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On a maximality problem for arithmetic subgroups of indefinite orthogonal groups of type \((D_{\ell})\). (Russian) Zbl 0717.20034

The present paper is a natural continuation of the author’s paper [Mat. Sb., Nov. Ser. 127, No.1, 72-91 (1985; Zbl 0574.20034)]. The main result of the paper is an explicit classification of the maximal arithmetic subgroups for K-point groups of indefinite orthogonal groups of type (\({\mathcal D}_{\ell})\), where K is a field of algebraic numbers. The sets of lattices of the corresponding quadratic K-spaces, the stabilizers of which are all the maximal arithmetic subgroups of K-point groups of indefinite orthogonal groups of type (\({\mathcal D}_{\ell})\) up to local G- conjugacy are described. As a preliminary result a local analogue of the maximality problem is considered and necessary and sufficient conditions are determined under which, for groups of the type indicated, the local maximality implies the global one.
A version of the description of the maximal arithmetic subgroups in simple K-split groups in terms of parahoric subgroups (the Bruhat-Tits theory) is obtained by J. Rohlfs [Math. Ann. 244, No.3, 219-231 (1979; Zbl 0426.20030)].
Reviewer’s remark. A description of the maximal arithmetic subgroups in terms of parahoric subgroups is in principle possible for arbitrary simple indefinite K-groups. However some observations indicate that the connection between the local and global cases in the nonsplit case is much deeper and subtle than in the split one. In particular this observation is confirmed by the description of the maximal arithmetic subgroups in indefinite orthogonal groups. A uniform general method of describing the maximal arithmetic subgroups of simple indefinite K-groups has been obtained by the reviewer.
Reviewer: A.A.Bondarenko

MSC:

20G30 Linear algebraic groups over global fields and their integers
20E28 Maximal subgroups
11F06 Structure of modular groups and generalizations; arithmetic groups
20H05 Unimodular groups, congruence subgroups (group-theoretic aspects)
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