5
$\begingroup$

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 $n' \in N_G(S)(k)$).

$\endgroup$
8
  • 3
    $\begingroup$ 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
    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
    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
    Sep 30, 2017 at 23:09
  • 1
    $\begingroup$ 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
    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
    Oct 1, 2017 at 2:48

1 Answer 1

1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.