$\newcommand{\Rep}{\operatorname{Rep}}$ $\newcommand{\mo}{\operatorname{-mod}}$ $\renewcommand{\hat}{\widehat}$

I apologize in advance if this is a naive question but my background in algebraic geometry is fairly superficial. I mostly care about global quotients $X/G$ where $X$ is an affine scheme over $\mathbb C$ and $G$ a complex connected affine algebraic (reductive if you like) group. My understanding of those is pretty much limited to the fact that we have an equivalence of symmetric monoidal categories $$QC(X/G)\simeq O(X)\mo_{\Rep G}$$ where $O(X)$ is the algebra of global functions, and $\Rep G$ the category of $O(G)$-comodules.

Let $x \in X$ be a fixed point of the $G$ action. In a nutshell my question is:

What is the correct definition of the formal completion of $X/G$ at $x$ ? In particular what is its category of quasi-coherent sheaves thinking of it as an "ordinary" rather than formal stack (f that makes sense) ?

A basic observation is that $\hat O(X)$, the completion of $O(X)$ by the ideal of functions vanishing at $x$, is **not** an object in $\Rep G$. Now it seems there are different things one can do:

- Look at the category $\hat O(X)\mo_{\Rep \mathfrak g}$, which I guess should be like quasi-coherent sheaves on $\hat X/\hat G$
- Think of $\hat O(X)$ as a topological algebra, hence as an object in a certain category of topological $G$-representation (say the pro-completion of the category of finite dimensional $G$-modules).
- We can look at the coalgebra $C(X)$ of "distributions supported at $x$", i.e. the coalgebra which satisfies $C(X)^*=\hat O(X)$, which is a a coalgebra in $\Rep G$ so that you take take comodules over it.. This is the idea that formal affine scheme are the same as "cospectrum" of cocommutative coalgebras, and I think the category you get is equivalent to the one in 2 by taking duals.
- Although $\hat O(X)$ is not an object in $\Rep G$, it still makes sense to look at modules for this algebra that happens to be in there, i.e. $\hat O(X)\mo_{\Rep G}$ do makes sense.

Is any of those the, or a, correct definition ? Any insight or reference would be much appreciated.