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Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group. (English) Zbl 0778.33009

Let $G$ be a finite Coxeter group acting on the Euclidean space $𝔞$ 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 $\xi \in {𝔞}_{ℂ}$ one can associate a differential-difference operator ${T}_{\xi }\left(k\right)$ on ${𝔞}_{ℂ}$, the so called Dunkl operator. The Dunkl operators commute so one can extend the construction to arbitrary polynomials $\xi$ on ${𝔞}_{ℂ}^{*}$. When $\xi$ is $G$-invariant the restriction ${D}_{\xi }$ of ${T}_{\xi }\left(k\right)$ of the $G$-invariant polynomials on ${𝔞}_{ℂ}$ 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 $ℂ\left[{𝔞}_{ℂ}^{*}\right]\to S\left(k\right)$ whose inverse is denoted by $\gamma \left(k\right)$. Thus for $\lambda \in {𝔞}_{ℂ}^{*}$ one has the eigenvalue problem

$\left(D-\gamma \left(k\right)\left(D\right)\left(\lambda \right)\right)f=0\phantom{\rule{1.em}{0ex}}\forall D\in S\left(k\right)$

on the space of $G$-invariant polynomials on ${𝔞}_{ℂ}$. This system of equations is called the Bessel equations on $G\setminus {𝔞}_{ℂ}$. When restricted to the regular points in ${𝔞}_{ℂ}$ 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 Macdonald’s concerning the evaluation of certain integrals involving the discriminant ${I}_{G}$ of $G$.

##### MSC:
 33C80 Connections of hypergeometric functions with groups and algebras 20F55 Reflection groups; Coxeter groups