Cohomology and geometry of Deligne–Lusztig varieties for GLₙ

Abstract We give a description of the cohomology groups of the structure sheaf on smooth compactifications X (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 X (w) X ¯ (w). Moreover, using our result for X (w) X ¯ (w) and a spectral sequence associated to a stratification of X (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.