Generalized Fourier transforms F_{k, a}

We construct a two-parameter family of actions ω_{k, a} of the Lie algebra sl (2, R) by differential-difference operators on R^N. 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. The action ω_{k, a} lifts to a unitary representation of the universal covering of SL (2, R), and can even be extended to a holomorphic semigroup Ω_{k, 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 Ω_{k, a} provides us with (k, a)-generalized Fourier transformsF_{k, a}, which includes the Dunkl transform D_k (a = 2) and a new unitary operator H_k (a = 1) as a Dunkl-type generalization of the classical Hankel transform. To cite this article: S. Ben Saïd et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).