  We give an explicit description of the Mirkovic-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 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.

Algebraic Geometry;Representation Theory