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On the integrable and square-integrable representations of $ {\rm Spin}(1, 2m)$


Author: Ernest Thieleker
Journal: Trans. Amer. Math. Soc. 230 (1977), 1-40
MSC: Primary 22E43
DOI: https://doi.org/10.1090/S0002-9947-1977-0453925-7
MathSciNet review: 0453925
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Abstract: All the unitary equivalence classes of irreducible integrable and square-integrable representations of the groups $ {\text{Spin}}(1,2m),m \geqslant 2$, are determined. The method makes use of some elementary results on differential equations and the classification of irreducible unitary representations of these groups. In the latter classification, certain ambiguities resulting from possible equivalences not taken into account in a previous paper, are cleared up here.


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  • [1] N. Bourbaki, Éléments de mathématique. Groupes et algèbres de Lie, Chaps. 4, 5, 6, Actualités Sci. Indust., no. 1337, Hermann, Paris, 1968. MR 39 #1590. MR 0240238 (39:1590)
  • [2] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi et al., Higher transcendental functions, Vol. I, McGraw-Hill, New York, 1953.
  • [3] J. Dixmier, Algèbres enveloppantes, Cahiers Scientifiques, fasc. 37, Gauthier-Villars, Paris, 1974. MR 0498737 (58:16803a)
  • [4] -, Représentations intégrables du groupe de De Sitter, Bull. Soc. Math. France 89 (1961), 9-41. MR 25 #4031. MR 0140614 (25:4031)
  • [5] A. M. Gavrilik and U. A. Klimyk, Analysis of the representations of the Lorentz and Euclidean groups of nth order, Acad. Sci. Ukranian SSR, Institute for Theoretical Physics, Kiev, 1975 (preprint). MR 0480891 (58:1040)
  • [6] 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)
  • [7] -, Representations of semisimple Lie groups. VI, Amer. J. Math. 78 (1956), 564-628. MR 18, 490. MR 0082056 (18:490d)
  • [8] -, Spherical functions on a semisimple Lie group, I, Amer. J. Math. 80 (1958), 241-310. MR 20 #925. MR 0094407 (20:925)
  • [9] -, Discrete series for semisimple Lie groups. II, Acta Math. 116 (1966), 1-111. MR 36 #2745. MR 0219666 (36:2745)
  • [10] -, The theory of the Eisenstein integral, Harmonic Analysis (Proc. Conf., College Park, Md., 1971), Lecture Notes in Math., vol. 266, Springer-Verlag, Berlin and New York, 1972. MR 0399355 (53:3200)
  • [11] S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York and London, 1962. MR 26 #2986. MR 0145455 (26:2986)
  • [12] H. Kraljević, Representations of the universal covering group of the group $ SU(n,1)$, Glasnik Mat. Ser. III 8 (28) (1973), 23-72. MR 48 #8692. MR 0330355 (48:8692)
  • [13] N. Jacobson, Lie algebras, Interscience, New York, 1962. MR 26 # 1345. MR 0143793 (26:1345)
  • [14] S. Lang, $ S{L_2}(R)$, Addison-Wesley, Reading, Mass., 1975. MR 0430163 (55:3170)
  • [15] E. Thieleker, On the quasi-simple irreducible representations of the Lorentz groups, Trans. Amer. Math. Soc. 179 (1973), 465-505. MR 48 #4202. MR 0325856 (48:4202)
  • [16] -, The unitary representations of the generalized Lorentz groups, Trans. Amer. Math. Soc. 199 (1974), 327-367. MR 0379754 (52:659)
  • [17] N. Wallach, Harmonic analysis on homogeneous spaces, Dekker, New York, 1973. MR 0498996 (58:16978)
  • [18] G. Warner, Harmonic analysis on semisimple Lie groups. Vols. I, II, Springer-Verlag, Berlin and New York, 1972.
  • [19] D. P. Želobenko, Compact Lie groups and their representations, ``Nauka", Moscow, 1970; English transl., Transl. Math. Monographs, vol. 40, Amer. Math. Soc., Providence, R.I., 1973.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1977-0453925-7
Keywords: Discrete series of $ {\text{Spin}}\;(1,2m)$, discrete series of generalized Lorentz groups, unitary representations of rank 1 real semisimple Lie groups, unitary representations of $ {\text{Spin}}\;(1,2m)$, Eisenstein integrals of $ {\text{Spin}}(1,2m)$
Article copyright: © Copyright 1977 American Mathematical Society

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