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."

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    $\begingroup$ 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 '18 at 7:42

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