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

Modular representations of reductive groups and geometry of affine Grassmannians