Cohomological degree conjecture for Deligne–Lusztig varieties

Let $\ell\neq p$ be primes, let $q$ be a power of $p$, let $d$ be the multiplicative order of $q$ modulo $\ell$, and let $G=G(q)$ be a finite group of Lie type with abelian Sylow $\ell$-subgroups. For coprime integers $d,\kappa\geq 1$, let $Y_{\kappa/d}$ be the associated Deligne–Lusztig variety. If $\chi$ is a unipotent character of $\bar{\Q}_\ell G$, let $\pi_{\kappa/d}(\chi)$ be the invariant defined from its generic degree and the generic degree of the associated $d$-cuspidal pair. Cohomological degree conjecture. $\pi_{\kappa/d}(\chi)$ is the unique degree of the cohomology of

$$H^\bullet(Y_{\kappa/d},\bar{\Q}_\ell)$$

in which $\chi$ appears. This conjecture predicts how unipotent characters occur in the cohomology of Deligne–Lusztig varieties and is intended to provide the geometric form of Broué's conjecture; the source gives no resolution of it.