On the W-action on B-sheets in positive characteristic

Let G be a connected reductive group defined over an algebraically closed base field of characteristic p ≥ 0, let B ⊆ G be a Borel subgroup, and let X be a G-variety. We denote the (finite) set of closed B-invariant irreducible subvarieties of X that are of maximal complexity by B0(X). The first named author has shown that for p = 0 there is a natural action of the Weyl group W on B0(X) and conjectured that the same construction yields a W-action whenever p ≠ 2. In the present paper, we prove this conjecture.