Mackey induction classifies regular Euclidean dual actions

Since $N$ is second-countable, locally compact, Hausdorff, and abelian, its dual has the compact-open character topology of [F2] and is locally compact Hausdorff by [F2]. The dual action is continuous: evaluation $(\gamma ,a)\mapsto \gamma (a)$ is continuous because for ${\gamma }_0(a_0)$ in an open $W\subseteq \mathbb{T}$, one may choose a compact neighborhood $C_0$ of $a_0$ inside ${\gamma }_0^{-1}(W)$, and then $S(C_0,W)\times \mathrm{int}(C_0)$ is a neighborhood mapped into $W$. Therefore $(k,\gamma ,a)\mapsto \gamma ({\alpha }_{k^{-1}}(a))$ is continuous. For compact $C\subseteq N$ and open $W\subseteq \mathbb{T}$, the tube lemma on $C$ shows that the inverse image of the compact-open subbasic set $S(C,W)$ under $(k,\gamma )\mapsto k\cdot \gamma$ is open. Hence this dual action is continuous, and the assumed invariant family makes its orbit relation regular. The stabilizer $K_{\chi }$ is closed as the inverse image of $\{\chi \}$ under the continuous orbit map. It inherits a countable basis from $K$; if $V$ is a relatively compact open neighborhood in $K$, then $V\cap K_{\chi }$ is relatively open in $K_{\chi }$ and its closure lies in the compact set $\bar{V}\cap K_{\chi }$, so $K_{\chi }$ is locally compact. Thus $K_{\chi }$ is second-countable locally compact and $H_{\chi }$ is closed by [F1]. The product-topology group $G=N\times K$ is second-countable locally compact, so [F3] applies. Applying [F3] to an irreducible representation $\pi$ gives an orbit $O=K\cdot \chi$ and an irreducible strongly continuous $\theta :K_{\chi }\to U(V)$ with $\pi \simeq {\mathrm{Ind}}_{H_{\chi }}^G(\chi \otimes \theta )$. To verify the separability needed below, let $\sigma$ be either $\pi$ or $\theta$ and let $v\mathrm{\ne }0$ in its carrier. The closed span of $\sigma (J)v$, where $J=G$ or $K_{\chi }$ respectively, is a nonzero invariant subspace and hence the whole carrier by irreducibility. Each $J$ is second-countable; [F9] and [A1] give a countable dense subset $D_J\subset J$. Strong continuity makes $\{\sigma (d)v:d\in D_J\}$ dense in the orbit $\sigma (J)v$, and finite $\mathbb{Q}(i)$-linear combinations of these vectors form a countable dense subset of the carrier by [F10]. Thus both $\pi$ and $\theta$ have separable carriers.

For $h\in K_{\chi }$, $\chi ({\alpha }_h(a))=(h^{-1}\cdot \chi )(a)=\chi (a)$. Therefore, using [F1], $(\chi \otimes \theta )((a,h)(b,j))=\chi (a+{\alpha }_h(b))\theta (hj)=\chi (a)\chi (b)\theta (h)\theta (j),$ so $\chi \otimes \theta$ is a representation of $H_{\chi }$. It is strongly continuous because both factors are, and $H_{\chi }$ is a closed subgroup.

The orbit $O$ is Borel: it is the intersection, over the countable separating invariant family, of each set containing $\chi$ and each complement of a set not containing $\chi$; points in $O$ have the same memberships as $\chi$, and any point outside $O$ is separated by one set. The orbit map $K\mathrm{/}{K}_{\chi }\to O$, $k{K}_{\chi }\mapsto k\cdot \chi$, is a continuous bijection. Its inverse is Borel: every open subset of the second-countable locally compact quotient is a countable union of compact subsets, whose images under the orbit map are compact and hence closed in the Hausdorff dual. The map $X=G\mathrm{/}{H}_{\chi }\to K\mathrm{/}{K}_{\chi }$, $(a,k)H_{\chi }\mapsto k{K}_{\chi }$, is a homeomorphism: translations lie in $H_{\chi }$, and $(a,k{)}^{-1}(b,\mathrm{\ell })\in H_{\chi }$ exactly when $k^{-1}\mathrm{\ell }\in K_{\chi }$, since the first coordinate is in $N\le H_{\chi }$. Thus $O$ and $X$ are Borel-isomorphic, as required to use the transitive system on $G\mathrm{/}{H}_{\chi }$ in [F4]. Composing this Borel identification with [F6] gives a Borel section $s:O\to K$ with $s(x)\cdot \chi =x$. In section coordinates, $f(x)=F((0,s(x)))$; its inverse is the covariant extension $F((0,s(x))h)=(\chi \otimes \theta )(h{)}^{-1}f(x)$. These are Borel inverse maps and preserve the quotient norm. By [F5] choose a full-support quasi-invariant Radon measure on $X$ and transport it across the Borel identification to $O$; write $D_g$ for its transported Radon–Nikodym cocycle. The section form of induction is $(U(a,k)f)(x)=D_{(a,k)}(x{)}^{1\mathrm{/}2}x(a)\theta \text{\hspace{0.17em}⁣}(s(x{)}^{-1}ks(k^{-1}\cdot x))f(k^{-1}\cdot x),$ where $D_g$ is the Radon–Nikodym cocycle from [F5]. To see the fibre factor, write $q=k^{-1}\cdot x$, $r=s(q{)}^{-1}{k}^{-1}s(x)\in K_{\chi }$, and $b=-{\alpha }_{s(q{)}^{-1}{k}^{-1}}(a)$; then $(a,k{)}^{-1}(0,s(x))=(0,s(q))(b,r)$. Covariance $F(yh)=(\chi \otimes \theta )(h{)}^{-1}F(y)$ gives $\chi (-b)=x(a)$ by the dual action definition, and $\theta (r{)}^{-1}=\theta (s(x{)}^{-1}ks(q))$. The little-group factor lies in $K_{\chi }$ because $s(x{)}^{-1}ks(k^{-1}\cdot x)\cdot \chi =\chi$. The formula is Borel, and [F5] gives its unitary extension and strong continuity.

The section model is the measurable Hilbert-field realization of $L^2(X,\mu ;V)$, with base $X=G\mathrm{/}{H}_{\chi }$ and measure as in step 2.1. By [F9], $X$ has a countable Borel generator and an increasing compact exhaustion $C_n$; the Radon measure is finite on each $C_n$. Let $\mathcal{A}$ be the countable algebra generated by a countable basis together with the exhaustion. On each finite-measure $C_n$, the class of Borel subsets approximable in symmetric-difference measure by $\mathcal{A}\cap C_n$ is a sigma algebra: complements are taken relative to $C_n$, finite unions preserve approximation, and countable unions are reduced to finite unions by continuity from below of the finite measure. It contains the restricted generating algebra, so it contains every Borel subset of $C_n$. Therefore simple functions from this countable algebra are dense in scalar $L^2(C_n,\mu )$; truncation to $C_n$ shows their union is dense in $L^2(X,\mu )$. Since $V$ is separable by step 1.1, finite simple functions with values in a countable dense subset of $V$ are dense in $L^2(X,\mu ;V)$; this set has a countable dense rational-complex span. The induced Hilbert space is therefore separable, and its closed subspaces $W,W^{\perp }$ are separable as required by [F4].

The section cocycle obeys $\kappa (k,x)\kappa (\mathrm{\ell },k^{-1}\cdot x)=\kappa (k\mathrm{\ell },x)$ by cancellation of $s(k^{-1}\cdot x)$, while $(k^{-1}\cdot x)(b)=x({\alpha }_k(b))$ by the dual action definition. Together with the Radon–Nikodym cocycle identity in [F5], these identities verify the semidirect-product multiplication in the displayed action. If $\mu$ is invariant, $D_g=1$; with a merely quasi-invariant $\mu$, the square-root factor is essential for unitarity.

Setting $k=e$ in the section formula shows that each translation $a\in N$ acts by multiplication by $x(a)$. The unique spectral PVM [F7] for this multiplication representation is therefore multiplication by the indicators of Borel subsets of $O$ in orbit coordinates; its value on $O$ is the identity.

Suppose the induced representation has a nonzero proper closed invariant subspace $W$. Its orthogonal complement is nonzero and invariant: for $u\in W^{\perp }$, $w\in W$, and $g\in G$, $⟨U(g)u,w⟩=⟨u,U(g^{-1})w⟩=0$. Hence the orthogonal projection onto $W$ commutes with $G$, and in particular with every translation operator; uniqueness in [F7] makes it commute with the translation PVM. Thus $W$ and $W^{\perp }$ inherit nonzero PVMs on $O$. Transport these PVMs along the equivariant Borel isomorphism of step 2.1 to $X=G\mathrm{/}{H}_{\chi }$. For either transported restriction $Q_W$, define $S_W=\{x\in X:Q_W(V)\mathrm{\ne }0\text{ for every open }V\ni x\}$. This set is closed and $G$-invariant by covariance. It is nonempty: otherwise the countable basis of $X$ from [F9] would cover $X$ by sets with zero projection, contradicting $Q_W(X)=I_W$ by countable additivity. Transitivity on $X$ gives $S_W=X$. The restrictions therefore give nonzero transitive systems of imprimitivity on $G\mathrm{/}{H}_{\chi }$, with separable carriers by step 3.1. This argument uses the quotient topology on $X$, not any unproved topological identification with $O$.

If two induced representations are equivalent, an intertwiner $T$ makes $TP{T}^{-1}$ a regular PVM for the other translation representation, so uniqueness [F7] makes it that representation's spectral PVM. Each PVM has value $I$ on its inducing orbit by step 3.3 and vanishes on every disjoint orbit; hence equivalence forces the inducing orbits to agree. At a fixed representative $\chi$, their transitive systems on $G\mathrm{/}{H}_{\chi }$ are equivalent through the identification in step 2.1, and uniqueness [F4] implies their little-group representations are equivalent. Conversely an equivalence of little-group representations induces an equivalence of the covariant models by acting with the fibre unitary pointwise. If ${\chi }^{\mathrm{\prime }}=k\cdot \chi$ is another representative, then $K_{{\chi }^{\mathrm{\prime }}}=k{K}_{\chi }{k}^{-1}$ and conjugation by $(0,k)$ carries $\chi \otimes \theta$ to ${\chi }^{\mathrm{\prime }}\otimes {\theta }^{\mathrm{\prime }}$, where ${\theta }^{\mathrm{\prime }}(h^{\mathrm{\prime }})=\theta (k^{-1}{h}^{\mathrm{\prime }}k)$; the corresponding induced models are equivalent by the same change of variables. Thus the orbit and little-group class determine exactly one irreducible representation class.

Put $\sigma =\chi \otimes \theta$. Any closed $\sigma$-invariant subspace is $\theta$-invariant because $\sigma (0,h)=\theta (h)$, so $\sigma$ is irreducible. The induced carrier is nonzero: full support and the Radon property give a Borel set $E\subset X$ with $0<\mu (E)<\mathrm{\infty }$, and ${\mathbf{1}}_{E}v$ is a nonzero square-integrable section for $v\mathrm{\ne }0$ in $V$. If $W$ exists as in step 4.1, apply [F4] to reconstruct its two restricted systems from strongly continuous representations ${\sigma }_W$ and ${\sigma }_{W^{\perp }}$ of $H_{\chi }$ on separable fibres $V_W$ and $V_{W^{\perp }}$. Both fibres are nonzero, since a zero fibre induces the zero system. Realize their canonical systems with the same quotient measure and section. Pointwise splitting of $V_W\oplus V_{W^{\perp }}$-valued functions is unitary and intertwines both the induced actions and the multiplication PVMs; hence the canonical system of ${\sigma }_W\oplus {\sigma }_{W^{\perp }}$ is their direct sum, which is the original system on $W\oplus W^{\perp }$. Uniqueness in [F4] gives $\sigma \simeq {\sigma }_W\oplus {\sigma }_{W^{\perp }}$, contradicting irreducibility of $\sigma$. Therefore the induced representation is irreducible.

Step 1.1 yields every irreducible as one of the induced models. Steps 1.2–3.3 establish the covariant model and its orbit PVM; steps 4.1 and 5.1 prove irreducibility; step 4.2 proves the equivalence criterion. Therefore the stated map is a bijection.