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 Zap. Nauchn. Sem. POMI, 2012, Volume 400, Pages 50–69 (Mi znsl5611)

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

K. O. Batalkin, N. A. Vavilov

Saint-Petersburg State University, Saint-Petersburg, Russia

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 $G=\mathrm{SO}(2l,R)$, $l\ge3$, 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.

Key words and phrases: split orthogonal group, orthogonal transvections, parabolic subgroups, relative elementary subgroup, Dedekind ring of arithmetic type.

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English version:
Journal of Mathematical Sciences (New York), 2013, 192:2, 154–163

Bibliographic databases:

Document Type: Article
UDC: 513.6

Citation: K. O. Batalkin, N. A. Vavilov, “Parabolic subgroups of $\mathrm{SO}_{2l}$ over a Dedekind ring of arithmetic type”, Problems in the theory of representations of algebras and groups. Part 23, Zap. Nauchn. Sem. POMI, 400, POMI, St. Petersburg, 2012, 50–69; J. Math. Sci. (N. Y.), 192:2 (2013), 154–163

Citation in format AMSBIB
\Bibitem{BatVav12}
\by K.~O.~Batalkin, N.~A.~Vavilov
\paper Parabolic subgroups of $\mathrm{SO}_{2l}$ over a~Dedekind ring of arithmetic type
\inbook Problems in the theory of representations of algebras and groups. Part~23
\serial Zap. Nauchn. Sem. POMI
\yr 2012
\vol 400
\pages 50--69
\publ POMI
\mathnet{http://mi.mathnet.ru/znsl5611}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3029565}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2013
\vol 192
\issue 2
\pages 154--163
\crossref{https://doi.org/10.1007/s10958-013-1381-y}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84884981119}