Skip to main content

Stein–Sahi Complementary Series and Their Degenerations

  • 1726 Accesses

Abstract

The paper is an introduction to the Stein–Sahi complementary series and to unipotent representations. We also discuss some open problems related to these objects. For the sake of simplicity, we consider only the groups U(n, n).

Keywords

Mathematics Subject Classification (2010):42B35, 22D10

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

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

    Article  MATH  Google Scholar 

  2. Bargmann, V. Irreducible unitary representations of the Lorentz group.Ann. Math, 48 (1947), 568–640

    Article  MATH  MathSciNet  Google Scholar 

  3. Berezin, F.A., Quantization in complex symmetric spaces.Izv. Akad. Nauk SSSR, Ser. Math., 39, 2, 1362–1402 (1975); English translation: Math USSR Izv. 9 (1976), No 2, 341–379(1976)

    Google Scholar 

  4. Berezin, F. A. The connection between covariant and contravariant symbols of operators on classical complex symmetric spaces.Sov. Math. Dokl. 19 (1978), 786–789

    MATH  Google Scholar 

  5. Branson, Th., Olafsson, G., Orsted, B. Spectrum generating operators and intertwining operators for representations induced from a maximal parabolic subgroup,J. Funct. Anal., 135, 163–205.

    Google Scholar 

  6. Delorme, P. Formule de Plancherel pour les espaces symmétrique reductifs.Ann. Math., 147 (1998), 417–452

    Article  MATH  MathSciNet  Google Scholar 

  7. van Dijk, G., Molchanov, V.F. The Berezin form for rank one para-Hermitian symmetric spaces.J. Math. Pure. Appl., 78 (1999), 99–119.

    MATH  Google Scholar 

  8. Dvorsky, A., Sahi, S. Explicit Hilbert spaces for certain unipotent representations. II.Invent. Math. 138 (1999), no. 1, 203–224.

    Article  MATH  MathSciNet  Google Scholar 

  9. Dvorsky, A., Sahi, S. Explicit Hilbert spaces for certain unipotent representations. III.J. Funct. Anal. 201(2003), no. 2, 430–456.

    Google Scholar 

  10. Faraut, J., Koranyi, A., Analysis in symmetric cones.Oxford Univ.Press, (1994)

    Google Scholar 

  11. Flensted-Jensen, M. Discrete series for semisimple symmetric spaces.Ann. of Math. (2) 111 (1980), no. 2, 253–311.

    Google Scholar 

  12. Friedrichs, K. O. Mathematical aspects of the quantum theory of fields.Interscience Publishers, Inc., New York, 1953.

    MATH  Google Scholar 

  13. Gelfand, I. M., Gindikin, S. G. Complex manifolds whose skeletons are semisimple Lie groups and analytic discrete series of representations.Funct. Anal. Appl., 11 (1978), 258–265

    Article  Google Scholar 

  14. Gelfand, I.M., Naimark, M.I., Unitary representations of classical groups.Unitary representations of classical groups.Trudy MIAN., t.36 (1950); German translation: Gelfand I.N., Neumark M.A., Unitare Darstellungen der klassischen gruppen., Akademie-Verlag, Berlin, 1957.

    Google Scholar 

  15. Gradshtein, I.S., Ryzhik, I.M. Tables of integrals, sums and products.Fizmatgiz, 1963; English translation: Acad. Press, NY, 1965

    Google Scholar 

  16. Groenevelt, W., Koelink, E., Rosengren, H. Continuous Hahn polynomials and Clebsch–Gordan coefficients.Theory and applications of special functions, 221–284, Dev. Math., 13, Springer, New York, 2005.

    Google Scholar 

  17. Harish-Chandra, Representations of semisimple Lie groups IV,Amer. J. Math., 743–777 (1955). Reprinted in Harish-Chandra Collected papers,v.2.

    Google Scholar 

  18. Higher transcendental functions, v.1., McGraw-Hill book company, 1953

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  20. Kadell, K. The Selberg–Jack symmetric functions.Adv. Math., 130 (1997), 33-102

    Article  MATH  MathSciNet  Google Scholar 

  21. Kirillov, A.A. Elements of representation theory,Moscow, Nauka, 1972; English transl.: Springer, 1976.

    Google Scholar 

  22. Krattenthaler, C. Advanced determinant calculus.The Andrews Festschrift (Maratea, 1998). Sem. Lothar. Combin. 42 (1999), Art. B42q, 67 pp. (electronic).

    Google Scholar 

  23. Molchanov, V. F.Tensor products of unitary representations of the three-dimensional Lorentz group.Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 4, 860–891, 967. English transl. in Izvestia.

    Google Scholar 

  24. Molchanov, V. F. Quantization on the imaginary Lobachevsky plane.Funct. Anal. Appl., 14 (1980), 162–144

    Article  MathSciNet  Google Scholar 

  25. Neretin, Yu. A. The restriction of functions holomorphic in a domain to a curve lying in the boundary, and discrete \({\mathrm{SL}}_{2}(\mathbb{R})\) -spectra.Izvestia: Mathematics, 62:3(1998), 493–513

    Google Scholar 

  26. Neretin, Yu.A., Matrix analogs of B-function and Plancherel formula for Berezin kernel representations, Mat. Sbornik, 191, No.5 (2000), 67–100;

    Google Scholar 

  27. Neretin, Yu.A., Plancherel formula for Berezin deformation of L 2 on Riemannian symmetric space, J. Funct. Anal. (2002), 189(2002), 336–408.

    Article  MATH  MathSciNet  Google Scholar 

  28. Neretin, Yu.A. Matrix balls, radial analysis of Berezin kernels, and hypergeometric determinants,Moscow Math. J., v.1 (2001), 157–221.

    MATH  MathSciNet  Google Scholar 

  29. Neretin, Yu.A. Notes Sahi–Stein representations and some problems of non-L 2 harmonic analysis., J. Math. Sci., New York, 141 (2007), 1452–1478

    Google Scholar 

  30. Neretin, Yu. A. Notes on matrix analogs of Sobolev spaces and Stein–Sahi representations.Preprint, http://arxiv.org/abs/math/0411419

  31. Neretin, Yu. A. Lectures on Gaussian integral operators and classical groups,to appear.

    Google Scholar 

  32. Neretin, Yu. A. Some continuous analogs of expansion in Jacobi polynomials and vector-valued orthogonal bases.Funct. Anal. Appl., 39 (2005), 31–46.

    Article  MathSciNet  Google Scholar 

  33. Neretin, Yu.A., Olshanskii, G.I., Boundary values of holomorphic functions,singular unitary representations of groups O(p,q) and their limits as q →∞.Zapiski nauchn. semin. POMI RAN 223, 9–91(1995); English translation: J.Math.Sci., New York, 87, 6 (1997), 3983–4035.

    Google Scholar 

  34. Olshanskij, G.I., Complex Lie semigroups, Hardy spaces, and Gelfand–Gindikin program.Deff. Geom. Appl., 1 (1991), 235–246

    Article  MATH  Google Scholar 

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

    Google Scholar 

  36. Pukanszky, L., On the Kronecker products of irreducible unitary representations of the 2 × 2 real unimodular group.Trans. Amer. Math. Soc., 100 (1961), 116–152

    Article  MATH  MathSciNet  Google Scholar 

  37. Pukanzsky, L. Plancherel formula for universal covering group of \(\mathrm{SL}(2, \mathbb{R})\) .Math. Ann., 156 (1964), 96–143

    Article  MathSciNet  Google Scholar 

  38. Ricci, F., Stein, E. M. Homogeneous distributions on spaces of Hermitean matrices.J. Reine Angew. Math. 368 (1986), 142–164.

    MATH  MathSciNet  Google Scholar 

  39. Rosengren, H. Multilinear Hankel forms of higher order and orthogonal polynomials.Math. Scand., 82 (1998), 53-88

    MATH  MathSciNet  Google Scholar 

  40. Sahi, S. A simple construction of Stein’s complementary series representations.Proc. Amer. Math. Soc. 108 (1990), no. 1, 257–266.

    MATH  MathSciNet  Google Scholar 

  41. Sahi, S., Unitary representations on the Shilov boundary of a symmetric tube domain,Contemp. Math. 145 (1993) 275–286.

    MathSciNet  Google Scholar 

  42. Sahi, S. Jordan algebras and degenerate principal series, J. Reine Angew.Math. 462 (1995) 1–18.

    Article  MATH  MathSciNet  Google Scholar 

  43. Sahi, S. Explicit Hilbert spaces for certain unipotent representations.Invent. Math. 110 (1992), no. 2, 409–418.

    Article  MATH  MathSciNet  Google Scholar 

  44. Sahi, S., Stein, E. M. Analysis in matrix space and Speh’s representations.Invent. Math. 101 (1990), no. 2, 379–393.

    Article  MATH  MathSciNet  Google Scholar 

  45. Sally, P. J., Analytic continuations of irreducible unitary representations of the universal covering group of \(\mathrm{SL}(2, \mathbb{R})\) .Amer. Math. Soc., Providence, 1967

    Google Scholar 

  46. Stein, E. M. Analysis in matrix spaces and some new representations ofSL (N, C).Ann. of Math. (2) 86 1967 461–490.

    Google Scholar 

  47. Unterberger, A., Upmeier, H., The Berezin transform and invariant differential operators. Comm.Math.Phys.,164, 563–597(1994)

    Google Scholar 

  48. Vogan, D. A., The unitary dual ofGL (n) over an Archimedean field.Invent. Math. 83 (1986), no. 3, 449–505.

    Article  MATH  MathSciNet  Google Scholar 

  49. Weyl, H. The Classical Groups. Their Invariants and Representations.Princeton University Press, Princeton, N.J., 1939.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Neretin, Y.A. (2012). Stein–Sahi Complementary Series and Their Degenerations. In: Krötz, B., Offen, O., Sayag, E. (eds) Representation Theory, Complex Analysis, and Integral Geometry. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4817-6_7

Download citation

Publish with us

Policies and ethics