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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.

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