Components of affine Springer fibers

TL;DR

This paper proves that the number of components of certain affine Springer fibers associated with a reductive group over a local field equals the order of the Weyl group, using $p$-adic orbital integral methods.

Contribution

It establishes a precise count of affine Springer fiber components in terms of Weyl group order, linking geometric and algebraic structures.

Findings

Number of affine Springer fiber components equals Weyl group order.

Methodology involves $p$-adic orbital integrals.

Results hold for topologically nilpotent regular semisimple elements.

Abstract

Let $\mathbf{G}$ be a connected split reductive group over a field of characteristic zero or sufficiently large characteristic, $\gamma_0\in(\operatorname{Lie}\mathbf{G})((t))$ 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.