Lifting Involutions in a Weyl Group to the Normalizer of the Torus

TL;DR

This paper constructs a specific section of the Weyl group in a split reductive group over a finite field, demonstrating a property of involutions under the Frobenius map that relates to the normalizer of a maximal torus.

Contribution

It provides a novel construction of a section of the Weyl group satisfying braid relations, with a key property relating lifts of involutions to the Frobenius map.

Findings

Constructed a section of the Weyl group satisfying braid relations.

Showed the lift of an involution under Frobenius is its inverse.

Established a connection between involutions and the normalizer of the torus.

Abstract

Let N be the normalizer of a maximal torus T in a split reductive group over F_q, and let w be an involution in the Weyl group N/T. We construct a section of W satisfying the braid relations, such that the image of the lift n of w under the Frobenius map is equal to the inverse of n.