Laguerre semigroup and Dunkl operators

<jats:title>Abstract</jats:title><jats:p>We construct a two-parameter family of actions<jats:italic>ω</jats:italic><jats:sub><jats:italic>k</jats:italic>,<jats:italic>a</jats:italic></jats:sub>of the Lie algebra 𝔰𝔩(2,ℝ) by differential–difference operators on ℝ<jats:sup><jats:italic>N</jats:italic></jats:sup>∖{0}. Here<jats:italic>k</jats:italic>is a multiplicity function for the Dunkl operators, and<jats:italic>a</jats:italic>>0 arises from the interpolation of the two 𝔰𝔩(2,ℝ) actions on the Weil representation of<jats:italic>Mp</jats:italic>(<jats:italic>N</jats:italic>,ℝ) and the minimal unitary representation of O(<jats:italic>N</jats:italic>+1,2). We prove that this action<jats:italic>ω</jats:italic><jats:sub><jats:italic>k</jats:italic>,<jats:italic>a</jats:italic></jats:sub>lifts to a unitary representation of the universal covering of<jats:italic>SL</jats:italic>(2,ℝ) , and can even be extended to a holomorphic semigroup Ω<jats:sub><jats:italic>k</jats:italic>,<jats:italic>a</jats:italic></jats:sub>. In the<jats:italic>k</jats:italic>≡0 case, our semigroup generalizes the Hermite semigroup studied by R. Howe (<jats:italic>a</jats:italic>=2) and the Laguerre semigroup studied by the second author with G. Mano (<jats:italic>a</jats:italic>=1) . One boundary value of our semigroup Ω<jats:sub><jats:italic>k</jats:italic>,<jats:italic>a</jats:italic></jats:sub>provides us with (<jats:italic>k</jats:italic>,<jats:italic>a</jats:italic>) -<jats:italic>generalized Fourier transforms</jats:italic> ℱ<jats:sub><jats:italic>k</jats:italic>,<jats:italic>a</jats:italic></jats:sub>, which include the Dunkl transform 𝒟<jats:sub><jats:italic>k</jats:italic></jats:sub>(<jats:italic>a</jats:italic>=2) and a new unitary operator ℋ<jats:sub><jats:italic>k</jats:italic></jats:sub> (<jats:italic>a</jats:italic>=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<jats:sub><jats:italic>k</jats:italic>,<jats:italic>a</jats:italic></jats:sub>. We also find kernel functions for Ω<jats:sub><jats:italic>k</jats:italic>,<jats:italic>a</jats:italic></jats:sub>and ℱ<jats:sub><jats:italic>k</jats:italic>,<jats:italic>a</jats:italic></jats:sub>for<jats:italic>a</jats:italic>=1,2 in terms of Bessel functions and the Dunkl intertwining operator.</jats:p>