Lifting of elements of Weyl groups

TL;DR

This paper investigates the conditions under which elements of Weyl groups lift to their normalizers with the same order, focusing on regular, elliptic, and twisted elements, revealing subtle distinctions in their lifting properties.

Contribution

It provides a detailed analysis of when Weyl group elements lift to the normalizer with preserved order, including special cases like regular, elliptic, and twisted elements.

Findings

Weyl group elements can lift with order equal to or double their original order.

Elliptic elements have conjugate lifts with identical order.

Conditions for lifting in twisted cases are characterized.

Abstract

Suppose $G$ is a reductive algebraic group, $T$ is a Cartan subgroup, $N=\text{Norm}(T)$, and $W=N/T$ is the Weyl group. If $w\in W$ has order $d$, it is natural to ask about the orders lifts of $w$ to $N$. It is straightforward to see that the minimal order of a lift of $w$ has order $d$ or $2d$, but it can be a subtle question which holds. We first consider the question of when $W$ itself lifts to a subgroup of $N$ (in which case every element of $W$ lifts to an element of $N$ of the same order). We then consider two natural classes of elements: regular and elliptic. In the latter case all lifts of $w$ are conjugate, and therefore have the same order. We also consider the twisted case.