My current research requires some knowledge on the eigenvectors of elements (of infinite order) of Coxeter groups view as reflections in their geometric representation.

After some reading, my impression is that many has been done for the spectrum of "Coxeter elements" or "Coxeter transformations", which are (if I understand correctly) the product of a permutation of the generators. However I find few result on the spectrum of other elements (of infinite order).

It's possible that this impression is wrong and I missed something.

Question: Did I miss any reference? If not, why didn't "non-Coxeter elements" interest people? Is the eigenvalues too obvious to study? or is it too complicated?