Selecta Mathematica

, Volume 24, Issue 5, pp 3973–4039 | Cite as

Parabolic induction in characteristic p

  • Rachel OllivierEmail author
  • Marie-France Vignéras


Let \(\mathrm{F}\) (resp. \(\mathbb F\)) be a nonarchimedean locally compact field with residue characteristic p (resp. a finite field with characteristic p). For \(k=\mathrm{F}\) or \(k=\mathbb F\), let \(\mathbf {G}\) be a connected reductive group over k and R be a commutative ring. We denote by \(\mathrm{Rep}( \mathbf G(k)) \) the category of smooth R-representations of \( \mathbf G(k) \). To a parabolic k-subgroup \({\mathbf P}=\mathbf {MN}\) of \(\mathbf G\) corresponds the parabolic induction functor \(\mathrm{Ind}_{\mathbf P(k)}^{\mathbf G(k)}:\mathrm{Rep}( \mathbf M(k)) \rightarrow \mathrm{Rep}( \mathbf G(k))\). This functor has a left and a right adjoint. Let \({{\mathcal {U}}}\) (resp. \({\mathbb {U}}\)) be a pro-p Iwahori (resp. a p-Sylow) subgroup of \( \mathbf G(k) \) compatible with \({\mathbf P}(k)\) when \(k=\mathrm{F}\) (resp. \(\mathbb F\)). Let \({H_{ \mathbf G(k)}}\) denote the pro-p Iwahori (resp. unipotent) Hecke algebra of \( \mathbf G(k) \) over R and \(\mathrm{Mod}({H_{ \mathbf G(k)}})\) the category of right modules over \({H_{ \mathbf G(k)}}\). There is a functor \(\mathrm{Ind}_{{H_{ \mathbf M(k)}}}^{{H_{ \mathbf G(k)}}}: \mathrm{Mod}({H_{ \mathbf M(k)}}) \rightarrow \mathrm{Mod}({H_{ \mathbf G(k) }})\) called parabolic induction for Hecke modules; it has a left and a right adjoint. We prove that the pro-p Iwahori (resp. unipotent) invariant functors commute with the parabolic induction functors, namely that \(\mathrm{Ind}_{\mathbf P(k)}^{\mathbf G(k)}\) and \(\mathrm{Ind}_{{H_{ \mathbf M(k)}}}^{{H_{ \mathbf G(k)}}}\) form a commutative diagram with the \({{\mathcal {U}}}\) and \({{\mathcal {U}}}\cap \mathbf M(\mathrm{F})\) (resp. \({\mathbb {U}}\) and \({\mathbb {U}}\cap \mathbf M(\mathbb F) \)) invariant functors. We prove that the pro-p Iwahori (resp. unipotent) invariant functors also commute with the right adjoints of the parabolic induction functors. However, they do not commute with the left adjoints of the parabolic induction functors in general; they do if p is invertible in R. When R is an algebraically closed field of characteristic p, we show that an irreducible admissible R-representation of \( \mathbf G(\mathrm{F}) \) is supercuspidal (or equivalently supersingular) if and only if the \({H_{ \mathbf G(\mathrm{F})}}\)-module \({\mathfrak {m}}\) of its \({{\mathcal {U}}}\)-invariants admits a supersingular subquotient, if and only if \({\mathfrak {m}}\) is supersingular.


Representations of p-adic groups Hecke algebras Parabolic induction 

Mathematics Subject Classification

11E95 20G25 20C08 22E50 


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We thank Noriyuki Abe for suggesting the counter example of Prop. 4.12 and for generously sharing his recent results with us. We are also thankful to Guy Henniart for his continuous interest and helpful remarks. Our work was carried out at the Institut de Mathematiques de Jussieu – Paris 7, the University of British Columbia and the Mathematical Sciences Research Institute. We would like to acknowledge the support of these institutions. The first author is partially funded by NSERC Discovery Grant.


  1. 1.
    Abe, N.: On a classification of irreducible admissible modulo \(p\) representations of a \(p\)-adic split reductive group. Compos. Math. 149(12), 2139–2168 (2013)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Abe, N.: Modulo p parabolic induction of pro-p-Iwahori Hecke algebra. J. Reine Angew. Math.,
  3. 3.
    Abe, N.: Parabolic inductions for pro-p-Iwahori Hecke algebras (2016) arXiv:1612.01312
  4. 4.
    Abe, N., Henniart, G., Herzig, F., Vignéras, M.-F.: A classification of admissible irreducible modulo \(p\) representations of reductive \(p\)-adic groups. J. Am. Math. Soc. 30(2), 495–559 (2016)CrossRefGoogle Scholar
  5. 5.
    Abe, N., Henniart, G., Vignéras, M.-F.: Mod \(p\) representations of reductive \(p\)-adic groups: functorial properties. To appear in Trans. Am. Math. Soc. (2018)Google Scholar
  6. 6.
    Abe, N., Henniart, G., Vignéras, M.-F.: On pro-p-Iwahori invariants of R-representations of p-adicgroups. Represent. Theory (2018) (to appear) Google Scholar
  7. 7.
    Bell, A., Farnsteiner, R.: On the theory of Frobenius extensions and its application to Lie superalgebras. Trans Am.Math. Soc. 335(1), 407–424 (1993)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Benson, D.J.: Representations and Cohomology I, Basic Representation Theory of Finite Groups and Associative Algebras. Cambridge Studies in Advanced Mathematics (1991)Google Scholar
  9. 9.
    Bernstein, J., Zelevinski, A.: Induced representations of \(p\)-adic groups I. Ann. Sci. Ecole Norm. Sup. (4) 10(4), 441–472 (1977)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Borel, A.: Admissible representations of a semisimple group with vectors fixed under anIwahori subgroup. Invent. Math. 35, 233–259 (1976)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bourbaki, N.: Éléments de mathématiques. Algèbre, Chap. 10. Algèbre homologique. Springer, Berlin (2006)zbMATHGoogle Scholar
  12. 12.
    Bourbaki, N.: Elements of Mathematics. Lie Groups and Lie Algebras, Chap. 4–6. Springer, Berlin (2002)CrossRefGoogle Scholar
  13. 13.
    Breuil, C.: Sur quelques représentations modulaires et p-adiques de \({\rm GL}_2(\mathbb{Q}_p)\) I. Compos. Math. 138, 165–188 (2003)CrossRefGoogle Scholar
  14. 14.
    Breuil, C., Paskunas, V.: Towards a modulo \(p\) Langlands correspondence for GL(2). Mem. Am. Math. Soc. 216 (2012)Google Scholar
  15. 15.
    Bruhat, F., Tits, J.: Groupes réductifs sur un corps local. I. Données radicielles valuées. Publ. Math. IHES 41, 5–251 (1972)CrossRefGoogle Scholar
  16. 16.
    Bruhat, F., Tits, J.: Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée. Publ. Math. IHES 60, 5–184 (1984)CrossRefGoogle Scholar
  17. 17.
    Bushnell, C.J., Kutzko, P.C., structure theory via types: Smooth representations of reductive p-adic groups. Proc. Lond. Math. Soc. 77, 582–634 (1998)CrossRefGoogle Scholar
  18. 18.
    Cabanes, M.: Extension groups for modular Hecke algebras. J. Fac. Sci. Univ. Tokyo 36(2), 347–362 (1989)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Cabanes, M.: A criterion of complete reducibility and some applications. In: Cabanes, M. (ed.), Représentations linéaires des groupes finis, Astérisque, 181–182 pp. 93–112 (1990)Google Scholar
  20. 20.
    Cabanes, M., Enguehard, M.: Representation Theory of Finite Reductive Groups. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  21. 21.
    Carter, R.: Finite Groups of Lie Type. Wiley Interscience, Hoboken (1985)zbMATHGoogle Scholar
  22. 22.
    Carter, R.W., Lusztig, G.: Modular representations of finite groups of Lie type. Proc. London Math. Soc. 32, 347–384 (1976)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Chuang, J., Rouquier, R.: Derived equivalences for symmetric groups and \({\mathfrak{sl}}_2\)-categorification. Ann. Math. 167(1), 245–298 (2008)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Colmez, P.: Representations de \(GL_2(\mathbb{Q}_p)\) et \((\varphi, \Gamma )\)-modules. Astérisque 330, 281–509 (2010)Google Scholar
  25. 25.
    Dat, J-Fr: Finitude pour les reprÃ\(\copyright \)sentations lisses des groupes p-adiques. J. Inst. Math. Jussieu 8(1), 261–333 (2009)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics 150. Springer, Berlin (2008)Google Scholar
  27. 27.
    Emerton, M.: Ordinary parts of admissible representations of reductive \(p\)-adic groups II. Asterisque 331, 383–438 (2010)zbMATHGoogle Scholar
  28. 28.
    Hilton, P.J., Stammbach, U.: A Course in Homological Algebra. Graduate Texts in Mathematics 4. Springer, Berlin (1971)CrossRefGoogle Scholar
  29. 29.
    Henniart, G., Vignéras, M.-F.: The Satake isomorphism modulo \(p\) with weight. J Für Reine Angew. Math. 701, 33–75 (2015)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Henniart, G., Vignéras, M.-F.: Comparison of compact induction with parabolic induction. Special issue to the memory of J. Rogawski. Pac. J. Math. 260(2), 457–495 (2012)CrossRefGoogle Scholar
  31. 31.
    Herzig, F.: The classification of admissible irreducible modulo \(p\) representations of a \(p\)-adic \(GL_{n}\). Invent. Math. 186, 373–434 (2011)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Iwahori, N., Matsumoto, H.: On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups. Publ. Math., Inst. Hautes Étud. Sci. 25, 5–48 (1965)MathSciNetCrossRefGoogle Scholar
  33. 33.
    James, G.: The irreducible representations of the finite general linear groups. Proc. Lond. Math. Soc. 52, 236–268 (1986)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Kashiwara, M., Shapira, P.: Categories and Sheaves. Grundlehren des Mathematischen Wissenschaften, vol. 332. Springer, Berlin (2006)Google Scholar
  35. 35.
    Khovanov, M.: Heisenberg algebra and a graphical calculus. Fundam. Math. 225, 169–210 (2014)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Kottwitz, R.: Isocrystals with additional structure II. Compos. Math. 109, 255–309 (1997)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Koziol, K.: Pro-\(p\)-Iwahori invariants for \({\rm SL}_2\) and \(L\)-packets of Hecke modules. Int. Math. Res. Not. 4, 1090–1125 (2016)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Lusztig, G.: Affine Hecke algebras and their graded version. J. Am. Math. Soc. 2(3), 599–635 (1989)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Nagao, H., Tsushima, Y.: Representations of Finite Groups. Academic Press, New York (1989)zbMATHGoogle Scholar
  40. 40.
    Ollivier, R.: Le foncteur des invariants sous l’action du pro-\(p\) Iwahori de \({\rm GL}_2(\mathbb{Q}_p)\). J. für dir reine und angewandte Mathematik 635, 149–185 (2009)zbMATHGoogle Scholar
  41. 41.
    Ollivier, R.: Parabolic Induction and Hecke modules in characteristic \(p\) for \(p\)-adic \({\rm GL}_n\). ANT 4(6), 701–742 (2010)CrossRefGoogle Scholar
  42. 42.
    Ollivier, R.: Compatibility between Satake and Bernstein isomorphisms in characteristic \(p\). ANT 8(5), 1071–1111 (2014)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Ollivier, R., Schneider, P.: Iwahori Hecke algebras are Gorenstein. J. Inst. Math. Jussieu 13(4), 753–809 (2014)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Ollivier, R., Sécherre, V.: Modules universels en caractéristique naturelle pour un groupe réductif fini. Ann. Inst. Fourier 65(1), 397–430 (2015)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Ollivier, R., Sécherre, V.: Modules universels de GL(3) sur un corps p-adique en caractéristique \(p\). Preprint (2011).
  46. 46.
    Paškūnas, V.: Coefficient systems and supersingular representations of \({\rm GL}_2(F)\), Mém. Soc. Math. Fr. (NS) 99 (2004)Google Scholar
  47. 47.
    Sawada, H.: Endomorphism rings of split \((B, N)\)-pairs. Tokyo J. Math. 1(1), 139–148 (1978)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Schneider, P., Stuhler, U.: The cohomology of \(p\)-adic symmetric spaces. Invent. Math. 105(1), 47–122 (1991)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Schneider, P., Stuhler, U.: Representation theory and sheaves on the Bruhat-Tits building. Publ. Math. IHES 85, 97–191 (1997)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Serre, J.-P.: Cours d’arithmétique. Presses Universitaires de France, Paris (1970)zbMATHGoogle Scholar
  51. 51.
    Silberger, A .J.: Isogeny restrictions of irreducible admissible representations are finite direct sums of irreducible admissible representations. Proc. Am. Math. Soc. 93(2), 263–264 (1979)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Tinberg, N .B.: Modular representations of finite groups with unsaturated split \((B,N)\)-pairs. Can. J. Math. 32(3), 714–733 (1980)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Tits, J.: Reductive groups over local fields. In: Borel, C. (ed.), Proc. Symp. Pure Math., vol. 33, no. 1, pp. 29–69. Automorphic Forms,Representations, and \(L\)-FunctionsAmerican Math. Soc (1979)Google Scholar
  54. 54.
    Vignéras, M.-F.: Représentations \(\ell \)-modulaires d’un groupe réductif fini \(p\)-adique avec \(\ell \ne p\). Birkhauser Prog. Math. 137 (1996)Google Scholar
  55. 55.
    Vignéras, M.-F.: Induced representations of reductive p-adic groups in characteristic \(\ell \ne p\). Sel. Math. 4, 549–623 (1998)CrossRefGoogle Scholar
  56. 56.
    Vignéras, M.-F.: Representations modulo p of the p-adic group GL(2, F ). Compos. Math. 140, 333–358 (2004)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Vignéras, M.-F.: Série principale modulo \(p\) de groupes réductifs \(p\)-adiques. GAFA Geom. Funct. Anal. 17, 2090–2112 (2007)CrossRefGoogle Scholar
  58. 58.
    Vignéras, M.-F.: Pro-\(p\) Iwahori Hecke ring and supersingular \(\overline{\mathbb{F}}_{p}\)-representations. Math. Annalen 331, pp. 523–556 (2005). Erratum vol. 333(3), pp. 699–701Google Scholar
  59. 59.
    Vignéras, M.-F.: Représentations irréductibles de \(GL(2,F)\) modulo \(p\). In: Burns, D., Buzzard, K., Nekovar, J., (eds.) \(L\)-Functions and Galois representations, LMS Lecture Notes, vol. 320 (2007)Google Scholar
  60. 60.
    Vignéras, M.-F.: Algèbres de Hecke affines génériques. Represent. Theory 10, 1–20 (2006)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Vignéras, M.-F.: The right adjoint of the parabolic induction. Birkhauser series progress in mathematics Arbeitstagung Bonn 2013: In: Ballmann, W., Blohmann, C., Faltings, G., Teichner, P., Zagier, D. (eds.), Memory of Friedrich Hirzebruch (2013)Google Scholar
  62. 62.
    Vignéras, M.-F.: The pro-\(p\) Iwahori Hecke algebra of a reductive \(p\)-adic group I. Compos. Math. 152, 693–753 (2016)MathSciNetCrossRefGoogle Scholar
  63. 63.
    Vignéras, M.-F.: The pro-\(p\) Iwahori Hecke algebra of a reductive \(p\)-adic group II. Muenster J. of Math. 7, 363–379 (2014)MathSciNetzbMATHGoogle Scholar
  64. 64.
    Vignéras, M.-F.: The pro-\(p\)-Iwahori Hecke algebra of a reductive \(p\)-adic group III (spherical Hecke algebras and supersingular modules). J. Inst. Math. Jussieu 16(3), 571–608 (2015)MathSciNetCrossRefGoogle Scholar
  65. 65.
    Vignéras, M.-F.: The pro-\(p\) Iwahori Hecke algebra of a reductive \(p\)-adic group IV (Levi subgroup and central extension). In preparationGoogle Scholar
  66. 66.
    Vignéras, M.-F.: The pro-\(p\) Iwahori Hecke algebra of a reductive \(p\)-adic group V (parabolic induction). Pac. J. Math. 279, 499–529 (2015)MathSciNetCrossRefGoogle Scholar
  67. 67.
    Zelevinsky, A.V.: Induced representations of reductive p-adic groups. II. On irreducible representations of GL(n). Ann. Sci. Ecole Norm. Sup. (4) 13(2), 165–210 (1980)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.University of British ColumbiaVancouverCanada
  2. 2.Institut de Mathématiques de Jussieu, UMR 7586ParisFrance

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