Notes on "Einstein metrics on compact simple Lie groups attached to standard triples"

Abstract: In the paper "Einstein metrics on compact simple Lie groups attached to standard triples", the authors introduced the definition of standard triples and proved that every compact simple Lie group $G$ attached to a standard triple $(G,K,H)$ admits a left-invariant Einstein metric which is not naturally reductive except the standard triple $(\Sp(4),2\Sp(2),4\Sp(1))$. For the triple $(\Sp(4),2\Sp(2),4\Sp(1))$, we find there exists an involution pair of $\sp(4)$ such that $4\sp(1)$ is the fixed point of the pair, and then give the decomposition of $\sp(4)$ as a direct sum of irreducible $\ad(4\sp(1))$-modules. But $\Sp(4)/4\Sp(1)$ is not a generalized Wallach space. Furthermore we give left-invariant Einstein metrics on $\Sp(4)$ which are non-naturally reductive and $\Ad(4\Sp(1))$-invariant. For the general case $(\Sp(2n_1n_2),2\Sp(n_1n_2),2n_2\Sp(n_1))$, there exist $2n_2-1$ involutions of $\sp(2n_1n_2)$ such that $2n_2\sp(n_1))$ is the fixed point of these $2n_2-1$ involutions, and it follows the decomposition of $\sp(2n_1n_2)$ as a direct sum of irreducible $\ad(2n_2\sp(n_1))$-modules. In order to give new non-naturally reductive and $\Ad(2n_2\Sp(n_1)))$-invariant Einstein metrics on $\Sp(2n_1n_2)$, we prove a general result, i.e. $\Sp(2k+l)$ admits at least two non-naturally reductive Einstein metrics which are $\Ad(\Sp(k)\times\Sp(k)\times\Sp(l))$-invariant if $k<l$. It implies that every compact simple Lie group $\Sp(n)$ for $n\geq 4$ admits at least $2[\frac{n-1}{3}]$ non-naturally reductive left-invariant Einstein metrics.