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Let $G$ be a connected Lie group with a maximal torus $T$. Suppose $\sigma$ is an automorphism of $G$ so that $\sigma(T)=T$. Then can we conclude that $\sigma$ is an inner automorphism of $G$? (i.e. is there an element $g\in G$ so that $\sigma= Ad_g : G\rightarrow G$).

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    $\begingroup$ @Li: Did you try to find a counterexample by looking at $SO(2)$ (which is its own maximal torus) or $SO(3)$ in the nonabelian case? $\endgroup$
    – Misha
    Apr 27, 2013 at 4:08
  • $\begingroup$ Try $G=SL(2,{\mathbb R})$ and $\sigma(x)=x^{-t}$, so $sigma$ is the composition of taking inverses and transposition. $\endgroup$
    – user1688
    Apr 27, 2013 at 9:40
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    $\begingroup$ Assume $G$ compact. If your result was true then every automorphism would be inner. Indeed if $\mu$ was an automorphism you can find $g$ such that $Ad_g \mu (T) = T$. But $Aut(G)/Ad(G)$ is the non-trivial group of automorphisms of the Dynkin diagram. $\endgroup$ Apr 27, 2013 at 13:12
  • $\begingroup$ I interpret Michael Murray's answer as saying that the answer is "this is as false as possible". $\endgroup$ May 4, 2013 at 13:37

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