Nonstandard Lagrangian Submanifolds in CP^n

We use Hamiltonian actions to construct nonstandard (as opposed to ${\text{RP}}^n$ and $T^n$) Lagrangian submanifolds of ${\text{CP}}^n$. First of all, a quotient of ${\text{RP}}^3$ by the dihedral group $D_3$ is a Lagrangian submanifold of ${\text{CP}}^3$. Secondly, $\text{Su}(n)\mathrm{/}{\mathbb{Z}}_n$ are Lagrangian submanifolds of ${\text{CP}}^{n^2-1}$.