We review properties of reductive groups related to existence of a (B, N)-pair. For an abstract group, having a (B, N)-pair is a very strong condition; many of the theorems we will give for reductive groups follow from this single property. Definition 3.1.1 We say that two subgroups B and N of a group G form a (B, N)-pair (also called a Tits system) for G if: (i) B and N generate G and T := B ∩ N is normal in N. (ii) The group W := N/T is generated by a set S of involutions such that: (a) For s ∈ S, w ∈ W we have BsB.BwB ⊂ BwB ∪ BswB.

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We state a conjecture about centralizers of certain roots of central elements in braid groups, and check it for braid groups of type A, B, G (d, 1, r) and a couple of other cases. Our proof makes use of results from Birman-Ko-Lee, of which we give a new intrinsic account.

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We study the cohomology of Deligne-Lusztig varieties with aim the construction of actions of Hecke algebras on such cohomologies, as predicted by the conjectures of Broué, Malle and Michel ultimately aimed at providing an explicit version of the abelian defect conjecture. We develop the theory for varieties associated to elements of the braid monoid and partial compactifications of them. We are able to compute the cohomology of varieties associated to (possibly twisted) rank 2 groups and powers of the longest element w 0 (some indeterminacies remain for G 2 ). We use this to construct Hecke algebra actions on the cohomology of varieties associated to w 0 or its square, for groups of arbitrary rank. In the subsequent work [F. Digne, J. Michel, Endomorphisms of Deligne-Lusztig varieties, Nagoya J. Math. 183 (2006)], we construct actions associated to more general regular elements and we study their traces on cohomology.