Unitary group acts transitively on the Lagrangian Grassmannian

I am trying to prove that the unitary group associated to an $$\omega$$-compatible complex structure $$J$$ acts transitively on the Lagrangian Grassmannian $$\mathcal{L}(V)$$.

I know that for a symplectic space $$(V, \omega)$$, one can find an $$\omega$$-compatible complex structure $$J$$. Then, a hermitian structure on $$V$$ is defined by $$\begin{equation*} \langle \cdot, \cdot \rangle = g_J(\cdot, \cdot)+ \mathrm{i} \omega(\cdot, \cdot), \end{equation*}$$ where $$g_J(x, y)= \omega(x,Jy)$$ is an inner product.

The unitary group $$U(V)$$ consists of linear transformations $$T\in GL(V)$$ which preserve the Hermitian structure. Then the intersection of $$Sp(V)$$ and $$O(V)$$ equals $$U(V)$$.

Thus, for any $$L_1,L_2\in \mathcal{L}(V)$$, we begin with an orthogonal transformation $$A:L_1\to L_2$$. The proof from the book is as follows. From $$A:L_1\to L_2$$, we construct a symplectromorphism from $$L_1\oplus L_1^*$$ to $$L_2\oplus L_2^*$$. Then we can generate a unitary transformation $$L_1\oplus JL_1\to L_2\oplus JL_2$$ which maps $$L_1$$ to $$L_2$$.

I cannot understand the proof. What I know is that $$L_i^*$$ is isomorphic to $$JL_i$$. But I have no idea why we consider $$L_1\oplus JL_1\to L_2\oplus JL_2$$?

Any advice on explaining the proof or new ideas are appreciated. Thanks in advance.

• I realize that for $x_1+y_1, x_2+y_2 \in L_1\oplus JL_1$, we have $g_J(x_1+y_1,x_2+y_2)=\omega(x_1+y_1,J(x_2+y_2))=g_J(x_1,x_2)+g_J(y_1,y_2)$. Moreover, $\omega(x_1+y_1,x_2+y_2)=\omega(x_1,y_2)-\omega(x_2,y_1)$ which is associated with the symplectic form on $L\oplus L^*$. I think that I misunderstood the meaning of orthogonal. The orthogonality is associated with the inner product $g_J$. Then we construct $A\otimes A^{-1}*$ which is the symplectromorphism. Thus, we construct a tranformation which preserves $g_J$ and $\omega$. Am I right? – Hu ju yuan Mar 25 at 10:23