Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-27T23:02:28.210Z Has data issue: false hasContentIssue false
  • English
  • Français

Coxeter Orbits and Modular Representations

Published online by Cambridge University Press:  11 January 2016

Cédric Bonnafé  and
Raphaël Rouquier
Cédric Bonnafé
Affiliation:
Céedric BonnaféeUniv. de Franche-Comtée Déepartement de Mathéematiques (CNRS UMR 6623)16 Route de Gray 25000 Besan¸[email protected]://www-math.univ-fcomte.fr/pp_Annu/CBONNAFE/
Raphaël Rouquier
Affiliation:
Institut de Mathématiques de Jussieu — CNRSUFR de Mathématiques, Université Denis Diderot2, place Jussieu, 75005Paris France and Department of Pure MathematicsUniversity of LeedsLeeds LS2 [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the modular representations of finite groups of Lie type arising in the cohomology of certain quotients of Deligne-Lusztig varieties associated with Coxeter elements. These quotients are related to Gelfand-Graev representations and we present a conjecture on the Deligne-Lusztig restriction of Gelfand-Graev representations. We prove the conjecture for restriction to a Coxeter torus. We deduce a proof of Brouée’s conjecture on equivalences of derived categories arising from Deligne-Lusztig varieties, for a split group of type An and a Coxeter element. Our study is based on Lusztig’s work in characteristic 0 [Lu2].

Keywords

20G0518E3020G40
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 183 , 2006 , pp. 1 - 34
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2006

References

[Bon] Bonnafé, C., Mackey Formula in Type A, Proc. London Math. Soc, 80 (2000), 545–574.Google Scholar
[BonRou1] Bonnafé, C. and Rouquier, R., Catégories dérivées et variétés de Deligne-Lusztig, Publ. Math. I.H.E.S., 97 (2003), 1–59.Google Scholar
[BonRou2] Bonnafé, C. and Rouquier, R., On the irreducibility of Deligne-Lusztig varieties, Comptes Rendus Math. Acad. Sci. Paris, 343 (2006), 37–39.Google Scholar
[Bor] Borel, A., Linear Algebraic Groups, Springer-Verlag, 1991.CrossRefGoogle Scholar
[Br1] Broué, M., Isométries parfaites, types de blocs, catégories dérivées, Astérisque, 181–182 (1990), 61–92.Google Scholar
[Br2] Broué, M., Reflection Groups, Braid Groups, Hecke Algebras, Finite Reductive Groups, Current Developments in Mathematics, 2000, International Press (2001), pp. 1–103.Google Scholar
[Ca] Carter, R. W., Finite groups of Lie type, Wiley, 1985.Google Scholar
[Cu] Curtis, C. W., On the Gelfand-Graev representations of a reductive group over a finite field, J. of Alg., 157 (1993), 517–533.Google Scholar
[CuSho] Curtis, C. W. and Shoji, T., A norm map for endomorphism algebras of Gelfand-Graev representations, Progr. Math. 141, Birkhäauser (1997), pp. 185–194.Google Scholar
[DeLu] Deligne, P. and Lusztig, G., Representations of reductive groups over finite fields, Ann. of Math., 103 (1976), 103–161.Google Scholar
[DiLeMi1] Digne, F., Lehrer, G. and Michel, J., The characters of the group of rational points of a reductive group with non-connected centre, J. reine. angew. Math., 425 (1992), 155–192.Google Scholar
[DiLeMi2] Digne, F., Lehrer, G. and Michel, J., On Gel’fand-Graev characters of reductive groups with disconnected centre, J. reine. angew. Math., 491 (1997), 131–147.Google Scholar
[DiMi1] Digne, F. and Michel, J., Fonctions L des variétés de Deligne-Lusztig et descente de Shintani, Mém. SMF, 20 (1985).Google Scholar
[DiMi2] Digne, F. and Michel, J., Representations of Finite Groups of Lie Type, London Math. Soc. Student Texts 21, Cambridge University Press, 1991.Google Scholar
[DiMiRou] Digne, F., Michel, J. and Rouquier, R., Cohomologie des variétés de Deligne-Lusztig, preprint, math.RT/0410454 (2004), to appear in Adv. Math.Google Scholar
[Dip] Dipper, R., On quotients of Hom-functors and representations of finite general linear groups II, J. Algebra, 209 (1998), 199–269.Google Scholar
[GePf] Geck, M. and Pfeiffer, G., Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Math. Soc. Monographs, Oxford University Press, 2000.Google Scholar
[Lau] Laumon, G., Majorations de sommes trigonométriques (d’après P. Deligne et N. Katz), Astérisque, 83 (1981), 221–258.Google Scholar
[Lu1] Lusztig, G., On the finiteness of the number of unipotent classes, Inv. Math., 34 (1976), 201–214.Google Scholar
[Lu2] Lusztig, G., Coxeter orbits and eigenspaces of Frobenius, Inv. Math., 38 (1976), 101–159.Google Scholar
[Lu3] Lusztig, G., Representations of Finite Chevalley Groups, Amer. Math. Soc., 1978.Google Scholar
[Ri1] Rickard, J., Finite group actions and étale cohomology, Publ. Math. I.H.E.S., 80 (1994), 81–94.Google Scholar
[Ri2] Rickard, J., Splendid equivalences: derived categories and permutation modules, Proc. London Math. Soc., 72 (1996), 331–358.Google Scholar
[Rou1] Rouquier, R., Block theory via stable and Rickard equivalences, Modular representation theory of finite groups, de Gruyter (2001), pp. 101–146.Google Scholar
[Rou2] Rouquier, R., Complexes de chaܐınes étales et courbes de Deligne-Lusztig, J. of Alg., 257 (2002), 482–508.Google Scholar
[St] Steinberg, R., Endomorphisms of linear algebraic groups, Memoirs of the American Math. Soc. 80, 1968.Google Scholar