Weight functions and ordered group-algebra coefficients

Let $(W,S)$ be as in Arbitrary-rank Coxeter presentations, length, reflections, and Bruhat order. A totally ordered abelian group $(\mathrm{\Gamma },+,0,<)$ is an abelian group with a total order invariant under translation. Its group algebra $A=\mathbb{Z}[\mathrm{\Gamma }]$ consists of finite sums ${\sum }_{\gamma \in \mathrm{\Gamma }}{a}_{\gamma }{v}^{\gamma }$, where $a_{\gamma }\in \mathbb{Z}$, with $v^{\gamma }{v}^{\delta }=v^{\gamma +\delta }$. Define the ring involution $\overline{v^{\gamma }}=v^{-\gamma }$, and set

$$A_{<0}=\underset{\gamma <0}{⨁}\mathbb{Z}{v}^{\gamma },A_{\le 0}=\underset{\gamma \le 0}{⨁}\mathbb{Z}{v}^{\gamma }\mathrm{.}$$

A weight function is a map $L:W\to \mathrm{\Gamma }$ such that $L(xy)=L(x)+L(y)$ whenever $\mathrm{\ell }(xy)=\mathrm{\ell }(x)+\mathrm{\ell }(y)$. Its simple-generator parameters are $L(s)$ for $s\in S$. The exact criterion for prescribing those values on the simple generators is proved in Simple-generator weights extend uniquely to a weight function ↗. No sign restriction on $L(s)$ is part of this definition.

The coefficient ordering used on the weighted Kazhdan–Lusztig basis is the given order on $\mathrm{\Gamma }$, through $A_{<0}$. The group algebra elements $v^{L(s)}$ are units for every $s\in S$.