A Classification of Unitary Highest Weight Modules

  • Thomas Enright
  • Roger Howe
  • Nolan Wallach
Part of the Progress in Mathematics book series (PM, volume 40)


Let G be a simply connected, connected simple Lie group with center Z. Let K be a closed maximal subgroup of G with K/Z compact and let g be the Lie algebra of G. A unitary representation (π,H) of G such that the underlying (ℊK) — module is an irreducible quotient of a Verma module for ℊ is called a unitary highest weight module. Harish-Chandra ([4],[5]) has shown that G admits nontrivial unitary highest weight modules precisely when (G,K) is a Hermitian symmetric pair. In this paper we give a complete classification of the unitary highest weight modules.


Root System Simple Root Verma Module Trivial Representation High Weight Vector 
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  1. 1.
    N. Bourbaki, “Groupes et Algèbres de Lie,” Chap. IV,V,VI, Act. Sci. Ind. 1337, Hermann, Paris, 1968.Google Scholar
  2. 2.
    T. J. Enright and R. Parthasarathy, A proof of a conjecture of Kashiwara and Vergne, “Non Commutative Harmonic Analysis and Lie Groups” Lecture Notes in Mathematics, 880, Springer Verlag, 1981.Google Scholar
  3. 3.
    H. Garland and G. Zuckerman, On unitarizable highest weight modules of Hermitian pairs, J. Fac. Sci. Tokyo, 28 (1982), 877–889.Google Scholar
  4. 4.
    Harish-Chandra, Representations of semisimple Lie groups, IV, Amer. J. Math., 77 (1955), 743–777.CrossRefGoogle Scholar
  5. 5.
    -, Representations of semisimple Lie groups V, Amer. J. Math., 78 (1956), 1–41.CrossRefGoogle Scholar
  6. 6.
    S. Helgason, “Differential Geometry and Symmetric Spaces”, Academic Press, New York, 1962.Google Scholar
  7. 7.
    R. Howe, Remarks on Classical Invariant Theory, preprint.Google Scholar
  8. 8.
    -, On a notion of rank for unitary representation of classical groups, Proceedings Cime Conference on Non Abelian Harmonic Analysis, Cortona, July, 1980.Google Scholar
  9. 9.
    H. Jakobsen, On singular holomorphic representations, Inv. Math. 62(1980), 67–78.CrossRefGoogle Scholar
  10. 10.
    -, The last possible place of unitarity for certain highest weight modules, preprint.Google Scholar
  11. 11.
    H. Jakobsen, Hermitian symmetric spaces and their unitary highest weight modules, preprint.Google Scholar
  12. 12.
    J. C. Jantzen, Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren, Math. Ann. 226 (1977), 53–65.CrossRefGoogle Scholar
  13. 13.
    -, “Moduln mit einem höchsten Gewicht”, Lecture Notes in Mathematics, No. 750, Springer-Verlag, Berlin/Heidelberg/New York, 1979.Google Scholar
  14. 14.
    M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representations and harmonic polynomials, Inv. Math. 44 (1978), 1–47.CrossRefGoogle Scholar
  15. 15.
    R. Parthasarathy, Criteria for the unitarizability of some highest weight modules, Proc. Indian Acad. Sci. 89 (1980), 1–24.CrossRefGoogle Scholar
  16. 16.
    J. Rawnsley, W. Schmid and J. Wolf, Singular unitary representations and indefinite harmonic theory, preprint, 1981.Google Scholar
  17. 17.
    N. R. Wallach, The analytic continuation of the discrete series I, II, T.A.M.S. 251 (1979), 1–17, 19–37.Google Scholar

Copyright information

© Birkhäuser Boston, Inc. 1983

Authors and Affiliations

  • Thomas Enright
  • Roger Howe
  • Nolan Wallach

There are no affiliations available

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