A positivity conjecture for the Alvis-Curtis dual of the intersection cohomology of a Deligne-Lusztig variety

We formulate a strong positivity conjecture on characters afforded by the Alvis-Curtis dual of the intersection cohomology of Deligne-Lusztig varieties. This conjecture provides a powerful tool to determine decomposition numbers of unipotent $\ell$-blocks of finite reductive groups.