Cartan motion groups and dual topology

Let $G$ be a connected reductive Lie group with Lie algebra $\mathfrak{g}$, and let $\overline{G}$ be the analytic subgroup corresponding to $[\mathfrak{g, g}]$. Assume $\overline{G}$ has finite center. Let $K$ be a maximal compact subgroup of $G$ and let $\mathfrak{g = k + s}$ be the corresponding Cartan decomposition. Then $K$ acts on $\mathfrak{s}$ by the adjoint representation $(k.X = Ad_K (k) X)$. The Cartan motion group $H$ (associated to $G$) is the semidirect product $H = K \ltimes \mathfrak{s}$. In this paper, we prove that the unitary dual $\hat{H}$ of $H$ is homeomorphic to the space $\mathfrak{h}^\ddagger / H$ of all admissible coadjoint orbits of $H$.

Keywords

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

2010 Mathematics Subject Classification

22D10, 22E27, 22E45

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To the memory of Majdi Ben Halima

Published 5 November 2019