## S. Ben Saïd, T. Kobayashi, and B. Ørsted, *Generalized
Fourier transforms **F*_{k,a}, C. R. Math. Acad. Sci. Paris, **347** (2009), 1119-1124
(published online first, on 21 August 2009)..

We construct a two-parameter family of actions ω_{k,a} of the Lie algebra *sl*(2,*R*)
by dierential-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. We prove that this 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. We establish a generalization of the Plancherel theorem, and the Heisenberg
uncertainty principle for *F*_{k,a}. We also find explicit kernel functions for
Ω_{k,a} and *F*_{k,a} for
*a* = 1,2 by means of Bessel functions and the Dunkl intertwining operator.

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© Toshiyuki Kobayashi