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An induced representation carries a canonical system of imprimitivity on $G / H$

Statement

Assume AC. Let $G$ be a second-countable locally compact Hausdorff topological group, $H \leq G$ closed, and $σ : H \to U (V)$ a strongly continuous unitary representation on a separable Hilbert space $V$ . Let $Π_{σ} = {Ind}_{H}^{G} σ$ be the induced representation on the covariant completion $H_{σ}$ with rho-measure $μ_{ρ}$ . For a Borel set $E \subseteq G / H$ define $P (E)$ on the covariant model by $(P (E) F) (x) = 1_{E} (x H) F (x)$ . Then $P$ is a projection-valued measure on $G / H$ , $P (E)$ is well defined on the completed space of measurable covariant sections, $Π_{σ} (g) P (E) Π_{σ} (g)^{- 1} = P (g E)$ for all $g \in G$ and Borel $E$ , and $(Π_{σ}, P)$ is a system of imprimitivity on $G / H$ with $P (G / H) = I$ . If $H = G$ the base is one point and $P ({G}) = I$ ; if $H = {e}$ one may normalize so that the system is the multiplication system on $L^{2} (G; V)$ with the left regular action $F \mapsto F (g^{- 1} \cdot)$ .

Facts & Assumptions

Given: AC, the second-countable LCH group $G$ , closed $H \leq G$ , a strongly continuous unitary $σ : H \to U (V)$ on separable $V$ , and the induced representation $Π_{σ}$ on the covariant completion with rho-measure $μ_{ρ}$ .

[F1]

The covariant model consists of (classes of) functions $F : G \to V$ with $F (x h) = σ (h)^{- 1} F (x)$ , compactly supported modulo $H$ , with the norm obtained by integrating the descended pointwise norm against $μ_{ρ}$ ; the dense subspace of continuous covariant sections with compact support modulo $H$ generates the completion, and continuous compactly supported covariant generators are dense (Continuous covariant model and measurable completion, Density of averaged covariant generators, Well-defined induced inner product).

[F2]

The induced action is $(Π_{ρ} (g) F) (x) = D_{g} (x H)^{1 / 2} F (g^{- 1} x)$ with $D_{g} (x H) = ρ (g^{- 1} x) / ρ (x)$ ; it preserves the inner product, satisfies $Π_{ρ} (g_{1}) Π_{ρ} (g_{2}) = Π_{ρ} (g_{1} g_{2})$ , and extends to a unitary on the completion (Unitary cocycle-corrected left action, Unitary induction from a closed subgroup, Continuous quotient translation cocycle).

[F3]

$μ_{ρ}$ is a full-support strongly quasi-invariant Radon measure, so the descended norm integral is a genuine $L^{2}$ integral over the standard Borel $G$ -space $G / H$ ; multiplication by the indicator of a Borel set of finite $μ_{ρ}$ -measure is a bounded self-adjoint idempotent on the completed space, and dominated convergence gives strong countable additivity (Existence of rho-functions and quotient measure classes, Quasi-invariant Radon measure on G/H, Second-countable locally compact Hausdorff spaces are Polish, and homogeneous quotients are standard Borel, Bounded borel pvm integral).

[F4]

The induced representation is strongly continuous and the system of imprimitivity axioms require the covariance identity $U_{g} P (E) U_{g}^{- 1} = P (g E)$ (Strongly continuous unitary representations, invariant linear subspaces and intertwiners, Systems of imprimitivity for a Borel $G$ -space).

[F5]

For $H = G$ the quotient is a point and the covariant model is $V$ with the action $σ$ ; for $H = {e}$ the rho-measure may be taken to be Haar measure, covariant functions are unconstrained, and the induced space is $L^{2} (G; V)$ with action $F \mapsto F (g^{- 1} \cdot)$ (Left and right cosets $g H$ and $H g$ of a subgroup, Unitary induction from a closed subgroup, Borel cross-sections for closed subgroups of second-countable locally compact Hausdorff groups).

[F6]

AC is the standing hypothesis, inherited through the rho-measure and induction suppliers (The Axiom of Choice).

Proof

technique · direct

Given: AC, the group, subgroup, representation $σ$ and the induced model of [F1].

1.1F1F2F3

Identify the covariant completion with the square-integrable measurable covariant sections using the density of the continuous covariant generators in [F1]. Thus a Borel-indicator multiple of a section remains in the completed model. On this measurable model, $(P (E) F) (x) = 1_{E} (x H) F (x)$ is covariant: $(P (E) F) (x h) = 1_{E} (x H) F (x h) = σ (h)^{- 1} (P (E) F) (x)$ , since $x h H = x H$ . It is idempotent and self-adjoint for the induced inner product because $1_{E}^{2} = 1_{E} = \overline{1_{E}}$ pointwise, and it is a contraction: the pointwise norm of $P (E) F$ is at most that of $F$ everywhere. Hence $P (E)$ extends uniquely to a bounded self-adjoint idempotent on $H_{σ}$ .

2.1F1F3step 1.1

$P (\emptyset) = 0$ , $P (G / H) = I$ , and $P (E) P (F) = P (E \cap F)$ follow pointwise from the same identities for indicators, hence hold on the completion by density; strong countable additivity holds because for a disjoint union $E = ⨆ E_{n}$ the partial sums converge pointwise to $1_{E}$ and are bounded, so dominated convergence in the $L^{2}$ -integral gives $P (⋃_{n \leq N} E_{n}) ξ \to P (E) ξ$ for every $ξ$ . Thus $P$ is a projection-valued measure on the Borel $σ$ -algebra of $G / H$ .

2.2F2step 1.1

Covariance: by [F2], $(Π (g) P (E) Π (g)^{- 1} F) (x) = D_{g} (x)^{1 / 2} (P (E) Π (g)^{- 1} F) (g^{- 1} x) = D_{g} (x)^{1 / 2} 1_{E} (g^{- 1} x) (Π (g)^{- 1} F) (g^{- 1} x) = 1_{E} (g^{- 1} x) F (x) = 1_{g E} (x) F (x)$ , so $Π (g) P (E) Π (g)^{- 1} = P (g E)$ for all $g$ and Borel $E$ , first on the dense model and then everywhere by continuity.

3.1F4step 2.1step 2.2

Consequently $(Π_{σ}, P)$ is a system of imprimitivity: $P$ is a PVM by [step 2.1], $Π_{σ}$ is a strongly continuous unitary representation, and the covariance identity is [step 2.2], with $P (G / H) = I$ .

3.2F5step 2.1step 2.2

Boundary cases of the statement: if $H = G$ then $G / H$ is a singleton and the only Borel sets are $\emptyset$ and the point, so $P ({G}) = P (G / H) = I$ and the system is the given representation with the trivial base. If $H = {e}$ then $G / H = G$ , covariant functions are arbitrary, and with the Haar normalization $ρ \equiv 1$ the induced action is $F \mapsto F (g^{- 1} \cdot)$ on $L^{2} (G; V)$ while $P (E)$ is pointwise multiplication by $1_{E}$ ; this is the multiplication system of the statement.

4.1step 2.1step 2.2step 3.1step 3.2F6∎

Steps 2.1, 3.1 and 3.2 prove that $P$ is a well-defined projection-valued measure on the completed space, that the pair $(Π_{σ}, P)$ is a system of imprimitivity on $G / H$ , and the two boundary identifications.

Depends on

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Sources

G. W. Mackey, Imprimitivity for Representations of Locally Compact Groups I, PNAS 35 (1949) 537-545 (Internet Archive capture of the PubMed Central scan) (standard reference, not scraped)
V. S. Sunder, Notes on the Imprimitivity Theorem (ISIBangalore/IMSc lecture notes, 22 pp.) (standard reference, not scraped)
G. Misra, E. K. Narayanan and C. Varughese, Mackey Imprimitivity and commuting tuples of homogeneous normal operators, arXiv:2402.15737 (standard reference, not scraped)

An induced representation carries a canonical system of imprimitivity on G/HG/H