Parabolic subgroups of $\mathrm{SL}_n$ and $\mathrm{Sp}_{2l}$ over a Dedekind ring of arithmetic type

A. V. Alexandrov   ; N. A. Vavilov

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 19, Tome 375 (2010), pp. 5-21

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 $G=\mathrm{SL}(n,R)$, $n\ge3$, or in $G=\mathrm{Sp}(2l,R)$, $l\ge2$, 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 $G$. Bibl. – 30 titles.

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