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 transforms F 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).