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

TL;DR

This paper proves that for the group GL_n and anisotropic elements, the affine Springer fibers are cohomologically pure, confirming a key hypothesis in geometric representation theory.

Contribution

It confirms the purity hypothesis for affine Springer fibers in the case of GL_n and anisotropic elements, advancing understanding in geometric representation theory.

Findings

Affine Springer fibers for GL_n are cohomologically pure.

The purity hypothesis of Goresky, Kottwitz, and MacPherson is verified.

Results support conjectures relating to the geometry of affine Springer fibers.

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.