On the cohomology of the compactification of the Deligne-Lusztig varieties
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 \(K\).
scientific article; zbMATH DE number 6387332
scientific article