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On effective equidistribution for quotients of SL(d,ℝ)

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We prove the first case of polynomially effective equidistribution of closed orbits of semisimple groups with nontrivial centralizer. The proof relies on uniform pectral gap, builds on, and extends work of Einsiedler, Margulis, and Venkatesh.

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  1. [1]

    A. Borel, Linear Algebraic Groups, Graduate Texts in Mathematics, Vol. 126, Springer, New York, 1991.

  2. [2]

    A. Borel and J. Tits, Groupes réductifs, Institut des Hautes Etudes Scientifiques. Publications Mathématiques 27 (1965), 55–150.

  3. [3]

    M. Burger and P. Sarnak, Ramanujan duals. II, Inventiones mathematicae 106 (1991), 1–11.

  4. [4]

    L. Clozel, Démonstration de la conjecture t, Inventiones mathematicae 151 (2003), 297–328.

  5. [5]

    S. G. Dani and G. A. Margulis, Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces, Indian Academy of Sciences. Proceedings. Mathematical Sciences 101 (1991), 1–17.

  6. [6]

    M. Einsiedler, G. Margulis and A. Venkatesh, Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces, Inventiones Mathematicae 177 (2009), 137–212.

  7. [7]

    M. Einsiedler, G. Margulis, A. Mohammadi and A. Venkatesh, Effective equidistribution and property (t), Journal of the American Mathematical Society, to appear,

  8. [8]

    A. Gorodnik, F. Maucourant and H. Oh, Manin’s and Peyre’s conjectures on rational points and adelic mixing, Annales Scientifiques de l’École Normale Supérieure 41 (2008), 383–435.

  9. [9]

    H. Jacquet and R. P. Langlands, Automorphic Forms on GL (2), Lecture Notes in Mathematics, Vol. 114, Springer, Berlin-New York, 1970.

  10. [10]

    D. A. Kazhdan, Connection of the dual space of a group with the structure of its close subgroups, Functional Analysis and its Applications 1 (1967), 63–65.

  11. [11]

    D. Y. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, in Sinaį’s Moscow Seminar on Dynamical Systems, American Mathematical Society Translations, Series 2, Vol. 171, American Mathematical Society, Providence, RI, 1996, pp. 141–172.

  12. [12]

    D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and diophantine approximation on manifolds, Annals of mathematics 148 (1998), 339–360.

  13. [13]

    D. Y. Kleinbock and G. A. Margulis, On effective equidistribution of expanding translates of certain orbits in the space of lattices, in Number Theory, Analysis and Geometry, Springer, New York, 2012, pp. 385–396.

  14. [14]

    G. Margulis, Discrete subgroups and ergodic theory, in Number Theory, Trace Formulas and Discrete Groups (Oslo, 1987), Academic Press, Boston, MA, 1989, pp. 377–398.

  15. [15]

    H. Oh, Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants, Duke Mathematical Journal 113 (2002), 133–192.

  16. [16]

    M. Ratner, Invariant measures for unipotent translations on homogeneous spaces, Proceedings of the National Academy of Sciences of the United States of America 87 (1990), 4309–4311.

  17. [17]

    M. Ratner, On measure rigidity of unipotent subgroups of semisimple groups, ActaMathematica 165 (1990), 229–309.

  18. [18]

    M. Ratner, Strict measure rigidity for unipotent subgroups of solvable groups, Inventiones Mathematicae 101 (1990), 449–482.

  19. [19]

    M. Ratner, Distribution rigidity for unipotent actions on homogeneous spaces, Bulletin of the American Mathematical Society 24 (1991), 321–325.

  20. [20]

    M. Ratner, On Raghunathan’s measure conjecture, Annals of Mathematics 134 (1991), 545–607.

  21. [21]

    A. Selberg, On the estimation of fourier coefficients of modular forms, in Theory of Numbers, Proceedings of Symposia in Pure Mathematics, Vol. 8, American Mathematical Society, Providence, RI, 1965, pp. 1–15.

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Author information

Correspondence to Manfred Einsiedler.

Additional information

M. A. acknowledges the support of ISEF, and SNF Grant 200021-152819.

M. E. acknowledges the support of the SNF Grant 200021-152819 and 200020-178958.

H. L. acknowledges support by Simons Foundation (426090) and NSF (DMS 1700109).

A. M. acknowledges support by the NSF (DMS 1724316, 1764246, 1128155) and Alfred P. Sloan Research Fellowship.

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Aka, M., Einsiedler, M., Li, H. et al. On effective equidistribution for quotients of SL(d,ℝ). Isr. J. Math. (2020).

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