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A topological criterion for group decompositions

Published online by Cambridge University Press:  24 October 2008

J. Hebda  and
P. Moylan
J. Hebda
Affiliation:
Department of Mathematics, St Louis University, St Louis, Missouri 63103
P. Moylan
Affiliation:
Department of Science and Mathematics, Parks College of Saint Louis University, Cahokia, Illinois 62206
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Abstract

Given a connected Lie group G and a closed connected subgroup H of G we prove a necessary and sufficient condition that G decomposes into the Cartesian product of H with G/H is that a similar decomposition holds for the maximal compact subgroups of G and H. Our criterion is applied to the three series of groups for which G/H is SO0(p, q)/SO0(p, q − 1), SU(q + 1, q + 1)/S[U(q + 1, q) × U(1)], and SU(q + 1, q + 1)/SL(n, ℂ) ⋊ H(n) (p, q ≥ 1), and we list the values of p and q for which G ≅ H × G/H in each of the three cases. We describe certain decompositions for some of the groups. We show the usefulness of our criterion in obtaining characterization of the space of differentiable vectors for a unitary induced group representation, and, finally, we show by example of SU(2, 2), how the asymptotic properties of certain function spaces for induced group representations are readily obtained using our results. Our results should be of interest to those working in de Sitter and conformal field theories.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 103 , Issue 2 , March 1988 , pp. 285 - 298
Copyright
Copyright © Cambridge Philosophical Society 1988

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