# Kazhdan-Lusztig positivity of monomials in the Hecke algebra of a Coxeter System

In 1990, Deodhar [3] showed that the non-negativity of Kazdan-Lusztig polynomials implies the expansion of any monomial of (dual) Kazhdan-Lusztig basis elements $$C'_{s_{i_1}}\cdots C'_{s_{i_r}}=\sum_x L_xC'_x$$ with $$s_{i_j}\in S$$ relative to the (dual) Kazhdan-Lusztig basis $$\{C'_w:w\in W\}$$ of the Hecke algebra of the Coxeter system $$(W,S)$$ has non-negative coefficients, i.e. that $$L_x\in\mathbb N[q^{\frac12}+q^{-\frac12}]$$. In fact, though he does not state it, Deodhar's work implies that the non-negativity conjecture is equivalent to these monomial expansions having non-negative $$L_x$$.

Now, given that cases of non-negativity conjecture have been settled by interpreting the coefficients of the Kazhdan-Lusztig polynomials in terms of intersection cohomology, do the coefficients of $$L_x$$ also have such an interpretation in those cases?

EDIT: As Alexader Woo pointed out, the answer is yes and one reference is Springer [2], who actually shows that $$C'_w\cdot C'_x=\sum_y L_{w,x;y}C'_y$$ with the coefficients of $$L_{w,x;y}\in\mathbb N[q^{\frac12},q^{-\frac12}]$$ coming from intersection cohomology. This motivates me to change my question to whether non-negativity of the Kazhdan-Lusztig polynomials alone implies that all products of dual Kazhdan-Lusztig basis elements lie in the cone spanned by linear combinations of dual Kazhdan-Lusztig basis elements with non-negative coefficients.

Relatedly perhaps, I should ask for clarification on the relationship between the Hecke algebra and the quantized enveloping algebra equipped with a canonical basis consisting of isomorphism classes of certain perverse sheaves that Lusztig associated to any graph in 1993 [1]. As far as I understand (which is not far at all), the two algebras have formally similar properties, but in the latter non-negativity comes for free since any monomial of special basis elements $$F_i^{(a)}$$ has an expansion $$F_{i_1}^{(a_1)}\cdots F_{i_m}^{(a_m)}=\sum_{b\in B;d\in\mathbb Z}M(b,i,a,d)v^{d+d_0}b$$, where $$M(b,i,a,d)$$ is the number of times the shifted pervese sheaf b[d] appears in a direct sum decomposition of the direct image of a constant sheaf under a proper morphism depending only on the sequences $$i$$ and $$a$$, and $$d_0$$ is likewise a integer depending only on $$i$$ and $$a$$.

[1]: G. Lusztig. Tight monomials in quantized enveloping algebras. *Quantum deformations of algebras and their representations. ed. A.Joseph et al., Isr.Math.Conf.Proc.7(1993), Amer.Math.Soc. 117-132.

[2]: T.A. Springer. Quelques applications de la cohomologie d'intersection. Asterisque, tome 92-93 (1982), Séminaire Bourbaki. p. 249-273.

[3]: Vinay V. Deodhar. A Combinatorial Setting for Questions in Kazhdan-Lusztig Theory. Geometriae Dedicata, 36(1):95-119, 1990.

• IIRC, you can basically answer your first question from Springer's Asterisque paper (in French) from 1981. Dec 25 '18 at 19:08
• @AlexanderWoo thank you for the reference! I have edited the question in response. Dec 25 '18 at 20:56
• The natural place to look for a hint (or possibly an answer) to your new question is the 2014 Annals paper of Elias and Williamson, where they show non-negativity of K-L polynomials holds for arbitrary Coxeter groups (not just Weyl groups where there are flag varieties, perverse sheaves, et c). Dec 26 '18 at 1:18