Measures invariant under horospherical subgroups in positive characteristic

Amir Mohammadi

doi:10.3934/jmd.2011.5.237

Article Contents

2011, Volume 5, Issue 2: 237-254. Doi: 10.3934/jmd.2011.5.237

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Measures invariant under horospherical subgroups in positive characteristic

Amir Mohammadi^1,

1.
Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL 60637

Received: June 30, 2010

Revised: January 31, 2011

Published: June 2011

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Abstract

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.

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Mathematics Subject Classification: Primary: 37D40, 37A17; Secondary: 22E40.

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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.
[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.