Abstract
Let \(G\) be a type I connected and simply connected generalized diamond Lie group defined as the semidirect product of a \(d\)-dimensional Abelian Lie group \(N\) with \((2n+1)\)-dimensional Heisenberg Lie group \(\mathbb{H}_{2n+1}\) for some \((n,d)\in(\mathbb{N}^*)^2\). Let \(\mathfrak{g}^*/G\) denote the set of coadjoint orbits of \(G\), where \(\mathfrak{g}^*\) is the dual vector space of the Lie algebra \(\mathfrak{g}\) of \(G\). In this paper, we address the problem of separation of coadjoint orbits of \(G\). We first specify the setting where \(d=1\); we prove that the closed convex hull of coadjoint orbit \(\mathcal{O}\) in \(\mathfrak{g}^*\) does characterize \(\mathcal{O}\). Whenever \(d\ge2\), we provide a separating overgroup \(G^+\) of \(G\). More precisely, we extend the group \(G\) to an overgroup denoted by \(G^+\), containing \(G\) as a subgroup, and we give an injective map \(\varphi\) from \(\mathfrak{g}^*\) into \((\mathfrak{g}^+)^*\), the dual vector space of Lie algebra \(\mathfrak{g}^+\) of \(G^+\) sending each \(G\)-orbit in \(\mathfrak{g}^*\) to the \(G^+\)-orbit in \((\mathfrak{g}^+)^*\) in such a manner that the closed convex hull of \(\varphi(\mathcal{O})\) does characterize \(\mathcal{O}\), where \(\mathcal{O}\) is a \(G\)-orbit in \(\mathfrak{g}^*\).
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Funding
This work was supported by the D. G. R. S. R. T Research Laboratory: LR 11 ES 35.
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Translated from Matematicheskie Zametki, 2022, Vol. 111, pp. 643–662 https://doi.org/10.4213/mzm13509.
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Abdelmoula, L., Bouaziz, Y. Separation of Coadjoint Orbits of Generalized Diamond Lie Groups. Math Notes 111, 659–675 (2022). https://doi.org/10.1134/S0001434622050017
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DOI: https://doi.org/10.1134/S0001434622050017