Let $G$ be a connected simple complex algebraic group and $\mathcal{O}(G)$ be its ring of regular functions. What are the invertible elements of $\mathcal{O}(G)$? Or is it true that $\mathcal{O}(G)^* = \mathbb{C}^\ast$?
It seems to be a basic question, but I couldn't find an answer in literature. The best I was able to find about $\mathcal{O}(G)$ is the following: if $G$ is simply-connected, then $\mathcal{O}(G)$ is factorial.
Any thoughts or references on the rings of regular functions on algebraic groups would be appreciated!