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SPECTRAL UNIQUENESS OF BI-INVARIANT METRICS ON SYMPLECTIC GROUPS

  • EMILIO A. LAURET
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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. []AYY13
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  2. []BD
    T. Bröcker, T. tom Dieck, Representations of Compact Lie Groups, Grad. Texts in Math., Vol. 98, Springer-Verlag, New York, 1995.Google Scholar
  3. []CS
    J. H. Conway, N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed., Grundlehren Math. Wiss., Vol. 290, Springer-Verlag, New York, 1999.Google Scholar
  4. []GGS17
    D. Goldstein, R. Guralnick, R. Stong, A lower bound for the dimension of a highest weight module, Represent. Theory 21 (2017), 611–625.MathSciNetCrossRefGoogle Scholar
  5. []GSS10
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  6. []GS10
    C. Gordon, C. Sutton, Spectral isolation of naturally reductive metrics on simple Lie groups, Math. Z. 266 (2010), no. 4, 979–995.MathSciNetCrossRefzbMATHGoogle Scholar
  7. []Kn
    A. W. Knapp, Lie Groups Beyond an Introduction, Progr. Math., Vol. 140, Birkhäuser Boston, BOston, MA, 2002.Google Scholar
  8. []La18
    E. A. Lauret, The smallest Laplace eigenvalue of homogeneous 3-spheres, arXiv:1801.04259 (2018).Google Scholar
  9. []Pr05
    E. Proctor, Isospectral metrics and potentials on classical compact simple Lie groups, Michigan Math. J. 53 (2005), no. 2, 305–318.MathSciNetCrossRefzbMATHGoogle Scholar
  10. []SS14
    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. []Sch01
    D. Schueth, Isospectral manifolds with different local geometries, J. Reine Angew. Math. 534 (2001), 41–94.MathSciNetzbMATHGoogle Scholar
  12. []Su02
    C. J. Sutton, Isospectral simply-connected homogeneous spaces and the spectral rigidity of group actions, Comment. Math. Helv. 77 (2002), no. 4, 701–717.MathSciNetCrossRefzbMATHGoogle Scholar
  13. []Ta73
    S. Tanno, Eigenvalues of the Laplacian of Riemannian manifolds, Tǒhoku Math. J. (2) 25 (1973), no. 3, 391–403.Google Scholar
  14. []Ur79
    H. Urakawa, On the least positive eigenvalue of the Laplacian for compact group manifolds, J. Math. Soc. Japan 31 (1979), no. 1, 209–226.MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CIEM–FaMAF (CONICET)Universidad Nacional de CórdobaCórdobaArgentina

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