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Borel-de Siebenthal theory can be thought of as an algorithm that, given a semisimple compact Lie group $G$, gives all semisimple compact Lie subgroups whose root systems have the same rank as $G$’s.

How can this be generalized to arbitrary semisimple algebraic groups?

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    $\begingroup$ If you indeed want to get an answer, I would recommend you to add details. $\endgroup$
    – Mikhail Borovoi
    Commented Nov 6, 2021 at 21:27
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    $\begingroup$ Are you asking for an analogue over other topologized fields, such as $p$-adic fields? $\endgroup$
    – Jason Starr
    Commented Nov 6, 2021 at 21:42
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    $\begingroup$ Indeed: what is Borel–de Siebenthal theory to you? To me, it is a statement about arbitrary semisimple (even reductive) algebraic groups. $\endgroup$
    – LSpice
    Commented Nov 6, 2021 at 22:17
  • $\begingroup$ Since the Borel-de Siebenthal theory is about root systems, it works verbatim for semisimple groups over an algebraically closed field of char. 0. If the base field is not closed, you should consider the Galois action and maybe Galois cohomology (1st and 2nd). $\endgroup$
    – Mikhail Borovoi
    Commented Nov 7, 2021 at 7:24
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    $\begingroup$ It also works for semisimple groups in good characteristic, and there is a generalization for all characteristics. But it is unclear from the question what kind of generalization you are looking for. $\endgroup$
    – spin
    Commented Nov 8, 2021 at 11:26

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