Bulletin of the Belgian Mathematical Society - Simon Stevin

Lipsman mapping and dual topology of semidirect products

Aymen Rahali

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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

Subjects
Primary: 22D10: Unitary representations of locally compact groups 22E27: Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.) 22E45: Representations of Lie and linear algebraic groups over real fields: analytic methods {For the purely algebraic theory, see 20G05}

Keywords
Lie groupssemidirect product unitary representations coadjoint orbits symplectic induction

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


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