Parabolic subgroups of SL n and Sp2l over a Dedekind ring of arithmetic type

Abstract

Let R be a commutative ring all of whose proper factor rings are finite and in which a unit of infinite order exists. It is shown that for a subgroup P in G =SL(n, R), n ≥3,or in G =Sp(2l, R), l ≥2, containing the standard Borel subgroup B, the following alternative holds: either P contains a relative elementary subgroup EI for some ideal I≠0 or H is contained in a proper standard parabolic subgroup. For Dedekind rings of arithmetic type, this allows one, under some mild additional assumptions on units, to completely describe the overgroups of B in G. Bibliography: 30 titles.