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Cohomology of Deligne-Lusztig varieties. (Cohomologie des variétés de Deligne-Lusztig.) (French. English summary) Zbl 1118.20006

This paper constructs actions of Iwahori-Hecke algebras on étale cohomology of certain generalized Deligne-Lusztig varieties, with the ultimate aim of getting a more elaborate version of the Abelian defect conjecture of Broué.
Let \(\mathbf G\) be a connected reductive group defined over the algebraic closure of the prime field \(\mathbb{F}_p\). Let \(F\) be an isogeny of \(\mathbf G\) with a power \(F^\delta\) that is the Frobenius with respect to a split form of \(\mathbf G\) over some \(\mathbb{F}_{p^m}\). Let \(\mathcal B\) be the flag variety of \(\mathbf G\). The \(\mathbf G\)-orbits in \(\mathcal B\times\mathcal B\) correspond with the elements of the Weyl group \(W\). One may also let the elements of the Weyl group correspond with the \(\mathbf G\)-orbit closures in \(\mathcal B\times\mathcal B\).
Mixing these two alternatives leads to variations on the Deligne-Lusztig varieties that are partial compactifications of the original ones. More specifically, in addition to the Artin-Tits braid monoid \(B^+\) one defines a bigger monoid \(\underline B^+\) that contains one copy of \(w\in W\) for each of the two alternatives. To every element \(\mathbf t\) of \(\underline B^+\) is attached a generalized Deligne-Lusztig variety \(\mathbf X(\mathbf t)\). Take a prime \(\ell\) different from \(p\). One is interested in the \((\overline\mathbb{Q}_\ell\mathbf G^F)\)-module structure of the \(\ell\)-adic cohomology with proper support \(H^i_c(\mathbf X(\mathbf t),\overline\mathbb{Q}_\ell)\).
First the paper develops machinery similar to the one that already exists for the case \(\mathbf t\in B^+\). Let \(\mathbf w_0\) be the lift to \(B^+\) of minimal length of the longest element \(w_0\) in \(W\). For the rank two groups, except for split \(G_2\), the module structure of \(H^i_c(\mathbf X(\mathbf t),\overline\mathbb{Q}_\ell)\) is completely determined for certain \(\mathbf t\in\underline B^+\), including all powers of \(\mathbf w_0\). Then these rank two results are used to construct Hecke algebra actions on the cohomology of \(\mathbf X(\mathbf w_0)\) and \(\mathbf X(\mathbf w_0^2)\) in the higher rank cases.

MSC:

20C08 Hecke algebras and their representations
20G10 Cohomology theory for linear algebraic groups
20G05 Representation theory for linear algebraic groups
20C11 \(p\)-adic representations of finite groups
14L30 Group actions on varieties or schemes (quotients)
14F42 Motivic cohomology; motivic homotopy theory
14F20 Étale and other Grothendieck topologies and (co)homologies
20G40 Linear algebraic groups over finite fields
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