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