Tempered dual of a locally compact group - Imported Knowlpedia Content

Definition

Let $G$ be a locally compact group, and let $\widehat G$ be its unitary dual. The tempered dual $\widehat G_{\mathrm{temp}}$ is the set of classes $[\pi]\in\widehat G$ for which $\pi$ is tempered, equivalently

\pi\prec\lambda_G,

where $\lambda_G$ is the left regular representation and $\prec$ denotes weak containment. It carries the topology inherited from the Fell topology on $\widehat G$ , hence the subspace topology. This definition does not require $G$ to be type I, although type I hypotheses are important for measurable decomposition and Plancherel theory.

Reduced-algebra interpretation

The class $[\pi]$ lies in $\widehat G_{\mathrm{temp}}$ exactly when the integrated form of $\pi$ factors through the reduced group $C^*$ -algebra $C_r^*(G)$ . Thus the tempered dual is also called the reduced dual: it is the irreducible representation spectrum seen by $C_r^*(G)$ , rather than by the full group $C^*$ -algebra Dixmier, §18.8.

Role in Plancherel theory

For a second-countable unimodular type I group, the Plancherel measure is supported on the tempered dual. The direct-integral decomposition of the regular representation therefore detects only tempered irreducibles, even when $\widehat G_{\mathrm{temp}}$ is a proper subset of $\widehat G$ . For real reductive groups, this subset includes the discrete series and the tempered principal series Knapp, Chapter XIV.

Conventions and scope

Some authors use “tempered dual” only for a specified class of reductive Lie groups, while the weak-containment definition applies to every locally compact group. “Plancherel-supported dual” can also mean the support of a particular representative of Plancherel measure; the intrinsic object here is the reduced dual, defined before any choice of Haar measure.

References

Jacques Dixmier, $C^*$ -Algebras, North-Holland Mathematical Library 15, North-Holland, 1977. Publisher record. Relevant: §18.8 on the reduced dual and Plancherel theory.
Anthony W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, 1986. Author-maintained record. Relevant: Chapter XIV on tempered representations and the Plancherel theorem.