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On the quasi-simple irreducible representations of the Lorentz groups


Author: Ernest Thieleker
Journal: Trans. Amer. Math. Soc. 179 (1973), 465-505
MSC: Primary 22E43
DOI: https://doi.org/10.1090/S0002-9947-1973-0325856-0
MathSciNet review: 0325856
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Abstract: For $ n \geq 2$, let $ G(n)$ denote the generalized homogeneous Lorentz group of an $ n + 1$-dimensional real vector space; that is, $ G(n)$ is the identity component of the orthogonal group of a real quadratic form of index $ ( + , - - \ldots - )$. Let $ \hat G(n)$ denote a two-fold covering group of $ G(n)$, and let $ \hat S(n)\hat M(n)$ be a parabolic subgroup of $ \hat G(n)$. We consider the induced representations of $ \hat G(n)$, induced by the finite-dimensional irreducible representations of $ \hat S(n)\hat M(n)$. By an extension of the methods used in a previous paper, we determine precise criteria for the topological irreducibility of these representations. Moreover, in the exceptional cases when these representations fail to be irreducible, we determine the irreducible subrepresentations of these induced representations. By means of some general results of Harish-Chandra together with the main results of this paper, we obtain a complete classification, up to infinitesimal equivalence, of the quasi-simple irreducible representations of the groups $ \hat G(n)$.


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  • [1] V. Bargmann, Irreducible unitary representations of the Lorentz group, Ann. of Math. (2) 48 (1947), 568-640. MR 9, 133. MR 0021942 (9:133a)
  • [2] H. Boerner, Darstellungen von Gruppen. Mit Berücksichtigung der Bedürfnisse der modernen Physik, Zweite, überarbeitete Auflage, Die Grundlehren der math. Wissenschaften, Band 74, Springer-Verlag, Berlin and New York, 1967. MR 37 #5307. MR 0229733 (37:5307)
  • [3] R. Brauer, Sur la multiplication des charactéristiques des groupes continus et semisimples, C. R. Acad. Sci. Paris 204 (1937), 1784-1786.
  • [4] F. Bruhat, Sur les représentations induites des groupes de Lie, Bull. Soc. Math. France 84 (1956), 97-205. MR 18, 907. MR 0084713 (18:907i)
  • [5] C. Chevalley, Theory of Lie groups. I, Princeton Math. Series, vol. 8, Princeton Univ. Press, Princeton, N.J., 1946. MR 7, 412. MR 0082628 (18:583c)
  • [6] J. Dixmier, Représentations intégrables du groupe de De Sitter, Bull. Soc. Math. France 89 (1961), 9-41. MR 25 #4031. MR 0140614 (25:4031)
  • [7] J. M. G. Fell, Non-unitary dual spaces of groups, Acta Math. 114 (1965), 267-310. MR 32 #4210. MR 0186754 (32:4210)
  • [8] R. Godement, A theory of spherical functions. I, Trans. Amer. Math. Soc. 73 (1952), 496-556. MR 14, 620. MR 0052444 (14:620c)
  • [9a] Harish-Chandra, Representations of a semisimple Lie group on a Banach space. I, Trans. Amer. Math. Soc. 75 (1953), 185-243. MR 15, 100. MR 0056610 (15:100f)
  • [9b] -, Representations of a semisimple Lie group. II, Trans. Amer. Math. Soc. 76 (1954), 26-65. MR 15, 398.
  • [9c] -, Representations of a semisimple Lie group. III, Trans. Amer. Math. Soc. 76 (1954), 234-253. MR 16, 11. MR 0062747 (16:11e)
  • [9d] -, The Plancherel formula for complex semisimple Lie groups, Trans. Amer. Math. Soc. 76 (1954), 485-528. MR 16, 111. MR 0063376 (16:111f)
  • [10] S. Helgason, Differential geometry and symmetric spaces, Pure and Appl. Math., vol. 12, Academic Press, New York, 1962. MR 26 #2986. MR 0145455 (26:2986)
  • [11a] T. Hirai, On infinitesimal operators of irreducible representations of the Lorentz group of n-th order, Proc. Japan Acad. 38 (1962), 83-87. MR 25 #2146. MR 0138703 (25:2146)
  • [11b] -, On irreducible representations of the Lorentz group of n-th order, Proc. Japan Acad. 38 (1962), 258-262. MR 32 #8844. MR 0191436 (32:8844)
  • [11c] T. Hirai, The characters of irreducible representations of the Lorentz group of n-th order, Proc. Japan Acad. 41 (1965), 526-531. MR 33 #4185. MR 0195989 (33:4185)
  • [12] M. A. Naĭmark, Linear representations of the Lorentz group, Fizmatgiz, Moscow, 1958; English transl., Macmillan, New York, 1964. MR 21 #4995; MR 30 #1211. MR 0170977 (30:1211)
  • [13] N. Jacobson, Lie algebras, Interscience Tracts in Pure and Appl. Math., no. 10, Interscience, New York, 1962. MR 26 #1345. MR 0143793 (26:1345)
  • [14] B. Kostant, On the existence and irreducibility of certain series of representations, Bull. Amer. Math. Soc. 75 (1969), 627-642. MR 39 #7031. MR 0245725 (39:7031)
  • [15] A. W. Knapp and E. M. Stein, Intertwining operators for semisimple groups, Ann. of Math. (2) 93 (1971), 489-578. MR 0460543 (57:536)
  • [16] P. J. Sally, Jr., Intertwining operators and the representations of $ {\text{SL}}(2,{\mathbf{R}})$, J. Functional Analysis 6 (1970), 441-453. MR 0299727 (45:8775)
  • [17] R. Takahashi, Sur les représentations unitaires des groupes de Lorentz généralisés, Bull. Soc. Math. France 91 (1963), 289-433. MR 0179296 (31:3544)
  • [18] E. Thieleker, On the irreducibility of nonunitary induced representations of certain semidirect products, Trans. Amer. Math. Soc. 164 (1972), 353-369. MR 0293017 (45:2097)
  • [19] A. Weil, L'intégration dans les groupes topologiques et ses applications, 2ième éd., Actualités Sci. Indust., no. 869, Hermann, Paris, 1951. MR 3, 198.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0325856-0
Keywords: Quasi-simple representations, topologically completely irreducible representations, irreducibility criteria for induced representations, classification of quasi-simple representations, representations of real semisimple Lie groups, representations of generalized homogeneous Lorentz groups, representations of Lorentz groups, representations of the deSitter group, nonunitary representations on a Banach space, unitary representations of generalized homogeneous Lorentz groups, infinitesimal equivalence, Naimark equivalence, modules over universal enveloping algebras
Article copyright: © Copyright 1973 American Mathematical Society

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