The Segal–Bargmann Transform on Compact Symmetric Spaces and Their Direct Limits

Part of the Progress in Mathematics book series (PM, volume 306)


This article studies the limit of the Segal-Bargman transform on inductive limits of compact symmetric spaces.


Heat equation Segal–Bargmann transform Compact Riemannian symmetric spaces Direct limits of compact symmetric spaces 



Both authors were supported by NSF grant DMS-0801010.


  1. [1]
    D.N. Akhiezer and S. Gindikin, On Stein extensions of real symmetric spaces, Math. Ann. 286 (1990), 1–12.Google Scholar
  2. [2]
    J-P. Anker and P. Ostellari, The Heat Kernel on Noncompact Symmetric Spaces, in: Lie groups and symmetric spaces: In memory of F.I. Karpelevich, S. Gindikin(ed.), Amer. Math. Soc. Tranl. (2)210, Amer. Math. Soc. (2004), 27–46.Google Scholar
  3. [3]
    V. Bargmann, On Hilbert spaces of analytic functions and an associated integral transform, Comn. Pure Appl. Math. 14 (1961), 187–214.Google Scholar
  4. [4]
    T. Branson, G. Ólafsson and A. Pasquale, The Paley-Wiener Theorem and the local Huygens’ principle for compact symmetric spaces: The even multiplicity case, Indag. Mathem., N.S., 16 (2005), 393–428.Google Scholar
  5. [5]
    J. Faraut, Espaces Hilbertiens invariant de fonctions holomorphes, Semin. Congr., Vol. 7, Soc. Math. de France, Paris, 2003, 101–167.Google Scholar
  6. [6]
     , Analysis on the crown of a Riemannian symmetric space, in: Lie Groups and Symmetric Spaces: In Memory of F.I. Karpelevich, S. Gindikin(ed.), Amer. Math. Soc. Transl. Ser. 2, Vol. 210, Amer. Math. Soc., Providence, RI, 2003, pp. 99–110.Google Scholar
  7. [7]
    M. Flensted-Jensen, Spherical functions on a real semisimple Lie group. A method of reduction to the complex case, J. Funct. Anal. 30 (1978), 106–146.Google Scholar
  8. [8]
     , Discrete series for semisimple symmetric spacdes, Ann of Math. 111 (1980), 253–311.Google Scholar
  9. [9]
    G. B. Folland, A Course in Abstract Harmonic Analysis, Studies in Advanced Mathematics, CRC Press, 1995.MATHGoogle Scholar
  10. [10]
    R. Goodman and N. R. Wallach, Representations and Invariants of the Classical Groups, Encyclopedia of Mathematics and its Applications 68, Cambridge University Press, Cambridge, Reprinted with corrections 2003.Google Scholar
  11. [11]
    M. Gordina, Holomorphic functions and the heat kernel measure on an infinite dimensional complex orthogonal group, Potential Analysis 12 (2000), 325–357.Google Scholar
  12. [12]
    B. C. Hall, The Segal-Bargmann transform for compact Lie groups, J. Funct. Anal. 143 (1994), 103–151.Google Scholar
  13. [13]
     , Holomorphic methods in analysis and mathematical physics, in: First Summer School in Analysis and Mathematical Physics, Contemp. Math., Vol. 260, Amer. Math. Soc., Providence, RI, 2000, 1–59.Google Scholar
  14. [14]
     , Harmonic analysis with respect to the heat kernel measure, Bull. Amer. Math. Soc. (N.S.) 38 (2001), 43–78.Google Scholar
  15. [15]
     , The range of the heat operator, in: Ed.: J. Jorgensen and L. Walling, The Ubiquitous Heat Kernel, 203–231, Contemp. Math., 398, AMS, 2006.Google Scholar
  16. [16]
    B. C. Hall and J. Mitchell, The Segal-Bargmann transform for noncompact symmetric spaces of the complex type, J. Funct. Anal. 227 (2005), 338–371.Google Scholar
  17. [17]
    B. C. Hall and A. N. Sengupta, The Segal-Bargmann transform for path-groups, J. Funct. Anal. 152 (1998), 220–254.Google Scholar
  18. [18]
    S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Amer. Math. Soc., Providence, RI, 2001.MATHGoogle Scholar
  19. [19]
     , Groups and Geometric Analysis, Amer. Math. Soc., Providence, RI, 2000.Google Scholar
  20. [20]
    J. Hilgert and G. Zhang, Segal-Bargmann and Weyl transforms on compact Lie groups, Monatsh. Math. 158 (2009), 285–305.Google Scholar
  21. [21]
    B. Krötz and R. Stanton, Holomorphic extension of representation (II): Geometry and harmonic analysis, Geom. Funct. Anal. 15 (2005), no. 1, 190–245.Google Scholar
  22. [22]
    B. Krötz, G. Ólafsson and R. Stanton, The Image of the Heat Kernel Transform on Riemannian Symmetric Spaces of the Noncompact Type, Int. Math. Res. Not. 22 (2005), 1307–1329.Google Scholar
  23. [23]
    M. Lassalle, Séries de Laurent des fonctions holomorphes dans la complexification d’un espace symétrique compact, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 2, 167–210.Google Scholar
  24. [24]
    L. Natarajan, E. Rodr\(\acute{i}\)quez-Carrington and J. A. Wolf, The Bott-Borel-Weil Theorem for direct limit Lie groups, Trans. Amer. Math. Soc. 353 (2001), 4583–4622.Google Scholar
  25. [25]
    K-H. Neeb, Holomorphy and convexity in Lie theory, de Gruyter Expositions in Mathematics, 28. Walter de Gruyter & Co., Berlin, 2000Google Scholar
  26. [26]
    G. Ólafsson, Analytic Continuation in Representation Theory and Harmonic Analysis, in: Global Analysis and Harmonic Analysis, ed. J. P. Bourguignon, T. Branson, and O. Hijazi. S\(\acute{e}\) minares et Congr, Vol 4, (2000), 201–233. The French Math. Soc.Google Scholar
  27. [27]
    G. Ólafsson and H. Schlichtkrull, The Segal-Bargmann transform for the heat equation associated with root systems, Adv. Math. 208 (1) (2007), 422–437.Google Scholar
  28. [28]
     , Representation theory, Radon transform and the heat equation on a Riemannian symmetric space. Group Representations, Ergodic Theory, and Mathematical Physics; A Tribute to George W. Mackey. in: Contemp. Math., 449 (2008), 315–344.Google Scholar
  29. [29]
     , Fourier transforms of spherical distributions on compact symmetric spaces. To appear in Math. Scand.Google Scholar
  30. [30]
    G. Ólafsson and B. Ørsted, Generalizations of the Bargmann transform, Lie theory and its applications in physics (Clausthal, 1995), 3–14, World Sci. Publ., River Edge, NJ, 1996.Google Scholar
  31. [31]
    G. Ólafsson and J. A. Wolf, Weyl group invariants and application to spherical harmonic analysis on symmetric spaces. Preprint, arXiv:0901.4765.Google Scholar
  32. [32]
     , Extension of Symmetric Spaces and Restriction of Weyl Groups and Invariant Polynomials, to appear in Contemporary Mathematics.Google Scholar
  33. [33]
    E. Opdam, Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995), 75–121.Google Scholar
  34. [34]
    I. E. Segal, Mathematical Problems of Relatvistic Physics, (Ed. M. Kac) Lectures in Applied Mathematics 2, AMS, 1963.Google Scholar
  35. [35]
    A. R. Sinton, The spherical transform on projective limits of symmetric spaces, J. Lie Theory 17 (2007), no. 4, 869–898.Google Scholar
  36. [36]
    H. Schlichtkrull, Hyperfunctions and Harmonic Analysis on Symmetric Spaces, Prog. Math. 49. Birkhäuser, Boston, 1984.Google Scholar
  37. [37]
    M. Stenzel, The Segal-Bargmann transform on a symmetric space of compact type, J. Funct. Anal. 165 (1999), 44–58.Google Scholar
  38. [38]
    M. Sugiura, Fourier series of smooth functions on compact Lie groups, Osaka Math. J. 8 (1971), 33–47.Google Scholar
  39. [39]
    S. Thangavelu, Holomorphic Sobolev spaces associated to compact symmetric spaces, J. Funct. Anal. 251 (2007), 438–462.Google Scholar
  40. [40]
    N. Wallach, Harmonic Analysis on Homogeneous Spaces, Marcel Dekker, 1973.Google Scholar
  41. [41]
    K. Wiboonton, The Segal-Bargmann Transform on Inductive Limits of Compact Symmetric Spaces, Ph.D. Thesis, LSU, 2009.Google Scholar
  42. [42]
    J. A. Wolf, Direct limits of principal series representations, Compositio Mathematica, 141 (2005), 1504–1530.MathSciNetMATHCrossRefGoogle Scholar
  43. [43]
     , Harmonic analysis on commutative spaces, Math. Surveys & Monographs Vol. 142, Amer. Math. Soc., 2007.Google Scholar
  44. [44]
     , Infinite Dimensional Multiplicity Free Spaces I: Limits of Compact Commutative Spaces, in: Developments and Trends in Infinite Dimensional Lie Theory, eds. K.-H. Neeb and A. Pianzola, Birkhäuser, to appear in 2009.Google Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsLouisiana State UniversityBaton RougeUSA
  2. 2.Department of Mathematics and Computer ScienceChulalongkorn UniversityBangkokThailand

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