Mackey machine for semidirect products

Statement

Let $A$ and $K$ be second-countable locally compact Hausdorff groups, with $A$ abelian, and let $G=A\rtimes K$ be their semidirect product. Suppose the $K$ -action on the Pontryagin dual $\widehat A$ is regular. For $\chi\in\widehat A$ , let $K_\chi$ be its stabilizer. The Mackey machine states that every irreducible unitary representation of $G$ is

\operatorname{Ind}_{A\rtimes K_\chi}^{G}(\widetilde\chi\otimes\sigma),

where $\widetilde\chi(a,k)=\chi(a)$ and $\sigma$ is an irreducible unitary representation of $K_\chi$ . Two such representations are equivalent exactly when their characters lie in the same $K$ -orbit and their little-group representations correspond under conjugation.

Why the classification works

Restricting a representation of $G$ to the abelian normal subgroup $A$ produces spectral data on $\widehat A$ . Under the regularity hypothesis, irreducibility concentrates that data on one $K$ -orbit. The resulting transitive system of imprimitivity is classified by the Mackey imprimitivity theorem, which recovers a representation of $K_\chi$ ; unitary induction then reconstructs the original representation.

Example: Euclidean motion groups

For $G=\mathbb R^n\rtimes SO(n)$ , the dual of the translation subgroup is $\widehat{\mathbb R^n}\cong\mathbb R^n$ . Nonzero orbits are spheres. The stabilizer of a nonzero covector is isomorphic to $SO(n-1)$ , so a radius together with an irreducible representation of $SO(n-1)$ determines an induced irreducible representation. The zero orbit has little group $SO(n)$ and gives representations trivial on translations.

Hypotheses and scope

Regularity may be expressed by the existence of a Borel cross-section for the orbit space; it prevents pathologies in which orbit data do not provide a usable measurable parametrization. The theorem above exploits both the abelianness of $A$ and the splitting $G=A\rtimes K$ . For a general normal subgroup, one replaces characters by irreducible representations and may encounter projective representations and a Mackey obstruction. “Mackey machine” is also used for that broader analysis.

References

George W. Mackey, “Unitary Representations of Group Extensions I,” Acta Mathematica 99 (1958), 265–311. Springer DOI record. Relevant: the orbit, stabilizer, and multiplier analysis for group extensions.
Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: §6.6, especially Theorem 6.43 on semidirect products with an abelian normal subgroup.