Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 146 / 1 / b / Solution

The unitary group acts on the Lagrangian Grassmannian by $A\cdot L=A(L)$. Every Lagrangian subspace has an orthonormal basis, and adjoining its $J_0$-image gives a unitary basis, so this action is transitive. The stabilizer of the standard real subspace ${\mathbb{R}}^n\subset {\mathbb{C}}^n$ consists exactly of real unitary matrices, namely $O(n)$. Henceis a continuous bijection from a compact space to a Hausdorff space and is therefore a homeomorphism.

For $n=1$, every line in ${\mathbb{R}}^2$ is Lagrangian. ThusConcretely, the line making angle $\theta$ with the real axis corresponds to $e^{2i\theta }$.