The trigonometric Dunkl intertwining operator and its dual associated with the Cherednik operators and the Heckman–Opdam theory

Abstract

In this paper we prove that there exists a unique topological isomorphism Vk from (the space of C∞-functions on ) onto itself which intertwines the Cherednik operators Tj, j = 1, 2, . . . , d, and the partial derivatives , j = 1, 2, . . . , d, called the trigonometric Dunkl intertwining operator (this name has been proposed by G. J. Heckman). To define and study the operator Vk we have introduced first the trigonometric Dunkl dual intertwining operator tVk. The operators Vk and tVk are the analogue in the Dunkl theory of the Dunkl intertwining operator and its dual (see [Dunkl, Can. J. Math. 43: 1213–1227, 1991, Trimèche, Integrals Transforms Special Funct. 12: 349–374, 2001]).