On the quasi-simple irreducible representations of the Lorentz groups

Thieleker, Ernest

doi:10.1090/S0002-9947-1973-0325856-0

On the quasi-simple irreducible representations of the Lorentz groups
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Trans. Amer. Math. Soc. 179 (1973), 465-505 Request permission

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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