Irreducible Characters and Semisimple Coadjoint Orbits

When $G_{\mathbb{R}}$ is a real, linear algebraic group, the orbit method predicts that nearly all of the unitary dual of $G_{\mathbb{R}}$ consists of representations naturally associated to orbital parameters $(\mathcal{O},\mathrm{\Gamma })$. If $G_{\mathbb{R}}$ is a real, reductive group and $\mathcal{O}$ is a semisimple coadjoint orbit, the corresponding unitary representation $\pi (\mathcal{O},\mathrm{\Gamma })$ may be constructed utilizing Vogan and Zuckerman’s cohomological induction together with Mackey’s real parabolic induction. In this article, we give a geometric character formula for such representations $\pi (\mathcal{O},\mathrm{\Gamma })$. Special cases of this formula were previously obtained by Harish-Chandra and Kirillov when $G_{\mathbb{R}}$ is compact and by Rossmann and Duflo when $\pi (\mathcal{O},\mathrm{\Gamma })$ is tempered.