>0$ . This “almost $L^2$ ” criterion is a major practical test for temperedness Cowling–Haagerup–Howe, Theorem 1.

Examples and boundary cases

The regular representation is tempered, as are its subrepresentations and weakly contained representations. Discrete-series representations of a semisimple group are tempered. The trivial representation is tempered exactly when $G$ is amenable; therefore it is a decisive non-example for nonamenable groups.

Role in harmonic analysis

Tempered irreducibles form the natural support of Plancherel theory: the regular representation decomposes over them, with measure described by nonabelian Plancherel measure. They also constitute the part of the unitary dual detected by $C_r^*(G)$ , explaining why temperedness connects representation theory with reduced group $C^*$ -algebras Knapp, Chapter VIII.

References

Michael Cowling, Uffe Haagerup, and Roger Howe, “Almost $L^2$ Matrix Coefficients,” Journal für die reine und angewandte Mathematik 387 (1988), 97–110. DOI record. Relevant: Theorem 1 and the $L^{2+\varepsilon}$ criterion.
Anthony W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, 1986. Author record. Relevant: Chapter VIII on tempered representations and Plancherel theory.