# American Institute of Mathematical Sciences

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.  Google Scholar [2] Y. Benoist and H. Oh, Effective equidistribution of $S$-arithmetic points on symmetric varieties,, Preprint., ().   Google Scholar [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. Google Scholar [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.  Google Scholar [5] A. Borel, "Linear Algebraic Groups," Second edition, Graduate Texts in Mathematics, 126, Springer-Verlag, New York, 1991.  Google Scholar [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.  Google Scholar [7] M. Burger, Horocycle flow on geometrically finite surfaces, Duke Math. J., 61 (1990), 779-803. doi: 10.1215/S0012-7094-90-06129-0.  Google Scholar [8] M. Burger and P. Sarnak, Ramanujan duals. II, Invent. Math., 106 (1991), 1-11. doi: 10.1007/BF01243900.  Google Scholar [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.  Google Scholar [10] B. Conrad, O. Gabber and G. Prasad, "Pseudo-Reductive Groups," New Mathematical Monographs, 17, Cambridge University Press, Cambridge, 2010.  Google Scholar [11] S. G. Dani, On invariant measures, minimal sets and a lemma of Margulis, Invent. Math., 51 (1979), 239-260. doi: 10.1007/BF01389917.  Google Scholar [12] S. G. Dani, Invariant measures and minimal sets of horospherical flows, Invent. Math., 64 (1981), 357-385. doi: 10.1007/BF01389173.  Google Scholar [13] S. G. Dani, Orbits of horospherical flows, Duke Math. J., 53 (1986), 177-188. doi: 10.1215/S0012-7094-86-05312-3.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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., ().   Google Scholar [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.  Google Scholar [21] A. Gorodnik and H. Oh, Rational points on homogeneous varieties and Equidistribution of Adelic periods, (with appendix by Mikhail Borovoi),, Preprint., ().   Google Scholar [22] G. Harder, Minkowskische Reduktionstheorie über Funktionenkörpern, (German), Invent. Math., 7 (1969), 33-54. doi: 10.1007/BF01418773.  Google Scholar [23] D. Kleinbock and G. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, Amer. Math. Soc. Transl. Ser. 2, 171 (1996), 144-172.  Google Scholar [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.  Google Scholar [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).  Google Scholar [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).  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [30] A. Mohammadi, Unipotent flows and isotropic quadratic forms in positive characteristic,, To appear in IMRN., ().   Google Scholar [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.  Google Scholar [32] M. Ratner, Horocycle flows: Joining and rigidity of products, Ann. Math. (2), 118 (1983), 277-313. doi: 10.2307/2007030.  Google Scholar [33] M. Ratner, Strict measure rigidity for unipotent subgroups of solvable groups, Invent. Math., 101 (1990), 449-482. doi: 10.1007/BF01231511.  Google Scholar [34] M. Ratner, On measure rigidity of unipotent subgroups of semi-simple groups, Acta Math., 165 (1990), 229-309. doi: 10.1007/BF02391906.  Google Scholar [35] M. Rather, Raghunathan topological conjecture and distributions of unipotent flows, Duke Math. J., 63 (1991), 235-280.  Google Scholar [36] M. Ratner, On Raghunathan's measure conjecture, Ann. of Math. (2), 134 (1992), 545-607.  Google Scholar [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.  Google Scholar [38] N. Shah, Uniformly distributed orbits of certain flows on homogeneous spaces, Math. Ann., 289 (1991), 315-334. doi: 10.1007/BF01446574.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [2] Y. Benoist and H. Oh, Effective equidistribution of $S$-arithmetic points on symmetric varieties,, Preprint., ().   Google Scholar [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. Google Scholar [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.  Google Scholar [5] A. Borel, "Linear Algebraic Groups," Second edition, Graduate Texts in Mathematics, 126, Springer-Verlag, New York, 1991.  Google Scholar [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.  Google Scholar [7] M. Burger, Horocycle flow on geometrically finite surfaces, Duke Math. J., 61 (1990), 779-803. doi: 10.1215/S0012-7094-90-06129-0.  Google Scholar [8] M. Burger and P. Sarnak, Ramanujan duals. II, Invent. Math., 106 (1991), 1-11. doi: 10.1007/BF01243900.  Google Scholar [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.  Google Scholar [10] B. Conrad, O. Gabber and G. Prasad, "Pseudo-Reductive Groups," New Mathematical Monographs, 17, Cambridge University Press, Cambridge, 2010.  Google Scholar [11] S. G. Dani, On invariant measures, minimal sets and a lemma of Margulis, Invent. Math., 51 (1979), 239-260. doi: 10.1007/BF01389917.  Google Scholar [12] S. G. Dani, Invariant measures and minimal sets of horospherical flows, Invent. Math., 64 (1981), 357-385. doi: 10.1007/BF01389173.  Google Scholar [13] S. G. Dani, Orbits of horospherical flows, Duke Math. J., 53 (1986), 177-188. doi: 10.1215/S0012-7094-86-05312-3.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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., ().   Google Scholar [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.  Google Scholar [21] A. Gorodnik and H. Oh, Rational points on homogeneous varieties and Equidistribution of Adelic periods, (with appendix by Mikhail Borovoi),, Preprint., ().   Google Scholar [22] G. Harder, Minkowskische Reduktionstheorie über Funktionenkörpern, (German), Invent. Math., 7 (1969), 33-54. doi: 10.1007/BF01418773.  Google Scholar [23] D. Kleinbock and G. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, Amer. Math. Soc. Transl. Ser. 2, 171 (1996), 144-172.  Google Scholar [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.  Google Scholar [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).  Google Scholar [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).  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [30] A. Mohammadi, Unipotent flows and isotropic quadratic forms in positive characteristic,, To appear in IMRN., ().   Google Scholar [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.  Google Scholar [32] M. Ratner, Horocycle flows: Joining and rigidity of products, Ann. Math. (2), 118 (1983), 277-313. doi: 10.2307/2007030.  Google Scholar [33] M. Ratner, Strict measure rigidity for unipotent subgroups of solvable groups, Invent. Math., 101 (1990), 449-482. doi: 10.1007/BF01231511.  Google Scholar [34] M. Ratner, On measure rigidity of unipotent subgroups of semi-simple groups, Acta Math., 165 (1990), 229-309. doi: 10.1007/BF02391906.  Google Scholar [35] M. Rather, Raghunathan topological conjecture and distributions of unipotent flows, Duke Math. J., 63 (1991), 235-280.  Google Scholar [36] M. Ratner, On Raghunathan's measure conjecture, Ann. of Math. (2), 134 (1992), 545-607.  Google Scholar [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.  Google Scholar [38] N. Shah, Uniformly distributed orbits of certain flows on homogeneous spaces, Math. Ann., 289 (1991), 315-334. doi: 10.1007/BF01446574.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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