Modular representations of reductive groups and geometry of affine Grassmannians

By the geometric Satake isomorphism of Mirkovic and Vilonen, decomposition numbers for reductive groups can be interpreted as decomposition numbers for equivariant perverse sheaves on the complex affine Grassmannian of the Langlands dual group. Using a description of the minimal degenerations of the affine Grassmannian obtained by Malkin, Ostrik and Vybornov, we are able to recover geometrically some decomposition numbers for reductive groups. In the other direction, we can use some decomposition numbers for reductive groups to prove geometric results, such as a new proof of non-smoothness results, and a proof that some singularities are not equivalent (a conjecture of Malkin, Ostrik and Vybornov). We also give counterexamples to a conjecture of Mirkovic and Vilonen stating that the stalks of standard perverse sheaves over the integers on the affine Grassmannian are torsion-free, and propose a modified conjecture, excluding bad primes.