Let $\mathcal{M}_{d}(G)$ be the moduli space of $G$-Higgs bundles. $\mathcal{M}_{d}(G)$ have a non-trivial $\mathbb{C}^{*}$-holomorphic action by multiplication of the Higgs field, $$ z\cdot (E, \varphi)=(E, z\varphi). $$ I'm interested in finding the fixed points of this $\mathbb{C}^*$-action.
It is known that the restriction of this action to $S^1\subset \mathbb{C}^{*}$ is Hamiltonian with a momentum map $\mu$. So, we know how to find the fixed points of the action of $S^{1}$ once they will be exactly the critical points of $\mu$.
In Florentino, Gothen, and Nozad - Homotopy type of moduli spaces of $G$-Higgs bundles and reducibility of the nilpotent cone, on page 5, the authors state that the sets of fixed points of the actions of $S^1$ and $\mathbb{C}^{*}$ coincide. One of the inclusions is clear. However, I can't see why the other one should also hold.
Does anyone have any suggestions of how can I prove this?