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On the Cortex of the Groups $𝕋^{n} ⋉ ℍ_{n}$

Aymen Rahali

Aymen Rahali

Université de Sfax, Faculté des Sciences Sfax, BP 1171, Sfax 3038, Tunisia

Laboratoire de recherche LAMHA, LR 11 ES 52, Tunisia

E-mail Address: aymen1.rahali@gmail.com

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https://doi.org/10.1142/S1793557121501497Cited by:0 (Source: Crossref)

Abstract

Let $ℍ_{n}$ be the $(2 n + 1)$ -dimensional Heisenberg group and let $𝕋^{n}$ be the $n$ -dimensional torus acting on $ℍ_{n}$ by automorphisms. We consider the semidirect product group $G_{n} : = 𝕋^{n} ⋉ ℍ_{n} .$ The cortex, $cor (G_{n}),$ of $G_{n}$ is the set of all unitary irreducible representations $π$ in the unitary dual $\hat{G_{n}}$ of $G_{n}$ that cannot be Hausdorff separated from the identity representation $1_{G_{n}}$ of $G_{n} .$ In this paper, we describe explicitly the cortex ( $cor (G_{n})$ ) of $G_{n}$ using the coadjoint orbits of the group.

Communicated by V. A. Artamonov

To the memory of Majdi Ben Halima

Keywords:

AMSC: 22D10, 22E27, 22E45

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