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Unitary representations of Lie groups with cocompact radical and applications


Author: L. Pukanszky
Journal: Trans. Amer. Math. Soc. 236 (1978), 1-49
MSC: Primary 22E30
DOI: https://doi.org/10.1090/S0002-9947-1978-0486313-9
MathSciNet review: 0486313
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Abstract: The paper gives a necessary and sufficient condition in order that a connected and simply connnected Lie group with cocompact radical be of type I. This result is then applied to a characterization of Lie groups, all irreducible unitary representations of which are completely continuous.


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  • [1] L. Auslander and B. Kostant, Polarization and unitary representations of solvable Lie groups, Invent. Math. 14 (1971), 255-354. MR 45 #2092. MR 0293012 (45:2092)
  • [2] L. Auslander and C. C. Moore, Unitary representations of solvable Lie groups, Mem. Amer. Math. Soc. No. 62 (1966), MR 34 #7723. MR 0207910 (34:7723)
  • [3] N. Bourbaki, Eléménts de mathématique. XXVI. Chap. 1: Algébres de Lie, Actualités Sci. Indust., no. 1285, Hermann, Paris, 1960. MR 24 # A2641.
  • [4] I. D. Brown, Dual topology of a nilpotent Lie group, Ann. Sci. École Norm. Sup. (4) 6 (1973), 407-411. MR 50 #4813. MR 0352326 (50:4813)
  • [5] J. Dixmier, Sur les representations unitaires des groupes de Lie nilpotents. V, Bull. Soc. Math. France 87 (1959), 65-79. MR 22 #5900a. MR 0115097 (22:5900a)
  • [6] -, Les $ {C^\ast}$ algébres et leurs représentations, Cahiers Scientifiques, fasc. 29, Gauthier-Villars, Paris, 1964. MR 30 # 1404. MR 0171173 (30:1404)
  • [7] -, Sur la représentation régulière d'un groupe localement compacte connexe, Ann. Sci. École Norm. Sup. (4) 2 (1969), 423-436. MR 41 #5553. MR 0260933 (41:5553)
  • [8] M. Duflo, Sur les extensions des représentations irréductibles des groupes de Lie nilpotents, Ann. Sci. École Norm. Sup. (4) 5 (1972), 71-120. MR 46 # 1966. MR 0302823 (46:1966)
  • [9] J. Glimm, Locally compact transformation groups, Trans. Amer. Math. Soc. 101 (1961), 124-138. MR 25 #146. MR 0136681 (25:146)
  • [10] Harish-Chandra, Representations of semisimple Lie groups. III, Trans. Amer. Math. Soc. 76 (1954), 234-253. MR 16, 11. MR 0062747 (16:11e)
  • [11] B. Kostant, Quantization and unitary representations. Lecture Notes in Math., vol. 170, Springer-Verlag, Berlin, 1970, pp. 87-208. MR 45 #3638. MR 0294568 (45:3638)
  • [12] R. Lipsman, The CCR property for algebraic groups, Amer. J. Math. 97 (1975), 741-752. MR 0390123 (52:10949)
  • [13] G. W. Mackey, Unitary representations of groups extensions. I, Acta Math. 99 (1958), 265-311. MR 20 #4789. MR 0098328 (20:4789)
  • [14] C. C. Moore and J. Rosenberg, Groups with $ {T_1}$ primitive ideal spaces, J. Functional Analysis 22 (1976), 204-224. MR 0419675 (54:7693)
  • [15] L. Pukańszky, On the characters and the Plancherel formula of nilpotent groups, J. Functional Analysis 1 (1967), 255-280. MR 37 #4236. MR 0228656 (37:4236)
  • [16] -, On the unitary representations of exponential groups, J. Functional Analysis 2 (1968), 73-113. MR 37 #4205. MR 0228625 (37:4205)
  • [17] -, Characters of algebraic solvable groups, J. Functional Analysis 3 (1969), 435-494. MR 40 #1539. MR 0248287 (40:1539)
  • [18] -, Unitary representations of solvable Lie groups, Ann. Sci. École Norm. Sup. 4 (1971), 457-608. MR 0439985 (55:12866)
  • [19] -, Action of algebraic groups of automorphisms on the dual of a class of type I groups, Ann. Sci. Ecole Norm. Sup. (4) 5 (1972), 379-395. MR 48 #466. MR 0322102 (48:466)
  • [20] -, Characters of connected Lie groups, Acta Math. 133 (1974), 81-137. MR 0409728 (53:13480)
  • [21] -, Lie groups with completely continuous representations, Bull. Amer. Math. Soc. 81 (1975), 1061-1063. MR 0422516 (54:10503)
  • [22] N. Wallach, Harmonic analysis on homogeneous spaces, Dekker, New York, 1973. MR 0498996 (58:16978)

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DOI: https://doi.org/10.1090/S0002-9947-1978-0486313-9
Article copyright: © Copyright 1978 American Mathematical Society

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