Affineness of Deligne–Lusztig varieties for minimal length elements

We prove that the Deligne–Lusztig varieties associated to elements of the Weyl group which are of minimal length in their twisted class are affine. Our proof differs from the proof of He and Orlik–Rapoport and it is inspired by the case of regular elements, which correspond to the varieties involved in Brouéʹs conjectures.