# Weyl group of a singular torus

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.

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I suppose the best one can do is to consider the group $W_{S,T} := (N(S) \cap N(T))/Z(T)$, where $S \subset T$ is a subtorus and $T$ maximal. This is a subgroup of $W_T$ that maps to $W$. A nice case is when $Z(S) = T$, then $W_{S,T} \to W$ is an isomorphism. (This should be discussed in standard books on alg. groups in case $G$ is defined over some subfield and $S$ is the max. split subtorus defined over it. The nice case is when $G$ is quasi-split. If the books don't help, try Borel-Tits in Pub. IHES, 1965.) –  fherzig Mar 7 '12 at 18:41
I think that this is what one often calls $W(G, M)$, the Weyl group in $G$ of the Levi $M = C_G(T)$. In addition to what Florian describes, one also sometimes has use for the space $W(S, T) = N_G(S, T)/C_G(S)$ described in \S5 of link.springer.com/article/10.1007%2FBF01429876 (van Dijk - Computation of certain induced characters of $p$-adic groups), where $N_G(S, T)$ is the transporter' of $S$ into $T$. –  L Spice Dec 1 '13 at 19:52

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.]

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EDITED: I'm not sure why "singular torus" appears in the header, since in the Borel-Chevalley structure theory this refers to a torus which lies in infinitely many Borel subgroups (e.g., the identity component in the kernel of a root).

Anyway, the foundational paper by Borel-Tits Groupes reductifs (IHES Publ. Math., 1965) includes among other things a description of all closed subgroups of a connected semisimple (or reductive) group $G$ which contain maximal tori of $G$. Here it's convenient to reserve the letter $T$ for such a maximal torus, while $S$ can be any subtorus. Now the centralizer $H:=Z_G(S)$ is connected, reductive, and contains $T$; more precisely, it is generated by $T$ along with some pairs of root subgroups centralizing $S$ (for pairs of positive and negative roots) and part of the Weyl group $W$ of $G$ relative to $T$. Then the Weyl group of $H$ relative to $T$ is naturally isomorphic to a subgroup of $W$.

This part of the structure theory is summarized briefly in Section 2.1 of my 1995 AMS book Conjugacy Classes in Semisimple Algebraic Groups (with references to Borel-Tits).

P.S. The terminology "Weyl group" here threw me off, so I was trying to answer a different question. The answer to the original question is basically no. Given a connected semisimple group $G$ (say with a maximal torus $T$, where the Weyl group of $G$ relative to $T$ is denoted $W$), the finite group $N_G(S)/Z_G(S)$ for an arbitrary subtorus $S$ of $T$ usually has no direct relationship with $W$. This group is always finite (rigidity of tori), but shouldn't be thought of as a "Weyl group". That notion is usually defined for a pair consisting of an algebraic group together with one of its maximal tori. For instance, the connected reductive group $H:=Z_G(S)$ has $T$ as a maximal torus and then the Weyl group of $H$ relative to $T$ is a subgroup of $W$ (as indicated in my corrected version above). But this is not the finite group in question here. As Florian indicates, in special cases there may still be an indirect connection between the group $N_G(S)/Z_G(S)$ and $W$.

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I don't think there's a natural injective homomorphism $N(S)/Z(S) \to N(T)/T$ in general. Concrete example: $G=GL_4$, $T$ diagonal, $S$ the subtorus of elements $diag(x,x,y,y)$ of rank 2. Then $Z(S)=GL_2 \times GL_2$ and $N(S)/Z(S)$ is of order two, with the non-trivial element interchanging the two blocks. To give a map to $W_T$ would mean to make a choice of a permutation on 4 letters (of order 2) that interchanges the subsets $\{1,2\}$ and $\{3,4\}$, wouldn't it? –  fherzig Mar 7 '12 at 21:07
In this example, what I called $W_{S,T}$ above has order 8 and it surjects onto $W_S$ of order 2. –  fherzig Mar 7 '12 at 21:09
@fherzig: I don't follow your example, since what you denote as $W_S$ will be a Klein 4-group generated by two transpositions but not involving interchange. In any case, the literature is not so helpful in this discussion, since for instance Borel-Tits focus more on fields of definition whereas the current discussion occurs over an algebraically closed field. It may clarify here to limit to the semisimple group $SL_4$ of type $A_3$ which has Weyl group $S_4$, while $Z(S)$ is a Levi subgroup of type $A_1×A_1$` with Weyl group of order 4. –  Jim Humphreys Mar 7 '12 at 23:21
@Jim Humphreys: by $W_S$ I meant $N_G(S)/Z_G(S)$, not the Weyl group of $T$ in $Z(S)$ (which would be $N_{Z_G(S)}(T)/T$). The example works equally well with the semisimple group $SL_4$, although I find it easier to think about $GL_4$ because there $Z(S)$ splits up as product. –  fherzig Mar 7 '12 at 23:47
@fherzig: Sorry for my misdirected comments. See my edited version. –  Jim Humphreys Mar 8 '12 at 13:26