Components of affine Springer fibers

Abstract

Let $\mathbf{G}$ be a connected split reductive group over a field of characteristic zero or sufficiently large characteristic, ${\gamma }_0\in (\mathrm{Lie}\mathbf{G})((t))$ be any topologically nilpotent regular semisimple element, and $\gamma =t{\gamma }_0$. Using methods from $p$-adic orbital integrals, we show that the number of components of the Iwahori affine Springer fiber over $\gamma$ modulo $Z_{\mathbf{G}((t))}(\gamma )$ is equal to the order of the Weyl group.

arXiv commentVarious proofs are expanded, and two preliminary sections as well as two appendixes are added in the revision thanks to the suggestion of referees