On subgroups of symplectic group containing a~subsystem subgroup

N. A. Vavilov

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 16, Tome 349 (2007), pp. 5-29

Résumé

Let $\Gamma=\operatorname{GSp}(2l,R)$ be the general symplectic group of rank $l$ over a commutative ring $R$ such, that $2\in R^*$, and $\nu$ be a symmetric equivalence relation on the index set $\{1,\ldots,l,-l,\ldots,1\}$, all of whose classes contain at least 3 elements. In the present paper we prove that if a subgroup $H$ of $\Gamma$ contains the group $E_{\Gamma}(\nu)$ of elementary block diagonal matrices of type $\nu$, then $H$ normalises the subgroup generated by all elementary symplectic transvections $T_{ij}(\xi)\in H$. Combined with the previous results, this completely describes overgroups of subsystem subgroups in this case. Similar results for subgroups of $\operatorname{GL}(n,R)$ were established by Z. I. Borewicz and the author in early 1980-ies, while for $\operatorname{GSp}(2l,R)$ and $\operatorname{GO}(n,R)$ they have been announced by the author in late 1980-ies, but the complete proof for the symplectic case has not been published before.