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Let $F_4$ be the connected, simply connected, simple, complex, linear algebraic group of type $\mathsf{F}_4$, with Dynkin diagram $$ \beta_1-\beta_2\Rightarrow\beta_3-\beta_4\,. $$ Let $P_{\{\beta_2\}}$ be the minimal parabolic subgroup $\neq B$ such that its set of smiple roots is $\{\beta_2\}$. Let $$ \mu\,\colon F_4\to\operatorname{Aut}(F_4/P_{\{\beta_2\}}) $$ be the obvious morphism which sends an element $g\in F_4$ to the translation $x\mapsto gx$.

Is the morphism $\mu$ surjective? Or are there other automorphisms which do not come from the group? If there are more, how can they be described?

This question and the reference given in the comments there seem to be related. But I can't answer myself.

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The following paper says that your map $\mu$ is indeed surjective; the author attributes the result over $\mathbb C$ to Tits.

Demazure, M. Invent Math (1977) 39: 179. https://doi.org/10.1007/BF01390108

I also add the link

Automorphism group of flag manifolds?

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  • $\begingroup$ Thank you very much, @Venkataramana. I only see a preview of the link, and it can't be Proposition 1 what I am looking for. Can you elaborate where precisely I find the result in this paper, and maybe extract the information. It would be great for me. $\endgroup$ – user66288 Sep 16 '18 at 14:51
  • $\begingroup$ I am talking about Theorem 1 ( Your case is $F_4$ and it is not exceptional in Demazure's Theorem 1). $\endgroup$ – Venkataramana Sep 16 '18 at 14:58
  • $\begingroup$ Right, thanks again, here an open source link: gdz.sub.uni-goettingen.de/id/PPN356556735_0039?tify={%22pages%22:[183],%22view%22:%22export%22} $\endgroup$ – user66288 Sep 16 '18 at 15:05
  • $\begingroup$ Thank you , The open source link is indeed better! $\endgroup$ – Venkataramana Sep 16 '18 at 15:07

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