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I am trying to read a paper by Bin Xu, "On a lifting problem of $L$-packets", but I am stuck on a detail in the first page.

The setting is: we have a local field $F$ of characteristic zero, and $G$ is an (affine) quasi-split connected reductive group over $F$.

In one of the last lines of the first page of the paper, the author claims the existence of a torus defined over $F$ and containing the center $Z_G$ of $G$ (but not necessarily a subtorus of $G$).

I have troubles understanding why this should be true.

What I see is that it exists a torus containing the identity component of the center, $Z_G^0$. Indeed, $Z_G^0$ is contained in the radical of $G$, and since $G$ is reductive, this says that $Z_G^0$ contains only semisimple elements. Then, affine + abelian + semisimple elements gives diagonalizable (by lemma 2.4.2 of Springer's "Linear Algebraic Groups", for instance). Also, it's a torus, since it is connected.

But I see no reason for which $Z_G$ should contain only semisimple elements. Can somebody give me some advice on this?

Andrea Agostini
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1 Answers1

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I found out. Let $T$ be a maximal torus of $G$ defined over $F$. The existence of such a torus is proved, for instance, in Humphreys' "Linear algebraic groups" in section 34.4. In the same book,in section 26, it is proved that the centralizer of a maximal torus in a reductive group is the torus itself. The center of $G$ is clearly contained in the centralizer of any maximal torus, hence in any maximal torus defined over $F$.

Andrea Agostini
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