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


1. $\mu^{\mathbb C}([\mathfrak p,\mathfrak p])=0$
2. $\dim \mathfrak g/\mathfrak g\_\mu=\dim \mathfrak g^{\mathbb C}/\mathfrak p$.
3. $\mathfrak g\_\mu\subset\mathfrak p$
4. $(\mathfrak p\oplus\bar{\mathfrak p})\cap \mathfrak g$ is a Lie subalgebra of $\mathfrak g$.


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



The set of leaves of the distribution $D$ on coadjoint orbit $O_{\mu}$
The leaves of $D$ are points and the leaf space $O\_\mu/D$ is $O\_\mu$ itself. 


Indeed, a $G$-invariant polarization $P$ (involutive lagrangian subbundle of $(TO\_\mu){}^\mathbf C$) is determined by the preimage $\mathfrak p\subset\mathfrak g^\mathbf C$ of its value $P\_\mu\subset (T\_\mu O\_\mu){}^\mathbf C=(\mathfrak g/\mathfrak g\_\mu){}^\mathbf C$. Likewise your $D$ is the $G$-invariant distribution whose value at $\mu$ is the subspace $D\_\mu=\mathfrak d/\mathfrak g\_\mu$ of $T\_\mu O\_\mu=\mathfrak g/\mathfrak g\_\mu$, where $\mathfrak d=\mathfrak p\cap\bar{\mathfrak p}\cap\mathfrak g$.


Now in your case ($G$ compact) the possible $\mathfrak p$ are known (parabolics containing $\mathfrak g\_\mu^\mathbf C$) and all *complex*, i.e. $\mathfrak p\cap\bar{\mathfrak p}\cap\mathfrak g=\mathfrak g\_\mu$. (E.g., if $G=\mathrm{SU}(2)$ and $O\_\mu=S^2$ then $\mathfrak p$ and $\bar{\mathfrak p}$ are opposite Borels (upper and lower triangulars in $\mathfrak{sl}(2,\mathbf C)$) intersecting in the diagonals.) So we have $\mathfrak d/\mathfrak g\_\mu=\{0\}$ and hence $D=\{0\}$.


