Irreducibility of the Laplacian eigenspaces of some homogeneous spaces

For a compact homogeneous space $G\mathrm{/}K$, 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.