Representation of the circle group

Every finite-dimensional continuous complex representation of $S^1$ is unitary and decomposes into one-dimensional weight spaces. Its continuous one-dimensional characters are $z\mapsto z^m$ for $m\in \mathbb{Z}$.

If a central subgroup is isomorphic to $S^1$, its weight spaces are invariant under the entire group, because every representing operator commutes with the central circle action.