The Lichnerowicz Laplacian $\Delta_L$ is an interesting differential operator on Riemannian manifolds, generalizing the Hodge-de Rham Laplacian on differential forms to tensors of arbitrary type. It features prominently in the study of the linear stability of Einstein metrics under the Einstein-Hilbert action. This is also closely related to the rigidity problem, i.e. whether a given Einstein metric on a compact manifold  is isolated in the Einstein moduli space.

In the 80s, Koiso studied the stability of symmetric spaces of compact type using harmonic analysis and utilizing the coincidence of $\Delta_L$ with a Casimir operator.

Motivated by his and also the $G$-stability results of Lauret-Lauret-Will, we generalize Koiso’s strategy to the more general setting of normal homogeneous spaces, where Casimir operators still occur naturally. Ultimately this approach is sufficient to provide many new non-symmetric examples of stable Einstein manifolds of positive scalar curvature.