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Let G be a complex simple Lie group of adjoint type. Then, it is well known that every such $G$ contains, unique up to conjugacy, an irreducibly embedded copy of $PSL(2,\mathbb{C}).$ This fact seems to play a critical role in why the theory of compact Riemann surfaces, in my sphere through the theory Higgs bundles and Hitchin representations, is so intricately tied to the geometry of these groups. In particular, every Fuchsian group uniformizing some Riemann surface can be seen to live in the split real form of $G,$ wherein it admits large families of deformations, and through this story the complex analysis of Riemann surfaces gets all tied up with the Lie theory of $G$ and the associated geometry of the symmetric space for $G.$

Of course, though I rarely think about this, there are plenty of algebraically closed fields of characteristic zero other than $\mathbb{C}$ out there, and from my scant knowledge, when you ask the question of how much the Lie theory of the associated algebraic groups is like that of their complex counterparts, or how much of the geometry of the associated homogeneous spaces reflects the complex case, the answer really depends on what question you might have asked.

My question is this: given some other algebraically closed field $\mathbb{K}$ (I'm not positive I should be asking it to be algebraically closed), and some reasonable class of algebraic groups defined over $\mathbb{K},$ are there objects associated to that situation that play the kind of role that Riemann surfaces (and their fundamental groups) play in the realm of complex simple Lie groups.

I can imagine that many people would point directly to the Langlands conjecture, and geometric Langlands, given this question, and although my knowledge is extremely limited regarding that story, I'd be more interested in hearing of other examples.

Perhaps another way to phrase my question is, given a suitable class of algebraic groups over some (algebraically closed) field of characteristic zero other than $\mathbb{C},$ is there some other group/object that plays an equally fundamental role as $PSL(2,\mathbb{C})$ and it's split real form $PSL(2,\mathbb{R})$ plays in the case of groups defined over $\mathbb{C}.$

If this question isn't appropriate for mathoverflow, I understand and apologize for causing any trouble.

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    $\begingroup$ In the theory of affine algebraic groups over arbitrary fields, embedded copies of $\operatorname{SL}_2$ play a key role (just like copies of $SU(2)$ inside a compact Lie group play a key role in its representation theory). I will leave it to the experts to explain this in more detail, though. $\endgroup$ Dec 7, 2014 at 1:10

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