## Abstract

We construct a two-parameter family of actions w _{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 ωk,a lifts to a unitary representation of the universal covering of SL(2, R), and can even be extended to a holomorphic semigroup 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 k,a provides 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 k,a and Fk,a for a = 1, 2 in terms of Bessel functions and the Dunkl intertwining operator.

Original language | English |
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Pages (from-to) | 1265-1336 |

Number of pages | 72 |

Journal | Compositio Mathematica |

Volume | 148 |

Issue number | 4 |

DOIs | |

Publication status | Published - Jul 2012 |

Externally published | Yes |

## Keywords

- Coxeter groups
- Dunkl operators
- Hankel transforms
- Heisenberg inequality
- Hermite semigroup
- Schrödinger model
- Weil representation
- generalized Fourier transform
- holomorphic semigroup
- minimal representation
- rational Cherednik algebra

## ASJC Scopus subject areas

- Algebra and Number Theory