Goresky–Kottwitz–Macpherson purity conjecture for fundamental domains

Let $F_{\gamma}$ be the fundamental domain for the action of the free abelian group generated by $\chi(\epsilon)$, with $\chi\in X_{*}(T)$, on the affine Springer fiber $\mathscr{X}_{\gamma}$ of a regular element $\gamma\in\mathfrak{t}(\mathcal{O})$. Goresky–Kottwitz–Macpherson purity conjecture for fundamental domains. The cohomology of $F_{\gamma}$ is pure in the sense of Deligne. By the paper's theorem, under the stated Levi-subgroup purity hypotheses, purity of $F_{\gamma}$ is equivalent to purity of $\mathscr{X}_{\gamma}$; the conjecture is known in some special cases but remains open in general.