Journal of Lie Theory

no. 2
On the Dual Topology of a Class of Cartan Motion Groups
Journal of Lie Theory, Volume 22 (2012) no. 2, pp. 491-503
\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 $G_{0}$.
DOI: 10.5802/jolt.678
Classification: 53C35, 22D05, 22D30, 53D05
Keywords: Symmetric space, motion group, induced representation, coadjoint orbit 
@article{JOLT_2012_22_2_a7,
     author = {M. Ben Halima and A. Rahali},
     title = {On the {Dual} {Topology} of a {Class} of {Cartan} {Motion} {Groups}},
     journal = {Journal of Lie Theory},
     pages = {491--503},
     year = {2012},
     volume = {22},
     number = {2},
     doi = {10.5802/jolt.678},
     zbl = {1244.53058},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.678/}
}
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%A A. Rahali
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%P 491-503
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%R 10.5802/jolt.678
%F JOLT_2012_22_2_a7
M. Ben Halima; A. Rahali. On the Dual Topology of a Class of Cartan Motion Groups. Journal of Lie Theory, Volume 22 (2012) no. 2, pp. 491-503. doi: 10.5802/jolt.678

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