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Let $(G,T,M)$ be a split reductive group (over say, the integers), with Lie algebra $(\mathfrak{g}, \mathfrak{t})$, and let $R$ be a commutative ring. When $R$ is an algebraically closed field, it is a well-known theorem (of Cartan for $R = \mathbb{C}$, and Chevalley in characteristic zero, and I think Humphreys in general) that all Cartan $R$-subalgebras of $\mathfrak{g}_R$ are $G(R)$-conjugate, but this fails for arbitrary $R$. For example, when $R$ is the field of real numbers and $G=\operatorname{SL}_2$, the Lie algebra of a non-split maximal torus is not $G(R)$-conjugate to $\mathfrak{t}_R$). Instead of this general conjugacy, I'd like to know how well the action of $G(R)$ approximates the action of the group of $R$-Lie algebra automorphisms of $\mathfrak{g}_R$.

Question 1: Does $G(R)$ act transitively on the $\operatorname{Aut}_{R\text{-Lie}} \mathfrak{g}_R$-orbit of $\mathfrak{t}_R$?

There is a group scheme analogue of this:

Question 2: Does $G(R)$ act transitively on the $\operatorname{Aut}_{R\text{-gp}.} G_R$-orbit of $T_R$?

It seems pretty clear that a positive answer to the first question implies a positive answer to the second, but I don't know if the reverse holds. As far as I can tell, the answer to both questions is positive when the canonical isogeny $Z(G) \times D(G) \to G$ is an isomorphism (that is, when $G$ is a direct product of a torus and an adjoint type group). The problem is that, while the adjoint actions of $G$ on $G_R$ and $\mathfrak{g}_R$ factor through the quotient $G/Z(G)$ by the center, the scheme-theoretic surjection $G \to G/Z(G)$ may not yield a surjection on rational points! There is a cokernel that lies in a flat $H^1$ group. Even so, I have not been able to construct an example giving a negative answer to either question.

Naturally, I would appreciate any insights even in the case of fields, or with restricted characteristic.

Edit 9 hours later: L. Spice has helpfully pointed out in a comment that when $R$ is a field, then the answer to question 2 is "yes". I now see that this is theorem 20.9(ii) in Borel's Linear algebraic groups. I haven't gone through the details yet, but it looks like the proof in the text can be extended to conjugacy that is Zariski local in $R$.

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    $\begingroup$ In $(G, T, M)$, I guess $G$ is the ambient reductive group and $T$ is a split maximal torus, but what is $M$? Anyway, over $R$ a field, all split tori are $G(R)$-conjugate, so the answer to Question 2 in that case is 'yes'. $\endgroup$
    – LSpice
    Mar 10 '20 at 18:33
  • $\begingroup$ (Based on my skimming of math.stanford.edu/~conrad/papers/luminysga3.pdf to try to answer your question, I guess that $M$ is the character lattice of $T$.) $\endgroup$
    – LSpice
    Mar 10 '20 at 20:21
  • $\begingroup$ (With $R$ field, $T$ nonsplit) For $G=\mathrm{SL}_2$, write $s=\mathrm{diag}(3,1)$ (so neither $s$ nor $-s$ is sum of two squares in $\mathbf{Q}$), $T$ the standard $\mathrm{SO}_2$. Then $sT_\mathbf{Q}s^{-1}$ is not in the $\mathrm{SL}_2(\mathbf{Q})$-orbit of $T_\mathbf{Q}$. Indeed, if it were equal to $s_1T_\mathbf{Q}s_1^{-1}$ with $s_1\in\mathrm{SL}_2(\mathbf{Q})$, then $t=s_1^{-1}s\in\mathrm{GL}_2(\mathbf{Q})$ normalizes $T_\mathbf{Q}$, so is in $\mathrm{O}(2)_\mathbf{Q}$ and has determinant $3$. But the determinant of a matrix in $\mathrm{O}(2)$ has determinant of the form $\pm (x^2+y^2)$. $\endgroup$
    – YCor
    Mar 10 '20 at 21:00
  • $\begingroup$ @YCor, I thought the question was asking specifically about the case where $T$ is split? Also, what do you mean by asking whether a diagonal matrix is or is not a sum of two scalar squares? $\endgroup$
    – LSpice
    Mar 10 '20 at 21:32
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    $\begingroup$ @LSpice it's a typo: I just mean neither $3$ or $-3$ is a sum of 2 squares (in $\mathbf{Q}$. Yes I didn't guess whether OP's exclusively interested in split $T$ although it seems so; anyway I did the comment. $\endgroup$
    – YCor
    Mar 10 '20 at 21:43
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When $R$ is a field, the answer to Question 1 is Yes (at least in char 0) with the same proof as for Question 2. For simplicity, we write $G$ for $G_R$, $T$ for $T_R$, etc.

Theorem. Let $G$ be a split (connected) reductive group over a field $R$ of characteristic 0, and let $T\subset G$ be a split maximal torus. Write ${\mathfrak g}={\rm Lie\,} G$ and ${\mathfrak t}={\rm Lie\,} T$. Let $a\in{\rm Aut\,} {\mathfrak t}$ be an automorphism of ${\mathfrak g}$, and set ${\mathfrak t}'=a({\mathfrak t})\subset {\mathfrak g}$. Then there exists $g\in G(R)$ such that ${\mathfrak t}'={\rm Ad}(g)({\mathfrak t})$.

Proof. First let us canonically decompose ${\mathfrak g}$ and ${\mathfrak t}$. Write ${\mathfrak z}$ for the center of ${\mathfrak g}$, and write ${\mathfrak g}^{\rm ss}=[{\mathfrak g},{\mathfrak g}]$, which is a semisimple Lie algebra. Then ${\mathfrak g}={\mathfrak z}\dotplus {\mathfrak g}^{\rm ss}$ (direct sum) and hence, $$ {\rm Aut\,} {\mathfrak g}={\rm Aut\,} {\mathfrak z}\times {\rm Aut\,}{\mathfrak g}^{\rm ss}.$$ Set ${\mathfrak t}^{\rm ss}={\mathfrak t}\cap{\mathfrak g}^{\rm ss}$; then ${\mathfrak t}={\mathfrak z}\dotplus{\mathfrak t}^{\rm ss}$. Write $$a=(a^{\mathfrak z}, a^{\rm ss})\in {\rm Aut\,} {\mathfrak z}\times {\rm Aut\,}{\mathfrak g}^{\rm ss}={\rm Aut\,} {\mathfrak g}.$$ Then it is clear that $$a({\mathfrak t})=a^{\mathfrak z}({\mathfrak z})\dotplus a^{\rm ss}({\mathfrak t}^{\rm ss})={\mathfrak z}\dotplus a^{\rm ss}({\mathfrak t}^{\rm ss})=a^{\rm ss}({\mathfrak t}).$$ Therefore, we may and shall assume that $a\in{\rm Aut\,}{\mathfrak g}^{\rm ss}$.

Now let us canonically decompose $G$. Write $Z^0=Z(G)^0$ (the identity component of the center of $G$). Then $Z^0$ is a torus with Lie algebra ${\rm Lie\,} Z^0={\mathfrak z}$. Write $G^{\rm ss}=[G,G]$ (the commutator subgroup of $G$), which is a connected semisimple group with Lie algebra ${\rm Lie\,} G^{\rm ss}={\mathfrak g}^{\rm ss}$. Let $G^{\rm sc}$ denote the universal cover of $G^{\rm ss}$. Then $G^{\rm sc}$ is a simply connected semisimple group with Lie algebra ${\rm Lie\,} G^{\rm sc}={\mathfrak g}^{\rm ss}$. We have a canonical homomorphism $$\rho\colon G^{\rm sc}\to G^{\rm ss}\hookrightarrow G$$ and a decomposition (in general not direct) $$G=Z^0\cdot\rho(G^{\rm sc}).$$

We can canonically decompose $T$. Set $T^{\rm ss}=T\cap G^{\rm ss}$, and let $T^{\rm sc}$ denote the preimage of $T^{\rm ss}$ in $G^{\rm sc}$. Then $T^{\rm ss}$ is a maximal torus in $G^{\rm ss}$, and $T^{\rm sc}$ is a maximal torus in $G^{\rm sc}$. We have $$ T=Z^0\cdot T^{\rm ss}=Z^0\cdot \rho(T^{\rm sc}).$$

We may and shall identify ${\rm Aut\,}{\mathfrak g}^{\rm ss}$ with ${\rm Aut\,} G^{\rm sc}$; then $a\in {\rm Aut\,} G^{\rm sc}$. Set $T'=Z^0\cdot \rho(a(T^{\rm sc}))\subset G$. Then $T'$ is a maximal torus in $G$ with Lie algebra $${\rm Lie\,} T'={\mathfrak z}\dotplus{\rm Lie\,} a(T^{\rm sc})={\mathfrak z}\dotplus a({\rm Lie\,} T^{\rm sc})={\mathfrak z}\dotplus a({\mathfrak t}^{\rm ss})=a({\mathfrak t})={\mathfrak t}'.$$ Since $T$ is a split torus, the tori $Z^0$, $T^{\rm ss}$, $T^{\rm sc}$, and $a(T^{\rm sc})$ are all split. Thus $T'$ is a split maximal torus in $G$. By Theorem 20.9(ii) in Borel's book Linear Algebraic Groups, there exists $g\in G(R)$ such that $T'=g\cdot T\cdot g^{-1}$. Then ${\mathfrak t}'={\rm Ad}(g)({\mathfrak t})$, as required.

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  • $\begingroup$ The decomposition $\mathfrak g = \mathfrak z \oplus \mathfrak g^{\text{ss}}$ can fail, for example, if $R = \mathbb F_p$ and $G = \operatorname{SL}_p$. (Also the equality $\operatorname{Lie} G^{\text{sc}} = \mathfrak g^{\text{ss}}$.) $\endgroup$
    – LSpice
    Mar 12 '20 at 17:48
  • $\begingroup$ Finally, why does $\operatorname{Aut} \mathfrak g^{\text{ss}}$ equal $\operatorname{Aut} G^{\text{sc}}$? For example, why can't there be some exotic Lie-algebra automorphisms in positive characteristic? $\endgroup$
    – LSpice
    Mar 12 '20 at 17:52
  • $\begingroup$ @LSpice: In my answer ${\rm char}\,R=0$ ! $\endgroup$ Mar 12 '20 at 18:18
  • $\begingroup$ Although you wrote that clearly, I skipped over it both in the parenthesis and in the statement of the theorem. Sorry! (Still, is it true algebraically, in characteristic 0, that automorphisms of a semisimple Lie algebra exponentiate to automorphisms of the simply connected group with that Lie algebra? I believe it, but don't know a reference.) $\endgroup$
    – LSpice
    Mar 12 '20 at 18:31

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