Bonnafé, Cédric; Rouquier, Raphaël Compactification of Deligne-Lusztig varieties. (Compactification des variétés de Deligne-Lusztig.) (French. English summary) Zbl 1167.14034 Ann. Inst. Fourier 59, No. 2, 621-640 (2009). 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 \(T^F\). Reviewer: Michael Bate (York) Cited in 1 Document MSC: 14M17 Homogeneous spaces and generalizations 14L30 Group actions on varieties or schemes (quotients) 20Gxx Linear algebraic groups and related topics Keywords:Deligne-Lusztig varieties; compactification; normalisation; monodromy PDF BibTeX XML Cite \textit{C. Bonnafé} and \textit{R. Rouquier}, Ann. Inst. Fourier 59, No. 2, 621--640 (2009; Zbl 1167.14034) Full Text: DOI Numdam EuDML References: [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.