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Boundary values of holomorphic functions, singular unitary representations ofO(p,q), and their limits asq→∞

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Abstract

Let Ω be a bounded circular domain in ℂN, let M be a submanifold in the boundary of Ω, and let H be a Hilbert space of holomorphic functions in Ω. We show that, under certain conditions stated in terms of the reproducing kernel of the space H, the restriction operator to the submanifold M is well defined for all functions from H. We apply this result to constructing a family of “singular” unitary representations of the groups SO(p,q). The singular representations arise as discrete components of the spectrum in the decomposition of irreducible unitary highest weight representations of the groups U(p,q) restricted to the subgroups SO(p,q). Another property of the singular representations is that they persist in the limit as q→∞. Bibliography: 70 titles.

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References

  1. A. B. Aleksandrov, “Function theory in the ball,”Sovremennye Problemy Matematiki. Fundamental'nye Napravleniya [in Russian]8, 115–190 (1985).

    MATH  Google Scholar 

  2. E. F. Bekenbach, R. Bellman,Inequalities, Ergeb. Math. (1960).

  3. F. A. Berezin,The Method of Second Quantization [in Russian], Nauka (1965).

  4. F. A. Berezin, “Quantization in complex symmetric spaces”,Izv. Akad. Nauk SSSR, Ser. Mat.,39. No. 2, 1362–1402 (1975).

    MathSciNet  Google Scholar 

  5. A. M. Vershik, I. M. Gel'fand, M. I. Graev, “Representations ofSL(2,R) whereR is a function ring”,Usp. Mat. Nauk. 28, No. 5, 83–128 (1973).

    Google Scholar 

  6. N. Ya. Vilenkin,Special Functions and Theory of Group Representation [in Russian], Nauka (1965).

  7. I. M. Gel'fand and G. E. Shilov,Generalized Functions [in Russian], Fizmatgiz (1958).

  8. Yu. V. Egorov and M. A. Shubin, “Linear partial differential equations. Elements of the Classical Theory,”Sov. Probl. Mat. Fund. Nap.,30 (1988).

  9. A. A. Kirillov,Elements of Representation Theory [in Russian], Nauka (1972).

  10. S. V. Kislyakov, “Exceptional sets in harmonic analysis”,Sov. Probl. Mat. Fund. Nap.,42, 199–228 (1988).

    MathSciNet  Google Scholar 

  11. M. G. Krein, “Hermitian-positive kernels on homogeneous spaces”,Ukr. Mat. Zhurn.,1, No. 4, 64–98 (1949).

    MATH  MathSciNet  Google Scholar 

  12. B. F. Molchanov, “Restriction of the complementary series representation to the pseudoorthogonal group of a lower dimension”,Dokl. Akad. Nauk SSSR,237, No. 4 (1977).

    Google Scholar 

  13. V. F. Molchanov, “Plancherel formula for hyperboloid”,Tr. Mat. Inst. Steklov,147, 65–85 (1980).

    MATH  MathSciNet  Google Scholar 

  14. V. F. Molchanov, “Harmonic analysis on homogeneous spaces”,Sov. Probl. Mat. Fund. Nap.,59, 5–144 (1990).

    MATH  MathSciNet  Google Scholar 

  15. M. A. Naimark, “Decomposition of the tensor product of irreducible representations of the proper Lorentz group into irreducible components III”,Tr. Mosk. Mat. Obshch.,10, 181–216 (1961).

    MathSciNet  Google Scholar 

  16. Yu. A. Neretin, “On discrete occurences of the complement series representations in a tensor product of unitary representations”,Funk. Anal. Pril.,20, No. 1, 79–80 (1986).

    MATH  MathSciNet  Google Scholar 

  17. Yu. A. Neretin, “Representations of the Virasoro algebra and of affine algebras”,Sov. Probl. Mat. Fund. Nap.,22, 163–224 (1988).

    MATH  MathSciNet  Google Scholar 

  18. Yu. A. Neretin, “On a semigroup of operators in the boson Fock space”,Funkt. Anal. Pril.,24, No. 2, 63–73 (1990).

    MATH  MathSciNet  Google Scholar 

  19. G. I. Ol'shanskii, “Unitary representations of infinite-dimensional classical groupsU(p, ∞),SO 0(p, ∞), Sp (p, ∞) in the corresponding motion groups”,Funkt. Anal. Pril.,12, No. 3, 32–44 (1978).

    MathSciNet  Google Scholar 

  20. G. I. Ol'shanskii, “Description of the highest weight unitary representations for groupsU (p, q)∼”,Funkt. Anal. Pril.,14, No. 3, 32–44 (1980).

    MATH  MathSciNet  Google Scholar 

  21. G. I. Ol'shanskii, “Infinite-dimensional classical groups of finiteR-rank: a description of representations and asymptotic theory”,Funkt. Anal. Pril.,18, No. 1, 28–42 (1984).

    MATH  MathSciNet  Google Scholar 

  22. G. I. Ol'shanskii, “Irreducible unitary represrntations of the groupsU(p, q) which admit the pass to the limit asq→∞”,Zap. Nauchn. Semin. LOMI,172, 114–120 (1989).

    MATH  Google Scholar 

  23. A. M. Perelomov,Generalized Coherent States and Their Applications [in Russian], Nauka (1987).

  24. I. I. Pyatetskij-Shapiro,Geometry of Classical Domains and the Theory of Automorphic Functions [in Russian], Fizmatgiz (1961).

  25. W. Rudin,Function Theory in the Unit Ball of ℂ N, Springer-Verlag, Berlin (1980).

    Google Scholar 

  26. M. Taylor,Pseudodifferential Operators, Springer-Verlag, Berlin (1980).

    Google Scholar 

  27. S. Helgasson,Differential Geometry, Lie Groups, and Symmetric Spaces Academic Press, New York (1962).

    Google Scholar 

  28. C. K. Hua Lo Ken,Harmonic Analysis of Functions in Several Complex Variables in Classical Domains [in Russian], Moscow (1959).

  29. B. V. Shabat,Introduction to Complex Analysis [in Russian],2, Nauka (1976).

  30. J. D. Adams, “Discrete spectrum of reductive dual pair (O(p, q), Sp (2m))”,Inv. Math.,74, 449–475 (1983).

    Article  MATH  Google Scholar 

  31. J. D. Adams, D. Barbash, and D. Vogan,The Langlands Classification and Irreducible Characters of Real Reductive Groups, Birkhauser (1992).

  32. V. Bargmann, “Irreducible unitary representations of Lorentz group”,Ann. Math.,48, 568–640 (1947).

    MATH  MathSciNet  Google Scholar 

  33. Ch. P. Boyer, “On complementary series ofSO 0(p, 1)”,J. Math. Phys.,14, No. 5, 609–617 (1973).

    MATH  Google Scholar 

  34. E. Cartan, “Sur les domaines bornés homogènes de l'espace den variables complexes”,Abhandl. Mat. Semin Univ. Hamburg, 116–162 (1936).

  35. T. J. Enright, R. Howe, and N. Wallach, “A classification of unitary highest weight modules”,Representation Theory of Reductive Groups, Birkhauser, Boston, 97–143 (1983).

    Google Scholar 

  36. M. Flensted-Jensen, “Discrete series for semisimple symmetric spaces”,Ann. Math.,111, 253–311 (1980).

    MATH  MathSciNet  Google Scholar 

  37. Harish-Chandra, “Representations of semisimple Lie groups IV”,Am. J. Math., 743–777 (1955).

  38. R. Howe, “On some results of Strichartz and of Rallis and Schiffmann”,J. Funct. Anal.,32, 297–303 (1979).

    MATH  MathSciNet  Google Scholar 

  39. M. Kashiwara and M. Vergne, “On the Segal-Shale-Weil representation and the harmonic polynomials”,Inv. Math.,44, 1–47 (1978).

    Article  MathSciNet  Google Scholar 

  40. A. Knapp,Reperesentation Theory of Semisimple Groups, Princeton Univ. Press (1986).

  41. T. Kobayashi, “Singular unitary representations and discrete series for indefinite Stiefel manifoldsU(p, q;F)/U(p - m, q;F)”,Mem. AMS,95 (1992).

  42. Yu. A. Neretin, “Mantles, trains and representations of infinite dimensional groups”,First European Congress of Mathematics, Vol. 2, Birkhauser, Boston (1994), pp. 293–310.

    Google Scholar 

  43. Yu. A. Neretin, “Integral operators with Gauss kernels”,Berezin Memorial Collection (to appear).

  44. G. I. Ol'shanskii, “Unitary representation of infinite dimensional pairs (G, K) and the formalism of R. Howe”, in:Representations of Lie Groups and Related Topics, Gordon and breach, (1990), pp. 269–464.

  45. L. Pukanszky, “On the Kronecker products of irreducible representations of 2×2 real unimodular group I,”Trans. Am. Math. Soc.,100, No. 1, 116–152 (1961).

    MATH  MathSciNet  Google Scholar 

  46. J. Rawnsley, W. Schmid, and J. A. Wolf, “Singular unitary representations and indefinite harmonic theory,”J. Funct. Anal.,51, 1–114 (1983).

    Article  MathSciNet  Google Scholar 

  47. H. Rossi and M. Vergne, “Analytic continuation of the holomorphic discrete series of a semisimple Lie group,”Acta Math.,136, No. 1-2, 1–59 (1976).

    MathSciNet  Google Scholar 

  48. H. Schlichtkrull, “A series of unitary irreducible representations induced from symmetric subgroup of a semisimple Lie group,”Inv. Math.,68, 497–516 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  49. S. Strichartz, “Harmonic analysis on hyperboloids,”J. Funct. Anal.,12, 341–383 (1973).

    Article  MATH  MathSciNet  Google Scholar 

  50. Tsuchikawa, “The Plancherel transform onSL 2(k) and its application to tensor products of irreducible representations,”J. Math. Kyoto Univ.,22, No. 3, 369–437 (1982).

    MATH  MathSciNet  Google Scholar 

  51. N. R. Wallach, “Analytic continuation of discrete series,”Trans. Am. Math. Soc.,251, 19–37 (1979).

    MATH  MathSciNet  Google Scholar 

  52. H. Weyl, “Inequalities between two kinds of eigenvalues of linear transformation,”Proc. Nat. Acad. Sci.,35, 408–411 (1949).

    MATH  MathSciNet  Google Scholar 

  53. G. Zuckerman, “Tensor products of infinite dimensional and finite dimensional representations of semisimple Lie group,”Ann. Math.,106, 295–308 (1977).

    MATH  MathSciNet  Google Scholar 

  54. I. M. Gel'fand and M. A. Naimark, “Unitary representations of the Lorentz group,”Izv. Akad. Nauk SSSR, Ser. Mat.,11, 411–504 (1947).

    MathSciNet  Google Scholar 

  55. J. Faraut and A. Koranyi,Analysis on Symmetric Cones, Oxford Univ. Press (1994).

  56. S. G. Gindikin, “Invariant distributions in homogeneous domains,”Funkt. Anal. Pril.,9, No. 1, 56–58 (1975).

    MATH  MathSciNet  Google Scholar 

  57. H. P. Jakobsen, “The last possible place of unitarity for certain highest weight modules,”Math. Ann.,256, 439–447 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  58. A. Barut and R. Ronchka,Group Representation Theory and Its Applications [Russian translation]. Mir, Moscow (1980).

    Google Scholar 

  59. I. M. Gelfand and M. I. Graev, “Principal representations of the groupU(∞),” in:Representations of Lie Groups and Related Topics, A. M. Vershik and D. P. Zhelobenko (eds.), Gordon and Breach (1993), pp. 119–154.

  60. H. P. Jakobsen and M. Vergne, “Restrictions and expansions of holomorphic representations,”J. Funct. Anal.,34, 29–53 (1979).

    Article  MathSciNet  Google Scholar 

  61. J. Faraut, “Distributions spheriques sur les espaces hyperboliques,”J. Math. Pures Appl.,58, 369–444 (1979).

    MATH  MathSciNet  Google Scholar 

  62. V. F. Molchanov, “Representations of the pseudoorthogonal group related to a cone”,Mat. Sb.,81, 358–375 (1970).

    MATH  MathSciNet  Google Scholar 

  63. T. Kobayashi, “Discrete decomposability of the restriction ofA q(λ) with respect to reductive subgroups and its applications”,Inv. Math.,117, 181–205 (1994).

    Article  MATH  Google Scholar 

  64. R. S. Ismagilov, “On representations of the Lorentz group which are unitary in an indefinite metric”,Tr. MIEM,2, 492–504 (1966).

    Google Scholar 

  65. L. Carleson,Selected Problems on Exceptional Sets, Van Nostrand, Princeton (1967).

    Google Scholar 

  66. V. S. Vladimirov and A. G. Sergeev, “Complex analysis in the future tube”,Sov. Prob. Mat. Fund. Nap.,8, 191–266 (1985).

    MathSciNet  Google Scholar 

  67. T. J. Enright and A. Joseph, “An intrinsic analysis of unititarizable nighest weight modules”,Math. Ann.,288, 571–594 (1990).

    Article  MathSciNet  Google Scholar 

  68. A. Joseph, “Annihilators and associated weight modules”,Ann. Sci. Ecole Norm. Sup.,25, 1–45 (1992).

    MATH  MathSciNet  Google Scholar 

  69. B. Kostant, “On tensor product of finite and infinite dimensional representations”,J. Funct. Anal.,20, 257–285 (1975).

    Article  MATH  MathSciNet  Google Scholar 

  70. C. Fefferman, “Inequalities for Strongly Singular convolution operators”,Acta Math.,124, 9–36 (1970)

    MATH  MathSciNet  Google Scholar 

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Authors
  1. Yu. A. Neretin
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  2. G. I. Ol'shanskii
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Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 9–91.

Translated by B. Bekker.

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Neretin, Y.A., Ol'shanskii, G.I. Boundary values of holomorphic functions, singular unitary representations ofO(p,q), and their limits asq→∞. J Math Sci 87, 3983–4035 (1997). https://doi.org/10.1007/BF02355796

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