Parity sheaves on the affine Grassmannian and the Mirković-Vilonen conjecture

Pramod N. Achar, Laura Rider

Published 2013-05-08, updated 2015-09-17Version 4

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.