Irreducible Characters and Semisimple Coadjoint Orbits

Benjamin Harris¹ ; Yoshiki Oshima²

¹U.S.A.
²Dept. of Pure and Applied Mathematics, Grad. School of Information Science and Technology, Osaka University, Suita Osaka 565-0871, Japan

Journal of Lie Theory, Tome 30 (2020) no. 3, pp. 715-765

Résumé

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 $(\mathcal{O},\Gamma)$. If $G_{\mathbb{R}}$ is a real, reductive group and $\mathcal{O}$ is a semisimple coadjoint orbit, the corresponding unitary representation $\pi(\mathcal{O}, \Gamma)$ 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 $\pi(\mathcal{O},\Gamma)$. 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 $\pi(\mathcal{O},\Gamma)$ is tempered.

Détail

Classification : 22E46
Mots-clés : Semisimple orbit, coadjoint orbit, orbit method, Kirillov's character formula, cohomological induction, parabolic induction, reductive group