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

For the group ${\mathrm{G}\mathrm{L}}_n$ 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.