Japanese Journal of Mathematics

, Volume 3, Issue 2, pp 247–290 | Cite as

Covariant quantization: spectral analysis versus deformation theory

Article

Abstract.

Formal deformation or rather symbolic calculus? To which extent do these approaches complete each other in the study of symmetry-preserving quantization procedures for homogeneous spaces? The representation theory of underlying Lie groups shows that the answer is much more delicate than initially thought and that it cannot be always reduced to asymptotic expansions with respect to some Planck’s constant. The main goal of this survey is to give hints regarding the aims of each approach and, on the domain where these intersect, to compare the answers they lead to.

Keywords and phrases:

quantization unitary representations semisimple symmetric spaces 

Mathematics Subject Classification (2000):

81-02 22E46 43A85 47G30 16S80 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. Alekseev and E. Meinrenken, The non-commutative Weil algebra, Invent. Math., 139 (2000), 135–172MATHMathSciNetGoogle Scholar
  2. 2.
    A. Alekseev and E. Meinrenken, Poisson geometry and the Kashiwara–Vergne conjecture, C. R. Math. Acad. Sci. Paris, 335 (2002), 723–728MATHMathSciNetGoogle Scholar
  3. 3.
    A. Alekseev and E. Meinrenken, On the Kashiwara–Vergne conjecture, Invent. Math., 164 (2006), 615–634MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    M. Andler, A. Dvorsky and S. Sahi, Kontsevich quantization and invariant distributions on Lie groups, Ann. Sci. École Norm. Sup. (4), 35 (2002), 371–390MATHMathSciNetGoogle Scholar
  5. 5.
    M. Andler, S. Sahi and Ch. Torossian, Convolution of invariant distributions: proof of the Kashiwara–Vergne conjecture, Lett. Math. Phys., 69 (2004), 177–203MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    H. Aoki and T. Ibukiyama, Simple graded rings of Siegel modular forms, differential operators and Borcherds products, Internat. J. Math., 16 (2005), 249–279MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    J. Arazy and B. Ørsted, Asymptotic expansions of Berezin transforms, Indiana Univ. Math. J., 49 (2000), 7–30MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    J. Arazy and H. Upmeier, Weyl calculus for complex and real symmetric domains, In: Harmonic Analysis on Complex Homogeneous Domains and Lie Groups, Rome, 2001, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 13, 2002, pp. 165–181.Google Scholar
  9. 9.
    J. Arazy and H. Upmeier, Invariant symbolic calculi and eigenvalues of invariant operators on symmetric domains, In: Function Spaces, Interpolation Theory and Related Topics, Lund, 2000, de Gruyter, Berlin, 2002, pp. 151–211.Google Scholar
  10. 10.
    J. Arazy and H. Upmeier, A one-parameter calculus for symmetric domains, Math. Nachr., 280 (2007), 939–961MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337–404MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    M. Atiyah and W. Schmid, A geometric construction of the discrete series of semisimple Lie groups, Invent. Math., 42 (1977), 1–62MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    K. Ban, On Rankin–Cohen–Ibukiyama operators for automorphic forms of several variables, Comment. Math. Univ. St. Pauli, 55 (2006), 149–171MATHMathSciNetGoogle Scholar
  14. 14.
    D. Barbash, S. Sahi and B. Speh, Degenerate series representations for \(GL(2n, {\mathbb {R}})\) and Fourier analysis, In: Indecomposable Representations of Lie Groups and Their Physical Applications, Rome, 1988, (ed. V. Cantoni), Sympos. Math., 31, Academic Press, 1990, pp. 45–69.Google Scholar
  15. 15.
    F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Deformation theory and quantization, Ann. Physics, 111 (1978), 61–110MATHMathSciNetGoogle Scholar
  16. 16.
    S. Ben Saïd, Weighted Bergman spaces on bounded symmetric domains, Pacific J. Math., 206 (2002), 39–68MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    F.A. Berezin, General concept of quantization, Comm. Math. Phys., 40 (1975), 153–174MathSciNetCrossRefGoogle Scholar
  18. 18.
    F.A. Berezin, Connection between co- and contravariant symbols of operators on the classical complex symmetric spaces, Dokl. Akad. Nauk SSSR, 241 (1978), 15–17MathSciNetGoogle Scholar
  19. 19.
    S. Bergman, The Kernel Function and Conformal Mapping, Math. Surveys, 5, Amer. Math. Soc., 1950.Google Scholar
  20. 20.
    W. Bertram, Un théorème de Liouville pour les algèbres de Jordan, Bull. Soc. Math. France, 124 (1996), 299–328MATHMathSciNetGoogle Scholar
  21. 21.
    W. Bertram, Algebraic structures of Makarevič spaces. I, Transform. Groups, 3 (1998), 3–32MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    W. Bertram, The Geometry of Jordan and Lie Structures, Lecture Notes in Math., 1754, Springer-Verlag, 2000.Google Scholar
  23. 23.
    P. Bieliavsky and M. Pevzner,Symmetric spaces and star representations. II. Causal symmetric spaces, J. Geom. Phys., 41 (2002), 224–234.MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    P. Bieliavsky and M. Pevzner, Symmetric spaces and star representations. III. The Poincaré disc, In: Noncommutative Harmonic Analysis, Progr. Math., 220, Birkhäuser Boston, Boston, MA, 2004, pp. 61–77Google Scholar
  25. 25.
    P. Bieliavsky, X. Tang and Y. Yao, Rankin–Cohen brackets and formal quantization, Adv. Math., 212 (2007), 293–314MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Th. Branson, Group representations arising from Lorentz conformal geometry, J. Funct. Anal., 74 (1987), 199–291MATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Th. Branson, G. Ólafsson and B. Órsted, Spectrum generating operators and intertwining operators for representations induced from a maximal parabolic subgroup, J. Funct. Anal., 135 (1996), 163–205MATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    A. Cattaneo and G. Felder, A path integral approach to the Kontsevich quantization formula, Comm. Amth. Phys., 212 (2000), 591–611.MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    A. Cattaneo and Ch. Torossian, Quantification pour les paires symetriques et diagrammes de Kontsevich, Ann. Sci. Éc. Norm. Supér. (4), to appear (2008).Google Scholar
  30. 30.
    J.-L. Clerc, A generalized Hecke Identity, J. Fourier Anal. Appl., 6 (2000), 105–111MATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    H. Cohen, Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann., 217 (1975), 271–285MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    A. Connes and H. Moscovici, Rankin–Cohen brackets and the Hopf algebra of transverse geometry, Mosc. Math. J., 4 (2004), 111–130MATHMathSciNetGoogle Scholar
  33. 33.
    G. van Dijk, A new approach to Berezin kernels and canonical representations, In: Asymptotic Combinatorics with Application to Mathematical Physics, Proceedings of the NATO Advanced Study Institute, (eds. V.A. Malyshev et al.), NATO Sci. Ser. II Math. Phys. Chem., 77, Kluwer Acad. Publ., Dordrecht, 2002, pp. 279–305.Google Scholar
  34. 34.
    G. van Dijk and S. Hille, Canonical representations related to hyperbolic spaces, J. Funct. Anal., 147 (1997), 109–139MATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    G. van Dijk and S. Hille, Maximal degenerate representations, Berezin kernels and canonical representations, In: Lie Groups and Lie Algebras, Math. Appl., 433, Kluwer Acad. Publ., Dordrecht, 1998, pp. 285–298.Google Scholar
  36. 36.
    G. van Dijk and V.F. Molchanov, The Berezin form for rank one para-Hermitian symmetric spaces, J. Math. Pures Appl. (9), 77 (1998), 747–799MATHMathSciNetGoogle Scholar
  37. 37.
    G. van Dijk and V.F. Molchanov, Tensor products of maximal degenerate series representations of the group,\(SL(n, {\mathbb{R}})\)), J. Math. Pures Appl. (9), 78 (1999), 99–119Google Scholar
  38. 38.
    G. van Dijk and M. Pevzner, Berezin kernels on tube domains, J. Funct. Anal., 181 (2001), 189–208MATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    G. van Dijk and M. Pevzner, Matrix-valued Berezin kernels, In: Geometry and Analysis on Finite- and Infinite-dimensional Lie Groups, (eds. A. Strasburger et al.), Banach Center Publ., 55, Polish Acad. Sci. Inst. Math., Warszawa, 2002, pp. 269–288.Google Scholar
  40. 40.
    G. van Dijk and M. Pevzner, Berezin kernels and maximal degenerate representations associated with Riemannian symmetric spaces of Hermitian type, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 292 (2002), 11–21Google Scholar
  41. 41.
    G. van Dijk, M. Pevzner and S. Aparicio, Invariant Hilbert subspaces of the oscillator representation, Indag. Math. (N.S.), 14 (2003), 309–318.MATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    G. van Dijk and M. Pevzner, Ring structures for holomorphic discrete series and Rankin–Cohen brackets, J. Lie Theory, 17 (2007), 283–305MATHMathSciNetGoogle Scholar
  43. 43.
    G. van Dijk and M. Pevzner, H*-algebras and quantization of para-Hermitian spaces, Proc. Amer. Math. Soc., 136 (2008), 2253–2260MATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    P.A.M. Dirac, The Principles of Quantum Mechanics, 3d ed., Oxford, at the Clarendon Press, 1947.MATHGoogle Scholar
  45. 45.
    V. Dolgushev, Covariant and equivariant formality theorems, Adv. Math., 191 (2005), 147–177MATHMathSciNetCrossRefGoogle Scholar
  46. 46.
    M. Duflo, Opérateurs différentiels bi-invariants sur un groupe de Lie, Ann. Sci. École Norm. Sup. (4), 10 (1977), 265–288MATHMathSciNetGoogle Scholar
  47. 47.
    A. Dvorsky and S. Sahi, Explicit Hilbert spaces for certain unipotent representations. II, Invent. Math., 138 (1999), 203–224MATHMathSciNetCrossRefGoogle Scholar
  48. 48.
    A. Dvorsky and S. Sahi, Explicit Hilbert spaces for certain unipotent representations. III, J. Funct. Anal., 201 (2003), 430–456MATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    W. Eholzer and T. Ibukiyama, Rankin–Cohen type differential operators for Siegel modular forms, Internat. J. Math., 9 (1998), 443–463MATHMathSciNetCrossRefGoogle Scholar
  50. 50.
    A. El Gradechi, The Lie theory of the Rankin–Cohen brackets and allied bi-differential operators, Adv. Math., 207 (2006), 484–531MATHMathSciNetCrossRefGoogle Scholar
  51. 51.
    M. Engliš, A mean value theorem on bounded symmetric domains, Proc. Amer. Math. Soc., 127 (1999), 3259–3268MATHMathSciNetCrossRefGoogle Scholar
  52. 52.
    J. Faraut, Intégrales de Riesz sur un espace symétrique ordonné, In: Geometry and Analysis on Finite- and Infinite-dimensional Lie Groups, Bȩdlewo, 2000, Banach Center Publ., 55, Polish Acad. Sci., Warsaw, 2002, pp. 289–308.Google Scholar
  53. 53.
    J. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford Sci. Publ., 1994.Google Scholar
  54. 54.
    J. Faraut and G. Ólafsson, Causal semisimple symmetric spaces, the geometry and harmonic analysis, In: Semigroups in Algebra, Geometry and Analysis, (eds. K.H. Hofmann, J.D. Lawson and E.B. Vinberg), de Gruyter, Berlin, 1995.Google Scholar
  55. 55.
    J. Faraut and M. Pevzner, Berezin kernels and analysis on Makarevich spaces, Indag. Math. (N.S.), 16 (2005), 461–486MATHMathSciNetCrossRefGoogle Scholar
  56. 56.
    J. Faraut and E.G.F. Thomas, Invariant Hilbert spaces of holomorphic functions, J. Lie Theory, 9 (1999), 383–402MATHMathSciNetGoogle Scholar
  57. 57.
    M. Flensted-Jensen, Discrete series for semisimple symmetric spaces, Ann. of Math. (2), 111 (1980), 253–311MathSciNetCrossRefGoogle Scholar
  58. 58.
    I.M. Gelfand and L.A. Dikiy, A family of Hamiltonian structures connected with integrable nonlinear differential equations, Akad. Nauk SSSR Inst. Prikl. Mat. Preprint, 136 (1978), 41 p.Google Scholar
  59. 59.
    S. Gindikin, Fourier transform and Hardy spaces of \(\bar{\partial}\)-cohomology in tube domains, C. R. Acad. Sci. Paris Sér. I Math., 315 (1992), 1139–1143.Google Scholar
  60. 60.
    P. Gordan, Invariantentheorie, Teubner, Leipzig, 1887.Google Scholar
  61. 61.
    S. Gundelfinger, Zur der binären Formen, J. Reine Angew. Math., 100 (1886), 413–424Google Scholar
  62. 62.
    C.S. Herz, Bessel functions of matrix argument, Ann. of Math. (2), 61 (1955), 474–523MathSciNetCrossRefGoogle Scholar
  63. 63.
    S. Hille, Canonical representations, Ph.D thesis, Leiden Univ., 1999.Google Scholar
  64. 64.
    L. Hörmander, The Analysis of Linear Partial Differential Operators. III. Pseudo-differential Operators, Classics Math., Springer-Verlag, Berlin, 2007.Google Scholar
  65. 65.
    R. Howe, On some results of Strichartz and Rallis and Schiffman, J. Funct. Anal., 32 (1979), 297–303MATHMathSciNetCrossRefGoogle Scholar
  66. 66.
    R. Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc., 313 (1989), 539–570MATHMathSciNetCrossRefGoogle Scholar
  67. 67.
    R. Howe and S.T. Lee, Degenerate principal series representations of \(GL_n (\mathbb{C})\) and \(GL_n (\mathbb{R})\), J. Funct. Anal., 166 (1999), 244–309MATHMathSciNetCrossRefGoogle Scholar
  68. 68.
    R. Howe and E. Tan, Non-Abelian Harmonic Analysis. Applications of \({SL}(2,{\mathbb{R}})\) , Universi, Springer-Verlag, New York, 1992.Google Scholar
  69. 69.
    R. Howe and E. Tan, Homogeneous functions on light cones: the infinitesimal structure of some degenerate principal series representations, Bull. Amer. Math. Soc., 28 (1993), 1–74MATHMathSciNetCrossRefGoogle Scholar
  70. 70.
    L.K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, Transl. Math. Monogr., 6, Amer. Math. Soc., Providence, RI, 1963.Google Scholar
  71. 71.
    H.P. Jakobsen and M. Vergne, Restrictions and expansions of holomorphic representations, J. Funct. Anal., 34 (1979), 29–53MATHMathSciNetCrossRefGoogle Scholar
  72. 72.
    K. Johnson, Degenerate principal series and compact groups, Math. Ann., 287 (1990), 703–718MATHMathSciNetCrossRefGoogle Scholar
  73. 73.
    K. Johnson, Degenerate principal series on tube type domains, Contemp. Math., 138 (1992), 175–187Google Scholar
  74. 74.
    S. Kaneyuki and M. Kozai, Paracomplex structures and affine symmetric spaces, Tokyo J. Math., 8 (1985), 81–98MATHMathSciNetCrossRefGoogle Scholar
  75. 75.
    I.L. Kantor, Non-linear groups of transformations defined by general norms of Jordan algebras, Soviet Math Dokl., 8 (1967), 176–180MATHGoogle Scholar
  76. 76.
    M. Kashiwara and M. Vergne, On the Segal–Shale–Weil representations and harmonic polynomials, Invent. Math., 44 (1978), 1–47MATHMathSciNetCrossRefGoogle Scholar
  77. 77.
    M. Kashiwara and M. Vergne, The Campbell–Hausdorff formula and invariant hyperfunctions, Invent. Math., 47 (1978), 249–272MATHMathSciNetCrossRefGoogle Scholar
  78. 78.
    M. Kashiwara and M. Vergne, Functions on the Shilov boundary of the generalized half plane, In: Non-Commutative Harmonic Analysis, Lectures Notes in Math., 728, Springer-Verlag, 1979, pp. 136–176.Google Scholar
  79. 79.
    A.A. Kirillov, Invariant operators over geometric quantities, In: Current Problems in Mathematics, Akad. Nauk SSSR, 16, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1980, pp. 3–29.Google Scholar
  80. 80.
    A.A. Kirillov, Lectures on the Orbit Method, Grad. Stud. Math., 64, Amer. Math. Soc., Providence, RI, 2004.Google Scholar
  81. 81.
    T. Kobayashi, Discrete decomposability of the restriction of \(A_{\mathfrak q}(\lambda)\) with respect to reductive subgroups and its applications, Invent. Math., 117 (1994), 181–205MATHMathSciNetCrossRefGoogle Scholar
  82. 82.
    T. Kobayashi, The restriction of \(A_{\mathfrak q}(\lambda)\) to reductive subgroups. I, Proc. Japan Acad. Sér. A Math. Sci., 69 (1993), 262–267; II, ibid., 71 (1995), 24–26.Google Scholar
  83. 83.
    T. Kobayashi, Multiplicity-free theorem in branching problems of unitary highest weight modules, In: Proceedings of the Symposium on Representation Theory, Saga, Kyushu, (ed. K. Mimachi), 1997, pp. 9–17.Google Scholar
  84. 84.
    T. Kobayashi, Discrete decomposability of the restriction of \(A_{\mathfrak q}(\lambda)\) with respect to reductive subgroups. II. Microlocal analysis and asymptotic K-support, Ann. of Math. (2), 147 (1998), 709–729MATHMathSciNetCrossRefGoogle Scholar
  85. 85.
    T. Kobayashi, Discrete decomposability of the restriction of \(A_{\mathfrak q}(\lambda)\) with respect to reductive subgroups. III. Restriction of Harish-Chandra modules and associated varieties, Invent. Math., 131 (1998), 229–256MATHGoogle Scholar
  86. 86.
    T. Kobayashi, Discrete series representations for the orbit spaces arising from two involutions of real reductive Lie groups, J. Funct. Anal., 152 (1998), 100–135MATHMathSciNetCrossRefGoogle Scholar
  87. 87.
    T. Kobayashi, Theory of discretely decomposable restrictions of unitary representations of semisimple Lie groups and some applications, Sugaku Expositions, 18 (2005), 1–37MathSciNetMATHGoogle Scholar
  88. 88.
    T. Kobayashi, Multiplicity-free representations and visible actions on complex manifolds, Publ. Res. Inst. Math. Sci., 41 (2005), 497–549MATHMathSciNetCrossRefGoogle Scholar
  89. 89.
    T. Kobayashi, Visible actions on symmetric spaces, Transform. Groups, 12 (2007), 671–694MATHMathSciNetCrossRefGoogle Scholar
  90. 90.
    T. Kobayashi, Multiplicity-free theorems of the restrictions of unitary highest weight modules with respect to reductive symmetric pairs, In: Representation Theory and Automorphic Forms, Progr. Math., 255, Birkhäuser Boston, Boston, MA, 2007, pp. 45–109.Google Scholar
  91. 91.
    T. Kobayashi and B. Órsted, Analysis on the minimal representation of O(p,q). I. Realization via conformal geometry, Adv. Math., 180 (2003), 486–512MATHMathSciNetCrossRefGoogle Scholar
  92. 92.
    T. Kobayashi and B. Órsted, Analysis on the minimal representation of O(p,q). II. Branching laws, Adv. Math., 180 (2003), 513–550MATHMathSciNetCrossRefGoogle Scholar
  93. 93.
    T. Kobayashi and B. Órsted, Analysis on the minimal representation of O(p,q). III. Ultrahyperbolic equations on \({\mathbb{R}}^{p-1,q-1}\), Adv. Math., 180 (2003), 551–595Google Scholar
  94. 94.
    M. Koecher, Über eine Gruppe von rationalen Abbildungen, Invent. Math., 3 (1967), 136–171MATHMathSciNetCrossRefGoogle Scholar
  95. 95.
    M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys., 66 (2003), 157–216MATHMathSciNetCrossRefGoogle Scholar
  96. 96.
    B. Kostant, A branching law for subgroups fixed by an involution and a noncompact analogue of the Borel–Weil theorem, In: Noncommutative Harmonic Analysis, Progr. Math., 220, Birkhäuser Boston, Boston, MA, 2004, pp. 291–353.Google Scholar
  97. 97.
    B. Kostant and S. Sahi, The Capelli identity, tube domains, and the generalized Laplace transform, Adv. Math., 87 (1991), 71–92MATHMathSciNetCrossRefGoogle Scholar
  98. 98.
    B. Kostant and S. Sahi, Jordan algebras and Capelli identities, Invent. Math., 112 (1993), 657–664MATHMathSciNetCrossRefGoogle Scholar
  99. 99.
    B. Krötz, On Hardy and Bergman spaces on complex Ol’shanskiǐ semigroups, Math. Ann., 312 (1998), 13–52MATHMathSciNetCrossRefGoogle Scholar
  100. 100.
    P.D. Lax and R.S. Phillips, Scattering Theory for Automorphic Functions, Ann. of Math. Stud., 87, Princeton Univ. Press, 1976.Google Scholar
  101. 101.
    S. Lee, On some degenerate principal series representations of U(n,n), J. Funct. Anal., 126 (1994), 305–366MATHMathSciNetCrossRefGoogle Scholar
  102. 102.
    S. Lee, Degenerate principal series representations of \(Sp(2n, {\mathbb{R}})\) , Compositio Math., 103 (1996), 123–151MATHMathSciNetGoogle Scholar
  103. 103.
    J. Liouville, Théorème sur l’équation \(dx^2+dy^2+dz^2=\lambda(d\alpha^2+d\beta^2+d\gamma^2)\) , J. Math. Pures Appl., 15 (1850), p. 103Google Scholar
  104. 104.
    O. Loos, Symmetric Spaces. I, II, Benjamin, New York, 1969.Google Scholar
  105. 105.
    G.W. Mackey, Unitary Group Representations in Physics, Probability and Number Theory, Math. Lecture Note Ser., 55, Benjamin/Cummings Publ. Co., Inc., Reading, Mass., 1978; Second ed., Adv. Book Classics, Addison-Wesley Publ. Co., Rewood City, CA, 1989.Google Scholar
  106. 106.
    D. Manchon and Ch. Torossian, Cohomologie tangente et cup-produit pour la quantification de Kontsevich, Ann. Math. Blaise Pascal, 10 (2003), 75–106MATHMathSciNetGoogle Scholar
  107. 107.
    V.F. Molchanov, Tensor products of unitary representations of the three-dimensional Lorentz group, Math. USSR-Izv., 15 (1980), 113–143MATHCrossRefGoogle Scholar
  108. 108.
    V.F. Molchanov, Maximal degenerate series representations of the universal covering of the group SU(n,n), In: Lie Groups and Lie Algebras. Their Representations, Generalisations and Applications, (eds. B.P. Komrakov et al.), Math. Appl., 433, Kluwer Acad. Publ., 1998, pp. 313–336Google Scholar
  109. 109.
    Yu. Neretin, Matrix analogues of the B-function, and the Plancherel formula for Berezin kernel representations, Sb. Math., 191 (2000), 683–715MathSciNetCrossRefGoogle Scholar
  110. 110.
    Yu. Neretin, Matrix balls, radial analysis of Berezin kernels, and hypergeometric determinants, Mosc. Math. J., 1 (2001), 157–220MATHMathSciNetGoogle Scholar
  111. 111.
    Yu. Neretin, (2002) Plancherel formula for Berezin deformation of L 2 on Riemannian symmetric space, J. Funct. Anal., 189 336–408MATHMathSciNetCrossRefGoogle Scholar
  112. 112.
    T. Nomura, Berezin transforms and group representations, J. Lie Theory, 8 (1998), 433–440MATHMathSciNetGoogle Scholar
  113. 113.
    G. Ólafsson and B. Órsted, The holomorphic discrete series for affine symmetric spaces. I, J. Funct. Anal., 81 (1988), 126–159MATHMathSciNetCrossRefGoogle Scholar
  114. 114.
    G.I. Olshanski, Invariant cones in Lie algebras, Lie semigroups and holomorphic discrete series, Funct. Anal. Appl., 15 (1981), 275–285Google Scholar
  115. 115.
    P.J. Olver, Classical Invariant Theory, London Math. Soc. Stud. Texts, 44, Cambridge Univ. Press, 1999.Google Scholar
  116. 116.
    P.J. Olver and J.A. Sanders, Transvectants, modular forms, and the Heisenberg algebra, Adv. in Appl. Math., 25 (2000), 252–283MATHMathSciNetCrossRefGoogle Scholar
  117. 117.
    B. Órsted and B. Speh, Branching laws for some unitary representations of SL(4,\({\mathbb{R}}\)), SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008), 19 p.Google Scholar
  118. 118.
    B. Órsted and J. Vargas, Restriction of square integrable representations: discrete spectrum, Duke Math. J., 123, (2004) 609–633MathSciNetCrossRefGoogle Scholar
  119. 119.
    B. Órsted and G. Zhang, Generalized principal series representations and tube domains, Duke Math. J., 78 (1995), 335–357MathSciNetCrossRefGoogle Scholar
  120. 120.
    B. Órsted and G. Zhang, Capelli identity and relative discrete series of line bundles over tube domains, In: Geometry and Analysis on Finite- and Infinite-dimensional Lie Groups, Proceedings of the workshop on Lie groups and Lie algebras, Bedlewo, Poland, September 4-15, 2000,Banach Center Publ., 55, Polish Acad. Sci., Warszawa, 2002, pp. 349–357Google Scholar
  121. 121.
    T. Oshima and T. Matsuki, A description of discrete series for semisimple symmetric spaces, In: Group Representations and Systems of Differential Equations, Adv. Stud. Pure Math., 4, 1984, pp. 331–390Google Scholar
  122. 122.
    T. Oshima and T. Matsuki, A description of discrete series for semisimple symmetric spaces, In: Group Representations and Systems of Differential Equations, Adv. Stud. Pure Math., 4, 1984, pp. 331–390Google Scholar
  123. 123.
    E. Pedon, Harmonic analysis for differential forms on complex hyperbolic spaces, J. Geom. Phys., 32 (1999), 102–130MATHMathSciNetCrossRefGoogle Scholar
  124. 124.
    J. Peetre, Hankel forms of arbitrary weight over a symmetric domain via the transvectant, Rocky Mountain J. Math., 24 (1994), 1065–1085MATHMathSciNetCrossRefGoogle Scholar
  125. 125.
    L. Peng, and G. Zhang, Tensor products of holomorphic representations and bilinear differential operators, J. Funct. Anal., 210 (2004), 171–192MATHMathSciNetCrossRefGoogle Scholar
  126. 126.
    M. Pevzner, Espace de Bergman d’un semi-groupe complexe, C. R. Acad. Sci. Paris Sér. I Math., 322(1996), 635–640MathSciNetGoogle Scholar
  127. 127.
    M. Pevzner, Analyse conforme sur les algèbres de Jordan, Ph.D. thesis, Univ. of Paris VI, 1998.Google Scholar
  128. 128.
    M. Pevzner, Représentation de Weil associée à une représentation d’une algèbre de Jordan, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 463–468MATHMathSciNetGoogle Scholar
  129. 129.
    M. Pevzner, Analyse conforme sur les algèbres de Jordan, J. Aust. Math. Soc., 73 (2002), 279–299MATHMathSciNetCrossRefGoogle Scholar
  130. 130.
    M. Pevzner and Ch. Torossian, Isomorphisme de Duflo et la cohomologie tangentielle, J. Geom. Phys., 51 (2004), 486–505.MATHMathSciNetCrossRefGoogle Scholar
  131. 131.
    M. Pevzner and A. Unterberger, Projective pseudodifferential analysis and harmonic analysis, J. Funct. Anal., 242 (2007), 442–485MATHMathSciNetCrossRefGoogle Scholar
  132. 132.
    M. Pevzner, Rankin–Cohen brackets and associativity, Lett. Math. Phys., 85 (2008), 195–202MathSciNetCrossRefMATHGoogle Scholar
  133. 133.
    J. Repka, Tensor products of holomorphic discrete series representations, Canad. J. Math., 31 (1979), 836–844MATHMathSciNetGoogle Scholar
  134. 134.
    F. Rouvière, Invariant analysis and contractions of symmetric spaces. I, Compositio Math., 73 (1990), 241–270; II, Compositio Math., 80 (1991), 11–136.Google Scholar
  135. 135.
    F. Rouvière, Fibrés en droites sur un espace symétrique et analyse invariante, J. Funct. Anal., 124 (1994), 263–291MATHMathSciNetCrossRefGoogle Scholar
  136. 136.
    H. Rubenthaler, Une série dégénérée de représentations de \(SL_n({\mathbb{R}})\), Lecture Notes in Math., 739 (1979), 427–459MathSciNetCrossRefGoogle Scholar
  137. 137.
    S. Sahi, The Capelli identity and unitary representations, Compositio Math., 81 (1992), 247–260MATHMathSciNetGoogle Scholar
  138. 138.
    S. Sahi, Explicit Hilbert spaces for certain unipotent representations, Invent. Math., 110 (1992), 409–418MATHMathSciNetCrossRefGoogle Scholar
  139. 139.
    S. Sahi, Unitary representations on the Shilov boundary of a symmetric tube domain, In: Representation Theory of Groups and Algebras, Contemp. Math., 145, Amer. Math. Soc., Providence, RI, 1993, pp. 275–286.Google Scholar
  140. 140.
    S. Sahi, Jordan algebras and degenerate principal series, J. Reine Angew. Math., 462 (1995), 1–18MATHMathSciNetGoogle Scholar
  141. 141.
    S. Sahi and E.M. Stein, Analysis in matrix space and Speh’s representations, Invent. Math., 101 (1990), 379–393MATHMathSciNetCrossRefGoogle Scholar
  142. 142.
    I. Satake, Algebraic Structures of Symmetric Domains, Iwanami Shoten; Princeton Univ. Press, 1980.Google Scholar
  143. 143.
    M. Schlichenmaier, Berezin–Toeplitz quantization and Berezin transform, In: Long Time Behaviour of Classical and Quantum Systems, Proceedings of the Bologna APTEX international conference, Bologna, (ed. S. Graffi), Ser. Concr. Appl. Math., 1, World Sci. Publ., Singapore, 2001, pp. 271–287.Google Scholar
  144. 144.
    W. Schmid, Construction and classification of irreducible Harish-Chandra modules, In: Harmonic Analysis on Reductive Groups, Brunswick, ME, 1989, Progr. Math., 101, Birkhäuser Boston, Boston, MA, 1991, pp. 235–275.Google Scholar
  145. 145.
    L. Schwartz, Sous-espaces hilbertiens d’espaces vectoriels topologiques et noyaux associés, J. Analyse Math., 13 (1964), 115–256MATHMathSciNetCrossRefGoogle Scholar
  146. 146.
    H. Seppänen, Branching laws for minimal holomorphic representations, J. Funct. Anal., 251 (2007), 174–209MATHMathSciNetCrossRefGoogle Scholar
  147. 147.
    B. Shoikhet, On the Duflo formula for L -algebras and Q-manifolds, e-print, QA/9812009.Google Scholar
  148. 148.
    M.A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, 2001.MATHGoogle Scholar
  149. 149.
    R.S. Strichartz, Harmonic analysis on hyperboloids, J. Functional Analysis, 12 (1973), 341–383MATHMathSciNetCrossRefGoogle Scholar
  150. 150.
    E.G.F. Thomas, The theorem of Bochner–Schwartz–Godement for generalised Gelfand pairs, In: Functional Analysis: Surveys and Recent Results. III, Paderborn, 1983, North-Holland Math. Stud., 90, 1984, pp. 291–304.Google Scholar
  151. 151.
    E.G.F. Thomas, Integral representations in conuclear cones, J. Convex Anal., 1 (1994), 225–258MATHMathSciNetGoogle Scholar
  152. 152.
    P. Torasso, Méthode des orbites de Kirillov–Duflo et représentations minimales des groupes simples sur un corps local de caractéristique nulle,Duke Math. J., 90 (1997), 261–377MATHMathSciNetCrossRefGoogle Scholar
  153. 153.
    Ch. Torossian, Opérateurs différentiels invariants sur les espaces symétriques. I. Méthodes des orbites, J. Funct. Anal., 117 (1993), 118–173MATHMathSciNetCrossRefGoogle Scholar
  154. 154.
    Ch. Torossian, Opérateurs différentiels invariants sur les espaces symétriques. II. L’homomorphisme de Harish-Chandra généralisé, J. Funct. Anal., 117 (1993), 174–214MATHMathSciNetCrossRefGoogle Scholar
  155. 155.
    Ch. Torossian, Sur la conjecture combinatoire de Kashiwara–Vergne, J. Lie Theory, 12 (2002), 597–616MATHMathSciNetGoogle Scholar
  156. 156.
    Ch. Torossian, Méthodes de Kashiwara–Verge–Rouvière pour les espaces symétriques, In: Noncommutative Harmonic Analysis, Progr. Math., 220, Birkhäuser Boston, Boston, MA, 2004, pp. 459–486.Google Scholar
  157. 157.
    F. Trèves, Introduction to Pseudodifferential and Fourier Integral Operators. Vol. 1, Pseudodifferential operators, The University Series in Mathematics, Plenum Press, New York-London, 1980.Google Scholar
  158. 158.
    A. Unterberger, Quantization and Non-holomorphic Modular Forms, Lecture Notes in Math., 1742, Springer-Verlag, Berlin, 2000.Google Scholar
  159. 159.
    A. Unterberger and J. Unterberger, La série discrète de \({\rm SL}(2, {\mathbb {R}})\) et les opérateurs pseudo-différentiels sur une demi-droite, Ann. Sci. École Norm. Sup. (4), 17 (1984), 83–116MATHMathSciNetGoogle Scholar
  160. 160.
    A. Unterberger and J. Unterberger, Quantification et analyse pseudo-différentielle, Ann. Sci. École Norm. Sup. (4), 21 (1988), 133–158MATHMathSciNetGoogle Scholar
  161. 161.
    A. Unterberger and J. Unterberger, Algebras of symbols and modular forms, J. Anal. Math., 68 (1996), 121–143MATHMathSciNetCrossRefGoogle Scholar
  162. 162.
    A. Unterberger and H. Upmeier, The Berezin transform and invariant differential operators, Comm. Math. Phys., 164 (1994), 563–597MATHMathSciNetCrossRefGoogle Scholar
  163. 163.
    A.M. Vershik, I.M. Gelfand and M.I. Graev, Representations of the group SL(2, R), where R is a ring of functions, Uspehi Mat. Nauk, 28 (1973), 83–128Google Scholar
  164. 164.
    E.B. Vinberg, Invariant convex cones and orderings in Lie groups, Funct. Anal. Appl., 14 (1980), 1–13MATHMathSciNetGoogle Scholar
  165. 165.
    A. Weil, Sur certains groupes d’opérateurs unitaires, Acta Math., 111 (1964), 143–211MATHMathSciNetCrossRefGoogle Scholar
  166. 166.
    H. Weyl, Gruppentheorie und Quantenmechanik, 2nd ed., S. Hirzel Verlag, Leipzig, 1931.Google Scholar
  167. 167.
    D. Zagier, Introduction to modular forms, In: From Number Theory to Physics, (eds. W. Waldschmidt, P. Moussa, J.-M. Luck and C. Itzykson), Springer-Verlag, Berlin, 1992, pp. 238–291.Google Scholar
  168. 168.
    D. Zagier, Modular forms and differential operators, Proc. Indian Acad. Sci. Math. Sci., 104 (1994), 57–75MATHMathSciNetCrossRefGoogle Scholar
  169. 169.
    G. Zhang, Jordan algebras and generalized principal series representations, Math. Ann., 302 (1995), 773–786MATHMathSciNetCrossRefGoogle Scholar
  170. 170.
    G. Zhang, Berezin transform on compact Hermitian symmetric spaces, Manuscripta Math., 97 (1998), 371–388MATHMathSciNetCrossRefGoogle Scholar
  171. 171.
    G. Zhang, Berezin transform on real bounded symmetric domains, Trans. Amer. Math. Soc., 353 (2001), 3769–3787MATHMathSciNetCrossRefGoogle Scholar
  172. 172.
    G. Zhang, Branching coefficients of holomorphic representations and Segal–Bargmann transform, J. Funct. Anal., 195 (2002), 306–349MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© The Mathematical Society of Japan and Springer Japan 2008

Authors and Affiliations

  1. 1.FRE 3111 du CNRSUniversité de ReimsReimsFrance

Personalised recommendations