Geodesic
Global Parametrization of Scalar Holomorphic Coadjoint Orbits of a Quasi-Hermitian Lie Group
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 52 (2013) no. 1, pp. 35-48.

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Let $G$ be a quasi-Hermitian Lie group with Lie algebra $\mathfrak g$ and $K$ be a compactly embedded subgroup of $G$. Let $\xi _0$ be a regular element of ${\mathfrak g}^{\ast }$ which is fixed by $K$. We give an explicit $G$-equivariant diffeomorphism from a complex domain onto the coadjoint orbit $\mathcal {O}({\xi _0})$ of $\xi _0$. This generalizes a result of [B. Cahen, Berezin quantization and holomorphic representations, Rend. Sem. Mat. Univ. Padova, to appear] concerning the case where ${\mathcal O}({\xi _0})$ is associated with a unitary irreducible representation of $G$ which is holomorphically induced from a unitary character of $K$. In particular, we consider the case $G=SU(p,q)$ and the case where $G$ is the Jacobi group.
Classification : 22E10, 22E15, 22E45, 32M05, 32M10, 32M15, 81S10
Keywords: quasi-Hermitian Lie group; coadjoint orbit; stereographic projection; Berezin quantization; unitary holomorphic representation; unitary group; Jacobi group 
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Cahen, Benjamin. Global Parametrization of Scalar Holomorphic Coadjoint Orbits of a Quasi-Hermitian Lie Group. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 52 (2013) no. 1, pp. 35-48. http://geodesic.mathdoc.fr/item/AUPO_2013__52_1_a2/