Weyl group covers for Brieskorn's resolutions in all characteristics and the integral cohomology of $G/P$

Abstract

We unify results of Artin, Brieskorn, Slodowy and others by showing that, in all characteristics, the Artin component of the deformation space of a rational surface singularity has a ramified cover where simultaneous resolution exists and the Galois group of this cover is the Weyl group determined by the configuration of $(-2)$-curves in the minimal resolution. This verifies a conjecture made by Burns and Rapoport. We use an integral version of this to show that certain actions of Weyl groups on polynomial rings over the integers give rings of invariants that are also polynomial, and deduce that the integral cohomology rings of complete flag varieties $G/B$ of types $A,D$ or $E$ can be described as the corresponding rings of co-invariants.