The inductive Lehrer–Solomon conjecture for parabolic subgroups

Let $W$ be a finite Coxeter group generated by $S$, let $L\subseteq S$, and let $W_L$ be the corresponding parabolic subgroup. Let $\mathcal{C}_L$ be the set of cuspidal conjugacy classes in $W_L$, with a fixed representative $w_C\in C$ for each $C\in\mathcal{C}_L$. Let $N_W(W_L)$ be the normalizer of $W_L$, and let $\widetilde{\Phi}_L$ and $\widetilde{\Psi}_L$ be the extensions to $N_W(W_L)$ of the top-component characters $\Phi_L$ and $\Psi_L$ of $\mathbb{C}W_L$ and $A(W_L)$. Inductive Lehrer–Solomon conjecture. For each $C\in\mathcal{C}_L$, there exist linear characters $\widetilde{\varphi}_{w_C}$ and $\widetilde{\psi}_{w_C}$ of $C_W(w_C)$ such that

$$\widetilde{\Phi}_L=\sum_{C\in\mathcal{C}_L}\operatorname{Ind}_{C_W(w_C)}^{N_W(W_L)}\widetilde{\varphi}_{w_C},$$ $$\widetilde{\Psi}_L=\sum_{C\in\mathcal{C}_L}\operatorname{Ind}_{C_W(w_C)}^{N_W(W_L)}\widetilde{\psi}_{w_C},$$

and $\widetilde{\psi}_{w_C}=\widetilde{\varphi}_{w_C}\epsilon_S\alpha_L$ for all $C\in\mathcal{C}_L$, where $\epsilon_S$ is the sign character and $\alpha_L$ is the relevant determinant character. This relative conjecture is designed to imply the global decomposition conjecture by induction and transitivity of induction; the paper reports the method as established for low-rank finite Coxeter groups, while the general inductive statement remains open.