density of lagrangian grassmannian in usual grassmannian.  Consider the canonical symplectic structure $(\omega, J)$ on $\mathbb{R}^{2n}$.
(i) What can be said about the density of the lagrangian grassmannian $L$ (i.e. those rank $n$ totally isotropic linear subspaces of $\mathbb{R}^{2n}$) in the usual Grassmannian $Gr=Gr_{n}(\mathbb{R}^{2n})$ of rank $n$ linear subspaces? 
(ii) How stable is $L$ within $Gr$. ie. which homeomorphisms of $Gr$ preserve $L$ (not necessarily pointwise)? 
(iii) In what sense can we see $Gr$ as ''swept out'' by $L$ ie. decompose $Gr$ as the orbits of $L$.
(iv) What projections $Gr \to L$ can we produce? 
 A: jmart has modified the question since I posted my original answer.  I'll now modify my answer to correspond:
(i) The Lagrangian Grassmannian ($L$ in your notation) is a closed submanifold of $Gr_n(\mathbb{R}^{2n})$, so it's not dense at all.  In fact, it has dimension $\frac12n(n{+}1)$.  
(ii) $L$ is homogeneous under the action of $\textrm{Sp}(n)\subset \textrm{GL}(2n,\mathbb{R})$, the subgroup that preserves the symplectic structure $\omega$.  (The $J$ is not needed to define $L$.)  Conversely, the subgroup of $\textrm{GL}(2n,\mathbb{R})$ that preserves $L$ in $Gr_n(\mathbb{R}^{2n})$ is easily seen to be the subgroup of $\textrm{GL}(2n,\mathbb{R})$ that preserves $\omega$ up to a multiple.
(iii)  I don't really know what you mean by 'the orbits of $L$', since $L$ is not a group.  Of course, $Gr_n(\mathbb{R}^{2n})$ is homogeneous under $\textrm{GL}(2n,\mathbb{R})$ so it's the $\textrm{GL}(2n,\mathbb{R})$-orbit of any point of $L$. Perhaps you actually want to know the orbits of $\textrm{Sp}(n)$ acting on $Gr_n(\mathbb{R}^{2n})$, with $L$ being the closed $\textrm{Sp}(n)$-orbit.  If you consider, for any $n$-plane $E\in Gr_n(\mathbb{R}^{2n})$, the rank $r(E)$ of the pullback of $\omega$ to $E$, then the $\textrm{Sp}(n)$-orbits are the level sets of $r$ (which takes values in nonnegative integers).
(iv)  I'm not sure what you mean by 'projections'.  If by 'projection', you mean a smooth map $\pi:Gr_n(\mathbb{R}^{2n})\to L$ that is the identity on $L$, such a thing does not exist.
