Let $k$ be a finite field and let $F = k( (t))$ with ring of integers $\mathfrak{o} = k[ [t]]$. Let $G$ be a connected linear algebraic $k$-group with Lie algebra $\mathfrak{g}$. Suppose that $\gamma\in \mathfrak{g}(F)$. Define the affine Springer fiber associated to $\gamma$ to be \begin{align*} X_\gamma = \{ g\in G(F)/G(\mathfrak{o}) : {\rm Ad}(g)^{-1}\gamma\in \mathfrak{g}(\mathfrak{o}) \}. \end{align*} In general, $X_\gamma$ is only a ind-scheme. If $\gamma$ is regular (=orbit of maximal dimension) and semisimple, then $X_\gamma$ is finite dimensional, and there is a nice formula for this dimension. One can find this an other facts nicely summarised in [1]. I'd like to be able to study this affine Springer fiber by taking points in various $k$-algebras $R$.

For instance, suppose $R = k'$ is a finite extension of the finite field $k$, so that we get a finite extension $E = k'( (t))$ of $F$. Then naively, I assume that one could write for any $k$-algebra $R$ \begin{align} X_\gamma(R) = \{ g\in G(F\otimes_k R)/G(\mathfrak{o}\otimes_k R) : {\rm Ad}(g)^{-1}\gamma\in \mathfrak{g}(\mathfrak{o}\otimes_k R)\} \end{align} (can I do this???) so that \begin{align*} X_\gamma(k') = \{ g\in G(E)/G(\mathfrak{o}_E) : {\rm Ad}(g)^{-1}\gamma\in\mathfrak{g}(\mathfrak{o}_E)\} \end{align*} where now $\mathfrak{o}_E$ is the ring of integers of $E$. I have a good idea on how to calculate the dimension of $X_\gamma(k')$ as the affine Springer fiber associated to $\gamma\in \mathfrak{g}(E)$ and $G_{k'}$, and I'd like to be able to say that $\dim X_\gamma(k') = \dim X_\gamma$ when $\gamma$ is regular semisimple.

Question: Can I calculate the dimension of the affine Springer fiber $X_\gamma$ in this way when $\gamma$ is regular semisimple?

Although I am comfortable with the functor of points for schemes, I am less so when it comes to ind-schemes or schemes whose definition includes ind-schemes, so I am not sure whether I am calculating the $R$-points for $X_\gamma$ correctly when $R$ is a $k$-algebra. I think I've read somewhere that the correct way to define the affine Springer fiber in general as a functor of points is to take the fpqc-sheafification of the presheaf $R\mapsto \{ g\in G(F\otimes_k R)/G(\mathfrak{o}\otimes_k R) : {\rm }Ad(g)^{-1}\in\mathfrak{g}(\mathfrak{o}\otimes_k R) \}$ though I am not comfortable with this construction yet or how to compute with it.

[1] Görtz, Ulrich. "Affine springer fibers and affine Deligne-Lusztig varieties." Affine Flag Manifolds and Principal Bundles. Springer Basel, 2010. 1-50.


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