  For quantum groups at a root of unity, there is a web of theorems (due to
Bezrukavnikov and Ostrik, and relying on work of Lusztig) connecting the
following topics: (i) tilting modules; (ii) vector bundles on nilpotent orbits;
and (iii) Kazhdan-Lusztig cells in the affine Weyl group. In this paper, we
propose a (partly conjectural) analogous picture for reductive algebraic groups
over fields of positive characteristic, inspired by a conjecture of Humphreys.
