Laguerre semigroup and Dunkl operators

TL;DR

This paper introduces a new family of differential-difference operators related to Dunkl operators, constructing a semigroup and generalized Fourier transforms that unify and extend classical harmonic analysis tools.

Contribution

It develops a two-parameter family of Lie algebra actions that generalize Hermite and Laguerre semigroups, incorporating Dunkl operators and extending harmonic analysis frameworks.

Findings

Constructed a unitary representation of the universal cover of SL(2,R).

Defined new generalized Fourier transforms including Dunkl-Hankel transform.

Established inversion, Plancherel, and uncertainty principles for the transforms.

Abstract

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.