Problem (The problem is from the book Groups and Representations by Alperin, page 56)
If $P$ is a parabolic subgroup of $G$ which is the stabilizer of the flag $(W_1,\ldots,W_r),$ then the $W_i$ are the only subspaces of $V_n(F)$ left invariant by $P$.
My attempt So $P$ is the parabolic subgroup of $G(n,F)$ which stabilizes the flag $(W_1,\ldots,W_r)$. That means $p(W_i)=W_i,\ \forall p\in P$ and $W_i \in (W_1,\ldots,W_r)$.
Hence we only need to show the uniqueness of $(W_1,\ldots,W_r)$. Assume by contradiction there exist some other subspace, say $W$ of $GL(n,F)$ which is left invariant by $P$. A flag on $V_n(F)$ is nested sequence of non-zero subspaces of $V_n(F)$. In other words a flag is a sequence $(W_1,\ldots,W_r)$ of subspaces of $V_n(F)$, where
$0 \subset W_1 \subset W_2 \subset \ldots \subset W_r = V_n(F)$
So $V$ is contained in some $W_i$, that is $W \subseteq W_i$. Hence there is a minimal $1\leq k\leq r$ such that $W \subseteq W_k$ and $W \not\subseteq W_{k-1}$. By the minimality of $k$, $W$ contains an element of the form $\sum_{j=1}^k \alpha_jv_j$. This element is sent to $w_{s+1},\ldots,w_k$(which are basis vectors of $W_k$ which are extensions of basis vectors of $W_{k-1}$ to obtain basis for $W_{k-1}$) by some $p's \in P$, which shows that $w_{s+1},\ldots,w_k \in W$.
And conversely, let $\sum_{j=1}^k \alpha_jv_j \in W_k-W_{k-1}$. This element is the image of vectors $w_{s+1},\ldots,w_k \in W$ under $p's \in P$ (which exactly is the part that I am missing) and hence lies in W.
Hence we can conclude that $W$ conatins $W_k-W_{k-1}$ and hence $W_k$. We conclude that $W=W_k$.
Question What would be those elements of $P$ that we can use to map those vectors to an element $\sum_{j=1}^k \alpha_jv_j$?