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Hello,

this question is a follow-up from Structure groups and a special class of L-functions
Let $\tilde{\phi}:\tilde{\mathcal{S}}\to\tilde{\mathcal{S}}$ be an automorphism of $\tilde{\mathcal{S}}$, and let's say that $\tilde{\phi}$ is isometric if there exists an affine isometry $\phi:\mathbb{C}\to\mathbb{C}$ such that for all $F\in\tilde{\mathcal{S}}$ and for all $s\in\mathbb{C},\ \ s\neq 1$, $(\tilde{\phi}(F))(s)=(\phi\circ F \circ \phi^{-1})(s)$.
Is it true that all automorphisms of $\tilde{\mathcal{S}}$ which restrict to identity on $\tilde{\mathcal{S}}_{\mathbb{R}}$ are isometric?
Thanks in advance.

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Why would $\phi \circ F \circ \phi^{-1}$ belong to $\tilde{\mathcal{S}}$? –  François Brunault Dec 2 '12 at 21:20
    
Hello François, there are examples of such $\phi$ that give rise to automorphisms of $\tilde{S}$: identity and complex conjugation. You can easily check that the maps $\tilde{\phi}$ induced by these affine isometries through my formula actually are automorphisms of $\tilde{S}$ which restrict to identity on $\tilde{\mathcal{S}}_{\mathbb{R}}$. My goal is to determine whether there are others or not. Please notice that there exist affine isometries that don't give rise to automorphisms of $\tilde{S}$, hence my definition of "isometric automorphisms" as peculiar objects. –  Sylvain JULIEN Dec 2 '12 at 22:51
    
@Sylvain. I have a hard time imagining there is any isometry other than the identity and complex conjugation (with your definition). So your question looks like a unnecessarily complicated way to ask whether the only automorphisms of S over S_R are identity and complex conjugation. –  François Brunault Dec 3 '12 at 8:15
    
Possibly. Would you have an answer to this question? –  Sylvain JULIEN Dec 3 '12 at 18:19
    
By the way, saying things like I did might be a way to answer the question I asked in "structure groups and a special class of L Functions", namely determining whether there is some kind of correspondence between automorphisms of $\tilde{\mathcal{S}}$ over $\tilde{\mathcal{S}}_{\mathbb{R}}$ and distinct affine isometries preserving $Z_{CS}(\tilde{\mathcal{S}})$ and restricting to identity on $Z_{CS}(\tilde{\mathcal{S}}_{\mathbb{R}})$, where "distinct" means that two affine isometries $f$ and $g$ should be considered as equal if $\forall s\in Z_{CS}(\tilde{\mathcal{S}})$ one has $f(s)=g(s)$. –  Sylvain JULIEN Dec 7 '12 at 17:57

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