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I have the following probably stupid question on the definition of $L-$groups.

We follows the book Introduction to automorphic Representations by Getz and Hahn. Let $F$ be a field and $F^s$ its separable closure. Let $G$ be a reductive group over $F$. The group $^LG$ is defined as $^LG=\widehat{G}(\mathbb{C})\rtimes \operatorname{Gal}_F$. The action of $\operatorname{Gal}_F$ is defined as the composition \begin{equation*} \operatorname{Gal}_F\to\operatorname{Aut}(G_{F^s})\to\operatorname{Aut}\Psi(G_{F^s},B,T)\cong\operatorname{Aut}\Psi(\widehat{G},\widehat{B},\widehat{T})\to\operatorname{Aut}(\widehat{G}) \end{equation*}

My question is that it seems to me the Galois action on $G_{F^s}$ is nonlinear, and I am not pretty sure what $\operatorname{Aut}(G_{F^s})$ mean here. Given an group scheme automorphism of $G_{F^s}$ that is not $F^s-$linear, I think we cannot define the map of root datum. Because by definition to induce a map $\operatorname{Hom}_{F^s}(T,\mathbb{G}_m)\to\operatorname{Hom}_{F^s}(T,\mathbb{G}_m)$ your map $T\to T$ should be $F^s-$linear.

Any help will be greatly appreciated!

Yuheng
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  • What is the definition of $\Psi$ here? – Suzet Oct 14 '25 at 10:28
  • It should be $Aut(\Psi(G_{F^s},B,T))$, the automorphism of the root data – Yuheng Oct 14 '25 at 10:51
  • The second arrow is defined as follows. Start with an automorphism of $f$ of $G_{F^s}$, choose an inner automorphism $g$ such that $gf$ fix the pinning, then it induce an morphism of the based root data – Yuheng Oct 14 '25 at 10:54

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