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](https://arxiv.org/abs/1805.10081), 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?



The fixed points set of the actions of $\mathbb{C}^*$ and $S^1$ on the Higgs bundle moduli space
Let me expand my comment "You can get the desired conclusion by putting the following facts together: (1) the moduli space $\mathcal{M}\_()$
is an algebraic variety, (2) the action of $\mathbb{C}^\*$
 on it is algebraic, (3) $S^1\subset \mathbb{C}^\*$ is a Zariski dense subgroup."


(1) and (2) follows from the construction. See for example, Simpson, *Moduli of representations of the fundamental group of a smooth projective variety I, II* for details of that. I don't have a reference for (3), but it's elementary:



> 
> Lemma. Suppose that $G$ is a (say) complex algebraic group acting algebraically on a variety $X$, and let $K\subset G$ be Zariski dense subgroup. Then a point of $X$ fixed by $K$ is fixed by $G$.
> 
> 
> 


Proof: Suppose that $x\in X$ is fixed by $K$. Its stabilizer is a Zariski closed subgroup $H\subseteq G$. But $K\subseteq H$, so $H=G$.


