We prove the Mirković–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.
Publication Source (Journal or Book title)
Achar, P., & Rider, L. (2015). Parity sheaves on the affine Grassmannian and the Mirković–Vilonen conjecture. Acta Mathematica, 215 (2), 183-216. https://doi.org/10.1007/s11511-016-0132-6