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Transactions of the American Mathematical Society

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



The dual topology for the principal and discrete series on semisimple groups

Author: Ronald L. Lipsman
Journal: Trans. Amer. Math. Soc. 152 (1970), 399-417
MSC: Primary 22.60
MathSciNet review: 0269778
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Abstract: For a locally compact group $ G$, the dual space $ \hat G$ is the set of unitary equivalence classes of irreducible unitary representations equipped with the hull-kernel topology. We prove three results about $ \hat G$ in the case that $ G$ is a semisimple Lie group: (1) the irreducible principal series forms a Hausdorff subspace of $ \hat G$; (2) the ``discrete series'' of square-integrable representations does in fact inherit the discrete topology from $ \hat G$; (3) the topology of the reduced dual $ {\hat G_r}$, that is the support of the Plancherel measure, is computed explicitly for split-rank 1 groups.

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Keywords: Semisimple Lie groups, irreducible unitary representations, dual space, hull-kernel topology, principal series, discrete series, Plancherel measure, universal enveloping algebra, characters, invariant eigendistributions, Cartan subgroups, root systems, Weyl group
Article copyright: © Copyright 1970 American Mathematical Society

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