On the groups of isometries of simple para-Hermitian symmetric spaces
The goal of this paper is to completely determine the group of isometries of a simple para-Hermitian symmetric space. This allows to provide a complete description of the group of affine transformations of such a space with respect to its canonical affine connection. Indeed, it is shown that the group of isometries coincides with the group of affine transformations. The structure of \(I(G/H,g)/I(G/H,g)_0\) (the quotient group of the group of isometry over its connected component) is given in an extensive table. The authors start by recalling the definition and some known facts about para-Hermitian symmetric spaces (see for example [\textit{S. Kaneyuki} and \textit{M. Kozai}, Tokyo J. Math. 8, 81--98 (1985; Zbl 0585.53029)]). They then discuss the interrelationship between the group of isometries and the group \(\mathrm{Aut}(\mathfrak{g},\sigma_*)\) of Lie algebra automorphisms that commute with the involution \(\sigma_*\). This is done by giving a description of \(\mathrm{Aut}(\mathfrak{g},\sigma_*)\) and its correlation with the group of affine transformations leading to an isomorphism between these two and the equality between the group of isometries and the group of affine transformations. They conclude by explaining the construction of the table giving the structure of the quotient \(I(G/H,g)/I(G/H,g)_0\) and provide some examples of the computations (for the classical type namely \(\mathfrak{sl}(n,{\mathbb R})\) and \(\mathfrak{so}(n,n)\) and for the exceptional types namely \(\mathfrak{e}_{6(6)}\) and \(\mathfrak{e}_{7(-25)}\)).
scientific article; zbMATH DE number 6802083