I'm currently going through this article: https://arxiv.org/pdf/1307.5522.pdf, and having some (a lot of!) trouble understanding the proof given to the following (this is Lemma 4 in the article):
Let $X$ be an irreducible variety (i.e. a space with functions having a finite open cover of affine sets) over a field of characteristic $0$, and let $G$ be a finite subgroup of $\mathrm{Aut}(X)$; the group of automorphisms of $X$. If $P\in X$ is fixed by $G$, then the natural action $\rho:G\to\mathrm{GL}(\mathfrak{m}/\mathfrak{m}^2)$ is faithful, where $\mathfrak{m}$ is the maximal ideal of the local ring $\mathscr{O}_{X,P}$.
It suffices to prove that ker$\rho$ acts trivially on $\mathfrak{m}$ for then it acts trivially on $\mathscr{O}_{X,P}\cong k \oplus \mathfrak{m}$, hence on its field of fractions $k(X)$ ($X$ is irreducible), hence on $X$. Then ker$\rho$ is trivial since it also acts faithfully on $X$.
My plan: Since $G$ is finite and the characteristic is zero, Maschke's theorem applies: there exists a subrepresentation $V$ of $\mathfrak{m}$ isomorphic to $\mathfrak{m}/\mathfrak{m}^2$. Then ker$\rho$ acts trivially on $V$.
The passage to the action on $\mathfrak{m}$ is where I'm stuck at the moment.
The idea of the next step is to show that the subspace $V^d$ generated by all products of $1$ to $d$ elements of $V$ surjects (in a $G$-equivariant way) into $\mathfrak{m}/\mathfrak{m}^{d+1}$. Now the kernel $K$ of $\rho$ acts trivially on $V$ so on $V^d$ too, and thus $K$ acts trivially on $\mathfrak{m}/\mathfrak{m}^{d+1}$.
Let $f \in \mathfrak{m}$, it follows from the above that, if $t \in K$, $t \cdot f-f \in \cap_{d \geq 1}{\mathfrak{m}^d}$ which is zero by Krull’s theorem. Thus $K$ acts trivially on $\mathfrak{m}$ and we are done.