In the book Combinatorics of Coxeter Groups by Björner and Brenti, chapter 5 exercise 13 a) asks the reader to prove the following identity:
$$\sum_{a\in [u,v]}P_{u,a}(q)P_{w_0v,w_0a}(q)=\delta_{u,v},$$
where $P_{u,v}(q)$ is the Kazhdan Lusztig polynomial. (The definition is involved and can be found in the book.)
After playing around with this for a while, I have yet to find any promising starts. I have tried applying the definition of $P_{u,v}$ from the first section of chapter 5. I have tried using some of the combinatorial interpretations from the 5th section of chapter 5. I have also noticed that exercise 11 seems related, but I have not found a way to make use of it. I would appreciate some hints about where to start!