Hiroshima Mathematical Journal

The Fourier transform of the Schwartz space on a semisimple Lie group

Masaaki Eguchi

Full-text: Open access

Article information

Source
Hiroshima Math. J., Volume 4, Number 1 (1974), 133-209.

Dates
First available in Project Euclid: 21 March 2008

Permanent link to this document
https://projecteuclid.org/euclid.hmj/1206137157

Digital Object Identifier
doi:10.32917/hmj/1206137157

Mathematical Reviews number (MathSciNet)
MR0357686

Zentralblatt MATH identifier
0286.43007

Subjects
Primary: 22E30: Analysis on real and complex Lie groups [See also 33C80, 43-XX]
Secondary: 43A30: Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.

Citation

Eguchi, Masaaki. The Fourier transform of the Schwartz space on a semisimple Lie group. Hiroshima Math. J. 4 (1974), no. 1, 133--209. doi:10.32917/hmj/1206137157. https://projecteuclid.org/euclid.hmj/1206137157


Export citation

References

  • [1] J. G. Arthur, Harmonic analysis of tempereddistributions on semisimple Lie groups of real rank one, Ph. D. Thesis, Yale University, 1970.
  • [2] F. Bruhat, Sur les representations induits des groupesde Lie, Bull. Soc. Math. France, 84 (1956), 97-205.
  • [3] M. Eguchi, Harmonic analysis on some types of semisimpleLie groups, (to appear in Proc. Japan Acad.).
  • [4] M. Eguchi and K. Okamoto, The Fourier transform of the Schwartz space on a symmetric space, (to appear in Proc. Japan Acad.).
  • [5] M. Eguchi, M. Hashizume and K. Okamoto, The Paley- Wiener theorem for distributions on symmetric spaces,Hiroshima Math. J., 3 (1973).
  • [6(a)] Harish-Chandra, Representations of a semisimple Lie group on a Banach spaceI, Trans. Amer. Math. Soc, 75 (1953), 185-243.
  • [6(b)] Harish-Chandra, Representations of semisimple Lie groupsII, Trans. Amer. Math. Soc, 76 (1954), 26-65.
  • [6(c)] Harish-Chandra, Representations of semisimple Lie groupsIII, Trans. Amer. Math. Soc, 76 (1954), 234- 253.
  • [6(d)] Harish-Chandra, Representations of semisimple Lie groups VI, Amer. J. Math., 78 (1956), 564-628.
  • [6(e)] Harish-Chandra, The characters of semisimple Lie groups, Trans. Amer. Math. Soc, 83 (1956), 98-163.
  • [6(f)] Harish-Chandra, Fourier transformson a semisimple Lie algebraI, Amer. J. Math., 79 (1957), 193-257.
  • [6(g)] Harish-Chandra, A formula for semisimple Lie groups, Amer. J. Math., 79 (1957), 733-760.
  • [6(h)] Harish-Chandra, Spherical functions on a semisimple Lie group I, Amer. J. Math., 80 (1958), 241-310.
  • [6(i)] Harish-Chandra, Spherical functions on a semisimple Lie group II, Amer. J. Math., 80 (1958), 553--613.
  • [6(j)] Harish-Chandra, Someresaults on an invariant integral on a semisimple Lie algebra,Ann. of Math., 80 (1964), 551-593.
  • [6(k)] Harish-Chandra, Invariant eigendistributions on a semisimple Lie group,Trans. Amer. Math. Soc, 119(1965), 457-508.
  • [6(1)] Harish-Chandra, Two theorems on semisimple Lie groups,Ann. of Math., 83 (1966), 74-128.
  • [6(m)] Harish-Chandra, Discrete seriesfor semisimple Lie groups II, Acta Math., 116 (1966), 1--111.
  • [6(n)] Harish-Chandra, Harmonic analysis on semisimple Lie groups,Bull. Amer. Math. Soc, 76 (1970), 529-551.
  • [6(o)] Harish-Chandra, On the theory of the Eisenstein integral, Lecture Notes in Mathematics, Springer-Verlag, 266 (1971), 123-149.
  • [7] S. Helgason, (a) Differential geometry and symmetric spaces, Academic Press, New York, 1962.
  • [7(b)] S. Helgason, Fundamental solutions of invariant differential operators on symmetric spaces, Amer. J. Math., 86 (1964), 565-601.
  • [7(c)] S. Helgason, A duality for symmetric spaces with applications to group representations, Advances in Math., 5 (1970), 1-154.
  • [7(d)] S. Helgason, Paley- Wiener theorems and surjectivityof invariant differential operators on symmetric spaces and Lie groups,Bull. Amer. Math. Soc, 79 (1973), 129-132.
  • [7(e)] S. Helgason, Function theory on symmetric spaces, Lecture notes, Summer institute on harmonic analysis on homogeneous space, 1972 (preprint).
  • [8] N. Jacobson, Lie algebra, Interscience, 1962.
  • [9] G. W. Mackey, (a) Induced representations on locally compactgroup I, Ann. of Math., 55 (1952), 101-139.
  • [9(b)] G. W. Mackey, Borel structure in groups and their duals, Trans. Amer. Math. Soc, 85 (1957), 134-165.
  • [10] G. D. Mostow, The extensibility of local Lie groups of transformations and groups on surfaces, Ann. of Math., 52 (1950), 606-636.
  • [11] H. Ozeki and M. Wakimoto, On polarizations of certain homogeneous spaces Hiroshima Math. J., 2.(1973),
  • [12] P. J. Sally and G. Warner, The Fourier transform of invariant distributions, Lecture Notes in Mathematics, Springer-Verlag, 266 (1971), 297-320.
  • [13] L. Schwartz, Theoriedes distributions, Hermann, Paris, 1967.
  • [14] M. Sugiura, Conjugate classes of Cartan subalgebras in realsemisimple Lie algebras,J. Math. Soc, Japan, 11 (1959), 374-434.
  • [15] P. Trombi, On the continuous spectrum for a semisimple Lie group, Lecture notes, Summer institute on harmonic analysis on homogeneous spaces, 1972 (preprint).
  • [16a] G. Warner, Harmonic analysis on semi-simple Lie groups I, Springer-Verlag, Berlin-Heidelberg-New York, 1972.
  • [16b] G. Warner, Harmonic analysis onsemi-simple Lie groups II, Springer-Verlag, Berlin-Heidelberg-New York, 1972.