Skip to main content
SpringerLink
Log in

Unitary representations of unimodular Lie groups in Bergman spaces

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

For an arbitrary unimodular Lie group G, we construct strongly continuous unitary representations in the Bergman space of a strongly pseudoconvex neighborhood of G in the complexification of its underlying manifold. These representation spaces are infinite-dimensional and have compact kernels. In particular, the Bergman spaces of these natural manifolds are infinite-dimensional.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arnal D., Ludwig J.: Q.U.P. and Paley–Wiener properties of unimodular, especially nilpotent, Lie groups. Proceed. Am. Math. Soc. 125, 1071–1080 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Atiyah M.F.: Elliptic operators, discrete groups, and von Neumann algebras. Soc. Math. de France, Astérisque 32(3), 43–72 (1976)

    MathSciNet  Google Scholar 

  3. Brudnyi A.: Representation of holomorphic functions on coverings of pseudoconvex domains in Stein manifolds via integral formulas on these domains. J. Funct. Anal. 231, 418–437 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brudnyi A.: On holomorphic L 2 functions on coverings of strongly pseudoconvex manifolds. Publ. Res. Inst. Math. Sci. 43, 963–976 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brudnyi A.: On holomorphic functions of slow growth on coverings of strongly pseudoconvex manifolds. J. Funct. Anal. 249, 354–371 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Duistermaat J.J., Kolk J.A.C.: Lie groups, Universitext. Springer, Berlin (2000)

    Book  Google Scholar 

  7. Engliš M.: Pseudolocal estimates for \({\bar\partial}\) on general pseudoconvex domains. Indiana Univ. Math. J. 50, 1593–1607 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Folland G.B.: Compact Heisenberg manifolds as CR manifolds. J. Geom. Anal. 14, 521–532 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Folland G.B., Kohn J.J.: The Neumann Problem for the Cauchy–Riemann Complex, Ann. Math. Studies, No. 75. Princeton University Press, Princeton (1972)

    Google Scholar 

  10. Folland G.B., Stein E.M.: Estimates for the \({\bar\partial_b}\) -complex and analysis on the Heisenberg group. Commun. Pure Appl. Math. 27, 429–522 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  11. Grauert H.: Bemerkenswerte pseudokonvexe Manninfaltigkeiten. Math. Z. 81, 377–391 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gromov M., Henkin G., Shubin M.: Holomorphic L 2 functions on coverings of pseudoconvex manifolds. Geom. Funct. Anal. 8, 552–585 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Heinzner, P., Huckleberry, A.T., Kutzschebauch, F.: Abels’ theorem in the real analytic case and applications to complexifications. In: Complex Analysis and Geometry. Lecture Notes in Pure and Applied Mathematics. Marcel Dekker, The Netherlands, pp. 229–273 (1995)

  14. Kohn J.J.: Harmonic Integrals on strongly pseudoconvex manifolds, I. Ann. Math. 78, 112–148 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kohn J.J.: Harmonic integrals on strongly pseudoconvex manifolds, II. Ann. Math. 79, 450–472 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kohn J.J., Nirenberg L.: Non-coercive boundary value problems. Commun. Pure Appl. Math. 18, 443–492 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lienau, C.: Analytic Dirac approximation for real linear algebraic groups. Math. Ann. http://dx.doi.org/10.1007/s00208-010-0607-2

  18. Margulis G.A.: Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 17. Springer, Berlin (1991)

    Google Scholar 

  19. Perez, J.J.: The G-Fredholm property of the \({\bar\partial}\) -Neumann problem. J. Geom. Anal. 19, 87–106 (2009). http://www.springerlink.com/content/n1wg14w451n4r653

  20. Perez, J.J.: The Levi problem on strongly pseudoconvex G-bundles. Ann. Glob. Anal. Geom. 37, 1–20 (2010). http://www.springerlink.com/content/d0lu28p4562q888m/

  21. Perez J.J.: A transversal Fredholm property for the \({\bar\partial}\) -Neumann problem on G-bundles. Contemp. Math. 535, 187–193 (2011)

    Article  Google Scholar 

  22. Todor R., Chiose I., Marinescu G.: L 2 holomorphic sections of bundles over weakly pseudoconvex coverings. Geom. Dedicata 91, 23–43 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Todor R., Chiose I., Marinescu G.: Morse inequalities for covering manifolds. Nagoya Math. J. 163, 145–165 (2001)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Universität Wien, Vienna, Austria

    Giuseppe Della Sala & Joe J. Perez

Authors
  1. Giuseppe Della Sala
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joe J. Perez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giuseppe Della Sala.

Additional information

GDS is supported by FWF grant Y377, Biholomorphic Equivalence: Analysis, Algebra and Geometry. JJP is supported by grants P19667 of the FWF, and the I382 joint project of the ANR-FWF.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Della Sala, G., Perez, J.J. Unitary representations of unimodular Lie groups in Bergman spaces. Math. Z. 272, 483–496 (2012). https://doi.org/10.1007/s00209-011-0945-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-011-0945-0

Mathematics Subject Classification (2000)

Search

Navigation