A unipotent representation associated with a minimal orbit: The \(so(4,3)\) case
\textit{D. Vogan} [Invent. Math. 116, 677-791 (1994; Zbl 0808.22003)] has proved the existence of an element \(\pi\) of the unitary dual of the universal covering of \(\text{SO}_0(4,3)\) of which the annihilator is the Joseph ideal. In this work, we present a new method for obtaining this result, which essentially uses a parametrization of the unitary dual of a real almost semialgebraic group, given by \textit{M. Duflo} [Acta Math. 149, 153-213 (1982; Zbl 0529.22011)]. Then, we give an explicit realization of \(\pi\) in \(L^2(\mathbb{R}^3 \times \mathbb{R}^*)\), we find the Langlands parameters and, at last, we study its restriction to \(G_2\).
scientific article; zbMATH DE number 880570