doi: 10.3934/dcds.2021183
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Equidistribution of translates of a homogeneous measure on the Borel–Serre compactification

Beijing International Center for Mathematical Research, Peking University, No. 5 Yiheyuan Road Haidian District, Beijing, 100871, China

Received  June 2021 Revised  September 2021 Early access November 2021

Let $ \mathit{\boldsymbol{\mathrm{G}}} $ be a semisimple linear algebraic group defined over rational numbers, $ \mathrm{K} $ be a maximal compact subgroup of its real points and $ \Gamma $ be an arithmetic lattice. One can associate a probability measure $ \mu_{ \mathrm{H}} $ on $ \Gamma \backslash \mathrm{G} $ for each subgroup $ \mathit{\boldsymbol{\mathrm{H}}} $ of $ \mathit{\boldsymbol{\mathrm{G}}} $ defined over $ \mathbb{Q} $ with no non-trivial rational characters. As G acts on $ \Gamma \backslash \mathrm{G} $ from the right, we can push forward this measure by elements from $ \mathrm{G} $. By pushing down these measures to $ \Gamma \backslash \mathrm{G}/ \mathrm{K} $, we call them homogeneous. It is a natural question to ask what are the possible weak-$ * $ limits of homogeneous measures. In the non-divergent case this has been answered by Eskin–Mozes–Shah. In the divergent case Daw–Gorodnik–Ullmo prove a refined version in some non-trivial compactifications of $ \Gamma \backslash \mathrm{G}/ \mathrm{K} $ for $ \mathit{\boldsymbol{\mathrm{H}}} $ generated by real unipotents. In the present article we build on their work and generalize the theorem to the case of general $ \mathit{\boldsymbol{\mathrm{H}}} $ with no non-trivial rational characters. Our results rely on (1) a non-divergent criterion on $ {\text{SL}}_n $ proved by geometry of numbers and a theorem of Kleinbock–Margulis; (2) relations between partial Borel–Serre compactifications associated with different groups proved by geometric invariant theory and reduction theory. 193 words.

Citation: Runlin Zhang. Equidistribution of translates of a homogeneous measure on the Borel–Serre compactification. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021183
References:
[1]

A. Borel, Introduction to Arithmetic Groups, University Lectures Series, vol. 73, American Mathematical Soc., 2019. doi: 10.1090/ulect/073.  Google Scholar

[2]

A. Borel and L. Ji, Compactifications of Symmetric and Locally Symmetric Spaces, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 2006.  Google Scholar

[3]

A. Borel and J.-P. Serre, Corners and arithmetic groups, Comment. Math. Helv., 48 (1973), 436-491.  doi: 10.1007/BF02566134.  Google Scholar

[4]

S. G. Dani and G. A. Margulis, Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces, Proc. Indian Acad. Sci. Math. Sci., 101 (1991), 1-17.  doi: 10.1007/BF02872005.  Google Scholar

[5]

S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, I. M. Gel$\overset{'}{\mathop{f}}$and Seminar, Adv. Soviet Math., Amer. Math. Soc., Providence, RI, 16 (1993), 91-137.   Google Scholar

[6]

C. DawA. Gorodnik and E. Ullmo, Convergence of measures on compactifications of locally symmetric spaces, Math. Z., 297 (2021), 1293-1328.  doi: 10.1007/s00209-020-02558-w.  Google Scholar

[7]

C. Daw, A. Gorodnik, E. Ullmo and J. Li, The Space of Homogeneous Probability Measures on $\overline{\Gamma\backslash X}^S_{ \rm max }$ is Compact, e-prints, 2019, arXiv: 1910.04568. Google Scholar

[8]

W. DukeZ. Rudnick and P. Sarnak, Density of integer points on affine homogeneous varieties, Duke Math. J., 71 (1993), 143-179.  doi: 10.1215/S0012-7094-93-07107-4.  Google Scholar

[9]

A. Eskin and C. McMullen, Mixing, counting, and equidistribution in Lie groups, Duke Math. J., 71 (1993), 181-209.  doi: 10.1215/S0012-7094-93-07108-6.  Google Scholar

[10]

A. EskinG. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math., 147 (1998), 93-141.  doi: 10.2307/120984.  Google Scholar

[11]

A. EskinS. Mozes and N. Shah, Unipotent flows and counting lattice points on homogeneous varieties, Ann. of Math., 143 (1996), 253-299.  doi: 10.2307/2118644.  Google Scholar

[12]

A. EskinS. Mozes and N. Shah, Non-divergence of translates of certain algebraic measures, Geom. Funct. Anal., 7 (1997), 48-80.  doi: 10.1007/PL00001616.  Google Scholar

[13]

D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math., 148 (1998), 339-360.  doi: 10.2307/120997.  Google Scholar

[14]

B. Klingler and A. Yafaev, The André-Oort conjecture, Ann. of Math., 180 (2014), 867-925.  doi: 10.4007/annals.2014.180.3.2.  Google Scholar

[15]

G. D. Mostow, Self-adjoint groups, Ann. of Math., 62 (1955), 44-55.  doi: 10.2307/2007099.  Google Scholar

[16]

S. Mozes and N. Shah, On the space of ergodic invariant measures of unipotent flows, Ergodic Theory Dynam. Systems, 15 (1995), 149-159.  doi: 10.1017/S0143385700008282.  Google Scholar

[17]

D. Mumford, J. Fogarty and F. Kirwan, Geometric Invariant Theory, 3$^{rd}$ edition, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994.  Google Scholar

[18]

M. Ratner, On Raghunathan's measure conjecture, Ann. of Math., 134 (1991), 545-607.  doi: 10.2307/2944357.  Google Scholar

[19]

T. A. Springer, Linear Algebraic Groups, 2$^{nd}$ edition, Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. doi: 10.1007/978-0-8176-4840-4.  Google Scholar

[20]

E. Ullmo and A. Yafaev, Galois orbits and equidistribution of special subvarieties: Towards the André-Oort conjecture, Ann. of Math., 180 (2014), 823-865.  doi: 10.4007/annals.2014.180.3.1.  Google Scholar

[21]

R. Zhang, Translates of homogeneous measures associated with Observable Subgroups on some homogeneous spaces, arXiv e-prints, 2020, arXiv: 1909.02666. Google Scholar

show all references

References:
[1]

A. Borel, Introduction to Arithmetic Groups, University Lectures Series, vol. 73, American Mathematical Soc., 2019. doi: 10.1090/ulect/073.  Google Scholar

[2]

A. Borel and L. Ji, Compactifications of Symmetric and Locally Symmetric Spaces, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 2006.  Google Scholar

[3]

A. Borel and J.-P. Serre, Corners and arithmetic groups, Comment. Math. Helv., 48 (1973), 436-491.  doi: 10.1007/BF02566134.  Google Scholar

[4]

S. G. Dani and G. A. Margulis, Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces, Proc. Indian Acad. Sci. Math. Sci., 101 (1991), 1-17.  doi: 10.1007/BF02872005.  Google Scholar

[5]

S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, I. M. Gel$\overset{'}{\mathop{f}}$and Seminar, Adv. Soviet Math., Amer. Math. Soc., Providence, RI, 16 (1993), 91-137.   Google Scholar

[6]

C. DawA. Gorodnik and E. Ullmo, Convergence of measures on compactifications of locally symmetric spaces, Math. Z., 297 (2021), 1293-1328.  doi: 10.1007/s00209-020-02558-w.  Google Scholar

[7]

C. Daw, A. Gorodnik, E. Ullmo and J. Li, The Space of Homogeneous Probability Measures on $\overline{\Gamma\backslash X}^S_{ \rm max }$ is Compact, e-prints, 2019, arXiv: 1910.04568. Google Scholar

[8]

W. DukeZ. Rudnick and P. Sarnak, Density of integer points on affine homogeneous varieties, Duke Math. J., 71 (1993), 143-179.  doi: 10.1215/S0012-7094-93-07107-4.  Google Scholar

[9]

A. Eskin and C. McMullen, Mixing, counting, and equidistribution in Lie groups, Duke Math. J., 71 (1993), 181-209.  doi: 10.1215/S0012-7094-93-07108-6.  Google Scholar

[10]

A. EskinG. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math., 147 (1998), 93-141.  doi: 10.2307/120984.  Google Scholar

[11]

A. EskinS. Mozes and N. Shah, Unipotent flows and counting lattice points on homogeneous varieties, Ann. of Math., 143 (1996), 253-299.  doi: 10.2307/2118644.  Google Scholar

[12]

A. EskinS. Mozes and N. Shah, Non-divergence of translates of certain algebraic measures, Geom. Funct. Anal., 7 (1997), 48-80.  doi: 10.1007/PL00001616.  Google Scholar

[13]

D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math., 148 (1998), 339-360.  doi: 10.2307/120997.  Google Scholar

[14]

B. Klingler and A. Yafaev, The André-Oort conjecture, Ann. of Math., 180 (2014), 867-925.  doi: 10.4007/annals.2014.180.3.2.  Google Scholar

[15]

G. D. Mostow, Self-adjoint groups, Ann. of Math., 62 (1955), 44-55.  doi: 10.2307/2007099.  Google Scholar

[16]

S. Mozes and N. Shah, On the space of ergodic invariant measures of unipotent flows, Ergodic Theory Dynam. Systems, 15 (1995), 149-159.  doi: 10.1017/S0143385700008282.  Google Scholar

[17]

D. Mumford, J. Fogarty and F. Kirwan, Geometric Invariant Theory, 3$^{rd}$ edition, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994.  Google Scholar

[18]

M. Ratner, On Raghunathan's measure conjecture, Ann. of Math., 134 (1991), 545-607.  doi: 10.2307/2944357.  Google Scholar

[19]

T. A. Springer, Linear Algebraic Groups, 2$^{nd}$ edition, Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. doi: 10.1007/978-0-8176-4840-4.  Google Scholar

[20]

E. Ullmo and A. Yafaev, Galois orbits and equidistribution of special subvarieties: Towards the André-Oort conjecture, Ann. of Math., 180 (2014), 823-865.  doi: 10.4007/annals.2014.180.3.1.  Google Scholar

[21]

R. Zhang, Translates of homogeneous measures associated with Observable Subgroups on some homogeneous spaces, arXiv e-prints, 2020, arXiv: 1909.02666. Google Scholar

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