Cohomology and geometry of Deligne–Lusztig varieties for GLₙ

We give a description of the cohomology groups of the structure sheaf on smooth compactifications X̄(w) X ¯ ( w ) of Deligne–Lusztig varieties X ( w ) for GLₙ GL n , for all elements w in the Weyl group. As a consequence, we obtain the mod\ pᵐ mod p m and integral p -adic étale cohomology of X̄(w) X ¯ ( w ) . Moreover, using our result for X̄(w) X ¯ ( w ) and a spectral sequence associated to a stratification of X̄(w) X ¯ ( w ) , we deduce the mod\ pᵐ mod p m and integral p -adic étale cohomology with compact support of X ( w ). In our proof of the main theorem, in addition to considering the Demazure–Hansen smooth compactifications of X ( w ), we show that a similar class of constructions provide smooth compactifications of X ( w ) in the case of GLₙ GL n . Furthermore, we show in the appendix that the Zariski closure of X ( w ), for any connected reductive group G and any w , has pseudo-rational singularities.