Realizations of inner automorphisms of order four and fixed points subgroups by them on the connected compact exceptional Lie group E8, Part III

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 γ˜ of order four on G and H is a fixed points subgroup Gγ 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 E8. In the present article, we give the explicit form of automorphisms ω˜4, κ˜4 and ε˜4 of order four on E8 induced by the C-linear transformations ω4, κ4 and ε4 of the 248-dimensional C-vector space e8C, respectively. Further, we determine the structure of these fixed points subgroups (E8)ω4, (E8)κ4 and (E8)ε4 of E8. These amount to the global realizations of three spaces among seven Riemannian 4-symmetric spaces G/H above corresponding to the Lie algebras h=su(2)⊕iR⊕e6, iR⊕so(14) and h=su(2)⊕iR⊕so(12), where h=Lie(H). With this article, the all realizations of inner automorphisms of order four and fixed points subgroups by them have been completed in E8.