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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 $$T^F$$.

MSC:
 14M17 Homogeneous spaces and generalizations 14L30 Group actions on varieties or schemes (quotients) 20Gxx Linear algebraic groups and related topics
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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
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