in the direction of $x$ for every $x\in{\mathfrak{a}}$. We shall denote the algebra $\{\partial(p)\,:\,p\in S({\mathfrak{a}}_{\mathbb{C}})\}$ by the symbol $S({\mathfrak{a}}_{\mathbb{C}})$, which is justified since $p\mapsto\partial(p)$ is an algebra isomorphism. Let $\mathbb{D}(A_{\mathbb{C}}^{\mathrm{reg}})={\mathbb{C}}[A_{\mathbb{C}}^{\mathrm{reg}}]\otimes S({\mathfrak{a}}_{\mathbb{C}})$ denote the algebra of differential operators on $A_{\mathbb{C}}$ with coefficients in ${\mathbb{C}}[A_{\mathbb{C}}^{\mathrm{reg}}]$ and $\mathbb{D}(A_{\mathbb{C}}^{\mathrm{reg}})^W$ its subalgebra of $W$-invariant elements, where $W$ acts by $w(\varphi\otimes\partial(p))=(w\varphi)\otimes\partial(wp)$.

We introduce an associative algebra structure on $\mathbb{D}(A_{\mathbb{C}}^{\mathrm{reg}})\otimes{\mathbb{C}}[W]$ via $(D_1\otimes w_1)\cdot(D_2\otimes w_2)=D_1w_1(D_2)\otimes w_1w_2$, where $(wD)(wf):=w(Df)$. Elements of $\mathbb{D}(A_{\mathbb{C}}^{\mathrm{reg}})\otimes{\mathbb{C}}[W]$ are the differential-reflection operators on $A_{\mathbb{C}}^{\mathrm{reg}}$, and they act on functions $f$ on $A_{\mathbb{C}}^{\mathrm{reg}}$ by $(D\otimes w)f=D(wf)$.

The *Cherednik operator* $T_x\in\mathbb{D}(A_{\mathbb{C}}^{\mathrm{reg}})\otimes{\mathbb{C}}[W]$ associated with $x\in{\mathfrak{a}}$ is defined by $$T_x=\partial_x-\rho(x)+\sum_{\alpha\in\Sigma^+}m_\alpha\alpha(x)(1-e^{-2\alpha})^{-1}\otimes(1-r_\alpha),$$ where $2\rho=\sum_{\alpha\in\Sigma^+}m_\alpha\alpha\in{\mathfrak{a}}^*$.

It is a deep result that the operators $T_x$ commute, an important consequence of which is that the map $x\mapsto T_x$ extends uniquely to an algebra homomorphism $p\mapsto T_p$ of $S({\mathfrak{a}}_{\mathbb{C}})$ into $\mathbb{D}(A_{\mathbb{C}}^{\mathrm{reg}})\otimes{\mathbb{C}}[W]$.

Define $\Upsilon:\mathbb{D}(A_{\mathbb{C}}^{\mathrm{reg}})\otimes{\mathbb{C}}[W]\to\mathbb{D}(A_{\mathbb{C}}^{\mathrm{reg}})$ by $\Upsilon(\sum_j D_j\otimes w_j)=\sum_jD_j$. Then $\Upsilon(P)f=Pf$ for every $P\in\mathbb{D}(A_{\mathbb{C}}^{\mathrm{reg}})\otimes{\mathbb{C}}[W]$ and every $W$-invariant function $f$ on $A_{\mathbb{C}}^{\mathrm{reg}}$. In particular we can define $D_p:=\Upsilon(T_p)$ for every $p\in S({\mathfrak{a}}_{\mathbb{C}})$. One can show that $D_p$ belongs to $\mathbb{D}(A_{\mathbb{C}}^{\mathrm{reg}})^W$ whenever $p\in S({\mathfrak{a}}_{\mathbb{C}})^W$, and that $\mathbb{D}:=\{D_p\,:\,p\in S({\mathfrak{a}}_{\mathbb{C}})^W\}$ is a commutative subalgebra of $\mathbb{D}(A_{\mathbb{C}}^{\mathrm{reg}})^W$, see [@Narayanan-Pasquale-Pusti p. 232] for details. In the geometric case where $({\mathfrak{a}},\Sigma,m)$ corresponds to the 