Subgroups of a spinor group containing a maximal split torus. I (Q1179908)

Let \(K\) be a field of characteristic other than 2 and of cardinality \(>5\). Let \(G_ n=\text{Spin}(n,K)\), \(n>6\), be the spinor group and \(T\) a maximal split torus of \(G_ n\). The main goal of this paper is to describe groups \(H\) such that \(T\subset H\subset G\). Let \(\Phi\) be the set of roots of \(G\) as a Chevalley group. A subset \(\Psi\) of \(\Phi\) is called closed if for \(\alpha,\beta\in\Psi\) we have \(\alpha+\beta\in\Psi\) whenever \(\alpha+\beta\in\Phi\). Let \(G(\Psi)\) denote the subgroup of \(G\) generated by \(T\) and by all the root subgroups \(X_ \alpha\) (\(\alpha\in\Psi\)). Theorem. Let \(H\) be as above with \(n\) even. Then there exists a closed set of roots \(\Psi\) such that \(G(\Psi)\subseteq H\subseteq N\) where \(N=N_ G(\Psi)\) is the normalizer of \(G(\Psi)\) in \(G\). The first part of the paper contains a reduction of the proof to the case where \(H\) contains no root subgroups. The second part seems to be still unpublished.