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We prove the Mirkovi\'c-Vilonen conjecture: the integral local intersectioncohomology groups of spherical Schubert varieties on the affine Grassmannianhave no p-torsion, as long as p is outside a certain small and explicitly givenset of prime numbers. (Juteau has exhibited counterexamples when p is a badprime.) The main idea is to convert this topological question into an algebraicquestion about perverse-coherent sheaves on the dual nilpotent cone using theJuteau-Mautner-Williamson theory of parity sheaves.

Parity sheaves on the affine Grassmannian and the Mirkovi\'c-Vilonen conjecture