Truncated affine grassmannians and truncated affine Springer fibers for $\mathrm{GL}_{3}$

TL;DR

This paper proposes a conjecture on affine pavings for certain algebraic varieties with torus actions and provides explicit constructions for truncated affine grassmannians and Springer fibers for GL3, revealing cohomological purity.

Contribution

It introduces a conjecture on affine pavings for cohomologically pure varieties and constructs explicit affine pavings for truncated affine grassmannians and Springer fibers for GL3.

Findings

Conjecture that truncated affine grassmannians for GL_{r+1} admit affine pavings.

Explicit affine pavings constructed for GL_3 truncated affine grassmannians.

Identified a family of cohomologically pure truncated affine Springer fibers.

Abstract

We state a conjecture on how to construct affine pavings for cohomologically pure projective algebraic varieties, which admit an action of torus such that the fixed points and $1$-dimensional orbits are finite. Experiments on the affine grassmannian for $\mathrm{GL}_{3}$ under the guideline of this conjecture, together with the work of Berenstein-Fomin-Zelevinsky and Kamnitzer, have led to the conjecture that the truncated affine grassmannians for $\mathrm{GL}_{r+1}$ admit affine pavings. For the group $\mathrm{GL}_{3}$, we construct affine pavings for the truncated affine grassmannians, and we use it to study the affine Springer fibers. In particular, we find a family of truncated affine Springer fibers which are cohomologically pure in the sense of Deligne.