April  2011, 5(2): 237-254. doi: 10.3934/jmd.2011.5.237

Measures invariant under horospherical subgroups in positive characteristic

1. 

Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, United States

Received  July 2010 Revised  February 2011 Published  July 2011

We prove measure rigidity for the action of maximal horospherical subgroups on homogeneous spaces over a field of positive characteristic. In the case when the lattice is uniform we prove the action of any horospherical subgroup is uniquely ergodic.
Citation: Amir Mohammadi. Measures invariant under horospherical subgroups in positive characteristic. Journal of Modern Dynamics, 2011, 5 (2) : 237-254. doi: 10.3934/jmd.2011.5.237
References:
[1]

H. Behr, Finite presentability of arithmetic groups over global function fields, Groups-St. Andrews 1985, Proc. Edinburgh Math. Soc. (2), 30 (1987), 23-39. doi: 10.1017/S0013091500017934.

[2]

Y. Benoist and H. Oh, Effective equidistribution of $S$-arithmetic points on symmetric varieties, Preprint.

[3]

I. N. Bernstein and A. V. Zelevinski, Representation of the group $GL(n, F)$ where $F$ is a non-archimedean local field, Russ. Math. Surv., 313 (1976), 1-68.

[4]

A. Borel, "Introduction aux Groupes Arithmétiques," Publications de l'Institut de Mathématique de l'Université de Strasbourg, XV. Actualités Scientifiques et Industrielles, No. 1341 Hermann, Paris 1969.

[5]

A. Borel, "Linear Algebraic Groups," Second edition, Graduate Texts in Mathematics, 126, Springer-Verlag, New York, 1991.

[6]

A. Borel and T. A. Springer, Rationality properties of linear algebraic groups. II, Tôhoku Math. J., 20 (1968), 443-497. doi: 10.2748/tmj/1178243073.

[7]

M. Burger, Horocycle flow on geometrically finite surfaces, Duke Math. J., 61 (1990), 779-803. doi: 10.1215/S0012-7094-90-06129-0.

[8]

M. Burger and P. Sarnak, Ramanujan duals. II, Invent. Math., 106 (1991), 1-11. doi: 10.1007/BF01243900.

[9]

L. Clozel, Démonstration de la conjecture $\tau$, (French) [Proof of the $\tau$-conjecture], Invent. Math., 151 (2003), 297-328. doi: 10.1007/s00222-002-0253-8.

[10]

B. Conrad, O. Gabber and G. Prasad, "Pseudo-Reductive Groups," New Mathematical Monographs, 17, Cambridge University Press, Cambridge, 2010.

[11]

S. G. Dani, On invariant measures, minimal sets and a lemma of Margulis, Invent. Math., 51 (1979), 239-260. doi: 10.1007/BF01389917.

[12]

S. G. Dani, Invariant measures and minimal sets of horospherical flows, Invent. Math., 64 (1981), 357-385. doi: 10.1007/BF01389173.

[13]

S. G. Dani, Orbits of horospherical flows, Duke Math. J., 53 (1986), 177-188. doi: 10.1215/S0012-7094-86-05312-3.

[14]

S. G. Dani, G. A. Margulis, Values of quadratic forms at primitive integral points, Invent. Math., 98 (1989), 405-424. doi: 10.1007/BF01388860.

[15]

S. G. Dani and G. A. Margulis, Orbit closures of generic unipotent flows on homogeneous spaces of $SL(3,R)$, Math. Ann., 286 (1990), 101-128. doi: 10.1007/BF01453567.

[16]

S. G. Dani, 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.

[17]

S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, I. M. Gelfand Seminar, 91-137, Adv. Soviet Math., 16, Part 1, Amer. Math. Soc., Providence, RI, 1993.

[18]

M. Einsiedler and A. Ghosh, Rigidity of measures invariant under semisimple groups in positive characteristic, Proc. Lond. Math. Soc. (3), 100 (2010), 249-268. doi: 10.1112/plms/pdp029.

[19]

M. Einsiedler and A. Mohammadi, A joining classification and a special case of Raghunathan's conjecture in positive characteristic, (with an appendix by Kevin Wortman), To appear in J. d'Analyse.

[20]

A. Ghosh, Metric Diophantine approximation over a local field of positive characteristic, J. Number Theory, 124 (2007), 454-469. doi: 10.1016/j.jnt.2006.10.009.

[21]

A. Gorodnik and H. Oh, Rational points on homogeneous varieties and Equidistribution of Adelic periods, (with appendix by Mikhail Borovoi), Preprint.

[22]

G. Harder, Minkowskische Reduktionstheorie über Funktionenkörpern, (German), Invent. Math., 7 (1969), 33-54. doi: 10.1007/BF01418773.

[23]

D. Kleinbock and G. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, Amer. Math. Soc. Transl. Ser. 2, 171 (1996), 144-172.

[24]

D. Kleinbock and G. Tomanov, Flows on S-arithmetic homogeneous spaces and applications to metric Diophantine approximation, Comm. Math. Helv., 82 (2007), 519-581. doi: 10.4171/CMH/102.

[25]

G. A. Margulis, Indefinite quadratic forms and unipotent flows on homogeneous spaces, Proceed of "Semester on Dynamical Systems and Ergodic Theory" (Warsaw 1986), 399-409, Banach Center Publ., 23, PWN, Warsaw, (1989).

[26]

G. A. Margulis, On the action of unipotent groups in the space of lattices, In Gelfand, I.M. (ed.) Proc. of the summer school on group representations. Bolyai Janos Math. Soc., Budapest, 1971, 365-370. Budapest: Akademiai Kiado, (1975).

[27]

G. A. Margulis, Formes quadratiques indefinies et flots unipotents sur les espaces homogènes, [Indefinite quadratic forms and unipotent flows on homogeneous spaces], C.R. Acad. Sci., Paris, Ser. I, 304 (1987), 249-253.

[28]

G. A. Margulis, "On Some Aspects of the Theory of Anosov Systems," With a survey by Richard Sharp: Periodic orbits of hyperbolic flows. Translated from the Russian by Valentina Vladimirovna Szulikowska. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2004.

[29]

G. A. Margulis and G. Tomanov, Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math., 116 (1994), 347-392. doi: 10.1007/BF01231565.

[30]

A. Mohammadi, Unipotent flows and isotropic quadratic forms in positive characteristic, To appear in IMRN.

[31]

H. Oh, Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants, Duke Math. J., 113 (2002), 133-192. doi: 10.1215/S0012-7094-02-11314-3.

[32]

M. Ratner, Horocycle flows: Joining and rigidity of products, Ann. Math. (2), 118 (1983), 277-313. doi: 10.2307/2007030.

[33]

M. Ratner, Strict measure rigidity for unipotent subgroups of solvable groups, Invent. Math., 101 (1990), 449-482. doi: 10.1007/BF01231511.

[34]

M. Ratner, On measure rigidity of unipotent subgroups of semi-simple groups, Acta Math., 165 (1990), 229-309. doi: 10.1007/BF02391906.

[35]

M. Rather, Raghunathan topological conjecture and distributions of unipotent flows, Duke Math. J., 63 (1991), 235-280.

[36]

M. Ratner, On Raghunathan's measure conjecture, Ann. of Math. (2), 134 (1992), 545-607.

[37]

M. Ratner, Raghunathan's conjectures for Cartesian products of real and $p$-adic Lie groups, Duke Math. J., 77 (1995), 275-382. doi: 10.1215/S0012-7094-95-07710-2.

[38]

N. Shah, Uniformly distributed orbits of certain flows on homogeneous spaces, Math. Ann., 289 (1991), 315-334. doi: 10.1007/BF01446574.

[39]

T. A. Springer, Reduction theory over global fields, K. G. Ramanathan memorial issue. Proc. Indian Acad. Sci. Math. Sci., 104 (1994), 207-216. doi: 10.1007/BF02830884.

[40]

J. Tits, Reductive groups over $p$-adic fields, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, 29-69, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979.

[41]

G. Tomanov, Orbits on homogeneous spaces of arithmetic origin and approximations, Analysis on homogeneous spaces and representation theory of Lie groups, Okayama-Kyoto (1997), 265-297, Adv. Stud. Pure Math., 26, Math. Soc. Japan, Tokyo, 2000.

show all references

References:
[1]

H. Behr, Finite presentability of arithmetic groups over global function fields, Groups-St. Andrews 1985, Proc. Edinburgh Math. Soc. (2), 30 (1987), 23-39. doi: 10.1017/S0013091500017934.

[2]

Y. Benoist and H. Oh, Effective equidistribution of $S$-arithmetic points on symmetric varieties, Preprint.

[3]

I. N. Bernstein and A. V. Zelevinski, Representation of the group $GL(n, F)$ where $F$ is a non-archimedean local field, Russ. Math. Surv., 313 (1976), 1-68.

[4]

A. Borel, "Introduction aux Groupes Arithmétiques," Publications de l'Institut de Mathématique de l'Université de Strasbourg, XV. Actualités Scientifiques et Industrielles, No. 1341 Hermann, Paris 1969.

[5]

A. Borel, "Linear Algebraic Groups," Second edition, Graduate Texts in Mathematics, 126, Springer-Verlag, New York, 1991.

[6]

A. Borel and T. A. Springer, Rationality properties of linear algebraic groups. II, Tôhoku Math. J., 20 (1968), 443-497. doi: 10.2748/tmj/1178243073.

[7]

M. Burger, Horocycle flow on geometrically finite surfaces, Duke Math. J., 61 (1990), 779-803. doi: 10.1215/S0012-7094-90-06129-0.

[8]

M. Burger and P. Sarnak, Ramanujan duals. II, Invent. Math., 106 (1991), 1-11. doi: 10.1007/BF01243900.

[9]

L. Clozel, Démonstration de la conjecture $\tau$, (French) [Proof of the $\tau$-conjecture], Invent. Math., 151 (2003), 297-328. doi: 10.1007/s00222-002-0253-8.

[10]

B. Conrad, O. Gabber and G. Prasad, "Pseudo-Reductive Groups," New Mathematical Monographs, 17, Cambridge University Press, Cambridge, 2010.

[11]

S. G. Dani, On invariant measures, minimal sets and a lemma of Margulis, Invent. Math., 51 (1979), 239-260. doi: 10.1007/BF01389917.

[12]

S. G. Dani, Invariant measures and minimal sets of horospherical flows, Invent. Math., 64 (1981), 357-385. doi: 10.1007/BF01389173.

[13]

S. G. Dani, Orbits of horospherical flows, Duke Math. J., 53 (1986), 177-188. doi: 10.1215/S0012-7094-86-05312-3.

[14]

S. G. Dani, G. A. Margulis, Values of quadratic forms at primitive integral points, Invent. Math., 98 (1989), 405-424. doi: 10.1007/BF01388860.

[15]

S. G. Dani and G. A. Margulis, Orbit closures of generic unipotent flows on homogeneous spaces of $SL(3,R)$, Math. Ann., 286 (1990), 101-128. doi: 10.1007/BF01453567.

[16]

S. G. Dani, 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.

[17]

S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, I. M. Gelfand Seminar, 91-137, Adv. Soviet Math., 16, Part 1, Amer. Math. Soc., Providence, RI, 1993.

[18]

M. Einsiedler and A. Ghosh, Rigidity of measures invariant under semisimple groups in positive characteristic, Proc. Lond. Math. Soc. (3), 100 (2010), 249-268. doi: 10.1112/plms/pdp029.

[19]

M. Einsiedler and A. Mohammadi, A joining classification and a special case of Raghunathan's conjecture in positive characteristic, (with an appendix by Kevin Wortman), To appear in J. d'Analyse.

[20]

A. Ghosh, Metric Diophantine approximation over a local field of positive characteristic, J. Number Theory, 124 (2007), 454-469. doi: 10.1016/j.jnt.2006.10.009.

[21]

A. Gorodnik and H. Oh, Rational points on homogeneous varieties and Equidistribution of Adelic periods, (with appendix by Mikhail Borovoi), Preprint.

[22]

G. Harder, Minkowskische Reduktionstheorie über Funktionenkörpern, (German), Invent. Math., 7 (1969), 33-54. doi: 10.1007/BF01418773.

[23]

D. Kleinbock and G. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, Amer. Math. Soc. Transl. Ser. 2, 171 (1996), 144-172.

[24]

D. Kleinbock and G. Tomanov, Flows on S-arithmetic homogeneous spaces and applications to metric Diophantine approximation, Comm. Math. Helv., 82 (2007), 519-581. doi: 10.4171/CMH/102.

[25]

G. A. Margulis, Indefinite quadratic forms and unipotent flows on homogeneous spaces, Proceed of "Semester on Dynamical Systems and Ergodic Theory" (Warsaw 1986), 399-409, Banach Center Publ., 23, PWN, Warsaw, (1989).

[26]

G. A. Margulis, On the action of unipotent groups in the space of lattices, In Gelfand, I.M. (ed.) Proc. of the summer school on group representations. Bolyai Janos Math. Soc., Budapest, 1971, 365-370. Budapest: Akademiai Kiado, (1975).

[27]

G. A. Margulis, Formes quadratiques indefinies et flots unipotents sur les espaces homogènes, [Indefinite quadratic forms and unipotent flows on homogeneous spaces], C.R. Acad. Sci., Paris, Ser. I, 304 (1987), 249-253.

[28]

G. A. Margulis, "On Some Aspects of the Theory of Anosov Systems," With a survey by Richard Sharp: Periodic orbits of hyperbolic flows. Translated from the Russian by Valentina Vladimirovna Szulikowska. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2004.

[29]

G. A. Margulis and G. Tomanov, Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math., 116 (1994), 347-392. doi: 10.1007/BF01231565.

[30]

A. Mohammadi, Unipotent flows and isotropic quadratic forms in positive characteristic, To appear in IMRN.

[31]

H. Oh, Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants, Duke Math. J., 113 (2002), 133-192. doi: 10.1215/S0012-7094-02-11314-3.

[32]

M. Ratner, Horocycle flows: Joining and rigidity of products, Ann. Math. (2), 118 (1983), 277-313. doi: 10.2307/2007030.

[33]

M. Ratner, Strict measure rigidity for unipotent subgroups of solvable groups, Invent. Math., 101 (1990), 449-482. doi: 10.1007/BF01231511.

[34]

M. Ratner, On measure rigidity of unipotent subgroups of semi-simple groups, Acta Math., 165 (1990), 229-309. doi: 10.1007/BF02391906.

[35]

M. Rather, Raghunathan topological conjecture and distributions of unipotent flows, Duke Math. J., 63 (1991), 235-280.

[36]

M. Ratner, On Raghunathan's measure conjecture, Ann. of Math. (2), 134 (1992), 545-607.

[37]

M. Ratner, Raghunathan's conjectures for Cartesian products of real and $p$-adic Lie groups, Duke Math. J., 77 (1995), 275-382. doi: 10.1215/S0012-7094-95-07710-2.

[38]

N. Shah, Uniformly distributed orbits of certain flows on homogeneous spaces, Math. Ann., 289 (1991), 315-334. doi: 10.1007/BF01446574.

[39]

T. A. Springer, Reduction theory over global fields, K. G. Ramanathan memorial issue. Proc. Indian Acad. Sci. Math. Sci., 104 (1994), 207-216. doi: 10.1007/BF02830884.

[40]

J. Tits, Reductive groups over $p$-adic fields, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, 29-69, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979.

[41]

G. Tomanov, Orbits on homogeneous spaces of arithmetic origin and approximations, Analysis on homogeneous spaces and representation theory of Lie groups, Okayama-Kyoto (1997), 265-297, Adv. Stud. Pure Math., 26, Math. Soc. Japan, Tokyo, 2000.

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