Broué–Malle–Michel conjecture on Deligne–Lusztig cohomology

Let $\mathbf G$ be a reductive group over a finite field, let $d$ be a positive integer, and let $({\mathcal L},\lambda)\in {\mathcal{Cus}}^d({\mathcal G})$ with ${\mathcal L}=(q,\mathbf L,F)$. Introduce the Deligne–Lusztig varieties and cohomology modules associated with a parabolic subgroup $\mathbf P$ admitting $\mathbf L$ as a Levi complement, and let $W_{\mathcal G}({\mathcal L},\lambda)$ be the associated complex reflection group.

Broué–Malle–Michel conjecture. There exists such a parabolic subgroup $\mathbf P$ for which the cohomology modules in distinct degrees have no common irreducible constituent, and for some parameter $k_{{\mathcal L},\lambda}$,

$$\operatorname{End}\bigl({\mathcal R}_{\mathbf L\subset\mathbf P}^{\mathbf G}(\lambda)\bigr)\simeq {\mathcal H}_{k_{{\mathcal L},\lambda}}\bigl(W_{\mathcal G}({\mathcal L},\lambda),\zeta_d^{-1}q\bigr).$$

This conjecture seeks a uniform Hecke-algebra explanation of $d$-Harish–Chandra theory through the $\ell$-adic cohomology of Deligne–Lusztig varieties. The source states that it is true for $d=1$ but far from proved in general.