Nonexistence of twisted Hecke algebras (Q1816606)

Let \((W,S)\) be a Coxeter system with a finite set \(S\neq\emptyset\). The group ring \(\mathbb{C}[W]\) can be deformed into the twisted group ring \(\mathbb{C}[W]'\) by a cocycle or to the \(q\)-deformation of the group ring \(H[q,w]\) [the Iwahori Hecke algebra] where \(q=\{q_w\}_{w\in W}\) is a family of non-zero complex numbers such that \(q_xq_y=q_{xy}\) if \(\ell(xy)=\ell(x)+\ell(y)\) where \(\ell\) is the length function on \(W\). The algebra \(H[q,w]\) has \(\mathbb{C}\)-basis \(\{T_w\}_{w\in W}\) and multiplication given by \[ T_sT_w=\begin{cases} T_{sw} &\text{if \(sw>w\)}\\ q_sT_{sw}+(q_{s^{-1}})T_w &\text{if \(sw<w\)}\end{cases} \] where \(\leq\) is the Bruhat order. The author proves that, in a sense, the \(q\)-deformation of the twisted group ring does not exist.