The Deligne–Lusztig cohomology concentration conjecture

Let $G=\b G^F$ be a finite group of Lie type, let $B$ be a unipotent $\ell$-block with abelian defect group, let $b$ be its Brauer correspondent, and let $Y$ be the associated Deligne–Lusztig variety. Let $K$ be the fraction field of the coefficient ring, and let $\chi_1,\dots,\chi_s$ be the unipotent ordinary characters of $B$, with perversities $\pi(\chi_i)$. The Deligne–Lusztig cohomology concentration conjecture. In the cohomology $H^\bullet(Y,K)$, the character $\chi_i$ appears in degree $\pi(\chi_i)$ and no other degree. This conjecture describes the cohomological realization of the expected geometric form of Broué's conjecture, in which the complex of $Y$ induces a perverse equivalence between $B$ and $b$.