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Unitary derived functor modules with small spectrum. (English) Zbl 0568.22007
This article has to do with the problem of finding the unitary dual of a real semisimple Lie group G. In order to describe all irreducible unitary representations of G, one needs Zuckerman’s derived functor parabolic induction. The difficulty is that this construction yields irreducible representations which may or may not be unitary. The main general results of this paper give sufficient conditions so that derived functor parabolic induction yields a unitary representation. These results were announced earlier by the same authors [Proc. Natl. Acad. Sci. USA 80, 7047-7050 (1983; Zbl 0527.22007)].
More than a third of the article consists of applications of the general theory to specific settings. These applications include: (1) The authors construct a unitary representation of SO(p,q) with \(p+q\) even. It is multiplicity free as a \(k\)-module and the highest weights of the \(k\)- submodules all lie along a single line. For this reason it is called a ladder representation. Here \(k\) is the complexification of the Lie algebra of the subgroup K which is the maximal connected subgroup of G whose image in G/center is compact. (2) They construct a family of ladder representations for each of the groups Sp(r,s). (3) Several families of unitary reprsentations of Sp(n,\({\mathbb{R}})\) are constructed. (4) The main results are used to give an alternate proof of unitarity for the Speh representations of SL(2n,\({\mathbb{R}})\). Analogous representations of \(SU^*(2n)\) are also described. (5) Unitarity results are obtained for the analytic continuation of certain discrete series representations of G.
Reviewer: Th.Farmer

MSC:
22E46 Semisimple Lie groups and their representations
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
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[1] Adams, J., Some results on the dual pair (O(p, q), Sp(2m)). Yale University thesis, May 1981.
[2] Boe, B., Homomorphisms between generalized Verma modules. Preprint. · Zbl 0568.17004
[3] Bourbaki, N.,Groupes et Algébres de Lie. Chapter VI, Hermann, 1968. · Zbl 0186.33001
[4] 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. · Zbl 0417.17005
[5] 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 · Zbl 0531.22012
[6] Enright, T. J. &Wallach, N. R., Notes on homological algebra and representations of Lie algebras.Duke Math. J., 47 (1980), 1–15. · Zbl 0429.17012
[7] 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.
[8] 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. · Zbl 0527.22007
[9] Enright, T. J. & Wolf, J. A., Continuation of unitary derived functor modules out of the canonical chamber. To appear inMemoires Math. Soc. France, 1984. · Zbl 0582.22013
[10] Flensted-Jensen, M., Discrete series for semisimple symmetric spaces.Ann. of Math., 111 (1980), 253–311. · Zbl 0462.22006
[11] Garland, H. &Zuckerman, G., On unitarizable highest weight modules of Hermitian parirs.J. Fac. Sci. Univ. Tokyo, 28 (1982), 877–889. · Zbl 0499.17004
[12] Jakobsen, H., Hermitian symmetric spaces and their unitary highest weight modules.J. Funct. Anal., 52 (1983), 385–412. · Zbl 0517.22014
[13] Jantzen, J. C., Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren.Math. Ann., 226 (1977), 53–65. · Zbl 0372.17003
[14] Kashiwara, M. &Vergne, M., On the Segal-Shale-Weil representations and harmonic polynomials.Invent. Math., 44 (1978), 1–47. · Zbl 0375.22009
[15] Matsuki, T. & Oshima, T., A description of discrete series for semisimple symmetric spaces. To appear inAdvanced Studies in Pure Mathematics. · Zbl 0577.22012
[16] Parthasarathy, R., An algebraic construction of a class of representations of a semisimple Lie algebra.Math. Ann., 226 (1977), 1–52. · Zbl 0374.17002
[17] –, A generalization of the Enright-Varadarajan modules.Compositio Math., 36 (1978), 53–73. · Zbl 0384.17005
[18] –, Criteria for unitarizability of some highest weight modules.Proc. Indian Acad. Sci., 89 (1980), 1–24. · Zbl 0434.22011
[19] Rawnsley, J., Schmid, W. & Wolf, J., Singular unitary representations and indefinite harmonic theory. To appear inJ. Funct. Anal., 1983. · Zbl 0511.22005
[20] Schlichtkrull, H., A series of unitary irreducible representations induced from a symmetric subgroup of a semisimple Lie group.Invent. Math., 68 (1982), 497–516. · Zbl 0501.22019
[21] Schmid, W., Die Randwerte holomorpher Functionen auf hermitesch symmetrischen Räumen.Invent. Math., 9 (1969), 61–80. · Zbl 0219.32013
[22] –, Some properties of square integrable representations of semisimple Lie groups.Ann. of Math., 102 (1975), 535–564. · Zbl 0347.22011
[23] 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. · Zbl 0283.17001
[24] Speh, B., Unitary representations ofGL(n,R) with non-trivial (g, K)-cohomology.Invent. Math., 71 (1983), 443–465. · Zbl 0505.22015
[25] Vogan, Jr, D.,Representations of real reductive Lie groups. Birkhäuser, Boston-Basel-Stuttgart, 1981. · Zbl 0469.22012
[26] –, Singular unitary representations,Non Commutative Harmonic Analysis and Lie Groups. Lecture Notes in Mathematics, 880. Springer-Verlag, Berlin-Heidelberg-New York, 1981.
[27] Vogan, Jr, D., Unitarizability of certain series of representations. Preprint. · Zbl 0561.22010
[28] Vogan, Jr, D. & Zuckerman, G., Unitary representations with non-zero cohomology. Preprint.
[29] Wallach, N., The analytic continuation of the discrete series I, II.Trans. Amer. Math. Soc., 251 (1979), 1–17, 19–37. · Zbl 0419.22017
[30] Weyl, H.,The Classical Groups. Princeton University Press, 1946. · Zbl 1024.20502
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