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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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The dual topology for the principal and discrete series on semisimple groups
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by Ronald L. Lipsman
Trans. Amer. Math. Soc. 152 (1970), 399-417
DOI: https://doi.org/10.1090/S0002-9947-1970-0269778-X

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.
References
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Bibliographic Information
  • © Copyright 1970 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 152 (1970), 399-417
  • MSC: Primary 22.60
  • DOI: https://doi.org/10.1090/S0002-9947-1970-0269778-X
  • MathSciNet review: 0269778