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Multipliers and a Paley-Wiener Theorem for Real Reductive Groups

  • James Arthur
Part of the Progress in Mathematics book series (PM, volume 40)

Abstract

The classical Paley-Wiener theorem is a description of the image of Cc (IR) under Fourier transform. The Fourier transform
$$\rm{\hat f (\wedge)\ =\ \int^\infty_{-\infty}\ f(x)e^{\wedge x}\ dx}$$
is defined a priori for purely imaginary numbers ⋀, but if f has compact support \(\hat f\) will extend to an entire function on the complex plane.

Keywords

Entire Function Parabolic Subgroup Eisenstein Series Spherical Function Discrete Series 
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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References

  1. [1]
    Arthur, J., A Paley-Wiener theorem for real reductive groups, to appear in Acta Math.Google Scholar
  2. [2]
    Campoli, O., The complex Fourier transform on rank one semi-simple Lie groups, Thesis, Rutgers University, 1977.Google Scholar
  3. [3]
    Delorme, P., Théorème de type Paley-Wiener pour les groupes de Lie semisimples réels avec une seule classe de conjugaison de sous-groupes de Cartan, preprint.Google Scholar
  4. [4](a)
    Ehrenpreis, L., and Mautner, F. I., (a) Some properties of the Fourier transform on semisimple Lie groups I, Ann. of Math. 61 (1955), 406–439.CrossRefGoogle Scholar
  5. (b).
    Some properties of the Fourier transform on semisimple Lie groups, II, T.A.M.S. 84 (1957), 1–55.Google Scholar
  6. [5]
    Gangolli, R., On the Plancherel formula and the Paley-Wiener theorem for spherical functions on semisimple Lie groups, Ann. of Math. 93 (1971), 150–165.CrossRefGoogle Scholar
  7. [6](a)
    Harish-Chandra, (a) Harmonic analysis on real reductive groups I, J. Funct. Anal. 19 (1975), 104–204.CrossRefGoogle Scholar
  8. (b).
    Harmonic analysis on real reductive groups III, Ann. of Math. 104 (1976), 117–201.CrossRefGoogle Scholar
  9. [7](a)
    Helgason, S., An analog of the Paley-Wiener theorem for the Fourier transform on certain symmetric spaces, Math. Ann. 165 (1966), 297–308.CrossRefGoogle Scholar
  10. (b).
    A duality for symmetric spaces with applications to group representations, Advances in Math. 5 (1970), 1–154.CrossRefGoogle Scholar
  11. (c).
    The surjectivity of invariant differential operators on symmetric spaces I, Ann. of Math. 98 (1973), 451–479.CrossRefGoogle Scholar
  12. (d).
    A duality for symmetric spaces with applications to group representations II. Differential equations and eigenspace representations, Advances in Math. 22 (1976), 187–219.CrossRefGoogle Scholar
  13. [8](a)
    Kawazoe, T., An analogue of Paley-Wiener theorem on SU(2,2), Tokyo J. Math.Google Scholar
  14. (b).
    An analogue of Paley-Wiener theorem on semi-simple Lie groups and functional equations for Eisenstein integrals, preprint.Google Scholar
  15. [9](a)
    Langlands, R. P., Eisenstein series, Proc. Sympos. Pure Math., vol. 9, Amer. Math. Soc, Providence, R.I. (1966), 235–252.Google Scholar
  16. (b).
    On the functional equations satisfied by Eisenstein series, Lecture Notes in Math., 544 (1976).Google Scholar
  17. [10]
    Zelobenko, D. P., Harmonic analysis on complex semisimple Lie groups, Proc. Int. Cong. Math., Vancouver, 1974, Vol. II, 129–134.Google Scholar

Copyright information

© Birkhäuser Boston, Inc. 1983

Authors and Affiliations

  • James Arthur

There are no affiliations available

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