Irreducibility of the Laplacian eigenspaces of some homogeneous spaces
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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.
KeywordsHomogeneous spaces Symmetric spaces Laplacian spherical representations
Mathematics Subject ClassificationPrimary 53C30 Secondary 58J50 22E46
The authors are supported by the Research Training Group 1463 “Analysis, Geometry and String Theory” of the DFG and the first author is supported as well by the GNSAGA of INdAM. Moreover, they would like to thank Fabio Podestà for valuable feedback and his interest in this work and Emilio Lauret for pointing out an inaccuracy in an earlier version of this article. Finally, they would like to thank the referee for the careful review and several valuable comments and suggestions.
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