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Compactification of Deligne-Lusztig varieties. (Compactification des variétés de Deligne-Lusztig.) (French. English summary) Zbl 1167.14034
Let \(G\) be a connected reductive group defined over the algebraic closure of a finite field, and let \(F:G \to G\) denote an isogeny of \(G\) of which some power is a (standard) Frobenius morphism. Let \(T\) be a maximal \(F\)-stable torus of \(G\), and let \(W\) denote the Weyl group of \(G\). Then to any element \(w \in W\) we can associate two Deligne-Lusztig varieties \(X(w)\) and \(Y(w)\) and a finite étale morphism \(Y(w) \to X(w)\) making \(X(w)\) a quotient of \(Y(w)\) by the action of \(T^F\), [see P. Deligne and G. Lusztig, Ann. Math. (2) 103, 103–161 (1976; Zbl 0336.20029)]. In [loc. cit.], Deligne and Lusztig construct a smooth compactification \(\overline{X}(w)\) of \(X(w)\); the main purpose of this paper is to give an explicit construction of the normalization \(\overline{Y}(w)\) of \(\overline{X}(w)\) in \(Y(w)\). The explicit nature of this construction allows the authors to give many properties of \(\overline{Y}(w)\), as well as to deduce a new proof of Lemme 9.11 of [loc. cit.], which is a key part of the proof of the Macdonald conjectures associating an irreducible representation of \(G^F\) to a character in general position of \(T^F\).

14M17 Homogeneous spaces and generalizations
14L30 Group actions on varieties or schemes (quotients)
20Gxx Linear algebraic groups and related topics
Full Text: DOI Numdam EuDML
[1] Bonnafé, C.; Rouquier, R., Catégories dérivées et variétés de Deligne-Lusztig, Publ. Math. I.H.E.S., 97, 1-59, (2003) · Zbl 1054.20024
[2] Borel, A., Linear algebraic groups, 126, (1991), Springer-Verlag · Zbl 0726.20030
[3] Deligne, P.; Lusztig, G., Representations of reductive groups over finite fields, Ann. of Math., 103, 103-161, (1976) · Zbl 0336.20029
[4] Digne, F.; Michel, J.; Rouquier, R., Cohomologie de certaines variétés de Deligne-Lusztig, Adv. Math., 209, 749-822, (2007) · Zbl 1118.20006
[5] Grothendieck, A., SGA1 Revêtements étales et groupe fondamental, 224, (1971), Springer
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