Laguerre semigroup and Dunkl operators
Abstract
We construct a two-parameter family of actions ωk,a of the Lie algebra sl(2,R) by differential-difference operators on RN 0. Here, k is a multiplicity-function for the Dunkl operators, and a>0 arises from the interpolation of the Weil representation of Mp(N,R) and the minimal unitary representation of O(N+1,2) keeping smaller symmetries. 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 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 includes the Dunkl transform Dk (a=2) and a new unitary operator Hk (a=1), namely a Dunkl-Hankel transform. We establish the inversion formula, and a generalization of the Plancherel theorem, the Hecke identity, the Bochner identity, and a Heisenberg uncertainty inequality 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.