New homogeneous Einstein metrics on quaternionic Stiefel manifolds

We consider invariant Einstein metrics on the quaternionic Stiefel manifolds $V_p\mathbb{H} ^n$ of all orthonormal $p$-frames in $\mathbb{H}^n$. This manifold is diffeomorphic to the homogeneous space $\mathrm{Sp}(n) / \mathrm{Sp}(n-p)$ and its isotropy representation contains equivalent summands. We obtain new Einstein metrics on $V_p\mathbb{H}^n \cong \mathrm{Sp}(n)/\mathrm{Sp}(n-p)$, where $n = k_1 + k_2 + k_3$ and $p = n-k_3$. We view $V_p\mathbb{H}^n$ as a total space over the generalized Wallach space $\mathrm{Sp}(n) / (\mathrm{Sp}(k_1) \times \mathrm{Sp}(k_2) \times \mathrm{Sp}(k_3))$ and over the generalized flag manifold $\mathrm{Sp}(n) / (\mathrm{U}(p) \times \mathrm{Sp}(n-p))$.