On the fundamental domain of affine Springer fibers

TL;DR

The paper introduces fundamental domains for affine Springer fibers associated with reductive groups, reduces the purity conjecture to these domains, and proves a rationality conjecture for the case of GL(3).

Contribution

It defines fundamental domains for affine Springer fibers, establishes a reduction method for purity conjecture, and proves the rationality conjecture for GL(3).

Findings

Fundamental domains can be paved in affine spaces.

Purity conjecture reduces to properties of fundamental domains.

Rationality conjecture is proved for GL(3).

Abstract

For $G$ a connected reductive group, $\gamma\in \kg(F)$ semisimple regular integral, we introduce a fundamental domain $F_{\gamma}$ for the affine Springer fibers $\xx_{\gamma}$. There is a beautiful way to reduce the purity conjecture of $\xx_{\gamma}$ to that of $F_{\gamma}$, we call it the Arthur-Kottwitz reduction. When restricted to the unramified case, it turns out that these fundamental domains behave well in family. We formulate a rationality conjecture about a generating series of their Poincar\'e polynomials. We then study them in detail for the group $\gl_{3}$. In particular, we pave them in affine spaces and we prove the rationality conjecture.