Another proof for the continuity of the Lipsman mapping

A.  Messaoud; A.  Rahali

doi:10.37863/umzh.v72i7.548

A. Messaoud
A. Rahali Univ. Sfax, Tunisia

DOI: https://doi.org/10.37863/umzh.v72i7.548

Keywords: Lie groups, semidirect product, unitary representations, coadjoint orbits, symplectic induction

Abstract

UDC 515.1

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 $\langle , \rangle$. By $\hat G $ we denote the unitary dual of $G$ and by ${\mathfrak{g}^{ \ddagger} /} G$ 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 $\hat{G} $ and ${\mathfrak{g}^{ \ddagger} /} G$ is bijective. Under some assumption on $G$, we give another proof for the continuity of the orbit mapping (Lipsman mapping)
$$\Theta : {\mathfrak{g}^{ \ddagger} /} G - \rightarrow \hat{G} .$$

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