Let $B$ is a nondegenerate symmetric bilinear form on $\mathbb{C}^n$ then the corresponding complex orthogonal group is $\{g : GL(n, \mathbb{C}): B(gx, gy) =(x,y) \}$
In particular we use $$B (x,y) = \sum\limits_{1\leq l \leq m}x^ly^l - \sum\limits_{m \leq l \leq n}x^ly^l, \; n=2m \; \text{or}\; 2m+1,$$ and denote $$O(n, \mathbb{C}) = \{g : GL(n, \mathbb{C}): B(gx, gy) =(x,y) \}.$$
A linear subspace $E \subset \mathbb{C}^n$ is totally isotropic if $B(E,E) =0.$ The parabolic subgroups of $O(n, \mathbb{C})$ are the $$P_{E_1, \cdots, E_k} = \{ g \in O(n,\mathbb{C}): gE_l =E_l \; \text{for}\; 1 \leq l \leq k\}$$
where $0 \neq E_1 \subset \cdots \subset E_k$ is a sequence of totally isotropic subspaces.
Now the maximal parabolic subgroups of $O(n, \mathbb{C})$ are the $$P_E = \{ g \in O(n, \mathbb{C}) : gE=E\},$$ where $E$ is nonzero totally isotropic in $\mathbb{C}^n.$
We know that $O(n, \mathbb{C})/P_E$ is a projective variety. In particular, that should be a flag variety.
My question is:1) What should be the exact presentation of the variety $O(n, \mathbb{C})/P_E?$
2) Can we do the similar thing for $O(n, \mathbb{R})$ or $SO(n, \mathbb{R})?$
Any help we be appreciated.
Thank you in advance
1) If $m\leq n/2$ is the dimension of $E$, the $O(n,\mathbb C)/P_E$ is the variety of all $m$-dimensional isotropic subspaces of $\mathbb C^n$. (It is easy to see that $O(n,\mathbb C)$ acts transitively on the set of such subspaces and $P_E$ is the stabilizer of one of them.)
2) This depends on what you mean by "the similar thing". On the one hand, $O(n,\mathbb R)$ and $SO(n,\mathbb R)$act transitively on the space of all $k$-dimensional subspaces of $\mathbb R^n$. Thus, you can view the Grassmannian $Gr(k,\mathbb R^n)$ as a homogeneous space of $O(n,\mathbb R)$ and $SO(n,\mathbb R)$. This leads to the well known presentation $ Gr(k,\mathbb R^n)=O(n)/(O(k)\times O(n-k))$ and similar for $SO$.
However, $O(n)$ and $SO(n)$ do not have parabolic subgroups since they are compact. If you want a real analog of those (and something that is closer to the complex case), you have to go to the indefinite orthogonal groups $O(p,q)$ for $p+q=n$. For those you have isotropic subspaces of dimensions $1\leq k\leq min(p,q)$ and the stabilizer of such an isotropic subspace is a parabolic subgroup $P_k\subset O(p,q)$. The generalized flag variety $O(p,q)/P_k$ then is the variety of $k$-dimensional isotropic subspaces of $\mathbb R^{p+q}$. The closest analogy to the complex case is obtained for the spilt real forms $O(m,m)$ and $O(m,m+1)$.
Edit (in view of your comment): What you write there about the complex case is not quite correct. In the definition of $W_m$ you have to add the condition that $B(v_1,v_j)=0$ for all $i$ and $j$ to make sure that the span of your vectors is totally isotropic. Moreover, the group $O(m,\mathbb C)$ should not be used here. The parabolic subgroup $P_E$ has a natural quotient (known as the Levi-factor) isomorphic to $GL(m,\mathbb C)\times O(n-2m,\mathbb C)$, and the bundle $W_m$ is associated to the standard representation of $GL(m,\mathbb C)$, so it is just a $GL(m,\mathbb C)$ vector bundle.
In the real cases and for a definite bilinear form, the situation is easier. There you have $Gr(k,\mathbb R^n)\cong O(n)/(O(k)\times O(n-k))$ and the analog of $W_m$ now really is formed by $k$-tuples of linearly independent vectos and this is an $O(k)$ bundle (so it carries a natural bundle metric). For $SO$ and $SU$ the situation is very similar.