The crystal structure on the set of Mirković–Vilonen polytopes

In an earlier work, we proved that MV polytopes parameterize both Lusztig's canonical basis and the Mirković–Vilonen cycles on the affine Grassmannian. Each of these sets has a crystal structure (due to Kashiwara–Lusztig on the canonical basis side and due to Braverman–Finkelberg–Gaitsgory on the MV cycles side). We show that these two crystal structures agree. As an application, we consider a conjecture of Anderson–Mirković which describes the BFG crystal structure on the level of MV polytopes. We prove their conjecture for sln and give a counterexample for sp6. Finally we explain how Kashiwara data can be recovered from MV polytopes.