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](http://www.numdam.org/article/ASENS_1957_3_74_2_85_0.pdf).


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



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



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> **Question 2.** *Is classification of involutions **of** simple finite groups of Lie type known?*
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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](https://projecteuclid.org/download/pdf_1/euclid.nmj/1118795948).



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



Symmetric subgroups of simple algebraic groups over finite fields
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. 


