Skip to main content
Log in

Differential operators on homogeneous spaces

  • Published:
Acta Mathematica

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

Access this article

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

Instant access to the full article PDF.

References

  1. Ásgeirsson, L., Über eine Mittelwertseigenschaft von Lösungen homogener linearer partieller Differentialgleichungen 2. Ordnung mit konstanten Koefficienten.Math. Ann. 113 (1936), 321–346.

    Article  MATH  Google Scholar 

  2. Berezin, F. A. &Gelfand, I. M., Some remarks on the theory of spherical functions on symmetric Riemannian manifolds.Trudy Moskov. Mat. Obšč. 5 (1956), 311–351.

    MathSciNet  MATH  Google Scholar 

  3. Cartan, E., Groupes simples clos et ouverts et géométrie riemannienne.J. Math. Pures Appl. 9, vol. 8 (1929), 1–33.

    MathSciNet  Google Scholar 

  4. —, Sur certaines formes riemanniennes remarquables des géométries à groupe fondamental simple.Ann. Ec. Normale 44 (1927), 345–467.

    MathSciNet  Google Scholar 

  5. —, Leçons sur la géométrie des espaces de Riemann. Deuxième édition. Paris 1951.

    MATH  Google Scholar 

  6. Cartier, P. &Dixmier, J., Vecteurs analytiques dans les représentations de groupes de Lie.Amer. J. Math. 80 (1958), 131–145.

    Article  MATH  MathSciNet  Google Scholar 

  7. Chevalley, C.,Theory of Lie Groups I, Princeton, 1946.

  8. Chevalley, C., The Betti numbers of the exceptional simple Lie Groups.Proc. Int. Congress of Math. Cambridge, Mass. 1950, Vol. 2, 21–24.

  9. —, Invariants of finite groups generated by reflections.Amer. J. Math. 77 (1955), 778–782.

    Article  MATH  MathSciNet  Google Scholar 

  10. Feller, W., Über die Lösungen der linearen partiellen Differentialgleichungen zweiter Ordnung vom elliptischen Typus.Math. Ann. 102 (1930), 633–649.

    Article  MathSciNet  MATH  Google Scholar 

  11. Gelfand, I. M., Spherical functions in symmetric Riemannian spaces.Doklady Akad. Nauk. SSSR (N.S.) 70 (1950), 5–8.

    MathSciNet  Google Scholar 

  12. —, The center of an infinitestimal group ring.Mat. Sbornik N.S. 26 (68) (1950), 103–112.

    MathSciNet  Google Scholar 

  13. Gelfand, I. M. &Graev, M. I., An analogue of the Plancherel formula for the classical groups,Trudy Moskov. Mat. Obšč, 4 (1955), 375–404.

    MathSciNet  Google Scholar 

  14. Godement, R., A theory of spherical functions I.Trans. Amer. Math. Soc. 73 (1952), 496–556.

    Article  MATH  MathSciNet  Google Scholar 

  15. —, Une généralisation du théorème de la moyenne pour les fonctions harmoniques.C. R. Acad. Sci. Paris 234 (1952), 2137–2139.

    MathSciNet  Google Scholar 

  16. Günther, P., Über einige spezielle probleme aus der Theorie der linearen partiellen Differentialgleichungen 2. Ordnung.Ber. Verh. Sächs. Akad. Wiss. Leipzig 102 (1957), 1–50.

    Google Scholar 

  17. Hadamard, J., Lès surfaces à courbures opposées et leur lignes géodésiques,J. Math. Pures Appl. 4 (1898), 27–73.

    Google Scholar 

  18. Hadamard, J.,Lectures on Cauchy's problem in linear partial differential equations. Yale Univ. Press 1923. Reprinted by Dover, New York 1952.

  19. Harish-Chandra, On Representations of Lie algebras,Ann. of Math. 50 (1949), 900–915.

    Article  MathSciNet  Google Scholar 

  20. —, Representations of Lie groups on a Banach space I.Trans. Amer. Math. Soc. 75 (1953), 185–243.

    Article  MATH  MathSciNet  Google Scholar 

  21. —, On the Plancherel formula for the right-invariant functions on a semi-simple Lie group.Proc. Nat. Acad. Sci. 40 (1954), 200–204.

    Article  MATH  MathSciNet  Google Scholar 

  22. —, The characters of semi-simple Lie groups.Trans. Amer. Math. Soc. 83 (1956), 98–163.

    Article  MATH  MathSciNet  Google Scholar 

  23. —, Fourier transforms on a semi-simple Lie algebra I.Amer. J. Math. 79 (1957), 193–257.

    Article  MATH  MathSciNet  Google Scholar 

  24. Helgason, S., On Riemannian curvature of homogeneous spaces.Proc. Amer. Math. Soc. 9 (1958), 831–838.

    Article  MathSciNet  Google Scholar 

  25. Helgason, S., Partial differential equations on Lie Groups. Scand. Math. Congress XIII, Helsinki 1957, 110–115.

  26. Herglotz, G., Über die Integration linearer partieller Differentialgleichungen mit konstanten Koefficienten.Ber. Math. Phys. Kl. Sächs. Akad. Wiss. Leipzig Part I 78 (1926), 41–74, Part II 78 (1926), 287–318, Part III 80 (1928), 69–114.

    Google Scholar 

  27. John, F.,Plane waves and spherical means applied to partial differential equations. New York 1955.

  28. Lichnerowicz, A. &Walker, A. G., Sur les espaces Riemanniens harmoniques de type hyperbolique normal.C. R. Acad. Sci. Paris 221 (1945), 394–396.

    MATH  MathSciNet  Google Scholar 

  29. Mostow, G. D., A new proof ofE. Cartan's theorem on the topology of semi-simple groups.Bull. Amer. Math. Soc. 55 (1949), 969–980.

    Article  MATH  MathSciNet  Google Scholar 

  30. Myers, S. B. &Steenrod, N. E., The group of isometries of a Riemannian manifold.Ann. of Math. 40 (1939), 400–416.

    Article  MathSciNet  Google Scholar 

  31. Nomizu, K., Invariant affine connections on homogeneous spaces.Amer. J. Math. 76 (1954), 33–65.

    Article  MATH  MathSciNet  Google Scholar 

  32. Riesz, M., L'intégrale de Riemann-Liouville et le problème de Cauchy.Acta Math. 81 (1949), 1–223.

    Article  MathSciNet  Google Scholar 

  33. Schwartz, L.,Théorie des distributions Vol. I, II. Paris 1950–1951.

  34. Selberg, A., Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series.J. Ind. Math. Soc. 20 (1956), 47–87.

    MathSciNet  MATH  Google Scholar 

  35. Tits, J., Sur certaines classes d'espaces homogènes de groupes de Lie.Acad. Roy. Belg. Cl. Sci. Mém. Coll. in 8° 29 (1955), no. 3.

  36. Wang, H. C., Two-point homogeneous spaces.Ann. of Math. 55 (1952), 177–191.

    Article  MathSciNet  Google Scholar 

  37. Willmore, T. J., Mean value theorems in harmonic Riemannian spaces.J. London Math. Soc. 25 (1950), 54–57.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Helgason, S. Differential operators on homogeneous spaces. Acta Math. 102, 239–299 (1959). https://doi.org/10.1007/BF02564248

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02564248

Keywords

Navigation