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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 ξ𝔞 one can associate a differential-difference operator T ξ (k) on 𝔞 , the so called Dunkl operator. The Dunkl operators commute so one can extend the construction to arbitrary polynomials ξ on 𝔞 * . When ξ is G-invariant the restriction D ξ of T ξ (k) of the G-invariant polynomials on 𝔞 is a partial differential operator. Let S(k) be the algebra of differential operators obtained in this way. The map ξD ξ is an isomorphism [𝔞 * ]S(k) whose inverse is denoted by γ(k). Thus for λ𝔞 * one has the eigenvalue problem

(D-γ(k)(D)(λ))f=0DS(k)

on the space of G-invariant polynomials on 𝔞 . This system of equations is called the Bessel equations on G𝔞 . When restricted to the regular points in 𝔞 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:
33C80Connections of hypergeometric functions with groups and algebras
20F55Reflection groups; Coxeter groups