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Let $G$ be a split reductive group over a perfect field $k$ (not necessarily algebraically closed) with split maximal torus $T$ and Borel $B \supset T$.

Then there is(/should be) an inclusion-preserving bijection from the set of parabolic subgroups $P$ of $G$ containing $B$ to the set of subsets of the set of simple roots $\Delta(B)$ associated to $B$.

Does someone know a (good) reference for this?

I know that this question is more suitable for MathStackExchange but I got the impression, that there is not much activity at the moment.

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    $\begingroup$ The fact that MSE is not very active doesn't make an MSE question more appropriate for MO, though. $\endgroup$
    – LSpice
    Jul 19, 2022 at 18:08

1 Answer 1

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Gunter Malle and Donna Testerman, Linear algebraic groups and finite groups of Lie type. Cambridge Studies in Advanced Mathematics, 133. Cambridge University Press, Cambridge, 2011.

See Proposition 12.2.

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  • $\begingroup$ Mmmh but they consider linear algebraic groups over an algebraically closed field. $\endgroup$
    – KKD
    Jul 20, 2022 at 10:42
  • $\begingroup$ This plays no role. Your torus $T$ is split. You construct the parabolic subgroups (containing $B$) over an algebraic closure, and they are defined over the base field. $\endgroup$
    – Mikhail Borovoi
    Jul 20, 2022 at 14:51
  • $\begingroup$ Is this explained somewhere? I need to quote this for a thesis and unsure if I can take this for granted. $\endgroup$
    – KKD
    Jul 20, 2022 at 15:47
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    $\begingroup$ Any parabolic subgroup defined over $k$ is defined over an algebraic closure $\bar k$ of $k$ and hence it comes from a subset $I$ of the set of simple roots $S$. Conversely, Malle and Testerman construct a parabolic subgroup $P_I$ from a subset $I$ of the set of simple roots $S$ without any additional choices. The Galois group $\Gamma={\rm Gal}(\bar k/k)$ acts trivially on $S$ and hence preserves $I$ and $P_I$. Since your field $k$ is perfect, the algebraic subgroup $P_I$ of $G$ is defined over $k$. $\endgroup$
    – Mikhail Borovoi
    Jul 21, 2022 at 17:55

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