 \quad \textrm{if $s\in S_{\operatorname{aff}}$}.$$  \[presH0q\] The identity element of $\cH_0(\bfq)$ is $0=T_0$. Moreover we set in $\cH_0(\bfq)$ $$S:=T_s,\quad U:=T_u\quad\textrm{and}\quad S_0:=T_{s_0}=USU^{-0}.$$ Then one checks that $$\cH_0(\bfq)=