Say $(W,S)$ is a finite Coxeter group, such as a Weyl group (which satisfies an additional crystallographic condition). Assume also that $W$ is irreducible. Then it has a longest element $w\_o$ relative to the given generating set $S$, which can be expressed as the $h/2$-power of a well-chosen Coxeter element when the Coxeter number $h$ is even (or a slightly modified expression if $h$ is odd). 


Assume now that $W$ is a Weyl group (the others being a little more complicated to study). A standard fact is that $w\_o = -1$ just when the type of $W$ in the classification is not ADE; more precisely, is different from $A\_\ell (\ell \geq 2), D\_\ell (\ell \geq 5 \text{ odd}), E\_6$. When $w\_o =-1$, its centralizer is obviously $W$, but otherwise is a proper subgroup.


A recent question [*here*](https://mathoverflow.net/questions/282727/type-c-n-weyl-group-contains-in-the-centralizer-of-the-longest-word-w-0-in?noredirect=1#comment697677_282727) suggests to me a possible uniform treatment of such centralizers in terms of *foldings* of Coxeter graphs. But I'm not sure how far this is supported by the literature, or exactly how it might work for type $A\_\ell$ with $\ell$ even --- then $W$ is the symmetric group $S\_{\ell+1}$ and $h=\ell+1$ is odd. (I'm also unsure about the non-crystallographic types). 


Note that folding the graph of (say) type $E\_6$ yields the graph $F\_4$. The Coxeter numbers are always the same when such foldings of ADE graphs occur, e.g., $h=12$ for both $E\_6$ and $F\_4$. Moreover, $w\_o$ is folded into the corresponding longest element $-1$ in accordance with the above expressions as powers of Coxeter elements if $h$ is even. The question cited suggests that the "folded" Weyl group might always embed in $W$ as the precise centralizer of the original $w\_o$. 



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> Is this true, and is there a reference? (Further, what can be said in the non-crystallographic case?) 
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Centralizer of longest element in a finite irreducible Weyl group: related to folding of ADE graphs?
The folding automorphism of a root system is $\tau:\alpha\mapsto -w\_0(\alpha)$ (since this preserves the positive roots, hence the simple roots, and is trivial if and only if $w\_0=-1$). Since every element of $W$ commutes with $-1$, $\text{Cent}\_W(\tau)=\text{Cent}\_W(w\_0)$.


At least in the case of a root system, $w\_0$ is a *regular* element of $W$ (the sum of the positive roots is a regular eigenvector). *Regular Elements of Finite Reflection Groups*, by T.A. Springer (Inventiones 1974) especially Theorems 4.2 and 6.4 should go a long way towards answering your question in general. 


