Skip to main content

Applications of Representation Theory to Harmonic Analysis of Lie Groups (and Vice Versa)

  • Chapter
Representation Theory and Complex Analysis

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1931))

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight 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.-Ph. Anker, Lp Fourier multipliers on Riemannian symmetric spaces of the noncompact type, Ann. of Math. 132 (1990), 597-628.

    Article  MathSciNet  Google Scholar 

  2. J.-Ph. Anker, The spherical Fourier transform of rapidly decreasing functions. A simple proof of a characterization due to Harish-Chandra, Helgason, Trombi, and Varadarajan, J. Funct. Anal. 96 (1991), 331-349.

    Article  MATH  MathSciNet  Google Scholar 

  3. J.-Ph. Anker, Sharp estimates for some functions of the Laplacian on noncom-pact symmetric spaces, Duke Math. J. 65 (1992), 257-297.

    Article  MATH  MathSciNet  Google Scholar 

  4. J.-Ph. Anker and L. Ji, Heat kernel and Green function estimates on noncom-pact symmetric spaces. I, Geom. Funct. Anal. 9 (1999), 1035-1091.

    Article  MATH  MathSciNet  Google Scholar 

  5. J.-Ph. Anker and L. Ji, Heat kernel and Green function estimates on noncom-pact symmetric spaces. II, pages 1-9 in: Topics in probability and Lie groups: boundary theory. CMS Conf. Proc. 28. Amer. Math. Soc., Providence, RI, 2001.

    Google Scholar 

  6. J.-Ph. Anker and P. Ostellari, The heat kernel on noncompact symmetric spaces, pages 27-46 in: Lie groups and symmetric spaces. Volume in mem-ory of F. I. Karpelevitch; S.G. Gindikin ed. Amer. Math. Soc. Transl. Ser. 2, 210. Amer. Math. Soc., Providence, RI, 2003.

    Google Scholar 

  7. F. Astengo, M. Cowling and B. Di Blasio, The Cayley transform and uniformly bounded representations, J. Funct. Anal. 213 (2004), 241-269.

    Article  MATH  MathSciNet  Google Scholar 

  8. A.D. Banner, Some properties of boundaries of symmetric spaces of rank one, Geom. Dedicata 88 (2001), 113-133.

    Article  MATH  MathSciNet  Google Scholar 

  9. M.E.B. Bekka, On uniqueness of invariant means, Proc. Amer. Math. Soc. 126 (1998),507-514.

    Article  MATH  MathSciNet  Google Scholar 

  10. M.E.B. Bekka and M. Cowling, Some unitary representations of G(K ) for a simple algebraic group over a field K, Math. Z. 241 (2002), 731-741.

    Article  MATH  MathSciNet  Google Scholar 

  11. M.E.B. Bekka and M. Cowling, Addendum to “Some unitary representations of G(K ) for a simple algebraic group over a field K ”, in preparation.

    Google Scholar 

  12. A. Borel and H. Garland, Laplacian and the discrete spectrum of an arithmetic group, Amer. J. Math. 105 (1983), 309-335.

    Article  MATH  MathSciNet  Google Scholar 

  13. T. Branson, G. Ólafsson and H. Schlichtkrull, Huyghens’ principle in Rie- mannian symmetric spaces, Math. Ann. 301 (1995), 445-462.

    Article  MATH  MathSciNet  Google Scholar 

  14. K.S. Brown, Buildings. Springer-Verlag, Berlin New York, 1989.

    MATH  Google Scholar 

  15. M. Burger, Kazhdan constants for SL(3, Z), J. reine angew. Math. 413 (1991), 36-67.

    MATH  MathSciNet  Google Scholar 

  16. M. Burger, J.-S. Li and P. Sarnak, Ramanujan duals and automorphic spec- trum, Bull. Amer. Math. Soc. (N.S.) 26 (1992), 253-257.

    Article  MATH  MathSciNet  Google Scholar 

  17. M. Burger and P. Sarnak, Ramanujan duals. II, Invent. Math. 106 (1991), 1-11.

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. O.A. Chalykh and A.P. Veselov, Integrability and Huygens’ principle on sym-metric spaces, Comm. Math. Phys. 178 (1996), 311-338.

    Article  MATH  MathSciNet  Google Scholar 

  20. J. Cheeger, M. Gromov, and M.E. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom. 17 (1982), 15-53.

    MATH  MathSciNet  Google Scholar 

  21. I. Cherednik, Macdonald’s evaluation conjectures and difference Fourier trans-form, Invent. Math. 122 (1995), 119-145.

    Article  MATH  MathSciNet  Google Scholar 

  22. I. Cherednik, Double affine Hecke algebras and Macdonald’s conjectures, Ann. of Math. 141 (1995), 191-216.

    Article  MATH  MathSciNet  Google Scholar 

  23. J.-L. Clerc and E.M. Stein, Lp multipliers for noncompact symmetric spaces, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 3911-3912.

    Article  MATH  MathSciNet  Google Scholar 

  24. J.-L. Clerc, Transformation de Fourier sphérique des espaces de Schwartz, J. Funct. Anal. 37 (1980), 182-202.

    Article  MATH  MathSciNet  Google Scholar 

  25. L. Clozel, Démonstration de la conjecture τ, Invent. Math. 151 (2003), 297-328.

    Article  MATH  MathSciNet  Google Scholar 

  26. L. Clozel, H. Oh and E. Ullmo, Hecke operators and equidistribution of Hecke points, Invent. Math. 144 (2001), 327-351.

    Article  MATH  MathSciNet  Google Scholar 

  27. T. Coulhon, Noyau de la chaleur et discrétisation d’une variété riemannienne, Israel J. Math. 80 (1992), 289-300.

    MATH  MathSciNet  Google Scholar 

  28. T. Coulhon and L. Saloff-Coste, Variétés riemanniennes isométriques à l’infini, Rev. Mat. Iberoamericana 11 (1995), 687-726.

    MATH  MathSciNet  Google Scholar 

  29. M. Cowling, The Kunze-Stein phenomenon, Ann. of Math. (2) 107 (1978), 209-234.

    Article  MathSciNet  Google Scholar 

  30. M. Cowling, Sur les coefficients des représentations unitaires des groupes de Lie simples, pages 132-178 in: Analyse harmonique sur les groupes de Lie II (Sém. Nancy-Strasbourg 1976-1978). Lecture Notes in Math. 739. Springer, Berlin, 1979.

    Google Scholar 

  31. M. Cowling, Unitary and uniformly bounded representations of some simple Lie groups, pages 49-128 in: Harmonic Analysis and Group Representations (C.I.M.E. II ciclo 1980). Liguori, Naples, 1982.

    Google Scholar 

  32. M. Cowling, Herz’s “principe de majoration” and the Kunze-Stein phenom- enon, pages 73-88 in: Harmonic analysis and number theory, Montreal 1996. S.W. Drury and M. Ram Murty (eds), CMS Conf. Proc. 21. Amer. Math. Soc., 1997.

    Google Scholar 

  33. M. Cowling, Measure theory and automorphic representations,, to appear. Bull. Kerala Math. Assoc. 3 (Special issue) (2006), 139-153.

    Google Scholar 

  34. M. Cowling and U. Haagerup, Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. Math. 96 (1989), 507-549.

    Article  MathSciNet  Google Scholar 

  35. M. Cowling, U. Haagerup and R.E. Howe, Almost L2 matrix coefficients, J. reine angew. Math. 387 (1988), 97-100.

    MATH  MathSciNet  Google Scholar 

  36. M. Cowling, S. Giulini and S. Meda, Lp -Lq estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces I, Duke Math. J. 72(1993),109-150.

    Article  MATH  MathSciNet  Google Scholar 

  37. M. Cowling, S. Giulini and S. Meda, Lp -Lq -estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces II, J. Lie Theory 5(1995),1-14.

    MATH  MathSciNet  Google Scholar 

  38. M. Cowling, S. Giulini and S. Meda, Lp -Lq -estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces III, Annales Inst. Fourier (Grenoble) 51 (2001), 1047-1069.

    MATH  MathSciNet  Google Scholar 

  39. M. Cowling, S. Giulini and S. Meda, Oscillatory multipliers related to the wave equation on noncompact symmetric spaces, J. London Math. Soc. 66 (2002), 691-709.

    Article  MATH  MathSciNet  Google Scholar 

  40. M. Cowling, S. Meda and A.G. Setti, Estimates for functions of the Laplace operator on homogeneous trees, Trans. Amer. Math. Soc. 352 (2000), 4271-4298.

    Article  MATH  MathSciNet  Google Scholar 

  41. M. Cowling, S. Meda and A.G. Setti, An overview of harmonic analysis on the group of isometries of a homogeneous tree, Expositiones Math. 16 (1998), 385-423.

    MATH  MathSciNet  Google Scholar 

  42. M. Cowling and A. Nevo, Uniform estimates for spherical functions on complex semisimple Lie groups, Geom. Funct. Anal. 11 (2001), 900-932.

    Article  MATH  MathSciNet  Google Scholar 

  43. C.W. Curtis and I. Reiner, Representation Theory of Finite Groups and Asso-ciative Algebras. Pure and Applied Mathematics 11. Interscience, New York, 1962.

    Google Scholar 

  44. P. Diaconis and L. Saloff-Coste, Random walks on finite groups: a survey of analytic techniques, pages 44-75 in: Probability Measures on Groups and Re-lated Structures, XI (Oberwolfach, 1994). World Sci. Publishing, River Edge, NJ, 1995.

    Google Scholar 

  45. A.H. Dooley, Heisenberg-type groups and intertwining operators, J. Funct. Anal. 212 (2004), 261-286.

    Article  MATH  MathSciNet  Google Scholar 

  46. P. Eymard and N. Lohoué, Sur la racine carrée du noyau de Poisson dans les espaces symétriques et une conjecture de E. M. Stein, Ann. Sci. E´cole Norm. Sup. (4) 8 (1975), 179-188.

    MATH  Google Scholar 

  47. A. Figà-Talamanca and C. Nebbia, Harmonic Analysis and Representation Theory for Groups Acting on Homogeneous Trees. London Mathematical Soci-ety Lecture Notes 162. Cambridge University Press, Cambridge, 1991.

    Google Scholar 

  48. M. Flensted-Jensen, Spherical functions of a real semisimple Lie group. A method of reduction to the complex case, J. Funct. Anal. 30 (1978), 106-146.

    Article  MATH  MathSciNet  Google Scholar 

  49. 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.

    Article  MathSciNet  Google Scholar 

  50. R. Gangolli and V.S. Varadarajan, Harmonic Analysis of Spherical Functions on Real Reductive Groups. Ergeb. Math. Grenzgeb. 101. Springer-Verlag, 1988.

    Google Scholar 

  51. I.M. Gel’fand, M.I. Graev and I.I. Pyatetskii-Shapiro, Representation The- ory and Automorphic Functions. Translated from the Russian by K.A. Hirsch. W.B. Saunders Co., Philadelphia, PA, 1969.

    Google Scholar 

  52. H. Gunawan, A generalization of maximal functions on compact semisimple Lie groups, Pacific J. Math. 156 (1992), 119-134.

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  55. Harish-Chandra, Harmonic analysis on real reductive groups. III. The Maass-Selberg relations and the Plancherel formula, Ann. of Math. (2) 104 (1976), 117-201.

    Article  MathSciNet  Google Scholar 

  56. P. de la Harpe and A. Valette, La propriété (T ) de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger). Astérisque 175 1989.

    Google Scholar 

  57. W. Hebisch, The subalgebra of L1 (AN ) generated by the Laplacian, Proc. Amer. Math. Soc. 117 (1993), 547-549.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  59. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces. Pure and Applied Math. Academic Press, New York, 1978.

    Google Scholar 

  60. S. Helgason, Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators and Spherical Functions. Pure and Applied Math. Aca-demic Press, New York, 1984.

    Google Scholar 

  61. S. Helgason, Wave equations on homogeneous spaces, pages 254-287 in: Lie Group Representations III (College Park, Md., 1982/1983). Lecture Notes in Math. 1077. Springer, Berlin-New York, 1984.

    Google Scholar 

  62. S. Helgason, Geometric Analysis on Symmetric Spaces. Mathematical Surveys and Monographs 39. American Mathematical Society, Providence, RI, 1994.

    Google Scholar 

  63. C.S. Herz, Sur le phénomène de Kunze-Stein, C. R. Acad. Sci. Paris (Série A) 271 (1970), 491-493.

    MATH  MathSciNet  Google Scholar 

  64. B. Hoogenboom, Spherical functions and invariant differential operators on complex Grassmann manifolds, Ark. Mat. 20 (1982), 69-85.

    Article  MATH  MathSciNet  Google Scholar 

  65. R.E. Howe, On a notion of rank for unitary representations of the classi-cal groups, pages 223-331 in: Harmonic Analysis and Group Representations (C.I.M.E. II ciclo 1980). Liguori, Naples, 1982.

    Google Scholar 

  66. A. Ionescu, An endpoint estimate for the Kunze-Stein phenomenon and related maximal operators, Ann. of Math. 152 (2000), 259-275.

    Article  MATH  MathSciNet  Google Scholar 

  67. K.D. Johnson and N.R. Wallach, Composition series and intertwining operators for the spherical principal series. I, Trans. Amer. Math. Soc. 229 (1977), 137-173.

    Article  MATH  MathSciNet  Google Scholar 

  68. P. Julg, La conjecture de Baum-Connes à coefficients pour le groupe Sp(n, 1), C. R. Math. Acad. Sci. Paris 334 (2002), 533-538.

    MATH  MathSciNet  Google Scholar 

  69. M. Kanai, Rough isometries, and combinatorial approximations of geometries of noncompact Riemannian manifolds, J. Math. Soc. Japan 37 (1985), 391-413.

    Article  MATH  MathSciNet  Google Scholar 

  70. D.A. Kazhdan, Connection of the dual space of a group with the structure of its closed subgroups, Funct. Anal. Appl. 1 (1967), 63-65.

    MATH  Google Scholar 

  71. A.W. Knapp, Representation Theory of Semisimple Groups. An Overview Based on Examples. Princeton Mathematical Series 36. Princeton University Press, Princeton, N.J., 1986.

    Google Scholar 

  72. T.H. Koornwinder, Jacobi functions and analysis on noncompact semisimple Lie groups, pages 1-85 in: Special Functions: Group Theoretical Aspects and Applications. R.A. Askey et al. (eds). Reidel, 1984.

    Google Scholar 

  73. B. Kostant, On the existence and irreducibility of certain series of representa- tions, Bull. Amer. Math. Soc. 75 (1969), 627-642.

    Article  MATH  MathSciNet  Google Scholar 

  74. B. Kostant, On the existence and irreducibility of certain series of representa-tions, pages 231-329 in: Lie Groups and Their Representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971). Halsted, New York, 1975.

    Google Scholar 

  75. J.-S. Li, The minimal decay of matrix coefficients for classical groups, pages 146-169 in: Harmonic Analysis in China. M.D. Cheng et al. (eds), Math. Appl. 327. Kluwer, 1995.

    Google Scholar 

  76. J.-S. Li, On the decay of matrix coefficients for exceptional groups, Math. Ann. 305(1996),249-270.

    Article  MATH  MathSciNet  Google Scholar 

  77. N. Lohoué and Th. Rychener, Die Resolvente von Δ auf symmetrischen Räumen vom nichtkompakten Typ, Comment. Math. Helv. 57 (1982), 445-468.

    Article  MATH  MathSciNet  Google Scholar 

  78. N. Lohoué and Th. Rychener, Some function spaces on symmetric spaces re-lated to convolution operators, J. Funct. Anal. 55 (1984), 200-219.

    Article  MATH  MathSciNet  Google Scholar 

  79. A. Lubotzky, Discrete Groups, Expanding Graphs and Invariant Measures. Birkhäuser-Verlag, Basel, 1994.

    MATH  Google Scholar 

  80. I.G. Macdonald, Spherical Functions on a Group of p-adic Type. Publications of the Ramanujan Institute 2. Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Madras, 1971.

    Google Scholar 

  81. G.A. Margulis, Explicit constructions of expanders, Problemy Peredachi Infor- matsii 9 (1973), 71-80.

    MATH  MathSciNet  Google Scholar 

  82. G.A. Margulis, Some remarks on invariant means, Monatsh. Math. 90 (1980), 233-235.

    Article  MATH  MathSciNet  Google Scholar 

  83. G.A. Margulis, Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators, Problemy Peredachi Informatsii 24 (1988), 51-60.

    MathSciNet  Google Scholar 

  84. C.C. Moore, Exponential decay of correlation coefficients for geodesic flows, pages 163-181 in: Group Representations, Ergodic Theory, Operator Algebras, and Mathematical Physics (Berkeley, Calif., 1984). Math. Sci. Res. Inst. Publ., 6. Springer, New York, 1987.

    Google Scholar 

  85. H. Oh, Tempered subgroups and representations with minimal decay of matrix coefficients, Bull. Soc. Math. France 126 (1998), 355-380.

    MATH  Google Scholar 

  86. H. Oh, Uniform pointwise bounds for matrix coefficients of unitary represen-tations and applications to Kazhdan constants, Duke Math. J. 113 (2002), 133-192.

    Article  MATH  Google Scholar 

  87. M. Ronan, Lectures on Buildings. Perspectives in Mathematics 7. Academic Press Inc., Boston, MA, 1989.

    Google Scholar 

  88. J.M. Rosenblatt, Uniqueness of invariant means for measure-preserving trans-formations, Trans. Amer. Math. Soc. 265 (1981), 623-636.

    Article  MATH  MathSciNet  Google Scholar 

  89. P. Sarnak, Some Applications of Modular Forms. Cambridge Tracts in Math-ematics 99. Cambridge University Press, Cambridge, 1990.

    Google Scholar 

  90. I. Satake, Spherical functions and Ramanujan conjecture, pages 258-264 in: Algebraic Groups and Discontinuous Subgroups. Proc. Symp. Pure Math.,Boulder, Colo., 1965. Amer. Math. Soc., Providence, R.I., 1966.

    Google Scholar 

  91. P. Sawyer, The heat equation on the spaces of positive definite matrices, Can. J. Math. 44 (1992), 624-651.

    MATH  MathSciNet  Google Scholar 

  92. P. Sawyer, On an upper bound for the heat kernel on SU (2n)/Sp(n), Can. Math. Bull. 37 (1994), 408-418.

    MATH  MathSciNet  Google Scholar 

  93. R. Scaramuzzi, A notion of rank for unitary representations of general linear groups, Trans. Amer. Math. Soc. 319 (1990), 349-379.

    Article  MATH  MathSciNet  Google Scholar 

  94. T.P. Schonbek, Lp -multipliers: a new proof of an old theorem, Proc. Amer. Math. Soc. 102 (1988), 361-364.

    Article  MATH  MathSciNet  Google Scholar 

  95. A.G. Setti, Lp and operator norm estimates for the complex time heat operator on homogeneous trees, Trans. Amer. Math. Soc. 350 (1998), 743-768.

    Article  MATH  MathSciNet  Google Scholar 

  96. R.J. Stanton and P. Tomas, Expansions for spherical functions on noncompact symmetric spaces, Acta Math. 140 (1978), 251-276.

    Article  MATH  MathSciNet  Google Scholar 

  97. J.-O. Strömberg, Weak type L1 estimates for maximal functions on non-compact symmetric spaces, Ann. of Math. 114 (1981), 115-126.

    Article  MathSciNet  Google Scholar 

  98. D. Sullivan, For n > 3 there is only one finitely additive rotationally invariant measure on the n-sphere defined on all Lebesgue measurable subsets, Bull. Amer. Math. Soc. (N.S.) 4 (1981), 121-123.

    Article  MATH  MathSciNet  Google Scholar 

  99. M.E. Taylor, Lp -estimates on functions of the Laplace operator, Duke Math. J. 58(1989),773-793.

    Article  MATH  MathSciNet  Google Scholar 

  100. P.C. Trombi and V.S. Varadarajan, Spherical transforms of semisimple Lie groups, Ann. of Math. 94 (1971), 246-303.

    Article  MathSciNet  Google Scholar 

  101. L. Vretare, Elementary spherical functions on symmetric spaces, Math. Scand. 39(1976),343-358.

    MathSciNet  Google Scholar 

  102. L. Vretare, On a recurrence formula for elementary spherical functions on sym- metric spaces and its applications to multipliers for the spherical Fourier trans- form, Math. Scand. 41 (1977), 99-112.

    MATH  MathSciNet  Google Scholar 

  103. S.P. Wang, The dual space of semi-simple Lie groups, Amer. J. Math. 91 (1969),921-937.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Cowling, M. (2008). Applications of Representation Theory to Harmonic Analysis of Lie Groups (and Vice Versa). In: Tarabusi, E.C., D'Agnolo, A., Picardello, M. (eds) Representation Theory and Complex Analysis. Lecture Notes in Mathematics, vol 1931. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-76892-0_1

Download citation

Publish with us

Policies and ethics