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On Unitarizability in the Case of Classical p-Adic Groups

  • Marko Tadić
Conference paper
Part of the Simons Symposia book series (SISY)

Abstract

In the introduction of this paper we discuss a possible approach to the unitarizability problem for classical p-adic groups. In this paper we give some very limited support that such approach is not without chance. In a forthcoming paper we shall give additional evidence in generalized cuspidal rank (up to) three.

Keywords

Non-archimedean local fields Classical p-adic groups Irreducible representations Unitarizability Parabolic induction 

2000 Mathematics Subject Classification

Primary 22E50 

Notes

Acknowledgements

This work has been supported by Croatian Science Foundation under the project 9364.

References

  1. 1.
    Arthur, J., The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups, American Mathematical Society Colloquium Publications 61, American Mathematical Society, Providence, RI, 2013.CrossRefGoogle 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. Amer. Math. Soc. 347 (1995), 2179–2189; Erratum, Trans. Amer. Math. Soc 348 (1996), 4687–4690.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Badulescu, A. I., On p-adic Speh representations, Bull. Soc. Math. France 142 (2014), 255–267.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Badulescu, A. I.; Henniart, G.; Lemaire, B. and Sécherre, V., Sur le dual unitaire de GL r(D), Amer. J. Math. 132 (2010), no. 5, 1365–1396.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Badulescu, A. I. and Renard, D. A., Sur une conjecture de Tadić, Glasnik Mat. 39 no. 1 (2004), 49–54.MathSciNetCrossRefGoogle Scholar
  6. 6.
    Badulescu, A. I. and Renard, D. A., Unitary dual of GL n at archimedean places and global Jacquet-Langlands correspondence, Compositio Math. 146, vol. 5 (2010), 1115–1164.Google Scholar
  7. 7.
    Baruch, D., A proof of Kirillov’s conjecture, Ann. of Math. (2) 158, no. 1 (2003), 207–252.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bernstein, J., P-invariant distributions on GL(N) and the classification of unitary representations of GL(N) (non-archimedean case), Lie Group Representations II, Lecture Notes in Math. 1041, Springer-Verlag, Berlin, 1984, 50–102.Google Scholar
  9. 9.
    Borel, A. and Wallach, N., Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Mathematics Studies, vol. 94, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980.Google Scholar
  10. 10.
    Casselman, W., Introduction to the theory of admissible representations of p-adic reductive groups, preprint (http://www.math.ubc.ca/~cass/research/pdf/p-adic-book.pdf).
  11. 11.
    Casselman, W., A new nonunitarity argument for p-adic representations, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28, no. 3 (1981), 907–928 (1982).Google Scholar
  12. 12.
    Gelfand, I. M. and Naimark, M. A., Unitary representations of semi simple Lie groups 1, Unitary representations of complex unimodular group, M. Sbornik 21 (1947), 405–434.Google Scholar
  13. 13.
    Gelfand, I. M. and Naimark, M. A., Unitäre Darstellungen der Klassischen Gruppen (German translation of Russian publication from 1950), Akademie Verlag, Berlin, 1957.Google Scholar
  14. 14.
    Hanzer, M., The unitarizability of the Aubert dual of the strongly positive discrete series, Israel J. Math. 169 (2009), no. 1, 251–294.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hanzer, M. and Tadić, M., A method of proving non-unitarity of representations of p-adic groups I, Math. Z. 265 (2010), no. 4, 799–816.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hanzer, M. and Jantzen, C., A method of proving non-unitarity of representations of p-adic groups, J. Lie Theory 22 (2012), no. 4, 1109–1124.MathSciNetzbMATHGoogle Scholar
  17. 17.
    Harris, M. and Taylor, R., On the geometry and cohomology of some simple Shimura varieties, Princeton University Press, Annals of Math. Studies 151, 2001.Google Scholar
  18. 18.
    Henniart, G., Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique, Invent. Math. 139 (2000), 439–455.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Howe, R. and Moore, C., Asymptotic properties of unitary representations, J. Funct. Anal. 32 (1979), no. 1, 72–96.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Jacquet, H., On the residual spectrum of GL(n), Lie Group Representations II, Lecture Notes in Math. 1041, Springer-Verlag, Berlin, 1984, 185–208.Google Scholar
  21. 21.
    Jantzen, C., On supports of induced representations for symplectic and odd-orthogonal groups, Amer. J. Math. 119 (1997), 1213–1262.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Jantzen, C., Discrete series for p-adic SO(2n) and restrictions of representations of O(2n), Canad. J. Math. 63, no. 2 (2011), 327–380.MathSciNetCrossRefGoogle Scholar
  23. 23.
    Jantzen, C., Tempered representations for classical p-adic groups, Manuscripta Math. 145 (2014), no. 3–4, 319–387.MathSciNetCrossRefGoogle Scholar
  24. 24.
    Kazhdan, D., Connection of the dual space of a group with the structure of its closed subgroups, Functional Anal. Appl. 1 (1967), 63–65.zbMATHGoogle Scholar
  25. 25.
    Kirillov, A. A., Infinite dimensional representations of the general linear group, Dokl. Akad. Nauk SSSR 114 (1962), 37–39; Soviet Math. Dokl. 3 (1962), 652–655.Google Scholar
  26. 26.
    Konno, T., A note on the Langlands classification and irreducibility of induced representations of p-adic groups, Kyushu J. Math. 57 (2003), no. 2, 383–409.MathSciNetCrossRefGoogle Scholar
  27. 27.
    Kudla, S. S., Notes on the local theta correspondence, lectures at the European School in Group Theory, 1996, preprint (http://www.math.toronto.edu/~skudla/castle.pdf).
  28. 28.
    Lapid, E. and Mínguez, A., On parabolic induction on inner forms of the general linear group over a non-archimedean local field, Selecta Math. (N.S.) to appear, arXiv:1411.6310.Google Scholar
  29. 29.
    Lapid, E., Muić, G. and Tadić, M., On the generic unitary dual of quasisplit classical groups, Int. Math. Res. Not. no. 26 (2004), 1335–1354.MathSciNetCrossRefGoogle Scholar
  30. 30.
    Laumon, G., Rapoport, M. and Stuhler, U., P-elliptic sheaves and the Langlands correspondence, Invent. Math. 113 (1993), 217–338.MathSciNetCrossRefGoogle Scholar
  31. 31.
    Matić, I. and Tadić, M., On Jacquet modules of representations of segment type, Manuscripta Math. 147 (2015), no. 3–4, 437–476.MathSciNetCrossRefGoogle Scholar
  32. 32.
    Mautner, F., Spherical functions over p-adic fields I, Amer. J. Math. 80 (1958). 441–457.MathSciNetCrossRefGoogle Scholar
  33. 33.
    Mœglin, C., Sur certains paquets d’Arthur et involution d’Aubert-Schneider-Stuhler généralisée, Represent. Theory 10 (2006), 86–129.MathSciNetCrossRefGoogle Scholar
  34. 34.
    Mœglin, C., Multiplicité 1 dans les paquets d’Arthur aux places p-adiques, in “On certain L-functions”, Clay Math. Proc. 13 (2011), 333–374.zbMATHGoogle Scholar
  35. 35.
    Mœglin, C., Paquets stables des séries discrètes accessibles par endoscopie tordue; leur paramètre de Langlands, Contemp. Math. 614 (2014), pp. 295–336.CrossRefGoogle Scholar
  36. 36.
    Mœglin, C. and Renard, D., Paquet d’Arthur des groupes classiques non quasi-déployś, preprint.Google Scholar
  37. 37.
    Mœglin, C. and Tadić, M., Construction of discrete series for classical p-adic groups, J. Amer. Math. Soc. 15 (2002), 715–786.MathSciNetCrossRefGoogle Scholar
  38. 38.
    Mœglin, C. and Waldspurger, J.-L., Sur le transfert des traces tordues d’un group linéaire à un groupe classique p-adique, Selecta mathematica 12 (2006), pp. 433–516.MathSciNetCrossRefGoogle Scholar
  39. 39.
    Muić, G. and Tadić, M., Unramified unitary duals for split classical p–adic groups; the topology and isolated representations, in “On Certain L-functions”, Clay Math. Proc. vol. 13, 2011, 375–438.zbMATHGoogle Scholar
  40. 40.
    Renard, D., Représentations des groupes réductifs p-adiques, Cours Spécialisés 17, Société Mathématique de France, Paris, 2010.Google Scholar
  41. 41.
    Rodier, F., Whittaker models for admissible representations, Proc. Sympos. Pure Math. AMS 26 (1983), pp. 425–430.CrossRefGoogle Scholar
  42. 42.
    Schneider, P. and Stuhler, U., Representation theory and sheaves on the Bruhat-Tits building, Publ. Math. IHES 85 (1997), 97–191.MathSciNetCrossRefGoogle Scholar
  43. 43.
    Sécherre, V., Proof of the Tadić conjecture (U0) on the unitary dual of GL m(D), J. reine angew. Math. 626 (2009), 187–203.MathSciNetzbMATHGoogle Scholar
  44. 44.
    Silberger, A., The Langlands quotient theorem for p-adic groups, Math. Ann. 236 (1978), no. 2, 95–104.MathSciNetCrossRefGoogle Scholar
  45. 45.
    Silberger, A., Special representations of reductive p-adic groups are not integrable, Ann. of Math. 111 (1980), 571–587.MathSciNetCrossRefGoogle Scholar
  46. 46.
    Speh, B., Unitary representations of \(GL(n, \mathbb R)\) with non-trivial (g, K)- cohomology, Invent. Math. 71 (1983), 443–465.Google Scholar
  47. 47.
    Tadić, M., Unitary representations of general linear group over real and complex field, preprint MPI/SFB 85–22 Bonn (1985), (http://www.mpim-bonn.mpg.de/preblob/5395).
  48. 48.
    Tadić M. Unitary dual of p-adic GL(n), Proof of Bernstein Conjectures, Bulletin Amer. Math. Soc. 13 (1985), 39–42.MathSciNetCrossRefGoogle Scholar
  49. 49.
    Tadić, M., Classification of unitary representations in irreducible representations of general linear group (non-archimedean case), Ann. Sci. École Norm. Sup. 19 (1986), 335–382.MathSciNetCrossRefGoogle Scholar
  50. 50.
    Tadić, M., Induced representations of GL(n, A) for p-adic division algebras A, J. reine angew. Math. 405 (1990), 48–77.MathSciNetzbMATHGoogle Scholar
  51. 51.
    Tadić, M., An external approach to unitary representations, Bull. Amer. Math. Soc. (N.S.) 28, no. 2 (1993), 215–252.MathSciNetCrossRefGoogle Scholar
  52. 52.
    Tadić, M., Representations of p-adic symplectic groups, Compositio Math. 90 (1994), 123–181.MathSciNetzbMATHGoogle Scholar
  53. 53.
    Tadić, M., Structure arising from induction and Jacquet modules of representations of classical p-adic groups, J. of Algebra 177 (1995), 1–33.MathSciNetCrossRefGoogle Scholar
  54. 54.
    Tadić, M. On regular square integrable representations of p-adic groups, Amer. J. Math. 120, no. 1 (1998), 159–210.MathSciNetCrossRefGoogle Scholar
  55. 55.
    Tadić, M., On reducibility of parabolic induction, Israel J. Math. 107 (1998), 29–91.MathSciNetzbMATHGoogle Scholar
  56. 56.
    Tadić, M., Square integrable representations of classical p-adic groups corresponding to segments, Represent. Theory 3 (1999), 58–89.MathSciNetCrossRefGoogle Scholar
  57. 57.
    Tadić, M., On classification of some classes of irreducible representations of classical groups, in book Representations of real and p-adic groups, Singapore University Press and World Scientific, Singapore, 2004, 95–162.zbMATHGoogle Scholar
  58. 58.
    Tadić, M., \(GL(n,\mathbb C)\hat {\ }\) and \(G L(n,\mathbb R)\hat {\ }\), in “Automorphic Forms and L-functions II, Local Aspects”, Contemp. Math. 489 (2009), 285–313.Google Scholar
  59. 59.
    Tadić, M., On reducibility and unitarizability for classical p-adic groups, some general results, Canad. J. Math. 61 (2009), 427–450.MathSciNetCrossRefGoogle Scholar
  60. 60.
    Tadić, M., On automorphic duals and isolated representations; new phenomena, J. Ramanujan Math. Soc. 25, no. 3 (2010), 295–328.MathSciNetzbMATHGoogle Scholar
  61. 61.
    Tadić, M., On tempered and square integrable representations of classical p-adic groups, Sci. China Math., 56 (2013), 2273–2313.MathSciNetCrossRefGoogle Scholar
  62. 62.
    Tadić, M., Remark on representation theory of general linear groups over a non-archimedean local division algebra, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 19(523) (2015), 27–53.MathSciNetzbMATHGoogle Scholar
  63. 63.
    Vogan, D. A., The unitary dual of GL(n) over an archimedean field, Invent. Math. 82 (1986), 449–505.MathSciNetCrossRefGoogle Scholar
  64. 64.
    Zelevinsky, A. V., Induced representations of reductive p-adic groups II. On irreducible representations of GL(n), Ann. Sci. École Norm. Sup. 13 (1980), 165–210.MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ZagrebZagrebCroatia

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