Pairs of microweight tori in $GL_n$

Abstract

In the present note we prove a reduction theorem for subgroups of the general linear group ${\operatorname{GL}}(n,T)$ over a skew-field $T$,
generated by a pair of microweight tori of the same type. It turns out, that any pair of tori of residue $m$ is conjugate to such
a pair in ${\operatorname{GL}}(3m,T)$, and the pairs that cannot be further reduced to ${\operatorname{GL}}(3m-1,T)$ form a single ${\operatorname{GL}}(3m,T)$-orbit.
For the case $m=1$ this leaves us with the analysis of ${\operatorname{GL}}(2,T)$, that was carried through some two decades ago by the second author,
Cohen, Cuypers and Sterk. For the next case $m=2$ this means that the only cases to be considered are ${\operatorname{GL}}(4,T)$ and ${\operatorname{GL}}(5,T)$.
In these cases the problem can be fully resolved by (direct but rather lengthy) matrix calculations, which are relegated to a
forthcoming paper by the authors.