Irreducibility of the Laplacian eigenspaces of some homogeneous spaces

Abstract

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.

Title changed, some inaccurate statements have been corrected and some arguments have been clarified. 25 pages, 2 tables. Comments are welcome