Journal of Lie Theory 22 (2012), No. 2, 491--503
Copyright Heldermann Verlag 2012
On the Dual Topology of a Class of Cartan Motion Groups
Majdi Ben Halima
Department of Mathematics, Faculty of Sciences, University of Sfax, Route de Soukra -- B.P.1171, 3000 Sfax, Tunisia
majdi.benhalima@yahoo.fr
Aymen Rahali
Department of Mathematics, Faculty of Sciences, University of Sfax, Route de Soukra -- B.P.1171, 3000 Sfax, Tunisia
[Abstract-pdf]
\def\g{{\frak g}} Let $(G,K)$ be a compact Riemannian symmetric pair, and let $G_{0}$ be the associated Cartan motion group. Under some assumptions on the pair $(G,K)$, we give a precise description of the set $(\widehat{G_{0}})_{\rm gen}$ of all equivalence classes of generic irreducible unitary representations of $G_{0}$. We also determine the topology of the space $(\g_{0}^{\ddagger}/G_{0})_{gen}$ of generic admissible coadjoint orbits of $G_{0}$ and we show that the bijection between $(\widehat{G_{0}})_{\rm gen}$ and $(\g_{0}^{\ddagger}/G_{0})_{\rm gen}$ is a homeomorphism. Furthermore, in the case where the pair $(G,K)$ has rank one, we prove that the unitary dual $\widehat{G_{0}}$ is homeomorphic to the space $\g_{0}^{\ddagger}/G_{0}$ of all admissible coadjoint orbits of $G_{0}$.
Keywords: Symmetric space, motion group, induced representation, coadjoint orbit.
MSC: 53C35, 22D05, 22D30, 53D05
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