(I appologize in advance if this question is too naive for experts, since I know very little about the geometry/combinatorics of Weyl/Coxeter groups.)
For simplicity, let $G$ be a connected reductive group over $\mathbb{C}$, so that there's no harm identifying an algebraic group with the group of its $\mathbb{C}$-points, and $G$ is split of course. We fix a Borel subgroup $B=TU$ of $G$ with its Levi decomposition ($T$ a maximal torus of $G$ and $U$ its unipotent radical), and then we get the root system $\Phi=\Phi(G,T)$ together with a set of positive roots $\Phi^+$ and simple roots $\Delta\subset \Phi^+$. Let $W:=W^G=N_G(T)/T$ be the Weyl group of $(G,T)$.
Let $P$ be a standard (i.e. $P\supset B$) parabolic subgroup of $G$ with its Levi decomposition $P=MN$ where $N$ is the unipotent radical of $P$ and $M\supset T$ is a Levi subgroup of $P$. Then $P$ corresponds to a subset $\theta$ of $\Delta$ where $(\Delta-\theta)$ is the set of positive simple roots that appear in ${\rm Lie}\,N$.
I want to understand the following subgroup:
$$\mathcal{W}(M):= \{ w\in W\ |\ wMw^{-1}=M \},$$
i.e. the "stabilizer" of the Levi factor $M$. Let $W^M$ be the Weyl group of $(M,T)$, then in the literature (of representation theory of reductive groups over local fields) I usually see the quotient group
$$W(M):=\mathcal{W}(M)/W^M.$$
But I have no idea what it looks like. I know that $W^M$ is the Weyl group of the root subsystem generated by $\theta\subset \Delta$, but what can we say about the subgroup $\mathcal{W}(M)$ and the quotient $W(M)$? Is there any related theorem? To be more concrete, I'm interested in questions:
- How to characterize $\mathcal{W}(M)$ and $W(M)$ geometrically (via root systems)?
- Can we compute the order of $W(M)$?
- Can we know anything more explicitly in the case when $P$ is a maximal parabolic ($\Delta-\theta=\{\alpha\}$)? For example, based on some examples, I guess that in the maximal case $|W(M)|$ can only be $1$ or $2$; is this right?
Thanks a lot in advance for any help/information!
Best Answer
As we discussed in the comments (example), $\mathcal W(M)$ is the subgroup of $W$ that preserves $\mathbb Z\theta$, and the map from $W(M)$ to the group of orthogonal transformations of $\mathbb Z\theta$, or of diagram automorphisms of the root system spanned by $\theta$, is an embedding.
You then asked whether $W(M)$ has order at most $2$ when $M$ is a maximal proper Levi subgroup of $G$, i.e., when $\theta$ misses exactly one simple root $\alpha$. As we discussed, every element $\overline w$ of $W(M)$ has a representative $w$ that preserves $\theta$. Since $w\Delta$ spans $\Phi(G, T)$, we have that $w\alpha$ belongs to $\pm\alpha + \mathbb Z\theta$. Suppose that $w\alpha$ belongs to $\alpha + \mathbb Z\theta$, hence to $\alpha + \mathbb Z_{\ge 0}\theta$. Then $w$ sends every root in $\Delta$ to a positive root, hence every positive root to a positive root, hence is the trivial element of $W$. Thus, any non-identity element of $ W(M)$ has a representative that preserves $\theta$, and carries $\alpha$ into $-\alpha + \mathbb Z\theta$; so any two non-identity elements have a quotient that preserves $\theta$, and carries $\alpha$ into $\alpha + \mathbb Z\theta$, and is therefore the identity.