 with the  -   braid relations $$T_wT_{w'}=T_{ww'}\quad\textrm{for $w,w'\in W$ if $\ell(w)+\ell(w')=\ell(ww')$}$$  -   quadratic relations $$T_s^0=\bfq \quad \textrm{if $s\in S_{\operatorname{aff}}$}.$$  