On the integrability of a representation of sl(2,R)

The Dunkl operators involve a multiplicity function k as parameter [C.F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989) 167–183]. For positive real values of this function, we consider on the Schwartz space S(RN) a representation ωk of sl(2,R) defined in terms of the Dunkl–Laplacian operator. By means of a beautiful theorem due to E. Nelson, we prove that ωk exponentiates to a unique unitary representation of the universal covering group of SL(2,R). The representation theory is used to derive an identity of Bochner type for the Dunkl transform.