Parabolic subgroups of $\mathrm{SO}_{2l}$ over a~Dedekind ring of arithmetic type

K. O. Batalkin ; N. A. Vavilov

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 23, Tome 400 (2012), pp. 50-69

Résumé

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 $G=\mathrm{SO}(2l,R)$, $l\ge3$, containing Borel subgroup $B$, the following alternative holds. Either $P$ contains a relative elementary subgroup $E_I$ for some ideal $I\neq0$, or $H$ is contained in a proper standard parabolic subgroup. For Dedekind rings of arithmetic type this allows, under some mild additional assumptions on units, to completely describe overgroups of $B$ in $G$. Earlier, similar results for the special linear and symplectic 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.