Let $\Phi$ be a root system and $\gamma \in \Phi$ a root. Let $W$ be the Weyl group and $\Delta$ a set of simple roots. Let $w \in W$ such that $w(\gamma)=\gamma$. Is it true that if $w=s_1\dots s_n$ with $s_i$ simple reflections, then $s_i(\gamma)=\gamma$ for $i=1\dots n$?

I can't see a reason why this shouldn't be true, but am unable to prove it.



This is definitely not true. For instance, already in $\Phi=B_2$, each root has a root orthogonal to it, so for every root there is some nontrivial element (in fact, a reflection) of the Weyl group fixing it. But e.g. a simple root in $B_2$ is not fixed by any simple reflection.

However, what you might want to know is the following. If we choose any point $v$ in the dominant chamber of our root system, then the stabilizer of $v$ is exactly the parabolic subgroup of $W$ generated by simple reflections that fix $v$. For a proof of this see Lemma 10.3B of Humphreys' "Introduction to Lie Algebras and Representation Theory" (https://www.springer.com/us/book/9780387900537).


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