Mirković-Vilonen cycles and polytopes

We give an explicit description of the Mirkovi-Vilonen cycles on the affine Grassmannian for arbitrary complex reductive groups. We also give a combinatorial characterization of the MV polytopes. We prove that a polytope is an MV polytope if and only if it is a lattice polytope whose defining hyperplanes are parallel to those of the Weyl polytopes and whose 2-faces are rank 2 MV polytopes. As an application, we give a bijection between Lusztig's canonical basis and the set of MV polytopes. Appendix A. Pseudo-Weyl polytopes A.1. Support functions A.2. Fans A.3. Support functions of -polytopes A.