Representation theory of $\mathbb{R}$? I am just learning a bit about representation theory (a representation being a homomorphism $\rho: G \to GL(V)$ for some complex vector space $V$) and I am trying to understand some examples. I get the representations of the finite cyclic groups and I think I get the representations of the integers. I am wondering about the representations of a group like $\mathbb{R}$ under addition. I know this group is Abelian, and so all the irreducible representations would be $1$-dimensional. If this is too hard I would be happy with a reference.

In the case of an infinite topological group, you probably want to consider only continuous characters. In the compact case, the Peter-Weyl theorem holds, and the situation is similar to the finite case. If you consider arbitrary representations of $\mathbb{R}$, then you can construct some uninteresting and pathological ones from a Hamel basis. In the abelian case, the characters are more or less determined by Pontryagin duality, assuming local compactness. For the particular case of $\mathbb{R}$, it follows that the continuous characters on $\mathbb{R}$ are all of the form $\chi(t) = e^{2\pi i \xi t}$ for some real $\xi$, with the correspondence induced by the Fourier transform.
