Laguerre semigroup and Dunkl operators

We construct a two-parameter family of actions omega(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 omega(k,a) lifts to a unitary representation of the universal covering of SL(2, R), and can even be extended to a holomorphic semigroup Omega(k,a). In the k equivalent to 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 Omega(k,a) provides us with (k, a)-generalized Fourier transforms F-k,F-a, which include the Dunkl transform D-k(a = 2) and a new unitary operator H-k(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 F-k,F-a. We also find kernel functions for Omega(k,a) and F-k,F-a for a = 1, 2 in terms of Bessel functions and the Dunkl intertwining operator.