Almost complex structures on \(k\)-symmetric spaces (Q5946129)

The notion of canonical almost complex structure on a homogeneous \(K\)-symmetric space of a Lie group \(G\) has been introcuced by \textit{J. Wolf} and \textit{A. Gray} [J. Differ. Geom. 2, 77-114, 115-159 (1968; Zbl 0169.24103)]. The author proves several important results. We shall quote here the following Proposition: Let \(G\) be a Lie group and \(H\) a subgroup of \(G\) such that \(G^t_0\subset H\subset G^t\). If \(G\) is semi-simple, then the canonical almost complex structure on \(G/H\) is integrable if and only if the canonical almost complex structure on the orbit at \({\mathcal J}\) with respect to the conjugation action of \(\Aut({\mathcal G})\) is integrable.