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We study the cohomology of parabolic Deligne-Lusztig varieties associated to unipotent blocks of GL n (q). We show that the geometric version of Broué's conjecture over Q , together with Craven's formula, holds for any unipotent block whenever it holds for the principal Φ 1 -block.doi:10.1090/s1088-4165-2013-00446-6 fatcat:gauhpsdepfhcja5wdjr7z4j5zy