In their seminal 1979 paper Representations of Coxeter groups and Hecke algebras (Invent. Math. 53, doi: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, arXiv: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 here and
here.
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
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.
Best Answer
Non-negativity is important in the proof of Lusztig's 15 conjectures (in fact, it is easy to be proved with the non-negativity property, like in the "split case" and "quasi-split case"). Although even when in unequal parameter setting, where non-negativity is no longer true, we still can't find any counterexample of the 15 conjectures. When all the conjectures hold, a lot of work can be done on the representation of Coxeter groups and their Hecke algebras. Everything is contained here. (I don't have enough reputation to add a comment, so I have to put my comment as an answer. I'm sorry if it is bad.)