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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the irreducibility of nonunitary induced representations of certain semidirect products

Author: Ernest Thieleker
Journal: Trans. Amer. Math. Soc. 164 (1972), 353-369
MSC: Primary 22E45
MathSciNet review: 0293017
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Abstract: Let G be a connected Lie group which is a semidirect product of a compact subgroup K and a normal solvable subgroup S. Let $ \Lambda $ be a character of S, and let $ {M_\Lambda }$ be the stabilizer of $ \Lambda $ in K. Let $ [H,{\Lambda _\mu }]$ be a finite-dimensional irreducible representation of the subgroup $ S{M_\Lambda }$ on the complex vector space H. In this paper we consider the induced representations of G on various Banach spaces, and study their topological irreducibility. The basic method used consists in studying the irreducibility of the Lie algebra representations which arise on the linear subspaces of K-finite vectors. The latter question then can be reduced to the problem of determining when certain modules over certain commutative algebras are irreducible. The method discussed in this paper leads to two theorems giving sufficient conditions on the character $ \Lambda $ that the induced representations be topologically irreducible. The question of infinitesimal equivalence of various induced representations is also discussed.

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  • [1] 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)
  • [2] J. M. G. Fell, Non-unitary dual spaces of groups, Acta Math. 114 (1965), 267-310. MR 32 #4210. MR 0186754 (32:4210)
  • [3] 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)
  • [4] 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)
  • [5] G. Hochschild and G. D. Mostow, Representations and representative functions of Lie groups, Ann. of Math. (2) 66 (1957), 495-542. MR 20 #5248. MR 0098796 (20:5248)
  • [6] G. Mackey, Induced representations of locally compact groups. I, Ann. of Math. (2) 55 (1952), 101-139. MR 13, 434. MR 0044536 (13:434a)
  • [7] -, Unitary representations of group extensions. I, Acta Math. 99 (1958), 265-311. MR 20 #4789. MR 0098328 (20:4789)
  • [8] E. Thieleker, On some infinite dimensional representations of Lie groups (to appear).
  • [9] A. Weil, L'intégration dans les groupes topologiques et ses applications, 2nd ed., Actualités Sci. Indust., no. 869, Hermann, Paris, 1951. MR 3, 198.

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Article copyright: © Copyright 1972 American Mathematical Society

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