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Let $G$ be a simply connected simple algebraic group over a field $k$. Let $\theta\colon G\to G$ be an involution of $G$ over $k$ (an automorphism of order 2). Let $H=(G^\theta)^0$, the identity component of the fixed point subgroup $G^\theta$ of $\theta$ in $G$. We say that $H$ is a symmetric subgroup of $G$ and that $(G,H)$ is a symmetric pair over $k$. What is known about classification of symmetric pairs pair over $k$?

If $k=\mathbf C$, the classification is given by Kac diagrams, see e.g., Helgason's book "Differential Geometry, Lie Groups, and Symmetric Spaces", Ch. X, in particular, Table V on page 518. (I think the answer was known to Élie Cartan in 1914.) Over $\mathbf R$, see Table II on pages 157-161 in Marcel Berger, Les espaces symétriques noncompacts.

What is known in positive characteristic? Is the classification over an algebraically closed field of characteristic $p$ different from the case of characteristic 0?

Question 1. Is classification of symmetric pairs $(G,H)$ over a finite field ${\mathbb F}_q$ known?

The question is related to a question on simple finite groups of Lie type. Let $k=\mathbb{F}_q$, let $G$ be a simply connected simple algebraic group over $k$ (which is quasi-split), and let $\mathcal{G}=G(k)/Z(G)(k)$ denote the corresponding simple finite group. We say that $\mathcal G$ is a simple finite group of Lie type.

Question 2. Is classification of involutions of simple finite groups of Lie type known?

In particular, we can consider involutions in $\mathcal G$, that is, elements of order dividing 2 in $\mathcal G$ (I mean, the conjugacy classes of involutions in $\mathcal G$). They are certainly known, but I don't know references, except for characteristic 2, where I know the paper Aschbacher and Seitz, Involutions in Chevalley groups over fields of even order.

Question 3. What are references to classification of involutions in Chevalley groups and twisted Chevalley groups over finite fields of odd order?

Comments and references are welcome!

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    $\begingroup$ Beware that the finite group community has the unfortunate habit to call "involution" an element of the group squaring to 2 (instead of an automorphism squaring to 2). $\endgroup$
    – YCor
    Commented Nov 29, 2018 at 7:48
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    $\begingroup$ It looks to me like there might be additional cases arising from conjugation by unipotent order-$2$ elements of $G$ in characteristic $2$. The fixed subgroup $H$ is typically not reductive (consider the case of $\textbf{SL}_2$ in characteristic $2$ where $H$ is an extension of $\mu_2$ by $\mathbb{G}_a$). It appears that all the examples on Berger's list have reductive fixed subgroup $H$. $\endgroup$
    – Jason Starr
    Commented Nov 29, 2018 at 14:39
  • $\begingroup$ @JasonStarr: Yes, the examples in Berger's list are with reductive fixed subgroup. $\endgroup$
    – Mikhail Borovoi
    Commented Nov 29, 2018 at 15:11
  • $\begingroup$ Maybe someone could translate the question into finite groups terminology? Anyway, things like automorphisms of finite groups of Lie type are well understood, and Q1 and Q2 have positive answers $\endgroup$
    – Dima Pasechnik
    Commented Nov 29, 2018 at 15:39
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    $\begingroup$ I believe that Atlas of Finite groups has what you need in the introductory chapter. Basically, there are no surprises for ranks big enough: automorphisms are either linear, or come from Frobenius automorphisms of the ground field, or from diagram automorphisms (all of this can be mixed, of course). And in the small ranks there are some exceptions, but still, all known (modulo classification of finite simple groups, I think). $\endgroup$
    – Dima Pasechnik
    Commented Nov 29, 2018 at 16:38

1 Answer 1

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The classification of automorphisms of order $2$ in finite and algebraic groups of Lie type over finite fields of odd characteristic, and the structure of their centralizers, are quite well-known. Calculations and tables are available in (chapters 2 and 4 especially of) D. Gorenstein, R. Lyons, and R. Solomon, {\it The Classification of the Finite Simple Groups. Number 3. Part I, Chapter A: Almost simple K-groups} Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1998. You will find there references to work of T. Springer and R. Steinberg, and N. Burgoyne and C. Williamson, and others, on whose work those chapters are based. It should be noted that these calculations do not depend on the classification of finite simple groups.

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  • $\begingroup$ Thank you, Richard! This is exactly what I need! $\endgroup$
    – Mikhail Borovoi
    Commented Nov 29, 2018 at 19:56
  • $\begingroup$ Now, concerning characteristic 2, where can I find tables, in addition to the paper of Aschbacher and Seitz? $\endgroup$
    – Mikhail Borovoi
    Commented Nov 29, 2018 at 19:58
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    $\begingroup$ I'm less sure of this. You might try M. Liebeck and G. Seitz, Surveys and Monographs 180, where they study all unipotent classes. $\endgroup$
    – Richard Lyons
    Commented Nov 29, 2018 at 20:11
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    $\begingroup$ Also R. Lawther, Liebeck, and Seitz, "Outer unipotent classes in automorphism groups of simple algebraic groups", Proc. London Math. Soc. (3) 109 (2014) no. 3, 553-595. $\endgroup$
    – Richard Lyons
    Commented Nov 29, 2018 at 20:22
  • $\begingroup$ For finite groups, the book of Burness and Giudici would also be worth looking at. They treat elements of prime order in almost simple groups in great detail. $\endgroup$
    – Nick Gill
    Commented Dec 13, 2018 at 12:28

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