The motion group of $\mathbb R^2$, noted by $G=M(2)$ is the semi-direct product of $\mathbb R^2$ with the special orthogonal group $K = SO(2)$. A well
known fact is that the unitary dual $\hat{G}$, of $G$ is precisely the collection [see, Sugiura, M.: Unitary Representations and Harmonic Analysis. Kodansha Scientific books, Tokyo (1975), p: 165]:
$$\{\pi\_\lambda,\, \lambda\in>0 \}\cup\{\chi\_n: \, n\in \mathbb Z \},$$ 
where $[\pi\_{\lambda}(z,e^{i\alpha})u](s) = e^{i\lambda \left<z,e^{is}\right>} u(s-\alpha)$ and where $\chi\_n(z,e^{i\alpha})=e^{in\alpha}$.


Now, the infinite dimensional, unitary irreducible representations of the motion group $M(n)$ of $\mathbb R^n$ is given by [see, N. I. Vilenkin, (1978). Special functions and the theory of group representations (Vol. 22). American Mathematical Soc]:


For each $\lambda>0$, a unitary representation $\pi\_{\lambda}$ of $M(n)$ realized on $L^2(S^{n-1})$ is
\begin{align\*}
 (\*) \quad [\pi\_{\lambda}(a,k)u](\xi) = e^{i\lambda \left<x,\xi\right>}\, u(k^{-1}\xi), 
\end{align\*}
for $(x,k)\in M(n)=\mathbb R^n \rtimes SO(n)$ and $u \in L^2(S^{n-1})$.


Apart from these family, we have another family $\{\chi\_m, m \in ? \}$, of one dimensional unitary representations of $M(n)$, which ? Is what it is $\chi\_n(x,k)=e^{i m.k}, m=(m\_1, \dots, m\_n) \in \mathbb Z^{n} $ with $m.k= m\_1 k\_1 + \dots + m\_n k\_n$. Or what ?


Thank you in advance



Unitary dual of the motion group $M(n)$, for $n> 2$
This is done by Mackey theory and discussed in many places, e.g. see Lipsman ([1974](http://ams.org/mathscinet-getitem?mr=372116), page [72](https://books.google.com/books?id=wzN7CwAAQBAJ&pg=PA72)) for an explicit list. In short, there are two series:


1. Your sought $\chi$’s: all (finite dimensional) irreps of $\mathrm{SO}(n)=M(n)/\mathbf R^n$ lifted from that quotient;
2. Representations induced from irreps $e^{i\langle\lambda,\cdot\rangle}\otimes\sigma$ of the stabilizer $\mathbf R^n\rtimes\mathrm{SO}(n-1)$ of a pole in the action of $M(n)$ on $S^{n-1}$ (via $\mathrm{SO}(n)$ again). For nontrivial $\sigma$ these act not in $L^2(S^{n-1})$ but in $L^2$ sections of (associated) vector bundles on $S^{n-1}$.


Best understood, in my opinion, from the point of view of Kirillov’s orbit method: (1) are attached to the compact coadjoint orbits of $M(n)$, which are themselves pulled back from the quotient $\mathrm{SO}(n)$; (2) to the noncompact orbits, which are themselves “symplectically induced” from coadjoint orbits of the stabilizer.


