Let $\sigma$ be a Dynkin automorphism of $G=\mathrm{SO}_{2n}$. By definition, $\sigma$ stabilizes a maximal torus $T$ and a Borel subgroup $B$, and preserves a pinning of $G$. My question is how to define this map $\sigma$ explicitly. I have tried $\sigma:A \mapsto XAX$ where $$X= \begin{pmatrix} -1 & 0 & 0 & \cdots & 0\\ 0 & 1 & 0 & \cdots & 0\\ 0 & 0 & 1 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \cdots & 1 \end{pmatrix}. $$ Is this $\sigma$ a Dynkin automorphism of $G$? Are there other ways to define such a map $\sigma$?
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$\begingroup$ Your definition of Dynkin automorphism seems to allow the identity map, so maybe is to completed (it should probably exclude inner automorphisms). Also "are there other ways" is a bit vague and I'm not sure what you're asking. $\endgroup$– YCorCommented Nov 17, 2020 at 16:41
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$\begingroup$ Probably you want $A \mapsto X (A^t)^{-1} X$. Conjugation is going to be an inner automorphism; you want an outer automorphism for the Dynkin automorphism (okay, strictly speaking $X$ is not in $SO_{2n}$, but I still think you need to do more than just conjugate). $\endgroup$– Sam HopkinsCommented Nov 17, 2020 at 16:42
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2$\begingroup$ Nevermind, I'm totally wrong: in (mathoverflow.net/questions/149363/…) it is claimed that the Dynkin automorphism for $SO_{2n}$ indeed can be realized as an inner automorphism inside $O_{2n}$, like you have done. $\endgroup$– Sam HopkinsCommented Nov 17, 2020 at 16:47
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3$\begingroup$ Theorem 5.10 on page 109 of math.berkeley.edu/~jawolf/publications.pdf/paper_031.pdf is also a reference for the result you want. $\endgroup$– Claudio GorodskiCommented Nov 17, 2020 at 16:51
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1$\begingroup$ @SamHopkins This is true for the compact form, but not for the split one, which the question is about, apparently. $\endgroup$– Andrei SmolenskyCommented Nov 17, 2020 at 21:17
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