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Transactions of the American Mathematical Society

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

 
 

 

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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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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