On the non-negativity of the first coefficient of Kazhdan-Lusztig polynomials (Q1904235)

Let \((W,S)\) be an arbitrary Coxeter system, and let \(P_{x,w}= \sum_{i\geq 0} p_i (x,w) q^i\in Z[q]\) be the Kazhdan-Lusztig polynomials for \(x,w\in W\). It was conjectured by \textit{D. Kazhdan} and \textit{G. Lusztig} [Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)]\ that all coefficients of Kazhdan-Lusztig polynomials are non-negative. This is still an open problem, but some special cases are verified. The conjecture is correct for finite Coxeter groups, affine Weyl groups and crystallographic Coxeter groups [cf. \textit{D. Alvis}, J. Algebra 107, 160-168 (1987; Zbl 0615.20019), \textit{D. Kazhdan} and \textit{G. Lusztig}, Proc. Symp. Pure Math. 36, 185-203 (1980; Zbl 0461.14015) and \textit{Z. Haddad}, ``Infinite dimensional flag varieties'' (Thesis, MIT, 1984)]. In this paper, the author shows the non-negativity of the first coefficient \(p_1 (x,w)\) of Kazhdan-Lusztig polynomials for arbitrary Coxeter systems.