Dunkl operators and Fischer decompositions

Abstract

In this thesis we will study the theory of Dunkl operators and Dunkl harmonic polynomials and have a look at some of the applications. We will also establish the existence of a certain class of Fischer decompositions of graded vector spaces. The decomposition of L^2(S,h^2d\omega) into Dunkl harmonics follows from a Fischer decomposition which belongs to this class. The similarities between Dunkl operators and partial derivatives can be expressed through a certain intertwining operator. The existence of this type of intertwining operator is explained from a general point of view.