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

TL;DR

This paper proves a conjecture that a Weyl group action on certain B-invariant subvarieties of a G-variety extends from characteristic zero to positive characteristic, except possibly when p=2.

Contribution

It establishes that the Weyl group action on maximal B-invariant subvarieties persists in positive characteristic, confirming a conjecture for all primes p ≠ 2.

Findings

Weyl group action extends to positive characteristic p ≠ 2

Confirms conjecture for B-invariant subvarieties in positive characteristic

Provides a new understanding of group actions in algebraic geometry over fields of positive characteristic

Abstract

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