Let $P_{x,w}$ and $Q_{x,w}$ be the Kazhdan Lusztig polynomial and the inverse Kazhdan Lusztig polynomial of Coxeter group $W$, respectively. i.e., $\sum_{x\le y\le z}(-1)^{\ell(y)-\ell(x)}P_{x,y}(q)Q_{y,z}(q)=\delta_{x,z}$.
Let $W$ be a crystallographic Coxeter group. Then the paper Anders Bjorner and Torsten Ekedahl --- On the shape of Bruhat intervals shows that:
This result has been proven by interpretating the coefficients of $P_{x,w}$ in terms of the dimension of the fibre of the cohomology of the intersection complex for $\overline{X_z}$ at a point of $X_x$.
When $W$ is finite, it is well-known that $Q_{x,w}(q)=P_{w_0w,w_0x}(q)$. So the Theorem 4.2 can be translated into the following:
Let $x\le y\le z$ in a finite crystallographic Coxeter group. Then $Q^{i}_{x,z}\ge Q^{i}_{x,y}$.
I would like to know whether we can drop the condition "finite". In other words, can we prove the monotonicity Theorem for $Q_{x,w}$ for infinite "Weyl groups", say, affine Weyl group, by using combinatorics or interpretation of dimension of the fibre of some cohomology or any other methods?
Personally, I prefer to have some elementary combinatorics method (e.g. using Theorem 4.2, together with some combinatorical relationship between $P_{x,w}$ and $Q_{x,w}$). But any doable methods are welcome.
