Parabolic subgroups of reductive group as stabilizers of flags

Question

$\DeclareMathOperator\GL{GL}$Let $G$ be a linear algebraic group (probably reductive will be needed). Consider a faithful representation $G \to \GL(V)$. Given a parabolic subgroup $P < G$, we can consider a composition series of $V$ as a $P$-module

$$ V_\bullet: 0 \subseteq V_1 \subseteq V_2 \subseteq \dotsb \subseteq V_s = V, $$ i.e. each $V_i$ is $P$-invariant and the flag is maximal in the sense that the quotients $V_i/V_{i-1}$ are irreducible representations for $P$.

I would really appreciate it if you could help me to shed some light on the following questions:

• (Main question) Is it true that $P$ is the stabilizer of this composition series, i.e. is $$ \operatorname{Stab}_G(V_\bullet) = \{g \in G \mid gV_i \subseteq V_i \text{ for all }i\} $$ exactly equal to $P$ (or conjugated to it)?
• In the case the answer is "yes": is it true that the Levi subgroup associated to $P$ is exactly the collection of $g \in G$ preserving the graded object attached to $V_\bullet$?
• In the case the answer is "no" for a general representation $V$: Is there a particular representation for which the previous sentences hold true?

Some comments

• If I'm not mistaken, the result is true for $G = \GL_n$ with the obvious representation. Indeed, the parabolic subgroups are “fat upper triangular” subgroups, and the steps in the staircase determine the flag. Actually, for “classic groups” such as $\GL_n$, $\operatorname{SL}_n$ or $\operatorname O_n$, I think the result holds true using an appropriate “isotropic flag”. My question is about a general reductive group.
• If $P=B$ is the Borel subgroup, the result (at least, the first question) is true. Indeed, it is typically the way you prove that $G/B$ is projective, by showing it is isomorphic to the variety of full flags. But the proof I know uses Lie–Kolchin's theorem, which requires the subgroup to be solvable.
• I'm aware that $G/P$ is usually referred to as the "partial flag variety" (which partially motivates by question), but I never saw that this variety can be actually understood as a genuine variety of (partial) flags. In particular, a related question would be: Is it possible to identify $G/P$ for a parabolic subgroup $P$ as a certain subvariety of the full flag variety $G/B$?

Answer 1

$\DeclareMathOperator\GL{GL}$Yes for reductive $G$, and this follows quickly from the dynamical characterization of parabolics: A subgroup $H$ of a reductive group $G$ is parabolic if and only if there is some $\mu \colon \mathbb G_m \to G$ such that $$H = \{ g \in G \mid \lim_{t\to 0} \mu(t) g \mu(t)^{-1} \text{ exists}\}.$$

A faithful representation $V$ of $G$ of dimension $n$ defines an embedding of $G$ into $\GL_n$ and this definition makes clear that every parabolic is the inverse image of a parabolic from $\GL_n$. The parabolic in $\GL_n$ is expressed as the stabilizer of a filtration, so $P$ is the stabilizer in $G$ of that filtration. Since $P$ stabilizes that filtration, that filtration is a coarsening of the composition series of $V$ as a $P$-module, so $P$ is also the stabilizer of the composition series.

One can directly see that this subgroup is the stabilizer of a filtration, without using the classification of parabolics of $\GL_n$ (or rather by reproving it) by choosing a basis of $V$ by eigenspaces of $\mu(t)$, noting that the limit exists if and only if the entries of $g$ corresponding to a pair of basis vectors where the first one has a larger eigenvalue all vanish, i.e. if $g$ preserves a filtration into sums of eigenspaces with eigenvalue less than or equal to a given one.

For the follow-up question, I don't think "the graded object attached to $V$" is well-defined. There are many possible ways of splitting a filtration, and some might have tiny stabilizers. However, any Levi certainly gives a splitting (via the decomposition of $V$ as a direct sum of irreducible representations of the Levi) and the Levi is the stabilizer of that splitting because of the characterization of the Levi as the centralizer of the image of a homomorphism $\mu \colon \mathbb G_m \to G$, hence the stabilizer of the splitting of $V$ into eigenspaces of $\mu$, and hence the stabilizer of its own decomposition into irreducible representations, which coarsens that one.

Why would you expect a partial flag variety to be a subvariety of the full flag variety. I don't think it usually is. "Partial" means information is forgotten, i.e. the partial flag variety admits a surjective map from the full flag variety. It doesn't refer to being a part of a full flag variety.