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Keywords:
quasi-Hermitian Lie group; coadjoint orbit; stereographic projection; Berezin quantization; unitary holomorphic representation; unitary group; Jacobi group
quasi-Hermitian Lie group; coadjoint orbit; stereographic projection; Berezin quantization; unitary holomorphic representation; unitary group; Jacobi group
Summary:
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.
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