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Orbits on homogeneous spaces of arithmetic origin and approximations. (English) Zbl 0960.22006
Kobayashi, Toshiyuki (ed.) et al., Analysis on homogeneous spaces and representation theory of Lie groups. Based on activities of the RIMS Project Research ’97, Okayama-Kyoto, Japan, during July and August 1997. Tokyo: Kinokuniya Company Ltd. Adv. Stud. Pure Math. 26, 265-297 (2000).
Let \(G\) be the group \({\mathbb G}(\oplus_{v\in S}K_v)\), where \(K\) is a number field, \({\mathbb G}\) is a \(K\) algebraic group and \(S\) is a finite set of valuations on \(K\) containing the Archimedean valuations. Let \(\Gamma\) be an \(S\)-arithmetic subgroup of \(G\). The main new result in the paper is about certain necessary conditions for the orbit of \(x\in G/\Gamma\) under a closed subgroup \(M\) of \(G\) to be a closed orbit with a finite \(M\)-invariant measure. Such closed orbits arise in a natural way in Ratner’s description of orbit-closures of subgroups generated by unipotent one-parameter subgroups, and the present result complements Ratner’s theorem in the case of \(S\)-arithmetic subgroups. The author applies the result to obtain a generalisation of the work on values of irrational quadratic forms at integral points, due to Margulis and also Borel and Prasad, to the general setting of Hermitian forms over division algebras with involutions of first or second kind. The result on closed orbits is also applied to give another proof of the strong approximation theorem for algebraic groups defined over number fields.
For the entire collection see [Zbl 0941.00016].
Reviewer: S.G.Dani (Mumbai)

22E40 Discrete subgroups of Lie groups
11J99 Diophantine approximation, transcendental number theory