Some Problems in Local Harmonic Analysis

Part of the Progress in Mathematics book series (PM, volume 101)


The purpose of this article is to discuss some questions in the harmonic analysis of real and p-adic groups. We shall be particularly concerned with the properties of a certain family of invariant distributions. These distributions arose naturally in a global context, as the terms on the geometric side of the trace formula. However, they are purely local objects, which include the ordinary invariant orbital integrals. One of our aims is to describe how the distributions also arise in a local context. They appear as the terms on the geometric side of a new trace formula, which is simpler than the original one, and is the solution of a natural question in local harmonic analysis. The local trace formula seems to be a promising tool. It might have implications for the difficult local problems which are holding up progress in automorphic forms.


Parabolic Subgroup Trace Formula Automorphic Form Invariant Distribution Levi Subgroup 
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  1. 1.
    J. Arthur, A theorem on the Schwartz space of a reductive Lie group, Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 4718–4719.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    J. Arthur, The trace formula in invariant form, Ann. of Math. (2) 114 (1981), 1–74.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    J. Arthur, The characters of supercuspidal representations as weighted orbital integrals, Proc. Indian Acad. Sci. 97 (1987), 3–19.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    J. Arthur, The invariant trace formula I. Local theory, J. Amer. Math. Soc. 1 (1988), 323–383.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    J. Arthur, The invariant trace formula II. Global theory, J. Amer. Math. Soc. 1 (1988), 501–554.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    J. Arthur, Intertwining operators and residues I. Weighted characters, J. Funct. Anal. 84 (1989), 19–84.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    J. Arthur, Intertwining operators and residues II. Invariant distributions, Compos. Math. 70 (1989), 51–99MathSciNetGoogle Scholar
  8. 8.
    J. Arthur, Unipotent automorphic representations: Conjectures, Astérisque 171-172 (1989), 13–71.Google Scholar
  9. 9.
    J. Arthur, Towards a local trace formula, Amer. J. Math, (to appear).Google Scholar
  10. 10.
    J. Arthur, A local trace formula, preprint.Google Scholar
  11. 11.
    J. Arthur and L. Clozel, Simple Algebras, Base Change and the Advanced Theory of the Trace Formula, Annals of Math. Studies, Vol. 120, Princeton University Press, Princeton, 1989.zbMATHGoogle Scholar
  12. 12.
    J. Bernstein, P. Deligne, and D. Kazhdan, Trace Paley-Wiener theorem for reductive p-adic groups, J. Analyse Math. 47 (1986), 180–192.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    L. Clozel, Orbital integrals on p-adic groups: A proof of the Howe conjecture, Ann. of Math. 129 (1989), 237–251.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    L. Clozel and P. Delorme, Le théoreme de Paley-Wiener invariant pour les groupes de Lie réductifs II, Ann. Scient. Ec. Norm. Sup. (to appear).Google Scholar
  15. 15.
    Harish-Chandra, Harmonic analysis on real reductive groups III. The Maass-Selberg relations and the Plancherel formula, Ann. of Math. 104 (1976), 117–201, reprinted in Collected Works, Springer-Verlag, Vol. IV, 259-343.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Harish-Chandra, The Plancherel formula for reductive p-adic groups, reproduced in Collected Works, Springer-Verlag, Vol. IV, pp. 353-367.Google Scholar
  17. 17.
    R. Herb, Discrete series characters and Fourier inversion on semisimple real Lie groups, Trans. Amer. Math. Soc. 277 (1983), 241–262.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    D. Kazhdan, Cuspidal geometry on p-adic groups, J. Analyse Math. 47 (1989), 1–36.MathSciNetCrossRefGoogle Scholar
  19. 19.
    R. Kottwitz and J. Rogawski, The distributions in the invariant trace formula are supported on characters, preprint.Google Scholar
  20. 20.
    R. Langlands, Base Change for GL(2), Annals of Math. Studies 96 (1980), Princeton University Press, Princeton.Google Scholar
  21. 21.
    R. Langlands and D. Shelstad, On the definition of transfer factors, Math. Ann. 278 (1987), 219–271.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    P. Mischenko, Invariant Tempered Distributions on the Reductive p-Adic Group GL n (F p ), Ph.D. Thesis, University of Toronto, Toronto, 1982.Google Scholar
  23. 23.
    J. Rogawski, The trace Paley-Wiener theorem in the twisted case, Trans. Amer. Math. Soc. 309 (1988), 215-229.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    J.-L. Waldspurger, Intégrates orbitales spheriques pour GL(N), Asterisque 171—172 (1989), 270–337.MathSciNetGoogle Scholar

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© Springer Science+Business Media New York 1991

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

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