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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)].

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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