Bulletin of the Belgian Mathematical Society - Simon Stevin

Lipsman mapping and dual topology of semidirect products

Aymen Rahali

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.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 26, Number 1 (2019), 149-160.

Dates
First available in Project Euclid: 20 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1553047234

Digital Object Identifier
doi:10.36045/bbms/1553047234

Mathematical Reviews number (MathSciNet)
MR3934086

Zentralblatt MATH identifier
07060321

Citation

Rahali, Aymen. Lipsman mapping and dual topology of semidirect products. Bull. Belg. Math. Soc. Simon Stevin 26 (2019), no. 1, 149--160. doi:10.36045/bbms/1553047234. https://projecteuclid.org/euclid.bbms/1553047234