he commutative ring $\cA_0(\bfq)$ is naturally a subring $$\cA_0(\bfq)\subset \operatorname{End}_{Z(\cH_0(\bfq))}(\cA_0(\bfq)),$$ an element $a\in \cA_0(\bfq)$ acting by multiplication $b\mapsto ab$ on $ \cA_0(\bfq)$.  \[sA0q\] There exists a unique $\bbZ[\bfq]$-algebra homomorphism $$\xymatrix{ \sA_0(\bfq):\cH_0(\bfq) \ar[r] & \operatorname{End}_{Z(\cH_0(\bfq))}(\cA_0(\bfq)) }$$ such that  -   $$\sA_0(\bfq)|_{\cA_0(\bfq)}=\textrm{ the natural inclusion $\cA_0(\bfq)\subset\operatorname{End}_{Z(\cH_0(\bfq))}(\cA_0(\bfq))$}$$  -   $$\sA_0(\bfq)(S)=