Let $(W,S)$ be a Coxeter system. For any subset $I\subseteq S$, we can have the *parabolic Kazhdan-Lusztig polynomial* $P\_{x,w}^I(q)$ with respect to $I$.


Now consider $I\subseteq J\subseteq S$. Both $(W,S)$, $(W\_J,J)$ are Coxeter systems.


Since $I\subseteq J$, we get the *parabolic Kazhdan-Lusztig polynomial* $\overline{P}\_{x,w}^I(q)$ with respect to $I$ when considering the Coxeter system $(W\_J,J)$.


Does $P\_{x,w}^I(q)=\overline{P}\_{x,w}^I(q)$ for all $x,w\in W\_J$? 



Parabolic Kazhdan-Lusztig polynomial coincide?
Yes, that's true. The standard recursive constructions will give you this fact easily, because the only group elements involved in $P\_{x,w}^I$ are those which are $\leq w$ w.r.t. the Bruhat order. If $w\in W\_J$, then all those elements are themselves contained in $W\_J$.


