Irreducibility of the Laplacian eigenspaces of some homogeneous spaces

TL;DR

This paper investigates when the Laplacian eigenspaces on compact homogeneous spaces are irreducible representations, showing that only rank one symmetric spaces have this property under the normal metric and providing existence results for other spaces.

Contribution

It characterizes the irreducibility of Laplacian eigenspaces on homogeneous spaces, highlighting the special case of rank one symmetric spaces and offering new existence results.

Findings

Normal metric of irreducible symmetric space is irreducible only in rank one.

Existence of such metrics is established for certain isotropy reducible spaces.

Provides a classification and new examples related to Laplacian eigenspaces.

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.