Refined Lehrer–Solomon conjecture for finite Coxeter groups

Let $W$ be a finite Coxeter group of rank $n$, let $\mathcal R$ be a set of representatives of its conjugacy classes, and let $C_W(w)$ be the centralizer of $w$. Let $\rho_W$ be the regular character, let $\omega_W^p$ be the character on $H^p(M)$ for the complement $M$ of the reflecting hyperplanes, and let $\omega_W=\sum_{p\geq0}\omega_W^p$. For $w\in\mathcal R$, let $\alpha_w$ be the composition of the determinant with restriction to the $1$-eigenspace of $w$; let $\epsilon$ be the sign character. Refined Lehrer–Solomon conjecture. For every $w\in\mathcal R$, there exists a linear character $\varphi_w$ of $C_W(w)$ such that

$$\rho_W=\sum_{w\in\mathcal R}\operatorname{Ind}_{C_W(w)}^W\varphi_w$$

and

$$\omega_W=\epsilon\sum_{w\in\mathcal R}\operatorname{Ind}_{C_W(w)}^W(\alpha_w\varphi_w).$$

If $\mathcal R_p$ consists of those $w\in\mathcal R$ for which the codimension of the $1$-eigenspace of $w$ in $V$ is $p$, then

$$\omega_W^p=\epsilon\sum_{w\in\mathcal R_p}\operatorname{Ind}_{C_W(w)}^W(\alpha_w\varphi_w).$$

This refinement simultaneously decomposes the regular and Orlik–Solomon characters, graded by eigenspace codimension. The paper verifies the conjecture computationally for all finite Coxeter groups of ranks three and four; the general conjecture remains unresolved in the source.