Let $G$ be a split reductive algebraic group (over a local field if you like), $B$ be a fixed Borel subgroup, and $P$ be a fixed standard parabolic subgroup. Let $W$ be the Weyl group of $G$. For $w\in W$, denote $C(w)=BwB$. 


Given Weyl elements $w>w'>w\_1$ (Bruhat order), if $C(w)\subset Pw\_1P$, do we know that $C(w')\subset Pw\_1P$? It's like to ask whether there are holes in $Pw\_1P$. On the other hand, this amounts to asking if $Pw\_1P\cap \Omega\_{w\_1}$ is closed in $\Omega\_{w\_1}$, where $$\Omega\_{w\_1}:=\coprod\_{w\ge w\_1}C(w).$$


Any comments and references are welcome. Thanks in advance.



There are no "holes" in the Bruhat decomposition of parabolic cell $Pw_1P$
Your condition $C(w)\subseteq Pw\_1P$ implies $PwP=Pw\_1P\Rightarrow\overline{PwP}=\overline{Pw\_1P}$.


Moreover $w\ge w'\Rightarrow\overline{BwB}\supseteq\overline{Bw'B}\Rightarrow\overline{PwP}\supseteq \overline{Pw'P}$.


Similarly $w'\ge w\_1\Rightarrow\overline{Bw'B}\supseteq\overline{Bw\_1B}\Rightarrow\overline{Pw'P}\supseteq \overline{Pw\_1P}$.


Together, one gets $\overline{Pw'P}=\overline{Pw\_1p}$.


But then $Pw'P=Pw\_1P$ and in particular $C(w')\subseteq Pw\_1P$.


Remark: One can show stronger that the set of $w$ with $C(w)\subseteq Pw\_1P$ is the double coset $W\_Iw\_1W\_I$ and that this double coset forms a Bruhat interval, i.e., there are $w\_{min},w\_{max}\in D$ such that $w\in D\Leftrightarrow w\_{min}\le w\le w\_{max}$. After a very cursory search I found it without proof in the paper "Parabolic double cosets in Coxeter groups" by Billey et al. Proposition 2.


