In their seminal 1979 paper <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002095319">Representations of Coxeter groups and Hecke algebras</a> (Invent. Math. **53**, doi:[10.1007/BF01390031](https://doi.org/10.1007/BF01390031)),
Kazhdan and Lusztig studied an arbitrary Coxeter group $(W,S)$ and the corresponding Iwahori-Hecke algebra.   In particular they showed how to pass from a standard basis of this algebra to a more canonical basis, with the change of basis coefficients involving polynomials indexed by pairs of elements of $W$ (in the Bruhat ordering) over $\mathbb{Z}$.   Even though the evidence at the time was quite limited, they conjectured following the statement of their Theorem 1.1 that the coefficients of these polynomials should always be non-negative.  (In very special cases this is true because the coefficients give dimensions of certain cohomology groups.)

Several decades later, Wolfgang Soergel worked out a coherent strategy for proving the non-negativity conjecture, in his paper 

* *Kazhdan–Lusztig-Polynome und unzerlegbare Bimoduln über Polynomringen.* J. Inst. Math. Jussieu 6 (2007), no. 3, 501–525, doi:[10.1017/S1474748007000023](https://doi.org/10.1017/S1474748007000023), arXiv:[math/0403496](https://arxiv.org/abs/math/0403496)

Now that his program seems to have been completed, it is natural to renew the question in the header:

> What if any implications would the non-negativity of coefficients of arbitrary Kazhdan-Lusztig polynomials have?

It has to be emphasized that in Soergel's formulation and the following work, the non-negativity is not itself the main objective.   Instead the combinatorial framework proposed was meant to provide a more self-contained conceptual setting for proof of the original Kazhdan-Lusztig conjecture on Verma module multipliities for a semisimple Lie algebra (soon a theorem) and further theorems in representation theory of a similar flavor.   But Coxeter groups form a vast general class of groups given by generators and relations, so it is surprising to encounter such strong constraints on the polynomials occurring in this generality.   

ADDED: There is some overlap with older questions related to Soergel's approach, posted <a href="https://mathoverflow.net/questions/29409/">here</a> and 
<a href="https://mathoverflow.net/questions/92221/">here</a>. 

UPDATE: It's been pointed out to me that older work by Jim Carrell and Dale Peterson involves the non-negativity condition, though their main goal is the study of singularities of Schubert varieties in classical cases.   See the short account (with a long title) 

* J.B. Carrell, *The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties.*
Algebraic groups and their generalizations: classical methods (University Park, PA, 1991), 53–61, Proc. Sympos. Pure Math., 56, Part 1, Amer. Math. Soc., Providence, RI, 1994. https://doi.org/10.1090/pspum/056.1 


The first section develops for an arbitrary Coxeter group some consequences of non-negativity of Kazhdan-Lusztig coefficients for the combinatorial study of Bruhat intervals.   For further details about the geometry, see 

* Carrell, J., Kuttler, J. _Smooth points of T-stable varieties in G/B and the Peterson map_. Invent. math. 151, 353–379 (2003). https://doi.org/10.1007/s00222-002-0256-5, arXiv:[math/0005025](https://arxiv.org/abs/math/0005025)

I'm still not sure whether such consequences of the 1979 K-L conjecture are enough to make the conjecture in itself "important".    But it's definitely been challenging to approach.