Realizations of inner automorphisms of order four and fixed points subgroups by them on the connected compact exceptional Lie group $E_8$, Part II

Abstract

The compact simply connected Riemannian 4-symmetric spaces were classified by J. A. Jiménez according to type of the Lie algebras. As homogeneous manifolds, these spaces are of the form $G/H$, where $G$ is a connected compact simple Lie group with an automorphism $\tilde{\gamma}$ of order four on $G$ and $H$ is a fixed points subgroup $G^\gamma$ of $G$. According to the classification by J. A. Jiménez, there exist seven compact simply connected Riemannian 4-symmetric spaces $G/H$ in the case where $G$ is of type $E_8$. In the present article, we give the explicit form of automorphisms $\tilde{w}_4$, $\tilde{\upsilon}_4$ and $\tilde{\mu}_4$ of order four on $E_8$ induced by the $C$-linear transformations $w_4, \upsilon_4$ and $\mu_4$ of the 248-dimensional vector space $𝔢^C_8$, respectively. Further, we determine the structure of these fixed points subgroups $(E_8)^{w_4}, (E_8)^{\upsilon_4}$ and $(E_8)^{\mu_4}$ of $E_8$. These amount to the global realizations of three spaces among seven Riemannian 4-symmetric spaces $G/H$ above corresponding to the Lie algebras $𝔥 = i \boldsymbol{R} \oplus 𝔰𝔲(8)$, $i\boldsymbol{R} \oplus 𝔢_7$ and $𝔥 = 𝔰𝔲(2) \oplus 𝔰𝔲(8)$, where $𝔥 = \mathrm{Lie}(H)$.