Cohomology and geometry of Deligne–Lusztig varieties for $$\textrm{GL}_n$$

Abstract

AbstractWe give a description of the cohomology groups of the structure sheaf on smooth compactifications $$\overline{X}(w)$$ X ¯ ( w ) of Deligne–Lusztig varieties X(w) for $$\textrm{GL}_n$$ GL n , for all elements w in the Weyl group. As a consequence, we obtain the $$\textrm{mod}\ p^m$$ mod p m and integral p-adic étale cohomology of $$\overline{X}(w)$$ X ¯ ( w ) . Moreover, using our result for $$\overline{X}(w)$$ X ¯ ( w ) and a spectral sequence associated to a stratification of $$\overline{X}(w)$$ X ¯ ( w ) , we deduce the $$\textrm{mod}\ p^m$$ 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 $$\textrm{GL}_n$$ 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.