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Let $G$ be a split reductive algebraic group over an arbitrary field $k$ Suppose we have a split maximal torus $T$. There is a short exact sequence of groups $$ 1\to \mathrm{Inn}(G)\to \mathrm{Aut}(G)\to \mathrm{Out}(G)\to 1. $$ where $\mathrm{Inn}(G) = G^{\mathrm{ad}}(k)$. This sequence splits in various ways, one choice for each pinning, which is a choice of a base $\Delta$ for the root system for $(G,T)$, along with isomorphisms $\mathbb{G}_a\to U_\alpha$ for each $\alpha\in\Delta$. After this choice, automorphisms in the image of the splitting map $\mathrm{Out}(G)\to\mathrm{Aut}(G)$ fix the torus $T$.

Now given an automorphism $\theta:G\to G$, we define the set $\mathrm{Aut}(G,\theta)$ to be the automorphisms of $G$ commuting with $\theta$, and let $\mathrm{Inn}(G,\theta)$ be the inner automorphisms in $G^{\mathrm{ad}}(k)$ that commute with $\theta$. Let's assume $T$ to be $\theta$-stable; i.e. $\theta(T) = T$. We get a short exact sequence $$ 1\to \mathrm{Inn}(G,\theta)\to \mathrm{Aut}(G, \theta)\to \mathrm{Out}(G,\theta)\to 1 $$

Question: Is it possible in general to choose a splitting (i.e. a section $\mathrm{Out}(G,\theta)\to \mathrm{Aut}(G,\theta)$) such that the image of this section consists of automorphisms that preserve the fixed maximal torus $T$?

(Note that for any $\varphi\in \mathrm{Aut}(G,\theta)$, the torus $\varphi(T)$ is also $\theta$-stable.) I feel like in special cases a splitting is possible, mainly when $\mathrm{Out}(G,\theta)$ is small, but I hope that there is some nice general answer to this question, or even something under more restrictive hypotheses on things such as: type of automorphism, order of automorphism, base field, etc.

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  • $\begingroup$ Please say that $T$ is assumed to be split (as split $G$ have plenty of non-split maximal $k$-tori when $k \ne k_s$) and that the root system you're speaking about is the one associated to $(G,T)$. Also, please say that Inn($G$) means $G^{\rm{ad}}(k)$ (as opposed to $G(k)/Z_G(k)$) and say whether Inn$(G,\theta)$ means the $k$-points of the identity component of the $\theta$-centralizer in $G^{\rm{ad}}$ or the group of all $k$-points of the $\theta$-centralizer in $G^{\rm{ad}}$ (a priori the latter could be bigger). $\endgroup$ – user76758 Mar 16 '14 at 5:31
  • $\begingroup$ @user76758: Thanks, I was a little vague. As for the last point, I'm not really sure which "inner automorphism group" is the correct notion for this. $\endgroup$ – Jason Polak Mar 17 '14 at 1:41

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