Corollary 32.1.11.label Let $G$ be a locally compact group. If $G$ is abelian, then every irreducible representation of $G$ is one-dimensional.
Proof, [Corollary 3.6, Fol16]. Let $(H, \pi)$ be an irreducible representation of $G$. Since $G$ is abelian, $\pi(G) \subset \pi(G)'$. By Schur’s Lemma, $\pi(G)' = \complex I$. As every one-dimensional subspace of $H$ is invariant, $H$ itself must be one-dimensional.$\square$
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