# Reference for the bijection of parabolic subgroups $P$ containing $B$ and subsets of the set of simple roots $\Delta(B)$

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.

• The fact that MSE is not very active doesn't make an MSE question more appropriate for MO, though. Jul 19 at 18:08

• 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. Jul 20 at 14:51
• 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 $k$. Jul 21 at 17:55