# Is there a "big open cell" analogue for parabolic subgroups?

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."

• 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. Mar 12, 2018 at 7:42
• @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. Jul 28 at 3:44

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.