Realizations of inner automorphisms of order four and fixed points subgroups by them on the connected compact exceptional Lie group \(E_8\). II (Q2176600)

Let \(G\) be a Lie group and \(H\) a compact subgroup of \(G\). A homogeneous space \(G/H\) with \(G\)-invariant Riemannian metric \(g\) is called a Riemannian 4-symmetric space if there exists an automorphism \(\widetilde\gamma\) of order 4 on \(G\) such that \((G^\gamma)_0\subset H\subset G^\gamma\), where \(G^\gamma\) and \((G^\gamma)_0\) are the fixed point subgroup and its identity component, respectively. \textit{J. A. Jiménez} [Trans. Am. Math. Soc. 306, No. 2, 715--734 (1988; Zbl 0647.53039)] classified compact simply connected Riemannian 4-symmetric spaces. According to the classification by Jiménez [loc. cit.], there exist seven compact simply connected Riemannian 4-symmetric spaces for the exceptional compact Lie group \(G\) of type \(E_8\). In the paper under review the author shows the group realization for three of the seven cases. He gives the explicit form of automorphisms \(\widetilde\omega_4\), \(\widetilde\nu_4\) and \(\widetilde\mu_4\) of order 4 on \(E_8\) induced by the \(\mathbb{C}\)-linear transformations \(\omega_4\), \(\nu_4\) and \(\mu_4\) of the 248-dimensional vector space \(e^C_8\), respectively. Furthermore, the author determines the structure of these fixed points subgroups \((E^{\omega_4}_8)\), \((E^{\nu_4}_8)\) and \((E^{\mu_4}_8)\), of \(E_8\). These amount to the global realizations of three spaces among Riemannian 4-symmetric spaces \(G/H\) of above corresponding to the Lie algebras \(\mathfrak{h}=i\mathbf{R} \oplus\mathfrak{s}\mathfrak{u}(8)\), \(\mathfrak{h}= i\mathbf{R}\oplus e_7\) and \(\mathfrak{h}= \mathfrak{s} \mathfrak{u}(2)\oplus\mathfrak{s}\mathfrak{u}(8)\), where \(\mathfrak{h}= Lie(H)\). In [Tsukuba J. Math. 41, No. 1, 91--166 (2017; Zbl 1379.53065)], the author showed the group realization for case \(\mathfrak{h}= \mathfrak{s}\mathfrak{o}(6)\oplus \mathfrak{s}\mathfrak{o}(10)\). The remaining three cases \(\mathfrak{h}= \mathfrak{s}\mathfrak{u}(2)\oplus i\mathbf{R}\oplus e_6\), \(\mathfrak{h}= i\mathbf{R}\oplus \mathfrak{s}\mathfrak{o}(14)\) and \(\mathfrak{h}= \mathfrak{s}\mathfrak{u}(2)\oplus i\mathbf{R}\oplus\mathfrak{s}\mathfrak{o}(12)\), will be shown in a forthcoming article by the author.