On algebraic normalisers of maximal tori in simple groups of Lie type

Abstract: Let $G$ be a finite simple group of Lie type and let $T$ be a maximal torus of $G$. It is well known that if the defining field of $G$ is large enough, then the normaliser of $T$ in $G$ is equal to the algebraic normaliser $N(G,T)$. We identify explicitly all the cases when $N_G(T)$ is not equal to $N(G,T).$