Letters in Mathematical Physics

, Volume 46, Issue 1, pp 1–47 | Cite as

An Introduction to p-adic Fields, Harmonic Analysis and the Representation Theory of SL2

  • Paul J. SallyJr.


In this expository article, we develop in considerable detail harmonic analysis on p-adic fields. This harmonic analysis is distinctly different from that on the real and complex numbers due to the nature of the underlying topology.

p-adic analysis harmonic analysis representation theory. 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bargmann, V.: Irreducible unitary representations of the Lorentz group, Ann. of Math. (2) 48(1947), 568-640.Google Scholar
  2. 2.
    Bourbaki, N.: Eléments de mathématique, livre 6, Intégration, Hermann, Paris, 1939.Google Scholar
  3. 3.
    Brekke, L. and Freund, P. G. O.: p-adic Numbers in Physics, Phys. Rep. 233(1) (1993).Google Scholar
  4. 4.
    Cassels, J. W. S.: Local Fields, London Mat. Soc. Student Texts 3, Cambridge University Press, Cambridge, 1986.Google Scholar
  5. 5.
    Gelfand, I. M. and Graev, M. I.: The group of matrices of second order with coefficients in a locally compact field and special functions on locally compact fields, Uspekhi Mat. Nauk 18(1963), 29-99.Google Scholar
  6. 6.
    Gelfand, I. M., Graev, M. I. and Piatetskii-Shapiro, I. I.: Representation Theory and Automorphic Functions, Saunders, 1969.Google Scholar
  7. 7.
    Gouvêa, F.: p-adic Numbers: An Introduction, Springer-Verlag, Berlin Heidelberg, 1993.Google Scholar
  8. 8.
    Katznelson, Y.: An Introduction to Harmonic Analysis, Dover, New York, 1976.Google Scholar
  9. 9.
    Krasner, M.: Nombre des extensions d'un degre d'un corps p-adique, Les tendances géom. en algèbre et théorie des nombres, Editions du Centre National de la Recherche Scientifique, Paris, 1966, pp. 143-169.Google Scholar
  10. 10.
    Naĩmark, M. A.: Linear representations of the Lorentz group, Trans. Amer. Math. Soc. (2) 6(1957), 379-458.Google Scholar
  11. 11.
    Pontryagin, L. S.: Topological Groups, translated from the second Russian edition by Arlen Brown, Gordon and Breach, New York, 1966.Google Scholar
  12. 12.
    Rudin, W.: Fourier Analysis on Groups, Wiley-Interscience, New York, 1962.Google Scholar
  13. 13.
    Sally, Jr., P. J.: Unitary and uniformly bounded representations of the two by two unimodular group over local fields, Amer. J. Math. 90(1968), 406-443.Google Scholar
  14. 14.
    Sally, Jr., P. J.: Harmonic analysis and group representations, Studies in Harmonic Analysis(Proc. Conf. DePaul Univ., Chicago, Ill, 1974), MAA Stud. Math. Vol. 13, Math. Assoc. Amer., Washington, D.C., 1976 pp. 224-256.Google Scholar
  15. 15.
    Sally, Jr. P. J. and Shalika, J.: Characters of the discrete series of representations of SL(2) over a local field, Proc. Nat. Acad. Sci. U.S.A. 61(1968), 1231-1237.Google Scholar
  16. 16.
    Sally, Jr. P. J. and Shalika, J.: The Plancherel formula for SL(2) over a local field, Proc. Nat. Acad. Sci. 63(1969), 661-667.Google Scholar
  17. 17.
    Sally, Jr. P. J. and Shalika, J.: The Fourier transform of orbital integrals on SL2 over a p-adic field, Lie Group Representations II(College Park, Md. 1982/1983), Lecture Notes in Math. 1041, Springer, Berlin, New York, 1984, pp. 303-340.Google Scholar
  18. 18.
    Sally, Jr. P. J. and Taibleson, M. H.: Special functions on locally compact fields, Acta. Math. 116(1966), 279-309.Google Scholar
  19. 19.
    Serre, J.-P.: A Course in Arithmetic, Springer-Verlag, New York, 1973.Google Scholar
  20. 20.
    Shalika, J.: Representations of the two by two unimodular group over local fields, Ph.D. thesis, The Johns Hopkins University (1966).Google Scholar
  21. 21.
    Tate, J. T.: Fourier analysis in number fields and Hecke's zeta-functions, in: J. W. S. Cassels and A. Fröhlich, (eds), Algebraic Number Theory, Academic Press, New York, 1967, pp. 305-347.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Paul J. SallyJr.
    • 1
  1. 1.Department of MathematicsUniversity of ChicagoChicagoU.S.A.

Personalised recommendations