Mirkovic–Vilonen cycle and affine-paving conjecture for truncated affine Grassmannians

Let $G$ be the group under consideration, let $\mathscr{X}$ be its affine Grassmannian, and for $x\in\mathscr{X}$ let $\mathrm{Ec}(x)$ be the associated convex polytope. Define

$$\mathscr{X}(\mathrm{Ec}(x)):=\{y\in\mathscr{X}\mid \mathrm{Ec}(y)\subset\mathrm{Ec}(x),\ \nu_{G}(y)=\nu_{G}(x)\}.$$

Let $W$ be the Weyl group of $G$, and let a Mirkovic–Vilonen cycle mean one of the cycles arising from the geometric Satake construction.

Mirkovic–Vilonen cycle and affine-paving conjecture. For every $x\in\mathscr{X}$, there exists $w\in W$ such that $w\cdot\mathscr{X}(\mathrm{Ec}(x))$ is a Mirkovic–Vilonen cycle. Moreover, the truncated affine Grassmannian $\mathscr{X}(\mathrm{Ec}(x))$ admits an affine paving and hence is cohomologically pure.

The conjecture is motivated by Lusztig parametrization and the theory of Mirkovic–Vilonen cycles. The paper proves the relevant assertion in the $\mathrm{GL}_{3}$ setting, while the general claim is open.