Paley Wiener Theorem on a Reductive Lie Group

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-22-2024-62
Published: Apr 8, 2024
Cite as:
Neantien Claudio Zoto Bi, Kangni Kinvi, Paley Wiener Theorem on a Reductive Lie Group, Int. J. Anal. Appl., 22 (2024), 62.

Main Article Content

Neantien Claudio Zoto Bi, Kangni Kinvi

Abstract

Let G be a locally compact group, K a maximal compact subgroup of G and δ on arbitrary class of irreducible unitary representations of K. The spherical Grassmannian Gp,δ is an equivalence class of spherical functions of type δ−positive of height p. In this work, we give an extension of orbital integral with respect to δ, when G is reductive Lie group. Moreover, if the discret serie is not empty, we give an extension of Paley-Wiener theorem using a compact Cartan subgroup of G.

Article Details

References

  1. J. Arthur, A Paley-Wiener Theorem for Real Reductive Groups, Acta Math. 150 (1983), 1–89.
  2. A. Bouaziz, Intégrales Orbitales sur les Groupes de Lie Réductifs, Ann. Sci. École Norm. Sup. 27 (1994), 573–609. https://doi.org/10.24033/asens.1701.
  3. H. Chandra, Harmonic Analysis on Real Reductive Groups, I. J. Funct. Anal. 19 (1975), 104–204.
  4. H. Chandra, Supertempered Distributions on Real Reductive Groups, Stud. App. Math. Adv. Math. 8 (1983), 139–153.
  5. P. Delorme, P. Mezo, A Twisted Invariant Paley-Wiener Theorem for Real Reductive Groups, Duke Math. J. 144 (2008), 341–380. https://doi.org/10.1215/00127094-2008-039.
  6. E.Y.F. N’Da, K. Kangni, On Some Extension of Paley Wiener Theorem, Concrete Oper. 7 (2020), 81–90. https://doi.org/10.1515/conop-2020-0006.
  7. I. Toure, K. Kangni, Spherical Functions of Type δ on Nilpotent Lie Groups, Elec. J. Math. Anal. Appl. 8 (2020), 309–315.
  8. K. Kangni, S.Toure, Transformation de Fourier Sphérique de Type δ, Ann. Math. Blaise Pascal. 3 (1996), 117–133.
  9. K. Kangni, Transformation de Fourier et Représentation Unitaire Sphérique de Type Delta, Thesis, Université d’Abidjan, 1994
  10. K. Kangni, S. Touré, Sur la Grassmannienne Sphérique, Applications aux Groupes de Lie Réductifs, Afr. Math. 18 (2007), 58–64.
  11. Y.Kraidi, K.Kangni, Reproducing Kernel Cartan Subalgebra, Afr. Math. Ann. 8 (2020), 7–15.
  12. C.M. Malanda, K. Kangni, the Quasi-Spherical Transformations, Afr. Math. Ann. 7 (2018), 65–71.
  13. P. Coulibaly, K. Kangni, Sur une Extension du Théorème de Bochner, Afr. Mat. 25 (2012), 411–416. https://doi.org/10.1007/s13370-012-0119-1.
  14. E.P. van den Ban, S. Souaifi, A Comparison of Paley-Wiener Theorems for Real Reductive Lie groups, J. Reine Angew. Math. 2014 (2014), 99–149. https://doi.org/10.1515/crelle-2012-0105.
  15. V.S. Varadarajan, Harmonic Analysis on Real Reductive Groups, Vol. 576, Lecture Notes in Mathematics, SpringerVerlag, Berlin, 1977.
  16. G.Warner, Harmonic Analysis on Semi-Simple Lie Group, Vol. I and Vol. II, Springer, New York, (1972).