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