Einstein metrics on aligned homogeneous spaces with two factors

Abstract

Given two homogeneous spaces of the form $G_1/K$ and $G_2/K$, where $G_1$ and $G_2$ are compact simple Lie groups, we study the existence problem for $G_1\times G_2$-invariant Einstein metrics on the homogeneous space $M=G_1\times G_2/K$. For the large subclass $C$ of spaces having three pairwise inequivalent isotropy irreducible summands (12 infinite families and 70 sporadic examples), we obtain that existence is equivalent to the existence of a real root for certain quartic polynomial depending on the dimensions and two Killing constants, which allows a full classification and the possibility to weigh the existence and nonexistence pieces of $C$.