New homogeneous Einstein metrics on Stiefel manifolds

We consider invariant Einstein metrics on the Stiefel manifold VqRnVqRn of all orthonormal q  -frames in RnRn. This manifold is diffeomorphic to the homogeneous space SO(n)/SO(n−q)SO(n)/SO(n−q) and its isotropy representation contains equivalent summands. We prove, by assuming additional symmetries, that V4RnV4Rn(n⩾6)(n⩾6) admits at least four SO(n)SO(n)-invariant Einstein metrics, two of which are Jensen's metrics and the other two are new metrics. Moreover, we prove that V5R7V5R7 admits at least six invariant Einstein metrics, two of which are Jensen's metrics and the other four are new metrics.