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Journal of Lie Theory 30 (2020), No. 3, 715--765
Copyright Heldermann Verlag 2020

Irreducible Characters and Semisimple Coadjoint Orbits

Benjamin Harris

Yoshiki Oshima
Dept. of Pure and Applied Mathematics, Grad. School of Information Science and Technology, Osaka University, Suita Osaka 565-0871, Japan


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.

Keywords: Semisimple orbit, coadjoint orbit, orbit method, Kirillov's character formula, cohomological induction, parabolic induction, reductive group.

MSC: 22E46.

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