Dunkl operator

Definition

Let $\R^n$ have the standard inner product $\langle \cdot, \cdot \rangle$. For $\alpha \in \R^n \setminus \set{0}$, let $\sigma_\alpha(x) = x - 2 \frac{\langle \alpha, x \rangle}{\langle \alpha, \alpha \rangle} \alpha$ be the reflection in the hyperplane orthogonal to $\alpha$. Let $R \subset \R^n \setminus \set{0}$ be a root system: a finite set such that $\sigma_\alpha(R) = R$ and $R \cap \R \alpha = \set{\alpha, -\alpha}$ for every $\alpha \in R$. Let $W$ be the finite reflection group generated by the $\sigma_\alpha$, and let $R_+ \subset R$ contain exactly one of $\alpha, -\alpha$ for each $\alpha \in R$. Let $k \colon R \to \C$ be a multiplicity function, that is, $k(w \alpha) = k(\alpha)$ for all $w \in W$ and $\alpha \in R$. For $\xi \in \R^n$, the Dunkl operator $T_\xi$ acts on smooth functions $f$ on $\R^n$ by $$T_\xi f(x) = \partial_\xi f(x) + \sum_{\alpha \in R_+} k(\alpha)\, \langle \alpha, \xi \rangle\, \frac{f(x) - f(\sigma_\alpha x)}{\langle \alpha, x \rangle},$$ where $\partial_\xi$ is the directional derivative along $\xi$.