$\DeclareMathOperator{\Ker}{Ker}$ Consider the standard symplectic structure on $\mathbb C^{2n}$ induced by the symplectic matrix $J$. We can write: $$\mathbb C^{2n}=\Ker(J-iI)\oplus \Ker(J+iI)$$ This symplectic decomoposition of $\mathbb C^{2n}$ gives a natural correspondence between Lagrangian subspaces of $\mathbb C^{2n}$ and the unitary group. Mainly, $L\subset\mathbb C^{2n}$ is Lagrangian if and only if there is some unitary map: $$U:\Ker(J-iI)\rightarrow \Ker(J+iI)$$ Such that $L$ is the graph of $U$. Meaning - $L=\{x+Ux\mid x\in \Ker(J-iI)\}$. The contrary also holds - the graph of such a unitary is always a Lagrangian subspace.
This allows one to identify the Lagrangian Grassmannian with $U(n)$ in some sense. This identification appears a lot in spectral theory, and in symplectic geometry in the context of the Maslov index. I wanted to ask a question regarding this identification:
I also know that the Lagrangian Grassmannian is diffeomorphic to the homogeneous space $Sp(n)/U(n)$. This means that the identification above of the Lagrangian Grassmannian with $U(n)$ is NOT a diffeomorphism, since one is a complex manifold and one is a real manifold.
Is it at least a homotopy equivalence? (I know that the fundamental groups of $U(n)$ and $Sp(n)/U(n)$ are both $\mathbb Z$)
If not, what is 'interesting' about this correspondence, other than being a bijection? Generally, I'd be happy to read a bit about the meaning of this identification and how one should think about it. I'd be happy to hear your answers to this or see references.
Thanks in advance.