On factorization of matrix of Kazhdan-Lusztig polynomials

Let $\mathcal{H} = \mathcal{H}(W,S)$ be the Hecke algebra of the Coxeter system $(W,S)$ over $\mathbb{Z}[q^{\pm1}]$, where $W$ is the Weyl group of a symmetrizable Kac-Moody algebra. In this paper, we show that the matrix of Kazhdan-Lusztig polynomials of $\mathcal{H}$ factorizes into a product of $|S|$ many matrices, each of which has entries as polynomials in $q$ with nonnegative coefficients. To achieve this goal, we use hybrid basis $TC^J$ for $J\subseteq S$ of $\mathcal{H}$, defined by Grojnowski-Haiman. The intermediate matrices in the aforementioned factorization turn out to be the transition matrices from $TC^J$-basis to $TC^I$-basis for $I\subset J$. Equivalently, these coefficients can be computed using a natural restriction map from $\mathcal{H}$ to the parabolic Hecke algebra $\mathcal{H}_J$. Moreover, following the ideas from Grojnowski-Haiman, we also give a geometric proof of the positivity of these coefficients.