Multiplicity function of a representation - Imported Knowlpedia Content

Definition

Let $G$ be a second-countable type I locally compact group and $U$ a unitary representation on a separable complex Hilbert space. Suppose $U$ has the direct-integral disintegration

U\cong\int_{\widehat G}^{\oplus} \bigl(\pi\otimes I_{\mathcal M_\pi}\bigr)\,d\mu(\pi)

over the unitary dual. The multiplicity function is the measurable function

m:\widehat G\longrightarrow\{0,1,2,\ldots,\infty\}

where $(\mathcal M_\pi)$ is a measurable field of multiplicity Hilbert spaces and $m(\pi)=\dim\mathcal M_\pi$ . It is defined only $\mu$ -almost everywhere relative to the chosen measure; normally $m(\pi)\geq1$ on the chosen carrier of $\mu$ . In the separable setting, $m(\pi)=\infty$ means $\mathcal M_\pi\cong\ell^2(\mathbb N)$ , not an algebraic space $\mathbb C^\infty$ .

What it classifies

Under the stated type I and separability hypotheses, the measure class of $\mu$ and the almost-everywhere equivalence class of $m$ determine $U$ up to unitary equivalence. Conversely, equivalent representations have the same spectral measure class and multiplicity function after identifying null sets. This uniqueness is the central classification feature of type I disintegration Folland, §7.4.

The value $m(\pi)=n$ means that the fiber contains $n$ copies of $\pi$ ; $m(\pi)=\infty$ means countably infinite multiplicity in the separable setting. Multiplicity is therefore fiberwise, rather than the measure of the set on which $\pi$ occurs.

Examples

For a finite Hilbert direct sum $U=\pi_1\oplus\pi_1\oplus\pi_2$ of pairwise inequivalent irreducibles, take counting measure on $\{\pi_1,\pi_2\}$ ; then $m(\pi_1)=2$ and $m(\pi_2)=1$ . For a multiplicity-free representation, $m=1$ almost everywhere. An irreducible representation has spectral measure concentrated at one point with multiplicity one.

Conventions and scope

The underlying Hilbert space is the direct integral of Hilbert spaces, and the representation acts fiberwise as specified by the direct-integral decomposition.

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: §7.4, multiplicity theory for type I direct integrals.
Jacques Dixmier, $C^*$ -Algebras, North-Holland, 1977. Publisher record. Relevant: §18.8, disintegration and multiplicities.