Highlights in Linear Algebraic Groups 13: Centralisers of Tori

In Highlights 12, we used some of the equivalent conditions for a connected algebraic group $G$ over a field $k=\overline{k}$ to have semisimple rank 1 in the study of reductive groups (these are the groups whose unipotent radical $R(G)_u$ is trivial).

Precisely, we showed that such a $G$ must have a semisimple commutator $[G,G]$ subgroup whose dimension is three, and that we can write

$G = Z(G)^\circ\cdot [G,G]$

where the $-\cdot-$ denotes that this is an almost direct product: in other words, the multiplication map $Z(G)^\circ\times[G,G]\to G$ is surjective with finite kernel.

Let $T\subseteq G$ be a maximal torus. We will show in this post that $C_G(T) = T$ for a connected reductive group $G$ of semisimple rank 1.
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