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.