Little groups for the real $ax+b$ group and its orientation-preserving subgroup

The dual action is $(k\cdot {\chi }_{\lambda })(b)=e^{i\lambda b\mathrm{/}k}$, so the parameter is $\lambda \mathrm{/}k$. For $K={\mathbb{R}}^{\times }$ the orbits are $\{0\}$ and ${\mathbb{R}}^{\times }$, with stabilizers $K$ and $\{1\}$; for $K={\mathbb{R}}_{>0}$ they are $\{0\}$, $(0,\mathrm{\infty })$, and $(-\mathrm{\infty },0)$, again with trivial nonzero stabilizers. Each partition is finite and Borel, hence regular, so [F3] makes the corresponding little-group inductions exhaustive.

At the zero character the inducing subgroup is $G$ and $N$ acts trivially. The irreducible quotient representations are one-dimensional by [F5]. Logarithm and the sign decomposition ${\mathbb{R}}^{\times }\cong {\mathbb{R}}_{>0}\times \{\pm 1\}$ give $\mathrm{\mid }a{\mathrm{\mid }}^{it}\mathrm{sign}(a{)}^{\delta }$ for the full group and $a^{it}$ for the positive group. Distinct parameters give distinct characters: vary $\log a$ and then the sign.

For $\lambda \mathrm{\ne }0$ use quotient coordinates $k\in K$ with section $s(k)=(0,k)$ and Haar measure $dk\mathrm{/}\mathrm{\mid }k\mathrm{\mid }$ (restricted to $k>0$ for the positive group). The induced action on $L^2(K,dk\mathrm{/}\mathrm{\mid }k\mathrm{\mid })$ is $({\pi }_{\lambda }(b,a)f)(k)=e^{i\lambda b\mathrm{/}k}f(k\mathrm{/}a)\mathrm{.}$ Indeed $s(k{)}^{-1}(b,a)s(k\mathrm{/}a)=(b\mathrm{/}k,1)$, and the quotient measure is invariant under $k\mapsto ak$, so its density factor is one. Finite scalar Borel measures on the real line are regular by [F4]. The spectral PVM of $N$ is multiplication by ${\mathbf{1}}_E(\lambda \mathrm{/}k)$: it is a regular PVM under the homeomorphism $k\mapsto \lambda \mathrm{/}k$ onto the nonzero orbit, and its character integral is the displayed translation action, so [F4]'s spectral uniqueness identifies it.

Let $A$ commute with ${\pi }_{\lambda }$. It commutes with all integrated $N$-operators, hence with their $C_0$ algebra by Fourier-transform density, and then with the spectral PVM. For the last inference, each unitary in the commutant conjugates $P$ to a regular PVM with the same integrated $N$-representation, hence preserves $P$ by spectral uniqueness. For a general commutant operator, its self-adjoint real and imaginary parts commute with $N$, and $e^{itS}$ for either part $S$ is a commuting unitary, by the norm-convergent power series. Differentiating that series at $t=0$ shows that $S$ commutes with $P$, and hence so does $A$. Thus $A$ commutes with all diagonal multiplications on $K$ and is $M_u$ for some bounded scalar $u$, by the diagonal commutant theorem. Commutation with ${\pi }_{\lambda }(0,a)$ makes $u(k\mathrm{/}a)=u(k)$ a.e. for every $a\in K$. Transitive ergodicity, applied to rational superlevel sets of the real and imaginary parts, makes $u$ constant a.e. Thus the commutant is scalar; an invariant closed subspace would have a commuting orthogonal projection, so ${\pi }_{\lambda }$ is irreducible. Disjoint intervals in $\log \mathrm{\mid }k\mathrm{\mid }$ give infinitely many nonzero orthogonal indicator sections, proving infinite dimension.

For $r\in K$ put $(R_{r}f)(k)=f(rk)$. Haar invariance makes $R_r$ unitary, and the explicit formula gives $R_r{\pi }_{\lambda }(b,a)R_r^{-1}={\pi }_{\lambda \mathrm{/}r}(b,a)$. Thus parameters in one orbit give equivalent representations. For the full group $r=-1$ equates $\lambda =1$ and $\lambda =-1$; for the positive group no $r$ changes sign, and the two classes are inequivalent because their spectral PVMs have disjoint supports, by [F4]'s uniqueness. They are also inequivalent to the quotient characters, whose $N$-spectrum is $\{0\}$.

Exhaustiveness from step 1.1, the character classification of step 2.1, irreducibility and infinite dimension from step 3.1, and the equivalences of step 4.1 give $\widehat{G}=(\mathbb{R}\times \{0,1\})\bigsqcup \{\pi \}$ and $\widehat{G_0}=\mathbb{R}\bigsqcup \{{\pi }_1,{\pi }_{-1}\}$.