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Zbl 0778.33009
Opdam, E.M.
Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group.
(English)
[J] Compos. Math. 85, No.3, 333-373 (1993). ISSN 0010-437X; ISSN 1570-5846/e

Let $G$ be a finite Coxeter group acting on the Euclidean space ${\germ a}$ 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 $\xi\in{\germ a}\sb \bbfC$ one can associate a differential-difference operator $T\sb \xi(k)$ on ${\germ a}\sb \bbfC$, the so called Dunkl operator. The Dunkl operators commute so one can extend the construction to arbitrary polynomials $\xi$ on ${\germ a}\sp*\sb \bbfC$. When $\xi$ is $G$-invariant the restriction $D\sb \xi$ of $T\sb \xi(k)$ of the $G$-invariant polynomials on ${\germ a}\sb \bbfC$ is a partial differential operator. Let $S(k)$ be the algebra of differential operators obtained in this way. The map $\xi\to D\sb \xi$ is an isomorphism $\bbfC[{\germ a}\sp*\sb \bbfC]\to S(k)$ whose inverse is denoted by $\gamma(k)$. Thus for $\lambda\in{\germ a}\sp*\sb \bbfC$ one has the eigenvalue problem $$(D-\gamma(k)(D)(\lambda))f=0 \quad \forall D\in S(k)$$ on the space of $G$-invariant polynomials on ${\germ a}\sb \bbfC$. This system of equations is called the Bessel equations on $G\setminus {\germ a}\sb \bbfC$. When restricted to the regular points in ${\germ a}\sb \bbfC$ 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 $I\sb G$ of $G$.
[J.Hilgert (Clausthal)]
MSC 2000:
*33C80 Connections of theory of special functions with groups and algebras
20F55 Coxeter groups

Keywords: Coxeter group; Dunkl operator; Bessel equations; monodromy representation

Cited in: Zbl 1230.33011

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