Stability of non-diagonal Einstein metrics on homogeneous spaces H × H/ΔK

Abstract

We consider the homogeneous space $M=H\times H/\Delta K$, where $H/K$ is an irreducible symmetric space and ΔK denotes diagonal embedding. Recently, Lauret and Will provided a complete classification of $H\times H$-invariant Einstein metrics on M. They obtained that there is always at least one non-diagonal Einstein metric on M, and in some cases, diagonal Einstein metrics also exist. We give a formula for the scalar curvature of a subset of $H\times H$-invariant metrics and study the stability of non-diagonal Einstein metrics on M with respect to the Hilbert action, obtaining that these metrics are unstable with different coindices for all homogeneous spaces M.