Parabolic Subgroups of SO2l Over a Dedekind Ring of Arithmetic Type

Abstract

Let R be a commutative ring all of whose proper factor rings are finite and such that there exists a unit of infinite order. We show that for a subgroup P in G = SO(2l, R), l ≥ 3, containing the Borel subgroup B, the following alternative holds: either P contains a relative elementary subgroup E I for some ideal I = 0 or H is contained in a proper standard parabolic subgroup. For Dedekind rings of arithmetic type this allows us, under some mild additional assumptions on units, to describe completely the overgroups of B in G. Earlier, similar results for the special linear and symplectiv groups were obtained by A. V. Alexandrov and the second author. The proofs in the present paper follow the same general strategy, but are noticeably harder, from a technical viewpoint. Bibliography: 34 titles.