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.


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
Full Text: DOI arXiv


[1] Beilinson, A.A.; Bernstein, J.; Deligne, P., Faisceaux pervers, Astérisque, 100, (1982) · Zbl 0536.14011
[2] Borel, A., Linear algebraic groups, Grad. texts in math., vol. 126, (1991), Springer · Zbl 0726.20030
[3] Bourbaki, N., Groupes et algèbres de Lie, chapitres 4, 5 et 6, (1981), Masson · Zbl 0483.22001
[4] Broué, M.; Malle, G.; Michel, J., Generic blocks of finite reductive groups, Astérisque, 212, 7-92, (1993) · Zbl 0843.20012
[5] Broué, M.; Michel, J., Sur certains éléments réguliers des groupes de Weyl et LES variétés de Deligne-Lusztig associées, (), 73-139 · Zbl 1029.20500
[6] C. Chevalley, Fondements de la géométrie projective, Paris, 1958 · Zbl 0087.35501
[7] Deligne, P., La conjecture de Weil II, Publ. math. inst. hautes études sci., 52, 138-252, (1979) · Zbl 0456.14014
[8] Deligne, P., Action du groupe des tresses sur une catégorie, Invent. math., 128, 159-175, (1997) · Zbl 0879.57017
[9] Deligne, P.; Lusztig, G., Characters of finite reductive groups, Ann. of math., 103, 103-161, (1976) · Zbl 0336.20029
[10] Digne, F.; Michel, J., Fonctions \(\mathcal{L}\) des variétés de Deligne-Lusztig et descente de shintani, Mém. soc. math. fr., 20, (1985) · Zbl 0608.20027
[11] Digne, F.; Michel, J., Representations of finite groups of Lie type, London math. soc. stud. texts, vol. 21, (1991), Cambridge Univ. Press · Zbl 0815.20014
[12] Digne, F.; Michel, J., Endomorphisms of Deligne-Lusztig varieties, Nagoya J. math., 183, (2006) · Zbl 1119.20008
[13] Fujiwara, K., Rigid geometry, Lefschetz-verdier trace formula and Deligne’s conjecture, Invent. math., 127, 489-533, (1997) · Zbl 0920.14005
[14] Geck, M.; Malle, G., Fourier transforms and Frobenius eigenvalues for finite Coxeter groups, J. algebra, 260, 162-193, (2003) · Zbl 1081.20050
[15] Geck, M.; Pfeiffer, G., On the irreducible characters of Hecke algebras, Adv. math., 102, 79-94, (1993) · Zbl 0816.20034
[16] Geck, M.; Pfeiffer, G., Characters of finite Coxeter groups and Iwahori-Hecke algebras, (2000), Oxford Univ. Press · Zbl 0996.20004
[17] Geck, M.; Kim, S.; Pfeiffer, G., Minimal length elements in twisted conjugacy classes of finite Coxeter groups, J. algebra, 229, 570-600, (2000) · Zbl 1042.20026
[18] Geisser, T., Motivic cohomology, K-theory and topological cyclic homology, (), 193-234 · Zbl 1113.14017
[19] Haastert, B., Die quasiaffinität der Deligne-Lusztig varietäten, J. algebra, 102-1, 186-193, (1986) · Zbl 0615.20021
[20] Kazhdan, D.; Lusztig, G., Representations of Coxeter groups and Hecke algebras, Invent. math., 53, 165-184, (1979) · Zbl 0499.20035
[21] Levine, M., Mixed motives, (1998), Amer. Math. Soc. · Zbl 0902.14003
[22] Lusztig, G., Finiteness of the number of unipotent classes, Invent. math., 34, 201-213, (1976) · Zbl 0371.20039
[23] Lusztig, G., Coxeter orbits and eigenspaces of Frobenius, Invent. math., 38, 101-159, (1976) · Zbl 0366.20031
[24] Lusztig, G., Representations of finite Chevalley groups, Cbms, vol. 39, (1978)
[25] Lusztig, G., Unipotent characters of the symplectic and odd orthogonal groups over a finite field, Invent. math., 64, 263-296, (1981) · Zbl 0477.20023
[26] Lusztig, G., Characters of reductive groups over finite fields, Ann. of math. stud., vol. 107, (1984), Princeton Univ. Press · Zbl 0572.20026
[27] Lusztig, G., Homology bases arising from reductive groups over a finite field, (), 53-72 · Zbl 0930.20041
[28] Michel, J., A note on words in braid monoids, J. algebra, 215, 366-377, (1999) · Zbl 0937.20017
[29] Milne, J.S., Étale cohomology, Princeton math. ser., vol. 33, (1980), Princeton Univ. Press · Zbl 0433.14012
[30] Ramanan, S.; Ramanathan, A., Projective normality of flag varieties and Schubert varieties, Invent. math., 79, 217-224, (1985) · Zbl 0553.14023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.