Weyl Group Covers for Brieskorn’s Resolutions in All Characteristics and the Integral Cohomology of G/P

Abstract

Given an affine surface X with rational singularities and minimal resolution ${\mathit{X}}^{\prime }$, the covering of the Artin component of the deformation space of X where simultaneous resolutions are achieved is Galois and the Galois group is the Weyl group W associated with the configuration of $(-2)$-curves on ${\mathit{X}}^{\prime }$. This gives the existence of actions of W on polynomial rings over $\mathbb{Z}$ where the ring of invariants is also polynomial. In turn, this leads to a description of the integral cohomology rings of flag varieties of type $\mathit{ADE}$ that extends the known description of the rational cohomology rings as rings of coinvariants for actions of W.