  Let $x$ be an eigenvector for an element of a finite irreducible reflection
group $W$. Let $W_x$ denote the subgroup of $W$ which stabilises $x$. We
provide an upper bound for the number of roots in the root system of $W_x$ .
This generalises a result of Kostant, who showed that every eigenvector with
eigenvalue a primitive $h^\mathrm{th}$ root of unity is regular, where $h$ is
the Coxeter number of $W$. We also give a Lie-theoretic interpretation of our
result in the study of semisimple conjugacy classes over Laurent series. In a
forthcoming paper, we use this result to establish a geometric analogue of a
conjecture of Gross and Reeder.
