Let $G=\text{SL}_{m+n}(\mathbb R)$ and $P=\left \{\begin{bmatrix} A & 0 \\ C & D \end{bmatrix}: A\in M_{m\times m}(\mathbb R),C\in M_{n\times m}(\mathbb R),D\in M_{n\times n}(\mathbb R) \right \}$ be a parabolic subgroup.
I saw in literature that the partial flag variety (also a $G$-homogeneous space) $G/P$ can be identified with the Grassmannian $Gr(m,m+n)$ the space of all $m$-dim subspaces of $\mathbb R^{m+n}$.
How do we establish this identification? Does the $G$ action on $G/P$ corresponds to any action on $Gr(m,m+n)$?
maybe better over $\mathbf{C}: $this has something to do with (partial) flag varieties:
a partial flag variety $X$ of type $(m,m+n)$ for a $\mathbf{C}$-vector space $V$ of dimension $m+n$ is the space parameterising chains of vector spaces $(V_1 \subset V_2=V)$, where $V_1$ is of dimension $m$. You might want to look here .
Now $G=\text{SL}_{m+n}(\mathbf{C})$ acts transitively on these chains and your block matrices $ \in P$ are precisely the stabilisers of this action, i.e. you get a homogenous space $G/P \cong X$. Now the latter is just $Gr(m,m+n)$ the spaces that parameterises the $m$ dimensional subspaces of an $m+n$ dimensional vectorspace $V$.
Remark: The weird expression partial flag variety comes from a deviation of the complete flag variety where chains have length $m+n$ (dim $V_i=i$)and $P$ is given by upper triangular matrices.