On one class of homogeneous compact Einstein manifolds (Q1577145)

It is well known that Wallach spaces (that are the flag manifolds over complex, quaternionic, and Cayley projective planes) admit Einstein metrics. All these spaces are compact homogeneous manifolds of the form \(G/H\). Consider the orthogonal decomposition with respect to the Killing form \(B\): \(g=h \oplus p\). In the case of Wallach spaces, there is an additional property: \(p=p_1 \oplus p_2 \oplus p_3\), where \(p_i\) are \(\text{ad}_h\)-irreducible and pairwise \(B\)-orthogonal modules, and \([p_i, p_i] \subset h\). The author proves that, for every compact homogeneous space \(G/H\) with semisimple Lie group \(G\), this property implies the existence of an Einstein metric of the form \(x_1 B|_{p_1} +x_2 B|_{p_2} +x_3 B|_{p_3}\) for some \(x_i < 0\). In particular, the spaces \(SO(a+b+c)/SO(a) \times SO(b) \times SO(c)\) and \(Sp(a+b+c)/Sp(a) \times Sp(b) \times Sp(c)\) admit homogeneous Einstein metrics.