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.

split(over some field which isn't important here). Also, it's helpful to give the Weyl group a label such as $W$; then "a Weyl element $w$" just means $w \in W$. The standard use of the symbol $w_0$ is for the "longest element" of $W$ (relative to some fixed set of simple reflections). Finally, it's enough to specify that $P$ is a standard parabolic $P_I$ for a subset $I$ of the simple reflections. $\endgroup$ – Jim Humphreys Apr 11 '17 at 16:08