Acta Mathematica

  • Acta Math.
  • Volume 154, Number 1-2 (1985), 105-136.

Unitary derived functor modules with small spectrum

T. J. Enright, R. Parthasarathy, N. R. Wallach, and J. A. Wolf

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The first, third and fourth authors have been supported in part by NSF Grants #MCS-8300793, #MCS-7903153 and #MCS-8200235 respectively.

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Acta Math., Volume 154, Number 1-2 (1985), 105-136.

Received: 3 November 1983
Revised: 10 January 1984
First available in Project Euclid: 31 January 2017

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1985 © Almqvist & Wiksell


Enright, T. J.; Parthasarathy, R.; Wallach, N. R.; Wolf, J. A. Unitary derived functor modules with small spectrum. Acta Math. 154 (1985), no. 1-2, 105--136. doi:10.1007/BF02392820.

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  • Adams, J., Some results on the dual pair (O(p, q), Sp(2m)). Yale University thesis, May 1981.
  • Boe, B., Homomorphisms between generalized Verma modules. Preprint.
  • Bourbaki, N., Groupes et Algébres de Lie. Chapter VI, Hermann, 1968.
  • Enright, T. J., On the fundamental series of a real semisimple Lie algebra: their irreducibility, resolutions and multiplicity formulae. Ann. of Math., 110 (1979), 1–82.
  • Enright, T. J., Unitary representations for two real forms of a semisimple Lie algebra: a theory of comparison. Lecture Notes in Mathematics, 1024. Springer-Verlag, 1983
  • Enright, T. J. & Wallach, N. R., Notes on homological algebra and representations of Lie algebras. Duke Math. J., 47 (1980), 1–15.
  • Enright, T. J., Howe, R. & Wallach, N. R., A classification of unitary highest weight modules. Representation Theory of Reductivy Groups (editor P. Trombi). Birkhäuser, Boston, 1982.
  • Enright, T. J., Parthasarathy, R., Wallach, N. R. & Wolf, J. A., Classes of unitarizable derived functor modules. Proc. Nat. Acad. Sci. U.S.A., 80 (1983), 7047–7050.
  • Enright, T. J. & Wolf, J. A., Continuation of unitary derived functor modules out of the canonical chamber. To appear in Memoires Math. Soc. France, 1984.
  • Flensted-Jensen, M., Discrete series for semisimple symmetric spaces. Ann. of Math., 111 (1980), 253–311.
  • Garland, H. & Zuckerman, G., On unitarizable highest weight modules of Hermitian parirs. J. Fac. Sci. Univ. Tokyo, 28 (1982), 877–889.
  • Jakobsen, H., Hermitian symmetric spaces and their unitary highest weight modules. J. Funct. Anal., 52 (1983), 385–412.
  • Jantzen, J. C., Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren. Math. Ann., 226 (1977), 53–65.
  • Kashiwara, M. & Vergne, M., On the Segal-Shale-Weil representations and harmonic polynomials. Invent. Math., 44 (1978), 1–47.
  • Matsuki, T. & Oshima, T., A description of discrete series for semisimple symmetric spaces. To appear in Advanced Studies in Pure Mathematics.
  • Parthasarathy, R., An algebraic construction of a class of representations of a semisimple Lie algebra. Math. Ann., 226 (1977), 1–52.
  • —, A generalization of the Enright-Varadarajan modules. Compositio Math., 36 (1978), 53–73.
  • —, Criteria for unitarizability of some highest weight modules. Proc. Indian Acad. Sci., 89 (1980), 1–24.
  • Rawnsley, J., Schmid, W. & Wolf, J., Singular unitary representations and indefinite harmonic theory. To appear in J. Funct. Anal., 1983.
  • Schlichtkrull, H., A series of unitary irreducible representations induced from a symmetric subgroup of a semisimple Lie group. Invent. Math., 68 (1982), 497–516.
  • Schmid, W., Die Randwerte holomorpher Functionen auf hermitesch symmetrischen Räumen. Invent. Math., 9 (1969), 61–80.
  • —, Some properties of square integrable representations of semisimple Lie groups. Ann. of Math., 102 (1975), 535–564.
  • Shapovalov, N. N., On a bilinear form on the universal enveloping algebra of a complex semisimple Lie algebra. Functional Anal. Appl., 6 (1972), 307–312.
  • Speh, B., Unitary representations of GL(n, R) with non-trivial (g, K)-cohomology. Invent. Math., 71 (1983), 443–465.
  • Vogan, Jr, D., Representations of real reductive Lie groups. Birkhäuser, Boston-Basel-Stuttgart, 1981.
  • —, Singular unitary representations, Non Commutative Harmonic Analysis and Lie Groups. Lecture Notes in Mathematics, 880. Springer-Verlag, Berlin-Heidelberg-New York, 1981.
  • Vogan, Jr, D., Unitarizability of certain series of representations. Preprint.
  • Vogan, Jr, D. & Zuckerman, G., Unitary representations with non-zero cohomology. Preprint.
  • Wallach, N., The analytic continuation of the discrete series I, II. Trans. Amer. Math. Soc., 251 (1979), 1–17, 19–37.
  • Weyl, H., The Classical Groups. Princeton University Press, 1946.