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Raghunathan’s conjectures for Cartesian products of real and $$p$$-adic Lie groups. (English) Zbl 0914.22016
Raghunathan conjectured that given a homogeneous space $$G/\Gamma$$ of a Lie group $$G$$ modulo a lattice $$\Gamma$$, the orbit of any unipotent subgroup of $$G$$ has a closure which is itself an orbit of a subgroup. He also noted the connection of his conjecture with a long-standing conjecture due to A. Oppenheim on values of quadratic forms at integral points. Oppenheim’s conjecture was proved by Margulis. He also conjectured a measure theoretic analog, namely that every Borel probability measure on $$G/\Gamma$$ invariant and ergodic under a subgroup $$U$$ generated by unipotent subgroups is algebraic, which means that there is a point $$p\in G/\Gamma$$ and a closed subgroup $$M$$ of $$G$$ such that $$Mp$$ is closed, the measure is concentrated on $$Mp$$ and is $$M$$-invariant. This measure rigidity result was proved by the author for real Lie groups in a series of papers. The paper under review deals with a more general class of groups, namely Cartesian products of real Lie groups with $$p$$-adic Lie groups. The author proves for a certain class of these groups Raghunathan’s conjecture, measure rigidity and a “uniform distribution” result, which says that for any bounded continuous function $$f$$ on $$G/\Gamma$$ and any one-parameter unipotent subgroup $$U$$ of $$G$$ the integral of $$f$$ with respect to any measure as above is the limit $$\tau\to\infty$$ of integrals of $$f$$ over symmetric intervals $$\{t:| t| \leq \tau\}$$ in the field parametrizing $$U$$. The paper is a detailed version of the announcement [M. Ratner, Math. Res. Not. 5, 141-146 (1993; Zbl 0801.22007)]. Very closely related results are contained in the following three papers by G. A. Margulis and G. M. Tomanov [Invent. Math. 116, 347-392 (1994; Zbl 0816.22004)] announced in [C. R. Acad. Sci., Paris, Sér. I 315, 1221-1226 (1992; Zbl 0789.22023)] and [J. Anal. Math. 69, 25-54 (1996; Zbl 0864.22005)].

##### MSC:
 22E40 Discrete subgroups of Lie groups 22E35 Analysis on $$p$$-adic Lie groups 22E30 Analysis on real and complex Lie groups 20G25 Linear algebraic groups over local fields and their integers
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##### References:
 [1] A. Borel, Linear Algebraic Groups , 2nd enl. ed., Graduate Texts in Math., vol. 126, Springer-Verlag, New York, 1991. · Zbl 0726.20030 [2] A. Borel and G. Prasad, Values of isotropic quadratic forms at $$S$$-integral points , Compositio Math. 83 (1992), no. 3, 347-372. · Zbl 0777.11008 [3] S. G. Dani, A simple proof of Borel’s density theorem , Math. Z. 174 (1980), no. 1, 81-94. · Zbl 0432.22008 [4] S. G. Dani, On orbits of unipotent flows on homogeneous spaces. II , Ergodic Theory Dynam. Systems 6 (1986), no. 2, 167-182. · Zbl 0601.22003 [5] S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms , I. M. Gelfand Seminar, Adv. Soviet Math, vol. 16, Amer. Math. Soc., Providence, 1993, pp. 91-137. · Zbl 0814.22003 [6] J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic Pro-$$p$$ Groups , London Math. Soc. Lecture Note Ser., vol. 157, Cambridge Univ. Press, Cambridge, 1991. · Zbl 0744.20002 [7] N. Jacobson, Lie Algebras , Dover, New York, 1979. · Zbl 0121.27504 [8] G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups , Ergeb. Math. Grenzgeb (3), vol. 17, Springer-Verlag, Berlin, 1991. · Zbl 0732.22008 [9] G. A. Margulis, Discrete subgroups and ergodic theory , Number Theory, Trace Formulas and Discrete Groups (Oslo, 1987), Academic Press, Boston, 1989, pp. 377-398. · Zbl 0675.10010 [10] G. A. Margulis and G. M. Tomanov, Invariant measures for actions of unipotent groups over local fields on homogeneous spaces , Invent. Math. 116 (1994), no. 1-3, 347-392. · Zbl 0816.22004 [11] G. Prasad, Ratner’s theorem in $$S$$-arithmetic setting , Workshop on Lie Groups, Ergodic Theory and Geometry, Math. Sci. Res. Inst. Publ., Springer-Verlag, New York, 1992, p. 53. [12] M. S. Raghunathan, Discrete Subgroups of Lie Groups , Ergeb. Math. Grenzgeb., vol. 68, Springer-Verlag, New York, 1972. · Zbl 0254.22005 [13] M. Ratner, Strict measure rigidity for unipotent subgroups of solvable groups , Invent. Math. 101 (1990), no. 2, 449-482. · Zbl 0745.28009 [14] M. Ratner, On measure rigidity of unipotent subgroups of semisimple groups , Acta Math. 165 (1990), no. 3-4, 229-309. · Zbl 0745.28010 [15] M. Ratner, On Raghunathan’s measure conjecture , Ann. of Math. (2) 134 (1991), no. 3, 545-607. JSTOR: · Zbl 0763.28012 [16] M. Ratner, Raghunathan’s topological conjecture and distributions of unipotent flows , Duke Math. J. 63 (1991), no. 1, 235-280. · Zbl 0733.22007 [17] M. Ratner, Invariant measures and orbit closures for unipotent actions on homogeneous spaces , Geom. Funct. Anal. 4 (1994), no. 2, 236-257. · Zbl 0801.22008 [18] M. Ratner, Raghunathan’s conjectures for $$p$$-adic Lie groups , Internat. Math. Res. Notices (1993), no. 5, 141-146. · Zbl 0801.22007 [19] J.-P. Serre, Lie Algebras and Lie Groups , Lectures given at Harvard University, vol. 1964, Benjamin, New York-Amsterdam, 1965. · Zbl 0132.27803 [20] T. Tamagawa, On discrete subgroups of $$p$$-adic algebraic groups , Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), Harper and Row, New York, 1965, pp. 11-17. · Zbl 0158.27801 [21] A. Tempelman, Ergodic Theorems for Group Actions , Math. Appl., vol. 78, Kluwer, Dordrecht, 1992. · Zbl 0753.28014 [22] B. L. Van der Waerden, Modern Algebra , Frederick Ungar Publ. Co., New York, 1953. [23] D. Witte, Zero-entropy affine maps on homogeneous spaces , Amer. J. Math. 109 (1987), no. 5, 927-961. JSTOR: · Zbl 0653.22005
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