A symplectic vector space $\left (\mathbb{R}^{2n}, \omega\right )$ can be thought of as having 4 fundamental types of subspaces: Lagrangian, isotropic, coisotropic, and symplectic.
If we consider the space $\mathcal{L}(n)$ of all Lagrangian subspaces, it has a homogeneous structure: $$ \mathcal{L}(n) \;\; = \;\; U(n)/O(n) $$
where if we think of the embedding $U(n)\hookrightarrow GL_{2n}(\mathbb{R})$ as $$ X+iY \;\; \to \;\; \left [ \begin{array}{cc} X & - Y \\ Y & X \\ \end{array} \right ] $$
then each Lagrangian subspace can be thought of as the span of the first $n$ columns of the above matrix, and these spans are invariant under right multiplication of $O(n)\times O(n)$.
My question is: do the other canonical subspaces hold homogeneous structures of this form?
For instance if we consider the set $\mathcal{I}(k)$ of isotropic subspaces of dimension $k$ in $\mathbb{R}^{2n}$, then I believe we can similarly form a unitary matrix of the type above where we pick $n\times k$ matrices $X$ and $Y$ such that $$ \left [ \begin{array}{c} X\\Y\\ \end{array} \right ] $$
forms a unitary frame for the subspace. Assuming it's possible to complete these to $n\times n$ matrices $\widetilde{X}, \widetilde{Y}$ such that $\widetilde{X}^T\widetilde{Y} = \widetilde{Y}^T\widetilde{X}$ then we can form a unitary matrix: $$ \left [ \begin{array}{cc} \widetilde{X} & - \widetilde{Y} \\ \widetilde{Y} & \widetilde{X} \\ \end{array} \right ] $$ and that a subspace can be represented by such a matrix mod right multiplication of the group $O(k)\times O(2n-k)$. Is it then possible to conclude that: $$ \mathcal{I}(k) \;\; =\;\; \frac{U(n)}{O(k)\times O(2n-k)}? $$
If this is true, then how do we go about constructing homogeneous structures for the set of coisotropic subspaces $\mathcal{C}(n+k)$, and the space of symplectic subspaces $\mathcal{S}(2k)$?
For reference, this question is based on a passage in McDuff & Salamon's Intro to Symplectic Topology where they prove the above claim for Lagrangian subspaces, but then pose an exercise where they ask the reader to prove "analogous statements" for the other types of subspaces.