Inventiones mathematicae

, Volume 52, Issue 1, pp 27–93 | Cite as

Spectra of compact locally symmetric manifolds of negative curvature

  • J. J. Duistermaat
  • J. A. C. Kolk
  • V. S. Varadarajan


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  1. 1.
    Araki, S.: On root systems and an infinitesimal classification of irreducible symmetric spaces. J. of Math., Osaka City Univ.13, 1–34 (1962)Google Scholar
  2. 2.
    Bérard, P. H.: On the wave equation on a compact Riemannian manifold without conjugate points. Math. Z.155, 249–276 (1977)Google Scholar
  3. 3.
    Borel, A.: Compact Clifford-Klein forms of symmetric spaces. Topology2, 111–122 (1963)Google Scholar
  4. 4.
    Colin de Verdière, Y.: Spectre conjoint d'opérateurs pseudo-différentiels qui commutent, I. Le cas non intégrable. Preprint, Institut Fourier, St. Martin d'Hères. (1978)Google Scholar
  5. 5.
    DeGeorge, D.L., Wallach, N.R.: Limit formulas for multiplicities inL 2 (Γ/G). Ann. of Math.107, 133–150 (1978)Google Scholar
  6. 6.
    Duistermaat, J.J., Guillemin, V.W.: The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math.29, 39–79 (1975)Google Scholar
  7. 7.
    Gangolli, R.: Asymptotic behaviour of spectra of compact quotients of certain symmetric spaces. Acta Math.121, 151–192 (1968)Google Scholar
  8. 8.
    Gangolli, R.: On the Plancherel formula and the Paley-Wiener theorem for spherical functions on semisimple Lie groups. Ann. of Math.93, 150–165 (1971)Google Scholar
  9. 9.
    Gangolli, R.: Spectra of discrete uniform subgroups of semisimple Lie groups. In: Symmetric Spaces. Short Courses Presented at Washington University, pp. 93–117. New York: Marcel Dekker 1972Google Scholar
  10. 10.
    Gangolli, R., Warner, G.: On Selberg's trace formula. J. Math. Soc. Japan27, 327–343 (1975)Google Scholar
  11. 11.
    Gel'fand, I.M.: Automorphic functions and the theory of representations. Proc. Int. Congr. of Math., Stockholm, 1962, pp. 74–85. Stockholm: Almquist & Wiksell 1963Google Scholar
  12. 12.
    Gel'fand, I.M., Graev, M.I.: Geometry of homogenoues spaces, representations of groups in homogeneous spaces, and related questions of integral geometry, I. Trudy. Moskov. mat. Obšč.8, 321–390 (1959)Google Scholar
  13. 13.
    Gindikin, S.G., Karpelevič, F.I.: Plancherel measure of Riemannian symmetric spaces of nonpositive curvature. Dokl. Akad. Nauk SSSR145, 252–255 (1962)Google Scholar
  14. 14.
    Harish-Chandra: On the Plancherel formula for the right-invariant functions on a semisimple Lie group. Proc. Nat. Acad. Sci. U.S.A.40, 200–204 (1954)Google Scholar
  15. 15.
    Harish-Chandra: The Plancherel formula for complex semisimple Lie groups. Trans. Amer. Math. Soc.76, 485–528 (1954)Google Scholar
  16. 16.
    Harish-Chandra: Differential operators on a semisimple Lie algebra. Amer. J. Math.79, 87–120 (1957)Google Scholar
  17. 17.
    Harish-Chandra: A formula for semisimple Lie groups. Amer. J. Math.79, 733–760 (1957)Google Scholar
  18. 18.
    Harish-Chandra: Spherical functions on a semisimple Lie group. I Amer. J. Math.80, 241–310 (1958)Google Scholar
  19. 19.
    Harish-Chandra: Spherical functions on a semisimple Lie group. II. Amer. J. Math.80, 553–613 (1958)Google Scholar
  20. 20.
    Harish-Chandra: Invariant eigendistributions on a semisimple Lie group. Trans. Amer. Math. Soc.119, 457–508 (1965)Google Scholar
  21. 21.
    Harish-Chandra: Discrete series for semisimple Lie groups, II. Explicit determination of the characters. Acta Math.116, 1–111 (1966)Google Scholar
  22. 22.
    Harish-Chandra: Harmonic analysis on real reductive groups, I. The theory of the constant term. J. of Functional Analysis19, 104–204 (1975)Google Scholar
  23. 23.
    Hejhal, D.A.: The Selberg Trace Formula for PSL(2,ℝ), Volume I. Lecture Notes in Math.548. Berlin-Heidelberg-New York: Springer 1976Google Scholar
  24. 24.
    Helgason, S.: Differential Geometry and Symmetric Spaces. New York: Academic Press 1962Google Scholar
  25. 25.
    Helgason, S., Johnson, K.: The bounded spherical functions on symmetric spaces. Advances in Math.3, 586–593 (1969)Google Scholar
  26. 26.
    Hörmander, L.: The spectral function of an elliptic operator. Acta Math.121, 193–218 (1968)Google Scholar
  27. 27.
    Kolk, J.: Formule de Poisson et distribution asymptotique du spectre simultane d'opérateurs différentiels. C.R. Acad. Sci. Paris Sér. A-B284, 1045–1048 (1977)Google Scholar
  28. 28.
    Kolk, J.A.C.: The Selberg Trace Formula and Asymptotic Behaviour of Spectra. Thesis, Rijksuniversiteit Utrecht, (1977)Google Scholar
  29. 29.
    Lax, P.D., Phillips, R.S.: Scattering Theory for Automorphic Functions. Ann. of Math. Studies No 87, Princeton: Princeton University Press 1976Google Scholar
  30. 30.
    Mackey, G.W.: The Theory of Unitary Group Representations. Chicago: The University of Chicago Press 1976Google Scholar
  31. 31.
    Menikoff, A., Sjöstrand, J.: On the eigenvalues of a class of hypoelliptic operators. Math. Annalen,235, 55–85 (1978)Google Scholar
  32. 32.
    Minakshisundaram, S., Pleijel, A.: Some properties of the eigenfunctions of the Laplace-operator on Riemannian maniforlds. Canadian J. Math.1, 242–256 (1949)Google Scholar
  33. 33.
    Mostow, G.D.: Intersections of discrete subgroups with Cartan subgroups. J. Indian Math. Soc.34, 203–214 (1970)Google Scholar
  34. 34.
    Mostow, G.D.: Strong Rigidity of Locally Symmetric Spaces. Ann. of Math. Studies No 78. Princeton: Princeton University Press 1973Google Scholar
  35. 35.
    Ozols, V.: Critical points of the displacement function of an isometry. J. Differential Geometry3, 411–432 (1969)Google Scholar
  36. 36.
    Prasad, G., Raghunathan, M.S.: Cartan subgroups and lattices in semi-simple groups. Ann. of Math.96, 296–317 (1972)Google Scholar
  37. 37.
    Randol, B.: On the analytic continuation of the Minakshisundaram-Pleijel zeta function for compact Riemann surfaces. Trans. Amer. Math. Soc.201, 241–246 (1975)Google Scholar
  38. 38.
    Randol, B.: The Riemann hypothesis for Selberg's zeta-function and the asymptotic behavior of eigenvalues of the Laplace operator. Trans. Amer. Math. Soc.236, 209–223 (1978)Google Scholar
  39. 39.
    Schiffmann, G.: Intégrales d'entrelacement et fonctions de Whittaker. Bull. Soc. Math. France99, 3–72 (1971)Google Scholar
  40. 40.
    Selberg, A.: Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc.20, 47–87 (1956)Google Scholar
  41. 41.
    Varadarajan, V.S.: Lie Groups, Lie Algebras and Their Representations. Englewood Cliffs: Prentice Hall 1974Google Scholar
  42. 42.
    Varadarajan, V.S.: Harmonic Analysis on Real Reductive Groups. Lecture Notes in Math.576. Berlin-Heidelberg-New York: Springer 1977Google Scholar
  43. 43.
    Wallach, N.R.: On the Selberg trace formula in the case of compact quotients. Bull. Amer. Math. Soc.82, 171–195 (1976)Google Scholar
  44. 44.
    Wallach, N.R.: An asymptotic formula of Gelfand and Gangolli for the spectrum of Γ/G. J. Differential Geometry11, 91–101 (1976)Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • J. J. Duistermaat
    • 1
  • J. A. C. Kolk
    • 2
  • V. S. Varadarajan
    • 2
  1. 1.Mathematisch InstituutRijksuniversiteitUtrechtNetherlands
  2. 2.Dept of MathematicsUniversity of CaliforniaLos AngelesUSA

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