Parabolic Kazhdan-Lusztig polynomial coincide?

Question

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 $x,w\in W_J$?

Answer 1

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 $W_J$.