Laguerre semigroup and Dunkl operators

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 ∖{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 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 . In the k ≡0 case, our semigroup generalizes the Hermite semigroup studied by R. Howe ( a =2) and the Laguerre semigroup studied by the second author with G. Mano ( a =1) . One boundary value of our semigroup Ω k , a provides us with ( k , a ) - generalized Fourier transforms ℱ k , a , which include 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, a generalization of the Plancherel theorem, the Hecke identity, the Bochner identity, and a Heisenberg uncertainty relation for ℱ k , a . We also find kernel functions for Ω k , a and ℱ k , a for a =1,2 in terms of Bessel functions and the Dunkl intertwining operator.