Skip to main content
SpringerLink
Log in

Eisenstein integrals and induction of relations

  • Chapter
Noncommutative Harmonic Analysis

Part of the book series: Progress in Mathematics ((PM,volume 220))

Abstract

A survey of joint work with Henrik Schlichtkrull on the induction of certain relations among (partial) Eisenstein integrals for the minimal principal series of a reductive symmetric space is given. The application of this principle of induction to the proof of the Fourier inversion formula in [11] and to the proof of the Paley-Wiener theorem in [15] is explained. Finally, the relation with the Plancherel decomposition is discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J. Arthur, A Paley-Wiener theorem for real reductive groups. Acta Math. 150 (1983), 1–89.

    Article  MathSciNet  MATH  Google Scholar 

  2. E.P. van den Ban, The principal series for a reductive symmetric space, I. H-fixed distribution vectors. Ann. scient. Éc. Norm. Sup. 21 (1988), 359–412.

    MATH  Google Scholar 

  3. E.P. van den Ban, The principal series for a reductive symmetric space II. Eisenstein integrals. J. Funct. Anal. 109 (1992), 331–441.

    Article  MathSciNet  MATH  Google Scholar 

  4. E.P. van den Ban, The action of intertwining operators on spherical vectors in the minimal principal series of a reductive symmetric space. Indag. Math. 145 (1997), 317–347.

    Google Scholar 

  5. E.P. van den Ban, J. Carmona and P. Delorme, Paquets d’ondes dans l’espace de Schwartz d’un espace symétrique réductif, J. Funct. Anal. 139 (1996), 225–243.

    Article  MathSciNet  MATH  Google Scholar 

  6. E.P. van den Ban and H. Schlichtkrull, Convexity for invariant differential operators on a semisimple symmetric space. Compos. Math. 89 (1993), 301–313.

    MATH  Google Scholar 

  7. E.P. van den Ban and H. Schlichtkrull, Expansions for Eisenstein integrals on semisimple symmetric spaces. Ark. Mat. 35 (1997), 59–86.

    Article  MathSciNet  MATH  Google Scholar 

  8. E.P. van den Ban and H. Schlichtkrull, Fourier transforms on a semisimple symmetric space. Invent. Math. 130 (1997), 517–574.

    Article  MathSciNet  MATH  Google Scholar 

  9. E.P. van den Ban and H. Schlichtkrull, The most continuous part of the Plancherel decomposition for a reductive symmetric space. Annals Math. 145 (1997), 267–364.

    Article  MATH  Google Scholar 

  10. E.P. van den Ban and H. Schlichtkrull, A residue calculus for root systems. Compositio Math. 123 (2000), 27–72.

    Article  MathSciNet  MATH  Google Scholar 

  11. E.P. van den Ban and H. Schlichtkrull, Fourier inversion on a reductive symmetric space. Acta Math. 182 (1999), 25–85.

    Article  MathSciNet  MATH  Google Scholar 

  12. E.P. van den Ban and H. Schlichtkrull, Analytic families of eigenfunctions on a reductive symmetric space. Represent. Theory 5 (2001), 615–712.

    Article  MathSciNet  MATH  Google Scholar 

  13. E.P. van den Ban and H. Schlichtkrull, The Plancherel decomposition for a reductive symmetric space, I. Spherical functions. arXiv.math.RT/0107063.

    Google Scholar 

  14. E.P. van den Ban and H. Schlichtkrull, The Plancherel decomposition for a reductive symmetric space, II. Representation theory. arXiv.math.RT/0111304.

    Google Scholar 

  15. E.P. van den Ban and H. Schlichtkrull, A Paley-Wiener theorem for reductive symmetric spaces. arXiv.math.RT/0302232.

    Google Scholar 

  16. O.A. Campoli, Paley-Wiener type theorems for rank-1 semisimple Lie groups, Rev. Union Mat. Argent. 29 (1980), 197–221.

    MathSciNet  MATH  Google Scholar 

  17. J. Carmona, Terme constant des fonctions tempérées sur un espace symétrique réductif, J. reine angew. Math. 491 (1997), 17–63.

    MathSciNet  MATH  Google Scholar 

  18. J. Carmona and P. Delorme, Base méromorphe de vecteurs distributions H-invariants pour les séries principales généralisées d’espaces symétriques réductifs: Equation fontionelle. J. Funct. Anal. 122 (1994), 152–221.

    Article  MathSciNet  MATH  Google Scholar 

  19. J. Carmona and P. Delorme, Transformation de Fourier sur l’espace de Schwartz d’un espace symétrique réductif, Invent. Math. 134 (1998), 59–99.

    Article  MathSciNet  MATH  Google Scholar 

  20. W. Casselman, Canonical extensions of Harish-Chandra modules to representations of G. Canad. J. Math. 41 (1989), 385–438.

    Article  MathSciNet  MATH  Google Scholar 

  21. W. Casselman and D. Miličić, Asymptotic behavior of matrix coefficients of admissible representations. Duke Math. J. 49 (1982), 869–930.

    Article  MathSciNet  MATH  Google Scholar 

  22. P. Delorme, Théorème de type Paley-Wiener pour les groupes de Lie semisimples réels avec une seule classe de conjugaison de sous groupes de Cartan. J. Funct. Anal. 47 (1982), 26–63.

    Article  MathSciNet  MATH  Google Scholar 

  23. P. Delorme, Intégrales d’Eisenstein pour les espaces symétriques réductifs: tempérance, majorations. Petite matrice B. J. Funct. Anal. 136 (1994), 422–509.

    Article  MathSciNet  Google Scholar 

  24. P. Delorme, Troncature pour les espaces symétriques réductifs, Acta Math. 179 (1997), 41–77.

    Article  MathSciNet  MATH  Google Scholar 

  25. P. Delorme, Formule de Plancherel pour les espaces symétriques réductifs. Annals Math. 147 (1998), 417–452.

    Article  MathSciNet  MATH  Google Scholar 

  26. M. Flensted-Jensen, Discrete series for semisimple symmetric spaces. Annals Math. 111 (1980), 253–311.

    Article  MathSciNet  MATH  Google Scholar 

  27. P. Harinck, Fonctions orbitales sur GC/GR• Formule d’inversion des integrals orbitales et formule de Plancherel. J. Funct. Anal. 153 (1998), 52–107.

    Article  MathSciNet  MATH  Google Scholar 

  28. Harish-Chandra, On the theory of the Eisenstein integral. Lecture Notes in Math. 266, 123–149, Springer-Verlag, New York, 1972. Also: Collected Papers, Vol 4, pp. 47–73, Springer-Verlag, New York, 1984.

    Google Scholar 

  29. Harish-Chandra, Harmonic analysis on real reductive groups I. The theory of the constant term. J. Funct. Anal. 19 (1975), 104–204. Also: Collected Papers, Vol 4, pp. 102–202, Springer-Verlag, New York, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  30. Harish-Chandra, Harmonic analysis on real reductive groups II. Wave packets in the Schwartz space. Invent. Math. 36 (1976), 1–55. Also: Collected papers, Vol 4, pp. 203–257, Springer-Verlag, New York, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  31. Harish-Chandra, Harmonic analysis on real reductive groups III. The Maass-Selberg relations and the Plancherel formula. Annals of Math. 104 (1976), 117–201. Also: Collected Papers, Vol 4, pp. 259–343, Springer-Verlag, New York, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  32. S. Helgason, Groups and Geometric Analysis. Academic Press, Orlando, FL, 1984.

    MATH  Google Scholar 

  33. R.P. Langlands, On the functional equations satisfied by Eisenstein series. Springer Lecture Notes 544, Springer-Verlag, Berlin, 1976.

    Google Scholar 

  34. T. Oshima, A realization of semisimple symmetric spaces and construction of boundary value maps. Adv. Studies in Pure Math. 14 (1988), 603–650.

    MathSciNet  Google Scholar 

  35. T. Oshima and T. Matsuki, A description of discrete series for semisimple symmetric spaces. Adv. Stud. Pure Math. 4 (1984), 331–390.

    MathSciNet  Google Scholar 

  36. N.R. Wallach, Real Reductive Groups I. Academic Press, Inc., San Diego, 1988.

    Google Scholar 

  37. N.R. Wallach, Real Reductive Groups II. Academic Press, Inc., San Diego, 1992.

    Google Scholar 

  38. D.P. Zhelobenko, Harmonic Analysis of functions on semisimple Lie groups. II. lzv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 1255–1295.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisch Instituut, Universiteit Utrecht, PO Box 80 010, 3508, TA, Utrecht, Netherlands

    E. P. van den Ban

Authors
  1. E. P. van den Ban
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut de Mathématiques de Luminy, UPR 9016 CNRS, 13288, Marseille Cedex 9, France

    Patrick Delorme

  2. Centre de Mathématiques, Ecole Polytechnique, 91128, Palaiseau Cedex, France

    Michèle Vergne

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer Science+Business Media New York

About this chapter

Cite this chapter

van den Ban, E.P. (2004). Eisenstein integrals and induction of relations. In: Delorme, P., Vergne, M. (eds) Noncommutative Harmonic Analysis. Progress in Mathematics, vol 220. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8204-0_18

Download citation

  • DOI: https://doi.org/10.1007/978-0-8176-8204-0_18

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6489-7

  • Online ISBN: 978-0-8176-8204-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation