On the fundamental domain of affine Springer fibers

Abstract

Let G be a connected reductive algebraic group over an algebraically closed field k, $\gamma \in \mathfrak {g}(k((\epsilon )))$ a semisimple regular element, we introduce a fundamental domain $F_{\gamma }$ for the affine Springer fibers ${\mathscr {X}}_{\gamma }$. We show that the purity conjecture of ${\mathscr {X}}_{\gamma }$ is equivalent to that of $F_{\gamma }$ via the Arthur–Kottwitz reduction. We then concentrate on the unramified affine Springer fibers for the group ${\mathrm {GL}}_{d}$. It turns out that their fundamental domains behave nicely with respect to the root valuation of $\gamma$. We formulate a rationality conjecture about a generating series of their Poincaré polynomials, and study them in detail for the group ${\mathrm {GL}}_{3}$. In particular, we pave them in affine spaces and we prove the rationality conjecture.