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.

This post shall conclude the detailed explanation of the concise proof of Corollary 25.3 in Humphrey’s book, which I encourage the reader to peruse for more information.

Centralisers of Tori in $ \mathrm{PGL}_2(k)$

The first step to show that $ C_G(T) = T$ is to show that this is true for $ G = \mathrm{PGL}_2(k)$. We have shown that $ \mathrm{PGL}_2(k)$ is the image of a connected group in the previous post so it is connected. We have the quotient morphism $ \mathrm{SL}_2(k)\to \mathrm{PGL}_2(k)$ of connected groups, and so each maximal torus of $ \mathrm{PGL}_2(k)$ is the image of some maximal torus in $ \mathrm{SL}_2(k)$. A couple lines of calculations will convince the reader that to show $ C_G(T) = T$ for $ G = \mathrm{PGL}_2(k)$, it is sufficient to show that $ C_G(T) = T$ for $ G = \mathrm{SL}_2(k)$ , which is also a short calculation.

Centralisers of Tori in Reductive Groups of SS Rank 1

We can now proceed with the general $ G$.

Let $ G$ be a connected reductive group of semisimple rank $ 1$ (i.e. the group $ G/R(G)$ has rank 1). If $ T\subseteq G$ is a maximal torus then $ C_G(T) = T$.
Proof. Fix a morphism $ \varphi:G\to\mathrm{PGL}_2(k)$ with $ \ker\varphi = R(G) = Z(G)^\circ$. From the previous section, we see that the torus $ T$ and its centraliser $ C_G(T)$ have the same image under $ \varphi$.

Hence we have the inclusions $ T\subseteq C_G(T) \subseteq T\ker\varphi = TR(G)$, since for any element $ x\in C_G(T)$ there is a $ t\in T$ such that $ \varphi(y) = \varphi(t)$. However, $ R(G)$ is a normal torus and hence contained in $ T$ so $ TR(G) = T$.

Notice that since $ Z(G)$ centralises any torus, $ Z(G)$ is contained in every torus.

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