System of imprimitivity - Imported Knowlpedia Content

Definition

Let a locally compact group $G$ act measurably on a standard Borel space $X$ . A system of imprimitivity for this action on a complex Hilbert space $\mathcal H$ is a pair $(U,P)$ , where $U$ is a strongly continuous unitary representation of $G$ on $\mathcal H$ and $P$ is a projection-valued measure on $X$ , such that

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

for every $g\in G$ and every Borel set $E\subseteq X$ . The equation is the covariance axiom linking the representation to the action on measurable subsets.

Function-algebra formulation

Integrating bounded measurable functions against $P$ gives a representation $M_P$ by operators on $\mathcal H$ . Covariance becomes

U(g)M_P(f)U(g)^{-1}=M_P(f\circ g^{-1}).

Thus a system of imprimitivity can equivalently be viewed as a covariant representation of the action on functions. The projection-valued and function-algebra formulations contain the same information.

Transitive actions and induction

When $X=G/H$ is a locally compact [[lie-groups/homogeneous-space|homogeneous space]] for a closed subgroup $H$ , Mackey's imprimitivity theorem classifies systems based on $G/H$ by unitary representations of $H$ . The associated representation of $G$ is induced from $H$ , and $P(E)$ acts by multiplication by the indicator of $E$ . This equivalence is the main bridge between induction and spectral localization Mackey, pp. 537–545.

Examples and non-examples

For the translation action of $G$ on itself, let $\mathcal H=L^2(G)$ , let $U$ be the left regular representation, and let $P(E)$ multiply by $1_E$ . This is a system of imprimitivity. Pairing the same $P$ with a representation that does not transport supports according to the action fails the covariance axiom.

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 theorem.
Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: §6.3, systems of imprimitivity and induced representations.