Let $G$ be a compact, connected Lie group and $K$ a closed, connected subgroup. If $K = T$ is a maximal torus, it is well known that $W := N_G(T)/Z_G(T) = N_G(T)/T$ is a finite reflection group, the Weyl group.
If $K = S$ is some submaximal torus, then $N := N_G(S)/Z_G(S)$ is a subquotient of $W$, given by those elements of the Weyl group which stabilize $S$ modulo those that fix it. It has a natural action on the tangent space $\mathfrak s$ to $S$ induced by the adjoint action of $G$
Under what conditions is this representation generated by reflections?
If we complexify, under what conditions is it generated by pseudoreflections (i.e., elements fixing a hyperplane)?