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Mathematische Zeitschrift

, Volume 291, Issue 1–2, pp 395–419 | Cite as

Irreducibility of the Laplacian eigenspaces of some homogeneous spaces

  • David PetreccaEmail author
  • Markus Röser
Article
  • 55 Downloads

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.

Keywords

Homogeneous spaces Symmetric spaces Laplacian spherical representations 

Mathematics Subject Classification

Primary 53C30 Secondary 58J50 22E46 

Notes

Acknowledgements

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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für DifferentialgeometrieLeibniz Universität HannoverHannoverGermany

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