# Diagonal automorphisms for twisted Chevalley groups

Let $$G$$ be a Chevalley group over a field $$k$$ of characteristic $$0$$. We know that a diagonal automorphism $$\phi_h$$ of $$G$$ is of the form $$g\mapsto hgh^{-1}$$, where $$h\in \hat H$$ and $$\hat H$$ normalizes $$G$$. Note that $$\hat H:=\{h(\chi)\mid \chi: \mathbb Z \Phi \rightarrow k^* \}$$, where $$\Phi$$ is the corresponding root system and $$\mathbb Z \Phi$$ is the root lattice.

Let $$G'$$ be the twisted Chevalley group over the field $$k$$. I would like to determine diagonal automorphisms for $$G'$$. So for this we have to determine the analogues group $$\hat H'$$ which normalizes $$G'$$.

I am following Robert Steinberg's Yale's lecture notes on "Lectures on Chevalley Groups". At page $$106$$ (before Theorem $$36$$), he wrote that for the twisted groups diagonal automorphisms can be defined analogous to the untwisted case. Also, I am following "Simple groups of Lie type" by Roger Carter.

I am not sure what is the precise group $$\hat H'$$?

If we define $$\hat H'$$ as follows: $$\hat H':=\hat H \cap N_{\mathrm{Aut}(\mathcal L_k)}(G'),$$ then by definition $$\hat H'$$ normalizes $$G'$$. Is it true that each of the diagonal automorphisms of $$G'$$ can be seen as a map
$$\psi_{h'}(g')=h'g'h'^{-1},$$ where $$h'\in \hat H'$$ and $$g'\in G'$$?

Thank you for your kind help.

First of all, I'd inquire what role the characteristic of the field plays here. It's true that the finite twisted groups rely on characteristics 2, 3 especiallu, but infinite twisted groups include the special unitary groups (etc.). which live over many fields.

While Steinberg's lecture notes are often quite useful, a more leisurely treatment (often with more detail) is found in Roger Carter's 1972 book Simple Groups of Lie Type here .

In any case, the groups $$\hat{H}$$ and the like are tori in the algebraic group sense and thus easy to describe abstractly. But in this context they take a little more work to pin down. The best strategy is probably to follow Carter's treatment for the specific case of unitary groups over an infinite field.

ADDED: First I'll mention some older papers, now freely available in PDF format for online use. These can give some idea of the arguments used as well as how the notation has shifted. First is Chevalley's important 1955 paper in the Tohoku Math. J. Next is Steinberg's 1959 paper Variations on a theme of Chevalley in the Pacific J. Math. (with independent contributons along the same line in a paper by J. Tits and a thesis by David Hertzig at Chicago directed I think by Andre Weil).

For Steinberg's 1960 treatment of automorphisms of Chevalley groups(etc.) over finite fields, see Canadian J. Math., while my follow-up paper on working over arbitrary infinite fields is in the same journal, here .

Aside from differences in notation, the algebraic torus $$\widehat{H}$$ for a Chevalley group should lead to the torus for a twisted group of type $$A_n, n \geq 2$$ or $$D_n$$ or $$E_6$$ given by the fixed points of the graph automorphism involved combined with a field automorphism of the same order.

• Thank you for your help. I have given a description of $\hat H'$. Could you please tell whether it is correct or not? – Sushil Sep 22 '19 at 12:57
• @Sushil: Sorry for the long delay in answering, but it's been a busy week. I did try to add to my answer but didn't succeed. Anyway, I think your formulation is correct. But it's unhelpful in terms of computability. More details to follow. – Jim Humphreys Sep 26 '19 at 21:42