Let $(W,S)$ be a Coxeter system (of finite rank) and $c \in W$ a Coxeter element.

If $W$ is finite, then the centralizer $C_W(c)$ is the cyclic group generated by $c$ (e.g. see the book "Reflection Groups and Invariant Theory" of Kane).

If $(W,S)$ is irreducible and not finite or affine, then the cyclic group $\langle c \rangle$ is of finite index in $C_W(c)$ (see Theorem 2.6 in the paper "Irreducible Coxeter Groups" by Luis Paris).

Are there more results in this direction? For affine Coxeter groups? Is it conjectured that $C_W(c) = \langle c \rangle$ holds in general? Or is there a counterexample?

**Update**: There is a paper called "On the centralizer of a Coxeter element" by Blokhina (seems like that he/ she was a student of Vinberg), originally in russian, but there is an english translation in Moscow Univ. Math. Bull. You find an entry
here
on mathscinet.

I just managed to get a copy of that paper (but because of the copyright I'm not sure if I'm allowed to make it available here). The following statement is proven in that paper (including all infinite affine families David is mentioning in his comment):

Let $(W,S)$ be a Coxeter system of finite rank and $c \in W$ a Coxeter element. If $(W,S)$ is of type $\widetilde{A}_n$ or if the Coxeter graph of $(W,S)$ is a tree **and** all of its labels are 3 (that is, it is simply-laced), then $C_W(c)= \langle c \rangle$.

(The arguments in the proof are somewhat similar to the proof of the statement for finite Coxeter groups given by Carter. Jim mentioned this reference implicitely in his comment. I add it here: It is Proposition 30 in Carter's paper "Conjugacy Classes in the Weyl group".)

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