Let $G$ be a compact connected Lie group and $O_{\mu}$ be a coadjoint orbit where $\mu\in \mathfrak{g}^*$ and $\mathfrak{g}^*$ is the dual of the Lie algebra of $\mathfrak{g}=\mathrm{Lie}(G)$. Let $P$ be an invariant polarization of the coadjoint orbit $O_{\mu}$ and $D=P\cap\bar P\cap TO_μ$ (here $\bar P$ i.e. complex conjugate of $P$ ) so how can we find the set of leaves of the distribution $D$ in $O_{\mu}$ .i.e. How can we find $O_{\mu}/D$?

ps: I need this for metaplectic correction. This question come from metaplectic correction on coadjoint orbits

Note that a polarization of the coadjoint orbit $G/G_\mu$, is given by the left invariant extension of complex Lie subalgebra $\mathfrak p\subset \mathfrak g^{\mathbb C}$ with the properties

$\mu^{\mathbb C}([\mathfrak p,\mathfrak p])=0$

$\dim \mathfrak g/\mathfrak g_\mu=\dim \mathfrak g^{\mathbb C}/\mathfrak p$.

$\mathfrak g_\mu\subset\mathfrak p$

$(\mathfrak p\oplus\bar{\mathfrak p})\cap \mathfrak g$ is a Lie subalgebra of $\mathfrak g$.

If we take $\mathfrak d=\mathfrak g\cap \mathfrak p$ then in complex polarization $\mathfrak d=\mathfrak g_\mu$

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