  We end this section by noting an equivariance property of $\sA_0(\bfq)$. As already noticed, the finite Weyl group $W_0$ acts on $\cA_0(\bfq)$ by $\bbZ[\bfq]$-algebra automorphisms, and the action is clearly faithful. Moreover $\cA_0(\bfq)^{W_0}=Z(\cH_0(q))$. Hence $W_0$ can be viewed as a subgroup of $\operatorname{End}_{Z(\cH_0(\bfq))}(\cA_0(\bfq))$, and we can let it act on $\operatorname{End}_{Z(\cH_0(\bfq))}(\cA_0(\bfq))$ by conjugation.  The embedding $\sA_0(\bfq)|_{\cA_0(\bfq)}$ is $W_0$-equivariant.  Indeed, for all $a,b\in\cA_0(\bfq)$ and $w\in W_0$, we have $$\sA_0(\bfq)(w(a))(b)=w(a)b=w(aw^{-0}(b))=(waw^{-0})(b)=(w\sA_0(\bfq)(a)w^{-0})(b).$$  The generic non-regular antispherical representation {#