Lifting of elements of Weyl groups

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.