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Invariant measures for actions of unipotent groups over local fields on homogeneous spaces. (English) Zbl 0816.22004
The study of dynamics of actions of unipotent subgroups on homogeneous spaces has been attracting considerable attention for the last 30 years. One of the main reasons for this was that some problems in number theory and, in particular, in Diophantine approximations can be reformulated in terms of such actions. M. S. Raghunathan observed that a long-standing conjecture due to A. Oppenheim on values of quadratic forms at integral points can be deduced form some results about actions of unipotent subgroups. More precisely, he conjectured that the closure of any orbit of a unipotent subgroup in the quotient of a Lie group \(G\) by a lattice \(\Gamma \subset G\) is an orbit of a bigger subgroup and noted the connection of his conjecture with Oppenheim’s conjecture.
Oppenheim’s conjecture was proved by Margulis. For a general survey of the area the reader is referred to [G. A. Margulis, Proc. Int. Congr. Math., Kyoto 1990, 193-215 (1991; Zbl 0747.58017)]. Major progress in the area was made by M. Ratner. She proved Raghunathan’s conjecture for a general Lie group \(G\), obtained a classification of all finite invariant measures for actions of unipotent groups \(U\) on \(G/\Gamma\) (measure rigidity) and proved uniform distribution for actions of one- parameter unipotent groups. The purpose of the paper under review is to give a proof of measure rigidity for a product of algebraic groups over local fields of characteristic zero. The authors also give applications concerning closures of orbits of unipotent groups, uniform distribution and values of families of quadratic forms.
It is worth mentioning that there is an announcement of closely related results by M. Ratner [Int. Math. Res. Not. 1993, No. 5, 141-146 (1993; Zbl 0801.22007) in Duke Math. J. 70, No. 2].

MSC:
22E40 Discrete subgroups of Lie groups
28D15 General groups of measure-preserving transformations
11H55 Quadratic forms (reduction theory, extreme forms, etc.)
20G25 Linear algebraic groups over local fields and their integers
22D40 Ergodic theory on groups
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