Annali di Matematica Pura ed Applicata

, Volume 86, Issue 1, pp 189–215 | Cite as

Local forms of invariant differential operators. I

  • Robert Carroll


We give a characterization of the local coefficients of invariant linear differential operators and indicate some of their properties.


Differential Operator Local Form Linear Differential Operator Local Coefficient Invariant Differential Operator 
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.


  1. [1]
    J. Adams,Lecture on Lie groups, Benjamin, New York, 1969.Google Scholar
  2. [2]
    M. Atiyah andI. Singer,The index of elliptic operators, I and III, Annals Math., 87 (1968), 484–530 and 546–604.CrossRefMathSciNetGoogle Scholar
  3. [3]
    M. Atiyah andG. Segal,The index of elliptic operators, II, Annals Math., 87 (1968), 531–545.CrossRefMathSciNetGoogle Scholar
  4. [4]
    V. Bargmann andE. Wigner,Group theoretical discussion of relativistic wave equations, Proc. Nat'l. Acad. Sciences, 34 (1948), 211–223.MathSciNetGoogle Scholar
  5. [5]
    R Bott,The index theorem for homogeneous differential operators, Differential and Combinational Topology, edited by S. Cairns, Princeton Univ. Press, 1965, 167–186.Google Scholar
  6. [6]
    R. Carroll,Abstract methods in partial differential equations, Harper-Row; New York, 1969.Google Scholar
  7. [7]
    A. Cerezo andF. Rouviere,Solution elémentaire d'un opérateur différentiel linéaire invariant à gauche sur un groupe de Lie réel compact et sur un espace homogène réductif compact, to appear.Google Scholar
  8. [8]
    C. Chevalley,Theory of Lie groups, Princeton Univ. Press, 1946.Google Scholar
  9. [9]
    L. Ehrenpreis, Proc. AMS Summer Inst. Global Analysis, Berkeley, 1968, to appear.Google Scholar
  10. [10]
    —— ——,Fourier analysis in several complex variables, Interscience-Wiley, New York, 1970.Google Scholar
  11. [11]
    I. Gelfand, R. Minlos, Z. Šapiro,Representations of the rotation and Lorentz groups, Moscow, 1958.Google Scholar
  12. [12]
    M. Hausner andJ. Schwartz,Lie groups; Lie algebras, Gordon and Breach, New York, 1968.Google Scholar
  13. [13]
    S. Helgason,Differential geometry and symmetric spaces, Academic Press, New York, 1962.Google Scholar
  14. [14]
    —— ——,Differential operators on homogeneous spaces, Acta Math., 102 (1959), 239–299.zbMATHMathSciNetGoogle Scholar
  15. [15]
    —— ——,Fundamental solutions of invariant differential operators on symmetric spaces, Amer, Jour. Math., 86 (1964), 505–601.MathSciNetGoogle Scholar
  16. [16]
    R. Hermann,Lie groups for physicists, Benjamin, New York, 1966.Google Scholar
  17. [17]
    G. Hochschild,The structure of Lie groups, Holden-Day, San Fransisco, 1965.Google Scholar
  18. [18]
    L. Hormander,Linear partial differential operators, Springer, Berlin, 1963.Google Scholar
  19. [19]
    D. Husemoller,Fibre bundles, McGraw-Hill, New York, 1966.Google Scholar
  20. [20]
    H. Jehle andW. Parke,Covariant spinor formulation of relativisitc wave equations under the homogeneous Lorentz group, Lectues Theoret. Physics, Vol. 7a, Univ. Colorado Press, 1965, 297–386.Google Scholar
  21. [21]
    F. Kamber andP. Tondeur,On the existence of certain types of invariant differential operators, to appear.Google Scholar
  22. [22]
    S. Kobayashi andK. Nomizu,Foundations of differential geometry, Vol. I, Interscience-Wiley, New York, 1963.Google Scholar
  23. [23]
    O. Loos,Symmetric spaces, Vols. I and II, Benjamin, New York, 1969.Google Scholar
  24. [24]
    G. Lyubarskij,The theory of groups and its applications in physics, Moscow, 1958.Google Scholar
  25. [25]
    B. Malgrange,Some remarks on the notion of convexity for differential operators, Colloq. Diff. Analysis, Bombay, 1964, 163–174 (Sem. Leray, 1962–63, pp. 190–223).Google Scholar
  26. [26]
    B. Malgrange,Sur les systemes différentiels à coefficients constants, Colloque CNRS, Paris, 1962, 113–122 (Sem. Leray, 1961–62, exposé 8).Google Scholar
  27. [27]
    A. McKerrell,Covariant wave equations for massless particles, Annals Physics, 40 (1966), 237–267.CrossRefGoogle Scholar
  28. [28]
    W. Miller,Lie theory and special functions, Academic Press, New York, 1968.Google Scholar
  29. [29]
    J. Milnor andJ. Moore,On the structure of Hopf algebras, Annals Math., 81 (1965), 211–264.CrossRefMathSciNetGoogle Scholar
  30. [30]
    M. Naimark,Linear representations of the Lorentz group, Moscow, 1958.Google Scholar
  31. [31]
    R. Palais,Seminar on the Atiyah-Singer index theorem, Princeton Univ. Press, 1965.Google Scholar
  32. [32]
    V. Palamodov,Linear differential operators with constant coefficients, Moscow, 1967.Google Scholar
  33. [33]
    M. Naimark,Linear representations of the Lorentz group, Moscow, 1958.Google Scholar
  34. [34]
    D. Quillen,Formal properties of over-determined systems of linear partial differential equations, Thesis, Harvard, 1964.Google Scholar
  35. [35]
    W. Rühl,Complete sets of solutions of linear Lorentz covariant field equations with an infinite number of field components, Comm. Math. Physics, 6 (1967), 312–342.CrossRefzbMATHGoogle Scholar
  36. [36]
    L. Schwartz,Théorie des distributions, édition 1966, Hermann, Paris.Google Scholar
  37. [37]
    —— ——,Théorie des distributions à valeurs vectorielles, Annales Inst. Fourier, 7–8 (1957–58), 1–141 and 1–209.Google Scholar
  38. [38]
    S. Schweber,An introduction to relativistic quantum field theory, Harper-Row, New York, 1962.Google Scholar
  39. [39]
    J. Serre,Algèbres de Lie semisimples complexes, Benjamin, New York, 1966.Google Scholar
  40. [40]
    D. Sims,Lie groups and quantum mechanics, Springer, Berlin, 1968.Google Scholar
  41. [41]
    W. Smoke,Invariant differential operators, Trans. Amer. Math. Soc., 127 (1967), 460–494.CrossRefzbMATHMathSciNetGoogle Scholar
  42. [42]
    W. Sweeney,The D-Neumann problem, Acta Math., 120 (1968), 223–277.CrossRefzbMATHMathSciNetGoogle Scholar
  43. [43]
    F. Treves,Linear partial differential equations with constant cofficients, Gordon-Breach, New York, 1966.Google Scholar

Copyright information

© Nicola Zanichelli Editore 1970

Authors and Affiliations

  • Robert Carroll
    • 1
  1. 1.U.S.A.

Personalised recommendations