Subgroups of the split orthogonal group. II

N. A. Vavilov

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 6, Tome 265 (1999), pp. 42-63

Résumé

In the first paper of the series, we proved standardness of subgroups containing a split maximal torus in the split orthogonal group $SO(n,R)$ over a commutative semilocal ring $R$ for the two following situations: 1) $n$ is even, 2) $n$ is odd and $R=K$ is a field. In the present paper we prove standardness of intermediate subgroups over a semilocal ring $R$ in the case of an odd $n$. Together with the preceeding papers by Z. I. Borewicz, the author, and E. V. Dybkova this paper completes description of the overgroups of split maximal tori in the classical groups over semilocal rings. The analysis of odd orthogonal groups turned out to be technically much more difficult than other classical cases. Unlike all preceeding papers the proof of the key step in the reduction to the field case relies on calculations with a class of semisimple elements which are neither microweight elements, nor semisimple root elements.