Israel Journal of Mathematics

, Volume 192, Issue 1, pp 83–120 | Cite as

Automorphic Plancherel density theorem

  • Sug Woo ShinEmail author


Let F be a totally real field, G a connected reductive group over F, and S a finite set of finite places of F. Assume that G(F ℝ) has a discrete series representation. Building upon work of Sauvageot, Serre, Conrey-Duke-Farmer and others, we prove that the S-components of cuspidal automorphic representations of \(G\left( {\mathbb{A}_F } \right)\) are equidistributed with respect to the Plancherel measure on the unitary dual of G(F S ) in an appropriate sense. A few applications are given, such as the limit multiplicity formula for local representations in the global cuspidal spectrum and a quite flexible existence theorem for cuspidal automorphic representations with prescribed local properties. When F is not a totally real field or G(F ℝ) has no discrete series, we present a weaker version of the above results.


Haar Measure Trace Formula Discrete Series Levi Subgroup Automorphic Representation 
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.


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  1. [AP05]
    A.-M. Aubert and R. Plymen, Plancherel measure for GL(n, F) and GL(m,D): explicit formulas and Bernstein decomposition, Journal of Number Theory 112 (2005), 26–66.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [Art88]
    J. Arthur, The invariant trace formula II. Global theory, Journal of the American Mathematical Society 1 (1988), 501–554.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [Art89]
    J. Arthur, The L 2 -Leftschetz numbers of Hecke operators, Inventiones Mathematicae 97 (1989), 257–290.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [BD84]
    J. Bernstein and P. Deligne, Le “centre” de Bernstein, in Represtntations of Reductive Group over a Local Field (P. Delinge, ed.), Travaux en Cours, Hermann, Paris, 1984, pp. 1–32.Google Scholar
  5. [BDK86]
    J. Bernstein, P. Deligne and D. Kazhdan, Trace Paley-Wiener theorem for reductive p-adic groups, Journal d’Analyse Mathématique 47 (1986), 180–192.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [Car84]
    H. Carayol, Représentations cuspidales du groupe linéaire, Annales Scientifiques de l’École Normale Supérieure 17 (1984), 191–225.MathSciNetzbMATHGoogle Scholar
  7. [CC09]
    G. Chenevier and L. Clozel, Corps de nombres peu ramifiés et formes automorphes autoduales, Journal of the American Mathematical Society 22 (2009), 467–519.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [CDF97]
    J. B. Conrey, W. Duke and D. W. Farmer, The distribution of the eigenvalues of Hecke operators, Acta Arithmetica 78 (1997), 405–409.MathSciNetzbMATHGoogle Scholar
  9. [CH]
    G. Chenevier and M. Harris, Construction of automorphic Galois representations, II,
  10. [Clo86]
    L. Clozel, On limit multiplicities of discrete series representations in spaces of automorphic forms, Inventiones Mathematicae 83 (1986), 265–284.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [CMS90]
    L. Corwin, A. Moy and P. Sally, Jr, Degrees and formal degress for division algebras and gl n over a p-adic field, Pacific Journal of Mathematics 141 (1990), 21–45.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [Fel60]
    J. Fell, The dual spaces of c*-algebras, Transactions of the American Mathematical Society 94 (1960), 365–403.MathSciNetzbMATHGoogle Scholar
  13. [Fer07]
    A. Ferrari, Théorème de l’indice et formule des traces, Manuscripta Mathematica 124 (2007), 363–390.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [GKM97]
    M. Goresky, R. Kottwitz and R. MacPherson, Discrete series characters and the Lefschetz formula for Hecke operators, Duke Mathematical Journal 89 (1997), 477–554.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [Gro]
    B. Gross, Irreducible cuspidal representations with prescribed local behavior, American Journal of Mathematics 133 (2011), 1231–1258.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [Gro97]
    B. Gross, On the motive of a reductive group, Inventiones Mathematicae 130 (1997), 287–313.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [Kot86]
    R. Kottwitz, Stable trace formula: Elliptic singular terms, Mathematische Annalen 275 (1986), 365–399.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [Kot88]
    R. Kottwitz, Tamagawa numbers, Annals of Mathematics 127 (1988), 629–646.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [KZ82]
    A. Knapp and G. Zuckerman, Classification of irreducible tempered representations of semisimple groups, Annals of Mathematics 116 (1982), 389–455.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [Lab]
    J.-P. Labesse, Changement de base CM et séries discrètes,
  21. [Lab99]
    J.-P. Labesse, Cohomologie, stabilisation et changement de base, Astérisque, Vol. 257, 1999.Google Scholar
  22. [RS87]
    J. Rohlfs and B. Speh, On limit multiplicities of representations with cohomology in the cuspidal spectrum, Duke Mathematical Journal 55 (1987), 199–211.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [San81]
    J.-J. Sansuc, Groupe de Brauer et arithmétique des groupes algébriques lineaires sur un corps de nombres, Journal für die Reine und Angewandte Mathematik 327 (1981), 12–80.MathSciNetzbMATHGoogle Scholar
  24. [Sar87]
    P. Sarnak, Statistical properties of eigenvalues of the Hecke operators, in Analytic Number Theory and Diophantine problems (Stillwater, OK, 1984), Progress in Mathematics, Vol. 70, Birkhäuser Boston, Boston, MA, 1987, pp. 321–331.CrossRefGoogle Scholar
  25. [Sar05]
    P. Sarnak, An Introduction to the Trace Formula, Clay Mathematics Monographs, Vol. 4, CMI/AMS, 2005, pp. 659–681.MathSciNetGoogle Scholar
  26. [Sau97]
    F. Sauvageot, Principe de densité pour les groupes réductifs, Compositio Mathematica 108 (1997), 151–184.MathSciNetzbMATHCrossRefGoogle Scholar
  27. [Ser97]
    J.-P. Serre, Répartition aymptotique des valeurs propres de l’opérateur de Hecke T p, Journal of American Mathematical Society 10 (1997), 75–102.zbMATHCrossRefGoogle Scholar
  28. [Shia]
    S. W. Shin, Galois representations arising from some compact Shimura varieties, Annals of Mathematics 173 (2011), 1645–1741.MathSciNetzbMATHCrossRefGoogle Scholar
  29. [Shib]
    S. W. Shin, On the cohomology of Rapoport-Zink spaces of EL-type, American Journal of Mathematics, to appear.Google Scholar
  30. [Wal84]
    N. Wallach, On the constant term of a square integrable automorphic form, in Operator Algebras and Group Representations. II, Monograph Studies in Mathematics, Vol. 18, Pitman, Boston, MA, 1984, pp. 227–237.Google Scholar
  31. [Wal03]
    J.-L. Waldspurger, La formule de Plancherel pour les groupes p-adiques d’après Harish-Chandra, Journal of the Institute of Mathematics of Jussieu 2 (2003), 235–333.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [Wei09]
    J. Weinstein, Hilbert modular forms with prescribed ramification, International Mathematics Research Notices (2009), 1388–1420.Google Scholar

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© Hebrew University Magnes Press 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA

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