Suppose you have a smooth irreducible representation (finite dimensional over $\mathbb{C}$) of $G = \mathbb{C}^{*}$, then does it have to be of the form $z.w = z^n w$ for some $n \in \mathbb{Z}$ and $w \in \mathbb{C}$?
note that the representation has dimension $1$ since $G$ is abelian. Note the result folds for $G=S^1$. I would be grateful for a reference.