Overgroups of elementary block-diagonal subgroups in hyperbolic unitary groups over quasi-finite rings: main results

A. V. Shchegolev

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 29, Tome 443 (2016), pp. 222-233

Résumé

Let $H$ be a subgroup of the hyperbolic unitary group $\operatorname U(2n,R,\Lambda)$ that contains the elementary block-diagonal subgroup $\operatorname{EU}(\nu,R,\Lambda)$ of type $\nu$. Assume that all self-conjugate blocks of $\nu$ are of size at least 6 (at least 4 if the form parameter $\Lambda$ satisfies the condition $R\Lambda+\Lambda R=R$) and that all non-self-conjugate blocks are of size at least 5. Then there exists a unique major exact form net of ideals $(\sigma,\Gamma)$ such that $\operatorname{EU}(\sigma,\Gamma)\le H\le\operatorname N_{\operatorname U(2n,R,\Lambda)}(\operatorname U(\sigma,\Gamma))$, where $\operatorname N_{\operatorname U(2n,R,\Lambda)}(\operatorname U(\sigma,\Gamma))$ stands for the normalizer in $\operatorname U(2n,R,\Lambda)$ of the form net subgroup $\operatorname U(\sigma,\Gamma)$ of level $(\sigma,\Gamma)$ and $\operatorname{EU}(\sigma,\Gamma)$ denotes the corresponding elementary form net subgroup. The normalizer $\operatorname N_{\operatorname U(2n,R,\Lambda)}(\operatorname U(\sigma,\Gamma))$ is described in terms of congruences.