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On the cohomology of the compactification of the Deligne-Lusztig varieties. (Sur la cohomologie de la compactification des variétés de Deligne-Lusztig.) (French. English summary) Zbl 1310.14041
Let \(\mathbf{G}\) be a connected reductive \(\mathbb{F}_q\)-group endowed with the Frobenius isogeny \(F\), where \(q\) is a power of a prime number \(p\). For an element \(w\) in the Weyl group of \(\mathbf{G}\), Deligne and Lusztig constructed a finite étale morphism \(\pi: X(w) \to Y(w)\) that is \(\mathbf{G}^F\)-equivariant, as well as a compactification \(j: X(w) \hookrightarrow \bar{X}(w) \) à la Demazure. When \(w\) is a Coxeter element, \(\bar{X}(w)\) admits a stratification indexed by the \(F\)-stable parabolic subgroups: let \(\mathbf{P}\) be an \(F\)-stable parabolic with unipotent radical \(\mathbf{U}\), the corresponding stratum is \[ i_{\mathbf{P}}: \bar{X}_{\mathbf{P}}(w) := \bar{X}(w)^{\mathbf{U}^F} \to \bar{X}(w). \] Put \(\Lambda = \mathbb{Z}/\ell^m\) for some prime number \(\ell \neq p\) and \(m\). The pull-back \[ R\Gamma(X(w), \pi_* \Lambda) = R\Gamma( \bar{X}(w), Rj_* (\pi_*\Lambda) ) \to R\Gamma( \bar{X}_{\mathbf{P}}(w), i^*_{\mathbf{P}} Rj_* (\pi_*\Lambda) ) \] then induces a natural morphism \[ R\Gamma(X(w), \pi_* \Lambda)^{\mathbf{U}^F} \to R\Gamma( \bar{X}_{\mathbf{P}}(w), i^*_{\mathbf{P}} Rj_* (\pi_*\Lambda) ). \]
In this paper, it is proved that the arrow above is an isomorphism when \(\mathbf{G} = \mathrm{GL}_d\). This result is motivated by the study of the cohomology of Drinfeld’s symmetric space over a \(p\)-adic field \(K\) with residual field \(\mathbb{F}_q\), for which \(Rj_*(\pi_* \Lambda)\) appears as certain nearby cycles. Such issues are pertinent in the geometric realization of the local Langlands conjecture and the Jacquet-Langlands correspondence over \(K\).

14L35 Classical groups (algebro-geometric aspects)
14F20 Étale and other Grothendieck topologies and (co)homologies
20G05 Representation theory for linear algebraic groups
20G10 Cohomology theory for linear algebraic groups
20G40 Linear algebraic groups over finite fields
20C20 Modular representations and characters
20C33 Representations of finite groups of Lie type
11S37 Langlands-Weil conjectures, nonabelian class field theory
Full Text: DOI
[1] Bonnafé, Cédric; Rouquier, Raphaël, Coxeter orbits and modular representations, Nagoya Math. J., 183, 1-34, (2006) · Zbl 1109.20038
[2] Bonnafé, Cédric; Rouquier, Raphaël, Compactification des variétés de Deligne-Lusztig, Ann. Inst. Fourier (Grenoble), 59, 2, 621-640, (2009) · Zbl 1167.14034
[3] Borel, Armand, Linear algebraic groups, 126, (1991), Springer-Verlag, New York · Zbl 0726.20030
[4] Dat, Jean-François, A lemma on vanishing cycles and its application to the tame Lubin-Tate space, Math. Res. Lett. (1), 19, 1-9, (2012) · Zbl 1281.11099
[5] Deligne, P.; Lusztig, G., Representations of reductive groups over finite fields, Ann. of Math. (2), 103, 1, 103-161, (1976) · Zbl 0336.20029
[6] Dudas, Olivier, Géométrie des variétés de Deligne-lusztig : décompositions, cohomologie modulo \(l\) et représentations modulaire, (2010)
[7] Grothendieck, Alexander, Revêtements étales et groupe fondamental, 224, (1971), Springer-Verlag, Berlin
[8] Grothendieck, Alexander, Théorie des topos et cohomologie étale des schémas. Tome 2, 270, (1972), Springer-Verlag, Berlin · Zbl 0234.00007
[9] Grothendieck, Alexander, Théorie des topos et cohomologie étale des schémas. Tome 3, 305, (1973), Springer-Verlag, Berlin
[10] Hartshorne, Robin, Algebraic geometry, 126, 52, (1977), Springer-Verlag, New York · Zbl 0531.14001
[11] Ito, Tetsushi, Weight-monodromy conjecture for \(p\)-adically uniformized varieties, Invent. Math, 159, 3, 607-656, (2005) · Zbl 1154.14014
[12] Liu, Qing, Algebraic geometry and arithmetic curves, 6, (2002), Oxford University Press, Oxford · Zbl 0996.14005
[13] Lusztig, G., On the finiteness of the number of unipotent classes, Invent. Math., 34, 3, 201-213, (1976) · Zbl 0371.20039
[14] Lusztig, G., Coxeter orbits and eigenspaces of Frobenius, Invent. Math., 38, 2, 101-159, (197677) · Zbl 0366.20031
[15] Springer, T. A., Linear algebraic groups, 9, (1998), Birkhäuser Boston Inc., Boston, MA · Zbl 0927.20024
[16] Wang, Haoran, L’espace de Drinfeld et correspondance de Langlands locale I · Zbl 1341.11021
[17] Wang, Haoran, L’espace de Drinfeld et correspondance de Langlands locale II · Zbl 1430.11064
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