Absolute continuity of the representing measures of the Dunkl intertwining operator and of its dual and applications

In this paper we prove the absolute continuity of the representing measures of the Dunkl intertwining operator and of its dual. Next we present some applications of this result. Key word : Dunkl intertwining operator and its dual. Absolute continuity of the representing measures. MSC (2000) : 33C80, 51F15, 44A15. Introduction We consider the differential-difference operators on R introduced by C.F.Dunkl in [4] and called Dunkl operators in the literature . These operators are very important in pure Mathematics and in Physics.They provide a useful tool in the study of special functions with root systems (see [3] [8]), and they are closely related to certain representations of degenerate affine Hecke algebras [2][16], moreover the commutative algebra generated by these operators has been used in the study of certain exactly solvable models of quantum mechanics, namely the Calogero-Sutherland-Moser models, which deal with systems of identical particles in a one dimensional spaces (see [10] [13] [14]).