In Woit's book chapter 20, he considers character of the group $(\mathbb{R}^d,+)$. Now he claims that irreducible representations of $(\mathbb{R}^d,+)$ are one dimensional and given by $\alpha_{\mathbf{p}}(\mathbf{a}) = e^{i\mathbf{p}\cdot \mathbf{a}} $. For $\mathbf{a} \in \mathbb{R}^d$, and $\mathbf{p}$ is in another copy of $\mathbb{R}^d$ . Think of it as position and momentum.
I can't justify this claim myself. I know that any irreducible of Abelian group is $1$-dimensional, so the irreps can be characterized by group characters as used above. But why is it of that form? I was thinking of using Pontryagin duality to get that form like in the case of $\mathbb{R}$ but I'm not entirely sure as I have limited background in Harmonic Analysis.