Invariant Einstein metrics on some homogeneous spaces of classical Lie groups

A Riemannian manifold $(M,\rho)$ is called Einstein if the metric $\rho$ satisfies the condition $\Ric (\rho)=c\cdot \rho$ for some constant $c$. This paper is devoted to the investigation of $G$-invariant Einstein metrics with additional symmetries, on some homogeneous spaces $G/H$ of classical groups. As a consequence, we obtain new invariant Einstein metrics on some Stiefel manifolds $SO(n)/SO(l)$, and on the symplectic analogues $Sp(n)/Sp(l)$. Furthermore, we show that for any positive integer $p$ there exists a Stiefel manifold $SO(n)/SO(l)$ and a homogenous space $Sp(n)/Sp(l)$ which admit at least $p$ SO(n) (resp. $Sp(n)$)-invariant Einstein metrics.