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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. Arthur, A Paley-Wiener theorem for real reductive groups. Acta Math. 150 (1983), 1–89.
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.
E.P. van den Ban, The principal series for a reductive symmetric space II. Eisenstein integrals. J. Funct. Anal. 109 (1992), 331–441.
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.
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.
E.P. van den Ban and H. Schlichtkrull, Convexity for invariant differential operators on a semisimple symmetric space. Compos. Math. 89 (1993), 301–313.
E.P. van den Ban and H. Schlichtkrull, Expansions for Eisenstein integrals on semisimple symmetric spaces. Ark. Mat. 35 (1997), 59–86.
E.P. van den Ban and H. Schlichtkrull, Fourier transforms on a semisimple symmetric space. Invent. Math. 130 (1997), 517–574.
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.
E.P. van den Ban and H. Schlichtkrull, A residue calculus for root systems. Compositio Math. 123 (2000), 27–72.
E.P. van den Ban and H. Schlichtkrull, Fourier inversion on a reductive symmetric space. Acta Math. 182 (1999), 25–85.
E.P. van den Ban and H. Schlichtkrull, Analytic families of eigenfunctions on a reductive symmetric space. Represent. Theory 5 (2001), 615–712.
E.P. van den Ban and H. Schlichtkrull, The Plancherel decomposition for a reductive symmetric space, I. Spherical functions. arXiv.math.RT/0107063.
E.P. van den Ban and H. Schlichtkrull, The Plancherel decomposition for a reductive symmetric space, II. Representation theory. arXiv.math.RT/0111304.
E.P. van den Ban and H. Schlichtkrull, A Paley-Wiener theorem for reductive symmetric spaces. arXiv.math.RT/0302232.
O.A. Campoli, Paley-Wiener type theorems for rank-1 semisimple Lie groups, Rev. Union Mat. Argent. 29 (1980), 197–221.
J. Carmona, Terme constant des fonctions tempérées sur un espace symétrique réductif, J. reine angew. Math. 491 (1997), 17–63.
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.
J. Carmona and P. Delorme, Transformation de Fourier sur l’espace de Schwartz d’un espace symétrique réductif, Invent. Math. 134 (1998), 59–99.
W. Casselman, Canonical extensions of Harish-Chandra modules to representations of G. Canad. J. Math. 41 (1989), 385–438.
W. Casselman and D. Miličić, Asymptotic behavior of matrix coefficients of admissible representations. Duke Math. J. 49 (1982), 869–930.
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.
P. Delorme, Intégrales d’Eisenstein pour les espaces symétriques réductifs: tempérance, majorations. Petite matrice B. J. Funct. Anal. 136 (1994), 422–509.
P. Delorme, Troncature pour les espaces symétriques réductifs, Acta Math. 179 (1997), 41–77.
P. Delorme, Formule de Plancherel pour les espaces symétriques réductifs. Annals Math. 147 (1998), 417–452.
M. Flensted-Jensen, Discrete series for semisimple symmetric spaces. Annals Math. 111 (1980), 253–311.
P. Harinck, Fonctions orbitales sur GC/GR• Formule d’inversion des integrals orbitales et formule de Plancherel. J. Funct. Anal. 153 (1998), 52–107.
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.
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.
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.
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.
S. Helgason, Groups and Geometric Analysis. Academic Press, Orlando, FL, 1984.
R.P. Langlands, On the functional equations satisfied by Eisenstein series. Springer Lecture Notes 544, Springer-Verlag, Berlin, 1976.
T. Oshima, A realization of semisimple symmetric spaces and construction of boundary value maps. Adv. Studies in Pure Math. 14 (1988), 603–650.
T. Oshima and T. Matsuki, A description of discrete series for semisimple symmetric spaces. Adv. Stud. Pure Math. 4 (1984), 331–390.
N.R. Wallach, Real Reductive Groups I. Academic Press, Inc., San Diego, 1988.
N.R. Wallach, Real Reductive Groups II. Academic Press, Inc., San Diego, 1992.
D.P. Zhelobenko, Harmonic Analysis of functions on semisimple Lie groups. II. lzv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 1255–1295.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights 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