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Nous construisons explicitement la normalisation de la compactification de Bott-Samelson-Demazure-Hansen des variétés de Deligne-Lusztig $\mathbf{X}\left(w\right)$ dans leur revêtement $\mathbf{Y}\left(w\right)$ et retrouvons ainsi un résultat de Deligne-Lusztig sur la monodromie locale autour des diviseurs de la compactification.

We construct a subalgebra ${\Sigma }^{\prime }\left(W_n\right)$ of dimension $2·3^{n-1}$ of the group algebra of the Weyl group $W_n$ of type $B_n$ containing its usual Solomon algebra and the one of ${𝔖}_n$: ${\Sigma }^{\prime }\left(W_n\right)$ is nothing but the Mantaci-Reutenauer algebra but our point of view leads us to a construction of a surjective morphism of algebras ${\Sigma }^{\prime }\left(W_n\right)\to \mathbf{Z}\mathrm{Irr}\left(W_n\right)$. Jöllenbeck’s construction of irreducible characters of the symmetric group by using the coplactic equivalence classes can then be transposed to $W_n$. In an appendix, P. Baumann and C. Hohlweg present in an explicit and...