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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 $$K$$.

##### MSC:
 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
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