Irreducibility of the Laplacian eigenspaces of some homogeneous spaces

For a compact homogeneous space G / K , we study the problem of existence of G -invariant Riemannian metrics such that each eigenspace of the Laplacian is a real irreducible representation of G . We prove that the normal metric of a compact irreducible symmetric space has this property only in rank one. Furthermore, we provide existence results for such metrics on certain isotropy reducible spaces.