Skip to main content
SpringerLink
Log in

The unitary dual of GL(n) over an archimedean field

  • Published:
Inventiones mathematicae Aims and scope

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.

References

  1. Arthur, J.: On some problems suggested by the trace formula In: Proceedings of the Special Year in Harmonic Analysis, University of Maryland, Herb, R., Lipsman, R., Rosenberg, J. (eds): Lect. Notes Math.1024, Berlin-Heidelberg-New York-Tokyo:Springer 1983

    Google Scholar 

  2. Barbasch, D., Vogan, D.: Unipotent representations of complex semisimple Lie groups. Ann. Math.121, 41–110 (1985)

    Google Scholar 

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

    Google Scholar 

  4. Beilinson, A., Bernstein, J.: Localisation de g-modules. C. R. Acad. Sci., Paris292, 15–18 (1981)

    Google Scholar 

  5. Bernstein, I.N., Gelfand, I. M., Gelfand, S.I.: Models of representations of compact Lie groups. Funct. Anal. Appl.9, 61–62 (1975)

    Google Scholar 

  6. Borho, W., Brylinski, J.-L.: Differential operators on homogeneous spaces I. Invent. Math.69, 437–476 (1982)

    Google Scholar 

  7. Conze-Berline, N., Duflo, M.: Sur les représentations induites des groupes semi-simples complexes. Compos. Math.34, 307–336 (1977)

    Google Scholar 

  8. Duflo, M.: Sur la classification des idéaux primitifs dans l'algèbre enveloppante d'une algèbre de Lie semi-simple. Ann. Math.105, 107–120 (1977)

    Google Scholar 

  9. Duflo, M.: Représentations unitaires des groupes semi-simples complexes. pp. 19–34. In: Group Theoretical Methods in Physics. Proc. Eighth Internat. Colloq., Kiryat Anavim, 1979. Ann. Israel Phys. Soc.3, Bristol: Hilger, 1980

    Google Scholar 

  10. Enright, T. J.: Relative Lie algebra cohomology and unitary representations of complex Lie groups. Duke Math. J.46, 513–525 (1979)

    Google Scholar 

  11. Enright, T.J., Wallach, N.R.: Notes on homological algebra and representations of Lie algebras. Duke Math. J.47, 1–15 (1980)

    Google Scholar 

  12. Gelfand, I.M.: Some aspects of functional analysis and algebra. pp. 253–276. In:Proceedings of the International Congress of Mathematicians 1954 (Amsterdam). Vol. I. Groningen: Erven P. Noordhoof, N. V., Amsterdam: North Holland Publishing Co., 1957

    Google Scholar 

  13. Gelfand, I.M., Naimark, M.A.: Unitary representations of the Lorentz group. Izv. Akad. Nauk SSSR11, 411–504 (1947)

    Google Scholar 

  14. Gelfand, I.M., Naimark, M.A.: Unitary representations of the classical groups, Trudy Mat. Inst. Steklov36 (1950). German translation; Berlin: Akademie-Verlag, 1957

    Google Scholar 

  15. Guillemonat, A.: Représentations sphériques singulières. In: Non-commutative Harmonic Analysis and Lie Groups, Carmona, J., Vergne, M. (eds), Lect. Notes Math.1020. Berlin-Heidelberg-New York-Tokyo: Springer-Verlag 1983

    Google Scholar 

  16. Harish-Chandra: Representations of semi-simple Lie groups I, Trans. Amer. Math. Soc.75, 185–243 (1953)

    Google Scholar 

  17. Helgason, S.: Groups and Geometric Analysis, Orlando-Florida: Academic Press 1984

    Google Scholar 

  18. Hirai, T.: On irreducible representations of the Lorentz group ofn-th order. Proc. Jap. Acad.38, 83–87 (1962)

    Google Scholar 

  19. Joseph, A.: Sur la classification des idéaux primitifs dans l'algèbre de sl (n+1,ℂ). C. R. Acad. Paris287, 302–306 (1978)

    Google Scholar 

  20. Joseph, A.: On the classification of primitive ideals in the enveloping algebra of a semisimple Lie algebra. In: Lie group representations I, Herb, R., Lipsman, R., Rosenberg, J. (eds): Lect. Notes Math.1024, Berlin-Heidelberg-New York-Tokyo: Springer 1983.

    Google Scholar 

  21. Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math.53, 165–184 (1979)

    Google Scholar 

  22. Knapp, A. W.: Representation theory of real semisimple groups: an overview based on examples (to appear)

  23. Knapp, A. W., Speh, B.: The role of basic cases in classification: Theorems about unitary representations applicable toSU (N, 2), 119–160. In: Non-commutative Harmonic Analysis and Lie Groups, Carmona, J., Vergne, M.: (eds.): Lect. Notes Math.1020. Berlin-Heidelberg-New York-Tokyo: Springer 1983

    Google Scholar 

  24. Kostant, B.: On the existence and irreducibility of certain series of representations. Bull. Am. Math. Soc.75, 627–642 (1969)

    Google Scholar 

  25. Kraft, H., Procesi, C.: Closures of conjugacy classes of matrices are normal. Invent. Math.53, 227–247 (1979)

    Google Scholar 

  26. Naimark, M.A.: On the description of all unitary representations of the complex classical groups I. Mat. Sb.35, 317–356 (1954)

    Google Scholar 

  27. Naimark, M.A.: On the description of all unitary representations of the complex classical groups II. Mat. Sb.37, 121–140 (1955)

    Google Scholar 

  28. Speh, B.: Some results on principal series for GL (n, ℝ). Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, Massachusetts, June, 1977

    Google Scholar 

  29. Speh, B.: Unitary representations ofSL (n, ℝ) and the cohomology of congruence subgroups. 483–505 in Non-commutative Harmonic Analysis and Lie groups, Carmona, J., Vergne, M. (eds.): Lect. Notes Math.880. Berlin-Heidelberg-New York: Springer 1981

    Google Scholar 

  30. Speh, B.: The unitary dual of GL (3, ℝ) and GL (4, ℝ). Math. Ann.258, 113–133 (1981)

    Google Scholar 

  31. Stein, E. M.: Analysis in matrix spaces and some new representations of SL (n,ℂ). Ann. Math.86, 461–490 (1967)

    Google Scholar 

  32. Tadic, M.: Unitary dual ofp-adic GL (n). Proof of Bernstein conjectures. Bull. Am., Math. Soc. (N.S.)13, 39–42 (1985)

    Google Scholar 

  33. Tsuchikawa, M.: On the representations of SL (3,ℂ),III.Proc. Jap. Acad.44, 130–132 (1968)

    Google Scholar 

  34. Vakhutinski, I.: Unitary representations of GL (3, ℝ). Mat. Sb.75, 303–320 (1968)

    Google Scholar 

  35. Vogan, D.: The algebraic structure of the representations of semisimple Lie groups I. Ann. Math.109, 1–60 (1979)

    Google Scholar 

  36. Vogan, D.: Representations of Real Reductive Lie Groups. Boston-Basel-Stuttgart: Birkh tuser 1981

    Google Scholar 

  37. Vogan, D.: Understanding the unitary dual. In: Proceedings of the Special Year in Harmonic Analysis, University of Maryland, Herb, R., Lipsman, R., Rosenberg, J. (eds.) Lect. Notes Math.1024, Berlin-Heidelberg-New York-Tokyo: Springer 1983

    Google Scholar 

  38. Vogan, D.: Unitarizability of certain series of representations. Ann. Math.120, 141–187 (1984)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, 02139, Cambridge, MA, USA

    David A. Vogan Jr.

Authors
  1. David A. Vogan Jr.
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Sloan Fellow. Supported in part by NSF grant MCS-8202127

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vogan, D.A. The unitary dual of GL(n) over an archimedean field. Invent Math 83, 449–505 (1986). https://doi.org/10.1007/BF01394418

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01394418

Search

Navigation