Mackey imprimitivity theorem - Imported Knowlpedia Content

Statement

Let $G$ be a second-countable locally compact Hausdorff group, let $H$ be a closed subgroup, and let $X=G/H$ . The Mackey imprimitivity theorem states that every system of imprimitivity $(U,P)$ for the transitive action of $G$ on $X$ is unitarily equivalent to one obtained from a strongly continuous unitary representation $\sigma$ of $H$ : the representation $U$ is $\operatorname{Ind}_H^G\sigma$ , and $P(E)$ acts on its section model by multiplication by $1_E$ . The representation $\sigma$ is determined up to unitary equivalence by $(U,P)$ .

From a stabilizer to a system

Starting with $\sigma$ , form the induced representation on $L^2$ -sections of $G\times_HV_\sigma\to G/H$ . Multiplication by bounded functions on $G/H$ , or equivalently by the projections $1_E$ , supplies a projection-valued measure. Translation of sections transports multiplication operators according to

U(g)P(E)U(g)^{-1}=P(gE),

so the pair is a system of imprimitivity.

Recovering the stabilizer representation

The converse is the substantive direction. Covariance forces the spectral data represented by $P$ to be spread transitively over $G/H$ . A measurable decomposition over the base isolates a fiber over the identity coset, and the stabilizer $H$ acts unitarily on that fiber. Re-inducing this $H$ -action reconstructs the original pair, including both the representation and its projection-valued measure Mackey, pp. 537–545.

Why the projection data matters

The theorem does not say that every unitary representation of $G$ is induced from $H$ . It classifies representations equipped with a compatible system based on the specified transitive $G$ -space. The same representation $U$ can admit inequivalent systems of imprimitivity, while a representation without such a system need not arise from that stabilizer.

Example

For the translation action of $G$ on $G/H$ , the system attached to the trivial representation of $H$ has $U$ equal to the quasi-regular representation and $P(E)$ equal to multiplication by $1_E$ . The theorem identifies the fiber representation at $eH$ as the trivial $H$ -action.

References

George W. Mackey, “Imprimitivity for Representations of Locally Compact Groups I,” Proceedings of the National Academy of Sciences 35 (1949), 537–545. DOI record. Relevant: the transitive imprimitivity classification.
Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: Chapter 6, Theorem 6.31 and the surrounding construction.