Purity of the anisotropic affine Springer fibers for ${\mathbf{GL}}_n$

Abstract

For the group ${\mathrm{GL}}_n$ and the anisotropic elements, we confirm the purity hypothesis of Goresky, Kottwitz and MacPherson, which states that the affine Springer fibers are cohomologically pure in the sense of Grothendieck-Deligne.

As pointed out by Zhiwei Yun, there is a gap in the final step of the proof of theorem 3.9. Indeed, when we make the intermediate extension from plane curve singularities to complete intersection curve singularities, extra singular support might appear