Lifting involutions in a Weyl group to the torus normalizer

TL;DR

This paper constructs explicit liftings of involutions in Weyl groups to the torus normalizer in split reductive groups, ensuring Frobenius map properties, advancing understanding of algebraic group symmetries.

Contribution

It provides an explicit construction of liftings of Weyl group involutions to the torus normalizer with Frobenius compatibility, a novel approach in algebraic group theory.

Findings

Explicit liftings of involutions constructed

Liftings satisfy Frobenius inverse property

Enhances understanding of Weyl group symmetries

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 explicitly a lifting n of w in N such that the image of n under the Frobenius map is equal to the inverse of n.