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I am curious about how Spaltenstein verify/construe such kind of diagrams? What's the algorithm to check $O_1 \subset \overline{O_2}$ without knowing the Hasse diagram a priori?
Thank you very much! You are right. The same computations of dimensions of null fiber shows that $m\geq 2n$ is the necessary conditions for the map $\phi$ to be faithfully flat.
Thank you very much for your answer! I have find a English book with a proof of faithfully flat: Section 5.7 in Jantzen's Representations of algebraic groups. For the locally trivial part. Yes, I means Zariski locally trivial as you said. I guess I may not be true in general. But it is quite interesting that although this is not true, by the faithfully flatness, the vector bundle $G\times_H V\to G/H$ is locally trivial, c.f. Secion~5.9 of Jantzen's book, where $V$ is a finite dimensional (over $k$) representation of $H$.