Hyperplane section $\mathbb{OP}^2_0$ of the complex Cayley plane as the homogeneous space $\mathrm{F_4/P_4}$

TL;DR

This paper demonstrates that the hyperplane section of the complex Cayley plane is a homogeneous space under the action of the exceptional Lie group F4, providing explicit constructions and identifying stabilizers.

Contribution

It offers a direct, constructive proof that F4 acts transitively on the hyperplane section of the complex Cayley plane and explicitly identifies the stabilizer as a specific parabolic subgroup.

Findings

F4 acts transitively on the hyperplane section of the complex Cayley plane

Explicit realization of vector and spin actions of Spin(9,C) within F4

Identification of the stabilizer as a parabolic subgroup P4

Abstract

We prove that the exceptional complex Lie group $F_4$ has a transitive action on the hyperplane section of the complex Cayley plane $\mathbb{OP}^2$. Our proof is direct and constructive. We use an explicit realization of the vector and spin actions of $\Spin(9,\C) \leq F_4$. Moreover, we identify the stabilizer of the $F_4$-action as a parabolic subgroup $P_4$ (with Levi factor $B_3T_1$) of the complex Lie group $F_4$. In the real case we obtain an analogous realization of $F_4^{(-20)}/P_4$.