Skip to main content
Log in

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

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

A Correction to this article was published on 13 March 2019

This article has been updated

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Change history

  • 13 March 2019

    Papers [1–6] have received funding from ERC under Grant Agreement No. 669655.

References

  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)

    Article  MathSciNet  Google Scholar 

  2. Arthur, J.: On the Fourier transforms of weighted orbital integrals. J. Reine Angew. Math. 452, 163–217 (1994)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  4. Bernstein, J., Deligne, P., Kazhdan, D.: Trace Paley-Wiener theorem for reductive p-adic groups. J. Anal. Math. 47, 180–192 (1986)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  6. Bernstein, J.: Representations of \(p\)-adic groups. Lectures at Harvard University, Fall. Notes by Karl E. Rumelhart (1992)

  7. DeBacker, S.: Homogeneity results for invariant distributions of a reductive \(p\)-adic group. Ann. Sci. École Norm. Sup. 35(3), 391–422 (2002)

    Article  MathSciNet  Google Scholar 

  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)

  9. Gelfand, I.M., Graev, M.I., Pyatetskii-Shapiro, I.I.: Generalized functions. Vol. 6. Representation theory and automorphic functions

  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)

  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)

  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)

    Article  MathSciNet  Google Scholar 

  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)

  14. Jacquet, H., Langlands, R.P.: Automorphic forms on \({{\rm GL}}(2)\). Lecture Notes in Mathematics 114. Springer, Berlin (1970)

  15. Kazhdan, D.: On Shalika germs. Sel. Math. 22(4), 1821–1824 (2016)

    Article  MathSciNet  Google Scholar 

  16. Kazhdan, D.: Cuspidal geometry of \(p\)-adic groups. J. Anal. Math. 47, 1–36 (1986)

    Article  MathSciNet  Google Scholar 

  17. Kazhdan, D.: Representations of groups over close local fields. J. Anal. Math. 47, 175–179 (1986)

    Article  MathSciNet  Google Scholar 

  18. Moy, A., Prasad, G.: Unrefined minimal \({K}\)-types for \(p\)-adic groups. Inv. Math. 116, 393–408 (1994)

    Article  MathSciNet  Google Scholar 

  19. Moy, A., Prasad, G.: Jacquet functors and unrefined minimal \({K}\)-types. Comment. Math. Helv. 71, 98–121 (1996)

    Article  MathSciNet  Google Scholar 

  20. Shalika, J.: A theorem on semi-simple \({\cal{P}}\)-adic groups. Ann. Math. 95, 226–242 (1972)

    Article  MathSciNet  Google Scholar 

  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)

  22. Waldspurger, J.-L.: Quelques resultats de finitude concernant les distributions invariantes sur les algèbres de Lie \(p\)-adiques, preprint (1993)

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to David Kazhdan.

Additional information

Dedicated to A. Beilinson on the occasion of his 60th birthday.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kazhdan, D. A spectral decomposition of orbital integrals for PGL(2, F) (with an appendix by S. Debacker). Sel. Math. New Ser. 24, 473–497 (2018). https://doi.org/10.1007/s00029-017-0321-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00029-017-0321-y

Mathematics Subject Classification

Navigation