Plancherel measure for a type I locally compact group

Definition

Let $G$ be a second-countable, unimodular, type I locally compact group, fix a left Haar measure $dg$ , and let $\widehat G$ be its unitary dual. For $f\in L^1(G)\cap L^2(G)$ , set $\pi(f)=\int_G f(g)\pi(g)\,dg$ . The Plancherel measure $\mu_{\mathrm{Pl}}$ is the unique positive Borel measure on $\widehat G$ for which

\lVert f\rVert_2^2 =\int_{\widehat G}\lVert\pi(f)\rVert_{\mathrm{HS}}^2\, d\mu_{\mathrm{Pl}}(\pi).

The transform extends to a unitary map from $L^2(G)$ onto the direct integral of the Hilbert–Schmidt operator spaces.

Regular-representation decomposition

Under the Plancherel transform, the left regular representation acts fiberwise by left multiplication with $\pi(g)$ . Consequently, the measure class of $\mu_{\mathrm{Pl}}$ is the spectral measure class of the regular representation in its direct-integral decomposition. The type I hypothesis supplies the standard measurable irreducible decomposition and its essential uniqueness Folland, Chapter 7, “The Plancherel Theorem”.

Examples and support

For an abelian $G$ , every irreducible representation is a character and $\mu_{\mathrm{Pl}}$ is the Haar measure on the Pontryagin dual normalized dually to $dg$ . For compact $G$ with Haar probability measure, it is discrete and assigns mass $\dim\pi$ to each irreducible class. In general, its support in the Fell topology is the tempered part of $\widehat G$ , which may be smaller than the full unitary dual.

Conventions and nonunimodular groups

Rescaling $dg$ rescales the representative Plancherel measure, although its null sets and hence its measure class are unchanged. For a nonunimodular group, the displayed Hilbert–Schmidt formula must be modified by a measurable field of positive, generally unbounded Duflo–Moore operators; the bare formula in the core is therefore not valid unchanged Duflo--Moore, abstract and main construction.

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. Publisher record. Relevant: Chapter 7, sections “Direct Integral Decompositions” and “The Plancherel Theorem.”
Michel Duflo and Calvin C. Moore, “On the Regular Representation of a Nonunimodular Locally Compact Group,” Journal of Functional Analysis 21 (1976), 209–243. DOI record. Relevant: the operator-valued correction to the nonunimodular Plancherel formula.