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


This is done by Mackey theory and discussed in many places, e.g. see Lipsman (1974, page 72) 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.

  • $\begingroup$ @ Francois Ziegler, I would like to know an explicit expression of the unitary dual of $M(n)$ as it is given for the case $M(2)$. $\endgroup$
    – Z. Alfata
    Jan 7 '18 at 19:01
  • $\begingroup$ I added details; Lipsman’s list is “explicit” to the extent that $\mathrm{SO}(n-1)\hat{\ }$ is. $\endgroup$ Jan 7 '18 at 19:41
  • $\begingroup$ Dear Francois Ziegler, I would like to realize it on $L^2(S^{n-1})$. Then, according to N. I. Vilenkin $\pi_\lambda$ is given by $(*)$ above and I'm looking for the expression of $\chi$ (always on $L^2(S^{n-1})$) ? $\endgroup$
    – Z. Alfata
    Jan 7 '18 at 21:57
  • 1
    $\begingroup$ @Z.Alfata Then you aren’t talking about the whole unitary dual. IIRC, Vilenkin limits himself to irreps with a $K$-fixed vector ($K=\mathrm{SO}(n)$); those are called various names and, correct me if I’m wrong, should here be: in (1) just the trivial representation; in (2) those with trivial $\sigma$. The latter act in $\smash{L^2(S^{n-1})}$ as you desire, with $K$-fixed vector the constant 1. $\endgroup$ Jan 7 '18 at 23:22
  • $\begingroup$ @ Francois Ziegler, in Bhowmik,Sen - JFAA 2017, p: 7-8 "the finite dimensional unitary representations of $SO(n)$ also yield finite dimensional unitary representations of $M(n)$'. Then, the expression the finite dimensional unitary representations of $SO(n)$ is a character or what ? $\endgroup$
    – Z. Alfata
    Jan 24 '18 at 12:33

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