Remarks concerning linear characters of reflection groups

Let $G$ be a finite group generated by unitary reflections in a Hermitian space $V$, and let $\zeta$ be a root of unity. Let $E$ be a subspace of $V$, maximal with respect to the property of being a $\zeta$-eigenspace of an element of $G$, and let $C$ be the parabolic subgroup of elements fixing $E$ pointwise. If $\chi$ is any linear character of $G$, we give a condition for the restriction of $\chi$ to $C$ to be trivial in terms of the invariant theory of $G$, and give a formula for the polynomial $\sum_{x\in G}\chi(x)T^{d(x,\zeta)}$, where $d(x,\zeta)$ is the dimension of the $\zeta$-eigenspace of $x$. Applications include criteria for regularity, and new connections between the invariant theory and the structure of $G$.