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We give a description of the cohomology groups of the structure sheaf on smooth compactifications X ¯ (w) of Deligne–Lusztig varieties X(w) for GL n , for all elements w in the Weyl group. As a consequence, we obtain the mod p m and integral p-adic étale cohomology of X ¯ (w) . Moreover, using our result for X ¯ (w) and a spectral sequence associated to a stratification of X ¯ (w) , we deduce the mod p m and integral p-adic étale cohomology with compact support of X(w). In our proof of the main theorem, in addition to considering the Demazure–Hansen smooth compactifications of X(w), we show that a similar class of constructions provide smooth compactifications of X(w) in the case of GL n . Furthermore, we show in the appendix that the Zariski closure of X(w), for any connected reductive group G and any w, has pseudo-rational singularities. [ABSTRACT FROM AUTHOR]

We give a combinatorial formula to calculate the Euler characteristic of an analogue of a Deligne–Lusztig variety, denoted Y w , g , which is attached to an element w in the Weyl group of G L n and g ∈ G L n . The main theorem of this paper states that the Euler characteristic of Y w , g only depends on the unipotent part of the Jordan decomposition of g and the conjugacy class of w . It generalizes the formula of the Euler characteristic of Springer fibers for type A . [ABSTRACT FROM AUTHOR]

Abstract: 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]. [Copyright &y& Elsevier]

Abstract: Let be an abelian prime-to-p Galois covering of smooth schemes over a perfect field k of characteristic . Let Y be a smooth compactification of such that 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 -lattice inside the latter; here W is the Witt vector ring of k. If a finite group G acts compatibly on X, and Y then our construction is G-equivariant. As an example we apply it to Deligne–Lusztig varieties. For a finite field k, if is a connected reductive algebraic group defined over k and a k-rational torus satisfying a certain standard condition, we obtain a meaningful equivariant -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. [Copyright &y& Elsevier]