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Irreducibility of the Laplacian eigenspaces of some homogeneous spaces

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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.

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References

  1. Bedulli, L., Gori, A.: A Hamiltonian stable minimal Lagrangian submanifold of projective space with non-parallel second fundamental form. Transform. Groups 12(4), 611–617 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bérard, P., Pesce, H.: Construction de Variétés Isospectrales Autour du Théorème de T. Sunada, Progress in Inverse Spectral Geometry. Trends in Mathematics. (S.I. Andersson and M.L. Lapidus, eds.), Birkhäuser, Basel, (1997)

  3. Berger, M., Gauduchon, P., Mazet, E.: Le spectre d’une variété riemannienne. Lecture Notes in Mathematics, vol. 194. Springer, Berlin-New York (1971)

  4. Besse, A.: Einstein Manifolds. Springer, Berlin (1987)

    Book  MATH  Google Scholar 

  5. Bezubik, A., Strasburger, A.: On spherical expansions of smooth \(\text{ SU }(n)\)-zonal functions on the unit sphere in \({\mathbb{C}}^n\). J. Math. Anal. Appl. 404, 570–578 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bröcker, T., tom Dieck, T.: Representations of Compact Lie Groups, Graduate Texts in Mathematics. Springer, Berlin Heidelberg (2003)

  7. Gindikin, S., Goodman, R.: Restricted roots and restricted form of the Weyl dimension formula for spherical varieties. J. Lie Theory 23(1), 257–311 (2013)

    MathSciNet  MATH  Google Scholar 

  8. Guillemin, V., Legendre, E., Sena-Dias, R.: Simple spectrum and Rayleigh quotients, Geometric and spectral analysis. Contemp. Math., vol. 630, Am. Math. Soc., Providence, RI, pp. 33–44 (2014)

  9. Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Elsevier Science, Pure and Applied Mathematics (1979)

    MATH  Google Scholar 

  10. Helgason, S.: Geometric analysis on symmetric spaces. American Mathematical Soc, Mathematical surveys and monographs (1993)

    MATH  Google Scholar 

  11. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. Interscience Publisher, Geneva (1963)

    MATH  Google Scholar 

  12. Krämer, M.: Eine Klassifikation bestimmter Untergruppen kompakter zusammenhängender Liegruppen. Comm. Algebra 3(8), 691–737 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  13. Krämer, M.: Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen. Compos. Math. 38, 129–153 (1979)

    MATH  Google Scholar 

  14. Nguyêñ, H.D.: Compact weakly symmetric spaces and spherical pairs. Proc. Am. Math. Soc. 128(11), 3425–3433 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Onishchik, A.L., Leites, D.A., Vinberg, E.B.: Lie Groups and Algebraic Groups. Springer Series in Soviet Mathematics. Springer, Berlin Heidelberg (2012)

    Google Scholar 

  16. Satake, I.: On representations and compactifications of symmetric Riemannian spaces. Ann. Math 2(71), 77–110 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  17. Schueth, D.: Generic irreducibilty of Laplace eigenspaces on certain compact Lie groups. Ann. Global Anal. Geom. 52(2), 187–200 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sepanski, M.R.: Compact Lie groups. Springer, Berlin (2007)

    Book  MATH  Google Scholar 

  19. Sugiura, M.: Representations of compact groups realized by spherical functions on symmetric spaces. Proc. Japan Acad. 38, 111–113 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  20. Takeuchi, M.: Modern spherical functions. Translations of mathematical monographs, American Mathematical Society (1994)

  21. Uhlenbeck, K.: Generic properties of eigenfunctions. Am. J. Math. 98(4), 1059–1078 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wolf, J.A.: The geometry and structure of isotropy irreducible homogeneous spaces. Acta Math. 120, 59–148 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  23. Yamaguchi, S.: Spectra of flag manifolds. Mem. Fac. Sci. Kyushu Univ. Ser. A 33(1), 95–112 (1979)

    MathSciNet  MATH  Google Scholar 

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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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  1. Institut für Differentialgeometrie, Leibniz Universität Hannover, Welfengarten 1, 30167, Hannover, Germany

    David Petrecca & Markus Röser

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  1. David Petrecca
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  2. Markus Röser
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Correspondence to David Petrecca.

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Petrecca, D., Röser, M. Irreducibility of the Laplacian eigenspaces of some homogeneous spaces. Math. Z. 291, 395–419 (2019). https://doi.org/10.1007/s00209-018-2088-z

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  • DOI: https://doi.org/10.1007/s00209-018-2088-z

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