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Unitary dual of the complex motion group $\mathbb C^2 \rtimes SU(2)$?

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 $G$ ...
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