# Automorphisms of homogeneous space $F_4/P_{\{\beta_2\}}$ over the exceptional group $F_4$

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.

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 am talking about Theorem 1 ( Your case is $F_4$ and it is not exceptional in Demazure's Theorem 1). – Venkataramana Sep 16 '18 at 14:58