Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T20:16:54.276Z Has data issue: false hasContentIssue false
  • English
  • Français

Lefschetz operator and local Langlands mod ℓ: The regular case

Published online by Cambridge University Press:  11 January 2016

Jean-François Dat
Jean-François Dat*
Affiliation:
Université Pierre et Marie Curie, Institut de Mathématiques de Jussieu, 4 place Jussieu, 75005 Paris, France, dat@math.jussieu.fr
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.

Let p and ℓ be two distinct primes. The aim of this paper is to show how, under a certain congruence hypothesis, the mod ℓ cohomology complex of the Lubin-Tate tower, together with a natural Lefschetz operator, provides a geometric interpretation of Vignéras’s local Langlands correspondence modulo ℓ for unipotent representations.

Keywords

11S3711F7014G35
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 208: Memorial Volume for Professor Hiroshi Saito , December 2012 , pp. 1 - 38
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2012

References

[1] Bonnafé, C. and Rouquier, R., Coxeter orbits and modular representations, Nagoya Math. J. 183 (2006), 1–34.CrossRefGoogle Scholar
[2] Borel, A. and Wallach, N. R., Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Ann. of Math. Stud. 94, Princeton University Press, Princeton, NJ, 1980.Google Scholar
[3] Boyer, P., Monodromie du faisceau pervers des cycles évanescents de quelques variétés de Shimura simples, Invent. Math. 177 (2009), 239–280.Google Scholar
[4] Broué, M. and Michel, J., Blocs et séries de Lusztig dans un groupe réductif fini, J. Reine Angew. Math. 395 (1989), 56–67.Google Scholar
[5] Dat, J.-F., Espaces symétriques de Drinfeld et correspondance de Langlands locale, Ann. Sci. Éc. Norm. Supér. (4) 39 (2006), 1–74.CrossRefGoogle Scholar
[6] Dat, J.-F., Théorie de Lubin-Tate non-abélienne et représentations elliptiques, Invent. Math. 169 (2007), 75–152.CrossRefGoogle Scholar
[7] Dat, J.-F., Finitude pour les représentations lisses de groupes p-adiques, J. Inst. Math. Jussieu 8 (2009), 261–333.CrossRefGoogle Scholar
[8] Dat, J.-F., A lemma on nearby cycles and its application to the tame Lubin-Tate space, Math. Res. Lett. 19 (2012), 1–9.CrossRefGoogle Scholar
[9] Dat, J.-F., Opérateur de Lefschetz sur les tours de Drinfeld et Lubin-Tate, Compos. Math. 148 (2012), 507–530.CrossRefGoogle Scholar
[10] Dat, J.-F., Théorie de Lubin-Tate non-abélienne l-entière, Duke Math. J. 161 (2012), 951-1010.CrossRefGoogle Scholar
[11] Dat, J.-F., Un cas simple de correspondance de Jacquet-Langlands modulo l, Proc. London Math. Soc. (3) 104 (2012), 690–727.CrossRefGoogle Scholar
[12] Deligne, P., La conjecture de Weil, II, Publ. Math. Inst. Hautes Etudes Sci. 52 (1980), 137–252.CrossRefGoogle Scholar
[13] Dipper, R., On quotients of Hom-functors and representations of finite general linear groups, II, J. Algebra 209 (1998), 199–269.CrossRefGoogle Scholar
[14] Dipper, R. and James, G., Identification of the irreducible modular representations of GLn(q), J. Algebra 104 (1986), 266–288.CrossRefGoogle Scholar
[15] Dudas, O., Coxeter orbits and Brauer trees, Adv. Math. 229 (2012), 3398–3435.CrossRefGoogle Scholar
[16] Fargues, L., “L’isomorphisme entre les tours de Lubin-Tate et de Drinfeld et applications cohomologiques” in L’isomorphisme entre les tours de Lubin-Tate et de Drinfeld, Progr. Math. 262, Birkhäuser, Basel, 2008, 1–321.CrossRefGoogle Scholar
[17] Lusztig, G., On the finiteness of the number of unipotent classes, Invent. Math. 34 (1976), 201–213.CrossRefGoogle Scholar
[18] Meyer, R. and Solleveld, M., Resolutions for representations of reductive p-adic groups via their buildings, J. Reine Angew. Math. 647 (2010), 115–150.Google Scholar
[19] Neeman, A., Triangulated Categories, Ann. of Math. Stud. 148, Princeton University Press, Princeton, NJ, 2001.Google Scholar
[20] Orlik, S., On extensions of generalized Steinberg representations, J. Algebra 293 (2005), 611–630.CrossRefGoogle Scholar
[21] Schneider, P. and Stuhler, U., The cohomology of p-adic symmetric spaces, Invent. Math. 105 (1991), 47–122.CrossRefGoogle Scholar
[22] Strauch, M., Deformation spaces of one-dimensional formal modules and their cohomology, Adv. Math. 217 (2008), 889–951.CrossRefGoogle Scholar
[23] Vignéras, M. F., Représentations l-modulaires d’un groupe p-adique avec l ≠ p, Progr. Math. 137, Birkhäuser, Boston, 1996.Google Scholar
[24] Vignéras, M. F., “À propos d’une conjecture de Langlands modulaire” in Finite Reductive Groups (Luminy, 1994), Progr. Math. 141, Birkhäuser, Boston, 1997, 415–452.Google Scholar
[25] Vignéras, M. F., Extensions between irreducible representations of p-adic GL(n), Pacific J. Math. 181 (1997), 349–357.CrossRefGoogle Scholar
[26] Vignéras, M. F., Induced R-representations of p-adic reductive groups, Selecta Math. (N.S.) 4 (1998), 549–623.CrossRefGoogle Scholar
[27] Vignéras, M. F., Correspondance de Langlands semi-simple pour GL(n, F) modulo l ≠ p, Invent. Math. 144 (2001), 177–223.CrossRefGoogle Scholar
[28] Yoshida, T., “On non-abelian Lubin-Tate theory via vanishing cycles” in Algebraic and Arithmetic Structures of Moduli Spaces (Sapporo, 2007), Adv. Stud. Pure Math. 58, Math. Soc. Japan, Tokyo, 2010, 361–402.CrossRefGoogle Scholar