The Deligne–Lusztig cohomology degree conjecture

Let $G=G(q)$ be a group of Lie type, let $\bar{\Q}_\ell$ be an algebraic closure of the $\ell$-adic numbers, and let $\ell$ be a prime dividing $\Phi_d(q)$. For a unipotent character $\chi$ of $G$, let $\pi(\chi_i)$ denote the rational number defined from the modified degree function $B_d$. The Deligne–Lusztig cohomology degree conjecture. The number $\pi(\chi_i)$ is the degree of the cohomology of a Deligne–Lusztig variety in which $\chi$ appears. This gives a general conjectural description of the degrees in which unipotent characters occur in the cohomology of the relevant Deligne–Lusztig varieties, extending earlier conjectures for $d=1$ and $d=2$.