On the cohomology of the compactification of the Deligne-Lusztig varieties
type: Item
instance of: scholarly article
title: On the cohomology of the compactification of the Deligne-Lusztig varieties
zbMATH Open document ID: 1310.14041
author: Haoran Wang
publication date: 2015-01-16T00:00:00Z
review text: 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 \(K\).
reviewed by: Wen-Wei Li
Mathematics Subject Classification ID: 14L35
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 11S37
zbMATH DE Number: 6387332
zbMATH Keywords: Deligne-Lusztig varieties
zbMATH Keywords: Coxeter element
zbMATH Keywords: compactification
MaRDI profile type: Publication
full work available at URL: aif.2905
OpenAlex ID: W2331633881
cites work: Coxeter Orbits and Modular Representations
cites work: Compactification of Deligne-Lusztig varieties
cites work: Q3999894
cites work: A lemma on nearby cycles and its application to the tame Lubin-Tate space
cites work: Representations of reductive groups over finite fields
cites work: Q2551490
cites work: Seminar of algebraic geometry du Bois-Marie 1963--1964. Topos theory and étale cohomology of schemes (SGA 4). Vol. 1: Topos theory. Exp. I--IV
cites work: Q3312347
cites work: Weight-monodromy conjecture for \(p\)-adically uniformized varieties
cites work: Q4530238
cites work: On the finiteness of the number of unipotent classes
cites work: Coxeter orbits and eigenspaces of Frobenius
cites work: Linear algebraic groups.
cites work: Symmetric Drinfeld space and local Langlands correspondence. I
cites work: Drinfeld symmetric space and local Langlands correspondence. II
DOI: 10.5802/AIF.2905
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published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 6387332