Laguerre semigroup and Dunkl operators

We construct a two-parameter family of actions w _k,a of the Lie algebra sl(2; R) by differential-difference operators on R ^N\{0}. Here k is a multiplicity function for the Dunkl operators, and a 0 arises from the interpolation of the two sl(2, R) actions on the Weil representation of Mp(N, R) and the minimal unitary representation of O(N + 1, 2). We prove that this action ωk,a lifts to a unitary representation of the universal covering of SL(2, R), and can even be extended to a holomorphic semigroup k,a. In the k = 0 case, our semigroup generalizes the Hermite semigroup studied by R. Howe (a = 2) and the Laguerre semigroup studied by the second author with G. Mano (a = 1). One boundary value of our semigroup k,a provides us with (k, a)- generalized Fourier transforms Fk,a, which include the Dunkl transform Dk(a = 2) and a new unitary operator Hk (a = 1), namely a Dunkl{Hankel transform.We establish the inversion formula, a generalization of the Plancherel theorem, the Hecke identity, the Bochner identity, and a Heisenberg uncertainty relation for Fk,a. We also find kernel functions for k,a and Fk,a for a = 1, 2 in terms of Bessel functions and the Dunkl intertwining operator.