2
$\begingroup$

Let $G$ be a reductive group over a $p$-adic field. Let $P$ be a parabolic subgroup of $G$ containing a minimal parabolic $P_0$. Let $S$ be a maximal split torus of $P_0$, and let $\Delta$ be a set of simple roots of $S$ corresponding to $P_0$, with $\theta \subseteq \Delta$ corresponding to $P$.

There is a unique $w$ in the Weyl group $W = N(S)/Z(S)$ such that $w$ maps $\theta$ into $\Delta$, and sends $\Delta - \theta$ to negative roots. Is the double coset $PwP$ always open in $G$?

Consider the special case where $G$ is quasisplit and $P = P_0$ is a Borel subgroup. In that case, $\theta = \emptyset$, and the $w$ described above is just the long element, in which case $PwP$ is open as the "big cell."

$\endgroup$
2
  • 6
    $\begingroup$ Sure, over any field $k$, via the "dynamic approach" to describing parabolic $k$-subgroups. Pick a 1-parameter $k$-subgroup $\lambda:\mathbf{G}_m\to P$ such that $P=P_G(\lambda)$, so $P=L \ltimes U$ for the smooth connected unipotent $U:=U_G(\lambda)$ and connected reductive $L:=Z_G(\lambda)$, so $U=\mathscr{R}_u(P)$ is the unipotent radical and $L$ is a Levi $k$-subgroup. The crucial point is that for $U^{-}:=U_G(-\lambda)$ the multiplication map $U^{-}\times L\times U=U^{-}\times P\to G$ is an open immersion. See section 2.1 of the book Pseudo-reductive Groups for a much wider context. $\endgroup$
    – nfdc23
    Mar 12, 2018 at 7:42
  • $\begingroup$ @nfdc23, I have converted your comment to an answer so that this question can be marked as answered (rather than just leaving a comment suggesting you do so, since I think you haven't been active on MO for a while). If you would like to leave the answer yourself, then please let me know so that I can delete mine. $\endgroup$
    – LSpice
    Jul 28 at 3:44

1 Answer 1

0
$\begingroup$

@nfdc23 answered this question in comments. Here is their comment as an answer, made CW to avoid reputation:

Sure, over any field $k$, via the "dynamic approach" to describing parabolic $k$-subgroups. Pick a 1-parameter $k$-subgroup $\lambda:\mathbf{G}_m\to P$ such that $P=P_G(\lambda)$, so $P=L \ltimes U$ for the smooth connected unipotent $U:=U_G(\lambda)$ and connected reductive $L:=Z_G(\lambda)$, so $U=\mathscr{R}_u(P)$ is the unipotent radical and $L$ is a Levi $k$-subgroup. The crucial point is that for $U^{-}:=U_G(-\lambda)$ the multiplication map $U^{-}\times L\times U=U^{-}\times P\to G$ is an open immersion. See section 2.1 of the book Pseudo-reductive Groups [by Conrad, Gabber, and Prasad] for a much wider context.

$\endgroup$

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.