Generalized Fourier transforms Fk,a

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). 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 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) 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 Omega(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 Said et al., C R. Acad. Sci. Paris, Ser. I 347 (2009). (C) 2009 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.