Kazhdan-Luzstig Polynomials and Lower Intervals in the Bruhat Order I have read in a number of places that the lower Bruhat interval $[e, w]$ is rank-symmetric if and only if the KL-polynomial $P_{e, w}(q) = 1$. All of the proofs I've come across use "rationally smooth Schubert varieties", which I don't really understand.
The KL polynomials can be defined purely in terms of the Iwahori-Hecke algebra of the Coxeter group, and satisfy a number of identities involving sums over Bruhat intervals. I would like to know then if there is a more direct way to prove that $[e, w]$ is rank symmetric iff $P_{e, w}(q) = 1$, using only the Hecke algebra (and Bruhat order).
 A: The statement does indeed hold in type H3; you can find the computation at http://sage.lacim.uqam.ca/home/pub/15/. The machine is still running (and I don't know how long it might take) doing type H4.
So, if no Coxeter theoretic proof is known, there is hope to find one...
A: The answer to the question is "yes", allowing for a generous interpretation of "direct way".   This will follow from the recently posted work of Ben Elias and Geordie Williamson on non-negativity of coefficients of Kazhdan-Lusztig polynomials for an arbitrary Coxeter group here.
See the Update to my MO question here,
which refers to the 1991 conference report by Jim Carrell (with Dale Peterson): in the first section, the equivalence you want is formulated for an arbitrary Coxeter group under the hypothesis that coefficients of relevant K-L polynomials are all non-negative.  (This may be one of the sources you are referring to.)
At first I had overlooked this type of answer to my own question.  (I'm still looking for other consequences of the non-negativity theorem, of course, but this one is interesting.)   Note that for general Coxeter groups, one needs an approach which doesn't involve the geometry of Schubert varieties.   What Elias and Williamson seem to do is avoid all that algebraic geometry by providing a sophisticated substitute.
A: "Singular Loci of Schubert Varieties" by Billey and Lakshmibai is by far the best reference I've found so far for this question. Chapter 6 in particular deals with the combinatorial consequences of $P_{x, w}(q) = 1$.
