# Lagrangian grassmannian of all lagrangian subspaces in $\mathbb{R}^n\times \mathbb{R}^n$ can be identified with $U(n,\mathbb{C})/O(n,\mathbb{R})$

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.

• 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 $\mathbb R^n \times \{0\}$. – Arctic Char Sep 25 at 4:39