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march 2019 Lipsman mapping and dual topology of semidirect products
Aymen Rahali
Bull. Belg. Math. Soc. Simon Stevin 26(1): 149-160 (march 2019). DOI: 10.36045/bbms/1553047234

Abstract

We consider the semidirect product $G = K \ltimes V$ where $K$ is a connected compact Lie group acting by automorphisms on a finite dimensional real vector space $V$ equipped with an inner product $\langle,\rangle$. We denote by $\widehat{G}$ the unitary dual of $G$ (note that we identify each representation $\pi\in\widehat{G}$ to its classes $[\pi]$) and by $\mathfrak{g}^\ddag/G$ the space of admissible coadjoint orbits, where $\mathfrak{g}$ is the Lie algebra of $G.$ It was pointed out by Lipsman that the correspondence between $\mathfrak{g}^\ddag/G$ and $\widehat{G}$ is bijective. Under some assumption on $G,$ we prove that the Lipsman mapping \begin{eqnarray*} \Theta:\mathfrak{g}^\ddag/G &\longrightarrow&\widehat{G}\\ \mathcal{O}&\longmapsto&\pi_\mathcal{O} \end{eqnarray*} is a homeomorphism.

Citation

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Aymen Rahali. "Lipsman mapping and dual topology of semidirect products." Bull. Belg. Math. Soc. Simon Stevin 26 (1) 149 - 160, march 2019. https://doi.org/10.36045/bbms/1553047234

Information

Published: march 2019
First available in Project Euclid: 20 March 2019

zbMATH: 07060321
MathSciNet: MR3934086
Digital Object Identifier: 10.36045/bbms/1553047234

Subjects:
Primary: 22D10, 22E27, 22E45

Rights: Copyright © 2019 The Belgian Mathematical Society

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Vol.26 • No. 1 • march 2019
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