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Let $G$ be a group over algebraically nonclosed field $k$ of characteristic $0$. And let $P$ be a minimal parabolic subgroup defined over $k$. Let $S$ be its maximal split torus and $T$ be its maximal torus defined over $k$ containing $S$ ($P\supset T \supset S$). Let $W(T)=N_G(T)/Z_G(T)$ and $W(S)=N_G(S)/Z_G(S)$ be the corresponding Weyl groups. Then we have Bruhat decomposition over $\overline{k}$ $G/P=\bigcup_{n\in N_G(T)(\overline{k})}PnP/P$ and by Borel-Tits we also have the decomposition for $k$ points i.e. $G(k)/P(k)=\bigcup_{w\in N_G(S)(k)}P(k)wP(k)/P(k)$ (which is different from above but each class $P(k)wP(k)/P(k)$ is Zariski dense in the corresponding class $PwP/P$ of the first decomposition). There is a remark 6.24 b) in the paper of Borel and Tits that it can happen that $PnP/P$ is defined over $k$ but does not contain $k$-points. I have the following questions about this issue:

0) What is the easiest example of this phenomenon?

1) Is there a reasonable condition on the field when this cannot happen?

2) Is there a reasonable condition on the field (Galois group) when this issue does not happen for the given minimal group $P$ and all possible parabolics of $G$ such that $Q\supset P$ and $Q/Rad Q$ is of split rank $1$? I.e. this issue does not happen for Bruhat decomposition of $Q/P$.

Remark (Introduced after a discussion): Given $n\in N_G(T)(k_s)$,there is a possibility that the smooth locally closed subscheme $P_{k_s}nP_{k_s}$ inside $G_{k_s}$ can be defined over $k$. (This of course happens if $^\gamma n\in N_{Z_G(S)}(T)n N_{Z_G(S)}(T)$ for all $\gamma \in {\rm{Gal}}(k_s/k)$, so the question for particular $n$ is reduced to the Galois action on the set $W(G,T)$ which is reduced to combinatorics of the root system. The question is about reasonable conditions on the field $k$ for the following to be true: if $P_{k_s}nP_{k_s}$ is defined over $k$ then it contains a $k$-point (or equivalently by the Bruhat decomposition for $G(k)$ with respect to $P(k)$, is necessarily equal to $P_{k_s}n'P_{k_s}$ for some $n' \in N_G(S)(k)$).

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    $\begingroup$ Writing $W_k$ to denote the relative Weyl group (which is not the group of $k$-points of what you call $W$) and simultaneously writing $H_k$ to denote the group of $k$-points of a $k$-group ($P_k, G_k, N_G(S)_k$, etc.) is rather confusing notation. Can you please denote the group of $k$-points in another way, such as $H(k)$? While doing that, please also replace $G_k$ with $G(k)/P(k)$. The subsets $(PwP)(\overline{k})$ for $w\in W(G,S)$ are pairwise disjoint, so can you please also clarify your reason for interest in the possibility that some $PwP$ for $w\in W(G,T)$ is defined over $k$? $\endgroup$
    – nfdc23
    Commented Sep 30, 2017 at 17:27
  • $\begingroup$ Thank you for noticing confusion in my notations. I can understand the reason for putting $P(k)$ instead of $P_k$, but what you can advise me to change $N_G(S)_k$ for? $N(G,S)(k)$? $\endgroup$
    – Vladimir_Z
    Commented Sep 30, 2017 at 18:14
  • $\begingroup$ I think that $N_G(S)(k)$ is reasonable notation. Since the finite etale $k$-group $W(G,S) = N_G(S)/Z_G(S)$ turns out to be constant with group of $k$-points always equal to $N_G(S)(k)/Z_G(S)(k)$ (a real miracle, in my opinion), one can also safely index those unions by the groups $W(G,T)(\overline{k})$ and $W(G,S)(k)$ respectively (where it is understood that $PwP$ means $P n_w P$ for a representative $n_w$ of $w$, the choice of which doesn't matter for the double coset). $\endgroup$
    – nfdc23
    Commented Sep 30, 2017 at 23:09
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    $\begingroup$ I think that the issue @nfdc23 brings up has not been addressed by the edit. You're still writing $G_k$ sometimes and $N_G(S)(k)$ sometimes. Do these both mean the group of rational points? $\endgroup$
    – LSpice
    Commented Oct 1, 2017 at 2:47
  • $\begingroup$ Also, what does your (2) mean? Do you mean to consider the same group $P$ as a parabolic inside several different groups $G$? I don't think that this can happen in any reasonable way. $\endgroup$
    – LSpice
    Commented Oct 1, 2017 at 2:48

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There is the following example when the remark of Borel and Tits is obviously true. Consider the group $G=SL(2, \Bbb D)$ over division algebra $\Bbb D$ for example $\Bbb H$ for $k=\Bbb R$. The parabolic subgroup is the following

\begin{pmatrix} \Bbb H^\times & \Bbb H \\ 0 & \Bbb H^\times \end{pmatrix}

Then it has split rank $1$, so the Bruhat decomposition for $k$-points is just $G_k=P_k\cup P_ksP_k$, where $s\in N_G(S)(k)$ (of course over $\Bbb C$ the cell $P/P$ is the point and the cell $PsP/P$ is open). Hovewer if we pass to the points of $G$ over $\Bbb C$ then $G/P(\Bbb C)$ is the Grassmanian $G(2,4)$ of $2$-dim vector subspaces of $4$-dim. Then there is a $P$-invariant open subset of the unique divisor which consists of $2$-dimensional vector spaces intersecting a given one that is stabilized by $P$. (This subspace correspond to the first block in the matrix and is clear that the condition on the intersection is preserved by the action of $P$). It is clear that being unique such divisor is Galois invariant, however it does not have the points defined over $k=\Bbb R$ on the dense open $P$-orbit.

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