Selecta Mathematica

, Volume 24, Issue 1, pp 473–497 | Cite as

A spectral decomposition of orbital integrals for PGL(2, F) (with an appendix by S. Debacker)

  • David KazhdanEmail author


Let F be a local non-archimedian field, G a semisimple F-group, dg a Haar measure on G and \(\mathcal {S}(G)\) be the space of locally constant complex valued functions f on G with compact support. For any regular elliptic congugacy class \(\Omega =h^G\subset G\) we denote by \(I_\Omega \) the G-invariant functional on \(\mathcal {S}(G)\) given by
$$\begin{aligned} I_\Omega (f)=\int _G f(g^{-1}hg)dg \end{aligned}$$
This paper provides the spectral decomposition of functionals \(I_\Omega \) in the case \(G={\text {PGL}}(2,F)\) and in the last section first steps of such an analysis for the general case.

Mathematics Subject Classification



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Many thanks for J.Bernstein, S. Debacker and Y. Flicker who corrected a number of imprecisions in the original draft and S. Debacker for writing an Appendix. I am partially supported by the ERC grant 669655-HAS.


  1. 1.
    Adler, J., DeBacker, S.: Some applications of Bruhat-Tits theory to harmonic analysis on the Lie algebra of a reductive \(p\)-adic group. Mich. Math. J. 50(2), 263–286 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Arthur, J.: On the Fourier transforms of weighted orbital integrals. J. Reine Angew. Math. 452, 163–217 (1994)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Aubert, A., Plymen, R.: Plancherel measure for \(GL(n, F)\) and \(GL(m, D)\): explicit formulas and Bernstein decomposition. Journal of Number Theory 112(1), 26–66 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bernstein, J., Deligne, P., Kazhdan, D.: Trace Paley-Wiener theorem for reductive p-adic groups. J. Anal. Math. 47, 180–192 (1986)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bernstein, I.N., Zelevinski, A.V.: Representations of the group \(GL(n,F)\), where \(F\) is a local non-Archimedean field (Russian). Uspehi Mat. Nauk 31(3), 5–70 (1976)MathSciNetGoogle Scholar
  6. 6.
    Bernstein, J.: Representations of \(p\)-adic groups. Lectures at Harvard University, Fall. Notes by Karl E. Rumelhart (1992)Google Scholar
  7. 7.
    DeBacker, S.: Homogeneity results for invariant distributions of a reductive \(p\)-adic group. Ann. Sci. École Norm. Sup. 35(3), 391–422 (2002)MathSciNetCrossRefGoogle Scholar
  8. 8.
    DeBacker, S.: Lectures on harmonic analysis for reductive p-adic groups, representations of real and p-adic groups, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 2, Singapore Univ. Press, Singapore, pp. 47–94 (2004)Google Scholar
  9. 9.
    Gelfand, I.M., Graev, M.I., Pyatetskii-Shapiro, I.I.: Generalized functions. Vol. 6. Representation theory and automorphic functionsGoogle Scholar
  10. 10.
    Harish-Chandra: Harmonic analysis on reductive p-adic groups. Notes by G. van Dijk. Lecture Notes in Mathematics 162. Springer, Berlin, pp iv+125 (1970)Google Scholar
  11. 11.
    Harish-Chandra: Admissible invariant distributions on reductive \(p\)-adic groups, preface and notes by Stephen DeBacker and Paul J. Sally, Jr., University Lecture Series, vol. 16, American Mathematical Society, Providence, RI (1999)Google Scholar
  12. 12.
    Howe, R.: The Fourier transform and germs of characters (case of \(\text{ Gl }_n\) over a \(p\)-adic field). Math. Ann. 208, 305–322 (1974)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Huntsinger, R.: Some aspects of invariant harmonic analysis on the Lie algebra of a reductive \(p\)-adic group. Ph.D. Thesis, The University of Chicago (1997)Google Scholar
  14. 14.
    Jacquet, H., Langlands, R.P.: Automorphic forms on \({{\rm GL}}(2)\). Lecture Notes in Mathematics 114. Springer, Berlin (1970)Google Scholar
  15. 15.
    Kazhdan, D.: On Shalika germs. Sel. Math. 22(4), 1821–1824 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kazhdan, D.: Cuspidal geometry of \(p\)-adic groups. J. Anal. Math. 47, 1–36 (1986)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kazhdan, D.: Representations of groups over close local fields. J. Anal. Math. 47, 175–179 (1986)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Moy, A., Prasad, G.: Unrefined minimal \({K}\)-types for \(p\)-adic groups. Inv. Math. 116, 393–408 (1994)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Moy, A., Prasad, G.: Jacquet functors and unrefined minimal \({K}\)-types. Comment. Math. Helv. 71, 98–121 (1996)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Shalika, J.: A theorem on semi-simple \({\cal{P}}\)-adic groups. Ann. Math. 95, 226–242 (1972)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Sally, P., Shalika, J.: The Fourier transform of orbital integrals on \({{\rm SL}}(2)\) over a \(p\)-adic field. Lie group representations, II (College Park, Md., 1982/1983), 303–340, Lecture Notes in Math., 1041. Springer, Berlin (1984)Google Scholar
  22. 22.
    Waldspurger, J.-L.: Quelques resultats de finitude concernant les distributions invariantes sur les algèbres de Lie \(p\)-adiques, preprint (1993)Google Scholar

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© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Einstein Institute of Mathematics, Edmond J. Safra CampusThe Hebrew University of JerusalemGivat Ram. JerusalemIsrael

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