S. Ben Saïd, T. Kobayashi, and B. Ørsted, Laguerre semigroup and Dunkl operators, Compositio Mathematica 148 (2012), 1265-1336, DOI: 10.1112/S0010437X11007445. arXiv:0907.3749 [math.RT]..

We construct a two-parameter family of actions ωk,a of the Lie algebra sl(2,R) by differential-difference operators on RN {0}. Here, k is a multiplicity-function for the Dunkl operators, and a>0 arises from the interpolation of the two sl(2,R) actions on the Weil representation of Mp(N,R) and the minimal unitary representation of O(N+1,2). We prove that this action ωk,a lifts to a unitary representation of the universal covering of SL(2,R), and can even be extended to a holomorphic semigroup Ωk,a. In the k 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 Ωk,a provides us with (k,a)-generalized Fourier transforms Fk,a, which includes the Dunkl transform Dk (a=2) and a new unitary operator Hk (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 relation for Fk,a. We also find kernel functions for Ωk,a and Fk,a for a=1,2 in terms of Bessel functions and the Dunkl intertwining operator.

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