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We prove that the Deligne–Lusztig variety associated to minimal length elements in any δ -conjugacy class of the Weyl group is affine, which was conjectured by Orlik and Rapoport in [S. Orlik, M. Rapoport, Deligne–Lusztig varieties and period domains over finite fields, arXiv: 0705.1646 ].

Let X→Y0 be an abelian prime-to-p Galois covering of smooth schemes over a perfect field k of characteristic p>0. Let Y be a smooth compactification of Y0 such that Y−Y0 is a normal crossings divisor on Y. We describe a logarithmic F-crystal on Y whose rational crystalline cohomology is the rigid cohomology of X, in particular provides a natural W[F]-lattice inside the latter; here W is the Witt vector ring of k. If a finite group G acts compatibly on X, Y0 and Y then our construction is G-equivariant. As an example we apply it to Deligne–Lusztig varieties. For a finite field k, if G is a connected reductive algebraic group defined over k and L a k-rational torus satisfying a certain standard condition, we obtain a meaningful equivariant W[F]-lattice in the cohomology (ℓ-adic or rigid) of the corresponding Deligne–Lusztig variety and an expression of its reduction modulo p in terms of equivariant Hodge cohomology groups.

Motivated by the Broué conjecture on blocks with abelian defect groups for finite reductive groups, we study ''parabolic'' Deligne-Lusztig varieties and construct on those which occur in the Broué conjecture an action of a braid monoid, whose action on their $\ell$-adic cohomology will conjecturally factor trough a cyclotomic Hecke algebra. In order to construct this action, we need to enlarge the set of varieties we consider to varieties attached to a ''ribbon category''; this category has a {\em Garside family}, which plays an important role in our constructions, so we devote the first part of our paper to the necessary background on categories with Garside families.