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The Plancherel Theorem for a Reductive Symmetric Space

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References

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

    MATH  MathSciNet  Google Scholar 

  2. J. Arthur, A local trace formula, Publ. Math. I.H.E.S. No. 73 (1991), 5–96.

    MATH  MathSciNet  Google Scholar 

  3. E. P. van den Ban, Invariant differential operators on a semisimple symmetric space and finite multiplicities in a Plancherel formula, Ark. Mat. 25 (1987), 175–187.

    MATH  MathSciNet  Google Scholar 

  4. E. P. van den Ban, Asymptotic behaviour of matrix coefficients related to reductive symmetric spaces, Nederl. Akad. Wetensch. Indag. Math. 49 (1987), 225–249.

    MATH  MathSciNet  Google Scholar 

  5. 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 

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

    MATH  MathSciNet  Google Scholar 

  7. E. P. van den Ban, Induced Representations and the Langlands classification, Proc. Edinburgh’ 96. Symposia in Pure Math. AMS.

    Google Scholar 

  8. 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 

  9. E. P. van den Ban, Eisenstein integrals and induction of relations, pp. 487–509 in: Noncommutative Harmonic Analysis, In Honor of Jacques Carmona, P. Delorme & M. Vergne, eds., Progress in Math. 220, Birkhäuser, Boston, 2004.

    Google Scholar 

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

    MathSciNet  Google Scholar 

  11. E. P. van den Ban, M. Flensted-Jensen, H. Schlichtkrull, Harmonic analysis on semisimple symmetric spaces: a survey of some general results, in Representation Theory and Automorphic Forms (Edinburgh, 1996), 191–217, Proc. Sympos. Pure Math., 61, Amer. Math. Soc., Providence, RI, 1997.

    Google Scholar 

  12. E. P. van den Ban, H. Schlichtkrull, Asymptotic expansions and boundary values of eigenfunctions on Riemannian symmetric spaces, J. Reine Angew. Math. 380 (1987), 108–165.

    MathSciNet  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  19. E. P. van den Ban, H. Schlichtkrull, Harmonic Analysis on reductive symmetric spaces, Proc. Third European Congress of Mathematics, 2000.

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  24. M. Berger, Les espaces symétriques non compacts, Ann. Sci. École Norm. Sup. (3) 74 (1957), 85–177.

    Google Scholar 

  25. J. N. Bernstein, On the support of Plancherel measure, J. Geom. Phys. 5 (1988), 663–710.

    Article  MATH  MathSciNet  Google Scholar 

  26. F. V. Bien, D-modules and spherical representations. Mathematical Notes, 39. Princeton University Press, Princeton, NJ, 1990.

    Google Scholar 

  27. N. Bopp, P. Harinck, Formule de Plancherel pour GL(n,C)/U(p, q), J. Reine Angew. Math. 428 (1992), 45–95.

    MathSciNet  Google Scholar 

  28. A. Borel, N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups. Second edition, Mathematical Surveys and Monographs, 67. American Mathematical Society, Providence, RI, 2000.

    Google Scholar 

  29. N. Bourbaki, Lie groups and Lie algebras. Chapters 4–6; Translated from the 1968 French original by Andrew Pressley. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 2002.

    Google Scholar 

  30. F. Bruhat, Sur les représentations induites des groupes de Lie, Bull. Soc. Math. France 84 (1956), 97–205.

    MATH  MathSciNet  Google Scholar 

  31. J.-L. Brylinski, P. Delorme, Vecteurs distributions H-invariants pour les séries principales généralisées d’espaces symétriques réductifs et prolongement méromorphe d’intégrales d’Eisenstein, Invent. Math. 109 (1992), 619–664.

    Article  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  41. P. Delorme, Harmonic analysis on real reductive symmetric spaces, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 545–554. Higher Ed. Press, Beijing, 2002.

    Google Scholar 

  42. J. Dixmier, Les C*-algèbres et leurs représentations. Reprint of the second (1969) edition, Les Grands Classiques Gauthier-Villars. Editions Jacques Gabay, Paris, 1996.

    Google Scholar 

  43. J. Faraut, Distributions sphériques sur les espaces hyperboliques, J. Math. Pures Appl. (9) 58 (1979), 369–444.

    MATH  MathSciNet  Google Scholar 

  44. M. Flensted-Jensen, Spherical functions of a real semisimple Lie group. A method of reduction to the complex case, J. Funct. Anal. 30 (1978), no. 1, 106–146.

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  46. M. Flensted-Jensen, Analysis on non-Riemannian symmetric spaces. CBMS Regional Conference Series in Mathematics, 61, AMS, Providence, RI, 1986.

    Google Scholar 

  47. R. Gangolli, On the Plancherel formula and the Paley-Wiener theorem for spherical functions on semisimple Lie groups, Ann. of Math. 93 (1971), 150–165.

    MATH  MathSciNet  Google Scholar 

  48. S.G. Gindikin, F.I. Karpelevič, Plancherel measure for symmetric Riemannian spaces of non-positive curvature, (Russian) Dokl. Akad. Nauk SSSR 145 (1962), 252–255.

    MathSciNet  Google Scholar 

  49. P. Harinck, Fonctions orbitales sur G/G. Formule d’inversion des intégrales orbitales et formule de Plancherel, J. Funct. Anal. 153 (1998), 52–107.

    MATH  MathSciNet  Google Scholar 

  50. Harish-Chandra, Spherical functions on a semisimple Lie group. I, Amer. J. Math. 80 (1958), 241–310.

    MathSciNet  Google Scholar 

  51. Harish-Chandra, Spherical functions on a semisimple Lie group. II, Amer. J. Math. 80 (1958), 553–613.

    MathSciNet  Google Scholar 

  52. Harish-Chandra, Differential equations and semisimple Lie groups. Unpublished manuscript (1960), in: Collected Papers Vol. 3, Springer-Verlag, New York, 1984.

    Google Scholar 

  53. Harish-Chandra, Discrete series for semisimple Lie groups. I. Construction of invariant eigendistributions, Acta Math. 113 (1965), 241–318.

    MathSciNet  Google Scholar 

  54. Harish-Chandra, Discrete series for semisimple Lie groups. II. Explicit determination of the characters, Acta Math. 116 (1966), 1–111.

    MathSciNet  Google Scholar 

  55. Harish-Chandra, On the theory of the Eisenstein integral, Lecture Notes in Math. Vol. 266, 123–149, Springer-Verlag, New York, 1972.

    Google Scholar 

  56. Harish-Chandra, Harmonic analysis on real reductive groups I. The theory of the constant term, J. Funct. Anal. 19 (1975), 104–204.

    MathSciNet  Google Scholar 

  57. Harish-Chandra, Harmonic analysis on real reductive groups II. Wave packets in the Schwartz space, Invent. Math. 36 (1976), 1–55.

    Article  MathSciNet  Google Scholar 

  58. Harish-Chandra, Harmonic analysis on real reductive groups III. The Maass-Selberg relations and the Plancherel formula, Annals of Math. 104 (1976), 117–201.

    MathSciNet  Google Scholar 

  59. G.J. Heckman, E.M. Opdam, Yang’s system of particles and Hecke algebras, Ann. of Math. 145 (1997), 139–173. Erratum: Ann. of Math. 146 (1997), 749–750.

    MathSciNet  Google Scholar 

  60. S. Helgason, An analogue of the Paley-Wiener theorem for the Fourier transform on certain symmetric spaces, Math. Ann. 165 (1966), 297–308.

    MATH  MathSciNet  Google Scholar 

  61. S. Helgason, A duality for symmetric spaces with applications to group representations, Advances in Math. 5 (1970), 1–154.

    Article  MATH  MathSciNet  Google Scholar 

  62. S. Helgason, A duality for symmetric spaces with applications to group representations. II. Differential equations and eigenspace representations, Advances in Math. 22 (1976), no. 2, 187–219.

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  64. L. Hörmander, Linear Partial Differential Operators, Springer Verlag, Berlin, 1963.

    Google Scholar 

  65. M. Kashiwara, A. Kowata, K. Minemura, K. Okamoto, T. Oshima, T. Tanaka, Eigenfunctions of invariant differential operators on a symmetric space, Ann. of Math. 107 (1978), 1–39.

    MathSciNet  Google Scholar 

  66. A. W. Knapp, E. M. Stein, Intertwining operators for semisimple groups. II, Invent. Math. 60 (1980), 9–84.

    Article  MathSciNet  Google Scholar 

  67. J. A. C. Kolk, V.S. Varadarajan, On the transverse symbol of vectorial distributions and some applications to harmonic analysis, Indag. Math. (N.S.) 7 (1996), 67–96.

    MathSciNet  Google Scholar 

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

    Google Scholar 

  69. T. Matsuki, The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan 31 (1979), 331–357.

    MATH  MathSciNet  Google Scholar 

  70. T. Matsuki, A description of discrete series for semisimple symmetric spaces. II, Representations of Lie groups, Kyoto, Hiroshima, 1986, 531–540, Adv. Stud. Pure Math., 14, Academic Press, Boston, MA, 1988.

    Google Scholar 

  71. C. Moeglin, J.-L. Waldspurger, Spectral Decomposition and Eisenstein Series, Cambridge Tracts in Mathematics, 113. Cambridge University Press, Cambridge, 1995.

    Google Scholar 

  72. V. F. Molčanov, An analog of Plancherel’s formula for hyperboloids, (Russian) Dokl. Akad. Nauk SSSR 183 (1968), 288–291.

    MathSciNet  Google Scholar 

  73. G.D. Mostow, Some new decomposition theorems for semi-simple groups, Mem. Amer. Math. Soc. no. 14, (1955), 31–54.

    MATH  MathSciNet  Google Scholar 

  74. G. Ólafsson, Fourier and Poisson transformation associated to a semisimple symmetric space, Invent. Math. 90 (1987), 605–629.

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  76. T. Oshima, A calculation of c-functions for semisimple symmetric spaces, in Lie Groups and Symmetric Spaces, 307–330, Amer. Math. Soc. Transl. Ser. 2, 210, Amer. Math. Soc., Providence, RI, 2003.

    Google Scholar 

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

    MathSciNet  Google Scholar 

  78. T. Oshima, J. Sekiguchi, Eigenspaces of invariant differential operators on an affine symmetric space, Invent. Math. 57 (1980), 1–81.

    Article  MathSciNet  Google Scholar 

  79. M. Poel, G. van Dijk, The Plancherel formula for the pseudo-Riemannian space SL(n, R)/GL(n − 1, R), Compositio Math. 58 (1986), 371–397.

    MathSciNet  Google Scholar 

  80. W. Rossmann, The structure of semisimple symmetric spaces, Canad. J. Math. 31 (1979), 157–180.

    MATH  MathSciNet  Google Scholar 

  81. W. Rossmann, Analysis on real hyperbolic spaces, J. Funct. Anal. 30 (1978), 448–477.

    Article  MATH  MathSciNet  Google Scholar 

  82. H. Schlichtkrull, Harmonic Analysis on semisimple symmetric spaces. Part II in: G. Heckman, H. Schlichtkrull, Harmonic Analysis and Special Functions on Symmetric Spaces; Persp. in Math., Vol. 16, Acad. Press, San Diego, 1994.

    Google Scholar 

  83. H. Schlichtkrull, Hyperfunctions and Harmonic Analysis on Symmetric Spaces, Progress in Mathematics, Vol. 49. Birkhäuser Boston, Inc., Boston, MA, 1984.

    Google Scholar 

  84. V.S. Varadarajan, Harmonic Analysis on Real Reductive Groups. Lecture Notes in Mathematics, Vol. 576. Springer-Verlag, New York, 1977.

    Google Scholar 

  85. D.A. Vogan, Irreducibility of discrete series representations for semisimple symmetric spaces, in Representations of Lie Groups, Kyoto, Hiroshima, 1986, 191–221, Adv. Stud. Pure Math., Vol. 14, Academic Press, Boston, MA, 1988.

    Google Scholar 

  86. D. A. Vogan, N. R. Wallach, Intertwining operators for real reductive groups, Adv. Math. 82 (1990), 203–243.

    MathSciNet  Google Scholar 

  87. N. R. Wallach, Asymptotic expansions of generalized matrix entries of representations of real reductive groups, in Lie Group Representations, I (College Park, Md., 1982/1983), 287–369, Lecture Notes in Math., Vol. 1024, Springer, Berlin, 1983.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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van den Ban, E.P. (2005). The Plancherel Theorem for a Reductive Symmetric Space. In: Anker, JP., Orsted, B. (eds) Lie Theory. Progress in Mathematics, vol 230. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4426-1_1

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