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The real motion group of $\mathbb R^2$, $M(2)$ 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 M. Sugiura: Unitary Representations and Harmonic Analysis. Kodansha Scientific books, Tokyo (1975), p: 165 $$\{\pi_\lambda,\, \lambda>0 \}\cup\{\chi_n: \, n\in \mathbb Z \},$$ where $[\pi_{\lambda}(z,e^{i\alpha})u](s) = e^{i\lambda \left<z,e^{is}\right>} u(s-\alpha)$ for $u\in L^2([0,2\pi])$ and where $\chi_n(z,e^{i\alpha})=e^{in\alpha}$.

Now, any references or someone can help me to write the analog of that in the cases of the complex motion group $G=\mathbb C^2 \rtimes SU(2)$ ? Thank you in advance $$................$$ Following the remark in Dual topology of the motion groups, p: 398, I tried to construct this representation!!

For each linear form $l$ on $\mathbb C^2$ and any irreducible unitary representation $\rho$ of the stabilizer $S_l$ of l in $SU(2)$, we have that $\, \sigma_{(\rho,l)}=\rho\otimes\chi_l \, $ is an irreducible unitary representation of $H_l = S_l\rtimes \mathbb C^2$ whose restriction to $\mathbb C^2$ is a multiple of the character $\chi_l$ of $\mathbb C^2$ given by $\chi_l(z) = e^{−i\left<l,z\right>}$ ($z\in \mathbb C^2$), and the induced representation $\pi_(\rho,l) := ind^{G}_{H_l} \sigma(\rho,l)$ is an irreducible representation of $G$.

If $r > 0$ is the radius of the sphere, we denote by $\chi_r$ the character associated with the linear form $l_r$ which is identified with the vector $(0, . . . , 0, r)^t$ .The stabilizer $S_{l_{r}}$ of $l_{r}$ is the subgroup $SU (1)$. Let us write $\rho_\mu$ instead of $\rho$ for the representation of $SU(1)$ with highest weight $\mu$ and $\pi(\mu,r)$ instead of $\pi(\rho_\mu,l_r)$. The representation $\pi(\mu,r)$ is realized on $L^2(SU(2))$ as follows; for all $(A, z)\in G$ and all $B \in SU(2)$ $$(*) \quad \pi(\mu,r)(A,z)F(B)=e^{i\left<B l_r, z\right>} F(A^{-1}B), \quad F\in L^2(SU(2)).$$ Finally, the unitary dual $\hat{G}$, of $G$ is precisely the collection: $$\{r>0, \rho\in\hat{SU(1)} \}\cup\{\pi \in\hat{SU(2)} \}.$$

Thank you in advance for any comments or suggestions!!

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    $\begingroup$ Same ref. as before. (Give a man a fish...) $\endgroup$
    – Francois Ziegler
    Commented Mar 14, 2018 at 15:33
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    $\begingroup$ You just need to learn to write induced representations in whatever way you regard as “explicit”. (Note that’s subjective and not a mathematically well-defined concept. But I can sympathize that some realizations are friendlier than others. If I remember correctly, Kirillov’s book is quite helpful for that.) $\endgroup$
    – Francois Ziegler
    Commented Mar 14, 2018 at 16:56
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    $\begingroup$ “$SU(1)$” (and attendant constructs) can be drastically simplified... $\endgroup$
    – Francois Ziegler
    Commented Mar 16, 2018 at 11:17
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    $\begingroup$ $SU(1) =\{1\}$. $\endgroup$
    – Francois Ziegler
    Commented Mar 16, 2018 at 11:24
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    $\begingroup$ Last comment sounds right. You should probably restore the original question, and make your proposed solution an answer. $\endgroup$
    – Francois Ziegler
    Commented Mar 16, 2018 at 17:50

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