Laguerre semigroup and Dunkl operators

Abstract

AbstractWe construct a two-parameter family of actionsωk,aof the Lie algebra 𝔰𝔩(2,ℝ) by differential–difference operators on ℝN∖{0}. Herekis a multiplicity function for the Dunkl operators, anda>0 arises from the interpolation of the two 𝔰𝔩(2,ℝ) actions on the Weil representation ofMp(N,ℝ) and the minimal unitary representation of O(N+1,2). We prove that this actionωk,alifts to a unitary representation of the universal covering ofSL(2,ℝ) , and can even be extended to a holomorphic semigroup Ωk,a. In thek≡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 Ωk,aprovides us with (k,a) -generalized Fourier transforms ℱk,a, which include the Dunkl transform 𝒟k(a=2) and a new unitary operator ℋk (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 ℱk,a. We also find kernel functions for Ωk,aand ℱk,afora=1,2 in terms of Bessel functions and the Dunkl intertwining operator.