Advertisement

Israel Journal of Mathematics

, Volume 199, Issue 1, pp 45–111 | Cite as

Smooth Fréchet globalizations of Harish-Chandra modules

  • Joseph Bernstein
  • Bernhard Krötz
Article

Abstract

We give an alternative proof of the Casselman-Wallach globalization theorem. The approach is based on lower bounds for matrix coefficients on a reductive group.

Keywords

Topological Vector Space Eisenstein Series Smooth Vector Discrete Series Sobolev Norm 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    E. van den Ban and P. Delorme, Quelques propriétés des représentations sphériques pour les espaces symétriques réductifs, Journal of Functional Analysis 80 (1988), 284–307.CrossRefMATHMathSciNetGoogle Scholar
  2. [2]
    J. Bernstein, Analytic continuation of generalized functions with respect to a parameter, Funkcional’nyi Analiz i ego Priloženija 6 (1972), 26–40.Google Scholar
  3. [3]
    F. Bruhat, Sur les représentations induites des groupes de Lie, Bulletin de la Société Mathématique de France 84 (1956), 97–205.MATHMathSciNetGoogle Scholar
  4. [4]
    J.-L. Brylinski and P. Delorme, Vecteurs distributions H-invariants pour les séries principales généralisées d’espaces symétriques réductifs et prolongement méromorphe d’intégrales d’Eisenstein, Inventiones Mathematicae 109 (1992), 619–664.CrossRefMATHMathSciNetGoogle Scholar
  5. [5]
    W. Casselman, Canonical extensions of Harish-Chandra modules to representations of G, Canadian Journal of Mathematics 41 (1989), 385–438.CrossRefMATHMathSciNetGoogle Scholar
  6. [6]
    J. Dixmier and P. Malliavin, Factorisations de fonctions et de vecteurs indéfiniment différentiables, Bulletin des Sciences Mathématiques 102 (1978), 307–330.MathSciNetGoogle Scholar
  7. [7]
    L. Garding, Vectors analytiques dans le représentations des groupes de Lie, Bulletin de la Société Mathématique de France 88 (1960), 73–93.MATHMathSciNetGoogle Scholar
  8. [8]
    H. Gimperlein, B. Krötz and C. Lienau, Analytic factorization of Lie group representations, Journal of Functional Analysis 262 (2012), 667–681.CrossRefMATHMathSciNetGoogle Scholar
  9. [9]
    R. Hamilton, The inverse function theorem of Nash and Moser, Bulletin of the American Mathematical Society 7 (1982), 65–222.CrossRefMATHGoogle Scholar
  10. [10]
    L. Hörmander, Linear Partial Differential Operators, Die Grundlehren der mathematischen Wissenschaften, Vol. 116, Springer-Verlag, New York-Heidelberg, 1963.CrossRefMATHGoogle Scholar
  11. [11]
    A. W. Knapp, Representation Theory of Semisimple Groups. An Overview Based on Examples, Reprint of the 1986 original, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2001.MATHGoogle Scholar
  12. [12]
    V. Milman and G. Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Mathematics, Vol. 1200, Springer-Verlag, Berlin, 1986.MATHGoogle Scholar
  13. [13]
    W. Schmid, Boundary value problems for group invariant differential equations, in The Mathematical Heritage of Élie Cartan (Lyon, 1984), Astérisque Numéro Hors Série, (1985), 311–321.Google Scholar
  14. [14]
    W. Soergel, An irreducible not admissible Banach representation of SL(2, R), Proceedings of the American Mathematical Society 104 (1988), 1322–1324.MATHMathSciNetGoogle Scholar
  15. [15]
    F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, 1967.MATHGoogle Scholar
  16. [16]
    N. Wallach, Asymptotic expansions of generalized matrix entries of representations of real reductive groups, in Lie Group Representations, I (College Park, Md., 1982/1983), Lecture Notes in Mathematics, Vol. 1024, Springer, Berlin, 1983, pp. 287–369.CrossRefGoogle Scholar
  17. [17]
    N. Wallach, Real Reductive Groups I, Academic Press, New York, 1988.MATHGoogle Scholar
  18. [18]
    N. Wallach, Real Reductive Groups II, Academic Press, New York, 1992.MATHGoogle Scholar
  19. [19]
    G. Warner, Harmonic Analysis on Semi-simple Lie Groups. I, Die Grundlehren der mathematischen Wissenschaften, Vol. 188, Springer-Verlag, New York-Heidelberg, 1972.CrossRefMATHGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2014

Authors and Affiliations

  1. 1.School of Mathematical SciencesTel Aviv University Ramat AvivTel AvivIsrael
  2. 2.Institut für MathematikUniversität PaderbornPaderbornGermany

Personalised recommendations