How to prove that, the unitary dual of $\{0\}$ and $\mathbb R$ are the trivial identity representation $id$ and the representation $\chi_x (y) = e^{i x y}; y\in \mathbb R$, respectively.
Thank you in advance
2026-04-06 21:02:48.1775509368
Unitary dual of $\{0\}$ and $\mathbb R$
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For a finite group $G$, the number of inequivalent irreducible unitary representations equals the number of conjugacy classes of $G$. The trivial group only has one conjugacy class. The second question on $\mathbb{R}$ has been answered at this MSE question.