Let $X$ be an irreducible locally noetherian $k$-scheme (for $k$ any field), $G$ an algebraic group acting on $X$ via $a:G \times X \to X$ and $x \in X$ a closed point, which is by Zariski's lemma equivalent to that there exist a finite field extension $L/k$ such that $x \in X(L)$, ie a $L$-valued point.
Assume $x $ is fixed by $G$. Following Brion's notes (see p 12) this is equivalent to the statement that the for ideal sheaf $\mathcal{I}_x \subset \mathcal{O}_X$ to pulls backs along $a, p_X:G \times X \to X$ satisfy $a^*(\mathcal{I}) \cdot \mathcal{O}_{G \times X} =(p_X)^*(\mathcal{I}) \cdot \mathcal{O}_{G \times X}$. Ie intuitively the ideal sheaf of $x$ is not $G$ sensitive.
In the same notes is next claimed that if $x \in X(k)$, ie $x$ rational, then if $x$ is fixed by $G$, then it induces naturally actions on the stalk of $x$ $\mathcal{O}_{X,x}$, and all infinitesimal thickerings $\mathcal{O}_{X,x}/ \mathfrak{m}_x^n$.
This induced action can be phrased in terms of natural trafo between group point functors as $G(R) \to \operatorname{Aut}(R \otimes_k (\mathcal{O}_{X,x}/ \mathfrak{m}_x^n)$ which in natural and group homo in every fin. gen. $k$-algebra $R$.
Question: Does the fixed point $x$ induce in exactly same manner the same actions on thickerings of it's residue field if $x$ is not rational? If not, where it precisely might run in troubles (eg with well definedness, naturality etc?) and where do we precisely above need the rationality assumption?
This question is contentually closed related to this one.
RMK.: above appears a potential notation problem/ notation abusion with notion for stalk $\mathcal{O}_{X,x}$ for $x \in X(L)$ non rational: keeping the notation precise by the stalk $\mathcal{O}_{X,x}$ is actually meant the stalk $\mathcal{O}_{X,\mathfrak{m}_x}$ which is as usually formed by localizing appropr ring $R$ corresponding to an affine chart of $X$ containg image of $x$ at (maximal) prime $\mathfrak{m}_x$ be the kernel of $x^*: R \to L$ corresponding to $x$.
