One regularly finds the following claim:
Every complex irreducible representation of an abelian group is one-dimensional.
In representation theory, one often only considers finite-dimensional representations. In this case, the claim is true by Schur's lemma and the fact that a linear operator on a finite-dimensional complex vector space will have an eigenvector.
I'm looking for an irreducible representation of $\mathbb{R}$ on an infinite-dimensional vector space.
Edit 1: related question
Edit 2: Inspired by this question write $\mathbb{R} = \bigoplus_{i \in I} span_{\mathbb{Q}}(e_i)$ for some basis $(e_i)_{i \in I}$, viewing it as a vector space over $\mathbb{Q}$. Define $V=\bigoplus_{i \in I} span_{\mathbb{C}}(e_i)$ and the representation $\rho$ of $\mathbb{R}$ on $V$ by $\rho(x)v = x + v$ where we use the natural inclusion of $\mathbb{R}$ in $V$. Is this representation irreducible?
