TITLE: There are no "holes" in the Bruhat decomposition of parabolic cell $Pw_1P$
QUESTION [7 upvotes]: 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.

REPLY [3 votes]: 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.