Cohomology and Geometry of Deligne—Lusztig varieties for the general linear group

Deligne–Lusztig varieties associated to a connected reductive group $G$ in positive characteristic are originally constructed for studying representations of $G(\mathbb{F}_q)$. In particular, Deligne and Lusztig (1976) showed that the irreducible representations are contained in the alternating sum of compactly supported $\ell$-adic cohomology groups of them. In our work, we adapt the double induction strategy by Orlik (2018), originally developed for studying the individual $\ell$-adic cohomology groups, to the cohomology of the structure sheaf of smooth compactifications of Deligne–Lusztig varieties for ${\rm GL}_n$. From our result we also obtained the compactly supported mod $p^r$ and $p$-adic cohomology for Deligne–Lusztig varieties for ${\rm GL}_n$.

Besides sketching the proof of the main theorem, we will also discuss possible directions for generalizing our result. For example, for arbitrary connected reductive $G$, we may already deduce that the Zariski closure of Deligne–Lusztig varieties in the flag variety $G/B$ have $F$-rational singularities. This allows us to apply a result of Kovács on birational equivalence inducing cohomological equivalence for the structure sheaf and the canonical sheaf.