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SPECTRAL UNIQUENESS OF BI-INVARIANT METRICS ON SYMPLECTIC GROUPS

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Abstract

In this short note, we prove that a bi-invariant Riemannian metric on Sp(n) is uniquely determined by the spectrum of its Laplace–Beltrami operator within the class of left-invariant metrics on Sp(n). In other words, on any of these compact simple Lie groups, every left-invariant metric which is not right-invariant cannot be isospectral to a bi-invariant metric. The proof is elementary and uses a very strong spectral obstruction proved by Gordon, Schueth and Sutton.

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References

  1. J. An, J.-K. Yu, J. Yu, On the dimension datum of a subgroup and its application to isospectral manifolds, J. Differential Geom. 94 (2013), no. 1, 59–85.

    Article  MathSciNet  Google Scholar 

  2. T. Bröcker, T. tom Dieck, Representations of Compact Lie Groups, Grad. Texts in Math., Vol. 98, Springer-Verlag, New York, 1995.

  3. J. H. Conway, N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed., Grundlehren Math. Wiss., Vol. 290, Springer-Verlag, New York, 1999.

  4. D. Goldstein, R. Guralnick, R. Stong, A lower bound for the dimension of a highest weight module, Represent. Theory 21 (2017), 611–625.

    Article  MathSciNet  Google Scholar 

  5. C. Gordon, D. Schueth, C. Sutton, Spectral isolation of bi-invariant metrics on compact Lie groups, Ann. Inst. Fourier (Grenoble) 60 (2010), no. 5, 1617–1628.

    Article  MathSciNet  Google Scholar 

  6. C. Gordon, C. Sutton, Spectral isolation of naturally reductive metrics on simple Lie groups, Math. Z. 266 (2010), no. 4, 979–995.

    Article  MathSciNet  Google Scholar 

  7. A. W. Knapp, Lie Groups Beyond an Introduction, Progr. Math., Vol. 140, Birkhäuser Boston, BOston, MA, 2002.

  8. E. A. Lauret, The smallest Laplace eigenvalue of homogeneous 3-spheres, arXiv:1801.04259 (2018).

  9. E. Proctor, Isospectral metrics and potentials on classical compact simple Lie groups, Michigan Math. J. 53 (2005), no. 2, 305–318.

    Article  MathSciNet  Google Scholar 

  10. B. Schmidt, C. Sutton, Detecting the moments of inertia of a molecule via its rotational spectrum, II, Preprint available at http://users.math.msu.edu/users/schmidt/ (Schmidt’s web page) (2014).

  11. D. Schueth, Isospectral manifolds with different local geometries, J. Reine Angew. Math. 534 (2001), 41–94.

    MathSciNet  MATH  Google Scholar 

  12. C. J. Sutton, Isospectral simply-connected homogeneous spaces and the spectral rigidity of group actions, Comment. Math. Helv. 77 (2002), no. 4, 701–717.

    Article  MathSciNet  Google Scholar 

  13. S. Tanno, Eigenvalues of the Laplacian of Riemannian manifolds, Tǒhoku Math. J. (2) 25 (1973), no. 3, 391–403.

    Article  MathSciNet  Google Scholar 

  14. H. Urakawa, On the least positive eigenvalue of the Laplacian for compact group manifolds, J. Math. Soc. Japan 31 (1979), no. 1, 209–226.

    Article  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. CIEM–FaMAF (CONICET), Universidad Nacional de Córdoba, Medina Allende s/n, Ciudad Universitaria, 5000, Córdoba, Argentina

    EMILIO A. LAURET

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  1. EMILIO A. LAURET
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Correspondence to EMILIO A. LAURET.

Additional information

This work was done during an Alexander von Humboldt Foundation Postdoctoral Fellowship at Humboldt-Universität zu Berlin between April 2017 and May 2018. Supported also by grants from CONICET and FONCyT.

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LAURET, E.A. SPECTRAL UNIQUENESS OF BI-INVARIANT METRICS ON SYMPLECTIC GROUPS. Transformation Groups 24, 1157–1164 (2019). https://doi.org/10.1007/s00031-018-9486-5

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  • DOI: https://doi.org/10.1007/s00031-018-9486-5

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