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Patterson–Sullivan distributions in higher rank

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Abstract

For a compact locally symmetric space X Γ of non-positive curvature, we consider sequences of normalized joint eigenfunctions which belong to the principal spectrum of the algebra of invariant differential operators. Using an h-pseudo-differential calculus on X Γ , we define and study lifted quantum limits as weak*-limit points of Wigner distributions. The Helgason boundary values of the eigenfunctions allow us to construct Patterson–Sullivan distributions on the space of Weyl chambers. These distributions are asymptotic to lifted quantum limits and satisfy additional invariance properties, which makes them useful in the context of quantum ergodicity. Our results generalize results for compact hyperbolic surfaces obtained by Anantharaman and Zelditch.

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References

  1. Anantharaman, N., Silberman, L.: A Haar component for quantum limits in locally symmetric spaces, (2010, preprint). arXiv:1009.492

  2. Anantharaman N., Zelditch S.: Patterson–Sullivan distributions and quantum ergodicity. Ann. Henri Poincaré 8, 361–426 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Anantharaman, N., Zelditch, S.: Intertwining the geodesic flow and the Schrödinger group on hyperbolic surfaces, (2010, preprint). arXiv:1010.0867

  4. Anker J.-P.: A basic inequality for scattering theory of Riemannian symmetric spaces of the noncompact type. Am. J. Math. 113, 391–398 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bourbaki N.: Groupes at Algèbres de Lie Chap. 4,5 et 6. Masson, Paris (1981)

    Google Scholar 

  6. Dimassi M., Sjöstrand J.: Spectral Asymptotics in the Semi-Classical Limit. Cambridge University Press, Cambridge (1999)

    Book  MATH  Google Scholar 

  7. Duistermaat J.J., Kolk J.A.C., Varadarajan V.S.: Spectra of compact locally symmetric manifolds of negative curvature. Invent. Math. 52, 27–93 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  8. Duistermaat J.J., Kolk J.A.C., Varadarajan V.S.: Functions, Flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groups. Compos. Math. 49, 309–398 (1983)

    MathSciNet  MATH  Google Scholar 

  9. Evans, L.E., Zworski, M.: Lectures on Semiclassical Analysis, University of California, Berkeley. http://math.berkeley.edu/~zworski/semiclasical.pdf

  10. Hansen, S.: Rayleigh-type surface quasimodes in general linear elasticity. Anal. PDE (2010, to appear). arXiv:1008.2930

  11. Heckman, G.: Projection of Orbits and Asymptotic Behaviour of Multiplicities for Compact Lie Groups. Dissertation, Utrecht University (1980)

    Google Scholar 

  12. Helgason, S.: Geometric Analysis on Symmetric Spaces. In: Mathematical surveys and monographs, American Mathematical Society, Providence (1994)

  13. Helgason, S.: Groups and geometric analysis. In: Mathematical surveys and monographs, American Mathematical Society, Providence (2000)

  14. Helgason, S.: Differential geometry, Lie Groups, and Symmetric Spaces. In: Graduate Studies in Mathematics, American Mathematical Society, Providence (2001)

  15. Hilgert, J., Schröder, M.: Patterson–Sullivan distributions for rank one symmetric spaces of the non-compact type. arXiv0909.2142

  16. Hilgert, J.: An Ergodic Arnold–Liouville Theorem for Locally Symmetric Spaces. In: Ali, S.T. et al. (eds.) Twenty Years of Bialowieza: A Mathematical Anthology. World Sci. Monogr. Ser. Math., vol. 8, World Science Publications, Hackensack (2005)

  17. Hörmander, L.: The Analysis of Linear Partial Differential Operators, vol. I–IV, Springer, Berlin (1983–1985)

  18. Kashiwara M., Kowata A., Minemura K., Okamoto K., Ōshima T., Tanaka M.: Eigenfunctions of invariant differential operators on a symmetric space. Ann. Math. 107, 1–39 (1978)

    Article  MATH  Google Scholar 

  19. Schlichtkrull H.: Hyperfunctions and Harmonic Analysis on Symmetric Spaces. Progress in Mathematics, vol. 49. Birkhäuser, Switzerland (1984)

    Book  Google Scholar 

  20. Schröder, M.: Patterson–Sullivan distributions for rank one symmetric spaces of the noncompact type. Dissertation, Paderborn University (2010)

    Google Scholar 

  21. Sharafutdinov, V.A.: Geometric symbol calculus for pseudo differential operators I and II. Siber. Adv. Math. 15(3), 81–125 (2005); 15(4), 71–95 (2005)

  22. Silberman L., Venkatesh A.: On quantum unique ergodicity for locally symmetric spaces. Geom. Funct. Anal. 17, 960–998 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Treves F.: Topological Vector Spaces, Distributions and Kernels. Academic Press, San Diego (1967)

    MATH  Google Scholar 

  24. Taylor M.E.: Pseudodifferential Operators. Princeton Univerity Press, Princeton (1981)

    MATH  Google Scholar 

  25. van den Ban E., Schlichtkrull H.: Asymptotic expansions and boundary values of eigenfunctions on Riemannian symmetric spaces. J. Reine Angew. Math. 380, 108–165 (1987)

    MathSciNet  MATH  Google Scholar 

  26. Wallach, N.R.: Real reductive groups 1. Pure and Applied Mathematics, Academic Press, San Diego (1988)

  27. Widom H.: A complete symbolic calculus for pseudodifferential operators. Bull. Sci. Math. 104(2), 19–63 (1980)

    MathSciNet  MATH  Google Scholar 

  28. Williams, F.L.: Lectures on the spectrum of L 2(Γ\G), Pitman Research Notes in Mathematics Series 242, Essex (1991)

  29. Wolpert S.: The modulus of continuity for Γ 0(m) semi-classical limits. Commun. Math. Phys. 216, 313–323 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zelditch S.: Pseudo-differential analysis on hyperbolic surfaces. J. Funct. Anal. 68, 72–105 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zelditch S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55, 919–941 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zelditch S.: The averaging method and ergodic theory for pseudo-differential operators on compact hyperbolic surfaces. J. Funct. Anal. 82, 38–68 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zelditch, S.: Local and global analysis of eigenfunctions on Riemannian manifolds. In: Handbook of Geometric Analysis, Advanced Lectures in Mathematics (ALM), vol. 7, International Press, Somerville (2008)

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  1. Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098, Paderborn, Germany

    Sönke Hansen, Joachim Hilgert & Michael Schröder

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  1. Sönke Hansen
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  2. Joachim Hilgert
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  3. Michael Schröder
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Correspondence to Joachim Hilgert.

Additional information

J. Hilgert: Part of this research was done at the Hausdorff Research Institute for Mathematics in the context of the trimester program “Interaction of Representation Theory with Geometry and Combinatorics”. M. Schröder: Partially supported by the DFG-IRTG 1133 “Geometry and Analysis of Symmetries”.

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Hansen, S., Hilgert, J. & Schröder, M. Patterson–Sullivan distributions in higher rank. Math. Z. 272, 607–643 (2012). https://doi.org/10.1007/s00209-011-0952-1

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