I know the following statement:
Theorem. Let $(V,\omega)$ be a finite dimensional symplectic linear space. Then the symplectic group $Sp(V)$ of $V$ transitively acts on the set $L=\lbrace W \subset V:\text{$W$ is Lagrange subspace of $V$} \rbrace$.
I think it is interesting, and have questions:
Does the set $L$ have a manifold structure in the same way as Grassmannian? If it is true, then is it a submanifold of the Grassmannian $\text{Gr}(n,V)$, where $2n=\text{dim}V$, and a homogeneous manifold just like Grassmannian?
For the simply example $\text{dim}(V)=2$, than $L$ is just the projective space $\text{P}\mathbb{R}^2$. Are there interesting examples for general $L$?
If $L$ is a manifold and the action of $Sp(V)$ is proper, then the quotient space $L/Sp(V)$ is a manifold. What is the topology of $L/Sp(V)$?
What is the corresponding symplectic geometry? Does the symplectic group of a symplectic manifold $M$ transitively act on the set of all Lagrange submanifolds of $M$?
The set $L$ is a homogeneous space, called the Lagrangian-Grassmannian. For a symplectic vector space of dimension $2n$, it is diffeomorphic to $U(n)/O(n)$. You can check this using a complex structure which is compatible with the symplectic form.
Also, since you already noted that $Sp(V)$ acts transitively on $L$, you know that its orbit space is trivial.
It is not true that the group of symplectomorphisms acts transitively on the set of Lagrangian submanifolds, because Lagrangian submanifolds need not be diffeomorphic. In fact, it is easy to show that any symplectic manifold admits infinitely many Lagrangian submanifolds that are tori. Remove a point from one, and it will still be a Lagrangian submanifold, which is not diffeomorphic to any of the tori. You can already see this is real dimension $2$.