# Computing affine Springer fibers

I'm having some trouble computing affine Springer fibers, even in simple cases. For example, consider the group $$G=SL_2$$ over $$\mathbb{C}$$ and let $$\mathcal{K}=\mathbb{C}((z))$$ and $$\mathcal{O}=\mathbb{C}[[z]]$$. Then the affine Grassmannian $$Gr_G$$ of $$G$$ parametrizes $$\mathcal{O}$$ lattices in $$\mathcal{K}^2$$. We then define the space $$S=\{(\gamma, L) \in \mathfrak{g}_{\mathcal{O}} \times Gr_G | \gamma L \subseteq L\}$$ together with the obvious projection $$\pi: S \to \mathfrak{g}_{\mathcal{O}}$$. The affine Springer fiber of $$\gamma \in \mathfrak{g}_{\mathcal{O}}$$ is then the fiber of $$\pi$$ over $$\gamma$$. Here $$\mathfrak{g}_{\mathcal{O}}$$ is the Lie algebra $$\mathfrak{g} \otimes \mathcal{O}$$.

For example, the reduced Springer fiber of $$\gamma= diag(x,-x)$$, for $$x \in \mathbb{C}-\{0\}$$ is an infinite discrete space parametrized by $$\mathbb{Z}$$. Letting $$x=z$$, we get an infinite chain of $$\mathbb{P}^1$$'s instead, where each one intersects the next in exactly one point.

In principle, to compute these things one takes an arbitrary matrix $$g$$ in $$SL_2((z))$$, conjugates $$\gamma$$ by $$g$$, and then works out conditions on the entries for this matrix so that the conjugate matrix $$g^{-1} \gamma g$$ is an element of $$\mathfrak{g}_{\mathcal{O}}$$, but this method is extremely tedious and difficult. Is there an elegant way to do this? If there isn't, could someone show me at least an efficient way to compute $$\pi^{-1}(\gamma)$$, say when $$\gamma=diag(z,-z)$$?

Lastly, assuming we have found that $$\pi^{-1}(\gamma)$$ is parametrized by the points of $$\mathbb{P}^1$$, say, how does one rigorously show that the scheme structure on the fiber agrees with that of $$\mathbb{P}^1$$? Thanks in advance.