## Bulletin of the Belgian Mathematical Society - Simon Stevin

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

### Lipsman mapping and dual topology of semidirect products

#### 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