*(English)*Zbl 0778.33009

Let $G$ be a finite Coxeter group acting on the Euclidean space $\U0001d51e$ and $R$ the corresponding root system. A complex valued $G$- invariant function on $R$ is called a multiplicity function. To each multiplicity function $k$ and $\xi \in {\U0001d51e}_{\u2102}$ one can associate a differential-difference operator ${T}_{\xi}\left(k\right)$ on ${\U0001d51e}_{\u2102}$, the so called Dunkl operator. The Dunkl operators commute so one can extend the construction to arbitrary polynomials $\xi $ on ${\U0001d51e}_{\u2102}^{*}$. When $\xi $ is $G$-invariant the restriction ${D}_{\xi}$ of ${T}_{\xi}\left(k\right)$ of the $G$-invariant polynomials on ${\U0001d51e}_{\u2102}$ is a partial differential operator. Let $S\left(k\right)$ be the algebra of differential operators obtained in this way. The map $\xi \to {D}_{\xi}$ is an isomorphism $\u2102\left[{\U0001d51e}_{\u2102}^{*}\right]\to S\left(k\right)$ whose inverse is denoted by $\gamma \left(k\right)$. Thus for $\lambda \in {\U0001d51e}_{\u2102}^{*}$ one has the eigenvalue problem

on the space of $G$-invariant polynomials on ${\U0001d51e}_{\u2102}$. This system of equations is called the Bessel equations on $G\setminus {\U0001d51e}_{\u2102}$. When restricted to the regular points in ${\U0001d51e}_{\u2102}$ its local holomorphic solutions form a locally constant sheaf of vector spaces and hence one has an associated monodromy representation. The paper under review gives a detailed study of this monodromy representation and, as an application solves a conjecture of Macdonaldâ€™s concerning the evaluation of certain integrals involving the discriminant ${I}_{G}$ of $G$.