Let $G$ be a semisimple algebraic group over an algebraically closed field, and let $T$ be a torus in $G$.


If $T$ is a maximal torus, then $N\_G(T)/Z\_G(T)=N\_G(T)/T$ is the Weyl group $W$ of $G$. If $T$ is not maximal, then what can be said aboug the "Weyl group" $W\_T=N\_G(T)/Z\_G(T)$ ? Is there a relation between $W\_T$ and $W$ ? I guess the answer is well-known, but I didn't find a relevant reference.



Weyl group of a singular torus
This answer combines my previous comments and adds a little more.


Let $G$ be connected reductive over an algebraically closed field. Let $S \subset T$ be a subtorus of a maximal torus.
I'll denote by $W\_S$ the Weyl group $N\_G(S)/Z\_G(S)$. In particular, $W\_T = N\_G(T)/T$, as $Z\_G(T) = T$.


In general neither of $N\_G(S)$ and $N\_G(T)$ is contained in the other (it suffices to consider $G = GL\_2$). I think the best comparison between $W\_S$ and
$W\_T$ is the following. Let $W\_{S,T} = (N\_G(S) \cap N\_G(T))/T$. This is clearly a subgroup of $W\_T$ and it naturally
maps to $W\_S$, as $T \subset Z\_G(S)$.


In fact, every Weyl group (in your sense) naturally lives in a maximal one. Note that $Z\_G(S)$ is a Levi
subgroup of a parabolic subgroup. (I happen to have Digne-Michel's nice book "Representations of finite groups of Lie type"
at hand, and they prove it in Prop. 1.21.) Moreover, it's clear that $N\_G(S)$ normalises even $Z\_G(S)$. Thus $W\_S$ is
in fact a subgroup of $W\_S' := N\_G(Z\_G(S))/Z\_G(S)$. I claim that $W\_S'$ is also a Weyl group. To see this, let $M = Z\_G(S)$,
a Levi subgroup and let $S' = Z(M)^0$ (the connected component of the centre of $M$). Then $W\_{S'} = N\_G(M)/M = N\_G(S')/Z\_G(S')$.
(It's a standard fact that $Z\_G(S') = M$, again see [DM], Prop. 1.21; for the second equality note again that $N\_G(S') = N\_G(Z\_G(S'))$.)


Thus the most interesting "Weyl groups" (in your sense) are the ones of the form $N\_G(M)/Z\_G(M)$ for $M$ a Levi subgroup.
Let $S' = Z(M)^0$ as above. I claim that in this case the natural map $W\_{S',T} \to W\_{S'}$ is surjective.
Suppose that $n \in N\_G(S')$. Then $n M n^{-1} = M$, so $n T n^{-1}$ is a maximal torus of $M$, hence it's of the form
$m T m^{-1}$ for some $m \in M$. It follows that $m^{-1} n \in N\_G(T)$, but it's also in $N\_G(S')$ (as $m \in Z\_G(S')$).
This proves the claim.


Finally, I think it's helpful to look at the simple example of $G = GL\_4$ (or $SL\_4$), $M$ the Levi with (2,2) blocks containing the
diagonal torus $T$ and $S'$ its centre, that I wrote down in the comments above. In this case $W\_{S'} = N\_G(M)/M$ is of order two (switching the two blocks),
$W\_{S',T}$ is of order 8 surjecting onto it and injecting into $W\_T \cong S\_4$.


[I didn't pay much attention to positive characteristic while writing this, but I think it should all be fine.]


