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

Peter Franek; Karel Pazourek; Vít Tuček

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arXiv · 1006.3407

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

Peter Franek·Karel Pazourek·Vít Tuček

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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 $F_4^{(-20)}/P_4$.

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Peter Franek, Karel Pazourek, Vít Tuček. 2010-06-17. Hyperplane section $\mathbb{OP}^2_0$ of the complex Cayley plane as the homogeneous space $\mathrm{F_4/P_4}$. https://arxiv.org/abs/1006.3407

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