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

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.