Integrable connections related to zonal spherical functions

Abstract

We define a system of differential equations of first order for a function valued in the group algebra of the Weyl group associated with an arbitrary root system. This is equivalent to the system of differential equations given by Heckman and Opdam which is a deformation of the system satisfied by the zonal spherical function of the Riemannian symmetric spaceG/K of non-compact type. When the root system is A n -type, our equation is related to the Knizhnik-Zamolodchikov equation in conformal field theory.