Let $G \subseteq GL_k(V)$ be an affine algebraic group. Hence $k$ is a field, $dim_k(V)=n$ and $G \subseteq GL_k(V)$ is a closed subgroup. A closed subgroup P is called parabolic if $G/P$ is a complete (and hence projective) variety. [Note $G/H$ is always quasi-projective]
If P is a parabolic subgroup of G, and Q is parabolic subgroup of P, show that Q is parabolic in G.
I was trying to chase this by definition and also from the perspective of the result that "P is parabolic in G iff it contains a Borel subgroup of G". But neither attempts led to a solution. Kindly help me here.
Question: "Hello. Can you provide a proof of the claim you made? That if the fibres are projective and we have a surjection to a projective variety then the domain is projective? – Angry_Math_Person"
Answer: I suspect it is true in greater generality, but do not have a precise reference. In
"Grothendieck, Alexander; Dieudonné, Jean A. Éléments de géométrie algébrique. I. (English) Zbl 0203.23301 Die Grundlehren der mathematischen Wissenschaften. 166."
Section 9.8 on relative Plucker embedings you may find some results in the case when the fiber $P/Q$ is a grassmannian or a flag variety. He constructs for any finite rank locally trivial sheaf $\mathcal{E}$ on a scheme $X$ a "Plucker embedding" (a closed immersion)
$$P: \mathbb{G}(k,\mathcal{E}) \rightarrow \mathbb{P}(\wedge^k \mathcal{E})$$
and if $X$ is projective (over a field $k$) it follows $\mathbb{P}(\wedge^k \mathcal{E})$ is projective, hence $\mathbb{G}(k,\mathcal{E})$ is projective. The classical grassmannian $\mathbb{G}(k,V)$ of a $k$-vector space $V$ of dimension $n$ may be constructed as $\mathbb{G}(k,V) \cong SL(V)/P$ where $P \subseteq SL(V)$ is the parabolic subgroup fixing a $k$-dimensional sub space $W \subseteq V$. The grassmannian bundle is a relative version of this: It has grassmannian varieties as fibers: There is a surjective map $\pi: \mathbb{G}(k,\mathcal{E}) \rightarrow X$ and for any point $x\in X$ it follows $\pi^{-1}(x) \cong \mathbb{G}(k,\mathcal{E}(x))$ where $\mathcal{E}(x)$ is the fiber at $x$. Hence $\mathbb{G}(k,\mathcal{E}(x))$ is the grassmannian variety parametrizing $k$-dimensional $\kappa(x)$-subspaces $W \subseteq \mathcal{E}(x)$.
There are similar constructions for flag varieties: There is a flag bundle $\pi: \mathbb{F}(d,\mathcal{E})\rightarrow X$ with the property that the fiber $\pi^{-1}(x) \cong \mathbb{F}(d,\mathcal{E}(x))$ is the flag variety of type $d$ of the fiber $\mathcal{E}(x)$. The fiber $\pi^{-1}(x)\cong SL(V)/P(d)$ where $P(d)$ is the parabolic subgroup fixing a flag of type $d$. There is a generalized Plucker embedding (a closed immersion)
$$P: \mathbb{F}(d,\mathcal{E}) \rightarrow \mathbb{P}:=\prod_i \mathbb{P}(\wedge^{d_i} \mathcal{E}),$$
and if $X$ is projective it follows $\mathbb{P}$ is projective, hence $\mathbb{F}(d,\mathcal{E})$ is projective since it is a closed subvariety of $\mathbb{P}$. Hence if $G$ is semi simple and $G/Q \cong \mathbb{F}(d,\mathcal{E})$ for some $\mathcal{E}$ the claim follows using the Plucker embedding.
If $G$ is semi simple it follows $P$ is parabolic iff $P$ is a subgroup stabilizing a flag in $V$. Hence in your case $Q$ is parabolic in $G$ (when $G$ is semi simple) iff $Q$ stablilize a flag in $V$. You find a suggested proof here as well:
$Q$ parabolic in $P$, $P$ parabolic in $G$ implies $Q$ parabolic in $G$.
The Springer book is readable.