Representations of the discrete Heisenberg group on distribution spaces of two-dimensional local fields

  • D. V. OsipovEmail author
  • A. N. Parshin


We study a natural action of the Heisenberg group of integer unipotent matrices of the third order on the distribution space of a two-dimensional local field for a flag on a two-dimensional scheme.


Irreducible Representation STEKLOV Institute Local Parameter Heisenberg Group Haar Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. A. Arnal’ and A. N. Parshin, “On irreducible representations of discrete Heisenberg groups,” Mat. Zametki 92 (3), 323–330 (2012) [Math. Notes 92, 295–301 (2012)].MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    I. V. Beloshapka and S. O. Gorchinskiy, “Irreducible representations of finitely generated nilpotent groups,” Mat. Sb. 207 (1), 45–72 (2016) [Sb. Math. 207, 41–46 (2016)]; arXiv:math/1508.06808 [math.RT].MathSciNetCrossRefGoogle Scholar
  3. 3.
    J. Bernstein, “Draft of: Representations of p-adic groups,” Lectures written by K.E. Rumelhart (Harvard Univ., 1992), http://wwwmathtauacil/~bernstei/Unpublished_texts/unpublished_texts/Bernstein93newharv. lectfrom-chicpdfGoogle Scholar
  4. 4.
    J. Dixmier, Les C -algèbres et leurs représentations (Gauthier-Villars, Paris, 1969).zbMATHGoogle Scholar
  5. 5.
    P. Kahn, “Automorphisms of the discrete Heisenberg group,” Preprint (Cornell Univ., Ithaca, 2005), http://wwwmathcornelledu/m/sites/default/files/imported/People/Faculty/HeisenpdfGoogle Scholar
  6. 6.
    M. Kapranov, “Semiinfinite symmetric powers,” arXiv: math/0107089 [math.QA].Google Scholar
  7. 7.
    D. V. Osipov, “n-Dimensional local fields and adeles on n-dimensional schemes,” in Surveys in Contemporary Mathematics, Ed. by N. Young and Y. Choi (Cambridge Univ. Press, Cambridge, 2008), LMS Lect. Note Ser. 347, pp. 131–164.Google Scholar
  8. 8.
    D. V. Osipov, “The discrete Heisenberg group and its automorphism group,” Mat. Zametki 98 (1), 152–155 (2015) [Math. Notes 98, 185–188 (2015)].MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    D. V. Osipov and A. N. Parshin, “Harmonic analysis on local fields and adelic spaces. I,” Izv. Ross. Akad. Nauk, Ser. Mat. 72 (5), 77–140 (2008) [Izv. Math. 72, 915–976 (2008)].MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    D. V. Osipov and A. N. Parshin, “Harmonic analysis on local fields and adelic spaces. II,” Izv. Ross. Akad. Nauk, Ser. Mat. 75 (4), 91–164 (2011) [Izv. Math. 75, 749–814 (2011)].MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    A. N. Parshin, “On holomorphic representations of discrete Heisenberg groups,” Funkts. Anal. Prilozh. 44 (2), 92–96 (2010) [Funct. Anal. Appl. 44, 156–159 (2010)].MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    A. N. Parshin, “Representations of higher adelic groups and arithmetic,” in Proc. Int. Congr. Math., Hyderabad, India, Aug. 19–27, 2010 (Hindustan Book Agency, New Delhi, 2010), Vol. 1, pp. 362–392.MathSciNetzbMATHGoogle Scholar
  13. 13.
    A. N. Parshin, “Notes on the Poisson formula,” Algebra Analiz 23 (5), 1–54 (2011) [St. Petersburg Math. J. 23, 781–818 (2012)].MathSciNetGoogle Scholar
  14. 14.
    A. Weil, “Fonction zêta et distributions,” in Séminaire Bourbaki 1965/66 (Soc. Math. France, Paris, 1995), Exp. 312, pp. 523–531.Google Scholar
  15. 15.
    A. Weil, Basic Number Theory, 3rd ed. (Springer, Berlin, 1974), Grundl. Math. Wiss. 144.CrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia
  2. 2.National University of Science and Technology MISiSMoscowRussia

Personalised recommendations