Advertisement

manuscripta mathematica

, Volume 156, Issue 1–2, pp 23–55 | Cite as

Jacquet modules and irrreducibility of induced representations for classical p-adic groups

  • Chris Jantzen
Article
  • 51 Downloads

Abstract

Let G be a classical p-adic group. If T is an irreducible tempered representation of such a group and \(\rho \) an irreducible unitary supercuspidal representation of a general linear group, we can form the parabolically induced representation \(\text{ Ind }_P^G (|det|^y \rho \otimes T)\). The main result in this paper is the determination for which \(y \in {\mathbb R}\) the induced representation is reducible. The key technical result in establishing this is the determination of a certain Jacquet module subquotient.

Mathematics Subject Classification

22E50 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arthur, J.: The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups. American Mathematical Society, Providence (2013)CrossRefzbMATHGoogle Scholar
  2. 2.
    Aubert, A.-M.: Dualité dans le groupe de Grothendieck de la catégorie des représentations lisses de longueur finie d’un groupe réductif \(p\)-adique. Trans. Am. Math. Soc. 347, 2179–2189 (1995) and Erratum. Trans. Am. Math. Soc. 348, 4687–4690 (1996)Google Scholar
  3. 3.
    Ban, D.: Parabolic induction and Jacquet modules of representations of \(O(2n, F)\). Glasnik. Mat. 34(54), 147–185 (1999)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ban, D., Jantzen, C.: Degenerate principal series for even orthogonal groups. Represent. Theory 7, 440–480 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ban, D., Jantzen, C.: Jacquet modules and the Langlands classification. Mich. Math. J. 56, 637–653 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bernstein, I., Zelevinsky, A.: Induced representations of reductive \(p\)-adic groups \(I\). Ann. Sci. École Norm. Sup. 10, 441–472 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Borel, A., Wallach, N.: Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups. Princeton University Press, Princeton (1980)zbMATHGoogle Scholar
  8. 8.
    Casselman, W.: Introduction to the theory of admissible representations of \(p\)-adic reductive groups, preprint www.math.ubc.ca/people/faculty/cass/research.html as “The \(p\)-adic notes”
  9. 9.
    Jantzen, C.: On supports of induced representations for symplectic and odd-orthogonal groups. Am. J. Math. 119, 1213–1262 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jantzen, C.: Jacquet modules of \(p\)-adic general linear groups. Represent. Theory 11, 45–83 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jantzen, C.: Discrete series for \(p\)-adic \(SO(2n)\) and restrictions of representations of \(O(2n)\). Can. J. Math. 63, 327–380 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jantzen, C.: Tempered representations for classical \(p\)-adic groups. Manuscr. Math. 145, 319–387 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Jantzen, C.: Duality for classical \(p\)-adic groups: the half-integral case, preprintGoogle Scholar
  14. 14.
    Konno, T.: A note on the Langlands classification and irreducibility of induced representations of \(p\)-adic groups. Kyushu J. Math. 57, 383–409 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Matić, I.: Strongly positive representations of metaplectic groups. J. Algebra 334, 255–274 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Matić, I.: On Jacquet modules of discrete series: the first inductive step. J. Lie Theory 26, 135–168 (2016)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Matić, I., Tadić, M.: On Jacquet modules of representations of segment type. Manuscr. Math. 147, 437–476 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mœglin, C.: Normalisation des opérateurs d’entrelacement et réductibilité des induites des cuspidales; le cas des groupes classiques \(p\)-adiques. Ann. Math. 151, 817–847 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mœglin, C.: Paquets stables des séries discrètes des groupes classiques p-adiques accessibles par endoscopie tordue; leur paramètre de Langlands. Contemp. Math. 614, 295–336 (2014)CrossRefGoogle Scholar
  20. 20.
    Mœglin, C., Tadić, M.: Construction of discrete series for classical \(p\)-adic groups. J. Am. Math. Soc. 15, 715–786 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Muić, G.: Composition series of generalized principal series; the case of strongly positive discrete series. Isr. J. Math. 140, 157–202 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Muić, G.: Reducibility of generalized principal series. Can. J. Math. 57, 616–647 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Schneider, P., Stuhler, U.: Representation theory and sheaves on the Bruhat-Tits building. Publ. Math. IHES 85, 97–191 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Shahidi, F.: A proof of Langlands conjecture on Plancherel measure; complementary series for \(p\)-adic groups. Ann. Math. 132, 273–330 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Shahidi, F.: Twisted endoscopy and reducibility of induced representations for \(p\)-adic groups. Duke Math. J. 66, 1–41 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Silberger, A.: The Langlands quotient theorem for \(p\)-adic groups. Math. Ann. 236, 95–104 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Silberger, A.: Special representations of reductive \(p\)-adic groups are not integrable. Ann. Math. 111, 571–587 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Tadić, M.: Structure arising from induction and Jacquet modules of representations of classical \(p\)-adic groups. J. Algebra 177, 1–33 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Tadić, M.: Representations of real and \(p\)-adic groups. In: Tan, E.-C., Zhu, C.-B. (eds.) On Classification of Some Classes of Irreducible Representations of Classical Groups. Lecture Notes Series. Institute for Mathematical Sciences. National University of Singapore, vol. 2, pp. 95–162. Singapore University Press, Singapore (2004)Google Scholar
  30. 30.
    Tadić, M.: On invariants of discrete series representations of classical \(p\)-adic groups. Manuscr. Math. 135, 417–435 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Tadić, M.: On tempered and square integrable representations of classical \(p\)-adic groups. Sci. China Math. 56, 2273–2313 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Waldspurger, J.-L.: La formule de Plancherel pour les groupes \(p\)-adiques d’après Harish-Chandra. J. Inst. Math. Jussieu 2, 235–333 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Zelevinsky, A.: Induced representations of reductive \(p\)-adic groups \(II\), On irreducible representations of \(GL(n)\). Ann. Sci. École Norm. Sup. 13, 165–210 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Zhang, Y.: L-packets and reducibilities. J. Reine Angew. Math. 510, 83–102 (1999)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsEast Carolina UniversityGreenvilleUSA

Personalised recommendations