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

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