We construct a two-parameter family of actions ωk,a of the Lie algebra sl(2,R) by dierential-difference operators on RN. Here, k is a multiplicity-function for the Dunkl operators, and a > 0 arises from the interpolation of the Weil representation and the minimal unitary representation of the conformal group. 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 Ωk,a. Our semigroup generalizes the Hermite semigroup studied by R. Howe (k ≡ 0, a = 2) and the Laguerre semigroup by T. Kobayashi and G. Mano (k ≡ 0, a = 1). The boundary value of our semigroup Ωk,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) as a Dunkl-type generalization of the classical Hankel transform. We establish a generalization of the Plancherel theorem, and the Heisenberg uncertainty principle for Fk,a. We also find explicit kernel functions for Ωk,a and Fk,a for a = 1,2 by means of Bessel functions and the Dunkl intertwining operator.
© Toshiyuki Kobayashi