## 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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### References:

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