New Non-Naturally Reductive Einstein Metrics on Sp(n)

In this paper, we consider a class of left invariant Riemannian metrics on Sp(n), which is invariant under the adjoint action of the subgroup Sp(n − 3) × Sp(1) × Sp(1) × Sp(1). Based on the related formulae in the literature, we show that the existence of Einstein metrics is equivalent to the existence of solutions of some homogeneous Einstein equations. Then we use a technique of the Grobner basis to get a sufficient condition for the existence, and show that this method will lead to new non-naturally reductive metrics.