by ${\mathbb{C}}$-linearity. Moreover $W$ acts via the left regular representation of functions on either one of these. The positive Weyl chamber ${\mathfrak{a}}^+$ is defined as the set of elements $x\in{\mathfrak{a}}$ for which $\alpha(x)>0$ for all $\alpha\in\Sigma^+$.

Let $P=\{\lambda\in{\mathfrak{a}}^*\,:\,\lambda_\alpha\in{\mathbb{Z}}\text{ for all }\alpha\in\Sigma\}$ denote the restricted weight lattice. To $\lambda\in P$ is associated the single-valued exponential $e^\lambda:A_{\mathbb{C}}\to{\mathbb{C}}$ given by $e^\lambda(h)=e^{\lambda(\log h)}$, and these are the characters of $A_{\mathbb{C}}$. It can be seen that $\mathrm{span}_{\mathbb{C}}\{e^\lambda\}$ is isomorphic to the ring ${\mathbb{C}}[A_{\mathbb{C}}]$ of regular functions on $A_{\mathbb{C}}$, the latter viewed as an algebraic variety, and $W$ acts on it by $w(e^\lambda):=e^{w\lambda}$ (which is well-defined since $P$ is $W$-invariant). The set of regular points for the $W$-action on $A_{\mathbb{C}}$ coincides with the set $A_{\mathbb{C}}^{\mathrm{reg}}=\{h\in{\mathbb{C}}\,:\,e^{2\alpha(\log h)}\neq 1\text{ for all }\alpha\in\Sigma\}$, and the algebra ${\mathbb{C}}[A_{\mathbb{C}}^{\mathrm{reg}}]$ of regular functions on $A_{\mathbb{C}}^{\mathrm{reg}}$ is the subalgebra of the quotient field of ${\mathbb{C}}[A_{\mathbb{C}}]$ generated by ${\mathbb{C}}[A_{\mathbb{C}}]$ and $\{\frac{1}{1-e^{-2\alpha}}\,:\,\alpha\in\Sigma^+\}$. We denote by ${\mathbb{C}}[A_{\mathbb{C}}^{\mathrm{reg}}]^W$ the subalgebra of $W$-invariant elements.

Let $S({\mathfrak{a}}_{\mathbb{C}})$ denote the symmetric algebra over ${\mathfrak{a}}_{\mathbb{C}}$ and let $S({\mathfrak{a}}_{\mathbb{C}})^W$ denote the subalgebra of its $W$-invariant elements. An element $p\in S({\mathfrak{a}}_{\mathbb{C}})$ gives rise to a constant coefficient differential operator $\partial(p)$ acting on functions $f$ on $A_{\mathbb{C}}$ such that $\partial(x)$ is the directional derivative 