Affine Springer fiber

Let $k$ be a field, put $F=k((t))$ and $\mathcal O=k[[t]]$ , and let $G$ be a reductive group. For a regular semisimple element $\gamma$ of its Lie algebra $\mathfrak g(F)$ whose Lie-algebra centralizer is a maximal torus (the Lie-algebra analogue of a strongly regular semisimple element), the affine Springer fiber is the subspace

\mathcal M_\gamma= \left\{ gG(\mathcal O)\in G(F)/G(\mathcal O): \operatorname{Ad}(g^{-1})\gamma\in\mathfrak g(\mathcal O) \right\}

of the affine Grassmannian.

Interpretation and group version

It parametrizes integral $G$ -lattices preserved by $\gamma$ .

There is also a group version in which $\gamma\in G(F)$ and the integrality condition is $g^{-1}\gamma g\in G(\mathcal O)$ .

Geometry

Affine Springer fibers are usually infinite-dimensional as ambient ind-scheme loci but have finite-dimensional reduced pieces under standard regularity hypotheses. They can be singular, reducible, and nonproper. A lattice in the loop centralizer of $\gamma$ acts on them, often with a projective quotient.

Orbital integrals

Point counts and compactly supported $\ell$ -adic cohomology of affine Springer fibers encode orbital integrals. The Hitchin fibration places these local fibers inside a global family, allowing geometric comparison with endoscopic fibers in the proof of the fundamental lemma.

References

David Kazhdan and George Lusztig, “Fixed point varieties on affine flag manifolds,” Israel Journal of Mathematics 62 (1988), 129–168.
Ngô Bảo Châu, “Survey on the fundamental lemma,” 2010, §§3–4. Author PDF.