  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.
