Subgroups of the spinor group containing a split maximal torus.~III

E. A. Filippova

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 9, Tome 289 (2002), pp. 287-299

Résumé

We describe subgroups of the spinor group $\operatorname{Spin}\,(2l+1, K)$ over a field $K$ such that $2\in K^*, |K|\ge9$ and $l\ge3$, which contain a split maximal torus. We prove that the description of these subgroups is standard in two cases: 1) $l$ is even; 2) $l$ is odd and $-1\in K^{*2}$. We show that as in the papers by N. A. Vavilov and V. Holubovsky, devoted to subgroups of the orthogonal group, one can reduce the odd case to the case of even $n=2l$. However, here the calculations are somewhat more involved since we can only use diagonal elements of $\operatorname{Spin}\,(2l+1,K)$. Furthermore, we strengthen the results of N. A. Vavilov pertaining to the even case by relaxing the condition on the field $K$ to $|K|\ge9$.