$(k, a)$-generalized Fourier transform with negative $a$

TL;DR

This paper explores the $(k, a)$-generalized Fourier transform for negative $a$, establishing a unitary transform linking it to the positive $a$ case, thus expanding understanding of its mathematical structure.

Contribution

It introduces a unitary transform that connects the known positive $a$ case with the new negative $a$ case of the $(k, a)$-generalized Fourier transform.

Findings

Established a unitary transform linking $a>0$ and $a<0$ cases.

Extended the mathematical framework of the $(k, a)$-generalized Fourier transform.

Provided insights into the deformation family related to Lie group representations.

Abstract

The $ (k, a) $-generalized Fourier transform $ \mathscr{F}_{k, a} $ introduced by Ben Sa\"id--Kobayashi--{\O}rsted is a deformation family of the classical Fourier transform with a Dunkl parameter $ k $ and a parameter $ a > 0 $ that interpolates minimal representations of two different simple Lie groups. In the present paper, we focus on the case $ a < 0 $. As a main result, we find a unitary transform that intertwines the known case $ a > 0 $ and the new case $ a < 0 $.