Duke Mathematical Journal

Représentations lisses modulo de GLm(D)

Alberto Mínguez and Vincent Sécherre

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Soit F un corps localement compact non archimédien de caractéristique résiduelle p, soit D une algèbre à division centrale de dimension finie sur F et soit R un corps algébriquement clos de caractéristique différente de p. Nous classons les représentations lisses irréductibles de GLm(D), m1, à coefficients dans R en termes de multisegments, ce qui généralise des travaux de Zelevinski, Tadić et Vignéras. Nous prouvons que toute R-représentation irréductible de GLm(D) a un unique support supercuspidal, et nous obtenons ainsi deux classifications: l’une par les multisegments supercuspidaux, qui classe les représentations en fonction de leur support supercuspidal; l’autre par les multisegments apériodiques, qui classe les représentations en fonction de leur support cuspidal. Ces constructions sont effectuées de façon purement locale, et font un usage substantiel de la théorie des types.

Abstract

Let F be a non-Archimedean locally compact field of residue characteristic p, let D be a finite-dimensional central division F-algebra, and let R be an algebraically closed field of characteristic different from p. We classify all smooth irreducible representations of GLm(D) for m1, with coefficients in R, in terms of multisegments, generalizing works by Zelevinski, Tadić, and Vignéras. We prove that any irreducible R-representation of GLm(D) has a unique supercuspidal support and thus get two classifications: one by supercuspidal multisegments, classifying representations with a given supercuspidal support, and one by aperiodic multisegments, classifying representations with a given cuspidal support. These constructions are made in a purely local way, with a substantial use of type theory.

Article information

Source
Duke Math. J., Volume 163, Number 4 (2014), 795-887.

Dates
First available in Project Euclid: 12 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1394630555

Digital Object Identifier
doi:10.1215/00127094-2430025

Mathematical Reviews number (MathSciNet)
MR3178433

Zentralblatt MATH identifier
0514.55001

Subjects
Primary: 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05]
Secondary: 20C20: Modular representations and characters

Citation

Mínguez, Alberto; Sécherre, Vincent. Représentations lisses modulo $\ell$ de $\mathrm{GL}_{m}(\mathrm{D})$. Duke Math. J. 163 (2014), no. 4, 795--887. doi:10.1215/00127094-2430025. https://projecteuclid.org/euclid.dmj/1394630555


Export citation

References

  • [1] S. Ariki, On the decomposition numbers of the Hecke algebra of $G(m,1,n)$, J. Math. Kyoto Univ. 36 (1996), 789–808.
  • [2] S. Ariki, Representations of Quantum Algebras and Combinatorics of Young Tableaux, Univ. Lecture Ser. 26, Amer. Math. Soc., Providence, 2002.
  • [3] S. Ariki et A. Mathas, The number of simple modules of the Hecke algebras of type $G(r,1,n)$, Math. Z. 233 (2000), 601–623.
  • [4] A. I. Badulescu, Un résultat d’irréductibilité en caractéristique non nulle, Tohoku Math. J. (2) 56 (2004), 583–592.
  • [5] A. I. Badulescu, G. Henniart, B. Lemaire et V. Sécherre, Sur le dual unitaire de $\operatorname{GL}_{r}(D)$, Amer. J. Math. 132 (2010), 1365–1396.
  • [6] J. Bernstein et A. Zelevinski, Representations of the group $\operatorname{GL} (n,F)$, where $F$ is a local non-Archimedean field, Uspehi Mat. Nauk 31 (1976), 5–70.
  • [7] J. Bernstein et A. Zelevinski, Induced representations of reductive $p$-adic groups, I, Ann. Sci. École Norm. Sup. (4) 10 (1977), 441–472.
  • [8] A. Borel et G. Harder, Existence of discrete cocompact subgroups of reductive groups over local fields, J. Reine Angew. Math. 298 (1978), 53–64.
  • [9] N. Bourbaki, Eléments de mathématique. 23. Première partie : Les structures fondamentales de l’analyse. Livre II : Algèbre. Chapitre 8 : Modules et anneaux semi-simples, Actualités Sci. Ind. 1261, Hermann, Paris, 1958.
  • [10] P. Broussous, V. Sécherre et S. Stevens, Smooth representations of $\operatorname{GL}(m,D)$, V: endo-classes, Documenta Math. 17 (2012), 23–77.
  • [11] N. Chriss et V. Ginzburg, Representation Theory and Complex Geometry, Birkhäuser, Boston, 1997.
  • [12] J.-F. Dat, $\nu$-tempered representations of $p$-adic groups, I: $l$-adic case, Duke Math. J. 126 (2005), 397–469.
  • [13] J.-F. Dat, Finitude pour les représentations lisses de groupes $p$-adiques, J. Inst. Math. Jussieu 8 (2009), 261–333.
  • [14] J.-F. Dat, Un cas simple de correspondance de Jacquet-Langlands modulo $\ell$, Proc. London Math. Soc. 104 (2012), 690–727.
  • [15] J.-F. Dat, Théorie de Lubin-Tate non abélienne $\ell$-entière, Duke Math. J. 161 (2012), 951–1010.
  • [16] P. Deligne, D. Kazhdan et M.-F. Vignéras, “Représentations des algèbres centrales simples $p$-adiques” dans Representations of Reductive Groups Over a Local Field, Travaux en Cours, Hermann, Paris, 1984, 33–117.
  • [17] R. Dipper, On quotients of Hom-functors and representations of finite general linear groups, I, J. Algebra 130 (1990), 235–259.
  • [18] R. Dipper et G. James, Identification of the irreducible modular representations of $\mathrm{GL}_{n}(q)$, J. Algebra 104 (1986), 266–288.
  • [19] G. Henniart et V. Sécherre, Types et contragrédientes, à paraître à Canad. J. Math., prépublication, 2013, http://lmv.math.cnrs.fr/annuaire/vincent-secherre.
  • [20] R. B. Howlett et G. I. Lehrer, On Harish-Chandra induction and restriction for modules of Levi subgroups, J. Algebra 165 (1994), 172–183.
  • [21] G. James, Representations of General Linear Groups, London Math. Soc. Lecture Note Ser. 94, Cambridge Univ. Press, Cambridge, 1984.
  • [22] G. James, The irreducible representations of the finite general linear groups, Proc. London Math. Soc. (3) 52 (1986), 236–268.
  • [23] I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2ème éd., Oxford Math. Monogr., Oxford Univ. Press, New York, 1995.
  • [24] A. Mathas, “Simple modules of Ariki-Koike algebras” dans Group Representations: Cohomology, Group Actions and Topology (Seattle, 1996), Proc. Sympos. Pure Math. 63, Amer. Math. Soc., Providence, 1998, 383–396.
  • [25] A. Mínguez, Sur l’irréductibilité d’une induite parabolique, J. Reine Angew. Math. 629 (2009), 107–131.
  • [26] A. Mínguez et V. Sécherre, Types modulo $\ell$ pour les formes intérieures de $\operatorname{GL}_{n}$ sur un corps local non archimédien, prépublication, 2014 http://lmv.math.cnrs.fr/annuaire/vincent-secherre.
  • [27] A. Mínguez et V. Sécherre, Représentations banales de $\operatorname{GL}(m,D)$, Compos. Math. 149 (2013), 679–704.
  • [28] A. Mínguez et V. Sécherre, Unramified $\ell$-modular representations of $\mathrm{GL}(n,F)$ and its inner forms, Int. Math. Res. Not. IMRN 2013, art. ID rns 278.
  • [29] C. Mœglin et J.-L. Waldspurger, Sur l’involution de Zelevinski, J. Reine Angew. Math. 372 (1986), 136–177.
  • [30] J. D. Rogawski, Representations of $\operatorname{GL}(n)$ over a $p$-adic field with an Iwahori-fixed vector, Israel J. Math. 54 (1986), 242–256.
  • [31] V. Sécherre, Représentations lisses de $\operatorname{GL}(m,D)$, III : Types simples, Ann. Scient. Éc. Norm. Sup. 38 (2005), 951–977.
  • [32] J.-P. Serre, Linear Representations of Finite Groups, Grad. Texts in Math. 42, Springer, New York, 1977.
  • [33] M. Tadić, Induced representations of $\mathrm{GL}(n,A)$ for $p$-adic division algebras $A$, J. Reine Angew. Math. 405 (1990), 48–77.
  • [34] M.-F. Vignéras, Représentations $l$-modulaires d’un groupe réductif $p$-adique avec $l\ne p$, Progr. Math. 137, Birkhäuser, Boston, 1996.
  • [35] M.-F. Vignéras, Induced $R$-representations of $p$-adic reductive groups, with an appendix by Alberto Arabia, Selecta Math. (N.S.) 4 (1998), 549–623.
  • [36] M.-F. Vignéras, “Irreducible modular representations of a reductive $p$-adic group and simple modules for Hecke algebras” in European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math. 201, Birkhäuser, Basel, 2001, 117–133.
  • [37] M.-F. Vignéras, “Modular representations of $p$-adic groups and of affine Hecke algebras” in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, 667–677.
  • [38] M.-F. Vignéras, “On highest Whittaker models and integral structures” in Contributions to Automorphic Forms, Geometry and Number Theory: Shalikafest 2002, Johns Hopkins Univ. Press, Baltimore, 2004, 773–801.
  • [39] A. Zelevinski, Induced representations of reductive $\mathfrak{p}$-adic groups, II: On irreducible representations of $\operatorname{GL}(n)$, Ann. Sci. École Norm. Sup. (4) 13 (1980), 165–210.