  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 \setminus {0}.
Here, k is a multiplicity-function for the Dunkl operators, and a>0 arises from
the interpolation of the Weil representation of Mp(N,R) and the minimal unitary
representation of O(N+1,2) keeping smaller symmetries.
  We prove that this 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}. In the k\equiv 0 case, our semigroup generalizes the
Hermite semigroup studied by R. Howe (a=2) and the Laguerre semigroup by the
second author with G. Mano (a=1).
  One 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), namely a Dunkl-Hankel
transform.
  We establish the inversion formula, and a generalization of the Plancherel
theorem, the Hecke identity, the Bochner identity, and a Heisenberg uncertainty
inequality for F_{k,a}. We also find kernel functions for \Omega_{k,a} and
F_{k,a} for a=1,2 in terms of Bessel functions and the Dunkl intertwining
operator.
