Action of the stabilizer group on the completed local ring Let $\mathcal{M}$ be a DM stack and $M$ its coarse moduli space. Then we have the map $\mathcal{M} \to M$. Choose a geometric point $x \in M$ and a geometric point $\bar{x} \in \mathcal{M}$ which lies over $x$. Denote by $\Gamma_{\bar{x}}$ the stabilizer group at the point $\bar{x}$. My question is: how to understand in general the action of $\Gamma_{\bar{x}}$ on the completed local ring $\hat{\mathcal{O}_{\mathcal{M}, \bar{x}}}$?
For example, if $\mathcal{M}$ is the moduli stack of elliptic curves over some field $k$, then $\bar{x}$ corresponds to some elliptic curve and $\Gamma_{\bar{x}}$ is its group of automorphisms. Let's say $\bar{x}$ corresponds to $y^2=x^3-x$ in which case $\Gamma_{\bar{x}} = \mathbb{Z}/6$. I think I should find some etale covering by a scheme to be able to compute anything. For example, I can take the Legendre family and it gives me an etale map $\operatorname{Spec} k[x][\frac{1}{x(x-1)}] \to \mathcal{M}$. Thus, I'm interested in the action of $\mathbb{Z}/6$ on the completed local ring of $k[t][\frac{1}{t(t-1)}]$ at $t=-1$ which is isomorphic to $k[[x]]$. However, I'm struggling to figure out how to write it down.
I think I also have the map $k[[j]] \to k[[t]]$ given by the $j$-invariant which should identify $k[[j]]$ with the ring of invariants $k[[t]]^{\mathbb{Z}/6}$ however I don't see how to write down the action.
 A: In general for any $x \in M$, there exists a pointed affine scheme $(U,u)$ and an action $\Gamma_\bar{x}$ on $U$ fixing $u$ and an étale map $[U/\Gamma_\bar{x}] \xrightarrow{f} \mathcal{M}$ such that $f(u) = x$. Then it suffices to compute the $\Gamma_\bar{x}$ on $\mathcal{O}_{U,u}$.
If $\mathcal{M}$ is presented as a global quotient $[V/G]$ where $G$ is reductive, then for every closed point $v \in V$, the Luna Slice Theorem allows one to construct a slice $(U,v)$ and an action of the stabilizer $G_v$ such that $[U/G_v]$ gives us such a chart.
This is the case for $\overline{\mathcal{M}}_{1,1}$ (away from characteristic $2,3$) which can be presented as the quotient $[\mathbb{A}^2/\mathbb{G}_m]$ with the action $g(a,b) = (g^4a, g^6b)$ and universal family $y^2 = x^3 - ax -b$. The point you want is $(1,0)$ which has stabilizer $\mu_4$ the group of fourth roots of unity. A Luna slice is given by $\{(1,t)\}$ with family $y^2 = x^3 - x - t$. The action (as it's induced by the restriction of the $G$ action on $V$) is simply $t \mapsto \zeta_4^6t= \zeta_4^2t$ and the completed local ring is $k[[t]]$ with this action of $\mu_4$.
It's not hard to see using the same method that at the point classifying $y^2 = x^3 - 1$, ${(t,1)}$ us a Luna slice and $t \mapsto \zeta_6^4t$, and at any other point the automorphism group $\mu_2$ acts trivially.
If the stack isn't given to you as a global quotient I don't see any general way to answer the question. You just need to compute a versal deformation and its automorphisms. One observation that helps when $\mathcal{M}$ is smooth though is that in this case we can arrange it so that there is an étale map $[U/\Gamma_\bar{x}] \to [T_x/\Gamma_\bar{x}]$ and so the question reduces to understanding the linear action of $\Gamma_\bar{x}$ on the tangent space. For example for the moduli of curves, we need to understand how $\mathrm{Aut}(C)$ acts on $H^1(C,T_C)$ which is not much more explicit but at least reduces the question to a cohomology computation.
