The cohomology of Deligne-Lusztig varieties for the general linear group

We propose two inductive approaches for determining the cohomology of Deligne-Lusztig varieties in the case of G = GL(n). The first one uses Demazure compactifications and analyzes the corresponding Mayer-Vietoris spectral sequence. This allows us to give an inductive formula for the Tate twist -1 contribution of the cohomology of a DL-variety. The second approach relies on considering more generally DL-varieties attached to hypersquares in the Weyl group. Here we give explicit formulas for the cohomology of height-one elements.