I've been trying to proof this using $U(n,\mathbb{C})$ action over all lagrangian subspaces of $\mathbb{R}^n\times \mathbb{R}^n$ but it didn't work. I mean, I got stuck and I didn't know what else to do. I found the excercise on Heckman's book symplectic geometry.

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    $\begingroup$ Um.... why doesn't it work? $U(n, \mathbb C)$ acts transitively, you need only to find which elements in $U(n, \mathbb C)$ fixes the standard Lagrangian $\mathbb R^n \times \{0\}$. $\endgroup$ – Arctic Char Sep 25 at 4:39

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