Geometry and cohomology of compactified Deligne--Lusztig varieties

Abstract: For connected reductive groups together with a Frobenius root $F$ that preserves the set of simple reflections, we show that for every tuple $(w_1,\ldots, w_r)$ of Weyl group elements there exists a minimal length element in an $F$-conjugacy class in $W$ such that the cohomology of the structure sheaf and the canonical sheaf for the associated compactified Deligne--Lusztig varieties are isomorphic respectively.