Let $G$ be a connected, reductive group over an algebraically closed field $k$. Let $T$ be a maximal torus of $G$, and $\Phi = \Phi(T,G)$ the set of roots of $T$ in $G$. The bases $\Delta$ of $\Phi$ are parameterized by the Borel subgroups of $G$ containing $T$.

Now assume that $k$ is not necessarily algebraically closed. Replace $T$ by a maximal $k$-split torus of $G$. Is there an analogous parameterization for the bases of $\Phi$? For example, are minimal $k$-parabolic subgroups containing $T$ in one to one correspondence with such bases?

My guess would be yes: if $P_0$ is a minimal parabolic, then $P_0$ contains $M_0 = Z_G(A_0)$, then I suppose $\mathscr R_u(P_0)$ comprises a system of positive roots. Conversely, I imagine relative root subgroups of a give system of positive roots $\Phi^+$ would comprise a connected unipotent subgroup $N_0$ for which $P_0 = M_0N_0$ is a minimal $k$-parabolic subgroup, for which we recover $N_0 = \mathscr R_u(P_0)$.

  • 2
    $\begingroup$ Yes, if $S$ is a maximal split $k$-torus in $G$ then the set of bases of $\Phi(G,S)$ is in bijection with the set of minimal parabolic $k$-subgroups of $G$ that contain $S$. The bijection goes by means of each set being in bijection with the set of positive systems of roots. If $\Phi^+$ is a positive system of roots then the (split) unipotent radical $U$ of the associated minimal parabolic $k$-subgroup $P$ is directly spanned in any order by the root groups relative to the non-divisible roots in $\Phi^+$ (and $P=Z_G(S)\ltimes U$). This is all part of the Borel-Tits structure theory. $\endgroup$ – nfdc23 Apr 22 '17 at 4:27
  • $\begingroup$ By the way, it is much better (e.g., in accordance with post-1950 algebraic geometry) to say "parabolic $k$-subgroup" rather than "$k$-parabolic subgroup". Also, not that it matters here, but what Springer's textbook calls a (connected) "$k$-reductive group" is a pseudo-reductive $k$-group. $\endgroup$ – nfdc23 Apr 22 '17 at 13:27
  • $\begingroup$ If you look at section 11 of the notes on the structure of reductive groups over fields that I indicated at an earlier question of yours (i.e., ams.org/open-math-notes/omn-view-listing?listingId=110663) then you'll find a complete discussion of this matter via principles different (though not entirely unrelated) from those in Borel's textbook. Also see Prop. V.3.3 (and Remark V.3.4) in there, which rest on Theorem 5.3.6 therein (avoiding various gritty commutator calculations that pervade the split case, useful since root groups can be non-commutative in the non-split case). $\endgroup$ – nfdc23 Apr 23 '17 at 14:59

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